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+{
+    "url": "http://arxiv.org/abs/2405.03690v2",
+    "title": "How Good is my Video LMM? Complex Video Reasoning and Robustness Evaluation Suite for Video-LMMs",
+    "abstract": "Recent advancements in Large Language Models (LLMs) have led to the\ndevelopment of Video Large Multi-modal Models (Video-LMMs) that can handle a\nwide range of video understanding tasks. These models have the potential to be\ndeployed in real-world applications such as robotics, AI assistants, medical\nsurgery, and autonomous vehicles. The widespread adoption of Video-LMMs in our\ndaily lives underscores the importance of ensuring and evaluating their robust\nperformance in mirroring human-like reasoning and interaction capabilities in\ncomplex, real-world contexts. However, existing benchmarks for Video-LMMs\nprimarily focus on general video comprehension abilities and neglect assessing\ntheir reasoning capabilities over complex videos in the real-world context, and\nrobustness of these models through the lens of user prompts as text queries. In\nthis paper, we present the Complex Video Reasoning and Robustness Evaluation\nSuite (CVRR-ES), a novel benchmark that comprehensively assesses the\nperformance of Video-LMMs across 11 diverse real-world video dimensions. We\nevaluate 9 recent models, including both open-source and closed-source\nvariants, and find that most of the Video-LMMs, especially open-source ones,\nstruggle with robustness and reasoning when dealing with complex videos. Based\non our analysis, we develop a training-free Dual-Step Contextual Prompting\n(DSCP) technique to enhance the performance of existing Video-LMMs. Our\nfindings provide valuable insights for building the next generation of\nhuman-centric AI systems with advanced robustness and reasoning capabilities.\nOur dataset and code are publicly available at:\nhttps://mbzuai-oryx.github.io/CVRR-Evaluation-Suite/.",
+    "authors": "Muhammad Uzair Khattak, Muhammad Ferjad Naeem, Jameel Hassan, Muzammal Naseer, Federico Tombari, Fahad Shahbaz Khan, Salman Khan",
+    "published": "2024-05-06",
+    "updated": "2024-05-08",
+    "primary_cat": "cs.CV",
+    "cats": [
+        "cs.CV"
+    ],
+    "label": "Original Paper",
+    "paper_cat": "Multi AND Modal AND LLM",
+    "gt": "Recent advancements in Large Language Models (LLMs) have led to the\ndevelopment of Video Large Multi-modal Models (Video-LMMs) that can handle a\nwide range of video understanding tasks. These models have the potential to be\ndeployed in real-world applications such as robotics, AI assistants, medical\nsurgery, and autonomous vehicles. The widespread adoption of Video-LMMs in our\ndaily lives underscores the importance of ensuring and evaluating their robust\nperformance in mirroring human-like reasoning and interaction capabilities in\ncomplex, real-world contexts. However, existing benchmarks for Video-LMMs\nprimarily focus on general video comprehension abilities and neglect assessing\ntheir reasoning capabilities over complex videos in the real-world context, and\nrobustness of these models through the lens of user prompts as text queries. In\nthis paper, we present the Complex Video Reasoning and Robustness Evaluation\nSuite (CVRR-ES), a novel benchmark that comprehensively assesses the\nperformance of Video-LMMs across 11 diverse real-world video dimensions. We\nevaluate 9 recent models, including both open-source and closed-source\nvariants, and find that most of the Video-LMMs, especially open-source ones,\nstruggle with robustness and reasoning when dealing with complex videos. Based\non our analysis, we develop a training-free Dual-Step Contextual Prompting\n(DSCP) technique to enhance the performance of existing Video-LMMs. Our\nfindings provide valuable insights for building the next generation of\nhuman-centric AI systems with advanced robustness and reasoning capabilities.\nOur dataset and code are publicly available at:\nhttps://mbzuai-oryx.github.io/CVRR-Evaluation-Suite/.",
+    "main_content": "Introduction Recently, Large Language Models (LLMs) [Touvron et al., 2023, Zheng et al., 2023, Jiang et al., 2024] have demonstrated impressive reasoning and planning capabilities while simultaneously handling a wide range of NLP tasks [Wei et al., 2022a, Brown et al., 2020]. Consequently, their integration with the vision modality, specifically for video understanding tasks, has given rise to Video Large Multi-modal Models (Video-LMMs) [Li et al., 2023b]. These models act as visual chatbots that accept both text and video as input and handle a diverse set of tasks, including video comprehension [Maaz et al., 2023], detailed video understanding [Lin et al., 2023], and action grounding [Zhang et al., 2023]. As these models directly capture video data, they hold substantial potential for deployment in real-world applications such as robotics, surveillance, medical surgery, and autonomous vehicles. However, as these models assume an expanding role in our everyday lives, assessing their performance in comprehending complex videos and demonstrating reliable reasoning and robustness capabilities arXiv:2405.03690v2  [cs.CV]  8 May 2024 \fBenchmark Textual Complex In the wild Contextual Multiple Temporal Order Robustness Reasoning (OOD) Dependency Actions & Fine-grained MSVD-QA [Xu et al., 2017] MSRVTT-QA [Xu et al., 2017] TGIF-QA [Jang et al., 2017] Activity Net-QA [Yu et al., 2019] VideoChat-GPT [Maaz et al., 2023] MVBench [Li et al., 2023c] SEED-Bench [Li et al., 2023a] CVRR-ES (ours) Table 1: Comparison of CVRR-ES with existing benchmarks for video QA. The CVRR-ES benchmark represents an initial effort to assess Video-LMMs in the context of their applicability and suitability in real-world applications. Non-existent actions with   non-existent scene depictions. 6.0% Multiple actions in   a single video. 13.25% Fine-grained   action understanding. 9.58% Partial actions. 8.58% Non-existent actions with   existent scene depictions. 5.75% Interpretation of   visual context. 11.38% Continuity and   Object Instance Count. 7.38% Unusual and Physically   Anomalous activities. 7.92% Interpretation of   social context. 11.67% Understanding of   emotional context. 12.17% Time order understanding. 6.33% CVRR   Evaluation Suite 0 20 40 60 80 100 Accuracy % (averaged over 11 video dimensions) Video LLaVa MovieChat LLaMA-VID Video-LLaMA-2 Video-ChatGPT VideoChat TimeChat Gemini-Pro GPT4V(ision) Human Video LMMs 15.92% 16.41% 16.46% 21.62% 24.96% 25.78% 32.89% 53.2% 70.78% 96.67% Figure 1: Left: CVRR-ES comprises of 11 diverse complex video evaluation dimensions encompassing a variety of complex, real-world contexts. Right: Overall performance of Video-LMMs on the CVRR-ES benchmark. Results for each Video-LMM are averaged across 11 video dimensions. across diverse real-world contexts becomes essential. Video-LMMs with such capabilities will be more effective when integrated into our daily lives for solving perception tasks and will be a promising step towards building human-centric AI-assistive systems. Several attempts in literature have been made to benchmark Video-LMMs. SEED-Bench [Li et al., 2023a] curated a MCQ-based benchmarking dataset including 3 evaluation dimensions for videos. Similarly, MV-Bench [Li et al., 2023c] constructed the Video-LMM benchmark and assembled 20 challenging video tasks for evaluating the spatial and temporal understanding of these models. While these methods aim at benchmarking Video-LMMs, they predominantly evaluate video and/or temporal comprehension abilities and overlook the complex reasoning aspects of Video-LMMs for real-world context, and their robustness towards user input text queries; both of which are crucial to ensure their responsible engagement with humans in various real-world situations in the wild. While some studies have explored similar areas such as hallucinations in image-based LLMs [Liu et al., 2023a, Qian et al., 2024], no such comprehensive study exists for the case of Video-LMMs. Motivated by the wide-scale applications of Video-LMMs and the lack of world-centric complex video benchmarking efforts, we present a new benchmark, Complex Video Reasoning and Robustness Evaluation Suite (CVRR-ES), to comprehensively assess the performance of Video-LMMs. As shown in Tab. 1, CVRR-ES evaluates Video-LMMs on key aspects of robustness and reasoning in videos, encompassing video domains that more accurately test models in real-world scenarios such as videos having contextual dependency and in-the-wild aspects. CVRR-ES is an open-ended video QA benchmark comprising 11 real-world video category dimensions (Fig. 1, left) that encompass diverse evaluation aspects. These dimensions span from context-dependent (e.g., social, emotional, etc.) categories to ones that often take place in the wild such as videos containing physically anomalous activities. We comprehensively evaluate a representative set of 9 recent Video-LMMs (Fig. 1, right) including both open-source and closed-source models on the CVRR-ES benchmark using a LLM-assisted automatic evaluation framework [Maaz et al., 2023, Cai et al., 2023]. The performance of Video-LMMs on the CVRR-ES benchmark reveals that these models struggle to correctly comprehend complex videos indicating their weak reasoning and lack of robustness to the textual user queries (Fig. 2). For instance, state-of-the-art Video-LLaVA [Lin et al., 2023] achieves only 15.92% performance averaged across 11 video dimensions of CVRR-ES. In contrast, closed-source models including GPT4V(vision) [OpenAI, 2023] and Gemini-Vision-Pro [Google, 2023] exhibit relatively stronger performance but still lag behind the performance of humans. Using CVRR-ES benchmark, we extensively perform quantitative and qualitative analysis formulating important insights into these Video-LMMs based on their failure cases and individual performances across the diverse video dimensions. 2 \f                      Based on the visual cues provided in the  sequence of images, it appears that four  individuals exit the vehicle.  Correct Response: Three people exit the  car to retrieve black bags from the trunk.  2. Two individuals exit the  vehicle to retrieve items  from the trunk. Two individuals exit the  vehicle to retrieve items  from the trunk. Two individuals exit the vehicle to retrieve items from the trunk. In the video, we can see a  young man and woman  exiting the car \u2026   How many individuals exit the vehicle to  retrieve items from the trunk?          Based on  \u2026, one person is performing a prank  on another individual. The prank seems to  involve pouring a yellow liquid from a bowl,  which is sneakily positioned behind the  unsuspecting person's back at waist level. Correct Response: one man uses his back  muscles to crack eggs placed by another  man into a bowl behind him. The process  is amusing and ends with a celebratory  turn by the man holding the eggs. \u2026 Another man is holding a bowl of food  behind the first man \u2026 the second man  appears to be pouring the contents of the  bowl down the first man's shorts. A man is seen standing in a  kitchen with his hands on his  hips. Another man is standing  behind him and pouring  something into a bowl. a guy is standing with his  back facing someone else.  The person standing behind  him pours water on his  back, causing him to shiver. In the video, a man is seen  standing in a kitchen and  washing dishes. He then  puts on a white robe and  continues washing dishes. In the video, we see a  young man pouring  water from a bucket to a  plastic bowl. What is happening in the video? Why is the golfer  celebrating in the  video? Based on \u2026 the golfer is celebrating because they  successfully hit the golf ball from a sand bunker  onto the green and possibly close to the hole\u2026 Correct Response: The golfer is not celebrating and  instead he is expressing his anger and frustration by  hitting the golf stick into the ground. The golfer is celebrating  because he made a hole-in-one.  A hole-in-one is \u2026 The golfer is celebrating  because he hit the ball into  hole on his first attempt. The golfer is celebrating because  he has just hit the ball into the  hole. The golfer is celebrating because  he has just made a hole in one, \u2026 The golfer is celebrating in the  video because he is  successfully teeing off and \u2026         Before the mug was moved, a tennis ball  became visible underneath it. Correct Response: No other object  was visible before the mug was  moved in the video.  A yellow tennis ball was visible in  the scene before the mug was  moved. A tennis  ball. Before the mug was moved, a tennis  ball was visible in the scene. In the beginning of the video, the person was  holding a mug with a tennis ball on a table.  Before the mug was moved, a white vase was  visible in the scene. Before the mug was moved, what other object  was visible in the scene? A tennis ball was visible in  the scene before the mug  was moved. GPT4(Vision) Gemini-Pro-V Video-LLaVA LLaMA-VID MovieChat TimeChat Figure 2: We observe that most Video-LMMs struggle to reason over complex videos (rows 1-3) and exhibit weak robustness and rectification capabilities when prompted to generate answers for user questions that can sometimes be confusing (row 4). The QA pairs in Comprehensive Video Reasoning and Robustness Evaluation Suite (CVRR-ES) benchmark assess the performance of Video-LMMs beyond general video comprehension. Based on our analysis, we observe that standard prompting of Video-LMMs struggles in steering their focus for complex video understanding. Additionally, their limitations in reasoning and robust video understanding of real-world scenarios are dominantly driven by the quality of textual inputs (i.e., user questions). Based on these insights, we develop a training-free Dual-Step Contextual Prompting (DSCP) technique, which effectively steers the model\u2019s behavior during inference to elicit video-specific reasoning and improved robustness within Video-LMMs. With DSCP, Video-LMMs show substantial improvements on our benchmark, suggesting the potential of prompting techniques for Video-LMMs. Our main contributions can be summarised as follows: \u2022 We present the Complex Video Robustness and Reasoning Evaluation suite (CVRR-ES), a Video Question Answering benchmark designed to assess the reasoning and robustness capabilities of Video-LMMs across 11 diverse world-centric complex video dimensions. \u2022 We comprehensively evaluate both open-source and closed-source Video-LMMs on the CVRR-ES benchmark and find that most models exhibit weak performance, highlighting their limited reasoning in complex videos and lack of robustness towards user text queries. \u2022 We conduct extensive analysis and formulate important conclusions about Video-LMMs based on their failure cases and performance on the CVRR-ES benchmark. Our findings provide valuable insights for building the next generation of human-centric AI systems with improved robustness and reasoning capabilities. \u2022 To improve Video-LMMs\u2019 reasoning and robustness abilities, we formulate a model-agnostic and training-free prompting technique that effectively enhances their performance. 3 \f2 Related Works Video Large Multi-modal models (Video-LMMs). Video-LMMs [Lin et al., 2023, Li et al., 2023d, Zhang et al., 2023] are advanced visual chatbots capable of performing a wide range of video understanding tasks, including video comprehension and captioning, video question-answering, and action grounding. These models accept both video and textual inputs and generate textual responses. From an architectural perspective, Video-LMMs typically combine pre-trained vision backbones [Radford et al., 2021, Fang et al., 2023, Wang et al., 2022b] with large language models [Touvron et al., 2023, Zheng et al., 2023] using connector modules such as MLP adapters, Q-former [Dai et al., 2023], and gated attention [Alayrac et al., 2022]. VideoChat [Li et al., 2023b] and VideoChat-GPT [Li et al., 2023d] presented initial open-source efforts in this direction and were trained with two stages of alignment and video-instruction following objectives. Recently, more advanced Video-LMMs have emerged in the field, with some models focusing on improving model architectures [Li et al., 2023d], expanding to new tasks [Munasinghe et al., 2023], and enabling support for long videos [Song et al., 2023, Ren et al., 2023]. In this work, we aim to develop a comprehensive benchmarking evaluation framework to assess the reasoning and robustness capabilities of Video-LMMs and develop a training-free prompting technique to improve their performance on these fronts. Benchmarking Video-LMMs. With the growing number of Video-LMMs emerging in the research community, several works have presented evaluation frameworks to assess and quantify these models for benchmarking and analysis purposes. SEED-Bench [Li et al., 2023a] evaluates the visual capabilities in both image and Video-LMMs across 12 unique dimensions. MV-Bench [Li et al., 2023c] curates 20 challenging video tasks to evaluate spatial and temporal understanding of VideoLMMs. Video-ChatGPT [Maaz et al., 2023] develops a quantitative evaluation framework to assess model understanding across five aspects of general video comprehension, such as the correctness and consistency of model captions. While these evaluation frameworks provide effective insights, their assessments do not extend beyond general video-comprehension metrics to more advanced aspects of reasoning and robustness, particularly for real-world context cases. In contrast, our work focuses on providing a complex video reasoning and robustness benchmark across 11 diverse real-world-centric evaluation types and offers a more thorough assessment of Video-LMMs in practical applications. Training-free Prompting Techniques. Steering model behavior at inference time using prompting has become a common paradigm in the NLP domain. Prompting [Wei et al., 2022b, Wang et al., 2022a] refers to the set of instructions given as a prefix to the language model to better align model responses with human intent without the need for task-specific fine-tuning. Prompting techniques can be as simple as a single sentence (e.g., \"Let\u2019s think step by step\") such as zero-shot chain of thought [Wei et al., 2022b] prompting, to more detailed techniques such as combining chain-ofthought prompting with few-shot learning [Brown et al., 2020] and self-consistency chain of thought prompting [Wang et al., 2022a]. Surprisingly, training-free prompting techniques for Video Large Multi-modal Models (Video-LMMs) have been minimally explored. In this work, we develop a dual-step prompting technique based on principled prompt instructions specifically designed to steer the model\u2019s behavior for improved reasoning and robustness over complex videos. 3 Complex Video Reasoning and Robustness Evaluation Suite As Video-LMMs are touching new real-world applications, it is essential to ensure that they robustly handle the user inputs, comprehend the visual world, and exhibit human-like reasoning capabilities. In this work, our goal is to establish a comprehensive benchmark that specifically assess the robustness and reasoning capabilities of Video-LMMs in a variety of complex and contextual videos covering diverse scenarios. To this end, we present Complex Video Reasoning and Robustness Evaluation Suite (CVRR-ES). We first provide a holistic overview of CVRR-ES benchmark below and detail the video evaluation dimensions in Sec. 3.1. Subsequently, we present the CVRR-ES creation process in Sec. 3.2. We provide details on the dataset quality and human evaluation in Appendix B. Overview of CVRR-ES Benchmark. CVRR-ES encompasses evaluation dimensions that cover diverse video categories related to real-world scenarios, ranging from context-dependent (e.g., social, emotional) categories to video types that often take place in the wild (e.g., anomalous activities). Specifically, we have compiled 11 video evaluation dimensions and curated 2,400 high-quality openended question-answer (QA) pairs, spanning 217 high-quality videos. The average video duration is 22.3 seconds, with maximum and minimum durations of 183 and 2 seconds, respectively. In Fig. 4 \fFigure 3: CVRR-ES Benchmark Statistics. Left: Frequency distribution of the type of questions. Right: Illustration of the most frequent keywords in the answer-set of CVRR-ES benchmark. 3 (left), we quantify the distribution of different question types present in our benchmark. This diverse set of questions aims to comprehensively capture the model\u2019s answering capabilities based on reasoning and robustness criteria. We show the word cloud plot based on the frequency of key words in the answer set of CVRR-ES in Fig. 3 (right). The frequent words correspond to objects and attributes with which Video-LMMs could most likely interact when deployed in practical scenarios. 3.1 CVRR-ES Video Category definitions. To assess the robustness and reasoning capabilities of Video-LMMs in the CVRR-ES benchmark, we carefully curate 11 diverse benchmark evaluation categories. As shown in Fig. 1 (left), these categories encompass a wide range of real-world complex and contextual videos within each category. Below, we define each video evaluation dimension of the CVRR-ES benchmark in detail. 1) Multiple actions in a single video. This category includes videos that contain multiple activities within a single video. The number of activities varies from 2 to 4 in these videos, mostly featuring humans performing multiple activities. We curate QA pairs in this category aiming to identify whether the model can reason over challenging questions concerning multiple actions and understand the interrelation between different actions within a video. 2) Fine-grained action understanding. We gather video samples with fine-grained actions. These actions encompass various fine-grained activities performed by humans, including pushing, opening, closing, spreading, sitting, etc. This category presents a challenge to the model\u2019s comprehension of subtle and fine-grained actions through carefully crafted questions. 3) Partial actions. Based on our observations that Video-LMMs predominantly generate content that may be contextually relevant and likely to co-occur with the depicted scene in the video, we compile videos featuring actions that have a high probability of being followed by subsequent actions but are not executed in the video. For instance, an action such as cracking an egg in a kitchen setting often anticipates the subsequent action of frying/cooking the egg. 4) Time order understanding. Accurately recognizing the temporal sequence of activities in videos is crucial for distinguishing between atomic actions, such as pushing and pulling. We collect videos of fine-grained actions occurring in a particular temporal direction and curate challenging questions. 5) Non-existent actions with existent scene depictions. This category examines the model\u2019s robustness and reasoning behavior in scenarios where we introduce non-existent activities into the video without altering the physical and spatial scenes or environmental details in it. 6) Non-existent actions with non-existent scene depictions. In this evaluation category, we make the QA task more challenging by creating questions that include both non-existent activities and non-existent scene comprehension. Non-existent scene comprehension involves changing the objects, attributes of objects, and background scene description. This evaluates the model\u2019s reliability to correct misleading questions and avoid generating imaginary content. 7) Continuity and object instance count. This category contains videos (both real and simulations) designed to test the models\u2019 ability to accurately recognize the number of instances of objects, people, etc., and distinguish between existing objects and new ones introduced in the same video scene. 8) Unusual and physically anomalous activities. This category consists of videos with unconventional activities and physical phenomena that seemingly defy the laws of physics. We meticulously 5 \fcollect relevant videos from various sources on the internet, focusing on capturing unusual activities such as a person floating in the air or driving a motorbike on a running river. We believe that assessing Video-LMMs in such scenarios is crucial, as it allows us to determine whether they can generalize to understand actions in out-of-distribution videos that can occur in practical situations. 9) Interpretation of social context. In the real world, human actions are often influenced by social context in their surroundings. For instance, a person might be helping an elderly individual cross the road. This category evaluates Video-LMMs on such scenarios to determine their ability to accurately infer the rationale behind actions based on the depicted social context. We gather diverse videos from the internet and create challenging questions that encompass the social context dimension. 10) Understanding of emotional context. Similar to social context, humans can accurately understand and interpret each other\u2019s actions by considering the emotional context. For example, a person being emotionally moved and crying in a gathering could be a happy moment if it is one stemming from success/joy. We collect videos and curate challenging reasoning questions aimed at recognizing the nature of actions solely based on emotional context for evaluating Video-LMMs. 11) Interpretation of visual context. This dimension focuses on assessing the model\u2019s reasoning abilities to recognize the actions by leveraging the overall visual contextual cues in the video. We curate specific videos containing actions where activity identification and reasoning require visual contextual cues. For example, to identify the number of people present based on the presence of shadows, one must utilize the visual context from the shadows to reason about the question. Qualitative Examples. Fig. 2 shows examples of collected videos for the CVRR-ES benchmark. The curated videos are carefully selected to be diverse and contain rich spatio-temporal content, aligned with the proposed video evaluation dimensions. 3.2 Building CVRR-ES Benchmark After defining the video evaluation dimensions, we now proceed toward building the CVRR-ES benchmark which consists of three stages. We present each stage in detail below. Stage 1: Data collection and Annotation. We first collect high-quality videos and annotate each video using human assistance. To ensure that each evaluation dimension captures the relevant attributes and information, we meticulously select videos that are representative of specific characteristics associated with that dimension. Across the 11 dimensions, 214 unique videos are selected for the benchmark with around 20 videos per evaluation category. Around 60% of these videos are collected from public academic datasets. To introduce diversity in the benchmark distribution, we incorporate video samples from multiple academic datasets including Something-Something-v2 [Goyal et al., 2017], CATER [Girdhar and Ramanan, 2020], Charades [Sigurdsson et al., 2016], ActivityNet [Caba Heilbron et al., 2015], HMDB51 [Kuehne et al., 2011], YFCC100M [Thomee et al., 2016]. The remaining 40% of videos are collected from the internet. Following the video collection process, two experienced human annotators are assigned to generate captions for each video. For videos where initial captions or metadata are available from academic datasets, the captions are generated by the annotators based on them. For videos collected from the internet, captions are entirely generated by human annotators. To ensure consistency and high quality, we provide annotation instructions to annotators, who generate captions accordingly. Personalized annotation guidelines are used for each video category. Refer to additional details in Appendix B. Stage 2: Question-Answer Generation. The first challenge is to select an evaluation setting to assess Video-LMMs. Humans typically engage in free-form conversation to interact with each other in day-to-day life. Inspired by this, we aim to simulate a similar style of interaction with Video-LMMs by curating open-ended QA pairs to evaluate these models for robustness and reasoning. We feed detailed ground-truth video captions to GPT-3.5 LLM, which are utilized to generate open-ended questions covering both reasoning and robustness aspects. Reasoning QA pairs: With Video-LMMs beginning to interact more directly with humans in our lives, it\u2019s crucial to validate the reasoning abilities of Video-LMMs for more reliable Human-AI interaction. When evaluating the reasoning capabilities of Video-LMMs, we aim to determine whether these models can understand the input video not only by analyzing spatial content but also by grasping the underlying rationale behind the occurring activities and their relationships with the surrounding context. This involves creating questions that go beyond simple video comprehension and scene 6 \fdescription and require the model to engage in complex logical inference, contextual understanding, and reasoning about counterfactual and hypothetical scenarios. Robustness QA pairs: In addition to evaluating the reasoning capabilities of LLMs, it is important to assess Video-LMMs to ensure their robust and responsible performance in real-world scenarios. In the context of Video-LMMs, robustness can be evaluated from both visual (video input) and textual interfaces. Our focus in this work lies on textual interface robustness by particularly testing the model\u2019s comprehension when posed with misleading or confusing questions. This scenario mirrors realistic situations where users, based on their expertise levels, may pose irrelevant, misleading, or confusing questions. It is crucial for models to demonstrate reliability and robustness in handling such queries and avoid generating unreal or hallucinated content for input videos. We curate specific prompts for each evaluation dimension to instruct LLM in generating QA pairs. Example prompts used as an instruction to LLMs for curating QA pairs for robustness and reasoning aspects are provided in Fig. 14 in the Appendix D. Stage 3: QA Pairs Filtration. After generating QA pairs, a manual filtration step is employed, with human assistance to verify each generated QA pair. Approximately 30% of the QA pairs generated by GPT-3.5 are found to be noisy, containing questions that are unrelated to the video evaluation dimensions or unanswerable based on the provided ground-truth captions. Additionally, many questions contain answers within the question itself. Therefore, an exhaustive filtering process is conducted which involves QA rectification and removing those samples which are not relevant to the video or evaluation type. This process results in a final set of 2400 high-quality QA pairs for the CVRR-ES benchmark. Examples of QA pairs are shown in Tab. 4 in the Appendix. Stage 4: Evaluation Procedure. Previous methods in the literature [Maaz et al., 2023, Cai et al., 2023, Liu et al., 2023a, Qian et al., 2024] have explored using LLM models as judges for quantifying results in open-ended QA benchmarks. We adopt a similar approach and instruct LLMs to act as teachers to assess the correctness of predicted responses from Video-LMMs compared to ground-truth answers. We generate open-ended predictions from Video-LMMs by providing video-question pairs as inputs and then present the model predictions and their corresponding ground-truth responses to the LLM Judge alongside the evaluation prompt. The Judge determines whether the prediction is correct or incorrect through a binary judgment, assigns a score from 1 to 5 representing the quality of the prediction, and provides a reasoning to explain its decision. Our ablative analysis in the Appendix. D demonstrates that reasoning-constrained LLM-based evaluation aligns well with human-based judgment. The evaluation prompt is shown in Fig. 13 in the Appendix D. 4 Dual-Step Contextual Prompting for Video-LMMs. Given their wide-scale potential in practical downstream applications, new Video-LMMs are frequently introduced by the research community. Despite the availability of numerous Video-LMMs, the majority of them are trained using only positive examples and video-conversational templates that are primarily limited to tasks such as video-captioning and video question answering. This leads to highly over-affirmative behavior and a lack of self-rectification abilities in these models (Sec. 5.4). Dual Step Contextual Prompting for Video-LMMs Retrieving Contextual reasoning information (Step 1) As an intelligent video comprehension model, focus on these guidelines: 1. Differentiate recurring objects, count accurately, and identify  movements and poses.  2. Understand directional movements and temporal order.  3. Pay attention to fine-grained actions with precision.  4. Assess incomplete actions without assuming completion.  5. Detect emotional, social, and visual cues.  6. Capture and analyze all relevant actions.  7. Identify unusual actions accurately.  8. Disagree with incorrect information given in question.  9. If you do not find the evidence in the frames, you can give a definite      answer by assuming that the asked action/attribute is not present.  10. Provide to the point and concise response.  Now, proceed with answering the following question faithfully while  keeping above guidelines in mind:  Question: What is happening in the video? Context conditioned question-answering (Step 2) Context for the given video is: {step 1 response}. Now answer a  question truthfully based on the video and the provided context.  Question: {User question} Figure 4: Principled prompt instructions in our DSCP method for improving reasoning and robustness in Video-LMMs. Additionally, the templates have minimal focus on enhancing reasoning and robustness capabilities through reasoning-based instruction-tuning pairs, resulting in weak performance of such models against robustness and reasoning QA evaluations in the CVRR-ES benchmark. Furthermore, curating reasoning-based instruction fine-tuning datasets requires meticulous data curation steps, and retraining these models is computationally expensive [Li et al., 2023d, Ren et al., 2023]. Alternatively, training-free prompting techniques in NLP literature have shown effectiveness in eliciting reasoning abilities in LLMs such as chain of thought and self-consistency prompting [Wei et al., 2022b, Wang et al., 2022a]. Inspired by these approaches, we introduce a prompting technique called Dual Step Contextual Prompting (DSCP), which aims to steer VideoLMM focus for enhanced reasoning while simultaneously encouraging the models to provide robust and 7 \fVideo-LLaVA LLaMA-VID MovieChat How many different wigs  with different colors did the  young girl try before settling  on the black one? I can see the young girl wearing a black wig,  and a brunette wig is in the background.  The girl is wearing a pink shirt in the  beginning of the video The young girl tried on three  different wigs with different  colors before settling on the black  one. The young girl tried on three different  wigs with different colors before settling  on the black one. Improved video reasoning and robustness with our DSCP Method The video does not provide information about  the number of different wigs the young girl  tried before settling on the black one. I cannot give an accurate answer as the video does not  show the different wigs that the young girl tried. The video does not provide information about the  number of wigs the young girl tried before settling  on the black one Is the atmosphere in the  dressing room tense and  uncomfortable as the girl  receives her new hair? Yes, the atmosphere in the dressing room is  tense and uncomfortable as the girl  receives her new hair. The video does not provide any  information about the  atmosphere in the dressing room. The video does not provide any  information about the atmosphere in the  dressing room. However, it is possible \u2026 Improved video reasoning and robustness with our DSCP Method No, the atmosphere in the dressing room  is not tense and uncomfortable. The  woman brushing the hair is smiling, and  the woman in the chair is smiling as \u2026 \u2026, but the video does not suggest that the girl is  uncomfortable or tense while receiving her new hair. No, the atmosphere in the dressing room is not tense  or uncomfortable. The girl is laughing and smiling \u2026,  indicating that she is enjoying the experience. Can we see visitors petting the  white tiger through the cage  bars? Sure, in this video we can see a baby  panda inside a cage with bars. There  are also other \u2026 Yes, we can see visitors petting  the white tiger through the cage  bars in the video. Yes, we can see visitors petting the white  tiger through the cage bars. Improved video reasoning and robustness with our DSCP Method No, we cannot see visitors petting the white  tiger through the cage bars in the video. No, we do not see visitors petting the white tiger  through the cage bars in the video. No, we cannot see visitors petting the white tiger  through the cage bars in the video. Does the chimpanzee use  tools to repair the zoo  vehicle in the video? Yes, the chimpanzee in the zoo is seen  using tools to fix the zoo vehicle. Yes, the chimpanzee is seen using  a tool to fix the vehicle in the  video. Yes, the chimpanzee is seen using a tool  to repair the zoo vehicle in the video. Improved video reasoning and robustness with our DSCP Method No, the chimpanzee does not use tools to  repair the zoo vehicle in the video. \u2026 the  chimpanzee driving the car \u2026 The chimpanzee does not use tools to repair the zoo  vehicle in the video. It can be seen driving the car \u2026 No, the chimpanzee does not use tools to repair the  zoo vehicle in the video. Figure 5: Qualitative results of DSCP prompting method. Using our DSCP approach, Video-LMMs demonstrate enhanced robustness and reasoning capabilities over complex videos. grounded answers. DSCP is a two-step prompting method that 1) ensures that the model comprehends the video while reasoning over crucial aspects of complex video understanding such as contextual information and decoding the complex relationships between objects and motions, etc., and 2) encourages robustness by generating the response against the question while conditioning both on video and the context retrieved in the first step. Below we discuss each step of DSCP in detail. Step 1: Reasoning over the video. We first guide Video-LMMs using principled prompts to interpret video content from a reasoning perspective. As shown in Fig. 4 (in blue), we formulate ten principled reasoning-based instructions for prompting, Preason, which directs Video-LMMs to not only comprehend the general video content but also steers them to reason over the rationale behind occurring activities and their relationships with the surrounding context. These prompt instructions include specific considerations like contextual priors, the temporal order of actions, instance count, and attributes. Additionally, the prompting technique incorporates instructions to ensure conciseness and factuality, aiming to mitigate hallucinations. Given a Video-LMM F and input video V, we retrieve contextual reasoning information Icontext by providing principled reasoning prompt Preason along with the video to the LMM, Icontext = F(Preason|V). The contextual information is utilized in the second step of DSCP to generate a more grounded response to the user question. Step 2: Context conditioned question answering. As discussed earlier, Video-LMMs are primarily trained with positive examples to answer questions, with limited emphasis on reasoning and robustness aspects. Consequently, enabling direct interaction of Video-LMMs with users in real-world scenarios can result in undesired responses when the user question is confusing and deceiving due to their extreme over-affirmative behavior. To address these challenges, we propose incorporating an additional inference step in Video-LMMs before answering the user\u2019s question. We note that Video-LMMs often possess factual knowledge about the video content but may become distracted and produce hallucinations when prompted with confusing or misleading questions (more details in Appendix C). Specifically, we devise a prompting method that conditions the model to first comprehend the video in detail without attending to the user question, thereby eliminating the influence of the question. The complex video comprehension information refers to Icontext formulated in step 1. Subsequently, we pose the user question in the second step using prompt Puser which combines user question and the contextual reasoning information (Fig. 4, in green) while conditioning the model on both the video and the contextual reasoning information Icontext. Concretely, Final response = F(Puser|V), where Puser = [question; Icontext]. 8 \fTable 2: Evaluation results of Video LLMs across various video-evaluation categories on the CVRR-ES benchmark. We present results for both open-source and closed-source models, alongside human evaluation results which serves as the upper bound on the benchmark. Benchmark Category Video-LLaMA-2 VideoChat Video-ChatGPT Video-LLaVA MovieChat LLaMA-VID TimeChat Gemini-V Pro GPT4V Human Multiple Actions in 16.98 23.90 27.67 15.72 12.58 17.92 28.30 43.08 57.55 93.40 single video. Fine-grained action 29.57 33.48 26.96 25.22 23.48 26.09 39.13 51.61 77.39 95.65 understanding. Partial 24.76 33.01 22.82 13.59 21.36 14.56 49.51 67.48 73.79 98.54 actions. Time order 16.45 31.58 27.63 21.05 16.45 19.74 34.21 45.39 57.89 97.37 understanding. Non-existent actions with 10.14 15.22 23.19 5.07 5.07 2.90 23.19 57.25 71.01 97.10 existent scene. Non-existent actions with 13.19 14.58 17.36 3.47 11.81 6.94 13.89 49.64 75.00 100.00 non-existent scene. Continuity and Object 28.25 24.29 28.41 21.47 19.77 24.86 34.46 36.16 62.71 96.49 instance Count. Unusual and Physically 18.95 18.42 18.95 15.79 17.89 16.32 27.37 60.00 74.74 96.84 Anomalous activities. Interpretation of 25.00 31.07 32.50 18.93 17.14 13.93 39.29 64.29 79.64 97.51 social context. Understanding of 21.92 23.63 21.23 15.07 13.70 14.73 27.40 47.26 66.44 95.55 emotional context. Interpretation of 32.60 34.43 27.84 19.78 21.25 23.08 45.05 63.00 82.42 94.87 visual context. Average 21.62 25.78 24.96 15.92 16.41 16.46 32.89 53.20 70.78 96.67 Intuitively, the factual content generated in the first step will guide the model towards a robust response in the second step to produce factual and correct responses, even in the presence of noisy/misleading user questions. We illustrate the qualitative results of the DSCP method in Fig. 5. This approach leads to responses that are better grounded with the actual video content and are robust against potential lesser-quality user queries. As we will later show, the DSCP technique effectively enhances the performance of Video-LMMs on the CVRR-ES benchmark. 5 Evaluation Experiments on CVRR-ES. Video-LMMs. Both open-source and closed-source models are selected for the evaluation. Among the open-source models, we evaluate 7 recent Video-LMMs, including Video-LLaVA [Lin et al., 2023], TimeChat [Ren et al., 2023], MovieChat [Song et al., 2023], LLaMA-ViD [Li et al., 2023d], VideoChat [Li et al., 2023b] Video-ChatGPT [Maaz et al., 2023], and Video-LLaMA-2 [Zhang et al., 2023]. For evaluating closed-source models, we use Gemini-Pro-Vision [Google, 2023] and GPT-4V(vision) [OpenAI, 2023]. Refer to the Appendix A for implementation details. 5.1 Main Experiments on CVRR-ES. In Tab. 2, we present the evaluation results of Video-LMMs on the 11 dimension categories of the CVRR-ES benchmark. Below, we present several key findings. Open Source Video-LMMs struggles on CVRR-ES benchmark. All open-source LMMs show inferior performance across the different evaluation dimensions of CVRR-ES. Interestingly, some of the earlier developed open-source Video-LMMs, like Video-LLaMA, VideoChat, and Video-ChatGPT, exhibit higher performance compared to more recent models such as Video-LLaVA, MovieChat, and LLaMA-VID. Overall, TimeChat achieves the highest performance of 32.89% averaged across the 11 evaluation dimensions among open-source LMMs, followed by VideoChat with a score of 25.78%. Humans rank highest in CVRR-ES benchmark. Human studies achieve the highest performance on the CVRR-ES benchmark, with over 95% accuracy across all evaluation dimensions. Furthermore, these results suggest that the CVRR-ES QA pairs are answerable and suitable for benchmarking. Closed source models perform competitively on CVRR-ES. As shown in Tab. 2, both Gemini and GPT4V surpass the performance of open-source models and achieve high gains across all evaluation dimensions. The competitive results of GPT4V and Gemini on complex video evaluation dimensions such as partial actions, non-existent action/scene depiction, and context-dependent categories show 9 \fPrompting Method VideoChat Video-LLaVA MovieChat LLaMA-VID TimeChat Standard prompting 25.78 15.92 16.41 16.46 32.89 Chain of Thought (CoT) prompting 22.44 25.87 15.89 29.68 39.57 DSCP (Stage 1) 38.07 32.12 28.05 25.13 33.04 DSCP (Both stages) 47.92 37.93 35.87 46.85 39.45 Table 3: Prompting methods. DSCP stage 1 uses only the principled instructions designed in step 1, while DSCP (Both stages) uses the complete dual-step prompting technique. that these models have a more sophisticated understanding of the complex visual contents of videos and have strong capabilities to rectify misleading and confusing user questions. Overall, GTP4V improves over Gemini by 17.58% and provides an average accuracy of 70.78% on CVRR-ES. 5.2 Effectiveness of DSCP method for improving Video-LMMs performance 0 10 20 30 40 50 60 Accuracy % (averaged over 11 video dimensions) Video LLaVa MovieChat LLaMA-VID Video-LLaMA-2 Video-ChatGPT VideoChat TimeChat Gemini-Pro Video LMMs with DSCP +22.01 +19.46 +30.39 +16.15 +8.93 +22.14 +6.56 +5.02 Figure 6: Video-LMMs with DSCP technique effectively improves their performance (gains are shown in green) on CVRR-ES benchmark. We next integrate DSCP technique with VideoLMMs and present results on the CVRR-ES benchmark in Fig. 6. The results indicate that DSCP improves the model\u2019s performance compared with models that use standard prompting (i.e., using only the question itself). These results suggest that prompting techniques in Video-LMMs can better guide models for improved reasoning and robustness. With DSCP, initially low-performing Video-LMMs such as Video-LLaVa, MovieChat, and LLaMA-Vid show much better relative gains and become competitive with other models. The highest relative gain of 184% is achieved by LLaMA-ViD, which moves from 7th place in the leaderboard to 2nd among the open-source models after utilizing DSCP prompting. We observe similar overall positive trends of using DSCP with closed-source model Gemini, which improves on the benchmark by an absolute overall gain of 5.02%. We provide more detailed results comparisons in Appendix C. 5.3 Different prompting techniques. We study the contribution of each step of DSCP and compare it with chain-of-thought prompting [Wei et al., 2022b]. The results for the top 5 performing Video-LMMs are shown in Tab. 3. Chainof-thought prompting improves over the standard prompting technique in 3 out of 5 Video-LMMs, suggesting that prompting techniques from NLP literature can effectively guide multi-modal VideoLMMs to enhance reasoning and robustness. Next, we ablate on the first step of DSCP prompting, which uses the principled instructions of DSCP step 1 as a prefix alongside the actual user question. Using the first step prompting technique of DSCP substantially improves model performance on all Video-LMMs, suggesting the effectiveness of the principled prompt instructions designed specifically for Video models. DSCP with both steps, which integrates an additional thinking step in the prompting step, further improves the results and provides the highest results on 4 out of 5 Video-LMMs. 5.4 Main findings and Qualitative Results Based on the results of Video-LMMs on CVRR-ES, we draw key findings and show qualitative results. These insights can serve as valuable guidance for developing the next generation of Video-LMMs, aiming to make them more robust and reliable when deployed in real-world applications. Models excelling at standard VQA benchmarks struggle on CVRR-ES benchmark. Our analysis in Sec. 5.1 reveals that the latest open-source Video-LMMs, such as Video-LLaVA, MovieChat, and LLaMA-VID, perform less effectively on the CVRR-ES benchmark compared to Video-LMMs that were introduced earlier in the community, such as VideoChat and Video-ChatGPT. Interestingly, the same recent models demonstrate superior performance on general video comprehension benchmarks. This discrepancy suggests that current VQA benchmarks, like ActivityNet-QA [Yu et al., 2019] and MSRVTT [Xu et al., 2017], do not adequately correlate with the complex video reasoning and robustness scenarios highlighted in our benchmark. Consequently, this also indicates that most newer Video-LMMs are heavily trained to excel on the general video comprehension benchmarks while reducing their generalizability, reasoning, and robustness capabilities. Over-affirmative behavior of open-source Video-LMMs. Another important observation about open-source models is their tendency to exhibit excessively positive and affirmative responses. As shown in Fig. 7, open-source Video-LMMs consistently respond with \"Yes\" even when faced with 10 \fconfusing questions that describe non-existent actions and objects. This highlights the vulnerability of these models when interacting with users in real-world scenarios. In our CVRR-ES benchmark, opensource models are particularly vulnerable to our evaluation dimensions of \"Non-existent actions with the existent scene\" and \"Non-existent actions with the non-existent scene\" compared to closed-source models. These models lack negation and self-rectification capabilities, especially when users provide misleading or confusing questions. We conjecture that such behavior arises due to the absence of negative instruction tuning pairs during the training of Video-LMMs. Tendency towards activity completion. Most open-source Video-LMMs have shown weak performance on the evaluation dimension of partial actions in CVRR-ES, which contains videos focusing on incomplete or atomic actions. To further analyze the models\u2019 behavior, we show qualitative results on such videos in Fig. 8. It can be observed that most open-source models tend to complete actions, even when only part of the action is provided in the video. For instance, Video-LLaVA struggles to reason over the video and describes the man as kicking the soccer ball, while the action in the video stops at the point of the man placing his foot beside the ball. We observe similar behavior in other Video-LMMs. Upon examining the fine-tuning strategies [Maaz et al., 2023, Liu et al., 2023b], we find that almost all models are trained on end-to-end actions-based instruction-tuning data, causing them to generate complete action descriptions at inference. This tendency highlights the vulnerability of Video-LMMs after deployment, as real-world scenarios often involve atomic, sub-atomic, and general actions alike. To improve the performance of Video-LMMs, it is crucial to incorporate diverse action types during training, including partial and incomplete actions. Weak Generalization to extreme OOD videos. The evaluation dimension of unusual and physically anomalous activities in CVRR-ES resembles extreme out-of-distribution video examples. With the exception of GPT4V and Gemini, Video-LMMs struggle with this dimension, indicating weak generalizability towards OOD videos containing the coexistence of unusual objects and activities that are extremely rare in typical videos. For instance, Video-LLaVA in Fig. 9 describes a person falling on the street, while the video actually shows the person performing an optical illusion. To be responsibly deployed in real-world applications, where OOD actions occur more frequently, Video-LMMs need to be trained to perform more robustly on OOD samples. This may involve incorporating diverse and atypical examples in the training data to improve the model\u2019s ability to handle unusual situations. Limited understanding of temporal order in complex videos. The CVRR-ES benchmark results show that Video-LMMs perform relatively better on the fine-grained action dimension compared to the time-order understanding dimension. While these models can accurately identify fine-grained actions, they struggle with comprehending the correct temporal order of these actions within a video. This limitation can lead to misinterpretations of the underlying information depending on temporal order. We present failure cases of this dimension in Fig. 10. For building more advanced world-centric Video-LMMs, it is crucial to enhance their ability to process and interpret event sequences accurately. Video-LMMs struggles in understanding the emotional and social context. For more reliable interaction between Video-LMMs and humans in practical scenarios, these models should comprehend the spatio-temporal scenes with social and contextual reasoning capabilities similar to humans. The lower performance of Video-LMMs on social and emotional contextual dimensions in CVRR-ES highlights their limitations and lack of understanding of scenes based on contextual cues. For instance, as shown in Fig. 11 (bottom row), GPT-4V struggles to comprehend a scene where a worker is attempting to prevent shoes from getting wet due to the rain by moving them under the shade. Instead, GPT-4V provides a response that contradicts the social cues present in the video. 6",
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+                "Muhammad Uzair Khattak",
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+                "Muhammad Ferjad Naeem",
+                "Yongqin Xian"
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+            [
+                "Muhammad Ferjad Naeem",
+                "Xiaohua Zhai"
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+            [
+                "Muhammad Ferjad Naeem",
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+                {
+                    "url": "http://arxiv.org/abs/2405.03690v2",
+                    "title": "How Good is my Video LMM? Complex Video Reasoning and Robustness Evaluation Suite for Video-LMMs",
+                    "abstract": "Recent advancements in Large Language Models (LLMs) have led to the\ndevelopment of Video Large Multi-modal Models (Video-LMMs) that can handle a\nwide range of video understanding tasks. These models have the potential to be\ndeployed in real-world applications such as robotics, AI assistants, medical\nsurgery, and autonomous vehicles. The widespread adoption of Video-LMMs in our\ndaily lives underscores the importance of ensuring and evaluating their robust\nperformance in mirroring human-like reasoning and interaction capabilities in\ncomplex, real-world contexts. However, existing benchmarks for Video-LMMs\nprimarily focus on general video comprehension abilities and neglect assessing\ntheir reasoning capabilities over complex videos in the real-world context, and\nrobustness of these models through the lens of user prompts as text queries. In\nthis paper, we present the Complex Video Reasoning and Robustness Evaluation\nSuite (CVRR-ES), a novel benchmark that comprehensively assesses the\nperformance of Video-LMMs across 11 diverse real-world video dimensions. We\nevaluate 9 recent models, including both open-source and closed-source\nvariants, and find that most of the Video-LMMs, especially open-source ones,\nstruggle with robustness and reasoning when dealing with complex videos. Based\non our analysis, we develop a training-free Dual-Step Contextual Prompting\n(DSCP) technique to enhance the performance of existing Video-LMMs. Our\nfindings provide valuable insights for building the next generation of\nhuman-centric AI systems with advanced robustness and reasoning capabilities.\nOur dataset and code are publicly available at:\nhttps://mbzuai-oryx.github.io/CVRR-Evaluation-Suite/.",
+                    "authors": "Muhammad Uzair Khattak, Muhammad Ferjad Naeem, Jameel Hassan, Muzammal Naseer, Federico Tombari, Fahad Shahbaz Khan, Salman Khan",
+                    "published": "2024-05-06",
+                    "updated": "2024-05-08",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Recently, Large Language Models (LLMs) [Touvron et al., 2023, Zheng et al., 2023, Jiang et al., 2024] have demonstrated impressive reasoning and planning capabilities while simultaneously handling a wide range of NLP tasks [Wei et al., 2022a, Brown et al., 2020]. Consequently, their integration with the vision modality, specifically for video understanding tasks, has given rise to Video Large Multi-modal Models (Video-LMMs) [Li et al., 2023b]. These models act as visual chatbots that accept both text and video as input and handle a diverse set of tasks, including video comprehension [Maaz et al., 2023], detailed video understanding [Lin et al., 2023], and action grounding [Zhang et al., 2023]. As these models directly capture video data, they hold substantial potential for deployment in real-world applications such as robotics, surveillance, medical surgery, and autonomous vehicles. However, as these models assume an expanding role in our everyday lives, assessing their performance in comprehending complex videos and demonstrating reliable reasoning and robustness capabilities arXiv:2405.03690v2  [cs.CV]  8 May 2024 \fBenchmark Textual Complex In the wild Contextual Multiple Temporal Order Robustness Reasoning (OOD) Dependency Actions & Fine-grained MSVD-QA [Xu et al., 2017] MSRVTT-QA [Xu et al., 2017] TGIF-QA [Jang et al., 2017] Activity Net-QA [Yu et al., 2019] VideoChat-GPT [Maaz et al., 2023] MVBench [Li et al., 2023c] SEED-Bench [Li et al., 2023a] CVRR-ES (ours) Table 1: Comparison of CVRR-ES with existing benchmarks for video QA. The CVRR-ES benchmark represents an initial effort to assess Video-LMMs in the context of their applicability and suitability in real-world applications. Non-existent actions with   non-existent scene depictions. 6.0% Multiple actions in   a single video. 13.25% Fine-grained   action understanding. 9.58% Partial actions. 8.58% Non-existent actions with   existent scene depictions. 5.75% Interpretation of   visual context. 11.38% Continuity and   Object Instance Count. 7.38% Unusual and Physically   Anomalous activities. 7.92% Interpretation of   social context. 11.67% Understanding of   emotional context. 12.17% Time order understanding. 6.33% CVRR   Evaluation Suite 0 20 40 60 80 100 Accuracy % (averaged over 11 video dimensions) Video LLaVa MovieChat LLaMA-VID Video-LLaMA-2 Video-ChatGPT VideoChat TimeChat Gemini-Pro GPT4V(ision) Human Video LMMs 15.92% 16.41% 16.46% 21.62% 24.96% 25.78% 32.89% 53.2% 70.78% 96.67% Figure 1: Left: CVRR-ES comprises of 11 diverse complex video evaluation dimensions encompassing a variety of complex, real-world contexts. Right: Overall performance of Video-LMMs on the CVRR-ES benchmark. Results for each Video-LMM are averaged across 11 video dimensions. across diverse real-world contexts becomes essential. Video-LMMs with such capabilities will be more effective when integrated into our daily lives for solving perception tasks and will be a promising step towards building human-centric AI-assistive systems. Several attempts in literature have been made to benchmark Video-LMMs. SEED-Bench [Li et al., 2023a] curated a MCQ-based benchmarking dataset including 3 evaluation dimensions for videos. Similarly, MV-Bench [Li et al., 2023c] constructed the Video-LMM benchmark and assembled 20 challenging video tasks for evaluating the spatial and temporal understanding of these models. While these methods aim at benchmarking Video-LMMs, they predominantly evaluate video and/or temporal comprehension abilities and overlook the complex reasoning aspects of Video-LMMs for real-world context, and their robustness towards user input text queries; both of which are crucial to ensure their responsible engagement with humans in various real-world situations in the wild. While some studies have explored similar areas such as hallucinations in image-based LLMs [Liu et al., 2023a, Qian et al., 2024], no such comprehensive study exists for the case of Video-LMMs. Motivated by the wide-scale applications of Video-LMMs and the lack of world-centric complex video benchmarking efforts, we present a new benchmark, Complex Video Reasoning and Robustness Evaluation Suite (CVRR-ES), to comprehensively assess the performance of Video-LMMs. As shown in Tab. 1, CVRR-ES evaluates Video-LMMs on key aspects of robustness and reasoning in videos, encompassing video domains that more accurately test models in real-world scenarios such as videos having contextual dependency and in-the-wild aspects. CVRR-ES is an open-ended video QA benchmark comprising 11 real-world video category dimensions (Fig. 1, left) that encompass diverse evaluation aspects. These dimensions span from context-dependent (e.g., social, emotional, etc.) categories to ones that often take place in the wild such as videos containing physically anomalous activities. We comprehensively evaluate a representative set of 9 recent Video-LMMs (Fig. 1, right) including both open-source and closed-source models on the CVRR-ES benchmark using a LLM-assisted automatic evaluation framework [Maaz et al., 2023, Cai et al., 2023]. The performance of Video-LMMs on the CVRR-ES benchmark reveals that these models struggle to correctly comprehend complex videos indicating their weak reasoning and lack of robustness to the textual user queries (Fig. 2). For instance, state-of-the-art Video-LLaVA [Lin et al., 2023] achieves only 15.92% performance averaged across 11 video dimensions of CVRR-ES. In contrast, closed-source models including GPT4V(vision) [OpenAI, 2023] and Gemini-Vision-Pro [Google, 2023] exhibit relatively stronger performance but still lag behind the performance of humans. Using CVRR-ES benchmark, we extensively perform quantitative and qualitative analysis formulating important insights into these Video-LMMs based on their failure cases and individual performances across the diverse video dimensions. 2 \f                      Based on the visual cues provided in the  sequence of images, it appears that four  individuals exit the vehicle.  Correct Response: Three people exit the  car to retrieve black bags from the trunk.  2. Two individuals exit the  vehicle to retrieve items  from the trunk. Two individuals exit the  vehicle to retrieve items  from the trunk. Two individuals exit the vehicle to retrieve items from the trunk. In the video, we can see a  young man and woman  exiting the car \u2026   How many individuals exit the vehicle to  retrieve items from the trunk?          Based on  \u2026, one person is performing a prank  on another individual. The prank seems to  involve pouring a yellow liquid from a bowl,  which is sneakily positioned behind the  unsuspecting person's back at waist level. Correct Response: one man uses his back  muscles to crack eggs placed by another  man into a bowl behind him. The process  is amusing and ends with a celebratory  turn by the man holding the eggs. \u2026 Another man is holding a bowl of food  behind the first man \u2026 the second man  appears to be pouring the contents of the  bowl down the first man's shorts. A man is seen standing in a  kitchen with his hands on his  hips. Another man is standing  behind him and pouring  something into a bowl. a guy is standing with his  back facing someone else.  The person standing behind  him pours water on his  back, causing him to shiver. In the video, a man is seen  standing in a kitchen and  washing dishes. He then  puts on a white robe and  continues washing dishes. In the video, we see a  young man pouring  water from a bucket to a  plastic bowl. What is happening in the video? Why is the golfer  celebrating in the  video? Based on \u2026 the golfer is celebrating because they  successfully hit the golf ball from a sand bunker  onto the green and possibly close to the hole\u2026 Correct Response: The golfer is not celebrating and  instead he is expressing his anger and frustration by  hitting the golf stick into the ground. The golfer is celebrating  because he made a hole-in-one.  A hole-in-one is \u2026 The golfer is celebrating  because he hit the ball into  hole on his first attempt. The golfer is celebrating because  he has just hit the ball into the  hole. The golfer is celebrating because  he has just made a hole in one, \u2026 The golfer is celebrating in the  video because he is  successfully teeing off and \u2026         Before the mug was moved, a tennis ball  became visible underneath it. Correct Response: No other object  was visible before the mug was  moved in the video.  A yellow tennis ball was visible in  the scene before the mug was  moved. A tennis  ball. Before the mug was moved, a tennis  ball was visible in the scene. In the beginning of the video, the person was  holding a mug with a tennis ball on a table.  Before the mug was moved, a white vase was  visible in the scene. Before the mug was moved, what other object  was visible in the scene? A tennis ball was visible in  the scene before the mug  was moved. GPT4(Vision) Gemini-Pro-V Video-LLaVA LLaMA-VID MovieChat TimeChat Figure 2: We observe that most Video-LMMs struggle to reason over complex videos (rows 1-3) and exhibit weak robustness and rectification capabilities when prompted to generate answers for user questions that can sometimes be confusing (row 4). The QA pairs in Comprehensive Video Reasoning and Robustness Evaluation Suite (CVRR-ES) benchmark assess the performance of Video-LMMs beyond general video comprehension. Based on our analysis, we observe that standard prompting of Video-LMMs struggles in steering their focus for complex video understanding. Additionally, their limitations in reasoning and robust video understanding of real-world scenarios are dominantly driven by the quality of textual inputs (i.e., user questions). Based on these insights, we develop a training-free Dual-Step Contextual Prompting (DSCP) technique, which effectively steers the model\u2019s behavior during inference to elicit video-specific reasoning and improved robustness within Video-LMMs. With DSCP, Video-LMMs show substantial improvements on our benchmark, suggesting the potential of prompting techniques for Video-LMMs. Our main contributions can be summarised as follows: \u2022 We present the Complex Video Robustness and Reasoning Evaluation suite (CVRR-ES), a Video Question Answering benchmark designed to assess the reasoning and robustness capabilities of Video-LMMs across 11 diverse world-centric complex video dimensions. \u2022 We comprehensively evaluate both open-source and closed-source Video-LMMs on the CVRR-ES benchmark and find that most models exhibit weak performance, highlighting their limited reasoning in complex videos and lack of robustness towards user text queries. \u2022 We conduct extensive analysis and formulate important conclusions about Video-LMMs based on their failure cases and performance on the CVRR-ES benchmark. Our findings provide valuable insights for building the next generation of human-centric AI systems with improved robustness and reasoning capabilities. \u2022 To improve Video-LMMs\u2019 reasoning and robustness abilities, we formulate a model-agnostic and training-free prompting technique that effectively enhances their performance. 3 \f2 Related Works Video Large Multi-modal models (Video-LMMs). Video-LMMs [Lin et al., 2023, Li et al., 2023d, Zhang et al., 2023] are advanced visual chatbots capable of performing a wide range of video understanding tasks, including video comprehension and captioning, video question-answering, and action grounding. These models accept both video and textual inputs and generate textual responses. From an architectural perspective, Video-LMMs typically combine pre-trained vision backbones [Radford et al., 2021, Fang et al., 2023, Wang et al., 2022b] with large language models [Touvron et al., 2023, Zheng et al., 2023] using connector modules such as MLP adapters, Q-former [Dai et al., 2023], and gated attention [Alayrac et al., 2022]. VideoChat [Li et al., 2023b] and VideoChat-GPT [Li et al., 2023d] presented initial open-source efforts in this direction and were trained with two stages of alignment and video-instruction following objectives. Recently, more advanced Video-LMMs have emerged in the field, with some models focusing on improving model architectures [Li et al., 2023d], expanding to new tasks [Munasinghe et al., 2023], and enabling support for long videos [Song et al., 2023, Ren et al., 2023]. In this work, we aim to develop a comprehensive benchmarking evaluation framework to assess the reasoning and robustness capabilities of Video-LMMs and develop a training-free prompting technique to improve their performance on these fronts. Benchmarking Video-LMMs. With the growing number of Video-LMMs emerging in the research community, several works have presented evaluation frameworks to assess and quantify these models for benchmarking and analysis purposes. SEED-Bench [Li et al., 2023a] evaluates the visual capabilities in both image and Video-LMMs across 12 unique dimensions. MV-Bench [Li et al., 2023c] curates 20 challenging video tasks to evaluate spatial and temporal understanding of VideoLMMs. Video-ChatGPT [Maaz et al., 2023] develops a quantitative evaluation framework to assess model understanding across five aspects of general video comprehension, such as the correctness and consistency of model captions. While these evaluation frameworks provide effective insights, their assessments do not extend beyond general video-comprehension metrics to more advanced aspects of reasoning and robustness, particularly for real-world context cases. In contrast, our work focuses on providing a complex video reasoning and robustness benchmark across 11 diverse real-world-centric evaluation types and offers a more thorough assessment of Video-LMMs in practical applications. Training-free Prompting Techniques. Steering model behavior at inference time using prompting has become a common paradigm in the NLP domain. Prompting [Wei et al., 2022b, Wang et al., 2022a] refers to the set of instructions given as a prefix to the language model to better align model responses with human intent without the need for task-specific fine-tuning. Prompting techniques can be as simple as a single sentence (e.g., \"Let\u2019s think step by step\") such as zero-shot chain of thought [Wei et al., 2022b] prompting, to more detailed techniques such as combining chain-ofthought prompting with few-shot learning [Brown et al., 2020] and self-consistency chain of thought prompting [Wang et al., 2022a]. Surprisingly, training-free prompting techniques for Video Large Multi-modal Models (Video-LMMs) have been minimally explored. In this work, we develop a dual-step prompting technique based on principled prompt instructions specifically designed to steer the model\u2019s behavior for improved reasoning and robustness over complex videos. 3 Complex Video Reasoning and Robustness Evaluation Suite As Video-LMMs are touching new real-world applications, it is essential to ensure that they robustly handle the user inputs, comprehend the visual world, and exhibit human-like reasoning capabilities. In this work, our goal is to establish a comprehensive benchmark that specifically assess the robustness and reasoning capabilities of Video-LMMs in a variety of complex and contextual videos covering diverse scenarios. To this end, we present Complex Video Reasoning and Robustness Evaluation Suite (CVRR-ES). We first provide a holistic overview of CVRR-ES benchmark below and detail the video evaluation dimensions in Sec. 3.1. Subsequently, we present the CVRR-ES creation process in Sec. 3.2. We provide details on the dataset quality and human evaluation in Appendix B. Overview of CVRR-ES Benchmark. CVRR-ES encompasses evaluation dimensions that cover diverse video categories related to real-world scenarios, ranging from context-dependent (e.g., social, emotional) categories to video types that often take place in the wild (e.g., anomalous activities). Specifically, we have compiled 11 video evaluation dimensions and curated 2,400 high-quality openended question-answer (QA) pairs, spanning 217 high-quality videos. The average video duration is 22.3 seconds, with maximum and minimum durations of 183 and 2 seconds, respectively. In Fig. 4 \fFigure 3: CVRR-ES Benchmark Statistics. Left: Frequency distribution of the type of questions. Right: Illustration of the most frequent keywords in the answer-set of CVRR-ES benchmark. 3 (left), we quantify the distribution of different question types present in our benchmark. This diverse set of questions aims to comprehensively capture the model\u2019s answering capabilities based on reasoning and robustness criteria. We show the word cloud plot based on the frequency of key words in the answer set of CVRR-ES in Fig. 3 (right). The frequent words correspond to objects and attributes with which Video-LMMs could most likely interact when deployed in practical scenarios. 3.1 CVRR-ES Video Category definitions. To assess the robustness and reasoning capabilities of Video-LMMs in the CVRR-ES benchmark, we carefully curate 11 diverse benchmark evaluation categories. As shown in Fig. 1 (left), these categories encompass a wide range of real-world complex and contextual videos within each category. Below, we define each video evaluation dimension of the CVRR-ES benchmark in detail. 1) Multiple actions in a single video. This category includes videos that contain multiple activities within a single video. The number of activities varies from 2 to 4 in these videos, mostly featuring humans performing multiple activities. We curate QA pairs in this category aiming to identify whether the model can reason over challenging questions concerning multiple actions and understand the interrelation between different actions within a video. 2) Fine-grained action understanding. We gather video samples with fine-grained actions. These actions encompass various fine-grained activities performed by humans, including pushing, opening, closing, spreading, sitting, etc. This category presents a challenge to the model\u2019s comprehension of subtle and fine-grained actions through carefully crafted questions. 3) Partial actions. Based on our observations that Video-LMMs predominantly generate content that may be contextually relevant and likely to co-occur with the depicted scene in the video, we compile videos featuring actions that have a high probability of being followed by subsequent actions but are not executed in the video. For instance, an action such as cracking an egg in a kitchen setting often anticipates the subsequent action of frying/cooking the egg. 4) Time order understanding. Accurately recognizing the temporal sequence of activities in videos is crucial for distinguishing between atomic actions, such as pushing and pulling. We collect videos of fine-grained actions occurring in a particular temporal direction and curate challenging questions. 5) Non-existent actions with existent scene depictions. This category examines the model\u2019s robustness and reasoning behavior in scenarios where we introduce non-existent activities into the video without altering the physical and spatial scenes or environmental details in it. 6) Non-existent actions with non-existent scene depictions. In this evaluation category, we make the QA task more challenging by creating questions that include both non-existent activities and non-existent scene comprehension. Non-existent scene comprehension involves changing the objects, attributes of objects, and background scene description. This evaluates the model\u2019s reliability to correct misleading questions and avoid generating imaginary content. 7) Continuity and object instance count. This category contains videos (both real and simulations) designed to test the models\u2019 ability to accurately recognize the number of instances of objects, people, etc., and distinguish between existing objects and new ones introduced in the same video scene. 8) Unusual and physically anomalous activities. This category consists of videos with unconventional activities and physical phenomena that seemingly defy the laws of physics. We meticulously 5 \fcollect relevant videos from various sources on the internet, focusing on capturing unusual activities such as a person floating in the air or driving a motorbike on a running river. We believe that assessing Video-LMMs in such scenarios is crucial, as it allows us to determine whether they can generalize to understand actions in out-of-distribution videos that can occur in practical situations. 9) Interpretation of social context. In the real world, human actions are often influenced by social context in their surroundings. For instance, a person might be helping an elderly individual cross the road. This category evaluates Video-LMMs on such scenarios to determine their ability to accurately infer the rationale behind actions based on the depicted social context. We gather diverse videos from the internet and create challenging questions that encompass the social context dimension. 10) Understanding of emotional context. Similar to social context, humans can accurately understand and interpret each other\u2019s actions by considering the emotional context. For example, a person being emotionally moved and crying in a gathering could be a happy moment if it is one stemming from success/joy. We collect videos and curate challenging reasoning questions aimed at recognizing the nature of actions solely based on emotional context for evaluating Video-LMMs. 11) Interpretation of visual context. This dimension focuses on assessing the model\u2019s reasoning abilities to recognize the actions by leveraging the overall visual contextual cues in the video. We curate specific videos containing actions where activity identification and reasoning require visual contextual cues. For example, to identify the number of people present based on the presence of shadows, one must utilize the visual context from the shadows to reason about the question. Qualitative Examples. Fig. 2 shows examples of collected videos for the CVRR-ES benchmark. The curated videos are carefully selected to be diverse and contain rich spatio-temporal content, aligned with the proposed video evaluation dimensions. 3.2 Building CVRR-ES Benchmark After defining the video evaluation dimensions, we now proceed toward building the CVRR-ES benchmark which consists of three stages. We present each stage in detail below. Stage 1: Data collection and Annotation. We first collect high-quality videos and annotate each video using human assistance. To ensure that each evaluation dimension captures the relevant attributes and information, we meticulously select videos that are representative of specific characteristics associated with that dimension. Across the 11 dimensions, 214 unique videos are selected for the benchmark with around 20 videos per evaluation category. Around 60% of these videos are collected from public academic datasets. To introduce diversity in the benchmark distribution, we incorporate video samples from multiple academic datasets including Something-Something-v2 [Goyal et al., 2017], CATER [Girdhar and Ramanan, 2020], Charades [Sigurdsson et al., 2016], ActivityNet [Caba Heilbron et al., 2015], HMDB51 [Kuehne et al., 2011], YFCC100M [Thomee et al., 2016]. The remaining 40% of videos are collected from the internet. Following the video collection process, two experienced human annotators are assigned to generate captions for each video. For videos where initial captions or metadata are available from academic datasets, the captions are generated by the annotators based on them. For videos collected from the internet, captions are entirely generated by human annotators. To ensure consistency and high quality, we provide annotation instructions to annotators, who generate captions accordingly. Personalized annotation guidelines are used for each video category. Refer to additional details in Appendix B. Stage 2: Question-Answer Generation. The first challenge is to select an evaluation setting to assess Video-LMMs. Humans typically engage in free-form conversation to interact with each other in day-to-day life. Inspired by this, we aim to simulate a similar style of interaction with Video-LMMs by curating open-ended QA pairs to evaluate these models for robustness and reasoning. We feed detailed ground-truth video captions to GPT-3.5 LLM, which are utilized to generate open-ended questions covering both reasoning and robustness aspects. Reasoning QA pairs: With Video-LMMs beginning to interact more directly with humans in our lives, it\u2019s crucial to validate the reasoning abilities of Video-LMMs for more reliable Human-AI interaction. When evaluating the reasoning capabilities of Video-LMMs, we aim to determine whether these models can understand the input video not only by analyzing spatial content but also by grasping the underlying rationale behind the occurring activities and their relationships with the surrounding context. This involves creating questions that go beyond simple video comprehension and scene 6 \fdescription and require the model to engage in complex logical inference, contextual understanding, and reasoning about counterfactual and hypothetical scenarios. Robustness QA pairs: In addition to evaluating the reasoning capabilities of LLMs, it is important to assess Video-LMMs to ensure their robust and responsible performance in real-world scenarios. In the context of Video-LMMs, robustness can be evaluated from both visual (video input) and textual interfaces. Our focus in this work lies on textual interface robustness by particularly testing the model\u2019s comprehension when posed with misleading or confusing questions. This scenario mirrors realistic situations where users, based on their expertise levels, may pose irrelevant, misleading, or confusing questions. It is crucial for models to demonstrate reliability and robustness in handling such queries and avoid generating unreal or hallucinated content for input videos. We curate specific prompts for each evaluation dimension to instruct LLM in generating QA pairs. Example prompts used as an instruction to LLMs for curating QA pairs for robustness and reasoning aspects are provided in Fig. 14 in the Appendix D. Stage 3: QA Pairs Filtration. After generating QA pairs, a manual filtration step is employed, with human assistance to verify each generated QA pair. Approximately 30% of the QA pairs generated by GPT-3.5 are found to be noisy, containing questions that are unrelated to the video evaluation dimensions or unanswerable based on the provided ground-truth captions. Additionally, many questions contain answers within the question itself. Therefore, an exhaustive filtering process is conducted which involves QA rectification and removing those samples which are not relevant to the video or evaluation type. This process results in a final set of 2400 high-quality QA pairs for the CVRR-ES benchmark. Examples of QA pairs are shown in Tab. 4 in the Appendix. Stage 4: Evaluation Procedure. Previous methods in the literature [Maaz et al., 2023, Cai et al., 2023, Liu et al., 2023a, Qian et al., 2024] have explored using LLM models as judges for quantifying results in open-ended QA benchmarks. We adopt a similar approach and instruct LLMs to act as teachers to assess the correctness of predicted responses from Video-LMMs compared to ground-truth answers. We generate open-ended predictions from Video-LMMs by providing video-question pairs as inputs and then present the model predictions and their corresponding ground-truth responses to the LLM Judge alongside the evaluation prompt. The Judge determines whether the prediction is correct or incorrect through a binary judgment, assigns a score from 1 to 5 representing the quality of the prediction, and provides a reasoning to explain its decision. Our ablative analysis in the Appendix. D demonstrates that reasoning-constrained LLM-based evaluation aligns well with human-based judgment. The evaluation prompt is shown in Fig. 13 in the Appendix D. 4 Dual-Step Contextual Prompting for Video-LMMs. Given their wide-scale potential in practical downstream applications, new Video-LMMs are frequently introduced by the research community. Despite the availability of numerous Video-LMMs, the majority of them are trained using only positive examples and video-conversational templates that are primarily limited to tasks such as video-captioning and video question answering. This leads to highly over-affirmative behavior and a lack of self-rectification abilities in these models (Sec. 5.4). Dual Step Contextual Prompting for Video-LMMs Retrieving Contextual reasoning information (Step 1) As an intelligent video comprehension model, focus on these guidelines: 1. Differentiate recurring objects, count accurately, and identify  movements and poses.  2. Understand directional movements and temporal order.  3. Pay attention to fine-grained actions with precision.  4. Assess incomplete actions without assuming completion.  5. Detect emotional, social, and visual cues.  6. Capture and analyze all relevant actions.  7. Identify unusual actions accurately.  8. Disagree with incorrect information given in question.  9. If you do not find the evidence in the frames, you can give a definite      answer by assuming that the asked action/attribute is not present.  10. Provide to the point and concise response.  Now, proceed with answering the following question faithfully while  keeping above guidelines in mind:  Question: What is happening in the video? Context conditioned question-answering (Step 2) Context for the given video is: {step 1 response}. Now answer a  question truthfully based on the video and the provided context.  Question: {User question} Figure 4: Principled prompt instructions in our DSCP method for improving reasoning and robustness in Video-LMMs. Additionally, the templates have minimal focus on enhancing reasoning and robustness capabilities through reasoning-based instruction-tuning pairs, resulting in weak performance of such models against robustness and reasoning QA evaluations in the CVRR-ES benchmark. Furthermore, curating reasoning-based instruction fine-tuning datasets requires meticulous data curation steps, and retraining these models is computationally expensive [Li et al., 2023d, Ren et al., 2023]. Alternatively, training-free prompting techniques in NLP literature have shown effectiveness in eliciting reasoning abilities in LLMs such as chain of thought and self-consistency prompting [Wei et al., 2022b, Wang et al., 2022a]. Inspired by these approaches, we introduce a prompting technique called Dual Step Contextual Prompting (DSCP), which aims to steer VideoLMM focus for enhanced reasoning while simultaneously encouraging the models to provide robust and 7 \fVideo-LLaVA LLaMA-VID MovieChat How many different wigs  with different colors did the  young girl try before settling  on the black one? I can see the young girl wearing a black wig,  and a brunette wig is in the background.  The girl is wearing a pink shirt in the  beginning of the video The young girl tried on three  different wigs with different  colors before settling on the black  one. The young girl tried on three different  wigs with different colors before settling  on the black one. Improved video reasoning and robustness with our DSCP Method The video does not provide information about  the number of different wigs the young girl  tried before settling on the black one. I cannot give an accurate answer as the video does not  show the different wigs that the young girl tried. The video does not provide information about the  number of wigs the young girl tried before settling  on the black one Is the atmosphere in the  dressing room tense and  uncomfortable as the girl  receives her new hair? Yes, the atmosphere in the dressing room is  tense and uncomfortable as the girl  receives her new hair. The video does not provide any  information about the  atmosphere in the dressing room. The video does not provide any  information about the atmosphere in the  dressing room. However, it is possible \u2026 Improved video reasoning and robustness with our DSCP Method No, the atmosphere in the dressing room  is not tense and uncomfortable. The  woman brushing the hair is smiling, and  the woman in the chair is smiling as \u2026 \u2026, but the video does not suggest that the girl is  uncomfortable or tense while receiving her new hair. No, the atmosphere in the dressing room is not tense  or uncomfortable. The girl is laughing and smiling \u2026,  indicating that she is enjoying the experience. Can we see visitors petting the  white tiger through the cage  bars? Sure, in this video we can see a baby  panda inside a cage with bars. There  are also other \u2026 Yes, we can see visitors petting  the white tiger through the cage  bars in the video. Yes, we can see visitors petting the white  tiger through the cage bars. Improved video reasoning and robustness with our DSCP Method No, we cannot see visitors petting the white  tiger through the cage bars in the video. No, we do not see visitors petting the white tiger  through the cage bars in the video. No, we cannot see visitors petting the white tiger  through the cage bars in the video. Does the chimpanzee use  tools to repair the zoo  vehicle in the video? Yes, the chimpanzee in the zoo is seen  using tools to fix the zoo vehicle. Yes, the chimpanzee is seen using  a tool to fix the vehicle in the  video. Yes, the chimpanzee is seen using a tool  to repair the zoo vehicle in the video. Improved video reasoning and robustness with our DSCP Method No, the chimpanzee does not use tools to  repair the zoo vehicle in the video. \u2026 the  chimpanzee driving the car \u2026 The chimpanzee does not use tools to repair the zoo  vehicle in the video. It can be seen driving the car \u2026 No, the chimpanzee does not use tools to repair the  zoo vehicle in the video. Figure 5: Qualitative results of DSCP prompting method. Using our DSCP approach, Video-LMMs demonstrate enhanced robustness and reasoning capabilities over complex videos. grounded answers. DSCP is a two-step prompting method that 1) ensures that the model comprehends the video while reasoning over crucial aspects of complex video understanding such as contextual information and decoding the complex relationships between objects and motions, etc., and 2) encourages robustness by generating the response against the question while conditioning both on video and the context retrieved in the first step. Below we discuss each step of DSCP in detail. Step 1: Reasoning over the video. We first guide Video-LMMs using principled prompts to interpret video content from a reasoning perspective. As shown in Fig. 4 (in blue), we formulate ten principled reasoning-based instructions for prompting, Preason, which directs Video-LMMs to not only comprehend the general video content but also steers them to reason over the rationale behind occurring activities and their relationships with the surrounding context. These prompt instructions include specific considerations like contextual priors, the temporal order of actions, instance count, and attributes. Additionally, the prompting technique incorporates instructions to ensure conciseness and factuality, aiming to mitigate hallucinations. Given a Video-LMM F and input video V, we retrieve contextual reasoning information Icontext by providing principled reasoning prompt Preason along with the video to the LMM, Icontext = F(Preason|V). The contextual information is utilized in the second step of DSCP to generate a more grounded response to the user question. Step 2: Context conditioned question answering. As discussed earlier, Video-LMMs are primarily trained with positive examples to answer questions, with limited emphasis on reasoning and robustness aspects. Consequently, enabling direct interaction of Video-LMMs with users in real-world scenarios can result in undesired responses when the user question is confusing and deceiving due to their extreme over-affirmative behavior. To address these challenges, we propose incorporating an additional inference step in Video-LMMs before answering the user\u2019s question. We note that Video-LMMs often possess factual knowledge about the video content but may become distracted and produce hallucinations when prompted with confusing or misleading questions (more details in Appendix C). Specifically, we devise a prompting method that conditions the model to first comprehend the video in detail without attending to the user question, thereby eliminating the influence of the question. The complex video comprehension information refers to Icontext formulated in step 1. Subsequently, we pose the user question in the second step using prompt Puser which combines user question and the contextual reasoning information (Fig. 4, in green) while conditioning the model on both the video and the contextual reasoning information Icontext. Concretely, Final response = F(Puser|V), where Puser = [question; Icontext]. 8 \fTable 2: Evaluation results of Video LLMs across various video-evaluation categories on the CVRR-ES benchmark. We present results for both open-source and closed-source models, alongside human evaluation results which serves as the upper bound on the benchmark. Benchmark Category Video-LLaMA-2 VideoChat Video-ChatGPT Video-LLaVA MovieChat LLaMA-VID TimeChat Gemini-V Pro GPT4V Human Multiple Actions in 16.98 23.90 27.67 15.72 12.58 17.92 28.30 43.08 57.55 93.40 single video. Fine-grained action 29.57 33.48 26.96 25.22 23.48 26.09 39.13 51.61 77.39 95.65 understanding. Partial 24.76 33.01 22.82 13.59 21.36 14.56 49.51 67.48 73.79 98.54 actions. Time order 16.45 31.58 27.63 21.05 16.45 19.74 34.21 45.39 57.89 97.37 understanding. Non-existent actions with 10.14 15.22 23.19 5.07 5.07 2.90 23.19 57.25 71.01 97.10 existent scene. Non-existent actions with 13.19 14.58 17.36 3.47 11.81 6.94 13.89 49.64 75.00 100.00 non-existent scene. Continuity and Object 28.25 24.29 28.41 21.47 19.77 24.86 34.46 36.16 62.71 96.49 instance Count. Unusual and Physically 18.95 18.42 18.95 15.79 17.89 16.32 27.37 60.00 74.74 96.84 Anomalous activities. Interpretation of 25.00 31.07 32.50 18.93 17.14 13.93 39.29 64.29 79.64 97.51 social context. Understanding of 21.92 23.63 21.23 15.07 13.70 14.73 27.40 47.26 66.44 95.55 emotional context. Interpretation of 32.60 34.43 27.84 19.78 21.25 23.08 45.05 63.00 82.42 94.87 visual context. Average 21.62 25.78 24.96 15.92 16.41 16.46 32.89 53.20 70.78 96.67 Intuitively, the factual content generated in the first step will guide the model towards a robust response in the second step to produce factual and correct responses, even in the presence of noisy/misleading user questions. We illustrate the qualitative results of the DSCP method in Fig. 5. This approach leads to responses that are better grounded with the actual video content and are robust against potential lesser-quality user queries. As we will later show, the DSCP technique effectively enhances the performance of Video-LMMs on the CVRR-ES benchmark. 5 Evaluation Experiments on CVRR-ES. Video-LMMs. Both open-source and closed-source models are selected for the evaluation. Among the open-source models, we evaluate 7 recent Video-LMMs, including Video-LLaVA [Lin et al., 2023], TimeChat [Ren et al., 2023], MovieChat [Song et al., 2023], LLaMA-ViD [Li et al., 2023d], VideoChat [Li et al., 2023b] Video-ChatGPT [Maaz et al., 2023], and Video-LLaMA-2 [Zhang et al., 2023]. For evaluating closed-source models, we use Gemini-Pro-Vision [Google, 2023] and GPT-4V(vision) [OpenAI, 2023]. Refer to the Appendix A for implementation details. 5.1 Main Experiments on CVRR-ES. In Tab. 2, we present the evaluation results of Video-LMMs on the 11 dimension categories of the CVRR-ES benchmark. Below, we present several key findings. Open Source Video-LMMs struggles on CVRR-ES benchmark. All open-source LMMs show inferior performance across the different evaluation dimensions of CVRR-ES. Interestingly, some of the earlier developed open-source Video-LMMs, like Video-LLaMA, VideoChat, and Video-ChatGPT, exhibit higher performance compared to more recent models such as Video-LLaVA, MovieChat, and LLaMA-VID. Overall, TimeChat achieves the highest performance of 32.89% averaged across the 11 evaluation dimensions among open-source LMMs, followed by VideoChat with a score of 25.78%. Humans rank highest in CVRR-ES benchmark. Human studies achieve the highest performance on the CVRR-ES benchmark, with over 95% accuracy across all evaluation dimensions. Furthermore, these results suggest that the CVRR-ES QA pairs are answerable and suitable for benchmarking. Closed source models perform competitively on CVRR-ES. As shown in Tab. 2, both Gemini and GPT4V surpass the performance of open-source models and achieve high gains across all evaluation dimensions. The competitive results of GPT4V and Gemini on complex video evaluation dimensions such as partial actions, non-existent action/scene depiction, and context-dependent categories show 9 \fPrompting Method VideoChat Video-LLaVA MovieChat LLaMA-VID TimeChat Standard prompting 25.78 15.92 16.41 16.46 32.89 Chain of Thought (CoT) prompting 22.44 25.87 15.89 29.68 39.57 DSCP (Stage 1) 38.07 32.12 28.05 25.13 33.04 DSCP (Both stages) 47.92 37.93 35.87 46.85 39.45 Table 3: Prompting methods. DSCP stage 1 uses only the principled instructions designed in step 1, while DSCP (Both stages) uses the complete dual-step prompting technique. that these models have a more sophisticated understanding of the complex visual contents of videos and have strong capabilities to rectify misleading and confusing user questions. Overall, GTP4V improves over Gemini by 17.58% and provides an average accuracy of 70.78% on CVRR-ES. 5.2 Effectiveness of DSCP method for improving Video-LMMs performance 0 10 20 30 40 50 60 Accuracy % (averaged over 11 video dimensions) Video LLaVa MovieChat LLaMA-VID Video-LLaMA-2 Video-ChatGPT VideoChat TimeChat Gemini-Pro Video LMMs with DSCP +22.01 +19.46 +30.39 +16.15 +8.93 +22.14 +6.56 +5.02 Figure 6: Video-LMMs with DSCP technique effectively improves their performance (gains are shown in green) on CVRR-ES benchmark. We next integrate DSCP technique with VideoLMMs and present results on the CVRR-ES benchmark in Fig. 6. The results indicate that DSCP improves the model\u2019s performance compared with models that use standard prompting (i.e., using only the question itself). These results suggest that prompting techniques in Video-LMMs can better guide models for improved reasoning and robustness. With DSCP, initially low-performing Video-LMMs such as Video-LLaVa, MovieChat, and LLaMA-Vid show much better relative gains and become competitive with other models. The highest relative gain of 184% is achieved by LLaMA-ViD, which moves from 7th place in the leaderboard to 2nd among the open-source models after utilizing DSCP prompting. We observe similar overall positive trends of using DSCP with closed-source model Gemini, which improves on the benchmark by an absolute overall gain of 5.02%. We provide more detailed results comparisons in Appendix C. 5.3 Different prompting techniques. We study the contribution of each step of DSCP and compare it with chain-of-thought prompting [Wei et al., 2022b]. The results for the top 5 performing Video-LMMs are shown in Tab. 3. Chainof-thought prompting improves over the standard prompting technique in 3 out of 5 Video-LMMs, suggesting that prompting techniques from NLP literature can effectively guide multi-modal VideoLMMs to enhance reasoning and robustness. Next, we ablate on the first step of DSCP prompting, which uses the principled instructions of DSCP step 1 as a prefix alongside the actual user question. Using the first step prompting technique of DSCP substantially improves model performance on all Video-LMMs, suggesting the effectiveness of the principled prompt instructions designed specifically for Video models. DSCP with both steps, which integrates an additional thinking step in the prompting step, further improves the results and provides the highest results on 4 out of 5 Video-LMMs. 5.4 Main findings and Qualitative Results Based on the results of Video-LMMs on CVRR-ES, we draw key findings and show qualitative results. These insights can serve as valuable guidance for developing the next generation of Video-LMMs, aiming to make them more robust and reliable when deployed in real-world applications. Models excelling at standard VQA benchmarks struggle on CVRR-ES benchmark. Our analysis in Sec. 5.1 reveals that the latest open-source Video-LMMs, such as Video-LLaVA, MovieChat, and LLaMA-VID, perform less effectively on the CVRR-ES benchmark compared to Video-LMMs that were introduced earlier in the community, such as VideoChat and Video-ChatGPT. Interestingly, the same recent models demonstrate superior performance on general video comprehension benchmarks. This discrepancy suggests that current VQA benchmarks, like ActivityNet-QA [Yu et al., 2019] and MSRVTT [Xu et al., 2017], do not adequately correlate with the complex video reasoning and robustness scenarios highlighted in our benchmark. Consequently, this also indicates that most newer Video-LMMs are heavily trained to excel on the general video comprehension benchmarks while reducing their generalizability, reasoning, and robustness capabilities. Over-affirmative behavior of open-source Video-LMMs. Another important observation about open-source models is their tendency to exhibit excessively positive and affirmative responses. As shown in Fig. 7, open-source Video-LMMs consistently respond with \"Yes\" even when faced with 10 \fconfusing questions that describe non-existent actions and objects. This highlights the vulnerability of these models when interacting with users in real-world scenarios. In our CVRR-ES benchmark, opensource models are particularly vulnerable to our evaluation dimensions of \"Non-existent actions with the existent scene\" and \"Non-existent actions with the non-existent scene\" compared to closed-source models. These models lack negation and self-rectification capabilities, especially when users provide misleading or confusing questions. We conjecture that such behavior arises due to the absence of negative instruction tuning pairs during the training of Video-LMMs. Tendency towards activity completion. Most open-source Video-LMMs have shown weak performance on the evaluation dimension of partial actions in CVRR-ES, which contains videos focusing on incomplete or atomic actions. To further analyze the models\u2019 behavior, we show qualitative results on such videos in Fig. 8. It can be observed that most open-source models tend to complete actions, even when only part of the action is provided in the video. For instance, Video-LLaVA struggles to reason over the video and describes the man as kicking the soccer ball, while the action in the video stops at the point of the man placing his foot beside the ball. We observe similar behavior in other Video-LMMs. Upon examining the fine-tuning strategies [Maaz et al., 2023, Liu et al., 2023b], we find that almost all models are trained on end-to-end actions-based instruction-tuning data, causing them to generate complete action descriptions at inference. This tendency highlights the vulnerability of Video-LMMs after deployment, as real-world scenarios often involve atomic, sub-atomic, and general actions alike. To improve the performance of Video-LMMs, it is crucial to incorporate diverse action types during training, including partial and incomplete actions. Weak Generalization to extreme OOD videos. The evaluation dimension of unusual and physically anomalous activities in CVRR-ES resembles extreme out-of-distribution video examples. With the exception of GPT4V and Gemini, Video-LMMs struggle with this dimension, indicating weak generalizability towards OOD videos containing the coexistence of unusual objects and activities that are extremely rare in typical videos. For instance, Video-LLaVA in Fig. 9 describes a person falling on the street, while the video actually shows the person performing an optical illusion. To be responsibly deployed in real-world applications, where OOD actions occur more frequently, Video-LMMs need to be trained to perform more robustly on OOD samples. This may involve incorporating diverse and atypical examples in the training data to improve the model\u2019s ability to handle unusual situations. Limited understanding of temporal order in complex videos. The CVRR-ES benchmark results show that Video-LMMs perform relatively better on the fine-grained action dimension compared to the time-order understanding dimension. While these models can accurately identify fine-grained actions, they struggle with comprehending the correct temporal order of these actions within a video. This limitation can lead to misinterpretations of the underlying information depending on temporal order. We present failure cases of this dimension in Fig. 10. For building more advanced world-centric Video-LMMs, it is crucial to enhance their ability to process and interpret event sequences accurately. Video-LMMs struggles in understanding the emotional and social context. For more reliable interaction between Video-LMMs and humans in practical scenarios, these models should comprehend the spatio-temporal scenes with social and contextual reasoning capabilities similar to humans. The lower performance of Video-LMMs on social and emotional contextual dimensions in CVRR-ES highlights their limitations and lack of understanding of scenes based on contextual cues. For instance, as shown in Fig. 11 (bottom row), GPT-4V struggles to comprehend a scene where a worker is attempting to prevent shoes from getting wet due to the rain by moving them under the shade. Instead, GPT-4V provides a response that contradicts the social cues present in the video. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2401.02418v1",
+                    "title": "Learning to Prompt with Text Only Supervision for Vision-Language Models",
+                    "abstract": "Foundational vision-language models such as CLIP are becoming a new paradigm\nin vision, due to their excellent generalization abilities. However, adapting\nthese models for downstream tasks while maintaining their generalization\nremains a challenge. In literature, one branch of methods adapts CLIP by\nlearning prompts using visual information. While effective, most of these works\nrequire labeled data which is not practical, and often struggle to generalize\ntowards new datasets due to over-fitting on the source data. An alternative\napproach resorts to training-free methods by generating class descriptions from\nlarge language models (LLMs) and perform prompt ensembling. However, these\nmethods often generate class specific prompts that cannot be transferred to\nother classes, which incur higher costs by generating LLM descriptions for each\nclass separately. In this work, we propose to combine the strengths of these\nboth streams of methods by learning prompts using only text data derived from\nLLMs. As supervised training of prompts is not trivial due to absence of\nimages, we develop a training approach that allows prompts to extract rich\ncontextual knowledge from LLM data. Moreover, with LLM contextual data mapped\nwithin the learned prompts, it enables zero-shot transfer of prompts to new\nclasses and datasets potentially cutting the LLM prompt engineering cost. To\nthe best of our knowledge, this is the first work that learns generalized\nprompts using text only data. We perform extensive evaluations on 4 benchmarks\nwhere our method improves over prior ensembling works while being competitive\nto those utilizing labeled images. Our code and pre-trained models are\navailable at https://github.com/muzairkhattak/ProText.",
+                    "authors": "Muhammad Uzair Khattak, Muhammad Ferjad Naeem, Muzammal Naseer, Luc Van Gool, Federico Tombari",
+                    "published": "2024-01-04",
+                    "updated": "2024-01-04",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction The Vision field is experiencing a new paradigm in its model-building approach with the emergence of foundational models [18, 23, 37, 47], which are large DNNs pretrained on web-scale data. Among these, Vision-Language models (VLMs) such as CLIP [37] stand out as the latest Method Do not require Transfer to images unseen datasets CoOp [50] \u2718 \u2713 Prompt learning CoCoOp [49] \u2718 \u2713 methods MaPLe [20] \u2718 \u2713 PromptSRC [21] \u2718 \u2713 DCLIP [29] \u2713 \u2718 Prompt ensembling WaffleCLIP-Concept [39] \u2713 \u2718 methods (LLM) CuPL [36] \u2713 \u2718 ProText (Ours) \u2713 \u2713 Table 1. Existing methods improve CLIP\u2019s generalization by learning prompts with image supervision or using non-transferable prompt ensembling with LLM knowledge. In contrast, our approach, ProText, effectively learns prompts with text-only supervision which are transferable to new datasets and classes. highlights which leverage contrastive pre-training on massive image-text pairs from the internet. During pre-training, CLIP learns to align image-text samples in a shared feature space. This allows CLIP to encode open-vocabulary concepts and generalize well to zero-shot recognition tasks. CLIP consists of two encoders to encode image and text inputs respectively. At inference, a hand-crafted prompt such as \u2018a photo of a CLS\u2019 is used as the text input. Text features of classes are compared with visual feature and class with highest similarity is assigned as predicted label. Improving the quality of text templates such as adding attributes [1], or class-specific details [19, 36] has shown to improve CLIP performance. However, designing highquality prompts that can best describe test image remains a key challenge, as image content is not known in advance. In literature, numerous techniques have been proposed to adapt CLIP for downstream recognition tasks. One branch of methods [6, 17, 27, 41, 49, 50] treat text prompts as learnable vectors and optimize them using task-specific objectives such as cross-entropy. As prompts are learned in the embedding space, this allows them to be used with classes and datasets beyond those on which they were trained on. While effective over the baseline CLIP, most of these methods require annotated image labels to optimize the prompts which is often impractical, especially in real-world scenarios such as medical imaging, remote sensing, security, arXiv:2401.02418v1  [cs.CV]  4 Jan 2024 \fAverage over 10 datasets 63 64 65 66 67 68 Performance (%) 65.15 65.15 63.88 65.74 65.81 66.3 67.23 CLIP CuPL CoOp CoCoOp PromptSRC MaPLe ProText (Ours) Figure 1. Without using any images for supervision, ProText with text-only training improves over CLIP, CuPL, and prior 16-shot image-supervised methods in challenging cross-dataset transfer settings. Prompt ensembling based CuPL performs same as CLIP as it cannot transfer class specific LLM templates to cross-datasets. surveillance, etc. Moreover, these methods tend to overfit on few-shot source samples and struggle to retain CLIP\u2019s generalization, especially in cross-dataset settings. Alternatively, several methods [29, 36] have adopted the training-free approach of prompt ensembling by leveraging the capabilities of Large Language Models (LLMs). Instead of using hand-crafted templates, these methods mine dataset or class specific descriptors and captions from LLMs to enrich text features. These enriched features aim to better represent content that could possibly occur in test images, leading to improvements over baseline CLIP. Although these methods do not require image information, the knowledge acquired from LLMs is mostly specific to each class and not directly transferable across unseen classes and datasets since no optimization is performed. Additionally, generating LLM descriptions for each concept separately incurs additional LLM serving and prompt engineering costs. In this work, we present a new paradigm to improve CLIP\u2019s generalization. Our motivation comes from combining the strengths of prompt learning and prompt ensembling approaches while effectively addressing their limitations. To this end, we introduce ProText: Prompt Learning with Text-Only Supervision. In contrast to previous methods, our approach instead proposes to learn prompts using text only data obtained from LLMs. As supervised training of prompts is not trivial due to image-free setting, we develop a novel training framework that allows prompts to learn and extract rich contextual knowledge from LLM data. Moreover, as LLM contextual knowledge is mapped within the learned prompts, it enables zero-shot transfer of prompts to new classes and datasets, potentially leading to a substantial reduction in LLM serving and prompt engineering cost. As shown in Tab. 1, our approach is different from prior methods as it does not require image samples to learn prompts, in addition the adapted CLIP transfers well to unseen classes and datasets, therefore addressing a key limitation of LLM-based prompt ensembling techniques. We demonstrate the effectiveness of ProText by performing extensive evaluations on 4 benchmarks. On challenging crossdataset transfer setting, ProText without using any visual information achieves an average gain of +2.08% over CLIP while surpassing the performance of previous best imagesupervised prompt learning method MaPLe [20] by +0.93% (Fig. 1). Further, ProText with text-only supervision performs competitively against prior methods in domain generalization, base-to-novel class, and text-only supervised setting. Our main contributions are summarized as follows: \u2022 We present a new approach for prompt learning in CLIP using text-only supervision. Our method harmonically combines the strengths of prompt learning and prompt ensembling methods to improve CLIP\u2019s generalization. \u2022 To optimize prompts with text-only data, we develop a training approach that allows prompts to learn a mapping by extracting rich contextual information from LLM data. \u2022 As LLM contextual knowledge is mapped within the learned prompts, this enables prompts to be directly used with new classes and datasets potentially cutting the additional LLM serving and prompt engineering cost. \u2022 We validate the effectiveness of our method through extensive experiments across four benchmarks. Our TextPro approach improves the generalization of CLIP across various settings and fares competitive to approaches that explicitly use labeled image samples for training. 2. Related Work Foundational Vision-Language models (VLMs). VLMs [18, 33, 37, 46\u201348] leverage joint image-text pretraining using internet-scale data in a self-supervised fashion. Representative VLMs like CLIP [37] and ALIGN [18] have utilized around 400M and 1B image-text pairs during their pre-training. Using the contrastive learning objective, VLMs learn rich multi-modal features by attracting together the features of paired images and texts while repelling un-paired image-text features in a joint feature space. The resulting model learns open-vocabulary concepts interpretable through natural language suitable for various downstream discriminative vision tasks such as open-vocabulary image classification [6, 20, 27, 31, 32, 50], detection [3, 10, 26, 30, 51], and segmentation [13, 24, 25]. Although promising, adapting VLMs effectively while maintaining their original generalization remains a crucial challenge. In this work, we propose a novel method to adapt CLIP with prompt learning through text modality supervision to improve its performance on vision modality tasks. Prompt Learning for VLMs. Prompt Learning [6, 9, 27, 40, 41, 49, 50] has emerged as an effective fine-tuning strategy to adapt large-scale models. This approach adds a small number of learnable embeddings along with model inputs which are optimized during training while the rest of the model is kept frozen. As the pre-trained model is unchanged during prompt learning, it has become particularly effective for VLMs such as CLIP, where maintaining the model\u2019s \foriginal generalizability is crucial. CoOp [50] is the pioneering prompt learning method for CLIP which learns text prompt embeddings to fine-tune CLIP. CoCoOp [49] improves CoOp\u2019s generalization by conditioning text prompts on visual features. MaPLe [20] proposes a multi-modal prompting framework to adapt both vision and language branches of CLIP. UPL [17] adopts an unsupervised prompt learning approach to finetune CLIP. PromptSRC [21] improves prompt learning from a regularization perspective by making use of additional loss functions during training. While these methods improve baseline CLIP performance, most of them require image samples with labels, which is less practical, and generating pseudo-labels is often less effective. In contrast, we present a novel prompt learning approach that improves CLIP generalization without relying on any visual samples during training. Training-Free Text Prompt Enhancement. With the emergence of LLMs such as GPT-3 [5], several approaches [29, 36, 39] have demonstrated their potential for improving zero-shot generalization of CLIP. Instead of using handcrafted templates for generating class features, these methods leverage LLMs to generate high-level concepts, class descriptions, and/or attributes which are used in one form or another to produce enriched text features. DCLIP [29] generates fine-grained per-class language descriptors and ensemble its similarity with image to produce classification scores. WaffleCLIP [39] matches DCLIP performance with random descriptors and show further gains by data-specific concepts generated via LLMs. CuPL [36] query LLMs to generate class-specific prompt descriptions for text prompt ensembling. Although effective, most of these approaches generate class-specific text data from LLMs which are not directly transferable to unseen classes and new datasets since no training is performed. On the other hand, we aim to leverage the same LLM data via novel text-only prompt learning technique which seamlessly allows the transfer of learned prompts toward unseen classes and new datasets. 3. Method Given the language interpretable nature of foundational VLMs such as CLIP [37], they are naturally suited for zeroshot recognition tasks. However, to achieve full potential of CLIP\u2019s generalization for downstream tasks, adaptation still appears to be necessary. Numerous approaches have since been proposed to adapt general knowledge of CLIP for userspecific downstream tasks. One line of methods adopts prompt learning [20, 27, 49, 50] to re-purpose CLIP features for downstream data. While effective, most of them require image samples with labels to learn the prompts, which is a hard requirement to meet. Another line of methods adopts training-free prompt ensembling techniques [29, 36, 39] with the help of LLMs. Although ensembling-based approaches do not require image information, the majority of these works generate class-specific LLM prompts that are not directly transferable to new classes and datasets. To this end, we present a new paradigm for learning generalized transferable prompts for VLMs using text-only supervision. Our proposed adaptation framework, ProText: Prompt Learning with Text only supervision aims to address the challenges of existing approaches by learning transferable prompts without relying on images. Fig. 2 shows our ProText framework. First, we curate text-only LLM template data using class names of a given dataset and a LLM such as GPT-3 [5]. As a text-supervised approach, ProText only requires CLIP text encoders during training. Specifically, we employ one frozen encoder with learnable prompts and a second frozen encoder without learnable prompts. Learnable prompts with class-name templates are input to the prompted text encoder to obtain the class-name template feature, and a frozen text encoder generates LLM template feature from its description obtained from LLM data. Next, we employ a contextual mapping training objective which maps class-name template feature to the LLM template feature. Contextual mapping allows the prompts to learn a mapping function that embeds rich contextual knowledge from LLM data within the prompt vectors. As prompts are learned in the embedding space, they are directly compatible with new classes and datasets. At inference, the learned prompts are shipped with CLIP model for standard zero-shot CLIP inference for visual recognition. Below we explain our proposed approach in detail. We first revisit CLIP and previous methods including Prompt Learning and Prompt Ensembling via LLMs in Sec. 3.1 and then we present our ProText approach in Sec. 3.2. 3.1. Preliminaries Contrastive Language-Image Pre-training (CLIP). CLIP consist of an image encoder f and a text encoder g which maps image and text input into visual and textual feature respectively. We denote CLIP parameters as \u03b8CLIP = {\u03b8f, \u03b8g} where \u03b8f and \u03b8g refer to the image and text encoder parameters, respectively. Input image X is divided into M patches which are linearly projected to produce patch tokens and a learnable class token CLS is prepended resulting in the final sequence as \u02dc X = {CLS, e1, e2, \u00b7 \u00b7 \u00b7 , eM}. The image encoder f encodes the input patches via multiple transformer blocks to produce a latent visual feature representation \u02dc f = f( \u02dc X, \u03b8f), where \u02dc f \u2208Rd. Next, the corresponding class label y is embedded in a text template, such as \u2018a photo of a [CLASS]\u2019 which can be formulated as \u02dc Y = {SOS, t1, t2, \u00b7 \u00b7 \u00b7 , tL, ck, EOS}. Here {tl|L l=1} and ck are the word embeddings corresponding to the text template and the label y, respectively while SOS and EOS are the learnable start and end token embeddings. The text encoder g encodes \u02dc Y via multiple transformer blocks to produce the latent text feature as \u02dc g = g( \u02dc Y , \u03b8g), where \u02dc g \u2208Rd. \fconcat Text Encoder Inference on visual domain Contextual Mapping LLM GPT-3 A persian cat is a \u00a0large, long-haired cat \u00a0with a broad face \u00a0and round eyes.\" \u00a0LLM Data How does a \u00a0persian cat\u00a0 look like? concat \"a photo of a \u00a0 persian cat\" \"a photo of a persian cat\" Class names Class name\u00a0 feature \u00a0LLM template feature Text prompts Frozen parameters Learnable parameters F5005F Text Encoder Text Encoder Image Encoder Training with text-only supervision Class name templates Figure 2. Overview of ProText framework. (Left) First, diverse captions are generated for training classes using LLM like GPT-3. During training, CLIP text encoders generate prompted class-name feature (\u02dc gp) from class-name templates with learnable prompts and frozen LLM template feature (\u02dc g) from LLM generated templates. Next, we employ contextual mapping loss to guide learnable prompts to learn a mapping from the prompted class-name feature to the LLM template feature containing more information about the class. This allows the learned prompts to exploit internal knowledge of text encoder complemented by LLM descriptions. (Right) At inference, learned prompts are used with class-name templates, and the standard zero-shot CLIP inference protocol is followed. Moreover, rich contextual information from LLM descriptions mapped within the learned prompts enables its transferability to new classes and datasets. For zero-shot inference, text features of text template with class labels {1, 2, \u00b7 \u00b7 \u00b7 , C} are matched with image feature \u02dc f as exp(sim(\u02dc g\u00b7 \u02dc f)\u03c4) PC i=1 exp(sim( \u02dc gi\u00b7 \u02dc f)\u03c4), where sim() denotes the cosine similarity and \u03c4 is the temperature. Prompt Learning with CLIP. Being a parameter efficient tuning method, prompt learning has emerged as a popular technique to adapt vision-language models like CLIP. Since most of the model is kept frozen during adaptation, prompt learning aims to reduce overfitting. Learnable prompts are appended either at the image side [2], text encoder side [49, 50], or both sides. In this work, we learn hierarchical prompts at the text encoder named Deep Language Prompting (DLP) [20] formulated as follows. T learnable language prompts Pt = {p1 t, p2 t, \u00b7 \u00b7 \u00b7 , pT t } are appended with text input tokens, resulting in \u02dc Yp = {SOS, Pt, t1, t2, \u00b7 \u00b7 \u00b7 , tL, ck, EOS}. The text encoder processes \u02dc Yp and prompted text feature is obtained as \u02dc gp = g( \u02dc Yp, \u03b8g). We use deep prompting which learns hierarchical prompts at subsequent transformer blocks of text encoder. Visual feature \u02dc f is obtained without utilizing learnable prompts. To adapt CLIP on image classification task on dataset D, prompts Pt are optimized in a supervised fashion using labeled image samples with cross-entropy loss, LCE. LCE = arg min Pt E(X,y)\u223cD L(sim( \u02dc f, \u02dc gp), y). (1) Prompt Ensembling with LLM descriptions. Several methods have recently proposed to adapt CLIP via trainingfree prompt ensembling techniques. The majority of these approaches leverage the capabilities of LLMs to mine rich descriptions, attributes, or high-level concepts of class names. The corresponding text features are either averaged [36] or the similarity score of each attribute with the image is calculated to obtain classification scores [39] [29]. In this work, we focus our comparison with a strong ensembling baseline CuPL [36]. Specifically, a Large Language Model F such as GPT-3 [5] is used to generate classspecific descriptions for class labels {1, 2, \u00b7 \u00b7 \u00b7 , C} using queries such as \u2018How does a CLASS look like\u2019. Text features of the same class description are averaged together, which serves as the ensembled text features. Finally, zero-shot inference is performed with those ensembled text features. 3.2. Prompt Learning with Text-Only Supervision While image-supervised prompt learning and LLM-based prompt ensembling methods have proven effective in adapting CLIP, they face notable challenges as outlined below. Visual data dependency. Existing prompt learning methods require visual samples with labels to optimize prompts using Eq. 1. However, collecting samples and labels is difficult in critical scenarios like medical images, remote sensing, and surveillance. Pseudo-labels alleviate label dependency but they are often less effective. Furthermore, these methods tend to overfit CLIP to source data distributions and compromise generalization across cross-datasets. For instance, CoOp utilizing labeled source samples reduces average CLIP performance by 1.27% on 10 cross-datasets. LLM Prompts transferabilty limitation. LLM-based prompt ensembling approaches like CuPL [36] generate class-specific LLM descriptions that cannot be directly transferred to unseen classes and datasets. While opensource LLMs exhibit lower performance, proprietary ones such as GPT-3 are required for generating data for new classes and datasets leading to additional serving costs. \fOur work aims to address the aforementioned limitations within a unified framework. Below we detail our strategy for curating text-to-text data via LLMs for training, followed by our text-only prompt learning framework. 3.2.1 Text-Only LLM data for Prompt Learning As discussed in Sec. 3.1, optimizing prompts for downstream datasets typically requires image-labels pairs. Since we explicitly aim to bypass this requirement, we first leverage LLMs to curate text data for prompt learning which consists of text inputs and text outputs. Given a set of classes {ci}C i=1, we prepare text inputs {Li inputs}C i=1 by wrapping each class name in a standard hand-written text template, Li inputs = \u2018a photo of a ci\u2019. Next, we prepare text outputs corresponding to the Linputs. Specifically, we query GPT-3 model to generate detailed descriptions for each class name ci. Similar to CuPL [36], we prompt GPT-3 with different queries Q conditioned on class names such as \u2018How does a ci look like?\u2019 and \u2018How can you identify a ci?\u201d to obtain text outputs, Li outputs = F(Q|ci). Similar to [36], we generate M text outputs per query Q and use N different queries, resulting in M \u00d7 N text outputs per class category. We associate all Loutputs with the corresponding single Linputs for each class ci. As LLMs are pre-trained on internet-scale text corpora, they possess the capability of generating very diverse and highquality descriptions and captions for different class categories which results in high-quality text outputs. Finally we combine Linputs and Loutputs to create LLM based text-to-text data for text only prompt learning, DPROMPT = {Li inputs, Li outputs}M\u00d7N\u00d7C i=1 . We refer the readers to supplementary for additional details on the choice of LLM prompts and examples of DPROMPT. 3.2.2 Contextual mapping with Prompt Learning To leverage LLM text-to-text data DPROMPT for learning generalized transferable prompts, we propose a contextual mapping strategy that effectively learns a mapping function that maps standard class name templates such as \u2018a photo of a ci\u2019 to the text feature generated from a LLM description which contains more information about the class ci. In other words, contextual mapping allows learnable prompts to map Linputs to Loutputs in the text feature space of CLIP. The mapping function is realized in the form of learnable prompt vectors, which we found to be more effective in our ablations as compared to other techniques such as adapters via linear projection and MLP. For an ith training sample from DPROMPT consisting of a text-to-text pair {Linputs, Loutputs}i, we obtain prompted class-name feature \u02dc gp for Li inputs using learnable prompts and frozen LLM feature \u02dc g for Li outputs without the prompt vectors within the pre-trained latent space of CLIP. We then impose a contextual mapping constraint between \u02dc gp and \u02dc g text features as follows, Lmapping = 1 d d X i=1 ||\u02dc gp \u2212\u02dc g||2 2. (2) As shown above, we utilize MSE loss objective to enforce contextual mapping from Li inputs to Li outputs. We study other choices of consistency objectives in our ablations (Sec. 4.7). Motivation for Lmapping. Contextual mapping objective allows learnable prompts to exploit internal knowledge of text encoder of CLIP to generate rich contextual features aligned with the LLM descriptions (Li outputs) for a given class. This strategy effectively learns prompts without using any visual information and when trained using all training classes together, it enables prompts to capture versatile and generalized context from the LLM descriptions. These contextaware prompts become adaptable for use with any dataset and effectively enable the transferability of class-specific LLM descriptions to unseen classes and datasets. Consequently, this substantially reduces the per-dataset overhead associated with LLM serving and prompt engineering. Inference. Once text prompt vectors are optimized through our TextPro framework in the text domain, they become ready to be shipped with CLIP for downstream visual domain inference with a standard zero-shot CLIP inference setup. As shown in Fig. 2 (right), the learned prompts Pt are fused with each given class name to produce prompted text features {\u02dc gp}C i=1. Finally, zero-shot inference is performed with the prompted text features and the input image feature \u02dc f to produce classification scores on test images. 4. Experiments 4.1. Evaluation settings We perform evaluations in 4 benchmark settings. Prompt ensembling methods and ProText utilize text-only LLM data for adapting CLIP while image-supervised prompt learning methods use image-label pairs for training. Base-to-Novel Generalization. This setting evaluates the generalization of methods within a dataset. Following previous methods [49, 50], we split each dataset into base and novel classes. Models are trained on base classes and evaluated on the test set of base and novel classes respectively. Cross-dataset transfer. This setting evaluates the generalization ability of models trained on ImageNet-1k [8] source dataset by directly transferring it on cross-datasets. Domain Generalization. We evaluate the robustness of different methods on out-of-distribution datasets. We train \fMethod ImageNet Acc. 1: CLIP (ICML\u201921) 66.72 2: CLIP-Attribute 67.60 3: CLIP-80 68.32 4: DCLIP (ICLR\u201923) 68.03 5: Waffle CLIP (ICCV\u201923) 68.34 6: CuPL (ICCV\u201923) 69.62 7: ProText-Attribute 68.05 8: ProText-80 68.48 9: ProText-CuPL 70.22 Table 2. With the same amount of text data, learning contextual prompts with text-only supervision improves CLIP performance in comparison to the prompt ensembling techniques. Dataset CLIP [37] CuPL [50] ProText (Ours) Base Novel HM Base Novel HM Base Novel HM ImageNet 72.43 68.14 70.22 74.30 68.14 71.09 75.00 71.38 73.14 Caltech101 96.84 94.00 95.40 97.22 94.00 95.58 98.06 95.63 96.83 OxfordPets 91.17 97.26 94.12 94.42 97.26 95.82 94.95 98.00 96.45 StanfordCars 63.37 74.89 68.65 63.54 74.89 68.75 64.54 76.08 69.84 Flowers102 72.08 77.80 74.83 74.36 77.80 76.04 74.36 78.44 76.35 Food101 90.10 91.22 90.66 89.93 91.22 90.57 90.20 91.98 91.08 Aircraft 27.19 36.29 31.09 30.61 36.29 33.21 30.91 34.13 32.44 SUN397 69.36 75.35 72.23 76.02 75.35 75.68 76.14 79.14 77.61 DTD 53.24 59.90 56.37 62.85 59.90 61.34 63.08 61.59 62.33 EuroSAT 56.48 64.05 60.03 59.64 64.05 61.77 59.71 80.97 68.73 UCF101 70.53 77.50 73.85 75.28 77.50 76.37 75.54 79.50 77.47 Average 69.34 74.22 71.70 72.56 74.22 73.38 72.95 76.98 74.91 Table 3. Base-to-novel setting. ProText enables the transferability of learned prompts to new classes and improves over CuPL [36]. models on the ImageNet-1k source dataset and evaluate its performance on four ImageNet variants with domain shifts. Supervised setting. We provide performance comparison of ProText with CuPL[36] with text-only data per dataset. Datasets. For the aforementioned benchmarks, we use same datasets as followed by previous works [20, 21, 49, 50]. For cross-dataset transfer, domain generalization, and base-to-novel generalization settings, we use 11 image datasets that cover multiple recognition tasks. These includes ImageNet [8] and Caltech101 [11] which contains generic objects; OxfordPets [35], StanfordCars [22], Flowers102 [34], Food101 [4], and FGVCAircraft [28] for finegrained classification, SUN397 [45] for scene recognition, UCF101 [42] for action recognition, DTD [7] for texture classification, and EuroSAT [14] for satellite images categorization. For domain generalization setting, we train models on ImageNet [8] as a source dataset and use ImageNetA [16], ImageNet-R [15], ImageNet-Sketch [44] and ImageNetV2 [38] for out of distribution dataset evaluation. Implementation details. We use a publically available pretrained ViT-B/16 CLIP model from OpenAI [37]. We train ProText with Deep Language Prompting in the first 9 transformer blocks of the CLIP text encoder. For cross-dataset transfer and domain generalization setting, we train ProText using T = 4 and T = 16 language prompts with 10 and 200 epochs respectively. Similar to [44], ProText and zeroshot CLIP use additional concepts where available with its prompts such as \u2018a photo of a CLS, a type of flower\u2019 for OxfordFlowers [34]. For base-to-novel and supervised text-only settings, ProText uses optimal prompt length and epoch configuration for each dataset. Optimal training configuration is obtained through hyper-parameter search on validation split of datasets. To generate text-only data, we utilize GPT-3 DaVinci-002 model [5] and generate classspecific descriptions using the LLM prompts provided by CuPL [36]. We use publicly available CuPL data and generate descriptions for datasets not provided by CuPL. AdamW optimizer is used with 5 warm-up epochs for training. We use a single 16-GB V100 to train our models. Refer to supplementary material for additional implementation details. 4.2. Effectiveness of Text-Only Supervision We first present an ablation to motivate our approach of learning prompts with text-only supervision. We train ProText with 3 types of text data and evaluate performance on ImageNet-1k [8]. ProText-Attribute uses 46 templates from [1] which corresponds to common image attributes such as rotation, blurriness, etc. ProText-80 is trained on standard 80 templates provided by CLIP [37] and ProText-CuPL is trained on class-specific LLM data employed by our main baseline CuPL [36] for its ensembling approach. In Tab. 2, we compare ProText with CLIP and recent LLM-based ensembling methods. Prompt ensembling with attribute templates and 80 templates improves over CLIP single template result. Among the LLM-based ensembling methods, CuPL provide highest performance of 69.62%. In contrast, ProText uses a learning-based approach and shows competitive performance against prompt ensembling methods using the same text data. ProText-Attribute provides gain of 0.45% over CLIP-Attribute while roughly maintaining its performance against CLIP-80. When equipped with CuPL LLM text-data, ProText surpasses CuPL by 0.60% leading to highest performance against all methods. These results motivate our approach that instead of prompt ensembling, one can achieve competitive results by utilizing the same available text data to learn prompts. Next, we demonstrate the generalization of ProText such that the learned prompts transfer well across new classes and datasets. 4.3. Base to novel class generalization We now present results in base-to-novel class generalization setting where training data for only base classes are available and the model is evaluated on both base and novel classes. For CuPL [36], we use base-class LLM templates for base classes and zero-shot CLIP results for its novel classes. For ProText, we use base-class LLM templates for training and transfer the learned prompts for novel classes. Results are shown in Tab. 3. CuPL outperforms zeroshot CLIP on base classes while maintaining its performance on novel classes as LLM prompts for new classes are not available. ProText shows consistent improvements over CuPL on base classes for 11 datasets. Furthermore, \fSource Target ImageNet Caltech101 OxfordPets StanfordCars Flowers102 Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average Methods utilizing labeled visual samples CoOp 71.51 93.70 89.14 64.51 68.71 85.30 18.47 64.15 41.92 46.39 66.55 63.88 Co-CoOp 71.02 94.43 90.14 65.32 71.88 86.06 22.94 67.36 45.73 45.37 68.21 65.74 MaPLe 70.72 93.53 90.49 65.57 72.23 86.20 24.74 67.01 46.49 48.06 68.69 66.30 PromptSRC 71.27 93.60 90.25 65.70 70.25 86.15 23.90 67.10 46.87 45.50 68.75 65.81 Zero-shot & Prompt ensembling methods CLIP 66.72 92.98 89.13 65.29 71.30 86.11 24.90 62.59 44.56 47.84 66.83 65.15 CuPL 69.62 92.98 89.13 65.29 71.30 86.11 24.90 62.59 44.56 47.84 66.83 65.15 Prompt learning with text-only supervision ProText (Ours) 69.80 94.81 91.01 66.00 72.35 86.66 24.72 67.34 47.93 51.86 69.60 67.23 Table 4. Crossdataset transfer setting. CuPL and CLIP perform same for cross-datasets as CuPL source data cannot transfer to cross-datasets. Image-based models are trained on 16-shot ImageNet samples. ProText employ same ImageNet data as CuPL for prompt learning. Source Target ImageNet -V2 -S -A -R Avg. Methods utilizing labeled visual samples CoOp 71.51 64.20 47.99 49.71 75.21 59.28 CoCoOp 71.02 64.07 48.75 50.63 76.18 59.91 MaPLe 70.72 64.07 49.15 50.90 76.98 60.27 Zero-shot & Prompt ensembling methods CLIP 66.72 60.83 46.15 47.77 73.96 57.18 CuPL 69.62 63.27 49.02 50.72 77.05 60.01 Prompt learning with text-only supervision ProText (Ours) 70.22 63.54 49.45 51.47 77.35 60.45 Table 5. Domain generalization. Prompt learning methods are trained on imageNet and evaluated on datasets with domain shifts. Dataset CLIP CuPL ProText \u2206 ImageNet 66.72 69.60 70.22 +0.62 Caltech101 92.98 94.32 95.29 +0.97 DTD 44.56 53.96 54.02 +0.06 EuroSAT 47.84 60.27 58.53 -1.74 StanfordCars 65.29 65.95 66.77 +0.82 Flowers102 71.30 73.85 74.42 +0.57 Aircraft 24.90 27.66 29.01 +1.35 SUN397 62.59 69.00 69.76 +0.76 OxfordPets 89.13 91.11 92.72 +1.61 UCF101 66.83 70.63 71.45 +0.82 Food101 86.11 86.11 86.68 +0.57 Average 65.15 69.31 69.90 +0.59 Table 6. ProText results with text supervision on each dataset. We compare ProText with CLIP and CuPL. Gains of ProText over CuPL are shown in blue. with the same LLM base-class data as CuPL, ProText effectively transfers learned prompts towards novel classes and improves CLIP and CuPL novel class performance by 2.76% averaged across 11 datasets. This shows the advantage of ProText prompts to benefit unseen class performance potentially reducing the LLM prompt serving cost by half. 4.4. Cross-dataset transfer In cross-dataset transfer setting, we compare ProText with CLIP [37], CuPL [36], and image-supervised prompt learning methods. Since class-specific ImageNet LLM prompts limit its transfer to other datasets in CuPL, we assign CLIP results to CuPL for cross-datasets. Image-supervised methods [20, 21, 49, 50] are trained with 16-shot ImageNet data. We show our main comparison results in Tab. 4. CuPL improves ImageNet performance of CLIP by ensembling ImageNet LLM prompts, while its cross-dataset results remain the same as CLIP. In contrast, ProText effectively addresses the transferability challenges of CuPL using generalized prompts trained with the same ImageNet LLM data. Since ProText allows generalization to unseen datasets, these learned prompts can directly be used with CLIP for cross-datasets leading to absolute average gains of +2.1% against CLIP and CuPL. With ProText, one can notably reduce proprietary LLM serving and prompt engineering costs as prompts learned on one dataset are effectively transferable to other datasets. We next compare ProText with strong 16-shot image-supervised methods. Without using any visual samples, ProText demonstrates effective generalization on cross-datasets and consistently surpasses previous state-of-the-art MaPLe on 9/10 datasets leading to the highest average accuracy of 67.23%. This highlights that text-only methods like ProText can lead to better generalization of CLIP as compared to image-supervised methods which tend to overfit on the source sample distributions. 4.5. Domain generalization experiments We present the results for domain generalization task in Table 5. As the domain shift variants of ImageNet share class names with ImageNet, CuPL employs prompt ensembling for each dataset and provides an average gain of +2.84% over CLIP. In contrast, ProText with learned prompts shows an additional gain of +0.44% against CuPL averaged over 4 datasets. Moreover, ProText fairs competitively with imagesupervised methods by showing consistent improvements over CoOp, CoCoOp, and MaPLe. These results suggest that text-only supervision methods like ProText can serve as an effective alternative to improve the robustness of VLMs when no visual information is available for training. \fFigure 3. Ablation: Prompt length (left) and prompt depth (right). Method ImageNet Top1. 1: ProText-contrastive loss 68.12 2: ProTextL1 loss 69.96 3: ProText-MSE loss 70.22 Table 7. Ablation of choice of loss for contextual mapping. MSE loss provides highest results. Method ImageNet Top1 1: ProText-80 templates 68.48 2: ProText-Alpaca 67.10 3: ProText-GPT-3 70.22 Table 8. Effect on performance with different text data for training. GPT-3 text data show highest results. Method ImageNet Top1. 1: Linear Adaptor 69.36 2: MLP Adaptor 69.24 3: Prompt Learning 70.22 Table 9. Ablation on the choice of mapping network. Prompt Learning shows optimal performance. Correct class confidence (%) \u2191Incorrect class confidence (%) \u2193 Method DTD SUN Caltech UFC DTD SUN Caltech UFC CLIP 30.5 49.3 84.5 56.4 1.51 0.13 0.16 0.44 ProText 33.1 54.2 89.1 59.5 1.45 0.12 0.11 0.40 Table 10. Confidence score analysis: ProText trained on ImageNet improves its logit confidence for correct classes in unseen datasets. 4.6. Supervised text-only training In this setting, we compare ProText with CuPL for each dataset trained on LLM template data and the results are shown in Tab. 6. While utilizing the same LLM data, ProText achieves consistent improvements over CuPL on 10/11 datasets with an average gain of +0.59%. This reflects the generalization of the ProText approach across various diverse image datasets where it better utilizes LLM data within the learned prompts. We also compare ProText with image-supervised methods and observe that ProText fares competitively with approaches utilizing up to 2-shot samples for training. This shows ProText as a potential alternative to image-supervised methods in extremely low-data regimes. Refer to supplementary for additional results. 4.7. Ablative analysis On understanding ProText prompts. In Table. 10, we present average confidence scores obtained from ProText logits trained on ImageNet-1k text data when applied to Figure 4. (Left) Effect of LLM data size on performance. (Right) Ablation on ensembling LLM descriptions for training ProText. cross-datasets. Compared to CLIP, ProText exhibits increased confidence scores for correct classes across various datasets, while marginally decreasing confidence scores for incorrect classes. This suggests that the prompts learned on ImageNet-1k provide complementary and transferable contextual cues, leading to improved results. We conjecture that ProText prompts potentially improve the classification of test samples situated near the decision boundary due to higher confidence for correct classes. Refer to the supplementary section for qualitative and additional analysis. Loss metric in contextual mapping. We ablate on choice of loss used for the contextual mapping module in Tab. 7. Distance-based losses improve over contrastive loss. We conjecture that contrastive loss treats samples of same class labels in a same batch as negatives leading to noisy training. Choice of LLM for generating text data. ProText by default uses GPT-3 [5] LLM to obtain text templates for training. Here we ablate on an open-source Alpaca [43] model as an alternative choice. As shown in Tab. 8, ProText with Alpaca templates performs worse than ProText-80 template and ProText-GPT-3. We observed that Alpaca templates are often noisy while GPT-3 descriptions contain more enriched class details which results in better performance. Prompt learning verses adapter. While ProText employs prompt learning to learn contextual mapping from LLM templates, here ablations on adapters in Tab. 9. Similar to [12], we attach adapter at the output of CLIP text encoder. Adapters perform lower as compared to prompting. We conjecture that adapter completely transforms text features and loses CLIP generalization. In contrast, prompt learning append learnable vectors with CLIP text input without significant replacement and learns effective mapping function. Training data size for text-supervision. To assess the effect of LLM template data size on ProText, we ablate on the number of descriptions per class in Fig. 4 (left). Increasing descriptions for each class consistently improves the results. This suggests that we could further boost ProText performance as quality and size of text data increases. Ensembling in ProText training. ProText uses multiple descriptions per class and enforce mapping of class-name template feature to feature of each LLM description for that class. We conduct an alternative experiment by ensembling \fa single feature from multiple LLM descriptions per class and enforce mapping on ensembled LLM feature. As shown in Fig. 4 (right), ProText-ensembled performs lower than ProText with individual samples. We conjecture that learning on each description allows the model to utilize additional context present in each description. Ensembling can potentially mask out less frequent details available in text. Prompt length and prompt depth. Fig. 3 (left) shows the effect of prompt length for training ProText. Setting prompt length to 16 leads to optimal performance. Fig. 3 (right) shows the effect of prompt depth on final performance where prompt depth of 9 shows optimal results. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2307.06948v2",
+                    "title": "Self-regulating Prompts: Foundational Model Adaptation without Forgetting",
+                    "abstract": "Prompt learning has emerged as an efficient alternative for fine-tuning\nfoundational models, such as CLIP, for various downstream tasks. Conventionally\ntrained using the task-specific objective, i.e., cross-entropy loss, prompts\ntend to overfit downstream data distributions and find it challenging to\ncapture task-agnostic general features from the frozen CLIP. This leads to the\nloss of the model's original generalization capability. To address this issue,\nour work introduces a self-regularization framework for prompting called\nPromptSRC (Prompting with Self-regulating Constraints). PromptSRC guides the\nprompts to optimize for both task-specific and task-agnostic general\nrepresentations using a three-pronged approach by: (a) regulating prompted\nrepresentations via mutual agreement maximization with the frozen model, (b)\nregulating with self-ensemble of prompts over the training trajectory to encode\ntheir complementary strengths, and (c) regulating with textual diversity to\nmitigate sample diversity imbalance with the visual branch. To the best of our\nknowledge, this is the first regularization framework for prompt learning that\navoids overfitting by jointly attending to pre-trained model features, the\ntraining trajectory during prompting, and the textual diversity. PromptSRC\nexplicitly steers the prompts to learn a representation space that maximizes\nperformance on downstream tasks without compromising CLIP generalization. We\nperform extensive experiments on 4 benchmarks where PromptSRC overall performs\nfavorably well compared to the existing methods. Our code and pre-trained\nmodels are publicly available at: https://github.com/muzairkhattak/PromptSRC.",
+                    "authors": "Muhammad Uzair Khattak, Syed Talal Wasim, Muzammal Naseer, Salman Khan, Ming-Hsuan Yang, Fahad Shahbaz Khan",
+                    "published": "2023-07-13",
+                    "updated": "2023-08-24",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Vision-Language (VL) models, such as CLIP [35] and ALIGN [20], have demonstrated remarkable generalization capabilities for downstream tasks. These VL models *Joint first authors. \u0000uzair.khattak@mbzuai.ac.ae are trained on large-scale web data with a contrastive loss, which allows them to encode open-vocabulary concepts by aligning pairs of images and texts in a shared embedding space. The resulting model is suited for downstream tasks such as open-vocabulary image recognition [23], object detection [11], and image segmentation [29]. Prompt learning has emerged as a more efficient alternative to fine-tuning large-scale models, as shown in recent studies [58, 59, 3, 17, 40, 28]. This approach introduces a few learnable prompt vectors to adapt models like CLIP for downstream tasks while keeping the pre-trained model weights fixed. However, since the prompts are optimized with respect to the task-specific objective [59], such as the cross-entropy loss for ImageNet [6] classification, the prompted model tends to overfit to the task-specific data distribution as the training progresses. This can result in the prompted model losing the original generalization capability of the frozen CLIP model towards new tasks. Therefore, learning prompts that can model both task-specific and task-agnostic representations remain a major challenge for adapting foundational VL models. This work seeks to self-regulate prompts to address the issue of prompt overfitting. To this end, we propose a selfregularizing framework that guides the prompts to jointly optimize for both task-specific and task-agnostic general representations using a three-pronged approach. a) Regulating via Mutual Agreement Maximization: We observe that generalizable zero-shot knowledge is preserved within frozen pre-trained VL model features but they lack taskspecific knowledge. In contrast, prompts achieve better adaptation to a given task but with reduced generalizability to new tasks. Therefore, we propose to regulate learned prompts by maximizing the agreement between prompted and frozen VL model features while adapting them to the downstream task. b) Regulating with the Self-ensemble: In the early epochs, prompts act are not mature to capture contextual information. As the training progresses, prompts tend to become more task-specific. Therefore we deploy a weighted prompt aggregation technique to prompts during training to regulate them using their self-ensemble over the arXiv:2307.06948v2  [cs.CV]  24 Aug 2023 \fFigure 1: (Left): Existing prompt learning approaches rely on task-specific objectives that restrict prompt learning to learn a feature space suitable only for downstream tasks and consequently lose the generalized knowledge of CLIP (shown in purple). Our self-regulating framework explicitly guides the training trajectory of prompts towards the closest point between two optimal solution manifolds (solid line) to learn task-specific representations while also retaining generalized CLIP knowledge (shown in green). (Middle): Averaged across 11 image recognition datasets, PromptSRC surpasses existing methods on the base-to-novel generalization setting. (Right): We evaluate our approach on four diverse image recognition benchmarks and it overall shows competitive results compared to the previous state-of-the-art. training phase. The weights are sampled from a Gaussian distribution which suitably aggregates the useful knowledge learned by prompts at different training epochs. c) Regulating with Textual Diversity: We note that unlike having multiple image samples per category for the vision encoder, there is only a single textual label available for each class. Therefore, imposing the mutual agreement constraints on multi-modal features results in sub-optimal performance due to the lack of diversity in text-side labels for the text encoder. We overcome this disparity and regulate the prompts through diverse text label templates for each class. Overall, our approach explicitly steers prompts to learn a representation space that maximizes its performance on downstream tasks without compromising pre-trained CLIP generalization (Fig. 1: Left). We demonstrate the effectiveness of PromptSRC on four representative tasks. On the base-to-novel generalization benchmark across 11 datasets (Fig. 1: Middle), our method achieves average gains of +1.42% in harmonic-mean over the state-of-the-art MaPLe [22] and +8.26% over CLIP. Further, PromptSRC achieves competitive results in cross-dataset transfer, domain generalization, and few-shot image recognition (Fig. 1:Right). In summary, our self-regulating prompt learning framework has the following main contributions: \u2022 We address the inherent problem of prompt overfitting for adapting foundational models through selfregularization. Our framework explicitly guides the prompts to jointly acquire both task-specific knowledge and task-agnostic generalized knowledge by maximizing the mutual agreement between prompted and frozen VL model features. (\u00a73.2.1) \u2022 We suggest a weighted self-ensembling strategy for prompts that captures their complementary features learned at different epochs during training and enhances their generalization performance. (\u00a73.2.2) \u2022 To overcome the significant diversity mismatch between the text and visual domains, we propose textside diversity which complements limited textual labels via multiple text augmentations and regularizes prompts to learn more generalized contexts. (\u00a73.2.3) 2. Related Work Vision Language models: Foundational vision-language (VL) models [35, 20, 54, 49, 51] leverage both visual and textual modalities to encode rich multi-modal representations. These models are pre-trained on a large corpus of image-text pairs available on the internet in a selfsupervised manner. For instance, CLIP [35] and ALIGN [20] utilize around 400M and 1B image-text pairs, respectively, to train their multi-modal networks. During pre-training, contrastive loss is commonly used as a selfsupervision loss. This loss pulls together the features of paired images and texts while pushing away the unpaired image-text features. VL models possess a strong understanding of open-vocabulary concepts, making them suitable for various downstream vision and vision-language applications [12, 56, 38, 30, 60, 13, 32, 53, 26, 36, 8]. However, transferring these foundational models for downstream tasks without compromising on their original generalization ability still remains a major challenge. Our work aims to address this problem by proposing a novel regularization framework to adapt VL models via prompt learning. Prompt learning: Prompt learning is an alternative finetuning method for transferring a model towards downstream tasks without re-learning the trained model parameters. This approach adapts a pre-trained model by adding a small number of new learnable embeddings at the input known as prompt tokens. Due to its efficiency in terms of parameters and convergence rate, prompt learning is found to be of great interest for adapting foundational models like CLIP for vision [21, 57, 45, 46] and vision-language tasks [59, 58, 61, 7]. CoOp [59] fine-tunes CLIP by optimizing a continuous set of prompt vectors in its language branch for few-shot image recognition. Bahng et al. [1] perform visual prompt tuning on CLIP by learning prompts \fon the vision branch. [3] and [28] propose to learn multiple sets of prompts for learning different contextual representations. CoCoOp [58] highlights the overfitting problem of CoOp and proposes to condition prompts based on visual features for improved performance on generalization tasks. MaPLe [22] proposes a multi-modal prompt learning approach by learning hierarchical prompts jointly at the vision and language branches of CLIP for better transfer. Our approach builds on a variant [37] where prompts are learned at both the vision and language encoder of CLIP. Network regularization: Incorporating regularization techniques in neural networks has been proven to enhance their generalization capabilities [25]. Regularization strategies can be broadly classified into two streams. The first stream consists of constraint-based regularization methods, such as weight decay [27] and adversarial training [50]. These techniques introduce additional constraints to the learning process, which helps to prevent overfitting. The second stream of regularization techniques involves modifying the inputs, model parameters, or annotations. This category includes methods such as data augmentations [52, 55, 5], dropout [42], model ensembling [18, 47], label smoothing [43] and batch normalization [19]. Our method aims to enhance the generalization performance of learned prompts via a multi-stage regularization framework, which takes inspiration from both streams of regularization techniques mentioned above. However, to the best of our knowledge, this is the first effort to regularize prompts during adaptation by jointly attending to the original VL model feature space, the training trajectory of prompts as well as the diversity of textual inputs for the multi-modal models. 3. Proposed Method Prompt learning aims to adapt the general knowledge of VL foundational models like CLIP without full fine-tuning [59, 58, 3]. Since prompts are the only learnable vectors, this strategy aims to retain the pretrained generalized feature representations of CLIP while re-purposing them for downstream task-specific data via prompts. Although effective, they are susceptible to overfitting on the supervised downstream task (see Fig. 2) and their generalization towards new classes and datasets reduces as compared to the original zero-shot pre-trained CLIP. Our work seeks to address the overfitting behavior of prompts. Unlike prior prompting approaches that improve generalization mainly from the model architecture perspective [58, 22], we motivate our work from the regularization perspective. As evidenced by the strong zero-shot performance, pre-trained CLIP features possess robust generalization characteristics. However, naively training prompts with the supervised task-specific loss struggles to retain these general attributes from the frozen CLIP. To this end, we propose a self-regularizing framework to explicitly guide the Figure 2: Naively training prompts with standard supervised objectives improves supervised class performance but leads to poor generalization as training schedule increases. Our PromptSRC method with explicit prompts consistency constraints improves on base classes as well as shows improvements on novel classes. training trajectory of prompts to maximize its interaction with the pre-trained knowledge stored in the frozen CLIP. Fig. 3 shows our overall methodology which optimizes the prompts as follows. a) Regularization through mutual agreement maximization: We impose an explicit consistency constraint between prompted features and the pretrained CLIP features within the CLIP embedding space. b) Regularization through prompt self-ensembling: To further reduce overfitting, we propose a Gaussian weighted average of the prompt vectors learned at different training epochs. This ensemble-level regularization aggregates information from learned prompts across different epochs for improved generalization. c) Regularization through textual diversity: Unlike having multiple images for each class, the text labels during fine-tuning are limited and bounded by the number of class categories. We incorporate textual augmentations by defining multiple text label templates for a given class. The ensemble of textual labels regularizes the prompts for better generalization during optimization. We now continue by explaining our methodology in detail. We first revisit CLIP and CLIP-based prompt learning in Sec. 3.1. This is followed by the explanation of our selfregulating prompt learning approach in Sec. 3.2. 3.1. Preliminaries We denote the CLIP image and text encoders as f and g, respectively and their pretrained parameters as \u03b8CLIP = {\u03b8f, \u03b8g} where \u03b8f and \u03b8g refer to the image and text encoder parameters, respectively. The input image X \u2208 RC\u00d7H\u00d7W is divided into M patches followed by a projection to produce patch tokens. Further, a learnable class token ecls is appended with the input patches as \u02dc X = {ecls, e1, e2, \u00b7 \u00b7 \u00b7 , eM}. The image encoder f encodes the input patches via multiple transformer blocks to produce a latent visual feature representation \u02dc f = f( \u02dc X, \u03b8f), where \u02dc f \u2208Rd. Next, the corresponding class label \fFigure 3: Our proposed PromptSRC framework for self-regulating prompt learning. CLIP encoders are used to generate prompted ( \u02dc fp, \u02dc gp) and pre-trained ( \u02dc f, \u02dc g) features at the image and text sides. First, we introduce textual diversity (\u00a73.2.3) and define textual augmentations to produce a diverse set of frozen VL textual features, which are averaged to represent the pre-trained VL text features (\u02dc g). Next, we employ Mutual Agreement Maximization constraints (LSCL) to regulate the prompts, which ensure that the prompted features align well with the pre-trained VL representations at both the feature and logit levels (\u00a73.2.1). As CLIP is frozen, we use the same VL encoders to obtain both types of features. Further, our prompt self-ensembling combines the strengths of prompts learned at different epochs (P1, P2 \u00b7 \u00b7 \u00b7 PE) during training via Gaussian weighted sampling (\u00a73.2.2). The ensembled visual and textual prompts are then used for the final inference. y is wrapped within a text template such as \u2018a photo of a {class label}\u2019 which can be formulated as \u02dc Y = {tSOS, t1, t2, \u00b7 \u00b7 \u00b7 , tL, ck, tEOS}. Here {tl|L l=1} and ck are the word embeddings corresponding to the text template and the class label, respectively while tSOS and tEOS are the learnable start and end token embeddings. The text encoder g encodes \u02dc Y via multiple transformer blocks to produce the latent textual feature as \u02dc g = g( \u02dc Y , \u03b8g), where \u02dc g \u2208Rd. For zero-shot inference, textual features of text template with class labels {1, 2, \u00b7 \u00b7 \u00b7 , C} are matched with image feature \u02dc f as exp(sim(\u02dc g\u00b7 \u02dc f)\u03c4) PC i=1 exp(sim( \u02dc gi\u00b7 \u02dc f)\u03c4), where sim() denotes the cosine similarity and \u03c4 is the temperature. Prompt Learning for CLIP: Prompt learning approaches append learnable prompt tokens at either the text [59, 58] encoder or image [1] encoder. We use a simple baseline method [37] that learns hierarchical prompt tokens on both the text and image encoders separately, named as Independent Vision-Language Prompting (IVLP). Specifically, we append learnable T language and V visual prompts given as Pt = {p1 t, p2 t, \u00b7 \u00b7 \u00b7 , pT t } and Pv = {p1 v, p2 v, \u00b7 \u00b7 \u00b7 , pV v } with the textual and visual input tokens, respectively. Therefore, the image encoder processes the following input tokens \u02dc Xp = {Pv, ecls, e1, e2, \u00b7 \u00b7 \u00b7 , eM} to generate prompted visual feature represented as \u02dc fp = f( \u02dc Xp, \u03b8f). Similarly, textual feature is obtained as \u02dc gp = g( \u02dc Yp, \u03b8g), where \u02dc Yp = {tSOS, Pt, t1, t2, \u00b7 \u00b7 \u00b7 , tL, ck, tEOS}. In contrast to shallow prompting where learnable prompts are introduced only at the first transformer block of the image and text encoders, our approach uses deep prompting which learns separate sets of prompts at every transformer block. The vision and language prompts are jointly represented as P = {Pv, Pt}. The feature representations obtained using these learnable prompts are referred to as prompted features. For image classification on a downstream dataset D, prompts P interact with pre-trained and frozen \u03b8f and \u03b8g and are optimized with the cross-entropy loss, LCE, as:   \\ l abe l { e q:LCE} \\ mathca l  {L _ {\\te xt {CE}}} = \\text {arg}&\\min _{\\bm {P}}\\mathbb {E}_{(\\bm {X}, {y})\\sim \\mathcal {D}} \\, \\mathcal {L} (\\text {sim}(\\bm {\\Tilde {f}_p},\\bm {\\Tilde {g}_p}), y). (1) 3.2. Self-Regularization for Prompt Learning The LCE objective employs ground truth labels to optimize the prompts for the downstream task. As a result, the prompts adapt and learn task-specific knowledge. During training, prompts interact with pre-trained and frozen CLIP tokens through self-attention layers in the transformer blocks. This interaction of prompts tokens with pre-trained CLIP weights \u03b8CLIP provides implicit regularization and encourages retaining the task-agnostic generalized knowledge within learned prompts. However, as shown in Fig. 2, prompts tend to overfit on the supervised task and drift away from the generalized CLIP space as the training schedule increases. Consequently, new task performance is degraded, despite the fact that CLIP image and text encoder weights \u03b8f and \u03b8g are kept frozen. As prompts undergo further training, the implicit generalization constraint becomes weaker against the task-specific LCE objective. One naive approach to address this issue is to reduce the training schedule to balance the performance between \fthe base and new tasks. However, training the prompts for fewer iterations to prevent losing generalization comes at the cost of relatively lower performance on the supervised task. Here, we present a prompt learning approach that maximizes supervised task performance without sacrificing performance on novel tasks and classes. We propose to anchor prompt training with self-regularization which constitutes three main components as discussed below. 3.2.1 Mutual agreement maximization As discussed above, the strong downstream dataset transfer constraint imposed by LCE causes the prompts to overfit on task-specific data and it struggles to effectively utilize the general information from the frozen CLIP. We propose to explicitly guide the training trajectory by imposing a constraint to maximize its mutual agreement between the prompted and the frozen CLIP features. We achieve this by explicitly conditioning the prompted features to be consistent with the CLIP features obtained without learnable prompts. As we do not require any second model for such conditioning, we call this regularizing constraint as a selfconsistency loss (SCL). For a given input sample and its corresponding textual label, we obtain visual features using learnable prompts and pre-trained visual features, \u02dc fp and \u02dc f within the frozen CLIP latent space. Similarly, we obtain textual features \u02dc gp and \u02dc g. We then impose a constraint on the prompted visual and text features to ensure their consistency with the CLIP pretrained features as follows,   \\label { e q : scl f ea t u res } \\mathca l   { L_{ \\t ex t   {SCL-image}}} = \\sum _{i=1}^{d}|\\bm {\\Tilde {f}_{p}} \\bm {\\Tilde {f}}|, \\; \\mathcal {L_{\\text {SCL-text}}} = \\sum _{i=1}^{d} |\\bm {\\Tilde {g}_{p}} \\bm {\\Tilde {g}}|. (2) As shown in Eq. 2, we utilize L1 loss to impose the feature level consistency. Note that our self-consistency constraint is also compatible with other variants of matching losses such as cosine similarity or MSE loss which we study in our ablations (Sec. 4.7). To further complement the regularization constraint and maximize the alignment between the general features and the prompted features, we impose logit level selfconsistency regularization and condition the prompted logits distribution on pre-trained CLIP logits distribution by minimizing the Kullback-Leibler divergence as follows,   \\label {e q :scl-log i ts}   \\mat hcal   {L _ {\\text {SCL-logits}}} = \\mathcal {D_{KL}} (\\text {sim}(\\bm {\\Tilde {f}_p},\\bm {\\Tilde {g}_{p}}), {\\text {sim}(\\bm {\\Tilde {f}},\\bm {\\Tilde {g}}))}. (3) Overall, the self-consistency training objectives guide the prompts to gain complementary knowledge from pretrained CLIP features, therefore providing strongly generalized prompts,   \\l a bel {eq:sclc ombined} \\m a thcal {L_{\\text {SCL}}} &= \\lambda _{1}\\mathcal {L_{\\text {SCL-image}}} + \\lambda _{2}\\mathcal {L_{\\text {SCL-text}}} + \\mathcal {L_{\\text {SCL-logits}}},s (4) where \u03bb1 and \u03bb2 are loss balancing hyper-parameters. Our overall training objective thus becomes,   \\lab e l { e q:final-loss} \\mathcal {L_{\\text {final}}} &= \\mathcal {L_{\\text {CE}}} + \\mathcal {L_{\\text {SCL}}} . (5) Discussion on Lfinal: LSCL loss guides the prompts to converge at solutions that are generalized. On the other hand, LCE guides the prompts to maximize performance on the downstream supervised tasks. The combination of these losses conditions the prompts to maximize their performance on supervised tasks and at the same time guides the prompts learning trajectory toward a weight space that is consistent with the CLIP zero-shot features. As shown in Fig. 2, our proposed methodology maximizes the supervised tasks\u2019 performance while also improving the generalization. This shows that the proposed training objectives for prompt learning setup are complementary to each other. 3.2.2 Regularization with prompt self-ensembling The second component in our self-regularizing framework enforces regularization using prompt self-ensembling. Model ensembling in the weight space has been shown to improve both the performance and generalization of a model [47, 18]. However, it has not been actively studied in the context of prompt learning, where prompts are only learnable parameters with frozen model parameters. To effectively utilize the prompts knowledge from the previous training iterations, we propose prompts aggregation for a generalizable solution. For a training schedule with E total epochs, prompts at every epoch are given by {P }E t=1. Aggregated prompts (AP) are then calculated as,   \\la b e l  {e q:ptu nin g}  \\{\\bm {P}\\}^{\\text {AP}} = \\sum _{t=1}^{E}{\\frac {w_{t} . \\bm {P}}{\\sum _{i=1}^{E}w_{i}}},  (6) where wi is the weight assigned to prompts at each epoch t. In the early epochs, prompts are not mature to capture contextual information due to their random initialization. During aggregation, they should be given less weight as they act as noise which is carried along with the input tokens. On the other hand, the prompts learned in the last few epochs are task specific and highly favours the supervised downstream task distribution. We propose to perform Gaussian weighted prompt aggregation (GPA), where small aggregation weights are given to prompts at initial epochs, higher weights to prompts at middle epochs, and relatively lower weights to prompts at final epochs, resulting in optimal prompt representations that improve generalization to downstream tasks. GPA provides optimal weight values wi by sampling from a Gaussian distribution wi \u223cN(\u00b5, \u03c32), where \u03c32 and \u00b5 are hyper-parameters and PE i=1 wi = 1. Gaussian distribution is defined over the epochs and its mean is dictated by the epoch number. We formulate this \fweighting as a moving average to avoid saving multiple copies of prompts by keeping one additional copy which is updated via aggregation at every epoch i,   \\l a b e l { eq:moving_average} {\\bm {P}}^{\\text {GPA}} = \\sum _{i=1}^{E}w_{i} . \\bm {P_{i}}.  (7) 3.2.3 Regulating prompts with textual diversity Through the LSCL loss, the visual prompted features to instill diverse generalized contexts from pre-trained CLIP visual features as multiple image samples are present for each label category. This provides a natural source of augmentations at the image side and promotes additional regularization. However, as opposed to having multiple images per category, we note that the text space during fine-tuning is limited, and prompted features are learned based on pretrained CLIP text features, with only one feature representation per category. This mismatch between the available diversity at the image and text side leads to sub-optimal learning of prompted textual features. To address the diversity mismatch, we incorporate textual diversity in the text encoder. Specifically, we use a pool of textual prompt templates {PT|N l=1}, containing N augmentations to form multiple text features per category. The pre-trained CLIP textual features are now obtained as an ensemble of multiple prompts templates \u02dc g = 1 N PN i=1 \u02dc gi. As pre-trained CLIP textual features are now represented by the ensemble of multiple augmentations for each label, the prompted textual features learn more diverse generalized contexts from the frozen CLIP. We note that the proposed textual diversity is different from the standard prompt ensembling technique explored by CLIP authors. CLIP uses ensemble of text prompts during inference for classification. In contrast, we utilize them during training for self-regularization by enforcing mutual agreement of ensembled features with prompted features, and prompted features are used at inference. Next, we show the efficacy of our proposed components via comprehensive experiments provided below. 4. Experiments 4.1. Evaluation settings We extensively evaluate our approach and present a comparison with other methods on four benchmark settings. Base-to-novel class generalization: In this setting, we equally split the datasets into base and novel classes. The model is trained on base classes and evaluated on both base classes and novel classes. This benchmark evaluates the generalization ability of a method within a dataset. Few-shot learning: We incorporate this setting to compare the learning capacity of the model under extremely limited supervision and verify if our approach learns complementary task-specific and task-agnostic knowledge. For each Dataset CLIP CoOp CoCoOp ProDA MaPLe PromptSRC \u2206 [35] [59] [58] [28] [22] (Ours) Average on 11 datasets Base 69.34 82.69 80.47 81.56 82.28 84.26 +2.0 Novel 74.22 63.22 71.69 72.30 75.14 76.10 +1.0 HM 71.70 71.66 75.83 76.65 78.55 79.97 +1.4 ImageNet Base 72.43 76.47 75.98 75.40 76.66 77.60 +0.9 Novel 68.14 67.88 70.43 70.23 70.54 70.73 +0.2 HM 70.22 71.92 73.10 72.72 73.47 74.01 +0.5 Caltech101 Base 96.84 98.00 97.96 98.27 97.74 98.10 +0.4 Novel 94.00 89.81 93.81 93.23 94.36 94.03 -0.3 HM 95.40 93.73 95.84 95.68 96.02 96.02 +0.0 OxfordPets Base 91.17 93.67 95.20 95.43 95.43 95.33 -0.1 Novel 97.26 95.29 97.69 97.83 97.76 97.30 -0.5 HM 94.12 94.47 96.43 96.62 96.58 96.30 -0.3 Stanford Cars Base 63.37 78.12 70.49 74.70 72.94 78.27 +5.3 Novel 74.89 60.40 73.59 71.20 74.00 74.97 +1.0 HM 68.65 68.13 72.01 72.91 73.47 76.58 +3.1 Flowers102 Base 72.08 97.60 94.87 97.70 95.92 98.07 +2.1 Novel 77.80 59.67 71.75 68.68 72.46 76.50 +4.1 HM 74.83 74.06 81.71 80.66 82.56 85.95 +3.4 Food101 Base 90.10 88.33 90.70 90.30 90.71 90.67 -0.1 Novel 91.22 82.26 91.29 88.57 92.05 91.53 -0.5 HM 90.66 85.19 90.99 89.43 91.38 91.10 -0.3 FGVC Aircraft Base 27.19 40.44 33.41 36.90 37.44 42.73 +5.3 Novel 36.29 22.30 23.71 34.13 35.61 37.87 +2.3 HM 31.09 28.75 27.74 35.46 36.50 40.15 +3.7 SUN397 Base 69.36 80.60 79.74 78.67 80.82 82.67 +1.9 Novel 75.35 65.89 76.86 76.93 78.70 78.47 -0.2 HM 72.23 72.51 78.27 77.79 79.75 80.52 +0.8 DTD Base 53.24 79.44 77.01 80.67 80.36 83.37 +3.0 Novel 59.90 41.18 56.00 56.48 59.18 62.97 +3.8 HM 56.37 54.24 64.85 66.44 68.16 71.75 +3.6 EuroSAT Base 56.48 92.19 87.49 83.90 94.07 92.90 -1.2 Novel 64.05 54.74 60.04 66.00 73.23 73.90 +0.7 HM 60.03 68.69 71.21 73.88 82.35 82.32 -0.1 UCF101 Base 70.53 84.69 82.33 85.23 83.00 87.10 +4.1 Novel 77.50 56.05 73.45 71.97 78.66 78.80 +0.1 HM 73.85 67.46 77.64 78.04 80.77 82.74 +2.0 Table 1: Accuracy comparison on Base-to-novel generalization of PromptSRC with previous methods. The prompts learned with our self-regularizing approach show overall consistent improvements on base classes, without losing generalization. Absolute gains over MaPLe [22] are shown in blue. dataset, we test the model\u2019s generalization for different Kshots per category, where K = 1, 2, 4, 8, 16. Domain generalization setting: We train a source model on ImageNet [6] and evaluate on out-of-distribution datasets to test performance under domain shifts. Cross-dataset evaluation: In cross-dataset transfer, we train the models on ImageNet [6] and directly evaluate it on other datasets without any data-specific fine-tuning. Datasets: For base to novel class generalization, fewshot setting and cross-dataset evaluation, we follow CoOp [59] and CoCoOp [58], and use 11 image recognition \fdatasets. The datasets cover multiple recognition tasks including ImageNet [6] and Caltech101 [10] which consists of generic objects; OxfordPets [34], StanfordCars [24], Flowers102 [33], Food101 [2], and FGVCAircraft [31] for fine-grained classification, SUN397 [48] for scene recognition, UCF101 [41] for action recognition, DTD [4] for texture classification, and EuroSAT [14] which consists of satellite images. For domain generalization benchmark, we use ImageNet [6] as a source dataset and use ImageNetA [16], ImageNet-R [15], ImageNet-Sketch [44] and ImageNetV2 [39] as out of distribution datasets. Implementation details: We use a ViT-B/16 based CLIP model in our experiments and report results averaged over 3 runs. We use deep prompting with V = T = 4 VL prompts and train for 50 epochs for few-shot setting and 20 epochs the rest of the 3 benchmarks respectively. For domain generalization and cross-dataset evaluation, we train the ImageNet source model on all classes with K = 16 shots using V = T = 4 VL prompts in the first 3 transformer layers. For few-shot and base-to-novel setting, prompts are learned in the first 9 transformer layers. Prompts are randomly initialized with a normal distribution except the text prompts of the first layer which are initialized with the word embeddings of \u201ca photo of a\u201d. We fix the learning rate to 0.0025. We set \u03bb1 = 10 and \u03bb2 = 25 to weight LSCL-image and LSCL-text respectively. The corresponding hyperparameters are fixed across all datasets and benchmarks. For textual diversity, we use a total of N = 60 standard prompt templates provided in [35]. For comparison with ProDA [28], we report their results produced by [7]. Refer to Appendix A for additional implementation details. 4.2. Effectiveness of Self-regulating Prompts We first disentangle the regularization components in our self-regulating prompting framework and show the individual contributions in Table 2. Baseline IVLP provides high base class performance but suffers from poor generalization (row-1). By enforcing mutual agreement through LSCL (row-2), novel class performance significantly increases by 3.95% while maintaining base class gains. This suggests that LSCL explicitly enforces the prompts to capture the generalizable features from frozen CLIP. Integrating GPA (row3) which suitably aggregates prompts across the training cycle further reduces overfitting and improves the novel class performance. Finally, combined with textual diversity to overcome the diversity mismatch between the text and visual domains (row-4), PromptSRC achieves improvements on both base and novel classes, leading to the average novel class and harmonic mean gains of +4.31% and +2.46% respectively. The averaged results on 11 datasets are summarized in Table 2. Note that even small improvements in these metrics correspond to significant gains. We refer the readers to Appendix B for results on individual datasets. Method Base Acc. Novel Acc. HM 1: Independent V-L prompting 84.21 71.79 77.51 2: + LSCL 84.21 75.38 79.55 3: + GPA 84.16 75.69 79.70 4: + Textual diversity 84.26 76.10 79.97 Table 2: Effect of our proposed regularization techniques. Results are averaged over 11 datasets. HM refers to harmonic mean. 4.3. Base-to-Novel Generalization We compare the performance of our approach with zeroshot CLIP [35], CoOp [59], CoCoOp [58], ProDA [28] and MaPLe [22], in Table 1. Overall, all existing approaches outperform zero-shot CLIP on base classes but show inferior performance on novel classes except MaPLe. This suggests that they overall tend to lose the generalizable features stored in the frozen CLIP model. In contrast, PromptSRC significantly improves base class performance while improving the zero-shot CLIP novel class accuracy by 1.88%. This shows the importance of explicit guidance provided by PromptSRC in learning complementary taskspecific and task-agnostic representations which aid base and novel classes respectively. CoOp is heavily trained on base classes and consequently compromises on its generalization. For instance, on EuroSAT [14], CoOp provides a substantial 92.19% base class accuracy and inferior novel class accuracy of 54.74%. On the other hand, PromptSRC which learns self-regulating prompts provides the highest base and novel class accuracies of 92.90% and 73.90% on EuroSAT respectively. In comparison to CoCoOp and ProDA, PromptSRC shows gains on the 10/11 datasets respectively. Against the recent MaPLe approach, PromptSRC improves performance on 8/11 datasets while using 77x less tunable parameters (3.55M of MaPLe vs 46K of PromptSRC). With respect to the averaged results, PromptSRC provides the best results of 84.26%, 76.10%, and 79.97% on the base class, novel class, and harmonic mean respectively. 4.4. Few-shot Experiments To explicitly verify if our regularization framework restricts the prompts to learn task-specific knowledge or not, we compare our few-shot results with existing methods in Fig. 4. In general, all prompt learning approaches perform better than the linear probe, especially in scenarios with lesser shots i.e., K = 1, 2, 4. PromptSRC overall provides consistent improvements on all shots in comparison with all existing methods. When compared with the existing best method MaPLe, PromptSRC consistently provides absolute gains of 3.05%, 2.72%, 2.59%, 1.80%, and, 1.07% on 1, 2, 4, 8, and 16 shots respectively which are averaged over 11 datasets. Furthermore, we note that our approach achieves relatively larger gains in minimal data cases such \fFigure 4: PromptSRC performance comparison in few-shot image recognition setting. All methods are trained on ViT-B/16 CLIP backbone using their best settings. PromptSRC demonstrates consistent improvements over existing methods specifically for lesser shots i.e. K = 1, 2, 4. On average, PromptSRC provides the highest performance gains for all shots. These results demonstrate that PromptSRC learns complementary task-agnostic general features from frozen CLIP without being restricted from learning downstream task representations. Source Target ImageNet Caltech101 OxfordPets StanfordCars Flowers102 Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average CoOp 71.51 93.70 89.14 64.51 68.71 85.30 18.47 64.15 41.92 46.39 66.55 63.88 Co-CoOp 71.02 94.43 90.14 65.32 71.88 86.06 22.94 67.36 45.73 45.37 68.21 65.74 MaPLe 70.72 93.53 90.49 65.57 72.23 86.20 24.74 67.01 46.49 48.06 68.69 66.30 PromptSRC 71.27 93.60 90.25 65.70 70.25 86.15 23.90 67.10 46.87 45.50 68.75 65.81 Table 3: Cross-dataset benchmark evaluation. PromptSRC achieves overall favourable performance. as for K = 1, 2 for almost all datasets. This demonstrates that PromptSRC regulates prompts against overfitting without restricting the prompts to learn task-specific knowledge. 4.5. Cross Dataset Evaluation We compare our cross-dataset performance with previous methods in Table 3. On the source dataset, PromptSRC performs comparably to other methods. In comparison with CoOp and CoCoOp, PromptSRC shows competitive performance and achieves better generalization in 8/10 and 7/10 datasets respectively. Compared with MaPLe, PromptSRC Source Target ImageNet -V2 -S -A -R Avg. CLIP 66.73 60.83 46.15 47.77 73.96 57.18 CoOp 71.51 64.20 47.99 49.71 75.21 59.28 Co-CoOp 71.02 64.07 48.75 50.63 76.18 59.91 MaPLe 70.72 64.07 49.15 50.90 76.98 60.27 PromptSRC 71.27 64.35 49.55 50.90 77.80 60.65 Table 4: Domain generalization. Prompt learning methods are trained on imageNet and evaluated on datasets with domain shifts. shows improved performance in 5/10 datasets while utilizing significantly less tunable parameters (46K vs 3.55M). 4.6. Domain Generalization Experiments Table 4 summarizes the results of PromptSRC and previous methods on out-of-distribution datasets. We directly evaluate our model trained on ImageNet. On target datasets, PromptSRC consistently outperforms all existing methods, with an overall highest average accuracy of 60.65%. This suggests that our self-regulating framework favors better generalization for datasets with domain shifts. \fMethod Base Acc. Novel Acc. HM 1: Independent V-L prompting (IVLP) 84.21 71.79 77.51 2: IVLP + Cosine similarity 84.47 74.51 79.17 3: IVLP + Mean square error (MSE) 84.59 74.68 79.33 4: IVLP + L1 84.42 74.99 79.43 Table 5: Effect of matching losses for LSCL-image and LSCL-image consistency objectives. L1 matching loss provides highest HM. Method Base Acc. Novel Acc. HM 1: Exponential moving average 83.09 76.15 79.47 2: Equal weighting (averaging) 83.50 76.47 79.83 3: GPA (Ours) 84.26 76.10 79.97 Table 6: Ablation on prompt ensembling techniques. Gaussian weighted prompt aggregation (GPA) provides better performance. Method GFLOP (train) GFLOP (test) Train time (min) FPS HM CoOp 162.5 162.5 10.08 1344 71.66 CoCoOp 162.5 162.5 39.53 15.08 75.83 IVLP 162.8 162.8 12.01 1380 77.51 PromptSRC 179.6 162.8 13.13 1380 79.97 Table 7: PromptSRC compute cost comparison using SUN397 dataset. Training time for all methods is calculated for 10 epochs on a single A100 GPU on SUN397 dataset. 4.7. Ablative Analysis Embedding consistency loss ablation: In Table 5, we ablate on the choice of matching loss metric used in our proposed feature level LSCL loss constraints. For simplicity, we only incorporate LSCL-image and LSCL-text on top of the IVLP baseline. Generally, distance-based matching metrics outperform the cosine similarity metric in terms of generalization as they impose a much harder constraint. Overall, the L1 matching metric provides the highest HM. Prompt ensembling: Table 6 shows ablation on various prompt ensembling techniques. Using equal weights for prompts reduces base class results as initial epoch prompts are not mature enough. In contrast, our proposed Gaussian weighted prompt aggregation results in the highest performance. Detailed ablation experiments for other hyperparameters are provided in Appendix C. Training and inference compute cost analysis: In Table 7, we show the compute cost analysis of our approach and compare it with other prompting methods. PromptSRC\u2019s overall training GFLOPs are only 0.13x higher than baseline IVLP, while it maintains the same GFLOPs and throughput during inference. Pre-trained CLIP textual features are pre-computed and a single additional forward pass is required through image encoder to compute pre-trained CLIP visual features for our mutual agreement maximization technique. Training time of PromptSRC is 9.3% longer than IVLP which is significantly lower than CoCoOp. We use 4 vision and text prompts similar to the IVLP. Figure 5: Ablation study on the number of textual prompts for textual diversity (left) and prompt token length (right) on ImageNet. Prompt Length: Fig. 5 (right) shows the effect of prompt token length on the harmonic mean. Overall, the performance increases as prompt length increases. Using 4 visionlanguage prompts provides the highest harmonic mean. No. of templates in textual diversity: In Fig. 5 (left), we ablate on the number of text prompt templates for textual diversity. We note that increasing the number of textual templates for textual diversity generally increases the performance. This suggests that adding textual diversity using multiple templates for pre-trained features provides more rich supervision for the learned prompted features. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2210.03117v3",
+                    "title": "MaPLe: Multi-modal Prompt Learning",
+                    "abstract": "Pre-trained vision-language (V-L) models such as CLIP have shown excellent\ngeneralization ability to downstream tasks. However, they are sensitive to the\nchoice of input text prompts and require careful selection of prompt templates\nto perform well. Inspired by the Natural Language Processing (NLP) literature,\nrecent CLIP adaptation approaches learn prompts as the textual inputs to\nfine-tune CLIP for downstream tasks. We note that using prompting to adapt\nrepresentations in a single branch of CLIP (language or vision) is sub-optimal\nsince it does not allow the flexibility to dynamically adjust both\nrepresentation spaces on a downstream task. In this work, we propose\nMulti-modal Prompt Learning (MaPLe) for both vision and language branches to\nimprove alignment between the vision and language representations. Our design\npromotes strong coupling between the vision-language prompts to ensure mutual\nsynergy and discourages learning independent uni-modal solutions. Further, we\nlearn separate prompts across different early stages to progressively model the\nstage-wise feature relationships to allow rich context learning. We evaluate\nthe effectiveness of our approach on three representative tasks of\ngeneralization to novel classes, new target datasets and unseen domain shifts.\nCompared with the state-of-the-art method Co-CoOp, MaPLe exhibits favorable\nperformance and achieves an absolute gain of 3.45% on novel classes and 2.72%\non overall harmonic-mean, averaged over 11 diverse image recognition datasets.\nOur code and pre-trained models are available at\nhttps://github.com/muzairkhattak/multimodal-prompt-learning.",
+                    "authors": "Muhammad Uzair Khattak, Hanoona Rasheed, Muhammad Maaz, Salman Khan, Fahad Shahbaz Khan",
+                    "published": "2022-10-06",
+                    "updated": "2023-04-01",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Foundational vision-language (V-L) models such as CLIP (Contrastive Language-Image Pretraining) [32] have shown excellent generalization ability to downstream tasks. Such models are trained to align language and vision modalities on web-scale data e.g., 400 million text-image pairs in CLIP. These models can reason about open-vocabulary visual concepts, thanks to the rich supervision provided by natural language. During inference, hand-engineered text prompts are used e.g., \u2018a photo of a <category>\u2019 as a query for text encoder. The output text embeddings are matched with the visual embeddings from an image encoder to predict the output class. Designing high quality contextual prompts have been proven to enhance the performance of CLIP and other V-L models [17,42]. Despite the effectiveness of CLIP towards generalization to new concepts, its massive scale and scarcity of training data (e.g., few-shot setting) makes it infeasible to \ufb01ne-tune the full model for downstream tasks. Such \ufb01ne-tuning can also forget the useful knowledge acquired in the large-scale pretraining phase and can pose a risk of over\ufb01tting to the downstream task. To address the above challenges, existing works propose language prompt learning to avoid manually adjusting the prompt templates and providing a mechanism to adapt the model while keeping the original weights frozen [14,25,29,48,49]. Inspired from Natural Language Processing (NLP), these approaches only explore prompt learning for the text encoder in CLIP (Fig. 1:a) while adaptation choices together with an equally important image encoder of CLIP remains an unexplored topic in the literature. Our motivation derives from the multi-modal nature of CLIP, where a text and image encoder co-exist and both contribute towards properly aligning the V-L modalities. We argue that any prompting technique should adapt the model completely and therefore, learning prompts only for the text encoder in CLIP is not suf\ufb01cient to model the adaptations needed for the image encoder. To this end, we set out to achieve completeness in the prompting approach and propose Multi-modal Prompt Learning (MaPLe) to adequately \ufb01ne-tune the text and image encoder representations such that their optimal alignment can be achieved on the downstream tasks (Fig. 1:b). Our extensive experiments on three key representative settings including base-to-novel generalization, cross-dataset evaluation, and domain generalization demonstrate the strength of MaPLe. On base-to-novel generalization, our proposed MaPLe outperforms existing prompt learning approaches across 11 diverse image recognition datasets (Fig. 1:c) and achieves absolute average gain of 3.45% on novel classes and 2.72% on harmonic-mean arXiv:2210.03117v3  [cs.CV]  1 Apr 2023 \fMaximize similarity Image Encoder Prompts \"Text input\" (a) Existing prompt tuning methods (Uni-modal) (b) Multi-modal Prompt Learning (MaPLe) (c) Performance comparison on base-to-novel generalization Text Encoder Maximize similarity Image Encoder Prompts \"Text input\" Text Encoder Prompts Prompts Prompts Prompts Prompts Figure 1. Comparison of MaPLe with standard prompt learning methods. (a) Existing methods adopt uni-modal prompting techniques to \ufb01ne-tune CLIP representations as prompts are learned only in a single branch of CLIP (language or vision). (b) MaPLe introduces branch-aware hierarchical prompts that adapt both language and vision branches simultaneously for improved generalization. (c) MaPLe surpasses state-of-the-art methods on 11 diverse image recognition datasets for novel class generalization task. over the state-of-the-art method Co-CoOp [48]. Further, MaPLe demonstrates favorable generalization ability and robustness in cross-dataset transfer and domain generalization settings, leading to consistent improvements compared to existing approaches. Owing to its streamlined architectural design, MaPLe exhibits improved ef\ufb01ciency during both training and inference without much overhead, as compared to Co-CoOp which lacks ef\ufb01ciency due to its image instance conditioned design. In summary, the main contributions of this work include: \u2022 We propose multi-modal prompt learning in CLIP to favourably align its vision-language representations. To the best of our knowledge, this is the \ufb01rst multimodal prompting approach for \ufb01ne-tuning CLIP. \u2022 To link prompts learned in text and image encoders, we propose a coupling function to explicitly condition vision prompts on their language counterparts. It acts as a bridge between the two modalities and allows mutual propagation of gradients to promote synergy. \u2022 Our multi-modal prompts are learned across multiple transformer blocks in both vision and language branches to progressively learn the synergistic behaviour of both modalities. This deep prompting strategy allows modeling the contextual relationships independently, thus providing more \ufb02exibility to align the vision-language representations. 2. Related Work Vision Language Models: The combined use of language supervision with natural images is found to be of great interest in the computer vision community. In contrast to models learned with only image supervision, these visionlanguage (V-L) models encode rich multimodal representations. Recently, V-L models like CLIP [32], ALIGN [15], LiT [45], FILIP [41] and Florence [43] have demonstrated exceptional performance on a wide spectrum of tasks including few-shot and zero-shot visual recognition. These models learn joint image-language representations in a selfsupervised manner using abundantly available data from the web. For example, CLIP and ALIGN respectively use \u223c400M and \u223c1B image-text pairs to train a multimodal network. Although these pre-trained V-L models learn generalized representations, ef\ufb01ciently adapting them to downstream tasks is still a challenging problem. Many works have demonstrated better performance on downstream tasks by using tailored methods to adapt V-L models for few-shot image-recognition [9,19,46], object detection [8,10,27,34,44,50], and segmentation [5,22,26,33]. In this work, we propose a novel multi-modal prompt learning technique to effectively adapt CLIP for few-shot and zeroshot visual recognition tasks. Prompt Learning: The instructions in the form of a sentence, known as text prompt, are usually given to the language branch of a V-L model, allowing it to better understand the task. Prompts can be handcrafted for a downstream task or learned automatically during \ufb01ne-tuning stage. The latter is referred to as \u2018Prompt Learning\u2019 which was \ufb01rst used in NLP [21,23,24] followed by the adaptation in V-L [48, 49, 51] and vision-only [16, 38, 39, 47] models. Similar to [16] our design also uses deep \u2018vision\u2019 prompting. However, ours is the \ufb01rst multi-modal prompting de\fText Encoder  Image Encoder  x1 Patch Embed Word Embed \"a photo of a  cat\" x1.y 1 x1.y 2 x1.y3 ... x1.yn x2.y 1 x2.y 2 x2.y3 ... x2.yn x3.y 1 x3.y 2 x2.y3 ... x3.yn ... ... ... ... xn.y1 xn.y2xn.y3 ... xn.yn ... ... ... Encoder Layer Encoder Layer ... Encoder Layer ... Encoder Layer ... Encoder Layer ... Encoder Layer x2 x3 ... xn y1 y2 y3 ... yn ..... ..... Figure 2. Overview of our proposed MaPLe (Multi-modal Prompt Learning) framework for prompt learning in V-L models. MaPLe tunes both vision and language branches where only the context prompts are learned, while the rest of the model is frozen. MaPLe conditions the vision prompts on language prompts via a V-L coupling function F to induce mutual synergy between the two modalities. Our framework uses deep contextual prompting where separate context prompts are learned across multiple transformer blocks. sign while [16] is uni-modal. Prompt Learning in Vision Language models: Full \ufb01netuning and linear probing [9] are two typical approaches to adapt a V-L model (i.e. CLIP) to the downstream tasks. The complete \ufb01ne-tuning results in degrading the previously learned joint V-L representation while linear probing limits the zero-shot capability of CLIP. To this end, inspired from prompt learning in NLP, many works have proposed to adapt V-L models by learning the prompt tokens in an end-to-end training. CoOp [49] \ufb01ne-tunes CLIP for fewshot transfer by optimizing continuous set of prompt vectors at its language branch. Co-CoOp [48] highlights the inferior performance of CoOp on novel classes and solves the generalization issue by explicitly conditioning prompts on image instances. [25] proposes to optimize multiple set of prompts by learning the distribution of prompts. [18] adapt CLIP by learning prompts for video understanding tasks. [1] perform visual prompt tuning on CLIP by prompting on the vision branch. We note that the existing methods follow independent uni-modal solutions and learn prompts either in the language or in the vision branch of CLIP, thus adapting CLIP partially. In this paper, we explore an important question: given the multimodal nature of CLIP, is complete prompting (i.e., in both language and vision branches) better suited to adapt CLIP? Our work is the \ufb01rst to answer this question by investigating the effectiveness of multi-modal prompt learning in order to improve alignment between vision and language representations. 3. Method Our approach concerns with \ufb01ne-tuning a pre-trained multimodal CLIP for better generalization to downstream tasks through context optimization via prompting. Fig. 2 shows the overall architecture of our proposed MaPLe (Multimodal Prompt Learning) framework. Unlike previous approaches [48, 49] which learn context prompts only at the language branch, MaPLe proposes a joint prompting approach where the context prompts are learned in both vision and language branches. Speci\ufb01cally, we append learnable context tokens in the language branch and explicitly condition the vision prompts on the language prompts via a coupling function to establish interaction between them. To learn hierarchical contextual representations, we introduce deep prompting in both branches through separate learnable context prompts across different transformer blocks. During \ufb01ne-tuning, only the context prompts along with their coupling function are learned while the rest of the model is frozen. Below, we \ufb01rst outline the pre-trained CLIP architecture and then present our proposed \ufb01ne-tuning approach. 3.1. Revisiting CLIP We build our approach on a pre-trained vision-language (VL) model, CLIP, which consists of a text and vision encoder. Consistent with existing prompting methods [48, 49], we use a vision transformer (ViT) [6] based CLIP model. CLIP encodes an image I \u2208RH\u00d7W \u00d73 and a corresponding text description as explained below. Encoding Image: Image encoder V with K transformer layers {Vi}K i=1, splits the image I into M \ufb01xed-size patches which are projected into patch embeddings E0 \u2208RM\u00d7dv. Patch embeddings Ei are input to the (i + 1)th transformer block (Vi+1) along with a learnable class (CLS) token ci \fCo-CooP: Base 77.0 Co-CooP: Novel  56.0 MaPLe: Base 80.4 MaPLe: Novel 59.2 (a) DTD (Texture Classification) (b) EuroSAT (Sattelite Imagery Recognition) (c) UCF101 (Action Recognition) Co-CooP: Base 87.5  Co-CooP: Novel  60.1 Co-CooP: Base 82.3 Co-CooP: Novel  73.5 MaPLe: Base 94.1 MaPLe: Novel 73.2 MaPLe: Base 83.0 M: Novel 78.7 Figure 3. t-SNE plots of image embeddings in uni-modal prompting method Co-CoOp, and MaPLe on 3 diverse image recognition datasets. MaPLe shows better separability in both base and novel classes. and sequentially processed through K transformer blocks, [ci, Ei] = Vi([ci\u22121, Ei\u22121]) i = 1, 2, \u00b7 \u00b7 \u00b7 , K. To obtain the \ufb01nal image representation x, the class token cK of last transformer layer (VK) is projected to a common V-L latent embedding space via ImageProj, x = ImageProj(cK) x \u2208Rdvl. Encoding Text: CLIP text encoder generates feature representations for text description by tokenizing the words and projecting them to word embeddings W0 = [w1 0, w2 0, \u00b7 \u00b7 \u00b7 , wN 0 ] \u2208RN\u00d7dl. At each stage, Wi is input to the (i + 1)th transformer layer of text encoding branch (Li+1), [Wi] = Li(Wi\u22121) i = 1, 2, \u00b7 \u00b7 \u00b7 , K. The \ufb01nal text representation z is obtained by projecting the text embeddings corresponding to the last token of the last transformer block LK to a common V-L latent embedding space via TextProj, z = TextProj(wN K) z \u2208Rdvl. Zero-shot Classi\ufb01cation: For zero-shot classi\ufb01cation, text prompts are hand-crafted with class labels y \u2208{1, 2, . . . C} (e.g., \u2018a photo of a <category>\u2019) having C classes. Prediction \u02c6 y corresponding to the image I having the highest cosine similarity score (sim(\u00b7)) is calculated with a temperature parameter \u03c4, p(\u02c6 y|x) = exp(sim(x, z\u02c6 y)/\u03c4) PC i=1 exp(sim(x, zi)) . 3.2. MaPLe: Multi-modal Prompt Learning To ef\ufb01ciently \ufb01ne-tune CLIP for downstream image recognition tasks, we explore the potential of multi-modal prompt tuning. We reason that prior works that have predominantly explored uni-modal approaches are less suitable as they do not offer the \ufb02exibility to dynamically adapt both language and vision representation spaces. Thus to achieve completeness in prompting, we underline the importance of multimodal prompting approach. In Fig. 3, we visualize and compare the image embeddings of MaPLe with recent stateof-the-art work, Co-CoOp. Note that the image embeddings of CLIP, CoOp and Co-CoOp will be identical as they do not learn prompts in the vision branch. The visualization shows that image embeddings of MaPLe are more separable indicating that learning vision prompts in addition to language prompts leads to better adaptation of CLIP. In addition to multi-modal prompting, we \ufb01nd that it is essential to learn prompts in the deeper transformer layers to progressively model stage-wise feature representations. To this end, we propose to introduce learnable tokens in the \ufb01rst J (where J < K) layers of both vision and language branches. These multi-modal hierarchical prompts utilize the knowledge embedded in CLIP model to effectively learn task relevant contextual representations (see Fig. 4). 3.2.1 Deep Language Prompting To learn the language context prompts, we introduce b learnable tokens {P i \u2208Rdl}b i=1, in the language branch of CLIP. The input embeddings now follow the form [P 1, P 2, \u00b7 \u00b7 \u00b7 , P b, W0], where W0 = [w1, w2, \u00b7 \u00b7 \u00b7 , wN] corresponds to \ufb01xed input tokens. New learnable tokens are further introduced in each transformer block of the language encoder (Li) up to a speci\ufb01c depth J, [ , Wi] = Li([Pi\u22121, Wi\u22121]) i = 1, 2, \u00b7 \u00b7 \u00b7 , J. (1) Here [\u00b7, \u00b7] refers to the concatenation operation. After Jth transformer layer, the subsequent layers process previous \flayer prompts and \ufb01nal text representation z is computed, [Pj, Wj] = Lj([Pj\u22121, Wj\u22121]) j = J + 1, \u00b7 \u00b7 \u00b7 , K, (2) z = TextProj(wN K). (3) When J = 1, the learnable tokens P are only applied at the input of \ufb01rst transformer layer, and this deep language prompting technique degenerates to CoOp [49]. 3.2.2 Deep Vision Prompting Similar to deep language prompting, we introduce b learnable tokens { \u02dc P i \u2208Rdv}b i=1, in the vision branch of CLIP alongside the input image tokens. New learnable tokens are further introduced in deeper transformer layers of the image encoder (V) up to depth J. [ci, Ei, ] = Vi([ci\u22121, Ei\u22121, \u02dc Pi\u22121]) i = 1, 2, \u00b7 \u00b7 \u00b7 , J, [cj, Ej, \u02dc Pj] = Vj([cj\u22121, Ej\u22121, \u02dc Pj\u22121]) j = J + 1, \u00b7 \u00b7 \u00b7 , K, x = ImageProj(cK). Our deep prompting provides the \ufb02exibility to learn prompts across different feature hierarchies within the ViT architecture. We \ufb01nd that sharing prompts across stages is better compared to independent prompts as features are more correlated due to successive transformer block processing. Thus, the later stages do not provide independently-learned complimentary prompts as compared to the early stages. 3.2.3 Vision Language Prompt Coupling We reason that in prompt tuning it is essential to take a multi-modal approach and simultaneously adapt both the vision and language branch of CLIP in order to achieve completeness in context optimization. A simple approach would be to naively combine deep vision and language prompting, where both the language prompts P, and the vision prompts \u02dc P, will be learned during the same training schedule. We name this design as \u2018Independent V-L Prompting\u2019. Although this approach satis\ufb01es the requirement of completeness in prompting, this design lacks synergy between vision and language branch as both branches do not interact while learning the task relevant context prompts. To this end, we propose a branch-aware multi-modal prompting which tunes vision and language branch of CLIP together by sharing prompts across both modalities. Language prompt tokens are introduced in the language branch up to Jth transformer block similar to deep language prompting as illustrated in Eqs. 1-3. To ensure mutual synergy between V-L prompts, vision prompts \u02dc P, are obtained by projecting language prompts P via vision-to-language projection which we refer to as V-L coupling function F(\u00b7), such that \u02dc Pk = Fk(Pk). The coupling function is implemented as a linear layer which maps dl dimensional inputs to dv. This acts as a bridge between the two modalities, thus encouraging mutual propagation of gradients. [ci, Ei, ] = Vi([ci\u22121, Ei\u22121, Fi\u22121(Pi\u22121)]) i = 1, \u00b7 \u00b7 \u00b7 , J [cj, Ej, \u02dc Pj] = Vj([cj\u22121, Ej\u22121, \u02dc Pj\u22121]) j = J + 1, \u00b7 \u00b7 \u00b7 , K x = ImageProj(cK) Unlike independent V-L prompting, explicit conditioning of \u02dc P on P helps learn prompts in a shared embedding space between the two branches, thus improving mutual synergy. 4. Experiments 4.1. Benchmark setting Generalization from Base-to-Novel Classes: We evaluate the generalizability of MaPLe, and follow a zero-shot setting where the datasets are split into base and novel classes. The model is trained only on the base classes in a few-shot setting and evaluated on base and novel categories. Cross-dataset Evaluation: To validate the potential of our approach in cross-dataset transfer, we evaluate our ImageNet trained model directly on other datasets. Consistent with Co-CoOp, our model is trained on all 1000 ImageNet classes in a few-shot manner. Domain Generalization: We evaluate the robustness of our method on out-of-distribution datasets. Similar to crossdataset evaluation, we test our ImageNet trained model directly on four other ImageNet datasets that contain various types of domain shifts. Datasets: For generalization from base-to-novel classes and cross-dataset evaluation, we follow [48, 49] and evaluate the performance of our method on 11 image classi\ufb01cation datasets which covers a wide range of recognition tasks. This includes two generic-objects datasets, ImageNet [4] and Caltech101 [7]; \ufb01ve \ufb01ne-grained datasets, OxfordPets [31], StanfordCars [20], Flowers102 [30], Food101 [2], and FGVCAircraft [28]; a scene recognition dataset SUN397 [40]; an action recognition dataset UCF101 [36]; a texture dataset DTD [3] and a satelliteimage dataset EuroSAT [11]. For domain generalization, we use ImageNet as source dataset and its four variants as target datasets including ImageNetV2 [35], ImageNetSketch [37], ImageNet-A [13] and ImageNet-R [12]. Implementation Details We use a few-shot training strategy in all experiments at 16 shots which are randomly sampled for each class. We apply prompt tuning on a pretrained ViT-B/16 CLIP model where dl = 512, dv = 768 and dvl = 512. For MaPLe, we set prompt depth J to 9 and the language and vision prompt lengths to 2. All models are trained for 5 epochs with a batch-size of 4 and a learning rate of 0.0035 via SGD optimizer on a single NVIDIA A100 GPU. We report base and novel class accuracies and their harmonic mean (HM) averaged over 3 runs. We initial\fMethod Base Acc. Novel Acc. HM GFLOPS 1: MaPLe shallow (J = 1) 80.10 73.52 76.67 167.1 2: Deep vision prompting 80.24 73.43 76.68 18.0 3: Deep language prompting 81.72 73.81 77.56 166.8 4: Independent V-L prompting 82.15 74.07 77.90 167.0 5: MaPLe (Ours) 82.28 75.14 78.55 167.0 Table 1. Comparison of MaPLe with different prompting designs in base-to-novel generalization. Results are averaged over 11 datasets. HM refers to harmonic mean. ize the language prompts of the \ufb01rst layer P0 with the pretrained CLIP word embeddings of the template \u2018a photo of a <category>\u2019, while for the subsequent layers they are randomly initialized from a normal distribution. For training MaPLe on all 1000 classes of ImageNet as a source model, prompt depth J is set to 3 and the model trained for 2 epochs with learning rate of 0.0026. Hyper-parameters for deep language prompting, deep vision prompting, and independent V-L prompting are detailed in Appendix A. The hyper-parameters are \ufb01xed across all datasets. 4.2. Prompting CLIP via Vision-Language Prompts Prompting Variants: We \ufb01rst evaluate the performance of different possible prompting design choices as an ablation for our proposed branch-aware multi-modal prompting, MaPLe. These variants include shallow MaPLe, deep language prompting, deep vision prompting and independent V-L prompting. In Table 1, we present the results averaged over the 11 image recognition datasets. Shallow MaPLe (row-1) provides consistant improvements over CoOp and Co-CoOp in terms of generalization. Deep language prompting (row-3) shows improvements over deep vision prompting (row-2), indicating that prompts learned at the language branch provide better adaptation of CLIP. Although separately combining the above two approaches (row-4) further improves the performance, it struggles to achieve comprehensive bene\ufb01ts from the language and vision branches. We hypothesize that this is due to the lack of synergy between the learned vision and language prompts as they do not interact with each other during training. Meanwhile, MaPLe tied with deep prompting (row-4) combines the bene\ufb01ts of prompting in both branches by enforcing interactions through explicit conditioning of vision prompts on the language prompts. It provides improvements on novel and base class accuracies which leads to the best HM of 78.55%. We explore other possible design choices and present the ablations in Appendix B. 4.3. Base-to-Novel Generalization Generalization to Unseen Classes: Table 3 presents the performance of MaPLe in base-to-novel generalization setting on 11 recognition datasets. We compare its performance with CLIP zero-shot, and recent prompt learning works including CoOp [49] and Co-CoOp [48]. In case of CLIP, we use hand-crafted prompts that are speci\ufb01cally designed for each dataset. In comparison with the state-of-the-art Co-CoOp, MaPLe shows improved performance on both base and novel categories on all 11 datasets with an exception of marginal reduction on only the base class performance of Caltech101. With mutual synergy from the branch-aware multi-modal prompting, MaPLe better generalizes to novel categories on all 11 datasets in comparison with Co-CoOp, and obtains an overall gain from 71.69% to 75.14%. When taking into account both the base and novel classes, MaPLe shows an absolute average gain of 2.72% over Co-CoOp. In comparison with CLIP on novel classes, Co-CoOp improves only on 4/11 datasets dropping the average novel accuracy from 74.22% to 71.69%. MaPLe is a strong competitor which improves accuracy over CLIP on novel classes on 6/11 datasets, with an average gain from 74.22% to 75.14%. Generalization and Performance on Base Classes: CoCoOp solves the poor generalization problem in CoOp by conditioning prompts on image instances and shows significant gains in novel categories. However on base classes, it improves over CoOp only on 3/11 datasets with an average drop in performance from 82.69% to 80.47%. Meanwhile, the completeness in prompting helps MaPLe improve over CoOp on base classes in 6/11 datasets maintaining the average base accuracy to around 82.28%, in addition to its improvement in generalization to novel classes. We \ufb01nd that the training strategies of Co-CoOp can be used to substantially boost the generalization performance of vanilla CoOp (6.8% gain in novel classes). We therefore compare our method with CoOp\u2020, which trains CoOp in CoCoOp setting (refer to Appendix A for more details). Base Novel HM CoOp 82.69 63.22 71.66 Co-CoOp 80.47 71.69 75.83 CoOp\u2020 80.85 70.02 75.04 MaPLe 82.28 75.14 78.55 Table 2. Generalization comparison of MaPLe with CoOp\u2020. Compare to CoOp\u2020, the vanilla CoOp model seems to over\ufb01t on base classes. When compared to CoOp\u2020 which attains an average base accuracy of 80.85%, MaPLe shows an improvement of 1.43% with the average base accuracy of 82.28% (Table 2). 4.4. Cross-Dataset Evaluation We test the cross-dataset generalization ability of MaPLe by learning multi-modal prompts on all the 1000 ImageNet classes and then transferring it directly on the remaining 10 datasets. Table 4 shows the performance comparison between MaPLe, CoOp and Co-CoOp. On the ImageNet source dataset, MaPLe achieves performance comparable to competing approaches but demonstrates a much stronger \f(a) Average over 11 datasets Base Novel HM CLIP 69.34 74.22 71.70 CoOp 82.69 63.22 71.66 Co-CoOp 80.47 71.69 75.83 MaPLe 82.28 75.14 78.55 +1.81 +3.45 +2.72 (b) ImageNet. Base Novel HM CLIP 72.43 68.14 70.22 CoOp 76.47 67.88 71.92 Co-CoOp 75.98 70.43 73.10 MaPLe 76.66 70.54 73.47 +0.68 +0.11 +0.37 (c) Caltech101 Base Novel HM CLIP 96.84 94.00 95.40 CoOp 98.00 89.81 93.73 Co-CoOp 97.96 93.81 95.84 MaPLe 97.74 94.36 96.02 -0.22 +0.55 +0.18 (d) OxfordPets Base Novel HM CLIP 91.17 97.26 94.12 CoOp 93.67 95.29 94.47 Co-CoOp 95.20 97.69 96.43 MaPLe 95.43 97.76 96.58 +0.23 +0.07 +0.15 (e) StanfordCars Base Novel HM CLIP 63.37 74.89 68.65 CoOp 78.12 60.40 68.13 Co-CoOp 70.49 73.59 72.01 MaPLe 72.94 74.00 73.47 +2.45 +0.41 +1.46 (f) Flowers102 Base Novel HM CLIP 72.08 77.80 74.83 CoOp 97.60 59.67 74.06 Co-CoOp 94.87 71.75 81.71 MaPLe 95.92 72.46 82.56 +1.05 +0.71 +0.85 (g) Food101 Base Novel HM CLIP 90.10 91.22 90.66 CoOp 88.33 82.26 85.19 Co-CoOp 90.70 91.29 90.99 MaPLe 90.71 92.05 91.38 +0.01 +0.76 +0.39 (h) FGVCAircraft Base Novel HM CLIP 27.19 36.29 31.09 CoOp 40.44 22.30 28.75 Co-CoOp 33.41 23.71 27.74 MaPLe 37.44 35.61 36.50 +4.03 +11.90 +8.76 (i) SUN397 Base Novel HM CLIP 69.36 75.35 72.23 CoOp 80.60 65.89 72.51 Co-CoOp 79.74 76.86 78.27 MaPLe 80.82 78.70 79.75 +1.08 +1.84 +1.48 (j) DTD Base Novel HM CLIP 53.24 59.90 56.37 CoOp 79.44 41.18 54.24 Co-CoOp 77.01 56.00 64.85 MaPLe 80.36 59.18 68.16 +3.35 +3.18 +3.31 (k) EuroSAT Base Novel HM CLIP 56.48 64.05 60.03 CoOp 92.19 54.74 68.69 Co-CoOp 87.49 60.04 71.21 MaPLe 94.07 73.23 82.35 +6.58 +13.19 +11.14 (l) UCF101 Base Novel HM CLIP 70.53 77.50 73.85 CoOp 84.69 56.05 67.46 Co-CoOp 82.33 73.45 77.64 MaPLe 83.00 78.66 80.77 +0.67 +5.21 +3.13 Table 3. Comparison with state-of-the-art methods on base-to-novel generalization. MaPLe learns multi-modal prompts and demonstrates strong generalization results over existing methods on 11 recognition datasets. Absolute gains over Co-CoOp are indicated in blue. Source Target ImageNet Caltech101 OxfordPets StanfordCars Flowers102 Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average CoOp 71.51 93.70 89.14 64.51 68.71 85.30 18.47 64.15 41.92 46.39 66.55 63.88 Co-CoOp 71.02 94.43 90.14 65.32 71.88 86.06 22.94 67.36 45.73 45.37 68.21 65.74 MaPLe 70.72 93.53 90.49 65.57 72.23 86.20 24.74 67.01 46.49 48.06 68.69 66.30 Table 4. Comparison of MaPLe with existing approaches on cross-dataset evaluation. Overall, MaPLe achieves competitive performance providing highest average accuracy, indicating better generalization. generalization performance by surpassing CoOp in 9/10 and Co-CoOp in 8/10 datasets. Overall, MaPLe shows competitive performance leading to the highest averaged accuracy of 66.30%. This suggests that the use of branch-aware V-L prompting in MaPLe facilitates better generalization. 4.5. Domain Generalization We show that MaPLe generalizes favourably on out-ofdistribution datasets as compared to CoOp and Co-CoOp. We evaluate the direct transferability of ImageNet trained model to various out-of-domain datasets, and observe that \fSource Target ImageNet ImageNetV2 ImageNet-S ImageNet-A ImageNet-R CLIP 66.73 60.83 46.15 47.77 73.96 CoOp 71.51 64.20 47.99 49.71 75.21 Co-CoOp 71.02 64.07 48.75 50.63 76.18 MaPLe 70.72 64.07 49.15 50.90 76.98 Table 5. Comparison of MaPLe with existing approaches in domain generalization setting. MaPLe shows consistant improvements on all target datasets. it consistently improves against all the existing approaches as indicated in Table 5. This indicates that utilizing multimodal branch-aware prompting helps MaPLe in enhancing the generalization and robustness of V-L models like CLIP. 4.6. Ablation Experiments Prompt Depth: In Fig. 4 (left), we illustrate the effect of prompt depth J for MaPLe and ablate on the depth of language and vision branch individually. In general, the performance improves as prompt depth increases. We note that performance sensitivity increases when randomly initialized prompts are inserted in the deeper layers of a frozen model where the model feature space is already mature. Similar trend is also reported by [16]. As earlier methods utilize shallow language prompting (J = 1), we compare our method with deep language prompting. Overall, MaPLe achieves better performance than deep language prompting and achieves maximum performance at a depth of 9. Prompt Length: Fig. 4 (right) shows the effect of prompt length for MaPLe. As the prompt length increases, the performance on base classes is generally maintained, while the novel class accuracy decreases. This indicates over-\ufb01tting which inherently hurts the generalization to novel classes. Effectiveness of Multi-modal Prompting: Fig. 5 shows the analysis of per class accuracy for selected datasets in the order of increasing domain shift. It indicates that the performance gains of MaPLe in comparison to Co-CoOp varies across different datasets. MaPLe provides signi\ufb01cant gains over Co-CoOp for datasets that have large distribution shifts from the pretraining dataset of CLIP, and vision concepts that are usually rare and less generic. Further detailed analysis is provided in Appendix C. Prompting complexity: Table 6 shows the computational Figure 4. Ablation on prompt depth (left) and prompt length (right) in MaPLe. We report average results on the held-out validation sets of all datasets. Figure 5. Percentage classes where MaPLe shows improved performance over Co-CoOp, which increases as dataset domain shift from generic categories increases (\u2192). complexity of MaPLe in comparison with other approaches. Although MaPLe utilizes multi-modal prompts, its overall FLOPS (Floating Point Operations) exceeds only by 0.1% over CoOp and Co-CoOp. The independent V-L prompting also provides comparable FLOP count. In terms of inference speed, Co-CoOp is signi\ufb01cantly slower and the FPS (Frames Per Second) remains constant as the batch size increases. In contrast, MaPLe has no such overhead and provides much better inference and training speeds. Further, MaPLe provides better convergence as it requires only half training epochs as compared to Co-CoOp (5 vs 10 epochs). MaPLe adds about 2.85% training parameters on top of CLIP. To study if the performance gain is mainly attributed to more parameters, we experiment with MaPLe\u2020, which uses a uni\ufb01ed V-L coupling function for all layer prompts. MaPLe\u2020 with about 9x lesser parameters than MaPLe also improves over existing methods. We also ablate by comparing MaPLe with heavier CoCoOp in Appendix D. Method Params Params FPS (with BS) HM % CLIP 1 4 100 CoOp 2048 0.002 13.8 55.3 1353.0 71.66 CoCoOp 35360 0.03 64.6 114.7 15.1 75.83 Independent V-L 31488 0.02 62.5 239.4 1383.8 77.90 MaPLe 3.55 M 2.85 60.2 239.0 1365.1 78.55 MaPLe\u2020 0.41 M 0.33 60.2 238.0 1365.0 78.11 Table 6. Comparison of computational complexity among different prompting methods. MaPLe\u2020 is a MaPLe version which utilizes a common V-L coupling function for all layers. 5."
+                }
+            ],
+            "Muzammal Naseer": [
+                {
+                    "url": "http://arxiv.org/abs/2302.12252v2",
+                    "title": "Boosting Adversarial Transferability using Dynamic Cues",
+                    "abstract": "The transferability of adversarial perturbations between image models has\nbeen extensively studied. In this case, an attack is generated from a known\nsurrogate \\eg, the ImageNet trained model, and transferred to change the\ndecision of an unknown (black-box) model trained on an image dataset. However,\nattacks generated from image models do not capture the dynamic nature of a\nmoving object or a changing scene due to a lack of temporal cues within image\nmodels. This leads to reduced transferability of adversarial attacks from\nrepresentation-enriched \\emph{image} models such as Supervised Vision\nTransformers (ViTs), Self-supervised ViTs (\\eg, DINO), and Vision-language\nmodels (\\eg, CLIP) to black-box \\emph{video} models. In this work, we induce\ndynamic cues within the image models without sacrificing their original\nperformance on images. To this end, we optimize \\emph{temporal prompts} through\nfrozen image models to capture motion dynamics. Our temporal prompts are the\nresult of a learnable transformation that allows optimizing for temporal\ngradients during an adversarial attack to fool the motion dynamics.\nSpecifically, we introduce spatial (image) and temporal (video) cues within the\nsame source model through task-specific prompts. Attacking such prompts\nmaximizes the adversarial transferability from image-to-video and\nimage-to-image models using the attacks designed for image models. Our attack\nresults indicate that the attacker does not need specialized architectures,\n\\eg, divided space-time attention, 3D convolutions, or multi-view convolution\nnetworks for different data modalities. Image models are effective surrogates\nto optimize an adversarial attack to fool black-box models in a changing\nenvironment over time. Code is available at https://bit.ly/3Xd9gRQ",
+                    "authors": "Muzammal Naseer, Ahmad Mahmood, Salman Khan, Fahad Khan",
+                    "published": "2023-02-23",
+                    "updated": "2023-04-04",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Deep learning models are vulnerable to imperceptible changes to the input images. It has been shown that for a successful attack, an attacker no longer needs to know the attacked target model to compromise its decisions (Naseer et al., 2019; 2020; Nakka & Salzmann, 2021). Adversarial perturbations suitably optimized from a known source model (a surrogate) can fool an unknown target model (Kurakin et al., 2016). These attacks are known as black-box attacks since the attacker is restricted to access the deployed model or compute its adversarial gradient information. Adversarial attacks are continuously evolving, revealing new blind spots of deep neural networks. Adversarial transferability has been extensively studied in image-domain (Akhtar & Mian, 2018; Wang & He, 2021; Naseer et al., 2022b; Malik et al., 2022). Existing works demonstrate how adversarial patterns can be generalized to models with different architectures (Zhou et al., 2018) and even different data domains (Naseer et al., 2019). However, the adversarial transferability between different architecture families designed for varying data modalities, e.g., image models to video models, has not been actively explored. Since the adversarial machine learning topic has gained maximum attention in the image-domain, it is natural to question if image models can help transfer 1 arXiv:2302.12252v2  [cs.CV]  4 Apr 2023 \fPublished as a conference paper at ICLR 2023 better to video-domain models. However, the image models lack dynamic temporal cues which are essential for transfer to the video models. We are motivated by the fact that in a real-world setting, a scene is not static but mostly involves various dynamics, e.g., object motion, changing viewpoints, illumination and background changes. Therefore, exploiting dynamic cues within an adversarial attack is essential to \ufb01nd blind-spots of unknown target models. For this purpose, we introduce the idea of encoding disentangled temporal representations within an image-based Vision Transformer (ViT) model using dedicated temporal prompts while keeping the remaining network frozen. The temporal prompts can learn the dynamic cues which are exploited during attack for improved transferability from image-domain models. Speci\ufb01cally, we introduce the proposed temporal prompts to three types of image models with enriched representations acquired via supervised (ViT (Dosovitskiy et al., 2020)), self-supervised (DINO (Caron et al., 2021)) or multi-modal learning (CLIP (Radford et al., 2021)). Our approach offers the bene\ufb01t that the attacks do not need to rely on specialized networks designed for videos towards better adversarial transferability. As an example, popular model designs for videos incorporate 3D convolutions, space-time attention, tube embeddings or multi-view information to be robust against the temporal changes (Bertasius et al., 2021; Arnab et al., 2021). Without access to such speci\ufb01c design choices, our approach demonstrates how an attacker can leverage regular image models augmented with temporal prompts to learn dynamic cues. Further, our approach can be easily extended to image datasets, where disentangled representations can be learned via tokens across a scale-space at varying image resolutions. In summary, the major contributions of this work include: \u2022 We demonstrate how temporal prompts incorporated with frozen image-based models can help model dynamic cues which can be exploited to fool deep networks designed for videos. \u2022 Our approach for dynamic cue modeling via prompts does not affect the original spatial representations learned by the image-based models during pre-training, e.g., fully-supervised, self-supervised and multi-modal models. \u2022 The proposed method signi\ufb01cantly improves transfer to black-box image and video models. Our approach is easily extendable to 3D datasets via learning cross-view prompts; and image-only datasets via modeling the scale-space. Finally, it enables generalization from popular plain ViT models without considering video-speci\ufb01c specialized designs. We analyse the adversarial space of three type of image models (fully-supervised, self-supervised, and text-supervised). A pre-trained ImageNet ViT with approximately 6 million parameters exhibits 44.6 and 72.2 top-1 (%) accuracy on Kinetics-400 and ImageNet validation sets using our approach, thereby signi\ufb01cantly improving the adversarial transferability on video-domain models. A similar trend exists with other image models. Our results indicate that the multi-modal CLIP can better adapt to video modalities than fully-supervised or self-supervised ViTs. However, CLIP adversaries are relatively less transferable as compared to fully-supervised ViT or self-supervised DINO model.As an example, a momentum based iterative attack launched from our DINO model can reduce the performance of TimesFormer (Bertasius et al., 2021) from 75.6% to 35.8% on Kinetics-400 dataset. 2 BOOSTING ADVERSARIAL TRANSFERABILITY USING DYNAMIC CUES Adversarial transferability refers to manipulating a clean sample (image, video, or 3D object rendered into multi-views) in a way that is deceiving for an unknown (black-box) model. In the absence of an adversarial perturbation, the same black-box model predicts the correct label for the given image, video, or a rendered view of a 3D object. A known surrogate model is usually used to optimize for the adversarial patterns. Instead of training the surrogate model from scratch on a given data distribution, an attacker can also adapt pre-trained image models to the new task. These image models can include supervised ImageNet models such as Deit (Touvron et al., 2020), self-supervised ImageNet models like DINO (Caron et al., 2021), and text-supervised large-scale multi-modal models e.g. CLIP (Radford et al., 2021). The adversarial attack generated from such pre-trained models with enriched representations transfer better in the black-box setting for image-to-image transfer task (Zhang et al., 2022; Naseer et al., 2022b; Aich et al., 2022). However, adversarial perturbations optimized from image models are not well suited to fool motion dynamics learned by a video model (Sec. 3). To cater for this, we introduce temporal cues to model motion dynamics within adversarial attacks through pre-trained image models. Our approach, therefore, models both spatial and temporal 2 \fPublished as a conference paper at ICLR 2023 Proposed training for dynamic cues within image models Image pre-trained  vision transformer MHSA Frozen model Layer Norm Layer Norm FFN Input Video (\ud835\udc65) Target Video / Image Models (Black-box) Source Image Model Random  Sampler Spatial  Tokens Tokenizer Transformation \ud835\udcaf MHSA Spatial  Pooling Tokenizer Learned Temporal  Tokens (\ud835\udc61) Perturbed Video  (\ud835\udc65\u2032) Multi-head Self-attention Feed-forward network \ud835\udc65 \ud835\udc65\u2032 Times Former \ud835\udc65 \ud835\udc65\u2032 ResNet3D \ud835\udc65 \ud835\udc65\u2032 BiT50 Crawling Jumping Crawling Playing Baby Whale Attack Spatial  head Temporal  head Figure 1: Overview of inducing dynamic cues within image models: Attackers can easily access freely available, pre-trained image models learned on large-scale image and language datasets to launch adversarial attacks. These models, however, lack temporal information. Therefore, adversarial attacks launched using image models have less success rate against a moving target such as in videos. We learn a transformation T (.) to convert a given video with t number of frames into t temporal tokens. Our transformation is based on self-attention thus it learns the motion dynamics between the video frames with global context during training. We randomly sample a single frame to represent the spatial tokens. The temporal and spatial tokens are concatenated and passed through the frozen model. The spatial tokens are ignored while the average of temporal tokens is processed through a temporal head for video recognition. The image/spatial class token learned on images (e.g., ImageNet) interacts with our temporal tokens (e.g., Kinetics) within the network through self-attention. After the training, image and video solutions are preserved within the spatial and temporal tokens and used in adversarial attacks. information during the attack (Sec. 2.1) and does not depend on specialized video networks e.g., with 3D convolutions (Tran et al., 2018), space-time divided attention (Bertasius et al., 2021) or even multi-branch models (Su et al., 2015). To this end, we \ufb01rst introduce temporal prompt adaptation of image models to learn dynamic cues on videos or multi-view information on rendered views of a 3D object, as explained next. 2.1 INTRODUCING DYNAMIC CUES THROUGH TEMPORAL PROMPTS Preliminaries: We consider an image model F pre-trained on the input samples x of size c \u00d7 h \u00d7 w, where c, h and w represent the color channels, height and width, respectively. We consider Vision Transformers (ViTs) (Dosovitskiy et al., 2020; Touvron et al., 2020) sequentially composed of n Transformer blocks comprising of multi-head self-attention (MHSA) and feed-forward layer (Dosovitskiy et al., 2020) i.e. F = (f1 \u25e6f2 \u25e6f3 \u25e6. . . fn), where fi represents a single Transformer block. ViTs divide an input sample x into N patches also called patch tokens, Pt \u2208RN\u00d7D, where D is the patch dimension. A class token1, Icls \u2208R1\u00d7D, is usually combined with these patch tokens within these image models (Dosovitskiy et al., 2020; Caron et al., 2021; Radford et al., 2021). We refer to these patch and class tokens collectively as \u2018spatial tokens\u2019 (Fig. 1). These tokens are then processed by the model F to produce an output of the same size as of input i.e. F(x) \u2208R(N+1)\u00d7D. In the case of image classi\ufb01cation, the re\ufb01ned class token (\u02dc Icls) is extracted from the model output and processed by MLP (gs), named spatial head (Fig. 1). It maps the re\ufb01ned class token from R1\u00d7D to R1\u00d7Cs, where Cs represents the number of image class categories. Motivation: The spatial class token in image models, Icls, is never optimized for temporal information. Our objective is to adapt the pre-trained and frozen image model for videos or multi-views by training a temporal class token It cls \u2208R1\u00d7D that is optimal for input samples of size t \u00d7 c \u00d7 h \u00d7 w, 1Hierarchical ViTs such as Swin Transformer (Liu et al., 2021c) or more simple designs without self-attention layers like MLP-Mixer (Tolstikhin et al., 2021) use average of patch tokens as the class token. 3 \fPublished as a conference paper at ICLR 2023 Figure 2: Mimicking dynamic cues on static images: We optimize our proposed transformation (Fig. 1) for different spatial scales to mimic changes. This improves model generalization e.g. Deit-T performance increases from 7.7% to 45% on ImageNet val. set at 96\u00d796. Attacking models with such prompts increases transferability. where t represents the number of temporal frames in a video or rendered views of a 3D object. We optimize the temporal token through MLP (gt), named temporal head (Fig. 1). It maps the temporal token from R1\u00d7D to R1\u00d7Ct, where Ct represents the number of video action categories. A trivial way is to concatenate all the additional frames of a video to generate t \u00d7 N patch tokens which can then be combined with the temporal token before forwadpass through the image model. We can then optimize temporal class token It cls and temporal head (gt) for video classi\ufb01cation and incorporate the dynamic cues within a frozen image model. This approach, however, has a drawback as the computational time complexity increases signi\ufb01cantly to O \u0000(t \u00d7 N)2 \u00d7 D \u0001 within the self-attention layers of ViTs. Another naive way is to either apply temporal pooling to convert t\u00d7N to only N spatial tokens or approximate a video via a single frame that is ti \u00d7c\u00d7h\u00d7w, where ti corresponds to a randomly sampled frame from the input video. These approaches are however sub-optimal as we are unable to optimize for motion dynamics available from different video frames. We induce the temporal information within image models without increasing the quadratic complexity within self-attention of a Vision Transformer. We achieve this by representing a given video by a randomly sampled frame while at the same time each frame in a video is modeled via a temporal prompt of size R1\u00d7D. The temporal prompts for different frames in a video are generated through our transformation T . Therefore, we only train transformation T , temporal class token It cls and head gt to learn motion dynamics within pre-trained and frozen image models. Our approach preserves the original image or spatial representation encoded within spatial class token Icls (Table 5). Both spatial and temporal tokens, Icls and It cls, are then used within our attack approach (Sec. 2.2). 2.1.1 TEMPORAL PROMPTS THROUGH TRANSFORMATION As mentioned above, the transformation T processes a video with t number of temporal frames and produces only t temporal prompts (Fig. 1). A video sample, x, is divided into patch tokens such that x \u2208Rt\u00d7N\u00d7D. The transformation T is a single self-attention block (Dosovitskiy et al., 2020) that interacts between the tokens and learns the relationship between the video frames. The transformation output is then pooled across spatial dimension N to produce t temporal tokens i.e., T (x) \u2208Rt\u00d7D. Attacking such temporal tokens allows \ufb01nding adversarial patterns capable of fooling the dynamic cues learned by the unknown black-box video model. Mimicking dynamic behavior on image datasets: Image samples are static and have no temporal dimension. So we adopt a simple strategy to mimic changing behavior for such static data. We consider images at different spatial scales to obtain a scale-space (Fig. 2) and learn prompts for different spatial scales. The dynamic cues within image models not only boost transferability of the existing adversarial attacks from image-to-video models but also increase the attack success rate in the black-box image models (Sec. 3). 2.1.2 IMAGE MODELS WITH TEMPORAL AND SPATIAL PROMPTS Our approach is motivated by the shallow prompt transfer learning (Jia et al., 2022). As discussed above, we divide a given a video x into video patches such that x \u2208Rt\u00d7N\u00d7D. These video patches are processed by our transformation to produce temporal tokens i.e. T (x) \u2208Rt\u00d7D. Further, we randomly sample a single frame xi \u223cx and divide it into patch tokens such that xi \u2208RN\u00d7D. These single frame tokens are then concatenated with the temporal prompts generated by T (x), a temporal class token It cls, and the image class token Icls to equip the pre-trained image representation of a model F with discriminative temporal information. F(x) = F \u0000\u0002 It cls, T (x), Icls, xi \u0003\u0001 (1) After the forwardpass through the image model (Eq. 1), we extract the re\ufb01ned temporal class token and temporal prompts from the model\u2019s output. The average of these re\ufb01ned temporal class tokens, 4 \fPublished as a conference paper at ICLR 2023 Figure 3: Visualizing the behavior of Adversarial Patterns across Time: We generate adversarial signals from our DINO model with temporal prompts using DIM attack (Xie et al., 2019). Observe the change in gradients across different frames (best viewed in zoom). We provide visual demos of adversarial patterns in Appendix K. and t temporal prompts is then projected through the temporal head gt for the new video classi\ufb01cation task. For the sake of brevity, we will refer to the average of our re\ufb01ned temporal class and prompts tokens collectively as \u02dc Itp cls. During backpass, we only update the parameters of our transformation, temporal class token and temporal head (Fig. 1). Training: Given a pre-trained ViT, we freeze all of its existing weights and insert our learnable transformation (T ), temporal class token It cls, and video-speci\ufb01c temporal head. We train for 15 epochs only using SGD optimizer with a learning rate of 0.005 which is decayed by a factor of 10 after the 11th and 14th epoch. We use batch size of 64 and train on 16 A100 GPUs for large-scale datasets such as Kinetics-400 (Kay et al., 2017) and only 2 A100 GPUs for other small datasets. We discuss the effect of our temporal prompts on temporal (video) and spatial (image) solutions in Table 5. Our method mostly retains the original image solution captured in image class token Icls while exhibiting dynamic temporal modeling. 2.2 TRANSFERABLE ATTACK USING SPATIAL AND TEMPORAL CUES Given an image model F adapted for motion dynamics via our approach (Fig. 1), our attack exploits the spatial and temporal cues to generate transferable perturbations for both image and video models. \u2022 Given a video and its corresponding label: Our attack uses the re\ufb01ned temporal class token \u02dc Itp cls output by the model in a supervised manner, while the spatial class token \u02dc Icls serves in an self-supervised objective. \u2022 Given an image and its corresponding label: Our attack uses the re\ufb01ned class tokens \u02dc Itp cls learned at different resolutions to mimic the dynamic behavior from images in a supervised manner, while the spatial class token \u02dc Icls serves in an self-supervised objective. We generate an adversarial example x\u2032 for a video or image sample x using the following objective: \ufb01nd x\u2032 such that F(x\u2032)argmax \u0338= y, and \u2225x \u2212x\u2032\u2225p \u2264\u03f5, (2) where y is the original label and \u03f5 represents a maximum perturbation budget within a norm distance p. We optimize Eq. 2 by maximizing the loss objective in Eq. 3 within existing adversarial attacks. maximize L = Ls \u2212Lss, (3) where Ls is a supervised loss function. It uses the labels of a given image/video through supervised MLP head. Ls can be optimized by any of the existing supervised losses proposed in transferable attacks literature; cross-entropy (Dong et al., 2018), relativistic cross-entropy (Naseer et al., 2019), or logit loss (Zhao et al., 2021). Unless otherwise mentioned, we use cross-entropy loss to demonstrate the effectiveness of our approach. Similarly, Lss is self-supervised loss that exploits pre-trained representation within spatial class tokens. We minimize the cosine similarity between the re\ufb01ned class tokens of clean and adversarial samples. Speci\ufb01cally, we de\ufb01ne Ls and Lss as follows: Ls = \u2212 Ct X j=1 yjlog \u0010 \u02dc I\u2032tp cls \u25e6gt \u0011 , Lss = \u02dc I\u2032\u22a4 cls\u02dc Icls \u2225\u02dc I\u2032cls\u2225\u2225\u02dc Icls\u2225 5 \fPublished as a conference paper at ICLR 2023 Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014) TimesFormer ResNet3D MVCNN Models (\u2193) Temporal Prompts UCF HMDB K400 SSv2 UCF HMDB K400 SSv2 Depth Shaded 90.6 59.1 75.6 45.1 79.9 45.4 60.5 26.7 92.6 94.8 Deit-T \u0017 75.1 32.9 57.8 23.5 25.7 9.8 23.3 14.7 16.0 79.0 \u0013 64.5(-26.1) 21.9(-37.2) 51.6(-24.0) 19.8(-25.3) 22.3(-57.6) 8.9(-36.5) 19.4(-41.1) 13.9(-12.8) 15.2(-77.4) 77.4(-17.4) Deit-S \u0017 74.7 33.5 58.2 23.0 25.4 9.7 23.9 15.9 17.2 79.8 \u0013 64.0(-26.6) 20.6(-38.5) 48.5(-27.1) 19.1(-26.0) 22.6(-57.3) 8.6(-36.8) 20.6(-39.9) 13.6(-13.1) 15.9(-76.7) 78.9(-15.9) Deit-B \u0017 75.5 31.7 57.5 22.6 25.0 9.2 22.4 15.5 17.3 80.6 \u0013 64.7(-25.9) 20.0(-39.1) 48.6(-27.0) 19.3(-25.8) 22.7(-57.2) 8.1(-37.3) 19.5(-41.0) 13.4(-13.3) 17.1(-75.5) 80.9(-13.9) DINO \u0017 71.0 33.7 54.2 23.8 32.0 12.4 24.3 15.2 15.2 72.7 \u0013 60.7(-29.9) 18.8(-40.3) 46.7(-28.9) 19.4(-25.7) 21.9(-58.0) 8.3(-37.1) 19.6(-40.9) 13.9(-12.8) 15.0(-77.6) 70.7(-24.1) CLIP \u0017 79.8 36.6 62.3 26.4 38.5 15.0 29.9 17.3 15.8 81.4 \u0013 77.5(-13.1) 29.3(-29.8) 59.5(-16.1) 25.0(-20.1) 35.6(-44.3) 11.2(-34.2) 28.8(-31.7) 16.5(-10.2) 15.6(-77.0) 81.2(-13.6) Projected Gradient Decent (PGD) (Madry et al., 2018) Deit-T \u0017 84.9 40.9 68.3 32.5 69.4 32.7 44.3 18.4 37.0 90.5 \u0013 78.1(-12.5) 32.0(-27.1) 61.5(-14.1) 27.8(-17.3) 65.9(-14.0) 29.5(-15.9) 39.1(-21.4) 17.6(-9.1) 34.6(-58.0) 90.2(-4.6) Deit-S \u0017 85.1 42.0 68.6 32.5 69.9 32.7 44.8 19.4 39.4 90.9 \u0013 74.1(-16.5) 26.1(-33.0) 58.3(-17.3) 26.1(-19.0) 65.4(-14.5) 28.2(-17.2) 28.9(-31.6) 17.1(-9.6) 36.5(-56.1) 90.1(-4.7) Deit-B \u0017 85.5 41.5 67.7 33.3 69.9 32.6 44.7 18.9 37.7 91.1 \u0013 73.6(-17.0) 25.0(-34.1) 56.3(-19.3) 24.4(-20.7) 64.7(-15.2) 26.6(-18.8) 37.0(-23.5) 16.6(-10.1) 36.7(-55.9) 90.5(-4.3) DINO \u0017 81.7 38.0 64.8 30.9 68.0 31.8 42.5 18.2 33.1 89.2 \u0013 64.9(-25.7) 14.8(-44.3) 51.5(-24.1) 22.4(-22.7) 63.1(-16.8) 25.9(-19.5) 35.2(-25.3) 16.6(-10.1) 28.9(-63.5) 87.5(-7.3) CLIP \u0017 86.9 44.5 70.7 35.4 71.9 35.1 46.5 20.0 39.8 91.3 \u0013 82.6(-8.0) 32.3(-26.8) 66.9(-8.7) 32.7(-12.4) 69.7(-10.2) 28.9(-16.5) 43.2(-17.3) 18.9(-7.8) 37.6(-55.0) 90.1(-4.7) Table 1: Adversarial Transferability from image to video models: Adversarial accuracy (%) is reported at \u03f5 \u226416. When attack is optimized from an image model without temporal prompts (\u0017), then we minimize the unsupervised adversarial objective (Eq. 3) i.e., cosine similarity between the feature embedding of spatial class token of clean and adversarial frames. We use 8 frames from a given video. Attacks transferability improves by a clear margin with temporal prompts. Clean accuracy (top1 % on randomly selected validation set of 1.5k samples) is highlighted with cell color. Single step FGSM and multi-step PGD attacks perform better with our proposed temporal prompts (e.g. FGSM reduces TimesFormer generalization from 90.6% to 64.5% on UCF by exploiting dynamic cues within the DeiT-T). A similar trend exists for ResNet3D and MVCNN. In this manner, any of the existing attacks can be combined with our approach to transfer adversarial perturbations as we demonstrate in Sec. 3. 2.2.1 CHANGING SPATIAL RESOLUTION FOR REGULARIZATION An adversarial attack can easily over\ufb01t the source models meaning it can have a 100% success rate on the source model but mostly fails to fool the unknown black-box model. Different heuristics like adding momentum (Dong et al., 2018; Lin et al., 2019; Wang & He, 2021), augmentations (Naseer et al., 2021; Xie et al., 2019), ensemble of different models (Dong et al., 2018) or even self-ensemble (Naseer et al., 2022b) at a target resolution are proposed to reduce such over\ufb01tting and increase adversarial transferability. Low-resolution pre-training followed by high-resolution \ufb01ne-tuning have a regularization effect on the generalization of neural networks (Touvron et al., 2022). Similarly, we observe that \ufb01ne-tuning scale-space prompts at different low resolutions (e.g., 96\u00d796) within a model pre-trained on high resolution (224\u00d7224) also has a complimentary regularization effect on the transferability of adversarial perturbations (Tables 3 and 4). Most publicly available image models are trained at a resolution of size 224\u00d7224 and their performance at low-resolution inputs is sub-optimal (Fig. 2). Learning at low-resolution with our approach (Fig. 1) also allows to mimic a changing scene over time from static datasets like ImageNet (Fig. 2). Our low-resolution scale-space prompts (Sec. 2.1) signi\ufb01cantly improves generalization at low-resolution inputs (Fig. 2). 3 EXPERIMENTS We generate l\u221eadversarial examples from the adapted image models and study their transferability to video and image models. We followed standard adversarial transferability protocol Dong et al. (2018); Naseer et al. (2022b); the attacker is unaware of the black-box video model but has access to its train data. Our temporal cues also boost cross-task adversarial transferability with no access 6 \fPublished as a conference paper at ICLR 2023 Momentum Iterative Fast Gradient Sign Method (MIM) (Dong et al., 2018) TimesFormer ResNet3D MVCNN Models (\u2193) Temporal Prompts UCF HMDB K400 SSv2 UCF HMDB K400 SSv2 Depth Shaded 90.6 59.1 75.6 45.1 79.9 45.4 60.5 26.7 92.6 94.8 Deit-T \u0017 77.0 30.1 58.5 25.2 42.0 16.2 29.1 16.0 19.2 83.0 \u0013 66.9(-23.7) 21.4(-37.7) 51.4(-24.2) 21.5(-23.6) 41.5(-38.4) 16.1(-29.3) 26.4(-34.1) 15.3(-11.4) 19.0(-73.6) 82.9(-11.9) Deit-S \u0017 78.7 31.8 59.3 25.1 42.2 16.5 31.0 16.5 20.6 83.6 \u0013 60.8(-29.8) 16.1(-43.0) 47.6(-28.0) 21.0(-24.1) 41.9(-38.0) 15.7(-29.7) 26.8(-33.7) 14.2(-12.5) 19.3(-73.3) 83.5(-11.3) Deit-B \u0017 78.7 32.0 59.7 25.9 43.1 16.2 29.5 16.8 20.6 84.4 \u0013 61.7(-25.9) 15.6(-43.5) 45.2(-30.4) 19.1(-26.0) 40.7(-39.2) 13.6(-31.8) 24.8(-35.7) 14.5(-12.2) 20.5(-72.1) 84.2(-10.6) DINO \u0017 71.3 28.0 54.6 24.1 46.0 18.8 30.3 15.9 19.7 80.4 \u0013 47.7(-42.9) 8.40(-50.7) 38.8(-36.8) 16.4(-28.7) 39.8(-40.1) 12.7(-32.7) 24.5(-36.0) 14.3(-12.4) 19.4(-73.2) 80.7(-14.1) CLIP \u0017 81.8 35.6 63.5 29.3 53.2 21.9 35.5 17.9 20.8 86.7 \u0013 78.2(-12.4) 28.2(-30.9) 61.3(-14.3) 27.2(-17.9) 45.9(-34.0) 17.8(-27.6) 32.9(-27.6) 16.5(-10.2) 20.2(-72.4) 84.7(-10.1) MIM with Input Diversity (DIM) (Xie et al., 2019) Deit-T \u0017 75.3 28.2 59.2 24.3 41.3 16.5 29.5 16.3 20.3 84.5 \u0013 62.6(-28.0) 16.9(-42.2) 48.1(-27.5) 20.5(-24.6) 37.8(-42.1) 13.4(-32.0) 24.2(-36.3) 14.3(-12.4) 19.6(-73.0) 85.0(-9.8) Deit-S \u0017 76.3 29.4 57.5 24.1 39.8 17.3 29.8 16.2 21.8 85.9 \u0013 54.1(-36.5) 12.3(-46.8) 42.8(-32.8) 17.7(-27.4) 36.8(-43.1) 12.7(-32.7) 22.3(-38.2) 14.6(-12.1) 21.0(-71.6) 85.7(-9.1) Deit-B \u0017 75.9 28.8 57.8 24.5 41.5 16.5 30.5 15.6 22.2 85.1 \u0013 55.5(-35.1) 12.6(-46.5) 40.6(-35.0) 16.8(-28.3) 37.3(-42.6) 11.4(-34.0) 21.5(-39.0) 13.3(-13.4) 20.5(-72.1) 86.3(-8.5) DINO \u0017 64.9 23.1 51.8 22.6 43.5 16.7 27.6 13.7 19.0 80.2 \u0013 45.2(-45.4) 7.30(-51.8) 35.8(-39.8) 15.8(-29.3) 32.2(-47.7) 10.6(-34.8) 18.3(-42.2) 12.0(-14.7) 18.9(-73.7) 78.6(-16.2) CLIP \u0017 78.0 29.5 61.0 26.9 50.0 19.6 32.8 16.9 20.8 87.8 \u0013 73.4(-17.2) 22.7(-36.4) 56.7(-18.9) 25.1(-20.0) 46.5(-33.4) 13.6(-31.8) 31.5(-29.0) 15.7(-11.0) 20.5(-72.1) 87.4(-7.4) Table 2: Adversarial Transferability from image to video models: Adversarial accuracy (%) is reported at \u03f5 \u226416. When attack is optimized from an Image model without temporal prompts (\u0017), then we minimize the unsupervised adversarial objective (Eq. 3), cosine similarity, between the feature embedding of spatial class token of clean and adversarial frames. We use 8 frames from a given video. Attacks transferability improves by a clear margin with temporal prompts. Clean accuracy (top1 % on randomly selected validation set of 1.5k samples) is highlighted with cell color. We observe that a small Image model (e.g. Deit-T with only 5 Million parameters) can signi\ufb01cantly reduce the generalization of TimesFormer with the presence of our temporal prompts. The attack performance with temporal prompts increases with network capacity (e.g. Deit-T to Deit-B). to the black-box video model, its training data, or label space ( Appendix B). The maximum pixel perturbation is set to \u03f5 \u226416 for pixel range of [0, 255]. We use standard image attacks to optimize adversarial examples through our adapted image models and boost adversarial transferability to video models by using the following protocols: Surrogate image models: We optimize adversaries from the known surrogate models. We study three types of surrogate image models trained via supervised (Touvron et al., 2020), self-supervised (Caron et al., 2021) and text-supervised (Radford et al., 2021) models. \u2212Supervised image models: We use Deit-T, DeiT-S, and DeiT-B with 5, 22, and 86 million parameters to video datasets. These models are pre-trained in a supervised manner on ImageNet. \u2212Self-Supervised image model: We use Vision Transformer trained in self-supervised fashion on ImageNet using DINO training framework (Caron et al., 2021). \u2212Text-Supervised image model: We adapt CLIP image encoder only trained on text-guided images. DeiT-B, DINO, and CLIP share the same network based on Vision Transformer. These models process images of size 3\u00d7224\u00d7224 that are divided into 196 patches with a patch size of 16. Each of these patch tokens has 768 embedding dimensions. Thus, the only difference between these models lies in their corresponding training frameworks. These models are adapted to different video datasets. Adversarial examples are then simply created using existing single and multi-step attacks (Goodfellow et al., 2014; Madry et al., 2018; Dong et al., 2018; Xie et al., 2019). Target video and image models: We transfer adversarial examples to unknown (black-box) video and image models. \u2212Target Video Models: We consider recently proposed TimesFormer (Bertasius et al., 2021) with divided space-time attention, 3D convolutional network (Tran et al., 2018), and multi-view convolutional network (Su et al., 2015). \u2212Target Image Models: We consider the same image models used in baseline by (Naseer et al., 2022b): BiT-ResNet50 (BiT50) (Beyer et al., 7 \fPublished as a conference paper at ICLR 2023 MIM with Input Diversity (DIM) (Xie et al., 2019) Convolutional Transformer Models (\u2193) Method BiT50 Res152 WRN DN201 ViT-L T2T24 TnT T2T-7 Deit-B Naseer et al. (2022b) 80.10 84.92 86.36 89.24 78.90 84.00 92.28 93.42 Deit-B Ours 86.64 90.88 92.14 93.82 95.64 95.74 98.48 94.74 DINO 84.36 93.74 95.16 96.20 96.48 89.68 94.74 96.18 CLIP 55.34 64.24 65.76 71.78 59.94 54.66 62.72 75.80 Table 3: Adversarial Transferability from image to image models: Fool Rate (%) on 5k ImageNet val. samples from Naseer et al. (2022b). We compare against the best results from Naseer et al. (2022b) that our baseline is a self-ensemble Deit-B with token re\ufb01nement blocks. To mimic the dynamics with static images, we optimize our proposed prompts at resolutions of 56\u00d756, 96\u00d794, 120\u00d7120, and 224\u00d7224. Adversaries are optimized at the target resolution of 224\u00d7224. Our method performs favorably well against self-ensemble with re\ufb01ned tokens. Models Temporal Prompts SR TimesFormer UCF HMDB DeiT-B \u0017 \u0017 75.9 28.8 \u0013 \u0017 55.5 12.6 \u0013 \u0013 53.9 11.8 Table 4: Effect of Change in Spatial Resolution (SR): Incorporating change in spatial resolution improves adversarial transferability (% accuracy, lower is better). Adversaries are created at target resolution of 224 from an ensemble of models adopted at 56\u00d756, 96\u00d796, and 224\u00d7224. 2021), ResNet152 (Res152) (He et al., 2016), Wide-ResNet-50-2 (WRN) (Zagoruyko & Komodakis, 2016), DenseNet201 (DN201) (Huang et al., 2017) and other ViT models including Token-to-Token transformer (T2T) (Yuan et al., 2021), Transformer in Transformer (TnT) (Mao et al., 2021). Adapting image models for videos using dynamic cues: We use UCF (Soomro et al., 2012), HMDB (Kuehne et al., 2011), K400 (Kay et al., 2017), and SSv2 (Goyal et al., 2017) training sets to learn temporal prompts and adapt image models to videos via our approach (Fig.1). HMDB has the smallest validation set of 1.5k samples. For evaluating robustness, we selected all validation samples in HMDB, while randomly selected 1.5k samples from UCF, K400, and SSv2 validation sets. We also use multi-view training samples rendered for 3D ModelNet40 (depth and shaded) for image models. We use validation samples of rendered multi-views for both modalities. Adapting image models to images mimicking dynamic cues: We use ImageNet training set and learn our proposed transformation and prompts at multiple spatial scales; 56\u00d756, 96\u00d796, 120\u00d7120, and 224\u00d7224. We optimize adversarial attacks at the target resolution of 224\u00d7224 by using the ensemble of our learned models at multiple resolutions. In this manner, our approach mimic the change over time by changing the spatial scale or resolution over time. We study our attack approach using the 5k samples from ImageNet validation set proposed by (Naseer et al., 2022b). Inference and metrics: We randomly sample 8 frames from a given video for testing. We report drop in top-1 (%) accuracy on adversarial and clean videos. We report fooling rate (%) of adversarial samples for which the predicted label is \ufb02ipped w.r.t the original labels) to evaluate on image datasets. Extending image attacks to videos: We apply image attacks such as single step fast gradient sign method (FGSM) (Goodfellow et al., 2014) as well as iterative attacks with twenty iterations including PGD (Madry et al., 2018), MIM (Dong et al., 2018) and input diversity (augmentations to the inputs) (DIM) (Xie et al., 2019) attacks to adapted image models and transfer their adversarial perturbations to black-box video models. We follow the attack settings used by (Naseer et al., 2022b). Our proposed transformation (T (.)) allows to model temporal gradients during attack optimization (Fig. 3). We simply aggregates the gradients from both branches of our model to the input video frames (Fig. 1). 3.1 ADVERSARIAL TRANSFER FROM IMAGE-TO-VIDEO MODELS The generalization of image models on videos is discussed in Table 5. We observe that image solution remains preserved in ViT backbones even after training with our proposed transformation and temporal prompts. Deit models retain their top-1 (%) accuracy on ImageNet when measured through image class token, while also exhibiting decent performance on video datasets and rendered mulit-views of ModelNet40. CLIP achieves the best performance on videos. The transferability of image based attacks from our adapted image models to video and multi-view models is presented in Tables 1 and 2. Following insights emerges from our analysis: a) The attack success to video models 8 \fPublished as a conference paper at ICLR 2023 Figure 4: Adversaries (DIM (Xie et al., 2019)) from an ensemble of image models with different networks transfer well and fool the black-box video models. First row shows transferablity from an ensemble of different networks, while second row shows attack results from an ensemble of similar networks but pre-trained with different training schemes. ImageNet ImageNet and Videos ImageNet and ModelNet40 Models IN IN \u2013 UCF IN \u2013 HMDB IN \u2013 K400 IN \u2013 SSv2 IN \u2013 Depth IN \u2013 Shaded Deit-T 72.2 72.0 \u2013 70.0 72.3 \u2013 36.2 72.2 \u2013 44.6 72.2 \u2013 11.2 72.2 \u2013 86.0 72.1 \u2013 81.0 Deit-S 79.9 79.8 \u2013 77.2 79.8 \u2013 44.6 79.9 \u2013 53.0 79.9 \u2013 15.3 78.8 \u2013 86.6 79.8 \u2013 86.2 Deit-B 81.8 81.9 \u2013 81.4 81.7 \u2013 47.7 81.9 \u2013 57.0 81.8 \u2013 17.5 81.8 \u2013 90.1 81.4 \u2013 88.2 DINO NA NA \u2013 79.5 NA \u2013 45.1 NA \u2013 57.4 NA \u2013 17.4 NA \u2013 90.1 NA\u2013 89.8 CLIP NA NA \u2013 86.0 NA \u2013 54.6 NA \u2013 67.3 NA \u2013 19.9 NA \u2013 89.5 NA \u2013 88.9 Table 5: Spatial along with Temporal Solutions within Image Models: Our approach successfully incorporates temporal information into pre-trained and frozen image models and increases their generalization (top-1 (%) at resolution 224) to high-dimensional datasets. The original image solution remains preserved in our approach. Ours Deit-B retains 81.9% top-1 accuracy on ImageNet validation set while exhibiting 81.4% on UCF. increases with size of the image models. The larger the image model (e.g. Deit-T to DeiT-B) the higher the attack success, b) The adversarial perturbations generated using self-supervised DINO transfer better than CLIP or supervised Deit-B, c) Divided space-time attention is more robust than 3D convolution, and d) MVCNN trained on shaded rendered views is more robust than depth. 3.1.1 ENSEMBLE ADVERSARIAL TRANSFER Adversarial attacks optimized from an ensemble of models (Dong et al., 2018; Xie et al., 2019) or self-ensemble (Naseer et al., 2022b) increases transferability. We study three types of image ensembles including a) Ensemble of Different Networks: We transfer attack from three image models of the Deit family (Touvron et al., 2020) including Deit-T, Deit-S and Deit-B. These models differ in architecture, b) Ensemble of Different Training Frameworks: We transfer attack from three image models including Deit-B, DINO and CLIP. These models share similar architecture but differ in their training, and c) Ensemble of Different Spatial Resolutions: We transfer attack from three image models with the same architecture (Deit-B) but prompt adapted to different spatial resolutions. Ensemble of different training frameworks performs favorably to boost attack transferability (Fig. 4). An attacker can adapt our approach at different resolutions to enhance transferability (Table 4). We provide generalization of Deit-B at varying spatial scales for video datasets in Appendix I. 3.2 ADVERSARIAL TRANSFER FROM IMAGE-TO-IMAGE MODELS Image datasets are static but we mimic the dynamic behaviour by learning our proposed transformation at different spatial scales (Fig. 2). Our approach exhibit signi\ufb01cant gains in adversarial transferability (Table 3). Refer to appendices B-J for extensive analysis on cross-task adversarial transferability, textual bias of CLIP on adversarial transferability, visualization of attention roll-outs and latent embeddings, and effect of the number of temporal prompts on adversarial transferability. 4"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2106.04169v3",
+                    "title": "On Improving Adversarial Transferability of Vision Transformers",
+                    "abstract": "Vision transformers (ViTs) process input images as sequences of patches via\nself-attention; a radically different architecture than convolutional neural\nnetworks (CNNs). This makes it interesting to study the adversarial feature\nspace of ViT models and their transferability. In particular, we observe that\nadversarial patterns found via conventional adversarial attacks show very\n\\emph{low} black-box transferability even for large ViT models. We show that\nthis phenomenon is only due to the sub-optimal attack procedures that do not\nleverage the true representation potential of ViTs. A deep ViT is composed of\nmultiple blocks, with a consistent architecture comprising of self-attention\nand feed-forward layers, where each block is capable of independently producing\na class token. Formulating an attack using only the last class token\n(conventional approach) does not directly leverage the discriminative\ninformation stored in the earlier tokens, leading to poor adversarial\ntransferability of ViTs. Using the compositional nature of ViT models, we\nenhance transferability of existing attacks by introducing two novel strategies\nspecific to the architecture of ViT models. (i) Self-Ensemble: We propose a\nmethod to find multiple discriminative pathways by dissecting a single ViT\nmodel into an ensemble of networks. This allows explicitly utilizing\nclass-specific information at each ViT block. (ii) Token Refinement: We then\npropose to refine the tokens to further enhance the discriminative capacity at\neach block of ViT. Our token refinement systematically combines the class\ntokens with structural information preserved within the patch tokens.",
+                    "authors": "Muzammal Naseer, Kanchana Ranasinghe, Salman Khan, Fahad Shahbaz Khan, Fatih Porikli",
+                    "published": "2021-06-08",
+                    "updated": "2022-03-03",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Transformers compose a family of neural network architectures based on the self-attention mechanism, originally applied in natural language processing tasks achieving state-of-the-art performance (Vaswani et al., 2017; Devlin et al., 2018; Brown et al., 2020). The transformer design has been subsequently adopted for vision tasks (Dosovitskiy et al., 2020), giving rise to a number of successful vision transformer (ViT) models (Touvron et al., 2020; Yuan et al., 2021; Khan et al., 2021). Due to the lack of explicit inductive biases in their design, ViTs are inherently different from convolutional neural networks (CNNs) that encode biases e.g., spatial connectivity and translation equivariance. ViTs process an image as a sequence of patches which are re\ufb01ned through a series of self-attention mechanisms (transformer blocks), allowing the network to learn relationships between any individual parts of the input image. Such processing allows wide receptive \ufb01elds which can model global context as opposed to the limited receptive \ufb01elds of CNNs. These signi\ufb01cant differences between ViTs and CNNs give rise to a range of intriguing characteristics unique to ViTs (Caron et al., 2021; Tuli et al., 2021; Mao et al., 2021; Paul & Chen, 2021; Naseer et al., 2021b). Adversarial attacks pose a major hindrance to the successful deployment of deep neural networks in real-world applications. Recent success of ViTs means that adversarial properties of ViT models 1 arXiv:2106.04169v3  [cs.CV]  3 Mar 2022 \fFigure 1: Left: Conventional adversarial attacks view ViT as a single classi\ufb01er and maximize the prediction loss (e.g., cross entropy) to fool the model based on the last classi\ufb01cation token only. This leads to sub-optimal results as class tokens in previous ViT blocks only indirectly in\ufb02uence adversarial perturbations. In contrast, our approach (right) effectively utilizes the underlying ViT architecture to create a self-ensemble using class tokens produced by all blocks within ViT to design the adversarial attack. Our self-ensemble enables to use hierarchical discriminative information learned by all class tokens. Consequently, an attack based on our self-ensemble generates transferable adversaries that generalize well across different model types and vision tasks. also become an important research topic. A few recent works explore adversarial robustness of ViTs (Shao et al., 2021; Mahmood et al., 2021; Bhojanapalli et al., 2021) in different attack settings. Surprisingly, these works show that large ViT models exhibit lower transferability in black-box attack setting, despite their higher parameter capacity, stronger performance on clean images, and better generalization (Shao et al., 2021; Mahmood et al., 2021). This \ufb01nding seems to indicate that as ViT performance improves, its adversarial feature space gets weaker. In this work, we investigate whether the weak transferability of adversarial patterns from high-performing ViT models, as reported in recent works (Shao et al., 2021; Mahmood et al., 2021; Bhojanapalli et al., 2021), is a result of weak features or a weak attack. To this end, we introduce a highly transferable attack approach that augments the current adversarial attacks and increase their transferability from ViTs to the unknown models. Our proposed transferable attack leverages two key concepts, multiple discriminative pathways and token re\ufb01nement, which exploit unique characteristics of ViT models. Our approach is motivated by the modular nature of ViTs (Touvron et al., 2020; Yuan et al., 2021; Mao et al., 2021): they process a sequence of input image patches repeatedly using multiple multi-headed self-attention layers (transformer blocks) (Vaswani et al., 2017). We refer to the representation of patches at each transformer block as patch tokens. An additional randomly initialized vector (class token1) is also appended to the set of patch tokens along the network depth to distill discriminative information across patches. The collective set of tokens is passed through the multiple transformer blocks followed by passing of the class token through a linear classi\ufb01er (head) which is used to make the \ufb01nal prediction. The class token interacts with the patch tokens within each block and is trained gradually across the blocks until it is \ufb01nally utilized by the linear classi\ufb01er head to obtain class-speci\ufb01c logit values. The class token can be viewed as extracting information useful for the \ufb01nal prediction from the set of patch tokens at each block. Given the role of the class token in ViT models, we observe that class tokens can be extracted from the output of each block and each such token can be used to obtain a class-speci\ufb01c logit output using the \ufb01nal classi\ufb01er of a pretrained model. This leads us to the proposed self-ensemble of models within a single transformer (Fig. 1). We show that attacking such a self-ensemble (Sec. 3) containing multiple discriminative pathways signi\ufb01cantly improves adversarial transferability from ViT models, and in particular from the large ViTs. Going one step further, we study if the class information extracted from different intermediate ViT blocks (of the self-ensemble) can be enhanced to improve adversarial transferability. To this end, we introduce a novel token re\ufb01nement module directed at enhancing these multiple discriminative pathways. The token re\ufb01nement module strives to re\ufb01ne the information contained in the output of each transformer block (within a single ViT model) and aligns the class tokens produced by the intermediate blocks with the \ufb01nal classi\ufb01er in order to maximize the discriminative power of intermediate blocks. Our token re\ufb01nement exploits the structural information stored in the patch tokens and fuses it with the class token to maximize the discriminative performance of each block. Both the re\ufb01ned tokens and self-ensemble ideas are combined to design an adversarial attack that is shown to signi\ufb01cantly boost the transferability of adversarial examples, thereby bringing out the true 1Average of patch tokens can serve as a class token in our approach for ViT designs that do not use an explicit class token such as Swin transformer (Liu et al., 2021) or MLP-Mixer (Tolstikhin et al., 2021) 2 \fgeneralization of ViTs\u2019 adversarial space. Through our extensive experimentation, we empirically demonstrate favorable transfer rates across different model families (convolutional and transformer) as well as different vision tasks (classi\ufb01cation, detection and segmentation). 2 BACKGROUND AND RELATED WORK Adversarial Attack Modeling: Adversarial attack methods can be broadly categorized into two categories, white-box attacks and black-box attacks. While the white-box attack setting provides the attacker full access to the parameters of the target model, the black-box setting prevents the attacker from accessing the target model and is therefore a harder setting to study adversarial transferability. White-box Attack: Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014) and Projected Gradient Descent (PGD) (Madry et al., 2018) are two initially proposed white-box attack methods. FGSM corrupts the clean image sample by taking a single step within a small distance (perturbation budget \u03f5) along the objective function\u2019s gradient direction. PGD corrupts the clean sample for multiple steps with a smaller step size, projecting the generated adversarial example onto the \u03f5-sphere around the clean sample after each step. Other state-of-the-art white-box attack methods include Jacobian-based saliency map attack (Papernot et al., 2016), Sparse attack (Modas et al., 2019), Onepixel attack (Su et al., 2019), Carlini and Wagner optimization (Carlini & Wagner, 2017), Elastic-net (Chen et al., 2018), Diversi\ufb01ed sampling (Tashiro et al., 2020), and more recently Auto-attack (Croce & Hein, 2020b). We apply white-box attacks on surrogate models to \ufb01nd perturbations that are then transferred to black-box target models. Black-box Attack and Transferability: Black-box attacks generally involve attacking a source model to craft adversarial signals which are then applied on the target models. While gradient estimation methods that estimate the gradients of the target model using black-box optimization methods such as Finite Differences (FD) (Chen et al., 2017; Bhagoji et al., 2018) or Natural Evolution Strategies (NES) (Ilyas et al., 2018; Jiang et al., 2019) exist, these methods are dependent on multiple queries to the target model which is not practical in most real-world scenarios. In the case of adversarial signal generation using source models, it is possible to directly adopt white-box methods. In our work, we adopt FGSM and PGD in such a manner. Methods like (Dong et al., 2018) incorporate a momentum term into the gradient to boost the transferability of existing white-box attacks, building attacks named MIM. In similar spirit, different directions are explored in literature to boost transferability of adversarial examples; a) Enhanced Momentum: Lin et al. (Lin et al., 2019) and Wang et al. (Wang & He, 2021) improve momentum by using Nesterov momentum and variance tuning respectively during attack iterations, b) Augmentations: Xie et al. (Xie et al., 2019) showed that applying differentiable stochastic transformations can bring diversity to the gradients and improve transferability of the existing attacks, c) Exploiting Features: Multiple suggestions are proposed in the literature to leverage the feature space for adversarial attack as well. For example, Zhou et al. (Zhou et al., 2018) incorporate the feature distortion loss during optimization. Similarly, (Inkawhich et al., 2020b;a; Huang et al., 2019) also exploit intermediate layers to enhance transferability. However, combining the intermediate feature response with \ufb01nal classi\ufb01cation loss is non-trivial as it might require optimization to \ufb01nd the best performing layers (Inkawhich et al., 2020b;a), and d) Generative Approach: Orthogonal to iterative attacks, generative methods (Poursaeed et al., 2018; Naseer et al., 2019; 2021a) train an autoencoder against the white-box model. In particular, Naseer et al. show that transferability of an adversarial generator can be increased with relativistic cross-entropy (Naseer et al., 2019) and augmentations (Naseer et al., 2021a). Ours is the \ufb01rst work to address limited transferability of ViT models. The Role of Network Architecture: Recent works exploit architectural characteristics of networks to improve the transferability of attacks. While Wu et al. (2020) exploit skip connections of models like ResNets and DenseNets to improve black-box attacks, Guo et al. (2020) build on similar ideas focused on the linearity of models. Our work similarly focuses on unique architectural characteristics of ViT models to generate more transferable adversarial perturbations with the existing white-box attacks. Robustness of ViTs: Adversarial attacks on ViT models are relatively unexplored. Shao et al. (2021) and Bhojanapalli et al. (2021) investigate adversarial attacks and robustness of ViT models studying various white-box and black-box attack techniques. The transferability of perturbations from ViT models is thoroughly explored in (Mahmood et al., 2021) and they conclude that ViT 3 \fFigure 2: Adversarial examples for ViTs have only moderate transferability. In fact transferabililty (%) of MIM (Dong et al., 2018) perturbations to target models goes down as the source model size increases such as from DeiT-T (Touvron et al., 2020) (5M parameters) to DeiT-B (Touvron et al., 2020) (86M parameters). However, the performance of the attack improves signi\ufb01cantly when applied on our proposed ensemble of classi\ufb01ers found within a ViT (MIME & MIMRE). models do not transfer well to CNNs, whereas we propose a methodology to solve this shortcoming. Moreover, Mahmood et al. (2021) explores the idea of an ensemble of CNN and ViT models to improve the transferability of attacks. Our proposed ensemble approach explores a different direction by converting a single ViT model into a collection of models (self-ensemble) to improve attack transferability. In essence, our proposed method can be integrated with existing attack approaches to take full advantage of the ViTs\u2019 learned features and generate transferable adversaries. 3 ENHANCING ADVERSARIAL TRANSFERABILITY OF VITS Preliminaries: Given a clean input image sample x with a label y, a source ViT model F and a target model M which is under-attack, the goal of an adversarial attack is generating an adversarial signal, x\u2032, using the information encoded within F, which can potentially change the target network\u2019s prediction (M(x\u2032)argmax \u0338= y). A set of boundary conditions are also imposed on the adversarial signal to control the level of distortion in relation to the original sample, i.e., \u2225x \u2212x\u2032\u2225p < \u03f5, for a small perturbation budget \u03f5 and a p-norm, often set to in\ufb01nity norm (\u2113\u221e). Motivation: The recent \ufb01ndings (Shao et al., 2021; Mahmood et al., 2021) demonstrate low black-box transferability of ViTs despite their higher parametric complexity and better feature generalization. Motivated by this behaviour, we set-up a simple experiment of our own to study the adversarial transferability of ViTs (see Fig. 2). We note that transferability of adversarial examples found via momentum iterative fast gradient sign method (Dong et al., 2018) (MIM) at \u2113\u221e\u226416 on DeiT (Touvron et al., 2020) does not increase with model capacity. In fact, adversarial transferability from DeiT base model (DeiT-B) on ResNet152 and large vision transformer (ViT-L (Dosovitskiy et al., 2020)) is lower than DeiT tiny model (DeiT-T). This is besides the fact that DeiT-B has richer representations and around 17\u00d7 more parameters than DeiT-T. We investigate if this behavior is inherent to ViTs or merely due to a sub-optimal attack mechanism. To this end, we exploit unique architectural characteristics of ViTs to \ufb01rst \ufb01nd an ensemble of networks within a single pretrained ViT model (self-ensemble, right Fig. 1). The class token produced by each self-attention block is processed by the \ufb01nal local norm and classi\ufb01cation MLP-head to re\ufb01ne class-speci\ufb01c information (Fig. 2). In other words, our MIME and MIMRE variants attack class information stored in the class tokens produced by all the self-attention blocks within the model and optimize for the adversarial example (Sec. 3.1 and 3.2). Exploring the adversarial space of such multiple discriminative pathways in a self-ensemble generates highly transferable adversarial examples, as we show next. 3.1 SELF-ENSEMBLE: DISCRIMINATIVE PATHWAYS OF VISION TRANSFORMER A ViT model (Dosovitskiy et al., 2020; Touvron et al., 2020), F, with n transformer blocks can be de\ufb01ned as F = (f1 \u25e6f2 \u25e6f3 \u25e6. . . fn) \u25e6g, where fi represents a single ViT block comprising of multi-head self-attention and feed-forward layers and g is the \ufb01nal classi\ufb01cation head. To avoid notation clutter, we assume that g consists of the \ufb01nal local norm and MLP-head (Touvron et al., 2020; Dosovitskiy et al., 2020). Self-attention layer within the vision transformer model takes a sequence of m image patches as input and outputs the processed patches. We will refer to the representations associated with the sequence of image patches as patch tokens, Pt \u2208Rm\u00d7d (where d is the dimensionality of each patch representation). Attention in ViT layers is driven by minimizing the empirical risk during training. In the case of classi\ufb01cation, patch tokens are further appended with the class token (Qt \u2208R1\u00d7d). These patch and class tokens are re\ufb01ned across multiple blocks (fi) and attention in these layers is guided such that the most discriminative information from patch tokens is preserved within the class token. The \ufb01nal class token is then projected to the number of classes by the classi\ufb01er, g. Due to the availability of class token at each transformer block, we can create an 4 \fFigure 3: Distribution of discriminative information across blocks of DeiT models. Note how multiple intermediate blocks contain features with considerable discriminative information as measured by top-1 accuracy on the ImageNet val. set. These are standard models pretrained on ImageNet with no further training. Each block (x-axis) corresponds to a classi\ufb01er Fk as de\ufb01ned in Equation 1. ensemble of classi\ufb01ers by learning a shared classi\ufb01cation head at each block along the ViT hierarchy. This provides us an ensemble of n classi\ufb01ers from a single ViT, termed as the self-ensemble: Fk =  k Y i=1 fi ! \u25e6g, where k = 1, 2, . . . , n. (1) We note that the multiple classi\ufb01ers thus formed hold signi\ufb01cant discriminative information. This is validated by studying the classi\ufb01cation performance of each classi\ufb01er (Eq. 1) in terms of top-1 (%) accuracy on ImageNet validation set, as demonstrated in Fig. 3. Note that multiple intermediate layers perform well on the task, especially towards the end of the ViT processing hierarchy. For an input image x with label y, an adversarial attack can now be optimized for the ViT\u2019s selfensemble by maximizing the loss at each ViT block. However, we observe that initial blocks (1-6) for all considered DeiT models do not contain useful discriminative information as their classi\ufb01cation accuracy is almost zero (Fig. 3). During the training of ViT models (Touvron et al., 2020; Yuan et al., 2021; Mao et al., 2021), parameters are updated based on the last class token only, which means that the intermediate tokens are not directly aligned with the \ufb01nal classi\ufb01cation head, g in our self-ensemble approach (Fig. 3) leading to a moderate classi\ufb01cation performance. To resolve this, we introduce a token re\ufb01nement strategy to align the class tokens with the \ufb01nal classi\ufb01er, g, and boost their discriminative ability, which in turn helps improve attack transferability. 3.2 TOKEN REFINEMENT As mentioned above, the multiple discriminative pathways within a ViT give rise to an ensemble of classi\ufb01ers (Eq. 1). However, the class token produced by each attention layer is being processed by the \ufb01nal classi\ufb01er, g. This puts an upper bound on classi\ufb01cation accuracy for each token which is lower than or equal to the accuracy of the \ufb01nal class token. Our objective is to push the accuracy of the class tokens in intermediate blocks towards the upper bound as de\ufb01ned by the last token. For this purpose, we introduced a token re\ufb01nement module to \ufb01ne-tune the class tokens. Our proposed token re\ufb01nement module is illustrated in Fig. 4. It acts as an intermediate layer inserted between the outputs of each block (after the shared norm layer) and the shared classi\ufb01er head. Revisiting our baseline ensemble method (Fig. 1), we note that the shared classi\ufb01er head contains weights directly trained only on the outputs of the last transformer block. While the class tokens of previous layers may be indirectly optimized to align with the \ufb01nal classi\ufb01er, there exists a potential for misalignment of these features with the classi\ufb01er: the pretrained classi\ufb01er (containing weights compatible with the last layer class token) may not extract all the useful information from the previous layers. Our proposed module aims to solve this misalignment by re\ufb01ning the class tokens in a way such that the shared (pretrained) classi\ufb01er head is able to extract all discriminative information Figure 4: Recent ViTs process 196 image patches, leading to 196 patch tokens. We rearranged these to create a 14x14 feature grid which is processed by a convolutional block to extract structural information, followed by average pooling to create a single patch token. Class token is re\ufb01ned via a MLP layer before feeding to the classi\ufb01er. Both tokens are subsequently merged. 5 \fFigure 5: Self-Ensemble for DeiT (Touvron et al., 2020): We measure the top-1 accuracy on ImageNet using the class-token of each block and compare to our re\ufb01ned tokens. These results show that \ufb01ne-tuning helps align tokens from intermediate blocks with the \ufb01nal classi\ufb01er enhancing their classi\ufb01cation performance. Thus token re\ufb01nement leads to strengthened discriminative pathways allowing more transferable adversaries. contained within the class tokens of each block. Moreover, intermediate patch tokens may contain additional information that is not at all utilized by the class tokens of those blocks, which would also be addressed by our proposed block. Therefore, we extract both patch tokens and the class token from each block and process them for re\ufb01nement, as explained next. \u2212Patch Token Re\ufb01nement: One of the inputs to the token re\ufb01nement module is the set of patch tokens output from each block. We \ufb01rst rearrange these patch tokens to regain their spatial relationships. The aim of this component within the re\ufb01nement module is to extract information relevant to spatial structure contained within the intermediate patch tokens. We believe that signi\ufb01cant discriminative information is contained within these patches. The obtained rearranged patch tokens are passed through a convolution block (standard ResNet block containing a skip connection) to obtain a spatially aware feature map, which is then average pooled to obtain a single feature vector (of same dimension as the class token). This feature vector is expected to extract all spatial information from patch tokens. \u2212Class Token Re\ufb01nement: By re\ufb01ning the class tokens of each block, we aim to remove any misalignment between the existing class tokens and the shared (pretrained) classi\ufb01er head. Also, given how the class token does not contain a spatial structure, we simply use a linear layer to re\ufb01ne it. We hypothesize that re\ufb01ned class token at each block would be much more aligned with the shared classi\ufb01er head allowing it to extract all discriminative information contained within those tokens. \u2212Merging Patch and Class Token: We obtain the re\ufb01ned class token and the patch feature vector (re\ufb01ned output of patch tokens) and sum them together to obtain a merged token. While we tested multiple approaches for merging, simply summing them proved suf\ufb01cient. \u2212Training: Given a ViT model containing k transformer blocks, we plugin k instances of our token re\ufb01nement module to the output of each block as illustrated in Figure 4. We obtain the pretrained model, freeze all existing weights, and train only the k token re\ufb01nement modules for only a single epoch on ImageNet training set. We used SGD optimizer with learning rate set to 0.001. Training \ufb01nishes in less than one day on a single GPU-V100 even for a large ViT model such as DeiT-B. As expected, the trained token re\ufb01nement module leads to increased discriminability of the class tokens, which we illustrate in Figure 5. Note how this leads to signi\ufb01cant boosting of discriminative power especially in the earlier blocks, solving the misalignment problem. We build on this enhanced discriminability of the ensemble members towards better transferability, as explained next. 3.3 ADVERSARIAL TRANSFER Our modi\ufb01cations to ViT models with respect to multiple discriminative pathways and token re\ufb01nement are exploited in relation to adversarial transfer. We consider black-box attack perturbations that are generated using a source (surrogate) ViT model. The source model is only pretrained on ImageNet, modi\ufb01ed according to our proposed approach and is subsequently \ufb01ne-tuned to update only the token re\ufb01nement module for a single epoch. We experiment with multiple white-box attacks, generating the adversarial examples using a joint loss over the outputs of each block. The transferability of adversarial examples is tested on a range of CNN and ViT models. Given input sample x and its label y, the adversarial object for our self-ensemble (Eq. 1) for the untargeted attack is de\ufb01ned as, 6 \fFast Gradient Sign Method (FGSM) (Goodfellow et al., 2014) Convolutional Transformers Source (\u2193) Attack BiT50 Res152 WRN DN201 ViT-L T2T-24 TnT ViT-S T2T-7 VGG19bn FGSM 23.34 28.56 33.92 33.22 13.18 10.78 12.96 25.08 29.90 MNAS FGSM 23.16 39.82 40.10 44.34 16.60 22.56 25.82 34.10 48.96 Deit-T FGSM 29.74 37.10 38.86 42.40 44.38 35.42 50.58 73.32 57.62 FGSME 30.34 39.60 41.42 45.58 48.34 35.08 51.00 80.74 62.82 FGSMRE 30.18(+0.44) 39.82(+2.7) 41.26(+2.4) 46.06(+3.7) 46.76(+2.4) 32.68(-2.7) 48.00(-2.6) 80.10(+6.8) 63.90(+6.3) Deit-S FGSM 25.44 31.04 33.58 36.28 36.40 33.41 41.00 58.78 43.48 FGSME 30.82 38.38 41.06 46.00 47.20 39.00 51.44 78.90 56.70 FGSMRE 34.84(+9.4) 43.86(+12.8) 46.26(+12.7) 51.88(+15.6) 47.92(+11.5) 39.86(+6.5) 55.7(+14.7) 82.00(+23.2) 66.20(+22.7) Deit-B FGSM 22.54 31.58 33.86 34.96 30.50 27.84 33.08 50.24 40.50 FGSME 31.12 41.46 43.02 47.12 42.28 35.40 46.22 73.04 57.32 FGSMRE 35.12(+12.6) 45.74(+14.2) 48.46(+14.6) 52.64(+17.7) 41.68(+11.2) 36.60(+8.8) 49.60(+16.5) 74.40(+24.2) 65.92(+25.4) Projected Gradient Decent (PGD) (Madry et al., 2018) VGG19bn PGD 19.80 28.56 33.92 33.22 5.94 10.78 12.96 13.08 29.90 MNAS PGD 19.44 36.28 36.22 40.20 8.04 18.04 21.16 19.60 41.70 Deit-T PGD 14.22 23.98 24.16 26.76 35.70 21.54 44.24 86.86 53.74 PGDE 14.42 24.58 25.46 28.38 39.84 21.86 45.08 88.44 53.80 PGDRE 22.46(+8.24) 34.64(+10.7) 37.62(+13.5) 40.56(+13.8) 58.60(+22.9) 26.58(+5.0) 55.52(+11.3) 96.34(+9.5) 66.68(+12.9) Deit-S PGD 18.78 24.96 26.38 30.38 37.84 33.46 60.62 84.38 47.14 PGDE 18.98 27.72 29.54 32.90 44.30 35.40 64.76 89.82 52.76 PGDRE 28.96(+10.2) 38.92(+14.0) 42.84(+16.5) 46.82(+16.4) 60.86(+23.0) 40.30(+6.8) 76.10(+15.5) 97.32(+12.9) 71.54(+24.4) Deit-B PGD 18.68 25.56 27.90 30.24 34.08 31.98 52.76 69.82 39.80 PGDE 23.64 32.84 35.40 38.66 43.56 37.82 64.20 82.32 51.68 PGDRE 37.92(+19.2) 49.10(+23.5) 53.38(+25.5) 56.96(+26.7) 56.90(+22.8) 45.70(+13.7) 79.56(+26.8) 94.10(+24.3) 74.78(+35.0) Table 1: Fool rate (%) on 5k ImageNet val. adversarial samples at \u03f5 \u226416. Perturbations generated from our proposed self-ensemble with re\ufb01ned tokens from a vision transformer have signi\ufb01cantly higher success rate. max x\u2032 k X i=1 [ [Fk(x\u2032)argmax \u0338= y] ], s.t. \u2225x \u2212x\u2032\u2225p \u2264\u03f5, k \u2208{1, 2, . . . , n} (2) where [ [\u00b7] ] is an indicator function. In the case of target attack, the attacker optimizes the above objective towards a speci\ufb01c target class instead of an arbitrary misclassi\ufb01cation. 4 EXPERIMENTS We conduct thorough experimentation on a range of standard attack methods to establish the performance boosts obtained through our proposed transferability approach. We create \u2113\u221eadversarial attacks with \u03f5 \u226416 and observe their transferability by using the following protocols: Source (white-box) models: We mainly study three vision transformers from DeiT (Touvron et al., 2020) family due to their data ef\ufb01ciency. Speci\ufb01cally, the source models are Deit-T, DeiT-S, and DeiT-B (with 5, 22, and 86 million parameters, respectively). They are trained without CNN distillation. Adversarial examples are created on these models using an existing white-box attack (e.g., FGSM (Goodfellow et al., 2014), PGD (Madry et al., 2018) and MIM (Dong et al., 2018)) and then transferred to the black-box target models. Target (black-box) models: We test the black-box transferability across several vision tasks including classi\ufb01cation, detection and segmentation. We consider convolutional networks including BiT-ResNet50 (BiT50) (Beyer et al., 2021), ResNet152 (Res152) (He et al., 2016), Wide-ResNet-50-2 (WRN) (Zagoruyko & Komodakis, 2016), DenseNet201 (DN201) (Huang et al., 2017) and other ViT models including Token-to-Token transformer (T2T) (Yuan et al., 2021), Transformer in Transformer (TnT) (Mao et al., 2021), DINO (Caron et al., 2021), and Detection Transformer (DETR) (Carion et al., 2020) as the black-box target models. Datasets: We use ImageNet training set to \ufb01ne tune our proposed token re\ufb01nement modules. For evaluating robustness, we selected 5k samples from ImageNet validation set such that 5 random samples from each class that are correctly classi\ufb01ed by ResNet50 and ViT small (ViT-S) (Dosovitskiy et al., 2020) are present. In addition, we conduct experiments on COCO (Lin et al., 2014) (5k images) and PASCAL-VOC12 (Everingham et al., 2012) (around 1.2k images) validation set. 7 \fMomemtum Iterative Fast Gradient Sign Method (MIM) (Dong et al., 2018) Convolutional Transformers Source (\u2193) Attack BiT50 Res152 WRN DN201 ViT-L T2T-24 TnT ViT-S T2T-7 VGG19bn MIM 36.18 46.98 54.04 57.32 12.80 21.84 25.72 28.44 47.74 MNAS MIM 34.78 54.34 55.40 64.06 18.88 34.54 38.70 40.58 60.02 Deit-T MIM 36.22 45.56 47.86 53.26 63.84 48.44 72.52 96.44 77.66 MIME 34.92 45.58 47.98 54.50 67.16 46.38 71.02 97.74 78.02 MIMRE 42.04(+5.8) 54.02(+8.5) 58.48(+10.6) 63.00(+9.7) 79.12(+15.3) 49.86(+1.4) 77.80(+5.3) 99.14(+2.7) 85.50(+7.8) Deit-S MIM 38.32 45.06 47.90 52.66 63.38 58.86 79.56 94.22 68.00 MIME 40.66 49.52 52.98 58.40 71.78 61.06 84.42 98.12 74.58 MIMRE 53.70(+15.4) 61.72(+16.7) 65.10(+17.2) 71.74(+19.1) 84.30(+20.9) 66.32(+7.5) 92.02(+12.5) 99.42(+5.2) 89.08(+21.1) Deit-B MIM 36.98 44.66 47.98 52.14 57.48 54.40 70.84 84.74 59.34 MIME 45.30 54.30 58.34 63.32 70.42 61.84 82.80 94.46 73.66 MIMRE 61.58(+24.6) 70.18(+25.5) 74.08(+26.1) 79.12(+27.0) 81.28(+23.8) 69.6(+15.2) 92.20(+21.4) 94.10(+9.4) 89.72(+30.4) MIM with Input Diversity (DIM) (Xie et al., 2019) VGG19bn DIM 46.90 62.08 68.30 73.48 16.86 30.16 34.70 35.42 58.62 MNAS DIM 43.74 62.08 68.30 73.48 25.06 42.92 47.24 52.74 71.98 Deit-T DIM 57.56 68.30 70.06 77.18 62.00 70.16 82.68 89.16 86.18 DIME 60.14 70.06 69.84 78.00 66.38 72.30 85.98 93.72 90.78 DIMRE 62.10(+4.5) 70.78(+2.5) 70.78(+0.7) 78.40(+1.2) 67.58(+5.6) 68.56(-1.6) 84.18(+1.5) 93.36(+4.2) 91.52(+5.3) Deit-S DIM 59.00 62.12 63.42 67.30 62.62 73.84 79.50 82.32 74.20 DIME 68.82 74.44 75.34 80.14 76.22 84.10 91.92 94.92 88.42 DIMRE 76.14(+17.1) 81.30(+19.18) 82.64(+19.22) 86.98(+19.68) 78.88(+16.3) 85.26(+11.4) 93.22(+13.7) 96.56(+14.2) 93.60(+19.4) Deit-B DIM 56.24 59.14 60.64 64.44 61.38 69.54 73.96 76.32 64.44 DIME 73.04 78.36 80.28 83.70 79.06 85.10 91.84 94.38 86.96 DIMRE 80.10(+23.9) 84.92(+25.8) 86.36(+25.7) 89.24(+24.8) 78.90(+17.5) 84.00(+14.5) 92.28(+18.3) 95.26(+18.9) 93.42(+28.9) Table 2: Fool rate (%) on 5k ImageNet val. adversarial samples at \u03f5 \u226416. Perturbations generated from our proposed self-ensemble with re\ufb01ned tokens from a vision transformer have signi\ufb01cantly higher success rate. Evaluation Metrics: We report fooling rate (percentage of samples for which the predicted label is \ufb02ipped after adding adversarial perturbations) to evaluate classi\ufb01cation. In the case of object detection, we report the decrease in mean average precision (mAP) and for automatic segmentation, we use the popular Jaccard Index. Given the pixel masks for the prediction and the ground-truth, it calculates the ratio between the pixels belonging to intersection and the union of both masks. Baseline Attacks: We show consistent improvements for single step fast gradient sign method (FGSM) (Goodfellow et al., 2014) as well as iterative attacks including PGD (Madry et al., 2018), MIM (Dong et al., 2018) and input diversity (transformation to the inputs) (DIM) (Xie et al., 2019) attacks. Iterative attacks ran for 10 iterations and we set transformation probability for DIM to default 0.7 (Xie et al., 2019). Our approach is not limited to speci\ufb01c attack settings, but existing attacks can simply be adopted to our self-ensemble ViTs with re\ufb01ned tokens. Refer to appendices A-J for extensive analysis with more ViT designs, attacks, datasets (CIFAR10 & Flowers), computational cost comparison, and latent space visualization of our re\ufb01ned token. 4.1 CLASSIFICATION In this section, we discuss the experimental results on adversarial transferability across black-box classi\ufb01cation models. For a given attack method \u2018Attack\u2019, we refer \u2018AttackE\u2019 and \u2018AttackRE\u2019 as self-ensemble and self-ensemble with re\ufb01ned tokens, respectively, which are the two variants of our approach. We observe that adversarial transferability from ViT models to CNNs is only moderate for conventional attacks (Tables 1 & 2). For example, perturbations found via iterative attacks from DeiT-B to Res152 has even lower transfer than VGG19bn. However, the same attacks when applied using our proposed ensemble strategy (Eq. 1) with re\ufb01ned tokens consistently showed improved transferability to other convolutional as well as transformer based models. We observe that models without inductive biases that share architecture similarities show higher transfer rate of adversarial perturbations among them (e.g., from DeiT to ViT (Dosovitskiy et al., 2020)). We further observe that models trained with the same mechanism but lower parameters are more vulnerable to black-box attacks. For example, ViT-S and T2T-T are more vulnerable than their larger counterparts, ViT-L and T2T-24. Also models trained with better strategies that lead to higher generalizability are less vulnerable to black-box attacks e.g., BiT50 is more robust than ResNet152 (Tables 1 and 2). 8 \fFigure 6: Ablative Study: Fooling rate of intermediate layers under MIM (white-box) attack using our self-ensemble approach. We obtain favorable improvements for our method. Source (\u2192) DeiT-T DeiT-S DeiT-B No Attack MIM MIMRE MIM MIMRE MIM MIMRE 38.5 24.0 19.7 23.7 19.0 22.9 16.9 DIM DIMRE DIM DIMRE DIM DIMRE 20.5 13.7 20.3 12.0 19.9 11.1 Table 3: Cross-Task Transferability (classi\ufb01cation\u2192detection) Object Detector DETR (Carion et al., 2020) is fooled. mAP at [0.5:0.95] IOU on COCO val. set. Our self-ensemble approach with re\ufb01ned token (RE) signi\ufb01cantly improves cross-task transferability. (lower the better) Source (\u2192) DeiT-T DeiT-S DeiT-B No Attack MIM MIMRE MIM MIMRE MIM MIMRE 42.7 32.5 31.6 32.5 31.0 32.6 30.6 DIM DIMRE DIM DIMRE DIM DIMRE 31.9 31.4 31.7 31.3 32.0 31.0 Table 4: Cross-Task Transferability (classi\ufb01cation\u2192segmentation) DINO (Caron et al., 2021) is fooled. Jaccard index metric is used to evaluate segmentation performance. Best adversarial transfer results are achieved using our method. (lower the better) Clean Image Adv Image Clean Image Adv Image Clean Image Adv Image Figure 7: Visualization of DETR failure cases for our proposed DIMRE attack generated from DeiT-S source model. (best viewed in zoom) The strength of our method is also evident by blockwise fooling rate in white-box setting (Fig. 6). It is noteworthy how MIM fails to fool the initial blocks of ViT, while our approach allows the attack to be as effective in the intermediate blocks as for the last class token. This ultimately allows us to fully exploit ViT\u2019s adversarial space leading to high transfer rates for adversarial perturbations. 4.2 CROSS-TASK TRANSFERABILITY Self-attention is the core component of transformer architecture regardless of the task; classi\ufb01cation (Dosovitskiy et al., 2020; Touvron et al., 2020; Yuan et al., 2021; Mao et al., 2021), object detection (Carion et al., 2020), or unsupervised segmentation (Caron et al., 2021). We explore the effectiveness of our proposed method on two additional tasks: object detection (DETR) (Carion et al., 2020) and segmentation (DINO) (Caron et al., 2021). We select these methods considering the use of transformer modules employing the self-attention mechanism within their architectures. While the task of object detection contains multiple labels per image and involves bounding box regression, the unsupervised model DINO is trained in a self-supervised manner with no traditional image-level labels. Moreover, DINO uses attention maps of a ViT model to generate pixel-level segmentations, which means adversaries must disrupt the entire attention mechanism to degrade its performance. We generate adversarial signals on source models with a classi\ufb01cation objective using their initial predictions as the label. In evaluating attacks on detection and segmentation tasks at their optimal setting, the source ViT need to process images of different sizes (e.g., over 896\u00d7896 pix for DETR). To cater for this, we process images in parts (refer appendix G) which allows generation of stronger adversaries. The performance degradation of DETR and DINO on generated adversaries are summarised in Tables 3 & 4. For DETR, we obtain clear improvements. In the more robust DINO model, our transferability increases well with the source model capacity as compared to the baseline. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2105.10497v3",
+                    "title": "Intriguing Properties of Vision Transformers",
+                    "abstract": "Vision transformers (ViT) have demonstrated impressive performance across\nvarious machine vision problems. These models are based on multi-head\nself-attention mechanisms that can flexibly attend to a sequence of image\npatches to encode contextual cues. An important question is how such\nflexibility in attending image-wide context conditioned on a given patch can\nfacilitate handling nuisances in natural images e.g., severe occlusions, domain\nshifts, spatial permutations, adversarial and natural perturbations. We\nsystematically study this question via an extensive set of experiments\nencompassing three ViT families and comparisons with a high-performing\nconvolutional neural network (CNN). We show and analyze the following\nintriguing properties of ViT: (a) Transformers are highly robust to severe\nocclusions, perturbations and domain shifts, e.g., retain as high as 60% top-1\naccuracy on ImageNet even after randomly occluding 80% of the image content.\n(b) The robust performance to occlusions is not due to a bias towards local\ntextures, and ViTs are significantly less biased towards textures compared to\nCNNs. When properly trained to encode shape-based features, ViTs demonstrate\nshape recognition capability comparable to that of human visual system,\npreviously unmatched in the literature. (c) Using ViTs to encode shape\nrepresentation leads to an interesting consequence of accurate semantic\nsegmentation without pixel-level supervision. (d) Off-the-shelf features from a\nsingle ViT model can be combined to create a feature ensemble, leading to high\naccuracy rates across a range of classification datasets in both traditional\nand few-shot learning paradigms. We show effective features of ViTs are due to\nflexible and dynamic receptive fields possible via the self-attention\nmechanism.",
+                    "authors": "Muzammal Naseer, Kanchana Ranasinghe, Salman Khan, Munawar Hayat, Fahad Shahbaz Khan, Ming-Hsuan Yang",
+                    "published": "2021-05-21",
+                    "updated": "2021-11-25",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction As visual transformers (ViT) attract more interest [1], it becomes highly pertinent to study characteristics of their learned representations. Speci\ufb01cally, from the perspective of safety-critical applications such as autonomous cars, robots and healthcare; the learned representations need to be robust and generalizable. In this paper, we compare the performance of transformers with convolutional neural networks (CNNs) for handling nuisances (e.g., occlusions, distributional shifts, adversarial and natural perturbations) and generalization across different data distributions. Our in-depth analysis is based on three transformer families, ViT [2], DeiT [3] and T2T [4] across \ufb01fteen vision datasets. For brevity, we refer to all the transformer families as ViT, unless otherwise mentioned. We are intrigued by the fundamental differences in the operation of convolution and self-attention, that have not been extensively explored in the context of robustness and generalization. While convolutions excel at learning local interactions between elements in the input domain (e.g., edges and contour information), self-attention has been shown to effectively learn global interactions (e.g., 35th Conference on Neural Information Processing Systems (NeurIPS 2021). arXiv:2105.10497v3  [cs.CV]  25 Nov 2021 \fFigure 1: We show intriguing properties of ViT including impressive robustness to (a) severe occlusions, (b) distributional shifts (e.g., stylization to remove texture cues), (c) adversarial perturbations, and (d) patch permutations. Furthermore, our ViT models trained to focus on shape cues can segment foregrounds without any pixel-level supervision (e). Finally, off-the-shelf features from ViT models generalize better than CNNs (f). relations between distant object parts) [5, 6]. Given a query embedding, self-attention \ufb01nds its interactions with the other embeddings in the sequence, thereby conditioning on the local content while modeling global relationships [7]. In contrast, convolutions are content-independent as the same \ufb01lter weights are applied to all inputs regardless of their distinct nature. Given the content-dependent long-range interaction modeling capabilities, our analysis shows that ViTs can \ufb02exibly adjust their receptive \ufb01eld to cope with nuisances in data and enhance expressivity of the representations. Our systematic experiments and novel design choices lead to the following interesting \ufb01ndings: \u2022 ViTs demonstrate strong robustness against severe occlusions for foreground objects, non-salient background regions and random patch locations, when compared with state-of-the-art CNNs. For instance, with a signi\ufb01cant random occlusion of up to 80%, DeiT [3] can maintain top-1 accuracy up to \u223c60% where CNN has zero accuracy, on ImageNet [8] val. set. \u2022 When presented with texture and shape of the same object, CNN models often make decisions based on texture [9]. In contrast, ViTs perform better than CNNs and comparable to humans on shape recognition. This highlights robustness of ViTs to deal with signi\ufb01cant distribution shifts e.g., recognizing object shapes in less textured data such as paintings. \u2022 Compared to CNNs, ViTs show better robustness against other nuisance factors such as spatial patch-level permutations, adversarial perturbations and common natural corruptions (e.g., noise, blur, contrast and pixelation artefacts). However, similar to CNNs [10], a shape-focused training process renders them vulnerable against adversarial attacks and common corruptions. \u2022 Apart from their promising robustness properties, off-the-shelf ViT features from ImageNet pretrained models generalize exceptionally well to new domains e.g., few-shot learning, \ufb01negrained recognition, scene categorization and long-tail classi\ufb01cation settings. In addition to our extensive experimental analysis and new \ufb01ndings, we introduce several novel design choices to highlight the strong potential of ViTs. To this end, we propose an architectural modi\ufb01cation to DeiT to encode shape-information via a dedicated token that demonstrates how seemingly contradictory cues can be modeled with distinct tokens within the same architecture, leading to favorable implications such as automated segmentation without pixel-level supervision. Moreover, our off-the-shelf feature transfer approach utilizes an ensemble of representations derived from a single architecture to obtain state-of-the-art generalization with a pre-trained ViT (Fig. 1). 2 Related Work CNNs have shown state-of-the-art performance in independent and identically distributed (i.i.d) settings but remain highly sensitive to distributional shifts; adversarial noise [11, 12], common image corruptions [13], and domain shifts (e.g., RGB to sketches) [14]. It is natural to ask if ViT, that processes inputs based on self-attention, offers any advantages in comparison to CNN. Shao et al. [15] analyze ViTs against adversarial noise and show ViTs are more robust to high frequency changes. Similarly, Bhojanapalli et al. [16] study ViT against spatial perturbations [15] and its robustness to removal of any single layer. Since ViT processes image patches, we focus on their robustness against patch masking, localized adversarial patches [17] and common natural corruptions. A concurrent work from Paul and Chen [18] also develops similar insights on robustness of ViTs but with a somewhat different set of experiments. Geirhos et al. [9] provide evidence that CNNs mainly exploit texture to make a decision and give less importance to global shape. This is further con\ufb01rmed by CNN ability to only use local features 2 \f[19]. Recently, [20] quanti\ufb01es mutual information [21] between shape and texture features. Our analysis indicates that large ViT models have less texture bias and give relatively higher emphasis to shape information. ViT\u2019s shape-bias approaches human-level performance when directly trained on stylized ImageNet [9]. Our \ufb01ndings are consistent with a concurrent recent work that demonstrates the importance of this trend on human behavioural understanding and bridging the gap between human and machine vision [22]. A recent work [23] shows that self-supervised ViT can automatically segment foreground objects. In comparison, we show how shape-focused learning can impart similar capability in the image-level supervised ViT models, without any pixel-level supervision. Zeiler et al. [24] introduce a method to visualize CNN features at different layers and study the performance of off-the-shelf features. In a similar spirit, we study the generalization of off-the-shelf features of ViT in comparison to CNN. Receptive \ufb01eld is an indication of network\u2019s ability to model long range dependencies. The receptive \ufb01eld of Transformer based models covers the entire input space, a property that resembles handcrafted features [25], but ViTs have higher representative capacity. This allows ViT to model global context and preserve the structural information compared to CNN [26]. This work is an effort to demonstrate the effectiveness of \ufb02exible receptive \ufb01eld and content-based context modeling in ViTs towards robustness and generalization of the learned features. 3 Intriguing Properties of Vision Transformers 3.1 Are Vision Transformers Robust to Occlusions? The receptive \ufb01eld of a ViT spans over the entire image and it models the interaction between the sequence of image patches using self-attention [26, 27]. We study whether ViTs perform robustly in occluded scenarios, where some or most of the image content is missing. Occlusion Modeling: Consider a network f, that processes an input image x to predict a label y, where x is represented as a patch sequence with N elements, i.e., x = {xi}N i=1 [2]. While there can be multiple ways to de\ufb01ne occlusion, we adopt a simple masking strategy, where we select a subset of the total image patches, M < N, and set pixel values of these patches to zero to create an occluded image, x\u2032. We refer to this approach as PatchDrop. The objective is then to observe robustness such that f(x\u2032)argmax = y. We experiment with three variants of our occlusion approach, (a) Random PatchDrop, (b) Salient (foreground) PatchDrop, and (c) Non-salient (background) PatchDrop. Random PatchDrop: A subset of M patches is randomly selected and dropped (Fig. 2). Several recent Vision Transformers [2, 3, 4] divide an image into 196 patches belonging to a 14x14 spatial grid; i.e. an image of size 224\u00d7224\u00d73 is split into 196 patches, each of size 16\u00d716\u00d73. As an example, dropping 100 such patches from the input is equivalent to losing 51% of the image content. Salient (foreground) PatchDrop: Not all pixels have the same importance for vision tasks. Thus, it is important to study the robustness of ViTs against occlusions of highly salient regions. We leverage a self-supervised ViT model DINO [23] that is shown to effectively segment salient objects. In particular, the spatial positions of information \ufb02owing into the \ufb01nal feature vector (class token) within the last attention block are exploited to locate the salient pixels. This allows to control the amount of salient information captured within the selected pixels by thresholding the quantity of attention \ufb02ow. We select the subset of patches containing the top Q% of foreground information (deterministic for \ufb01xed Q) and drop them. Note that this Q% does not always correspond to the pixel percentage, e.g., 50% of the foreground information of an image may be contained within only 10% of its pixels. Non-salient (background) PatchDrop: The least salient regions of the image are selected following the same approach as above, using [23]. The patches containing the lowest Q% of foreground information are selected and dropped here. Note this does not always correspond to the pixel percentage, e.g., 80% of the pixels may only contain 20% of the non-salient information for an image. Figure 2: An example image with its occluded versions (Random, Salient and NonSalient). The occluded images are correctly classi\ufb01ed by Deit-S [3] but misclassi\ufb01ed by ResNet50 [28]. Pixel values in occluded (black) regions are set to zero. Original Image Random PatchDrop Salient PatchDrop Non-Salient PatchDrop 3 \fFigure 3: Robustness against object occlusion in images is studied under three PatchDrop settings (see Sec 3.1). (left) We study the robustness of CNN models to occlusions, and identify ResNet50 as a strong baseline. (mid-left) We compare the DeiT model family against ResNet50 exhibiting their superior robustness to object occlusion. (mid-right) Comparison against ViT model family. (right) Comparison against T2T model family. Robust Performance of Transformers Against Occlusions: We consider visual recognition task with models pretrained on ImageNet [2]. The effect of occlusion is studied on the validation set (50k images). We de\ufb01ne information loss (IL) as the ratio of dropped and total patches (M / N). IL is varied to obtain a range of occlusion levels for each PatchDrop methodology. The results (Top-1 %) reported in Fig. 3 show signi\ufb01cantly robust performance of ViT models against CNNs. In the case of random PatchDrop, we report the mean of accuracy across 5 runs. For Salient and Non-Salient Patchdrop, we report the accuracy values over a single run, since the occlusion mask is deterministic. CNNs perform poorly when 50% of image information is randomly dropped. For example, ResNet50 (23 Million parameters) achieves 0.1% accuracy in comparison to DeiT-S (22 Million parameters) which obtains 70% accuracy when 50% of the image content is removed. An extreme example can be observed when 90% of the image information is randomly masked but Deit-B still exhibits 37% accuracy. This \ufb01nding is consistent among different ViT architectures [2, 3, 4]. Similarly, ViTs show signi\ufb01cant robustness to the foreground (salient) and background (non-salient) content removal. See Appendix A, B, C, D, and E for further results on robustness analysis. ViT Representations are Robust against Information Loss: In order to better understand model behavior against such occlusions, we visualize the attention (Fig. 4) from each head of different layers. While initial layers attend to all areas, deeper layers tend to focus more on the leftover information in non-occluded regions of an image. We then study if such changes from initial to deeper layers lead to token invariance against occlusion which is important for classi\ufb01cation. We measure the correlation coef\ufb01cient between features/tokens of original and occluded images by using corr(u, v) = P i \u02c6 ui \u02c6 vi n , where \u02c6 ui = ui\u2212E[ui] \u03c3(ui) , E[\u00b7] and \u03c3(\u00b7) are mean and standard deviation operations [29]. In our case, random variables u and v refer to the feature maps for an original and occluded image de\ufb01ned over the entire ImageNet validation set. In the case of ResNet50, we consider features before the logit layer and for ViT models, class tokens are extracted from the last transformer block. Class tokens from transformers are signi\ufb01cantly more robust and do not suffer much information loss as compared to ResNet50 features (Table 1). Furthermore, we visualize the correlation coef\ufb01cient across the 12 selected superclasses within ImageNet hierarchy and note that the trend holds across different class types, even for relatively small object types such as insects, food items and birds (Fig. 5). See Appendix F for attention visualizations and G for the qualitative results. Given the intriguing robustness of transformer models due to dynamic receptive \ufb01elds and discriminability preserving behaviour of the learned tokens, an ensuing question is whether the learned representations in ViTs are biased towards texture or not. One can expect a biased model focusing only on texture to still perform well when the spatial structure for an object is partially lost. 4 \fFigure 4: Attention maps (averaged over the entire ImageNet val. set) relevant to each head in multiple layers of an ImageNet pre-trained DeiT-B model. All images are occluded (Random PatchDrop) with the same mask (bottom right). Observe how later layers clearly attend to non-occluded regions of images to make a decision, an evidence of the model\u2019s highly dynamic receptive \ufb01eld. Model Correlation Coef\ufb01cient: Random PatchDrop 25% Dropped 50% Dropped 75% Dropped ResNet50 0.32\u00b10.16 0.13\u00b10.11 0.07\u00b10.09 TnT-S 0.83\u00b10.08 0.67\u00b10.12 0.46\u00b10.17 ViT-L 0.92\u00b10.06 0.81\u00b10.13 0.50\u00b10.21 Deit-B 0.90\u00b10.06 0.77\u00b10.10 0.56\u00b10.15 T2T-24 0.80\u00b10.10 0.60\u00b10.15 0.31\u00b10.17 Table 1: Correlation coef\ufb01cient b/w features/\ufb01nal class tokens of original and occluded images for Random PatchDrop. Averaged across the ImageNet val. set. Figure 5: Correlation b/w features/\ufb01nal tokens of original and occluded images for 50% Random Drop. Results are averaged across classes for each superclass. 3.2 Shape vs. Texture: Can Transformer Model Both Characteristics? Geirhos et al. [9] study shape vs. texture hypothesis and propose a training framework to enhance shape-bias in CNNs. We \ufb01rst carry out similar analysis and show that ViT models preform with a shape-bias much stronger than that of a CNN, and comparably to the ability of human visual system in recognizing shapes. However, this approach results in a signi\ufb01cant drop in accuracy on the natural images. To address this issue, we introduce a shape token into the transformer architecture that learns to focus on shapes, thereby modeling both shape and texture related features within the same architecture using a distinct set of tokens. As such, we distill the shape information from a pretrained CNN model with high shape-bias [9]. Our distillation approach makes a balanced trade-off between high classi\ufb01cation accuracy and strong shape-bias compared to the original ViT model. We outline both approaches below. Note that the measure introduced in [9] is used to quantify shape-bias within ViT models and compare against their CNN counterparts. Training without Local Texture: In this approach, we \ufb01rst remove local texture cues from the training data by creating a stylized version of ImageNet [9] named SIN. We then train tiny and small DeiT models [3] on this dataset. Typically, ViTs use heavy data augmentations during training [3]. However, learning with SIN is a dif\ufb01cult task due to less texture details and applying further augmentations on stylized samples distorts shape information and makes the training unstable. Thus, we train models on SIN without applying any augmentation, label smoothing or mixup. We note that ViTs trained on ImageNet exhibit higher shape-bias in comparison to similar capacity CNN models e.g., DeiT-S (22-Million params) performs better than ResNet50 (23-Million params) (Fig. 6, right plot). In contrast, the SIN trained ViTs consistently perform better than CNNs. Interestingly, DeiT-S [3] reaches human-level performance when trained on a SIN (Fig. 6, left plot). 5 \fFigure 6: Shape-bias Analysis: Shape-bias is de\ufb01ned as the fraction of correct decisions based on object shape. (Left) Plot shows shape-texture tradeoff for CNN, ViT and Humans across different object classes. (Right) classmean shape-bias comparison. Overall, ViTs perform better than CNN. The shape bias increases signi\ufb01cantly when trained on stylized ImageNet (SIN). Model Distilled Token Type ImageNet top-1 (%) Shape Bias DeiT-T-SIN \u0017 cls 40.5 0.87 DeiT-T-SIN \u0013 cls 71.8 0.35 DeiT-T-SIN \u0013 shape 63.4 0.44 DeiT-S-SIN \u0017 cls 52.5 0.93 DeiT-S-SIN \u0013 cls 75.3 0.39 DeiT-S-SIN \u0013 shape 67.7 0.47 Table 3: Performance comparison of models trained on SIN. ViT produces dynamic features that can be controlled by auxiliary tokens. \u2018cls\u2019 represents the class token. During distillation cls and shape tokens converged to vastly different solution using the same features as compared to [3]. Shape Distillation: Knowledge distillation allows to compress large teacher models into smaller student models [30] as the teacher provides guidance to the student through soft labels. We introduce a new shape token and adapt attentive distillation [3] to distill shape knowledge from a CNN trained on the SIN dataset (ResNet50-SIN [9]). We observe that ViT features are dynamic in nature and can be controlled by auxiliary tokens to focus on the desired characteristics. This means that a single ViT model can exhibit both high shape and texture bias at the same time with separate tokens (Table 3). We achieve more balanced performance for classi\ufb01cation as well as shape-bias measure when the shape token is introduced (Fig. 7). To demonstrate that these distinct tokens (for classi\ufb01cation and shape) indeed model unique features, we compute cosine similarity (averaged over ImageNet val. set) between class and shape tokens of our distilled models, DeiT-T-SIN and DeiT-S-SIN, which turns out to be 0.35 and 0.68, respectively. This is signi\ufb01cantly lower than the similarity between class and distillation tokens [3]; 0.96 and 0.94 for DeiT-T and Deit-S, respectively. This con\ufb01rms our hypothesis on modeling distinct features with separate tokens within ViTs, a unique capability that cannot be straightforwardly achieved with CNNs. Further, it offers other bene\ufb01ts as we explain next. Figure 7: Shape Distillation. Shape-biased ViT Offers Automated Object Segmentation: Interestingly, training without local texture or with shape distillation allows a ViT to concentrate on foreground objects in the scene and ignore the background (Table 4, Fig. 8). This offers an automated semantic segmentation for an image although the model is never shown pixel-wise object labels. That is, shape-bias can be used as self-supervision signals for the ViT model to learn distinct shape-related features that help localize the right foreground object. We note that a ViT trained without emphasis on shape does not perform well (Table 4). The above results show that properly trained ViT models offer shape-bias nearly as high as the human\u2019s ability to recognize shapes. This leads us to question if positional encoding is the key that helps ViTs achieve high performance under severe occlusions (as it can potentially allow later layers to recover the missing information with just a few image patches given their spatial ordering). This possibility is examined next. 6 \fModel Distilled Token Type Jaccard Index DeiT-T-Random \u0017 cls 19.6 DeiT-T \u0017 cls 32.2 DeiT-T-SIN \u0017 cls 29.4 DeiT-T-SIN \u0013 cls 40.0 DeiT-T-SIN \u0013 shape 42.2 DeiT-S-Random \u0017 cls 22.0 DeiT-S \u0017 cls 29.2 DeiT-S-SIN \u0017 cls 37.5 DeiT-S-SIN \u0013 cls 42.0 DeiT-S-SIN \u0013 shape 42.4 Table 4: We compute the Jaccard similarity between ground truth and masks generated from the attention maps of ViT models (similar to [23] with threshold 0.9) over the PASCAL-VOC12 validation set. Only class level ImageNet labels are used for training these models. Our results indicate that supervised ViTs can be used for automated segmentation and perform closer to the self-supervised method DINO [23]. DeiT-S DeiT-S-SIN DeiT-S-SIN (Distilled) Figure 8: Segmentation maps from ViTs. Shape distillation performs better than standard supervised models. 3.3 Does Positional Encoding Preserve the Global Image Context? Transformers\u2019 ability to process long-range sequences in parallel using self-attention [27] (instead of a sequential design in RNN [31]) is invariant to sequence ordering. For images, the order of patches represents the overall image structure and global composition. Since ViTs operate on a sequence of images patches, changing the order of sequence e.g., shuf\ufb02ing the patches can destroy the image structure. Current ViTs [2, 3, 4, 26] use positional encoding to preserve this context. Here, we analyze if the sequence order modeled by positional encoding allows ViT to excel under occlusion handling. Our analysis suggests that transformers show high permutation invariance to the patch positions, and the effect of positional encoding towards injecting structural information of images to ViT models is limited (Fig. 10). This observation is consistent with the \ufb01ndings in the language domain [32] as described below. Figure 9: An illustration of shuf\ufb02e operation applied on images used to eliminate their structural information. (best viewed zoomed-in) Sensitivity to Spatial Structure: We remove the structural information within images (spatial relationships) as illustrated in Fig. 9 by de\ufb01ning a shuf\ufb02ing operation on input image patches. Fig. 10 shows that the DeiT models [3] retain accuracy better than their CNN counterparts when spatial structure of input images is disturbed. This also indicates that positional encoding is not absolutely crucial for right classi\ufb01cation decisions, and the model does not \u201crecover\u201d global image context using the patch sequence information preserved in the positional encodings. Without encoding, the ViT performs reasonably well and achieves better permutation invariance than a ViT using position Figure 10: Models trained on 196 image patches. Top-1 (%) accuracy over ImageNet val. set when patches are shuf\ufb02ed. Note the performance peaks when shuf\ufb02e grid size is equal to the original number of patches used during training, since it equals to only changing the position of input patch (and not disturbing the patch content). Figure 11: DeiT-T [3] trained on different number of image patches. Reducing patch size decreases the overall performance but also increases sensitivity to shuf\ufb02e grid size. 7 \fTrained with Augmentations Trained without Augmentation DeiT-B DeiT-S DeiT-T T2T-24 TnT-S Augmix ResNet50 ResNet50-SIN DeiT-T-SIN DeiT-S-SIN 48.5 54.6 71.1 49.1 53.1 65.3 76.7 77.3 94.4 84.0 Table 4: mean Corruption Error (mCE) across common corruptions [13] (lower the better). While ViTs have better robustness compared to CNNs, training to achieve a higher shape-bias makes both CNNs and ViTs more vulnerable to natural distribution shifts. All models trained with augmentations (ViT or CNN) have lower mCE in comparison to models trained without augmentations on ImageNet or SIN. Figure 12: Robustness against adversarial patch attack. ViTs even with less parameters exhibit a higher robustness than CNN. Models trained on ImageNet are more robust than the ones trained on SIN. Results are averaged across \ufb01ve runs of patch attack over ImageNet val. set. Figure 13: Robustness against sample speci\ufb01c attacks including single step, FGSM [34], and multi-step, PGD [35]. ViTs even with less parameters exhibit a higher robustness than CNN. PGD ran for 5 iterations only. Attacks are evaluated under l\u221enorm and \u03f5 represents the perturbation budget by which each pixel is changed in the input image. Results are reported over the ImageNet val. set. encoding (Fig. 10). Finally, when the patch size is varied during ViT training, the permutation invariance property is also degraded along with the accuracy on unshuf\ufb02ed natural images (Fig. 11). Overall, we attribute the permutation invariance performance of ViTs to their dynamic receptive \ufb01eld that depends on the input patch and can adjust attention with the other sequence elements such that moderately shuf\ufb02ing the elements does not degrade the performance signi\ufb01cantly. The above analysis shows that just like the texture-bias hypothesis does not apply to ViTs, the dependence on positional encodings to perform well under occlusions is also incorrect. This leads us to the conclude that ViTs robustness is due to its \ufb02exible and dynamic receptive \ufb01eld (see Fig. 4) which depends on the content of an input image. We now delve further deep into the robustness of ViT, and study its performance under adversarial perturbations and common corruptions. 3.4 Robustness of Vision Transformers to Adversarial and Natural Perturbations After analyzing the ability of ViTs to encode shape information (Sec. 3.2), one ensuing question is: Does higher shape-bias help achieve better robustness? In Table 4, we investigate this by calculating mean corruption error (mCE) [13] on a variety of synthetic common corruptions (e.g., rain, fog, snow and noise). A ViT with similar parameters as CNN (e.g., DeiT-S) is more robust to image corruptions than ResNet50 trained with augmentations (Augmix [33]). Interestingly, CNNs and ViTs trained without augmentations on ImageNet or SIN are more vulnerable to corruptions. These \ufb01ndings are consistent with [10], and suggest that augmentations improve robustness against common corruptions. We observe similar performance against untargeted, universal adversarial patch attack [17] and sample speci\ufb01c attacks including single step, fast gradient sign method (FGSM) [34], and multi-step projected gradient attack known as PGD [35]. Adversarial patch attack [17] is unbounded that is it can change pixel values at certain location in the input image by any amount, while sample speci\ufb01c attacks [34, 35] are bounded by l\u221enorm with a perturbation budget \u03f5, where \u03f5 represents the amount by which each pixel is changed in the entire image. ViTs and CNN trained on SIN are signi\ufb01cantly 8 \fFigure 14: A single ViT model can provide a features ensemble since class token from each block can be processed by the classi\ufb01er independently. This allows us to identify the most discriminative tokens useful for transfer learning. Figure 15: Top-1 (%) for ImageNet val. set for class tokens produced by each ViT block. Class tokens from the last few layers exhibit highest performance indicating the most discriminative tokens. Blocks Class Patch CUB Flowers iNaturalist Tokens Tokens [37] [38] [39] Only 12th (last block) \u0013 \u0017 68.16 82.58 38.28 \u0013 \u0013 70.66 86.58 41.22 From 1st to 12th \u0013 \u0017 72.90 91.38 44.03 \u0013 \u0013 73.16 91.27 43.33 From 9th to 12th \u0013 \u0017 73.58 90.00 45.15 \u0013 \u0013 73.37 90.33 45.12 Table 5: Ablative Study for off-the-shelf feature transfer on three datasets using ImageNet pretrained DeiT-S [3]. A linear classi\ufb01er is learned on only a concatenation of class tokens or the combination of class and averaged patch tokens at various blocks. We note class token from blocks 9-12 are most discriminative (Fig. 15) and have the highest transferability in terms of Top-1 (%) accuracy. more vulnerable to adversarial attack than models trained on ImageNet (Figs. 12 and 13), due to the shape-bias vs. robustness trade-off [10]. Given the strong robustness properties of ViT as well as their representation capability in terms of shape-bias, automated segmentation and \ufb02exible receptive \ufb01eld, we analyze their utility as an off-the-shelf feature extractor to replace CNNs as the default feature extraction mechanism [36]. 3.5 Effective Off-the-shelf Tokens for Vision Transformer A unique characteristic of ViT models is that each block within the model generates a class token which can be processed by the classi\ufb01cation head separately (Fig. 14). This allows us to measure the discriminative ability of each individual block of an ImageNet pre-trained ViT as shown in Fig. 15. Class tokens generated by the deeper blocks are more discriminative and we use this insight to identify an effective ensemble of blocks whose tokens have the best downstream transferability. Transfer Methodology: As illustrated in Fig. 15, we analyze the block-wise classi\ufb01cation accuracy of DeiT models and determine the discriminative information is captured within the class tokens of the last few blocks. As such, we conduct an ablation study for off-the-shelf transfer learning on \ufb01ne-grained classi\ufb01cation dataset CUB [37], Flowers [38] and large scale iNaturalist [39] using DeiT-S [3] as reported in Table 5. Here, we concatenate the class tokens (optionally combined with average patch tokens) from different blocks and train a linear classi\ufb01er to transfer the features to downstream tasks. Note that a patch token is generated by averaging along the patch dimension. The scheme that concatenate class tokens from the last four blocks shows the best transfer learning performance. We refer to this transfer methodology as DeiT-S (ensemble). Concatenation of both class and averaged patch tokens from all blocks helps achieve similar performance compared to the tokens from the last four blocks but requires signi\ufb01cantly large parameters to train. We \ufb01nd some exception to this on the Flower dataset [38] where using class tokens from all blocks have relatively better improvement (only 1.2%), compared to the class tokens from the last four blocks (Table 5). However, concatenating tokens from all blocks also increases the number of parameters e.g., transfer to Flowers from all tokens has 3 times more learnable parameters than using only the last four tokens. We conduct further experimentation with DeiT-S (ensemble) across a broader range of tasks to validate our hypothesis. We further compare against a pre-trained ResNet50 baseline, by using features before the logit layer. Visual Classification: We analyze the transferability of off-the-shelf features across several datasets including Aircraft [40], CUB [37], DTD [41], GTSRB [42], Fungi [43], Places365 [44] and iNaturalist [39]. These datasets are developed for \ufb01ne-grained recognition, texture classi\ufb01cation, 9 \fFigure 16: Off-the-shelf ViT features transfer better than CNNs. We explore transferability of learned representations using generic classi\ufb01cation as well as few-shot classi\ufb01cation for out-of-domain tasks. In the case of classi\ufb01cation (left), the ImageNet pre-trained ViTs transfer better than their CNN counterparts across tasks. In the case of few-shot learning (right), ImageNet pre-trained ViTs perform better on average. traf\ufb01c sign recognition, species classi\ufb01cation and scene recognition with 100, 200, 47, 43, 1394, 365 and 1010 classes respectively. We train a linear classi\ufb01er on top of the extracted features over the train split of each dataset, and evaluate the performance on their respective test splits. The ViT features show clear improvements over the CNN baseline (Fig. 16). We note that DeiT-T, which requires about 5 times fewer parameters than ResNet50, performs better among all datasets. Furthermore, the model with the proposed ensemble strategy achieves the best results across all datasets. Few-Shot Learning: We consider meta-dataset [45] designed as a large-scale few-shot learning (FSL) benchmark containing a diverse set of datasets from multiple domains. This includes letters of alphabets, hand-drawn sketches, images of textures, and \ufb01ne-grained classes making it a challenging dataset involving a domain adaption requirement as well. We follow the standard setting of training on ImageNet and testing on all other datasets which are considered as the downstream tasks. In our experiments, we use a network pre-trained for classi\ufb01cation on ImageNet dataset to extract features. For each downstream dataset, under the FSL setting, a support set of labelled images is available for every test query. We use the extracted features to learn a linear classi\ufb01er over the support set for each query (similar to [46]), and evaluate using the standard FSL protocol de\ufb01ned in [45]. This evaluation involves a varying number of shots speci\ufb01c for each downstream dataset. On average, the ViT features transfer better across these diverse domains (Fig. 16) in comparison to the CNN baseline. Furthermore, we note that the transfer performance of ViT is further boosted using the proposed ensemble strategy. We also highlight the improvement in QuickDraw, a dataset containing hand-drawn sketches, which aligns with our \ufb01ndings on improved shape-bias of ViT models in contrast to CNN models (see Sec. 3.2 for elaborate discussion). 4 Discussion and"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2103.14641v2",
+                    "title": "On Generating Transferable Targeted Perturbations",
+                    "abstract": "While the untargeted black-box transferability of adversarial perturbations\nhas been extensively studied before, changing an unseen model's decisions to a\nspecific `targeted' class remains a challenging feat. In this paper, we propose\na new generative approach for highly transferable targeted perturbations\n(\\ours). We note that the existing methods are less suitable for this task due\nto their reliance on class-boundary information that changes from one model to\nanother, thus reducing transferability. In contrast, our approach matches the\nperturbed image `distribution' with that of the target class, leading to high\ntargeted transferability rates. To this end, we propose a new objective\nfunction that not only aligns the global distributions of source and target\nimages, but also matches the local neighbourhood structure between the two\ndomains. Based on the proposed objective, we train a generator function that\ncan adaptively synthesize perturbations specific to a given input. Our\ngenerative approach is independent of the source or target domain labels, while\nconsistently performs well against state-of-the-art methods on a wide range of\nattack settings. As an example, we achieve $32.63\\%$ target transferability\nfrom (an adversarially weak) VGG19$_{BN}$ to (a strong) WideResNet on ImageNet\nval. set, which is 4$\\times$ higher than the previous best generative attack\nand 16$\\times$ better than instance-specific iterative attack. Code is\navailable at: {\\small\\url{https://github.com/Muzammal-Naseer/TTP}}.",
+                    "authors": "Muzammal Naseer, Salman Khan, Munawar Hayat, Fahad Shahbaz Khan, Fatih Porikli",
+                    "published": "2021-03-26",
+                    "updated": "2021-08-13",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction We study the challenging problem of targeted transferability of adversarial perturbations. In this case, given an input sample from any source category, the goal of the adversary is to change the decision of an unknown model to a speci\ufb01c target class (e.g., misclassify any painting image to Fire truck, see Fig. 1). This task is signi\ufb01cantly more dif\ufb01cult than merely changing the decision to a ranNatural Images Source  Domain OR : Augmenter Paintings Fire Truck Target  Domain OR Parachute Target Latent Space 4 < Q Q 4 : Generator < : Discriminator Maximize Distribution Agreement Adversarial Augmented  Adversarial Figure 1: Attack Overview (TTP): Instead of \ufb01nding perturbations speci\ufb01c to a class-boundary information learned by a model, TTP seeks to match global distribution statistics between the source and the target domains. Speci\ufb01cally, our generator function is trained to maximize agreement between the perturbed source distribution, its augmented version and the target distribution in the feature space. Importantly, our attack can function in an unsupervised fashion and does not require source domain to be the same as target (e.g., perturbations can be learned from paintings to transfer on natural images). dom target class or any similar class (e.g., changing \u2018cat\u2019 to \u2018aeroplane\u2019 is more dif\ufb01cult than altering the decision to \u2018dog\u2019). Target transferability can therefore lead to goaldriven adversarial perturbations that provide desired control over the attacked model. However, target transferability remains challenging for the current adversarial attacks [26, 24, 4, 44, 15, 14, 13, 21, 42] that transfer adversarial noise in a black-box setting, where architecture and training mechanism of the attacked model remain unknown, and the attack is restricted within a certain perturbation budget. We observe that modest performance of existing metharXiv:2103.14641v2  [cs.CV]  13 Aug 2021 \fods on targeted transferability is due to their reliance on class-boundary information learned by the model which lacks generalizability. For example, iterative instancespeci\ufb01c attacks rely on the classi\ufb01cation score information to perturb a given sample, thereby ignoring the global classspeci\ufb01c information [26, 4, 44, 14]. Such adversarial directions also vary across different models [24], leading to poor target transferability [24, 4]. On the other hand, although universal and generative perturbations are designed to encode global noise patterns [27, 36, 32], they still exploit the class impressions learned by a neural network which alone are not fully representative of the target distribution, thereby achieving only modest black-box fooling rates [37]. Furthermore, they are dependent on the classi\ufb01cation information, necessitating a supervised pretrained model for generator\u2019s guidance and therefore cannot directly work with unsupervised features [1, 9]. Another group of techniques exploit intermediate features, but they either \ufb01nd untargeted perturbations by design [29, 38] or are limited in their capacity to transfer targeted perturbations [21, 15, 13, 14]. We introduce a novel generative training framework which maps a given source distribution to a speci\ufb01c target distribution by maximizing the mutual agreement between the two in the latent space of a pretrained discriminator. Our main contributions are: \u2022 Generative Targeted Transferability: We propose a novel generative approach to learn transferable targeted adversarial perturbations. Our unique training mechanism allows the generator to explore augmented adversarial space during training which enhances the transferability of adversarial examples during inference (Sec. 3.1). \u2022 Mutual Distribution Matching: Our training approach is based on maximizing the mutual agreement between the given source and the target distribution. Therefore, our method can provide targeted guidance to train the generator without the need of classi\ufb01cation boundary information. This allows an attacker to learn targeted generative perturbations from the unsupervised features [1, 9] and eliminate the cost of labelled data (Sec. 3.2). \u2022 Neighbourhood Similarity Matching: Alongside global distribution matching, we introduce batch-wise neighbourhood similarity matching objective between adversarial and target class samples to maximize the local alignment between the two distributions (Sec. 3.3). Our extensive experiments on various ImageNet splits and CNN architectures show state-of-the-art targeted transferability against naturally and adversarially trained models, stylized models and input-processing based defenses. The results demonstrate our bene\ufb01t compared to recent targeted instance-speci\ufb01c as well as other generative methods. Further, our attack demonstrates rapid convergence. 2. Related Work Iterative Instance-Speci\ufb01c Perturbations: After Szegedy et al. [41] highlighted the vulnerability of neural networks, many adversarial attacks have been introduced to study if the adversarial examples are transferable from one model to another, when a target model is unknown. Among these, iterative instance-speci\ufb01c attacks [4, 44, 5] perturb a given sample by iteratively using gradient information. Target transferability of such attacks is very poor [4, 24] (as shown in Sec. 4). Other attacks also use feature space either by maximizing the feature difference [45, 11, 22] or applying attention [43] or avoiding non-linearity while backpropagating gradients [8] or exploiting skip-connections [42]. However, these attacks are mainly designed to enhance non-targeted transferability which is an easier problem. Recently, different instance-speci\ufb01c (transferable) targeted attacks have been proposed including [21] which introduces a triplet loss to push adversarial examples towards the target label while increasing their distance from the original label. Inkawhich et al. [13, 14] proposed to exploit feature space [15] along with the classi\ufb01er information [14] to generate target adversaries that are shown to transfer relatively better than other instance-speci\ufb01c attacks. These attacks [13, 14] have the following limitations. a) They need access to a labeled dataset e.g., ImageNet [39] in order to train one-vs-all binary classi\ufb01ers for attacked target classes. b) They need to identify best performing single layer [13] or a combination of layers [14] which adds further complexity to attack optimization. c) Finally, the attack performance degrades signi\ufb01cantly with quality of features, e.g., it struggles to transfer target perturbations from VGG models [13]. Universal Perturbation: In contrast to instance-speci\ufb01c perturbations, [27] learns a single universal noise pattern which is representative of the entire data distribution and can fool a model on majority of samples. Li et al. [23] introduce gradient transformation module to \ufb01nd smooth universal patterns while [29] shows that such patterns can be found without any training data. Although universal perturbations [27, 28, 29, 23] based attacks are ef\ufb01cient (the attacker just needs to add the noise to any given sample at inference), they are limited in their capacity to yield transferable adversaries which can generalize across different data distributions and models [36, 32]. Generative Perturbations: Generative adversarial perturbations perform better than directly optimizing universal noise [38, 36, 32]. Poursaeed et al. [36] proposed the \ufb01rst generative approach to adapt perturbations to an input sample. Naseer et al. [32] improved this framework with relativistic training objective which also allows cross-domain transferability. Our method belongs to the generative category and can adapt to an input sample with a single forward pass. Unlike [38, 36, 32], we seek to fool the model by matching distributions of source and targets with distribu\ftion matching and neighbourhood similarity criteria. Our proposed framework does not require labeled source or target data and can extract target perturbations from a discriminator model trained in an unsupervised manner while previous generative methods are dependent on class-boundary information learned by the model. Further, our method converges faster (Sec. 4) and provides improved targeted transferability owing to its novel loss and training mechanism. 3. Generating Targeted Adversaries Our goal is to craft adversarial perturbations \u03b4 that can fool a model to misclassify any given input to a speci\ufb01c target class t. We assume access to source and target domain data represented by P and Q, from which the source and target class samples are obtained i.e., xs \u223cP, xt \u223cQ. The source and target domains are likely to be non-aligned i.e., P \u0338= Q, making it challenging to achieve targeted transferability of adversarial perturbations. We also consider a perturbed source data P \u2032 that comprises of adversarially manipulated samples x\u2032 s \u223cP \u2032 where x\u2032 s = xs +\u03b4. xs, x\u2032 s and xt represent source, adversarial and target domain samples while D\u03c8(xs), D\u03c8(x\u2032 s) and D\u03c8(xt) are their corresponding latent distributions. 3.1. Generative Model We propose a generative approach to perturb the source domain samples xs to a speci\ufb01ed target class. The framework (see Fig. 2) consists of a generator G\u03b8 and a discriminator D\u03c8 parameterized by \u03b8 and \u03c8, respectively. The generator function G\u03b8 learns a mapping from the source images to the target category such that the input images are minimally changed i.e., adversarial noise \u03b4 is strictly constrained under a norm distance l\u221e\u2264\u03f5. This is ensured by projecting the unbounded adversaries from G\u03b8 within \ufb01xed norm distance of xs using a differentiable clipping operation, x\u2032 s = clip \u0000min(xs + \u03f5, max(W \u2217G\u03b8(xs), xs \u2212\u03f5)) \u0001 , (1) where, W is a smoothing operator with \ufb01xed weights that reduces high frequencies without violating the l\u221edistance constraint. The smooth projection in Eq. 1 (denoted by P in Fig. 2) not only tightly bounds generator\u2019s output within l\u221enorm but also encourages avoiding redundant high frequencies [35] during the optimization process. This allows the generator to converge to a more meaningful solution. The existing generative designs for adversarial attacks [36, 32] leverage the decision space of the discriminator to craft perturbations. In such cases, the class-boundary information learned by the discriminator is used to fool DNN models (e.g. for ImageNet, discriminator is pretrained on 1k classes). This dependence is problematic since an attacker must have access to a discriminator trained on largescale labeled dataset [3]. Attacker then tries to learn target Algorithm 1 Generating TTP Require: Source data Xs, Target data Xt, pretrained discriminator D\u03c8, perturbation budget \u03f5, loss criteria LG. Ensure: Randomly initialize the generator, G\u03b8 1: repeat 2: Randomly sample mini-batches xs \u223cXs and xt \u223cXt 3: Create augmented copy of the source mini-batch \u02dc xs. 4: Forward-pass xs and \u02dc xs through the generator and generate unbounded adversaries; x\u2032 s, \u02dc x\u2032 s. 5: Bound the adversaries using Eq. 1 such that: \u2225x\u2032 s \u2212xs\u2225\u221e\u2264\u03f5 and \u2225\u02dc x\u2032 s \u2212\u02dc xs\u2225\u221e\u2264\u03f5 6: Forward pass x\u2032 s, \u02dc x\u2032 s and xt through D\u03c8. 7: Compute the matching losses; L, Laug and Lsim using Eq. 3, 4 and 8, respectively. 8: Compute the generator loss given in Eq. 9. 9: Backward pass and update G\u03b8 10: until G\u03b8 converges. class impressions using either cross-entropy (CE) [36] or relativistic CE [32]. Thus, the generated perturbations are directly dependent on the quality of the discriminator\u2019s classi\ufb01cation space. Furthermore, the generated adversaries are dependent on the input instance-speci\ufb01c features and do not model the global properties of the target distribution, resulting in only limited transferability. To address above limitations, our generative design models the target distribution Q and pushes the perturbed source distribution P \u2032 closer to Q using the latent space of D\u03c8, \u2225\u03b4 \u2225\u221e\u2264\u03f5, s.t., D\u03c8(x\u2032 s) \u2248D\u03c8(xt). (2) This global objective provides two crucial bene\ufb01ts. First, reducing mismatch between perturbed and target distributions provides an improved guidance to the generator. The resulting perturbations well align the input samples with the target distribution, leading to transferable adversaries. Second, the distributions alignment task makes us independent of the D\u03c8\u2019s classi\ufb01cation information. In turn, our approach can function equally well with a discriminator trained in a self-supervised manner on unlabelled data [1, 9]. In our case, we simply align the feature distributions from D\u03c8 to match P \u2032 and Q. Thus, for a given sample x, n-dimensional features are obtained i.e., D\u03c8(x) \u2208Rn. If D\u03c8 is trained in a supervised manner on ImageNet then n = 1000, and if D\u03c8 is trained in an unsupervised fashion then n is equal to the output feature dimension. 3.2. Distribution Matching We measure the mutual agreement between P \u2032 and Q using Kullback Leibler (KL) divergence de\ufb01ned on discrimi\fShared Weights Augmenter \ud835\udcab \ud835\udcab \ud835\udc65! \ud835\udc65 \"! Pretrained  Features Target Samples Unrestricted Adversaries Projection (\ud835\udcab) Generator,  \ud835\udca2! Discriminator \ud835\udc9f\" Source Samples \ud835\udc65\" ! \ud835\udc65 \"\" ! \ud835\udc65# KL  Divergence \u2112+ \u2112$%& Neighbourhood  Similarity \u2112!'( Distribution Matching Generator Loss \ud835\udc9f) \ud835\udc65\" ! \u2248\ud835\udc9f) \ud835\udc61 \ud835\udc9f) \ud835\udc65 \"\" ! \u2248\ud835\udc9f) \ud835\udc61 Figure 2: Targeted Transferable Perturbations: During training, TTP matches adversarial and augmented adversarial samples to a target domain within discriminator\u2019s latent space for improved transferability. The adversarial samples corresponding to original and augmented images are bounded (via projection) around their source samples to explore adversarial space around natural as well as augmented samples. nator features D\u03c8(x\u2032 s) and D\u03c8(xt), DKL(P \u2032\u2225Q) = 1 N N P i=1 n P j=1 \u03c3(D\u03c8(x\u2032,i s ))j log \u03c3(D\u03c8(x\u2032,i s ))j \u03c3(D\u03c8(xi t))j , where N represents the number of samples, n is the discriminator\u2019s output dimension, and \u03c3 denotes the softmax operation. In simple terms, KL divergence measures the difference between two distributions in terms of the average surprise in experiencing xt when we expected to see x\u2032 s. Since KL divergence is asymmetric i.e. DKL(P \u2032\u2225Q) \u0338= DKL(Q\u2225P \u2032), and not a valid distance measure, we de\ufb01ne our loss function for distribution matching [20] as follows: L = DKL(P \u2032\u2225Q) + DKL(Q\u2225P \u2032). (3) As a regularization measure, we add augmented versions of the source domain samples during distribution matching. This enables the generator to focus speci\ufb01cally on adding target class-speci\ufb01c patterns that are robust to input transformations. To this end, we randomly apply rotation, crop resize, horizontal \ufb02ip, color jittering or gray scale transformation to create augmented samples \u02dc xs from the original xs. The \u02dc xs \u223c\u02dc P are passed through the G\u03b8 and the perturbed augmented samples \u02dc x\u2032 s \u223c\u02dc P \u2032 are projected using Eq. 1 to stay close to the augmented samples i.e., \u2225\u02dc x\u2032 s \u2212\u02dc xs\u2225\u221e\u2264\u03f5. No augmentation is applied to the target domain samples. We then pass \u02dc x\u2032 s through the discriminator and compute the mutual agreement between D\u03c8(\u02dc x\u2032 s) and D\u03c8(xt) as follows: Laug = DKL( \u02dc P \u2032\u2225Q) + DKL(Q\u2225\u02dc P \u2032). (4) The impact of data augmentations and their effectiveness for our proposed targeted attack is studied in Sec. 4. 3.3. Neighbourhood Similarity Matching The above objective promotes alignment between the distributions but does not consider the local structure e.g., the relationship between a sample and its augmented versions. For a faithful alignment between perturbed source samples and the target class samples, we propose to also match the neighbourhood similarity distributions between the two domains. Speci\ufb01cally, consider a batch of target domain samples {xi t}N i=1 and a batch of perturbed source domain samples {x\u2032i s }N i=1. For the case of x\u2032 s, in a given training batch, we compute a similarity matrix Ss whose elements encode the cosine similarity between the original sample and its augmented version \u02dc x\u2032 s, i.e., Ss i,j = D\u03c8(x\u2032,i s ) \u00b7 D\u03c8(\u02dc x\u2032,j s ) \u2225D\u03c8(x\u2032,i s )\u2225\u2225D\u03c8(\u02dc x\u2032,j s )\u2225 . (5) In contrast, for the case of xt, we compute similarity between only the original target samples (no augmentations) as we need to model the local neighbourhood connectivity in the target domain. This choice is impractical for the source domain case where many categories co-exist, while for the target distribution, we assume a single category. Thus the target similarity matrix St is computed as, St i,j = D\u03c8(xi t) \u00b7 D\u03c8(xj t) \u2225D\u03c8(xi t)\u2225\u2225D\u03c8(xj t)\u2225 . (6) The resulting similarity matrices are normalized along the row dimension with softmax to obtain probability estimates, \u00af Si,j = exp(Si,j) P k exp(Si,k), where, S \u2208{Ss, St}. (7) Here, each term shows the probability with which the two sample pairs are related to each other. Given \u00af Ss and \u00af St, we \fcompute the KL divergence to enforce a loss term that seeks to match the local neighbourhood patterns between source and target domains, Lsim = X i,j \u00af St i,j log \u00af St i,j \u00af Ss i,j + X i,j \u00af Ss i,j log \u00af Ss i,j \u00af St i,j . (8) 3.4. Overall loss function Finally, the generator parameters are updated by minimizing the following loss (Algorithm 1): LG = L + Laug + Lsim. (9) This loss encourages the generator to perturb source samples that not only match the global characteristics of the target distribution (L + Laug), but also the local information based on neighbourhood connectivity (Lsim). 4. Experiments Our generator G\u03b8 is based on ResNet architecture [17], and outputs an adversarial sample with the same size as of input (Fig. 3). This generator architecture is the same as in the baseline generative attacks [36, 32]. Our discriminator D\u03c8 is pre-trained in a supervised or self-supervised manner. For training G\u03b8, we freeze D\u03c8. We use Adam optimizer [19] with a learning rate of 10\u22124 (\u03b21 = .5, \u03b22 = .999) for 20 epochs. For source domain data, we use 50k random images from ImageNet train set. Our method is not sensitive to the choice of source samples since it can learn transferable perturbations even from other domains e.g. Paintings. Similar to other generative methods [36, 32], we \ufb01x source data. For target domain data, we use 1300 images for each target collected from ImageNet training set (without their original labels). We used default settings or implementations as provided by the authors of baseline attacks. Similarly, we used open-sourced (pretrained) stylized [7], adversarial [40] and puri\ufb01er (NRP) [30] models to evaluate robustness. 4.1. Evaluation Settings We perform inference on ImageNet validation set (50k samples). No augmentations are applied at inference time. The perturbation budget is tightly bounded and clearly mentioned in each experiment following the standard practices l\u221e\u226416 [4, 15, 14] and l\u221e\u226432 [32, 23]. We perturb all the ImageNet val. samples (except the target samples) to the pre-de\ufb01ned target class. We repeat this process for all the given targets and report Top-1 (%) accuracy averaged across all targets. We compare our method under two main settings (10-Targets and 100-Targets), as described below. 10-Targets: We further consider two settings. (a) 10Targets (subset-source) which is consistent with [13] and has a subset of source classes at inference. (b) 10-Targets (all-source) which is a more challenging large-scale setting Naturally Trained (IN) Models Src. Attack VGG19BN Dense121 ResNet50 ResNet152 WRN-50-2 VGG19BN PGD [26] 95.67\u2217 0.31 0.30 0.20 0.25 MIM [4] 99.91\u2217 0.92 0.68 0.36 0.47 DIM [44] 99.38\u2217 3.10 2.08 1.02 1.29 DIM-TI [5] 89.71\u2217 1.08 0.66 0.42 0.45 Po-TRIP [21] 99.40\u2217 4.61 3.21 1.78 2.01 GAP [36] 98.23\u2217 16.19 15.83 5.89 7.78 CDA [32] 98.30\u2217 16.26 16.22 5.73 8.35 Ours-P 97.38\u2217 45.53 42.90 26.72 31.00 Ours 98.54\u2217 45.77 45.87 27.18 32.63 Vens Ours 97.34\u2217 71.41 71.68 50.78 48.03 Dense121 PGD [26] 1.28 97.40\u2217 1.78 1.01 1.37 MIM [4] 1.85 99.90\u2217 2.71 1.68 1.88 DIM [44] 7.31 98.81\u2217 9.06 5.78 6.29 DIM-TI [5] 0.91 88.59\u2217 1.18 0.77 0.86 Po-TRIP [21] 8.10 99.00\u2217 11.21 7.83 8.50 GAP [36] 39.01 97.30\u2217 47.85 39.25 34.79 CDA [32] 42.77 97.22\u2217 54.28 44.11 46.01 Ours-P 57.91 97.41\u2217 71.35 55.57 53.45 Ours 58.90 97.61\u2217 68.72 57.11 56.80 Dens Ours 76.96 96.25\u2217 88.81 83.48 81.85 ResNet50 PGD [26] 0.92 1.38 93.74\u2217 1.86 1.89 MIM [4] 1.58 3.37 98.76\u2217 3.39 3.17 DIM [44] 9.14 15.47 99.01\u2217 12.45 12.61 DIM-TI [5] 0.79 2.12 88.91\u2217 1.47 1.45 Po-TRIP [21] 12.01 19.43 99.22\u2217 14.41 15.10 GAP [36] 58.47 71.72 96.81\u2217 64.89 61.82 CDA [32] 64.58 73.57 96.30\u2217 70.30 69.27 Ours-P 73.09 84.76 96.63\u2217 76.27 75.92 Ours 78.15 81.64 97.02\u2217 80.56 78.25 Rens Ours 90.43 94.39 96.67\u2217 95.48\u2217 92.63 Table 1: Target Transferability: {10-Targets (all-source)} Top1 target accuracy (%) averaged across 10 targets with 49.95K ImageNet val. samples. Perturbation budget: l\u221e\u226416. Our method outperforms previous instance-speci\ufb01c as well as generative approaches by a large margin. \u2019*\u2019 indicates white-box attack. Ours-P represents TTP trained on Paintings. as source images can come from all the ImageNet classes except the target class. For consistency and direct comparison, the ten target classes are same as in [13]. \u221210-Targets (subset-source): Following [13], for each target class, 450 source samples belonging to remaining 9 classes (except target class) become inputs to G\u03b8 to be transferred to the selected target. \u221210-Targets (all-source): For each target class, samples of all 999 source classes (except the target class) in ImageNet val. set are considered i.e., for each target class, 49,950 samples of 999 classes become inputs to G\u03b8. 100-Targets (all-source): We divide ImageNet 1k classes into 100 mutually exclusive sets. Each set contains 10 classes. We randomly sample 1 target from each set to create 100 targets (see Appendix E for more details). Generators are trained against these targets and evaluated on ImageNet val. set in 100-Targets (all-source) setting with the same protocol as described for 10-Targets (all-source). \fIndigo Bunting Cardoon Impala Wood Rabbit Crane Elephant Fire Truck Fire Truck Fire Truck Fire Truck Fire Truck Fire Truck Figure 3: Targeted adversaries produced by a TTP generator learned to maximize the agreement with \u2019Fire Truck\u2019 distribution against Dense121 ImageNet model. 1st and 2nd rows show clean images and unrestricted outputs of the adversarial generator, respectively. 3rd row shows adversaries after valid projection. See Appendix F for more qualitative examples including comparisons between targeted patterns learned by TTP from different source models of a certain family of networks. Src. Attack VGG19BN Dense121 ResNet50 VGG19BN AA [15] \u2013 0.8 0.6 FDA-fd [13] \u2013 3.0 2.1 FDAN [14] \u2013 6.0 5.4 CDA [32] \u2013 17.82 17.09 Ours-P \u2013 48.56 44.47 Ours \u2013 48.29 47.07 Dense121 AA [15] 0.0 \u2013 0.0 FDA-fd [13] 34.0 \u2013 34.0 FDAN [14] 42.0 \u2013 48.3 CDA [32] 44.84 \u2013 53.73 Ours-P 59.81 71.32 Ours 61.75 \u2013 69.60 ResNet50 AA [15] 1.1 2.0 \u2013 FDA-fd [13] 16.0 21.0 \u2013 FDAN [14] 32.1 48.3 \u2013 CDA [32] 68.55 75.68 \u2013 Ours-P 75.18 85.71 Ours 79.04 84.42 \u2013 Table 2: Target Transferability:{10-Targets (sub-source)} Top1 accuracy (%) across 10 targets. Our method shows signi\ufb01cant improvements in trasfering target perturbations compared to generative as well as feature based instance-speci\ufb01c method [13, 14]. Perturbation budget:l\u221e\u226416. Only black-box attack results are shown. Ours-P represents TTP trained on Paintings. 4.2. Attack Protocols and Results We evaluate black-box target transferability in the following scenarios. (a) Unknown Target Model: Attacker has access to a pretrained discriminator trained on labeled data but has no knowledge about the architecture of the target model. (b) Unknown Decision Space: Attacker has access to the pre-trained discriminator trained on unlabeled data in an unsupervised manner but does not know about the architecture and the class-boundary information learned by the target model. (c) Unknown Defense: Attacker is unaware of the type of defense deployed at the target model, or if any defense is applied at all, e.g., the defense can be an input processing approach or a robust training mechanism such as adversarial training. 4.2.1 Unknown Target Model Natural Training: We evaluate naturally trained ImageNet models and show strong empirical results in Tables 1, 2 & 3 demonstrating that generative methods are far superior than sample-speci\ufb01c targeted attacks based on boundary information [21, 4, 44] or feature exploitation [15, 13, 14]. Our approach has signi\ufb01cantly higher target transferability rates than previous generative methods [32, 36]. To highlight an example from Table 2, our method achieves 47.07% transferability from VGG19BN to ResNet50 which is 175% and 771% better than the previous best generative [32] and sample-speci\ufb01c [14] target attacks, respectively. Ensemble Effect: We also train generators with our algorithm on the ensembles of same-family discriminators. Speci\ufb01cally, we de\ufb01ne the following ensembles: Vens:VGG{11,13,16,19}BN,Rens:ResNet{18,50,101, 152}, and Dens:DenseNet{121,161,169,201}. The purpose of such ensembles is to understand if the combination of weak individual models from the same family can provide strong learning for the target distributions. From Table 1, we observe that modeling target distribution from an ensemble provides signi\ufb01cantly better tranferability than any individual discriminator (see Appendix A for more analysis). This signi\ufb01es that an attacker can use multiple variants of the same network to boost the attack. Target Transferability and Model Disparity: We note that within a speci\ufb01c family, transferring targeted perturbations from a smaller model to a larger one (e.g. ResNet18 \u2192ResNet152 or VGG11BN \u2192VGG19) is dif\ufb01cult as we increase the size discrepancy. Interestingly, this trend remains the same even from larger to smaller models i.e., the attack strength will increase with the disparity between models rather than only depending upon the strength of target model. For example, target transferability ResNet152 \u2192ResNet50 is higher than ResNet152 \u2192ResNet18 even though ResNet18 is weaker than ResNet50 (Fig. 4). Similar behaviour can be observed within cross-family models i.e., \fVGG11 VGG13 VGG16 VGG19 VGG11BN VGG13BN VGG16BN VGG19BN Source Models 62.2 60.0 55.0 46.8 47.6 47.4 42.4 38.1 52.4 58.0 59.6 54.8 60.4 65.5 64.4 69.0 Ours VGG11 VGG13 VGG16 VGG19 VGG11BN VGG13BN VGG16BN VGG19BN 45.7 41.7 43.3 28.8 19.7 20.6 21.1 15.9 28.3 38.3 41.9 34.8 33.2 40.0 49.8 43.3 CDA VGG11 VGG13 VGG16 VGG19 VGG11BN VGG13BN VGG16BN VGG19BN 56.2 47.2 46.7 31.2 19.5 25.9 23.0 17.5 40.8 39.9 45.3 42.8 31.7 38.9 53.4 42.8 GAP ResNet18 ResNet50 ResNet101ResNet152 Target Models ResNet18 ResNet50 ResNet101 ResNet152 Source Models 97.6 65.5 49.1 47.6 75.2 97.0 82.0 80.6 71.0 76.1 96.1 85.0 75.0 86.7 87.1 96.4 ResNet18 ResNet50 ResNet101 ResNet152 Target Models ResNet18 ResNet50 ResNet101 ResNet152 98.0 64.5 47.3 45.4 59.0 96.3 70.6 70.3 58.7 82.8 95.2 79.5 58.9 78.0 79.6 95.8 ResNet18 ResNet50 ResNet101 ResNet152 Target Models ResNet18 ResNet50 ResNet101 ResNet152 97.9 59.1 46.0 41.4 55.4 96.8 68.5 64.9 57.6 79.6 92.7 76.8 61.1 81.3 79.6 95.0 Figure 4: Within Family Target Transferability: {10-Targets (all source) settings} These results indicate that our approach boosts target transferability within different models of the same family with or without batch-norm and favorably beats the previous generative approaches (GAP [36], CDA [32]) by a large margin. Each value is averaged across 10 targets (Sec. 4) with 49.95k ImageNet val. samples for each target. Perturbation budget is set to l\u221e= 16. GAP CDA Ours Attack 0 10 20 30 40 Target Accuracy (%) VGG19BN  Dense121 VGG19BN  Dense121 MoCo   Dense121 GAP CDA Ours Attack 0 5 10 15 20 25 30 35 40 Target Accuracy (%) Dense121   VGG19 Dense121   VGG19 MoCo   VGG19 GAP CDA Ours Attack 0 5 10 15 20 25 Target Accuracy (%) ResNet50   InceptionV3 ResNet50   InceptionV3 MoCo   InceptionV3 Figure 5: Target Transferability of Unsupervised Features: {10-Targets (all-source) settings}. Our approach when applied to unsupervised features, MoCo [9], surpasses GAP [36] and CDA [32] that are dependent on classi\ufb01cation layer by design. Perturbation budget is l\u221e= 16. target transferability from ResNet50 to Dense121 and vice versa is higher than VGG19BN as both models share skip connections (Table 1). See Appendix B for vulnerability of models with and without batch-norm [16]. 4.2.2 Unknown Decision Space Here we investigate the question, \u201cCan unsupervised features provide targeted adversarial perturbations?\u201d A unique property of our proposed approach is that it can be applied to feature space without any class boundary information to achieve target adversarial direction. This allows an attacker to bene\ufb01t from recently proposed unsupervised feature learning methods [9, 1]. Rather than using a discriminator trained on large scale labelled data, attack can be learned and launched from features of a discriminator trained purely in an unsupervised fashion. Therefore, our attack can eliminate the cost of label annotations. Results in Fig. 5 demonstrate that our method learned from unsupervised features, MoCo [9], not only provides target transferability but surpasses the previous generative methods which are dependent on the discriminator trained on labelled data. 4.2.3 Unknown Defense Mechanisms Input Processing as a Defense: We evaluate robustness of different input processing based adversarial defense methods in Fig. 6. We consider the following four representative defenses: a) JPEG with compression quality set to 50% [2], b) DNN-Oriented JPEG compression [25], c) Median Blur with window size set to 5\u00d75 [31], and d) Neural representaAttack VGG19BN Dense121 ResNet-152 WRN-50-2 SIN [7] GAP [36] 47.87 58.10 54.72 49.65 7.1 CDA [32] 53.41 60.34 57.67 51.23 7.6 Ours 69.55 77.48 75.74 74.61 31.0 Table 3: Target Transferability: {100-Targets (all-source)} Top1 target accuracy (%) averaged across 100 targets with 49.95K ImageNet val. samples per target . Generators are trained against ResNet50. Perturbation budget is l\u221e\u226416. \u03f5 Attack Augmix Stylized [7] Adversarial [40] [10] SIN-IN SIN l\u221e l2 \u03f5=.5 \u03f5=1 \u03f5=.1 \u03f5=.5 16 GAP [36] 51.57 76.92 12.96 1.88 0.34 23.41 0.92 CDA [32] 59.79 75.93 9.21 2.10 0.39 23.89 1.18 Ours 73.09 87.40 30.17 4.63 0.56 45.40 1.99 Oursens 88.79 92.96 57.75 14.23 1.24 74.95 7.62 32 GAP [36] 54.86 81.15 28.07 26.32 6.36 59.04 16.53 CDA [32] 63.18 76.81 19.65 27.60 6.74 57.54 16.07 Ours 78.66 91.27 41.52 46.82 16.35 75.97 30.94 Oursens 89.96 94.15 70.70 70.22 34.21 90.42 58.25 Table 4: Target Transferability: {10-Targets (all source) settings} Top-1 (%) target accuracy. Generators are trained against naturally trained ResNet50 or ResNet ensemble. Perturbation are then transferred to ResNet50 trained using different methods including Augmix [10], Stylized [7] or adversarial [40]. tion puri\ufb01er (NRP) [30] which is a state-of-the-art defense. Generators are trained against naturally trained ResNet50 and target perturbations are then transferred to VGG19BN and Dense121 which are protected by the input processing defenses. We observe (Fig. 6) that JPEG is the least effec\f16 20 24 28 32 Pertubation Budget ( ) 30 40 50 60 70 Target Accuracy (%) ResNet50   JPEG: VGG19BN GAP CDA Ours 16 20 24 28 32 Perturbation Budget ( ) 20 30 40 50 60 Target Accuracy (%) ResNet50   JPEG-DNN: VGG19BN GAP CDA Ours 16 20 24 28 32 Pertubation Budget ( ) 15 20 25 30 35 40 45 Target Accuracy (%) ResNet50   Blur: VGG19BN GAP CDA Ours 16 20 24 28 32 Pertubation Budget ( ) 0 10 20 30 40 50 60 Target Accuracy (%) ResNet50   NRP: VGG19BN GAP CDA Ours 16 20 24 28 32 Pertubation Budget ( ) 30 40 50 60 70 80 Target Accuracy (%) ResNet50   JPEG: Dense121 GAP CDA Ours 16 20 24 28 32 Perturbation Budget ( ) 20 25 30 35 40 45 50 55 60 Target Accuracy (%) ResNet50   JPEG-DNN: Dense121 GAP CDA Ours 16 20 24 28 32 Pertubation Budget ( ) 10 15 20 25 30 35 40 45 Target Accuracy (%) ResNet50   Blur: Dense121 GAP CDA Ours 16 20 24 28 32 Pertubation Budget ( ) 0 10 20 30 40 50 60 70 Target Accuracy (%) ResNet50   NRP: Dense121 GAP CDA Ours Figure 6: Target Transferability against Input Processing Defenses: {10-Targets (allsource) settings} Input processing including NRP [30] are broken under targeted blackbox attacks. Our method outperforms GAP [36] and CDA [32] on all the considered defenses including JPEG, JPEG-DNN [25], Median Blur and NRP [30]. Each point is an averaged across 10 targets (Sec. 4) with 49.95k ImageNet val. samples for each target. Generators are trained against ResNet50. tive method against target attacks while JEPG-DNN [25] performs relatively better than JPEG. Compared to JPEG, JPEG-DNN and Median blur, NRP shows better resistance to target attacks at l\u221e\u226416 but quickly breaks as perturbation is increased. Median blur shows more resistance than JPEG, JPEG-DNN and NRP at higher perturbation rates (l\u221e\u226432)1. Success rate of our method is much better than previous generative attacks [36, 32] even when the target model and the input processing remain unknown (Fig. 6). Robust Training Mechanism: Here we study the transferability of our approach against various robust training methods (augmented vs. stylized vs. adversarial) based defense strategies. Augmentation based training can make the model robust to natural corruptions [10] while training on stylized ImageNet [7] improves shape bias and training on adversarial examples can improve robustness against adversarial attacks at the cost of computation, clean accuracy, and generalization to global changes [6]. We evaluate the vulnerability of these training methods in Table 4. Generators are trained against naturally trained ResNet50 or ResNet ensemble and adversarial perturbations are then transferred to ResNet50 trained using Augmix [10], Stylized ImageNet (SIN) [7], mixture of Stylized and natural ImageNet (SININ) and adversarial examples [40]. Target transferability can easily be achieved against models trained on mixture (SIN-IN), however, the model trained on stylized images (SIN) shows higher resistance but remains vulnerable as our target attack (ensemble) achieves \u224871% success at perturbation of l\u221e= 32 (Table 4). Adversarially trained models using Madry\u2019s method [26] are more robust to target attacks. 4.3. Ablative Analysis In order to understand the effect of each component of our approach, we present an ablative study in Fig. 7. Target perturbations are transferred from ResNet50 to VGG16 (SIN) trained on stylized ImageNet which is a much harder task than transferring to naturally trained VGG16. We observe that training TTP on only distribution matching loss (Eq. 3) increases the transferability by more than 100% in 1Blur defense causes large drop in clean accuracy (see Appendix D). GAP CDA Ours Ours Ours Ours Attack 0 5 10 15 20 25 30 Target Accuracy (%) ResNet50   VGG16 (SIN) CE RCE  without   with  + aug + aug + sim 1 3 5 7 9 11 13 15 17 19 Epochs 0 5 10 15 20 25 30 Target Accuracy (%) ResNet50   VGG16 (SIN) GAP CDA Ours Figure 7: Ablation: We dissect effect of each component of our method including novel losses, augmentation, smooth projection and epochs. Results are presented with 10-Target (all source) settings. Perturbation budget is set to l\u221e= 16. comparison to GAP [36] (with cross-entropy) or CDA [32] (with relativistic cross-entropy). Adding smoothing operator W enhances the ef\ufb01ciency of TTP. W is a differentiable Gaussian kernel with size 3\u00d73. We then noticed a signi\ufb01cant jump in transferability when augmentations are introduced and TTP is trained using both distribution matching losses (Eq. 3 & 4) which is further complemented by neighbor similarity loss (Eq. 8). Our generator trained for only one epoch outperforms GAP and CDA trained for 20 epochs (Fig. 7) which highlights our rapid convergence rate. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2006.04924v1",
+                    "title": "A Self-supervised Approach for Adversarial Robustness",
+                    "abstract": "Adversarial examples can cause catastrophic mistakes in Deep Neural Network\n(DNNs) based vision systems e.g., for classification, segmentation and object\ndetection. The vulnerability of DNNs against such attacks can prove a major\nroadblock towards their real-world deployment. Transferability of adversarial\nexamples demand generalizable defenses that can provide cross-task protection.\nAdversarial training that enhances robustness by modifying target model's\nparameters lacks such generalizability. On the other hand, different input\nprocessing based defenses fall short in the face of continuously evolving\nattacks. In this paper, we take the first step to combine the benefits of both\napproaches and propose a self-supervised adversarial training mechanism in the\ninput space. By design, our defense is a generalizable approach and provides\nsignificant robustness against the \\textbf{unseen} adversarial attacks (\\eg by\nreducing the success rate of translation-invariant \\textbf{ensemble} attack\nfrom 82.6\\% to 31.9\\% in comparison to previous state-of-the-art). It can be\ndeployed as a plug-and-play solution to protect a variety of vision systems, as\nwe demonstrate for the case of classification, segmentation and detection. Code\nis available at: {\\small\\url{https://github.com/Muzammal-Naseer/NRP}}.",
+                    "authors": "Muzammal Naseer, Salman Khan, Munawar Hayat, Fahad Shahbaz Khan, Fatih Porikli",
+                    "published": "2020-06-08",
+                    "updated": "2020-06-08",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Adversarial training (AT) has shown great potential to safeguard neural networks from adversarial attacks [33, 40]. So far in literature, AT is performed in the model space i.e., a model\u2019s parameters are modi\ufb01ed by minimizing empirical risk for a given data distribution as well as the perturbed images. Such AT strategy results in the following challenges. (a) Task dependency: AT is task-dependent e.g. robust classi\ufb01cation models cannot directly be incorporated into an object detection or a segmentation pipeline, since the overall system would still require further training Purifier  Network Until Converge Perturbed Image Clean Image Purified Image Perceptual  Feature Space Maximize  Feature  Distortion \u2206G Minimize  Feature  Distortion \u2206H \u2206G SSP \u2206H Figure 1: Our main idea is to train a Puri\ufb01er Network in a selfsupervised manner. We generate perturbed images using our proposed Self-supervised Perturbation (SSP) attack that disrupts the deep perceptual features. The Puri\ufb01er Network projects back the perturbed images close to the perceptual space of clean images. This creates a training loop independent of the task or label space. with modi\ufb01ed task-dependant loss functions. (b) Computational cost: AT is computationally expensive [33] which restricts its applicability to high-dimensional and large-scale datasets such as ImageNet [39]. (c) Accuracy drop: models trained with AT lose signi\ufb01cant accuracy on the original distribution e.g. ResNet50 [19] accuracy on ImageNet validation set drops from 76% to 64% when robusti\ufb01ed against PGD attack [33] at a perturbation budget of only \u03f5 \u22642 (i.e. maximum change in each pixel can be 2/255). (d) Label leakage: supervised AT suffers from label leakage [26] which allows the model to over\ufb01t on perturbations thus affecting model generalization to unseen adversaries [56]. In comparison to AT, input processing methods [16, 50] for adversarial defense are scalable and can work across different tasks. However, they have been broken in white-box 1 arXiv:2006.04924v1  [cs.CV]  8 Jun 2020 \fsettings [2] and shown to be least effective in black-box settings. For example, [10] successfully transfer their attack against multiple input processing based defenses even when the backbone architecture is adversarially trained using [48]. Furthermore, input transformations (e.g., Gaussian smoothing and JPEG compression) can maximize the attack strength instead of minimizing it [37, 10]. Motivated by the complementary strengths of AT and input processing methods, we propose a self-supervised AT mechanism in the input space. Our approach (Fig. 1) uses a min-max (saddle point) formulation to learn an optimal input processing function that enhances model robustness. In this way, our optimization rule implicitly performs AT. The main advantage of our approach is its generalization ability, once trained on a dataset, it can be applied offthe-shelf to safeguard a completely different model. This makes it a more attractive solution compared to popular AT approaches that are computationally expensive (and thus less scalable to large-scale datasets). Furthermore, in comparison to previous pre-processing based defenses that are found to be vulnerable towards recent attacks, our defense demonstrates better robustness. Our main contributions are: \u2022 Task Generalizability: To ensure a task independent AT mechanism, we propose to adversarially train a purifying model named Neural Representation Puri\ufb01er (NRP). Once trained, NRP can be deployed to safeguard across different tasks, e.g., classi\ufb01cation, detection and segmentation, without any additional training (Sec. 3). \u2022 Self-Supervision: The supervisory signal used for AT should be self-supervised to make it independent of label space. To this end, we propose an algorithm to train NRP on adversaries found in the feature space in random directions to avoid any label leakage (Sec. 3.1). \u2022 Defense against strong perturbations: Attacks are continuously evolving. In order for NRP to generalize, it should be trained on worst-case perturbations that are transferable across different tasks. We propose to \ufb01nd highly transferable perceptual adversaries (Sec. 4.3). \u2022 Maintaining Accuracy: A strong defense must concurrently maintain accuracy on the original data distribution. We propose to train the NRP with an additional discriminator to bring adversarial examples close to original samples by recovering the \ufb01ne texture details (Sec. 4.2). 2. Related Work Defenses: A major class of adversarial defenses processes the input images to achieve robustness against adversarial patterns. For example, [16] used JPEG compression to remove high-frequency components that are less important to human vision using discrete cosine transform. A compressed sensing approach called Total Variation Minimization (TVM) was proposed in [16] to remove the small localized changes caused by adversarial perturbations. Xie et al. [51] introduced the process of Random Resizing and Padding (R&P) as a pre-processing step to mitigate the adversarial effect. A High-level representation Guided Denoiser (HGD) [29] framework was used as a pre-processing step to remove perturbations. NeurIPS 2017 Defense Competition Rank-3 (NeurIPS-r3) approach [47] introduced a two step prep-processing pipeline where the images \ufb01rst undergo a series of transformations (JPEG, rotation, zoom, shift and sheer) and then passed through an ensemble of adversarially trained models to obtain the weighted output response as a prediction. [41] proposed to recover adversaries using GAN and [35] super-resolve images to minimize adversarial effect. As compared to the above defenses, we design an input processing model that derives a selfsupervised signal from the deep feature space to adversarially train the defense model. Our results show signi\ufb01cantly superior performance to all so-far developed input processing based defenses. Attacks: The self-supervised perturbation signal obtained to adversarially train our proposed approach can also be used as an adversarial attack. Since the seminal work of Szegedy et al. [46], many adversarial attack algorithms [14, 15, 3, 9] have been proposed to show the vulnerability of neural networks against imperceptible changes to inputs. A single-step attack, called Fast Gradient Sign Method (FGSM), was proposed in [14]. In a follow-up work, Kurakin et al. [15] proposed a robust multi-step attack, called Iterative Fast Gradient Sign Method (I-FGSM) that iteratively searches the loss surface of a network under a given metric norm. To improve transferability, a variant of I-FGSM, called momentum iterative fast gradient sign method (MI-FGSM), was introduced [9], which signi\ufb01cantly enhanced the transferability of untargeted attacks on ImageNet dataset [39] under a l\u221enorm budget. More recently, [53] proposed a data augmentation technique named input diversity method (DIM) to further boost the transferability of these attack methods. In contrast to our selfsupervised attack approach, all of these methods are supervised adversarial attacks that rely on cross-entropy loss to \ufb01nd the deceptive gradient direction. 3. Neural Representation Puri\ufb01er Our defense aims to combine the bene\ufb01ts of adversarial training and input processing methods in a single framework that is computationally ef\ufb01cient, generalizable across different tasks and retains the clean image accuracy. The basic intuition behind our defense mechanism is to effectively use information contained in the feature space of deep networks to obtain an automatic supervisory signal. To this end, we design a Neural Representation Puri\ufb01er (NRP) model that learns to clean adversarially perturbed images based on the automatically derived (self) supervision. The objective is to recover the original benign image x \fFigure 2: Neural Representation Puri\ufb01er. Using a self-supervision signal, the proposed defense learns to purify perturbed images, such that their corresponding perceptual representation in deep feature space becomes close to clean natural images. given an input adversarial image x\u2032. We wish to remove the adversarial patterns by training a neural network P\u03b8 parameterized by \u03b8, which we refer as the puri\ufb01er network. The main objective is to be independent of the task-speci\ufb01c objective function, such that once trained, the proposed defense is transferable to other models (even across tasks). Towards this end, the network P\u03b8 is trained in an adversarial manner by playing a game with the critic network C\u03c6, and a feature extractor F\u03c8 (see Fig. 2). The function of the puri\ufb01er and critic networks is similar to generator and discriminator in a traditional Generative Adversarial Network (GAN) framework, with the key difference that in our case, P\u03b8 performs image restoration instead of image generation. The feature extractor, F\u03c8, is pretrained on ImageNet and remains \ufb01xed, while the other two networks are optimized during training. Adversarial examples x\u2032 are created by maximizing the F\u03c8\u2019s response in random directions de\ufb01ned by a distance measure (Algorithm 1), while at minimization step, P\u03b8 tries to recover the original sample x by minimizing the same distance (Algorithm 2). 3.1. Self-Supervision The automatic supervision signal to train NRP defense is obtained via a loss-agnostic attack approach. Below, we \ufb01rst outline why such a Self-Supervised Perturbation (SSP) is needed and then describe our approach. Motivation: Strong white-box attacks [15, 6], that are generally used for AT, consider already-known network parameters \u03b8 and perturb the inputs to create x\u2032, such that they are misclassi\ufb01ed by the target model, i.e. T (x\u2032; \u03b8) \u0338= y. Since the perturbations are calculated using gradient directions speci\ufb01c to \u03b8, the resulting perturbed images x\u2032 do not generalize well to other networks [9, 43, 9, 53, 58]. This dependency limits these attacks to a speci\ufb01c network and task. In contrast, our goal is to design a self-supervised perturbation mechanism that can generalize across networks and tasks, thus enabling a transferable defense approach. 2 3 4 5 6 7 8 9 10 Number of iterations 0 10 20 30 40 50 60 I-FGSM: Fooling Rate I-FGSM: Feature Distortion MI-FGSM: Fooling Rate  MI-FGSM: Feature Distortion Figure 3: Fooling rate of Inc-v4 and average feature distortion is shown for adversaries generated on Inc-v3 (black-box setting) by I-FGSM and MI-FGSM. As the number of iterations increases, fooling rate of I-FGSM decreases along with its feature distortion while MI-FGSM maintains its distortion as iterations increase. The self-supervised perturbation is based on the concept of \u2018feature distortion\u2019, introduced next. Feature Distortion: Given a clean image x and its perturbed counterpart x\u2032 that is crafted to fool the target model T (\u00b7), the feature distortion refers to the change that x\u2032 causes to the internal representations of a neural network F(\u00b7) relative to x. This can be represented by, \u2206(x, x\u2032) = d \u0000F(x; \u03b8)|n, F(x\u2032; \u03b8)|n \u0001 , (1) where, F(x; \u03b8)|n denotes the internal representation obtained from the nth layer of a pretrained deep network F(\u00b7) and d(\u00b7) is a distance metric which can be \u2113p [14], Wasserstein distance [1] or cosine similarity between the features of the original and perturbed sample. The reason why we base our self-supervised perturbation on feature distortion is its direct impact on the perturbation transferability. To show this, we conduct a proofof-concept experiment by generating adversarial examples \fAlgorithm 1 SSP: Self-Supervised Perturbation Require: A feature extractor F\u03c8, batch of clean samples x, input transformation R, perturbation budget \u03f5, step-size \u03ba, and number of iterations T. Ensure: Perturbed sample x\u2032 with \u2225x\u2032 \u2212x\u2225\u221e\u2264\u03f5. 1: g0 = 0; x\u2032 = R(x); 2: for t = 1 to T do 3: Forward pass x\u2032 t to F\u03c8 and compute \u2206using Eq. 1; 4: Compute gradients gt = \u2207x \u2206(xt, x\u2032); 5: Generate adversaries using; x\u2032 t+1 = x\u2032 t + \u03ba \u00b7 sign(gt); (2) 6: Project adversaries in the vicinity of x x\u2032 t+1 = clip(x\u2032 t+1, x \u2212\u03f5, x + \u03f5); (3) 7: end for 8: return x\u2032 = x\u2032 T . on ImageNet-NeurIPS [7]. We consider two popular attack methods, MI-FGSM [9] and I-FGSM [15], among which MI-FGSM has higher transferability compared to IFGSM. Interestingly, feature distortion strength of I-FGSM decreases as the number of attack iterations increases, compared to MI-FGSM (Fig. 3). MI-FGSM maintains its perturbation strength with increasing number of iterations. This indicates that feature distortion has a direct impact on transferability and therefore maximizing the objective in Eq. 1 (signifying feature-space distortion) can boost the transferability of adversarial examples without using any decision boundary information. Based on this observation, our proposed perturbation generation approach directly maximizes the distortion in deep feature space to create strong, highly generalizable and task-independent adversarial examples. Self-supervised Perturbation: Conventional black-box attacks operate in the logit-space of deep networks. The objective of \u2018logit-based\u2019 adversarial attacks is to change the target model\u2019s prediction for a clean image T (x) \u0338= T (x\u2032) such that x\u2032 is bounded: \u2225x \u2212x\u2032\u2225\u2264\u03f5. In contrast to these methods, we propose to \ufb01nd adversaries by maximizing the feature loss (Sec. 3.2) of neural networks. Our approach does not rely on decision-boundary information since our \u2018representation-based\u2019 attack directly perturbs the feature space by solving the following optimization problem: max x\u2032 \u2206(x, x\u2032) subject to: \u2225x \u2212x\u2032\u2225\u221e\u2264\u03f5, (4) Our proposed method to maximize feature distortion for a given input sample is summarized in Algorithm 1. We apply a transformation R to input x at the \ufb01rst iteration (Algorithm 1) to create a neural representation difference between an adversarial and benign example and then maximize the difference within a given perturbation budget. There can be different choices for R but in this work, R simply adds random noise to the input sample, i.e. our algorithm takes a random step at the \ufb01rst iteration. Algorithm 2 NRP: Neural Representation Puri\ufb01cation via Self-Supervised Adversarial Training Require: Training data D, Puri\ufb01er P\u03b8, feature extractor F\u03c8, critic network C\u03c6, perturbation budget \u03f5 and loss criteria L. Ensure: Randomly initialize P\u03b8 and C\u03c6. 1: repeat 2: Sample mini-batch of data, x, from the training set. 3: Find adversaries, x\u2032, at a given perturbation budget \u03f5 by maximizing distance, \u2206(Eq. 1), using Algorithm 1. 4: Forward-pass x\u2032 through P\u03b8 and calculate LP\u03b8 (Eq. 8). 5: Back-pass and update \u03b8 to minimize LP\u03b8 (Eq. 8). 6: Update C\u03c6 to classify x from P\u03b8(x\u2032). 7: until P\u03b8 converges. 3.2. NRP Loss functions We propose a hybrid loss function that is used to train the puri\ufb01er network (see Algorithm 2). This loss function consists of three terms that we explain below: Feature loss: The Self-supervised Perturbation (SSP) generated by Algorithm 1 is the direct result of increasing the feature loss function, \u2206, de\ufb01ned on the feature extractor F\u03c8. In order to learn the puri\ufb01er network, we must decrease this distance as follows: Lfeat = \u2206 \u0000F\u03c8(x), F\u03c8(P\u03b8(x\u2032)) \u0001 , (5) where, \u2206is formally de\ufb01ned in Eq. 1, and the distance measure used to compute \u2206is the mean absolute error (MAE). We empirically observe that removing Lfeat loss leads to a network that does not converge to a meaningful state and produces weaker defense (see Fig. 5). Pixel loss: Smoothing images can help in mitigating the adversarial effect since the perturbation patterns resemble to that of noise. Therefore, in order to encourage smoothness, we apply l2 loss in the image pixel space, Limg = \u2225P\u03b8(x\u2032) \u2212x\u22252. (6) Adversarial loss: Instead of using vanilla GAN objective, we use relativistic average GAN which has shown better convergence properties [23, 37]. For a given batch of original, x, and adversarial examples, x\u2032, the relativistic loss for the puri\ufb01er network P\u03b8 is given as: Ladv = \u2212log \u0000\u03c3 \u0000C\u03c6(P\u03b8(x\u2032)) \u2212C\u03c6(x) \u0001 \u0001 , (7) where \u03c3 represents the sigmoid layer. The overall loss objective for P\u03b8 is the combination of losses de\ufb01ned on pixel and feature spaces as well as the relativistic loss: LP\u03b8 = \u03b1 \u00b7 Ladv | {z } Adversarial loss + \u03b3 \u00b7 Limg | {z } P ixel loss + \u03bb \u00b7 Lfeat | {z } F eature loss . (8) The pixel and feature losses focus on restoring image content and style, while adversarial loss restores texture details. \fBlock3-Conv3 Fire8-Conv3 Before-Classifier 30 40 50 60 70 80 90 Fooling Rate Model-wise Comparison VGG16 SqueezeNet AlexNet Block2-Conv2 Block3-Conv3 Block4-Conv3 Block5-Conv3 30 40 50 60 70 80 90 Fooling Rate Layer-wise Comparison: VGG16 Figure 4: Fooling rate for Inc-v3 [26] on ImageNet-NeurIPS dataset. Adversaries are created by applying SSP (Algorithm 1) at different layers and best results for each model is selected. Perceptual adversaries found in VGG space has the highest transferability (further analysis is in supplementary material). 3.3. NRP Architecture Here, we outline the architecture of generator, feature extractor and discriminator blocks. Generator (P\u03b8): Our generator architecture is inspired by [27, 49]. It consists of a convolution layer followed by multiple \u201cbasic blocks\u201d. Each basic block is composed of 3 \u201cdense blocks\u201d and each dense block contains \ufb01ve convolutional layers followed by leaky-relu [54] and \ufb01nally a convolutional layer that has output with same dimension as input. Generally, adding a skip connection from input to generator\u2019s output helps in restoration tasks e.g., image super resolution [27] and deblurring [25]. However, in our case an important design criteria is to avoid such skip connection since our objective is to remove adversarial noise and a direct skip connection can potentially reintroduce harmful noise patterns. Feature Extractor (F\u03c8): It is a VGG [42] network pretrained on ImageNet. During training, F\u03c8 remains \ufb01xed while its response is maximized in random directions (adversary generation process) and minimized (puri\ufb01cation process) using a prede\ufb01ned distance metric. In our experiments, we demonstrate the effectiveness of VGG space for creating strong adversaries as compared to other deep architectures. Discriminator (C\u03c6): Our discriminator architecture is also based on VGG network [42]. It consists of \ufb01ve convolutional blocks containing convolutional layers followed by batch-norm and leaky-relu and then a fully connected layer. 3.4. On Suitable Perceptual Adversaries The intuition to train NRP on boundary-agnostic perceptual adversaries is based on the extensive study [57] that found correlation of deep features with human perception. Speci\ufb01cally, [57] compares three models i.e. VGG [42], AlexNet [24] and SqueezeNet [21]. Following [57], we study these models from adversarial perspective by applying feature distortion at different layers in Fig. 4. Our \ufb01ndings are as follows: (a) VGG\u2019s perceptual adversaries are more transferable than AlexNet and SqueezeNet (a detailed transferability analysis on seen/unseen perturbations of VGG is in supplementary material), (b) under same feature distortion settings, adversaries found at different layers are not equally transferable e.g. conv3.3 (block 3, layer 3) features offer better adversarial transferability than the rest of the network. We believe this is because the initial VGG layers learn low-level features while the deeper ones become too speci\ufb01c to the label space. Further, we found that increasing the representation loss at multiple network layers does not notably increase attack success rate and adds a signi\ufb01cant computational overhead. Since NRP training process is agnostic to the label-space of the source model i.e., it neither depends on a particular task-speci\ufb01c loss function (e.g., cross entropy) nor on the ground-truth labels, this makes it a generic algorithm, which can defend a totally unseen model. Furthermore, we demonstrate that perturbations discovered with our SSP approach offer high transferability across models trained on different datasets and tasks. 4. Experiments 4.1. Training Details Training is done on randomly selected 25k images from MS-COCO data set. These images are resized to 480 \u00d7 480 \u00d7 3. Adversaries created using SSP are fed as inputs to NRP with their corresponding clean images used as target labels. During training, we randomly crop images of 128 \u00d7 128 \u00d7 3. Batch size is set to 16 and training is done on four Tesla v100 GPUs. Learning rates for generator and discriminator are set to 10\u22124, with the value of \u03b1 = 5 \u00d7 10\u22123, \u03b3 = 1 \u00d7 10\u22122 and \u03bb = 1. We study eight models trained on the ImageNet [39]. Five of these models are naturally trained. These include Inceptionv3 (Inc-v3) [45], Inceptionv4 (Inc-v4), Inception Resnet v2 (IncRes-v2) [44], Resnet v2-152 (Res-152) [20] and VGG-19 [42]. The other three models including Adv-v3 [26], Inc-v3ens3 and IncRes-v2ens [48] are adversarially trained. The speci\ufb01c details about these models can be found in [26, 48]. 4.2. Defense Results and Insights (a) Generalizability Across Attacks: Figs. 6, 7 & 8 demonstrate generalization ability of NRP to recover images from strong adversarial noise. Quantitative analysis in Table 1 shows that compared to previously broken defenses [10], NRP achieves strong robustness against stateof-the-art attacks [53, 10], bringing down the effectiveness of the ensemble translation-invariant attack with input diversity (DIMT I) [10] from 79.8% to 31.9%. (b) NRP as Cross-task Defense: In order to measure the cross-task defense capabilities, we deploy NRP against cross-domain attack (CDA) [37], a state-of-the-art attack that generates diverse cross-domain adversarial perturbations. Results in Table 2 demonstrate that NRP successfully removes all unseen perturbations and proves a generic cross-task defense for classi\ufb01cation, object detection and in\fTable 1: Robustness of different defense methods against stateof-the-art black-box attacks (lower is better). IncRes-v2ens is used as backbone model following [10]. NRP signi\ufb01cantly reduces the attack success rate. Adversaries (\u03f5 \u226416) are created against Incv3, Inc-v4, IncRes-v2, Res-v2-152 and Ensemble. Defenses Attacks FGSM FGSMT I MIFGSM MIFGSMT I DIM DIMT I Inc-v3 JPEG [17] 19.9 25.5 20.3 28.2 30.7 37.0 TVM [17] 18.8 30.7 19.4 34.9 24.4 44.2 NIPS-r3 [47] 9.8 24.5 12.9 26.7 18.0 41.4 R&P [50] 6.5 19.8 8.7 23.9 13.3 36.8 HGD [28] 2.1 18.4 6.9 25.7 9.7 38.3 APE-GAN [41] 19.6 28.0 17.9 30.4 23.6 38.6 SR [35] 23.0 36.7 23.6 38.3 32.5 49.0 NRP 3.2 4.8 4.5 9.1 5.1 11.0 Inc-v4 JPEG [17] 21.8 27.9 26.0 31.6 38.6 43.5 TVM [17] 19.9 31.8 24.8 38.4 29.1 45.6 NIPS-r3 [47] 11.5 24.6 15.6 29.5 14.1 41.9 R&P [50] 7.9 21.6 12.1 28.0 17.2 39.3 HGD [28] 2.6 18.1 9.6 27.8 32.4 58.7 APE-GAN [41] 21.1 28.8 20.7 32.8 25.0 39.0 SR [35] 25.3 34.1 29.2 42.3 39.3 52.3 NRP 3.1 4.4 4.8 10.3 5.2 12.5 IncRes-v2 JPEG [17] 24.7 32.4 31.6 45.9 47.2 55.7 TVM [17] 23.4 38.5 34.4 55.4 41.7 66.2 NIPS-r3 [47] 13.3 31.4 22.7 46.2 37.6 61.5 R&P [50] 9.9 28.1 18.6 45.2 30.2 61.4 HGD [28] 3.9 25.4 19.6 45.1 32.4 58.7 APE-GAN [41] 24.7 36.8 30.4 50.5 36.3 60.5 SR [35] 27.6 42.4 42.6 62.1 54.3 72.2 NRP 3.5 6.9 7.6 18.7 7.5 20.8 Res-v2-152 JPEG [17] 24.0 32.7 31.2 38.3 42.4 50.8 TVM [17] 22.0 38.1 24.5 41.2 36.8 55.7 NIPS-r3 [47] 12.5 30.1 18.0 34.4 34.4 52.9 R&P [50] 8.6 27.4 14.6 31.1 26.4 50.4 HGD [28] 3.6 24.4 15.1 31.8 32.6 51.8 APE-GAN [41] 24.3 37.1 23.2 38.6 34.3 53.8 SR [35] 26.3 41.8 30.2 49.2 48.4 63.9 NRP 3.4 6.5 5.8 11.9 6.3 17.8 Ensemble JPEG [17] 38.1 43.3 67.7 77.2 82.5 83.4 TVM [17] 30.0 39.8 50.1 72.1 64.1 79.8 NIPS-r3 [47] 19.8 33.9 43.9 71.4 63.7 83.1 R&P [50] 13.8 31.2 32.8 68.3 51.7 81.4 HGD [28] 4.9 29.9 38.6 73.3 57.7 82.6 APE-GAN [41] 32.0 42.1 44.6 69.3 59.6 74.5 SR [35] 38.1 45.8 65.2 79.9 79.3 84.9 NRP 3.7 7.9 10.1 27.8 11.4 31.9 stance level segmentation against CDA. (c) Ablation: Fig. 5 thoroughly investigates the impact of different training mechanisms in combination with our defense, and provides the following insights: (i) Relativistic GAN loss offers a more robust solution than vanilla GAN, (ii) NRP performance decreases slightly without pixel loss, (iii) NRP without feature loss loses supervisory signal de\ufb01ned by perceptual-space boundary, hence the generator Table 2: NRP generalizability across different adversarial attacks. Classi\ufb01cation model is defended against CDA trained against Incv3 while detection and segmentation models are defended against CDA trained against Res-v2-152 (higher is better). (q=quantity, w=weights, win=window size) Classi\ufb01cation: Defending IncRes-v2ens [48] against CDA [37] Method No ImageNet Comics Paintings Attack l\u221e\u22648 l\u221e\u226416 l\u221e\u22648 l\u221e\u226416 l\u221e\u22648 l\u221e\u226416 No Defense 97.8 83.0 30.9 94.0 56.6 71.6 23.7 JPEG (q=75) 97.6 74.9 18.6 90.1 42.6 68.0 18.0 JPEG (q=50) 96.2 74.2 19.0 90.1 43.4 66.0 19.2 JPEG (q=20) 94.1 73.4 21.7 87.0 51.3 62.7 18.8 TVM (w=10) 93.1 82.3 30.2 91.0 77.2 72.7 27.4 TVM (w=30) 96.0 81.1 27.3 93.4 66.4 70.6 24.1 MF (win=3) 95.4 77.3 27.7 92.4 66.8 65.0 22.1 NRP 95.6 95.7 96.0 95.4 94.2 95.3 94.1 Detection: Defending Mask-RCNN [18] against CDA [37] No Defense 59.9 35.2 8.1 40.5 16.8 41.7 14.8 JPEG (q=75) 57.6 41.3 11.9 41.6 19.4 44.5 18.3 JPEG (q=50) 54.6 41.7 14.5 39.5 18.5 47.7 19.9 JPEG (q=20) 39.7 30.7 15.1 28.2 14.7 30.5 15.3 TVM (w=10) 54.1 32.1 14.3 40.5 28.9 37.6 21.5 TVM (w=30) 58.0 39.9 10.1 46.8 21.0 45.4 17.2 MF (win=3) 54.7 32.1 9.0 41.1 20.4 37.6 15.2 NRP 54.4 51.5 50.3 53.5 53.7 53.2 54.3 Segmentation: Mask-RCNN [18] defense against CDA [37] No Defense 56.8 32.4 7.3 37.6 15.5 39.1 13.8 JPEG (q=75) 54.4 38.5 11 38.5 17.8 41.7 16.9 JPEG (q=50) 51.5 38.9 13.4 36.6 17.3 40 18.2 JPEG (q=20) 37.1 28.8 14.0 26.3 13.8 28.3 14.3 TVM (w=10) 50.8 29.8 13.2 37.6 26.6 34.9 19.8 TVM (w=30) 54.4 37.1 9.3 43.7 19.3 42.3 15.9 MF (win=3) 51.5 29.8 8.3 36.0 18.8 34.9 13.9 NRP 51.3 48.4 47.3 50.3 50.8 50.2 51.4 Clean FGSMTI MIFGSMTI DIMTI 40 50 60 70 80 90 100 Accuray NRP (Proposed) NRP without Pixel Loss NRP with GAN Loss FGSP GNP NRP without Feature Loss Figure 5: Ablation. Proposed NRP is able to recover input samples from the strong black-box ensemble attack [10] as compared to GNP and FGSP. NRP trained without Lfeat performs poorly indicating the importance of perceptual loss. Top-1 accuracy (higher is better) is reported for IncRes-v2ens [48] on ImageNet-NeurIPS. does not converge to a meaningful state, (iv) Gaussian smoothing (Gaussian noise data augmentation) proves to be useful in reducing adversarial vulnerability of classi\ufb01er [8, 55]. Training NRP as a Gaussian denoiser, named Gaus\fTable 3: Success rate (lower is better) of BPDA [6] and DIMT I [10] attacks against NRP. Res-v2-152 [20] is combined with other puri\ufb01er networks (ResG [27], UNet [38]). Adversaries are then transferred to the naturally and adversarially trained models. NRP protects the backbone network even when the attacker tries to bypass using BPDA technique. (attack iterations: 10, \u03f5 \u226416) Source Attack NRP Inc-v3 Inc-v4 IncRes-v2 Adv-v3 Inc-v3ens3 IncRes-v2ens Res-v2-152 DIMT I \u0017 77.4 77.9 74.2 51.2 56.2 47.7 ResG \u2295Res-v2-152 DIMT I \u2295BPDA \u0013 29.7 26.2 19.6 22.3 22.1 16.1 UNet \u2295Res-v2-152 DIMT I \u2295BPDA \u0013 29.0 27.1 19.5 26.9 27.7 18.8 Afghan Hound (0.73, \u0017) Porcupine (0.64, \u0017) Erythrocebus Patas (0.53, \u0017) Guenon Monkey (0.77, \u0017) Crane (0.55, \u0017) Monarch Butter\ufb02y (0.65, \u0013) Dung Beetle (0.90, \u0013) Lycaenid (0.94, \u0013) Lorikeet (0.94, \u0013) Flamingo (0.90, \u0013) Figure 6: A visual illustration of NRP generalizability to different adversaries (\u03f5 \u226416) (top: attacked; bottom: puri\ufb01ed). Our method can clean challenging adversarial patterns resulting from SSP applied to adversarially robust model [12]. Previous denoising methods are not designed for this type of structured noise. IncRes-v2ens backbone is used here. (see supplementary material for more examples) sian Noise Puri\ufb01er (GNP) does not prove effective against translation-invariant attacks [10], and (v) Training NRP to stabilize FGSM adversaries (termed FGSP in Fig. 5) performs relatively better than GNP. (d) What if Attacker knows about the Defense: We study this dif\ufb01cult scenario with the following criteria: (i) attacker knows that the defense is deployed and has access to its training data and training mechanism, and (ii) attacker trains a local defense similar to NRP, and then uses BPDA [6] to bypass the defense. To simulate this attack, we train residual generator (ResG) [27] and UNet [38] with the same training mechanise as described in Sec. 4.1. We then combine BPDA [2] with translation-invariant attack to bypass NRP. Under these challenging settings, NRP shows a relative gain of 74% and 66% respectively for IncRes-v2, IncRes-v2ens (see Table 3). 4.3. Self Supervised Perturbation as an Attack Next, we evaluate the strength of SSP as an attack for the tasks of classi\ufb01cation, detection and segmentation. Classi\ufb01cation: Table 5 compares SSP with FGSM [14], RFGSM [48], I-FGSM [15], MI-FGSM [9], TAP [58] and DIM [53] using their standard hyper-parameters (see supTable 4: Cross-task SSP Attack: Pixel-level accuracy is shown for SegNet-Basic [4] on Camvid testset [5], while mAP (with IoU = 0.5) is reported for Mask-RCNN. Problem Method No Attack SSP (l\u221e\u22648) SSP (l\u221e\u226416) Semantic Seg. SegNet [4] 79.70 52.48 32.59 Instance Seg. Mask-RCNN [18] 56.8 29.4 8.8 Object Det. RetinaNet [30] 53.78 22.75 5.16 Mask-RCNN [18] 59.50 31.8 9.7 plementary material). The results in Table 5 provide the following insights. (i) SSP consistently demonstrates a strong black-box adversarial transferability on both naturally and adversarially trained models, bringing down top-1 accuracy of IncRes-v2 [44] from 100.0% to 14.1%, (ii) While MIFGSM [9] and DIM [53] perform slightly better on adversarially trained ensemble models [48] in terms of top-1 accuracy, SSP shows comparable top-1 rate and surpasses in terms of top-5 accuracy, and (iii) These results indicate that decision-boundary based attacks \ufb02ip the label of input sample to the near-by class category, while SSP being agnostic to decision-level information pushes the adversaries far from the original input category. Cross-task Adversarial \fDIM [53]: Welsh Springer (0.52, \u0017) Puri\ufb01ed: Pomeranian (0.88, \u0013) DIMT I [10]: Cocker (0.71, \u0017) Puri\ufb01ed: Pomeranian (0.86, \u0013) Figure 7: NRP successfully recovers diverse patterns from strongest black-box attacks (l\u221e\u226416). IncRes-v2ens used as backbone. CDA [37]: Adversarial Prediction for Adversarial Puri\ufb01ed Prediction for Puri\ufb01ed Figure 8: NRP successfully removes perturbation generated by CDA[37] (\u03f5 \u226416) and stabilizes Mask-RCNN [18] predictions. Table 5: SSP as an attack for Classi\ufb01cation. Top-1 (T-1) and Top-5 (T-5) accuracies are reported under untargeted l\u221e adversarial attacks on ImageNet-NIPS with perturbation budget l\u221e\u226416. \u2018\u2217\u2019 indicates white-box attacks. Naturally Trained Adv. Trained Attack Inc-v3 Inc-v4 Res-152 IncRes-v2 VGG-19 Adv-v3 Inc-v3ens3 IncRes-v2ens T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 Res-152 FGSM [14] 55.1 81.1 62.6 85.1 18.9\u2217 44.7\u2217 65.0 86.5 43.9 70.4 64.6 85.8 76.9 93.5 87.9 98.2 R-FGSM [48] 60.8 84.3 68.4 88.1 14.6\u2217 40.3\u2217 71.9 90.3 55.8 71.4 74.8 92.3 81.1 96.0 87.1 97.5 I-FGSM [15] 80.9 96.7 85.3 97.8 0.9\u2217 10.8\u2217 93.1 98.8 75.9 94.8 89.2 99.2 90.5 97.9 94.6 99.5 MI-FGSM [9] 38.9 72.7 44.8 76.5 0.6\u2217 2.9\u2217 47.7 79.6 42.1 71.8 67.0 89.9 69.4 93.3 81.5 96.4 TAP [58] 48.2 55.7 7.6\u2217 55.2 49.2 57.8 64.1 DIM [53] 15.9 44.0 17.3 48.4 0.8\u2217 3.0\u2217 20.0 50.2 25.6 56.3 55.8 82.8 54.9 84.2 71.5 93.1 VGG16 FGSM [14] 32.6 58.6 38.4 62.6 38.5 66.3 44.5 68.5 8.8 25.1 51.7 75.3 54.9 81.7 70.8 90.7 R-FGSM [48] 44.4 69.5 47.6 75.1 51.1 78.8 56.4 78.8 11.2 31.8 65.5 87.4 66.7 89.2 77.5 93.6 I-FGSM [15] 69.2 93.0 75.2 93.7 79.0 96.2 85.6 96.8 14.4 49.3 83.5 97.7 83.9 96.7 92.1 98.8 MI-FGSM [9] 20.4 45.0 19.7 43.2 25.2 53.8 26.8 53.8 1.5 12.1 43.0 70.9 42.0 72.7 62.0 86.8 TAP [58] 23.9 28.1 23.9 32.3 38.8 41.9 63.8 DIM [53] 14.7 38.8 16.6 39.0 21.0 48.0 21.5 45.7 0.6 7.6 35.8 65.8 31.8 60.8 53.7 79.5 FFF [34] 61.7 80.7 60.8 78.7 72.8 90.1 76.1 90.1 44.0 68.0 79.6 93.1 83.1 93.1 92.8 98.5 SSP 5.3 11.0 5.9 11.9 16.5 29.5 14.1 25.5 2.7 6.8 25.9 43.2 40.2 58.3 58.0 75.0 Attack: Since SSP is loss-agnostic, it enables attacks on altogether different tasks. Table 4 explores SSP for object detection and image segmentation. For Segmentation, the self-supervised perturbations created on CAMVID [5] in VGG-16 feature space are able to bring down the per pixel accuracy of Segnet-Basic by 47.11% within l\u221e\u226416. For object detection, on MS-COCO validation set [31], mean Average Precision (mAP) with 0.5 intersection over union (IOU) of RetinaNet [30] and Mask-RCNN [18] drop from 53.78% to 5.16% and 59.5% to 9.7%, respectively, under l\u221e\u226416. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1905.11736v5",
+                    "title": "Cross-Domain Transferability of Adversarial Perturbations",
+                    "abstract": "Adversarial examples reveal the blind spots of deep neural networks (DNNs)\nand represent a major concern for security-critical applications. The\ntransferability of adversarial examples makes real-world attacks possible in\nblack-box settings, where the attacker is forbidden to access the internal\nparameters of the model. The underlying assumption in most adversary generation\nmethods, whether learning an instance-specific or an instance-agnostic\nperturbation, is the direct or indirect reliance on the original\ndomain-specific data distribution. In this work, for the first time, we\ndemonstrate the existence of domain-invariant adversaries, thereby showing\ncommon adversarial space among different datasets and models. To this end, we\npropose a framework capable of launching highly transferable attacks that\ncrafts adversarial patterns to mislead networks trained on wholly different\ndomains. For instance, an adversarial function learned on Paintings, Cartoons\nor Medical images can successfully perturb ImageNet samples to fool the\nclassifier, with success rates as high as $\\sim$99\\% ($\\ell_{\\infty} \\le 10$).\nThe core of our proposed adversarial function is a generative network that is\ntrained using a relativistic supervisory signal that enables domain-invariant\nperturbations. Our approach sets the new state-of-the-art for fooling rates,\nboth under the white-box and black-box scenarios. Furthermore, despite being an\ninstance-agnostic perturbation function, our attack outperforms the\nconventionally much stronger instance-specific attack methods.",
+                    "authors": "Muzammal Naseer, Salman H. Khan, Harris Khan, Fahad Shahbaz Khan, Fatih Porikli",
+                    "published": "2019-05-28",
+                    "updated": "2019-10-14",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Albeit displaying remarkable performance across a range of tasks, Deep Neural Networks (DNNs) are highly vulnerable to adversarial examples, which are carefully crafted examples generated by adding a certain degree of noise (a.k.a. perturbations) to the corresponding original images, typically appearing quasi-imperceptible to humans [1]. Importantly, these adversarial examples are transferable from one network to another, even when the other network fashions a different architecture and possibly trained on a different subset of training data [2, 3]. Transferability permits an adversarial attack, without knowing the internals of the target network, posing serious security concerns on the practical deployment of these models. Adversarial perturbations are either instance-speci\ufb01c or instance-agnostic. The instance-speci\ufb01c attacks iteratively optimize a perturbation pattern speci\ufb01c to an input sample (e.g., [4, 5, 6, 7, 8, 9, 10, 11]). In comparison, the instance-agnostic attacks learn a universal perturbation or a function that \ufb01nds adversarial patterns on a data distribution instead of a single sample. For example, [12] 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada. arXiv:1905.11736v5  [cs.CV]  14 Oct 2019 \fFigure 1: Transferable Generative Adversarial Perturbation: We demonstrate that common adversaries exist across different image domains and introduce a highly transferable attack approach that carefully crafts adversarial patterns to fool classi\ufb01ers trained on totally different domains. Our generative scheme learns to reconstruct adversaries on paintings or comics (left) that can successfully fool natural image classi\ufb01ers with high fooling rates at the inference time (right). proposed universal adversarial perturbations that can fool a model on the majority of the source dataset images. To reduce dependency on the input data samples, [13] maximizes layer activations of the source network while [14] extracts deluding perturbations using class impressions relying on the source label space. To enhance the transferability of instance-agnostic approaches, recent generative models attempt to directly craft perturbations using an adversarially trained function [15, 16]. We observe that most prior works on crafting adversarial attacks suffer from two pivotal limitations that restrict their transferability to real-world scenarios. (a) Existing attacks rely directly or indirectly on the source (training) data, which hampers their transferability to other domains. From a practical standpoint, source domain can be unknown, or the domain-speci\ufb01c data may be unavailable to the attacker. Therefore, a true \"black-box\" attack must be able to fool learned models across different target domains without ever being explicitly trained on those data domains. (b) Instance-agnostic attacks, compared with their counterparts, are far more scalable to large datasets as they avoid expensive per-instance iterative optimization. However, they demonstrate weaker transferability rates than the instance-speci\ufb01c attacks. Consequently, the design of highly transferable instance-agnostic attacks that also generalize across unseen domains is largely an unsolved problem. In this work, we introduce \u2018domain-agnostic\u2019 generation of adversarial examples, with the aim of relaxing the source data reliance assumption. In particular, we propose a \ufb02exible framework capable of launching vastly transferable adversarial attacks, e.g., perturbations found on paintings, comics or medical images are shown to trick natural image classi\ufb01ers trained on ImageNet dataset with high fooling rates. A distinguishing feature of our approach is the introduction of relativistic loss that explicitly enforces learning of domain-invariant adversarial patterns. Our attack algorithm is highly scalable to large-scale datasets since it learns a universal adversarial function that avoids expensive iterative optimization from instance-speci\ufb01c attacks. While enjoying the ef\ufb01cient inference time of instance-agnostic methods, our algorithm outperforms all existing attack methods (both instance-speci\ufb01c and agnostic) by a signi\ufb01cant margin (\u223c86.46% average increase in fooling rate from naturally trained Inception-v3 to adversarially trained models in comparison to state-of-the-art [10]) and sets the new state-of-the-art under both white-box and black-box settings. Figure 1 provides an overview of our approach. 2 Related Work Image-dependent Perturbations: Several approaches target creation of image-dependent perturbations. [17] noticed that despite exhibiting impressive performance, neural networks can be fooled through maliciously crafted perturbations that appear quasi-imperceptible to humans. Following this 2 \f\ufb01nding, many approaches [4, 5, 6, 7, 8, 9] investigate the existence of these perturbations. They either apply gradient ascent in the pixel space or solve complex optimizations. Recently, a few methods [18, 10] propose input or gradient transformation modules to improve the transferability of adversarial examples. A common characteristic of the aforementioned approaches is their data-dependence; the perturbations are computed for each data-point separately in a mutually exclusive way. Further, these approaches render inef\ufb01ciently at inference time since they iterate on the input multiple times. In contrast, we resort to a data-independent approach based on a generator, demonstrating improved inference-time ef\ufb01ciency along with high transferability rates. Universal Adversarial Perturbation: Seminal work of [12] introduces the existence of Universal Adversarial Perturbation (UAP). It is a single noise vector which when added to a data-point can fool a pretrained model. [12] crafts UAP in an iterative fashion utilizing target data-points that is capable of \ufb02ipping their labels. Though it can generate image-agnostic UAP, the success ratio of their attack is proportional to the number of training samples used for crafting UAP. [13] proposes a so-called data-independent algorithm by maximizing the product of mean activations at multiple layers given a universal perturbation as input. This method crafts a so-called data-independent perturbation, however, the attack success ratio is not comparable to [12]. Instead, we propose a fully distribution-agnostic approach that crafts adversarial examples directly from a learned generator, as opposed to \ufb01rst generating perturbations followed by their addition to images. Generator-oriented Perturbations: Another branch of attacks leverage generative models to craft adversaries. [15] learns a generator network to perturb images, however, the unbounded perturbation magnitude in their case might render perceptible perturbations at test time. [33] trains conditional generators to learn original data manifold and searches the latent space conditioned on the human recognizable target class that is mis-classi\ufb01ed by a target classier. [19] apply generative adversarial networks to craft visually realistic perturbations and build distilled network to perform black-box attack. Similarly, [16, 14] train generators to create adversaries to launch attacks; the former uses target data directly and the latter relies on class impressions. A common trait of prior work is that they either rely directly (or indirectly) upon the data distribution and/or entail access to its label space for creating adversarial examples (Table 1). In contrast, we propose a \ufb02exible, distribution-agnostic approach inculcating relativistic loss to craft adversarial examples that achieves state-of-the-art results both under white-box and black-box attack settings. Method Data Type Transfer Label Cross-domain Strength Agnostic Attack FFF [13] Pretrained-net/data Low \u0013 \u0017 AAA [14] Class Impressions Medium \u0017 \u0017 UAP [12] ImageNet Low \u0017 \u0017 GAP [16] ImageNet Medium \u0017 \u0017 RHP [11] ImageNet Medium \u0017 \u0017 Ours Arbitrary (Paintings, Comics, Medical scans etc.) High \u0013 \u0013 Table 1: A comparison of different attack methods based on their dependency on data distribution and labels. 3 Cross-Domain Transferable Perturbations Our proposed approach is based on a generative model that is trained using an adversarial mechanism. Assume we have an input image xs belonging to a source domain Xs \u2208Rs. We aim to train a universal function that learns to add a perturbation pattern \u03b4 on the source domain which can successfully fool a network trained on source Xs as well as any target domain Xt \u2282Rt when fed with perturbed inputs x\u2032 t = xt + \u03b4. Importantly, our training is only performed on the unlabelled source domain dataset with ns samples: {xi s}ns i=1 and the target domain is not used at all during training. For brevity, in the following discussion, we will only refer the input and perturbed images using x and x\u2032 respectively and the domain will be clear from the context. The proposed framework consists of a generator G\u03b8(x) and a discriminator D\u03c8(x) parameterized by \u03b8 and \u03c8. In our case, we initialize discriminator with a pretrained network and the parameters \u03c8 are remained \ufb01xed while the G\u03b8 is learned. The output of G\u03b8 is scaled to have a \ufb01xed norm and it lies within a bound; x\u2032 = clip \u0000min(x + \u03f5, max(G\u03b8(x), x \u2212\u03f5)) \u0001 . The perturbed images x\u2032 as well as 3 \fFigure 2: The proposed generative framework seeks to maximize the \u2018fooling gap\u2019 that helps in achieving very high transferability rates across domains. The orange dashed line shows the \ufb02ow of gradients, notably only the generator is tuned in the whole pipeline to fool the pretrained discriminator. the real images x are passed through the discriminator. The output of the discriminator denotes the class probabilities D\u03c8(x, x\u2032) \u2208[0, 1]c, where c is the number of classes. This is different from the traditional GAN framework where a discriminator only estimate whether an input is real or fake. For an adversarial attack, the goal is to fool a network on most examples by making minor changes to its inputs, i.e., \u2225\u03b4 \u2225\u221e\u2264\u03f5, s.t., P \u0000argmaxj(D\u03c8(x\u2032)j) \u0338= argmaxj(D\u03c8(x)j) \u0001 > fr, (1) where, fr is the fooling ratio, y is the ground-truth label for the example x and the predictions on clean images x are given by, y = argmaxj(D\u03c8(x)j). Note that we do not necessarily require the ground-truth labels of source domain images to craft a successful attack. In the case of adversarial attacks based on a traditional GAN framework, the following objective is maximized for the generator to achieve the maximal fooling rate: \u03b8\u2217\u2190argmax \u03b8 CROSSENTROPY(D\u03c8(x\u2032), 1y), (2) where 1y is the one-hot encoded label vector for an input example x. The above objective seeks to maximize the discriminator error on the perturbed images that are output from the generator network. We argue that the objective given by Eq. 2 does not directly enforce transferability for the generated perturbations \u03b4. This is primarily due to the reason that the discriminator\u2019s response for clean examples is totally ignored in the conventional generative attacks. Here, inspired by the generative adversarial network in [20], we propose a relativistic adversarial perturbation (RAP) generation approach that explicitly takes in to account the discriminator\u2019s predictions on clean images. Alongside reducing the classi\ufb01er\u2019s con\ufb01dence on perturbed images, the attack algorithm also forces the discriminator to maintain a high con\ufb01dence scores for the clean samples. The proposed relativistic objective is given by: \u03b8\u2217\u2190argmax \u03b8 CROSSENTROPY(D\u03c8(x\u2032) \u2212D\u03c8(x), 1y). (3) The cross entropy loss would be higher when the perturbed image is scored signi\ufb01cantly lower than the clean image response for the ground-truth class i.e., D\u03c8(x\u2032)y < < D\u03c8(x)y. The discriminator basically seeks to increase the \u2018fooling gap\u2019 (D\u03c8(x\u2032)y \u2212D\u03c8(x)y) between the true and perturbed samples. Through such relative discrimination, we not only report better transferability rates across networks trained on the same domain, but most importantly show excellent cross-domain transfer rates for the instance-agnostic perturbations. We attribute this behaviour to the fact that once a perturbation pattern is optimized using the proposed loss on a source distribution (e.g., paintings, cartoon images), the generator learns a \"contrastive\" signal that is agnostic to the underlying distribution. As a result, when the same perturbation pattern is applied to networks trained on totally different domain (e.g., natural images), it still achieves the state-of-the-art attack transferability rates. Table 2 shows the gain in transferability when using relativistic cross-entropy (Eq. 3) in comparison to simple cross-entropy loss (Eq. 2). For an untargeted attack, the above mentioned objective in Eq. 2 and 3 suf\ufb01ces, however, for a targeted adversarial attack, the prediction for the perturbed image must match a given target class y\u2032 i.e., argmaxj(D\u03c8(x\u2032)j) = y\u2032 \u0338= y. For such a case, we employ the following loss function: \u03b8\u2217\u2190argmin \u03b8 CROSSENTROPY(D\u03c8(x\u2032), 1y\u2032) + CROSSENTROPY(D\u03c8(x), 1y). (4) The overall training scheme for the generative network is given in Algorithm 1. 4 \fAlgorithm 1 Generator Training for Relativistic Adversarial Perturbations 1: A pretrained classi\ufb01er D\u03c8, arbitrary training data distribution X, perturbation budget \u03f5, loss criteria L. 2: Randomly initialize generator network G\u03b8 3: repeat 4: Sample mini-batch of data from the training set. 5: Use the current state of the generator, G\u03b8, to generate unbounded adversaries. 6: Project adversaries, G\u03b8(x), within a valid perturbation budget to obtain x\u2032 such that \u2225x\u2032 \u2212x\u2225\u221e\u2264\u03f5. 7: Forward pass x\u2032 to D\u03c8 and compute loss given in Eq. (3)/Eq. (4) for targeted/untargeted attack. 8: Backward pass and update the generator, G\u03b8, parameters to maximize the loss. 9: until model convergence. 1 2 3 4 5 6 7 8 9 10 Number of iterations 0 1 2 3 4 5 Loss Loss Trend Over Iterations 1 2 3 4 5 6 7 8 9 10 Number of iterations 0 1 2 3 4 5 Taxicab Norm Gradients Trend Over Iterations Figure 3: Loss and gradients trend for CE and RCE loss functions. Results are reported with VGG16 network on 100 random images for MI-FGSM attack. Trends are shown in log scale. 4 Gradient Perspective of Relativistic Cross-Entropy Adversarial perturbations are crafted via loss function gradients. An effective loss function helps in the generation of perturbations by back-propagating stronger gradients. Below, we show that Relativistic Cross-Entropy (RCE) ensures this requisite and thus leads to better performance than regular Cross-Entropy (CE) loss. Suppose, the logit-space outputs from the discriminator (pretrained classi\ufb01er) corresponding to a clean image (x) and a perturbed image (x\u2019) are denoted by a and a\u2032, respectively. Then, CE(a\u2032, y)=\u2212log \u0000ea\u2032 y/P k ea\u2032 k\u0001 is the cross-entropy loss for a perturbed input x\u2032. For clarity, we de\ufb01ne p\u2032 y = ea\u2032 y/P k ea\u2032 k. The derivative of p\u2032 y w.r.t a\u2032 i is \u2202p\u2032 y/\u2202a\u2032 i = p\u2032 y([ [i=y] ] \u2212p\u2032 i). Using chain rule, the derivative of cross-entropy loss is given by: \u2202CE \u2202a\u2032 i = p\u2032 i \u2212[ [i=y] ]. (5) For the relativistic loss formulated as RCE(a\u2032, a, y)=\u2212log \u0000ea\u2032 y\u2212ay/P k ea\u2032 k\u2212ak\u0001 , we de\ufb01ne ry= \u0000ea\u2032 y\u2212ay/P k ea\u2032 k\u2212ak\u0001 . The derivative of ry w.r.t a\u2032 i is \u2202ry/\u2202a\u2032 i = ri([ [i=y] ] \u2212ry). From chain rule, RCE derivative w.r.t to a\u2032 i is given by: \u2202RCE \u2202a\u2032 i = ri \u2212[ [i=y] ]. (6) In light of above relation, RCE has three important properties: 1. Comparing (Eq.5) with (Eq.6) shows that RCE gradient is a function of \u2018difference\u2019 (a\u2032 y\u2212ay) as opposed to only scores a\u2032 y in CE loss. Thus, it measures the relative change in prediction as an explicit objective during optimization. 2. RCE loss back-propagates larger gradients compared to CE, resulting in ef\ufb01cient training and stronger adversaries (see Figure 3 for empirical evidence). Sketch Proof: We can factorize the denominator in (Eq. 6) as follows: \u2202RCE/\u2202a\u2032 i = \u0000ea\u2032 y\u2212ay/(ea\u2032 y\u2212ay + P k\u0338=y ea\u2032 k\u2212ak) \u0001 \u2212 [ [i=y] ]. Consider the fact that maximization of RCE is only possible when e(a\u2032 y\u2212ay) decreases 5 \fand P k\u0338=y e(a\u2032 k\u2212ak) increases. Generally, ay \u226bak\u0338=y for the score generated by a pretrained model and a\u2032 y \u226aa\u2032 k\u0338=y (here k denotes an incorrectly predicted class). Thus, \u2202RCE/\u2202a\u2032 i > \u2202CE/\u2202a\u2032 i since e(a\u2032 y\u2212ay) < e(a\u2032 y) and P k\u0338=y e(a\u2032 k\u2212ak) > P k\u0338=y e(a\u2032 k). In simple words, the gradient strength of RCE is higher than CE. 3. In case x is misclassi\ufb01ed by F(\u00b7), the gradient strength of RCE is still higher than CE (here noise update with the CE loss will be weaker since adversary\u2019s goal is already achieved i.e., x is misclassi\ufb01ed). Loss VGG-16 VGG-19 Squeeze-v1.1 Dense-121 Cross Entropy (CE) 79.21 78.96 69.32 66.45 Relativistic CE 86.95 85.88 77.81 75.21 Table 2: Effect of Relativistic loss on transferability in terms of fooling rate (%) on ImageNet val-set. Generator is trained against ResNet-152 on Paintings dataset. 5 Experiments 5.1 Rules of the Game We report results using following three different attack settings in our experiments: (a) White-box. Attacker has access to the original model (both architecture and parameters) and the training data distribution. (b) Black-box. Attacker has access to a pretrained model on the same distribution but without any knowledge of the target architecture and target data distribution. (c) Cross-domain Black-box. Attacker has neither access to (any) pretrained model, nor to its label space and its training data distribution. It then has to seek a transferable adversarial function that is learned from a model pretrained on a possibly different distribution than the original. Hence, this setting is relatively far more challenging than the plain black-box setting. Perturbation Attack VGG-19 ResNet-50 Dense-121 Fool Rate (\u2191) Top-1 (\u2193) Fool Rate (\u2191) Top-1 (\u2193) Fool Rate (\u2191) Top-1 (\u2193) l\u221e\u226410 Gaussian Noise 23.59 64.65 18.06 70.74 17.05 70.30 Ours-Paintings 47.12 46.68 31.52 60.77 29.00 62.0 Ours-Comics 48.47 45.78 33.69 59.26 31.81 60.40 Ours-ChestX 40.81 50.11 22.00 67.72 20.53 67.63 l\u221e\u226416 Gaussian Noise 33.80 57.92 25.76 66.07 23.30 66.70 Ours-Paintings 66.52 30.21 47.51 47.62 44.50 49.76 Ours-Comics 67.75 29.25 51.78 43.91 50.37 45.17 Ours-ChestX 62.14 33.95 34.49 58.6 31.81 59.75 l\u221e\u226432 Gaussian Noise 61.07 35.48 47.21 48.40 39.90 54.37 Ours-Paintings 87.08 11.96 69.05 28.77 63.78 33.46 Ours-Comics 87.90 11.17 71.91 26.12 71.85 26.18 Ours-ChestX 88.12 10.92 62.17 34.85 59.49 36.98 Table 3: Cross-Domain Black-box: Untargeted attack success (%) in terms of fooling rate on ImageNet val-set. Adversarial generators are trained against ChexNet on Paintings, Comics and ChestX datasets. Perturbation budget, l\u221e\u226410/16/32, is chosen as per the standard practice. Even without the knowledge of targeted model, its label space and its training data distribution, the transferability rate is much higher than the Gaussian noise. 5.2 Experimental Settings Generator Architecture. We chose ResNet architecture introduced in [21] as the generator network G\u03b8; it consists of downsampling, residual and upsampling blocks. For training, we used Adam optimizer [22] with a learning rate of 1e-4 and values of exponential decay rate for \ufb01rst and second moments set to 0.5 and 0.999, respectively. Generators are learned against the four pretrained ImageNet models including VGG-16, VGG-19 [23], Inception (Inc-v3) [24], ResNet-152 [25] and ChexNet (which is a Dense-121 [26] network trained to diagnose pneumonia) [27]. Datasets. We consider the following datasets for generator training namely Paintings [28], Comics [29], ImageNet and a subset of ChestX-ray (ChestX) [27]. There are approximately 80k samples in Paintings, 50k in Comics, 1.2 million in ImageNet training set and 10k in ChestX. 6 \fBee Eater Cardoon Impala Anemone Fish Crane Jigsaw Puzzle Jigsaw Puzzle Jigsaw Puzzle Jigsaw Puzzle Jigsaw Puzzle Figure 4: Untargeted adversaries produced by generator trained against Inception-v3 on Paintings dataset. 1st row shows original images while 2nd row shows unrestricted outputs of adversarial generator and 3rd row are adversaries after valid projection. Perturbation budget is set to l\u221e\u226410. Figure 5: Illustration of attention shift. We use [31] to visualize attention maps of clean (1st row) and adversarial (2nd row) images. Adversarial images are obtained by training generator against VGG-16 on Paintings dataset. Inference: Inference is performed on ImageNet validation set (val-set) (50k samples), a subset (5k samples) of ImageNet proposed by [11] and ImageNet-NeurIPS [30] (1k samples) dataset. Evaluation Metrics: We use the fooling rate (percentage of input samples for which predicted label is \ufb02ipped after adding adversarial perturbations), top-1 accuracy and % increase in error rate (the difference between error rate of adversarial and clean images) to evaluate our proposed approach. 5.2.1 Results Table 3 shows the cross-domain black-box setting results, where attacker have no access to model architecture, parameters, its training distribution or label space. Note that ChestX [27] does not have much texture, an important feature to deceive ImageNet models [32], yet the transferability rate of perturbations learned against ChexNet is much better than the Gaussian noise. Tables 4 and 5 show the comparison of our method against different universal methods on both naturally and adversarially trained models [34] (Inc-v3, Inc-v4 and IncRes-v2). Our attack success rate is much higher both in white-box and black-box settings. Notably, for the case of adversarially trained models, Gaussian smoothing on top of our approach leads to signi\ufb01cant increase in transferability. We provide further comparison with GAP [16] in the supplementary material. Figures 4 and 5 show the model\u2019s output and attention shift on example adversaries. 7 \fModel Attack VGG-16 VGG-19 ResNet-152 VGG-16 FFF 47.10\u2217 41.98 27.82 AAA 71.59\u2217 65.64 45.33 UAP 78.30\u2217 73.10 63.40 Ours-Paintings 99.58\u2217 98.97 47.90 Ours-Comics 99.83\u2217 99.56 58.18 Ours-ImageNet 99.75\u2217 99.44 52.64 VGG-19 FFF 38.19 43.60\u2217 26.34 AAA 69.45 72.84\u2217 51.74 UAP 73.50 77.80\u2217 58.00 Ours-Paintings 98.90 99.61\u2217 40.98 Ours-Comics 99.29 99.76\u2217 42.61 Ours-ImageNet 99.19 99.80\u2217 53.02 ResNet-152 FFF 19.23 17.15 29.78\u2217 AAA 47.21 48.78 60.72\u2217 UAP 47.00 45.5 84.0\u2217 Ours-Paintings 86.95 85.88 98.03\u2217 Ours-Comics 88.94 88.84 94.18\u2217 Ours-ImageNet 95.40 93.26 99.02\u2217 Table 4: White & Black-box Setting: Fool rate (%) of untargeted attack on ImageNet val-set. Perturbation budget is l\u221e\u226410. * indicates white-box attack. Our attack\u2019s transferability from ResNet-152 to VGG16/19 is even higher than other white-box attacks. Model Attack Inc-v3ens3 Inc-v3ens4 IncRes-v2ens Inc-v3 UAP 1.00/7.82 1.80/5.60 1.88/5.60 GAP 5.48/33.3 4.14/29.4 3.76/22.5 RHP 32.5/60.8 31.6/58.7 24.6/57.0 Inc-v4 UAP 2.08/7.68 1.94/6.92 2.34/6.78 RHP 27.5/60.3 26.7/62.5 21.2/58.5 IncRes-v2 UAP 1.88/8.28 1.74/7.22 1.96/8.18 RHP 29.7/62.3 29.8/63.3 26.8/62.8 Ours-Paintings 33.92/72.46 38.94/71.4 33.24/69.66 Ours-gs-Paintings 47.78/73.06 48.18/72.68 42.86/73.3 Ours-Comics 21.06/67.5 24.1/68.72 12.82/54.72 Ours-gs-Comics 34.52/70.3 56.54/69.9 23.58/68.02 Ours-ImageNet 28.34/71.3 29.9/66.72 19.84/60.88 Ours-gs-ImageNet 41.06/71.96 42.68/71.58 37.4/72.86 Table 5: Black-box Setting: Transferability comparison in terms of % increase in error rate after attack. Results are reported on subset of ImageNet (5k) with perturbation budget of l\u221e\u226416/32. Our generators are trained against naturally trained Inc-v3 only. \u2018gs\u2019 represents Gaussian smoothing applied to generator output before projection that enhances our attack strength. 5.2.2 Comparison with State-of-the-Art Finally, we compare our method with recently proposed instance-speci\ufb01c attack method [10] that exhibits high transferability to adversarially trained models. For the very \ufb01rst time in literature, we showed that a universal function like ours can attain much higher transferability rate, outperforming the state-of-the-art instance-speci\ufb01c translation invariant method [10] by a large average absolute gain of 46.6% and 86.5% (in fooling rates) on both naturally and adversarially trained models, respectively, as reported in Table 6. The naturally trained models are Inception-v3 (Inc-v3) [24], Inception-v4 (Inc-v4), Inception Resnet v2 (IncRes-v2) [35] and Resnet v2-152 (Res-152) [36]). The adversarially trained models are from [34]. Attack Naturally Trained Adversarially Trained Inc-v3 Inc-v4 IncRes-v2 Res-152 Inc-v3ens3 Inc-v3ens4 IncRes-v2ens Inc-v3 FGSM 79.6\u2217 35.9 30.6 30.2 15.6 14.7 7.0 TI-FGSM 75.5\u2217 37.3 32.1 34.1 28.2 28.9 22.3 MI-FGSM 97.8\u2217 47.1 46.4 38.7 20.5 17.4 9.5 TI-MI-FGSM 97.9\u2217 52.4 47.9 41.1 35.8 35.1 25.8 DIM 98.3\u2217 73.8 67.8 58.4 24.2 24.3 13.0 TI-DIM 98.5\u2217 75.2 69.2 59.2 46.9 47.1 37.4 IncRes-v2 FGSM 44.3 36.1 64.3\u2217 31.9 18.0 17.2 10.2 TI-FGSM 49.7 41.5 63.7\u2217 40.1 34.6 34.5 27.8 MI-FGSM 74.8 64.8 100.0\u2217 54.5 25.1 23.7 13.3 TI-MI-FGSM 76.1 69.5 100.0\u2217 59.6 50.7 51.7 49.3 DIM 86.1 83.5 99.1\u2217 73.5 41.2 40.0 27.9 TI-DIM 86.4 85.5 98.8\u2217 76.3 61.3 60.1 59.5 Res-152 FGSM 40.1 34.0 30.3 81.3\u2217 20.2 17.7 9.9 TI-FGSM 46.4 39.3 33.4 78.9\u2217 34.6 34.5 27.8 MI-FGSM 54.2 48.1 44.3 97.5\u2217 25.1 23.7 13.3 TI-MI-FGSM 55.6 50.9 45.1 97.4\u2217 39.9 37.7 32.8 DIM 77.0 77.8 73.5 97.4\u2217 40.5 36.0 24.1 TI-DIM 77.0 73.9 73.2 97.2\u2217 60.3 58.8 42.8 Ours-Paintings 100.0\u2217 99.7 99.8 98.9 69.3 74.6 64.8 Ours-gs-Paintings 99.9\u2217 98.5 97.6 93.6 85.2 83.9 75.9 Ours-Comics 99.9\u2217 99.8 99.8 98.7 39.3 46.8 23.3 Ours-gs-Comics 99.9\u2217 97.0 93.4 87.7 60.3 58.8 42.8 Ours-ImageNet 99.8\u2217 99.1 97.5 98.1 55.4 60.5 36.4 Ours-gs-ImageNet 98.9\u2217 95.4 90.5 91.8 78.6 78.4 68.9 Table 6: White-box and Black-box: Transferability comparisons. Success rate on ImageNetNeurIPS validation set (1k images) is reported by creating adversaries within the perturbation budget of l\u221e \u2264 16, as per the standard practice [10]. Our generators are learned against naturally trained Inceptionv3 only. \u2217indicates white-box attack. \u2018gs\u2019 is Gaussian smoothing applied to the generator output before projection. Smoothing leads to slight decrease in transferability on naturally trained but shows signi\ufb01cant increase against adversarially trained models. 8 \f1 3 5 7 9 0 20 40 60 80 100 Fool Rate (%) Transferability Vs. Epochs (a) Naturally Trained IncRes-v2 1 3 5 7 9 0 10 20 30 40 50 60 70 Fool Rate (%) Transferability Vs. Epochs (b) Adversarially Trained IncRes-v2 No Smoothing 3x3 5x5 7x7 9x9 11x11 0 20 40 60 80 100 Fool Rate (%) Transferability Vs. Kernel Size (c) Naturally Trained IncRes-v2 No Smoothing 3x3 5x5 7x7 9x9 11x11 0 10 20 30 40 50 60 Fool Rate (%) Transferability Vs. Kernel Size (d) Adversarially Trained IncRes-v2 Figure 6: Effect of Gaussian kernel size and number of training epochs is shown on the transferability (in %age fool rate) of adversarial examples. Generator is trained against Inception-v3 on Paintings, while the inference is performed on ImageNet-NeurIPS. Firstly, as number of epochs increases, transferability against naturally trained IncRes-v2 increases while decreases against its adversarially trained version. Secondly, as the size of Gaussian kernel increases, transferability against naturally as well as adversarially trained IncRes-v2 decreases. Applying kernel of size 3 leads to optimal results against adversarially trained model. Perturbation is set to l\u221e\u226416. 5.3 Transferability: Naturally Trained vs. Adversarially Trained Furthermore, we study the impact of training iterations and Gaussian smoothing [10] on the transferability of our generative adversarial examples. We report results using naturally and adversarially trained IncRes-v2 model [35] as other models exhibit similar behaviour. Figure 6 displays the transferability (in %age accuracy) as a function of the number of training epochs (a-b) and various kernel sizes for Gaussian smoothing (c-d). Firstly, we observe a gradual increase in the transferability of generator against the naturally trained model as the training epochs advance. In contrast the transferability deteriorates against the adversarially trained model. Therefore, when targeting naturally trained models, we train for ten epochs on Paintings, Comics, and ChestX datasets (although we anticipate better performance for higher epochs). When targeting adversarially trained models, we deploy an early stopping criterion to obtain the best trained generator since the performance drops on such models as epochs are increased. This fundamentally shows the reliance of naturally and adversarially trained models on different set of features. Our results clearly demonstrate that the adversarial solution space is shared across different architectures and even across distinct data domains. Since we train our generator against naturally trained models only, therefore it converges to a solution space on which an adversarially trained model has already been trained. As a result, our perturbations gradually become weaker against adversarially trained models as the training progress. A visual demonstration is provided in supplementary material. Secondly, the application of Gaussian smoothing reveals different results on naturally trained and adversarially trained models. After applying smoothing, adversaries become stronger for adversarially trained models and get weaker for naturally trained models. We achieve optimal results with the kernel size of 3 and \u03c3 = 1 for adversarially trained models and use these settings consistently in our experiments. We apply Gaussian kernel on the unrestricted generator\u2019s output, therefore as the kernel size is increased, generator\u2019s output becomes very smooth and after projection within valid l\u221erange, adversaries become weaker. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1811.09020v3",
+                    "title": "Task-generalizable Adversarial Attack based on Perceptual Metric",
+                    "abstract": "Deep neural networks (DNNs) can be easily fooled by adding human\nimperceptible perturbations to the images. These perturbed images are known as\n`adversarial examples' and pose a serious threat to security and safety\ncritical systems. A litmus test for the strength of adversarial examples is\ntheir transferability across different DNN models in a black box setting (i.e.\nwhen the target model's architecture and parameters are not known to attacker).\nCurrent attack algorithms that seek to enhance adversarial transferability work\non the decision level i.e. generate perturbations that alter the network\ndecisions. This leads to two key limitations: (a) An attack is dependent on the\ntask-specific loss function (e.g. softmax cross-entropy for object recognition)\nand therefore does not generalize beyond its original task. (b) The adversarial\nexamples are specific to the network architecture and demonstrate poor\ntransferability to other network architectures. We propose a novel approach to\ncreate adversarial examples that can broadly fool different networks on\nmultiple tasks. Our approach is based on the following intuition: \"Perpetual\nmetrics based on neural network features are highly generalizable and show\nexcellent performance in measuring and stabilizing input distortions. Therefore\nan ideal attack that creates maximum distortions in the network feature space\nshould realize highly transferable examples\". We report extensive experiments\nto show how adversarial examples generalize across multiple networks for\nclassification, object detection and segmentation tasks.",
+                    "authors": "Muzammal Naseer, Salman H. Khan, Shafin Rahman, Fatih Porikli",
+                    "published": "2018-11-22",
+                    "updated": "2019-03-26",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Transferability is a phenomenon where adversarial examples created for one network can fool others. The transferability of adversarial examples makes it challenging to deploy deep neural networks in security critical environments. This is of high concern because it gives attackers the \ufb02exibility to train a local network and transfer its attack against an already deployed network, without knowing its architecture or parameters (\u2018black-box attacks\u2019). Current Image Classification Adversary + Adversarial  Noise Deep  Neural  Network Classification Detection Segmentation Segmentation Detection Zebra Brain Coral Off-the-shelf  Deep  Features Figure 1: Similar to off-the-shelf deep features that are employed to boost the performance of different computer vision tasks, adversarial noise patterns found in deep features space are transferable across different tasks. (Noise pattern is magni\ufb01ed for better visualization) attack algorithms [8, 5] perform well when the network architecture and parameters are known (\u2018white-box setting\u2019); however, their strength signi\ufb01cantly decreases in the black box setting, as shown in [21]. Recent attempts on enhancing the transferability in black-box settings have been reported in [6, 26, 27]. Nevertheless, their dependency on a taskspeci\ufb01c loss function make them non-transferable across different tasks. For example, to fool classi\ufb01cation models, the attacker starts from the softmax cross-entropy to \ufb01nd a gradient direction that increases the model loss for a given sample. Examples found in this way are speci\ufb01c and do not generalize beyond their original task. We propose a novel approach to generate high strength adversarial examples that are transferable across different network architectures and, most importantly, across different vision tasks (e.g., image segmentation, classi\ufb01cation and object detection). Our approach is based on the following intuitions: (a) neural networks trained on ImageNet [18] (or other suf\ufb01ciently large image datasets) learn generic internal representations that are transferable to new tasks and datasets [10, 19]. As a result, it is common practice to use pre-trained classi\ufb01cation networks as the basic building block (network backbone) for a variety of different tasks 1 arXiv:1811.09020v3  [cs.CV]  26 Mar 2019 \f[13, 15] and, (b) a perceptual metric based on VGG internal representations aligns well with human perception [?] and can be used not only to measure the input distortion but also to stabilize it [11]. We hypothesize that adversarial examples based on perceptual distortion, under a given bound, e.g. l\u221e\u2264\u03f5, in the deep features space, are ideally suited to fool any deep network, whether designed for classi\ufb01cation, object detection, segmentation or other vision tasks. We present the \ufb01rst such algorithm, which creates adversarial examples by distorting the deep neural activations. This not only generates high-strength perturbations but also provides \ufb02exibility to work with any task, as the proposed attack does not use any task-dependent loss function. To the best of our knowledge, the closest to our approach is a decision-boundary free attack (called FFF) [17]. The idea is to train a single perturbation within a given metric norm to maximize the activation response of the network\u2019s internal layers. After training, the perturbation is added to the input images to make them adversarial. The problem with this approach is that it optimizes adversarial noise in a way that is independent of the data sample; hence noise, severely over\ufb01ts the network and has very low transferability. In contrast, we do not optimize a single noise pattern, instead we directly maximize the distortions in the network\u2019s internal representations for a given input sample. Zhou et al. [27] also proposed to maximize representation loss. However, their approach is speci\ufb01c to only the classi\ufb01cation task, since the gradient direction in their approach is dependent on cross-entropy loss, which requires labels for the task at hand. Thus, their attack algorithm is essentially a supervised adversarial attack. In contrast, we don\u2019t use any task dependent loss in our objective, so our attack method does not rely on any labels. Thus, it is an unsupervised adversarial attack. Furthermore, we focus on VGG networks for high-dimensional datasets. Remarkably, other networks (Inception/Resnet) do not offer enough distortion in a constrained optimization scenario to carry out this type of attack. One intriguing aspect of our approach is its simplicity and ef\ufb01ciency. For instance, we only use features from a single layer (conv3.3) of VGG-16 [20] (instead of multiple layers as in [17]) and calculate the mean squared difference between the original and adversarial examples to represent neural representation distortion (NRD). NRD is fully differentiable and its minimization can help in image restoration problems [11]. Here, we propose to maximize the NRD to construct adversarial examples. Finding adversarial examples based on feature representation makes our attack generalizable across different architectures for different tasks. Speci\ufb01cally, we show high inter-task and intra-task transferability for our approach on large-scale datasets, including ImageNet [18], MS-COCO [14] and CAMVID [4]. Our method is not restricted to the original backbone models trained on a speci\ufb01c benchmark. Most backbone models are \ufb01ne-tuned with additional training datasets to a speci\ufb01c task. As we elaborate in Sec. 6, our method can successfully be applied to any network that is pretrained on one benchmark, then \ufb01ne-tuned on another, e.g. RetinaNet [13] and SegNet [3]. Contributions: We study and highlight the importance of a neural network\u2019s internal representations (Fig. 1) in the context of adversarial attacks. Our major contributions are: \u2022 We propose a generalizable, black-box, untargeted adversarial attack algorithm on a neural network\u2019s internal representation. \u2022 We leverage generic representations learned by models (e.g. VGG-16 [20]) trained on large image datasets (e.g. ImageNet [18]) to construct transferable adversarial examples. \u2022 Our attack algorithm does not rely on a task-speci\ufb01c loss function or a speci\ufb01c set of input labels, therefore it demonstrates cross-network, cross-dataset, and cross-task transferability. \u2022 We provide state-of-the-art results for classi\ufb01cation networks and provide a robust benchmark to measure the robustness of any neural network based vision system against generic adversarial examples. 2. Related Work Since the seminal work of Szegedy et al. [24] many adversarial attack algorithms [7, 8, 2, 6] have been proposed to show the vulnerability of neural networks against imperceptible changes to inputs. A single-step attack, called fast gradient sign method (FGSM), was proposed by [7]. In a follow-up work, Kurakin et al. [8] proposed a robust multi-step attack, called iterative fast gradient sign methods (I-FGSM) that iteratively searches the loss surface of a network under a given metric norm. To improve transferability, a variant of I-FGSM, called momentum iterative fast gradient sign method (MI-FGSM), was introduced [6], which signi\ufb01cantly enhances the transferability of untargeted attacks on ImageNet [18] under a perturbation budget of l\u221e\u226416. Authors [6] associated the transferability of MI-FGSM with its ability to break local maxima as the number of attack iterations increase. Recently, [26] proposed a data augmentation technique to further boost the transferability of these attack methods. In contrast to ours, all of these methods are supervised adversarial attacks dependent on cross-entropy loss to \ufb01nd the harmful gradient direction. Interestingly, NRD of I-FGSM decreases as the number of attack iterations increases as compared to MI-FGSM as shown in Fig. 2. We generate adversarial examples on ImageNet [18] subset provided by the NIPS security challenge 2017. As can be seen, MI-FGSM maintains its NRD with increasing number of iterations. This also indicates that di2 \f2 3 4 5 6 7 8 9 10 Number of iterations 0 10 20 30 40 50 60 70 80 90 Accuracy and NRD I-FGSM: Accuracy I-FGSM: NRD MI-FGSM: Accuracy  MI-FGSM: NRD Figure 2: Accuracy of Inc-v4 and NRD is shown for adversarial examples generated on Inc-v3 by I-FGSM and MIFGSM. NRD is averaged over all examples. As the number of iterations increases, the accuracy of Inc-v4 on adversarial example found by I-FGSM increases, i.e., the transferability of I-FGSM decreases along with its NRD. rectly maximizing the NRD can boost the transferability of adversarial examples. 3. Adversarial Attacks In this section, we \ufb01rst provide our problem setting, followed by a brief background to adversarial attacks. We explain how popular attack mechanisms, such as FGSM [7], I-FGSM [8] and MI-FGSM [6], differ from each other. This background will form the basis of our proposed attack in Sec. 4. Problem Setting: In this paper, we speci\ufb01cally consider the transferability of untargeted attacks under the l\u221enorm constraint on perturbation strength. Untargeted attacks are considered because they have higher transferability compared to targeted attacks [6, 26]. Furthermore, to make sure that the benign and adversarial examples are close to each other, an attacker is constrained under a metric norm like l\u221e\u2264\u03f5, i.e., in the case of images the attacker can change each pixel intensity value by at maximum \u03f5 amount. 3.1. FGSM Adversarial examples can be formulated as a constrained optimization problem. Suppose we are given a classi\ufb01er function F that maps an input x to its ground-truth class y, a cost function J(x, y) that is used to train the classi\ufb01er and an allowed perturbation budget \u2018\u03f5\u2019. FGSM [7] \ufb01nds an adversarial example x\u2032 that satis\ufb01es \u2225x\u2032 \u2212x \u2225\u221e\u2264\u03f5 using the following formulation: x\u2032 = x + \u03f5 \u00b7 sign(\u2207xJ(x, y)), (1) where \u2207xJ(x, y) represent the gradient of the cost function w.r.t input x. A common choice for J is the cross-entropy loss. The problem with FGSM is that it is a single-step attack, which reduces the attack success rate due to under\ufb01tting the threat model. To overcome this dif\ufb01culty, an iterative version of FGSM was proposed [8]. 3.2. I-FGSM I-FGSM [8] iteratively applies FGSM with a small step size \u03b1 for a given number of iterations T. The step size \u03b1 can be calculated by dividing the perturbation budget \u03f5 with the number of iterations T, i.e., \u03b1 = \u03f5/T. I-FGSM can be represented as follows for steps t \u2208[1, T]: x\u2032 0 = x, x\u2032 t+1 = x\u2032 t + \u03b1 \u00b7 sign(\u2207xJ(x\u2032 t, y)). (2) The problem with I-FGSM is that it over\ufb01ts the threat model, reducing model accuracy to even 0%, while producing a small neural representation distortion (NRD) (See Fig. 2 for empirical evidence). One side effect of having low NRD is the reduced transferability of adversarial examples. This is what Dong et al. [6] built upon, proposing an attack algorithm that \ufb01nds adversarial examples iteratively, while maintaining the transferability rate. 3.3. MI-FGSM The work in [6] added momentum into the optimization objective of I-FGSM. It can be expressed as follows: x\u2032 0 =x, x\u2032 t+1 = x\u2032 t + \u03b1 \u00b7 sign(gt+1), t \u2208[1, T] gt+1 = \u00b5 \u00b7 gt + \u2207xJ(x\u2032 t, y) \u2225\u2207xJ(x\u2032 t, y)\u22251 . (3) The strength of MI-FGSM can be described by two of its control parameters, number of iterations and momentum. The number of attack iterations makes it strong in whitebox settings (like I-FGSM), while momentum allows it to maintain NRD, enhancing the attack success rate in blackbox settings. Based on the above observations, we build our framework and propose to enhance the NRD directly to create strong adversarial examples for black-box attacks. 4. Neural Representation Distortion The Problem: Strong white-box attack algorithms [8, 5] consider already-known network parameters \u03b8 and perturb the input to create x\u2032, such that the example is misclassi\ufb01ed, i.e., F(x\u2032; \u03b8) \u0338= y. Since the perturbations are calculated using gradient directions that are speci\ufb01c to \u03b8, the resulting perturbed images x\u2032 do not generalize well to other networks [6, 21]. The attacks presented in [6, 26, 27] show relatively better transferability, however, these attacks also perturb input images along gradient directions \u2207xJ that are dependent on the ground-truth label y and the de\ufb01nition of the loss function J. This dependency limits the crossnetwork and cross-task transferability of these attacks. 3 \fOriginal (a) Sports Car l\u221e\u226416 (b) Racer l\u221e\u226416 (c) Racer l\u221e\u226416 (d) Loggerhead l\u221e\u226416 (e) Quilt Figure 3: VGG-16 output is shown for sample images. (a) represents benign example, while (b), (c), (d) and (e) show adversarial examples generated by FGSM, MI-FGSM, DIM and NRDM, respectively, against VGG-16 . All adversarial examples have distance l\u221e\u226416 from the original seed (a). Our Solution: In this paper, we propose to directly maximize the perceptual metric based on representation loss of deep feature activations by solving the following optimization problem: max x\u2032 F(x\u2032)|k \u2212F(x)|k subject to: \u2225x \u2212x\u2032\u2225\u221e\u2264\u03f5, (4) where F is DNN based classi\ufb01er, k is the internal representation layer and \u03f5 is the allowed perturbation budget. We apply a transformation T to input x at the \ufb01rst iteration (Algorithm 1) to create a neural representation difference of an adversarial w.r.t a benign example and then maximize the mean squared error of this difference with in a given perturbation budget. There can be different choices for T but in this work T simply adds random noise to the input sample, i.e our algorithm takes a random step at the \ufb01rst iteration. Random noise is convenient to attain a difference at the starting point of our algorithm and it is preferable to heuristic transformations that may cause methodical bias. We use the VGG-16 [20] conv3.3 feature map as the neural representation distortion. This choice is based on observations, reported in the recent study [21], that adversarial examples found in VGG space have high transferability. This is also evident in our experimentation (Table 4). Increasing the representation loss at multiple network layers did not notably increase attack success and adds a signi\ufb01cant computational overhead. Our attack algorithm does not rely on the cross-entropy loss or input labels. This makes it a generic algorithm, which can be used to attack any system using off-the-shelf features in their pipeline. This makes several popular computer vision tasks vulnerable to adversarial attacks, e.g., object detection and segmentation. Furthermore, our proposed approach is complementary to recent best-performing attack methods, such as MI-FGSM [6] and DIM [26]. Therefore, we demonstrates that it can be used alongside them, which further boosts the strength of adversaries. Our proposed method to maximize NRD for a given input sample is summarized in Algorithm 1. Algorithm 1 Neural Representation Distortion Method Input: A classi\ufb01er F, input sample x, input transformation T , internal network layer k, perturbation budget \u03f5 and number of iterations T. Output: An adversarial example x\u2032 with \u2225x\u2032 \u2212x\u2225\u221e\u2264\u03f5. 1: g0 = 0; x\u2032 = x; 2: for t = 0 to T \u22121 do 3: if t = 0 then 4: x\u2032 = T (x) 5: end if 6: Forward pass x\u2032 t to F and compute L as follows; L = \u2225F(x\u2032)|k \u2212F(x)|k\u22252; (5) 7: Compute gradients gt = \u2207xL(x\u2032 t, x); 8: Apply the following equation; x\u2032 t+1 = x\u2032 t + \u03f5 \u00b7 sign(gt); (6) 9: Project adversary into the vicinity of x x\u2032 t+1 = clip(x\u2032 t+1, x \u2212\u03f5, x + \u03f5); (7) 10: end for 11: return x\u2032 = x\u2032 T . 5. Experiments 5.1. Evaluation Protocol In this section, we describe the datasets used for evaluation, network architectures under attack, and the parameter settings for each attack algorithm. 5.1.1 Datasets We use the MNIST, and CIFAR10 test sets and the ImageNet [18] subset provided by NIPS security challenge 2017 (ImageNet-NIPS) to validate the effectiveness of the proposed attack against classi\ufb01cation models. The MNIST and CIFAR10 test sets contain 10k samples each, while 4 \fImageNet-NIPS contains 1k image samples. For object detection, we used MS-COCO [14] validation set, which contains 40.5k images. This is a multi-task dataset popular for image segmentation, object detection and image captioning tasks. We report adversarial attack performance against object detection, however adversarial examples found on this dataset can be used to fool other related tasks e.g., visual question answering. For segmentation, we use the CAMVID [4] test set to measure segmentation robustness against NRDM (Algorithm 1). This dataset contains 233 image samples extracted from video sequences of driving scenes. model-m model-c conv2d(32, 3x3) 2\u2217{conv2d(96, 3x3)} maxpool(2x2) conv2d(96, 3x3, s=2) conv2d(64, 3x3) 2\u2217{conv2d(192, 3x3)} maxpool(2x2) conv2d(96, 3x3, s=2) conv2d(64, 3x3) 2\u2217{conv2d(192, 3x3)} fc(64) conv2d(10, 3x3) softmax(10) avg-pool softmax(10) Table 1: Architectures of naturally trained convolutional networks for MNIST (model-m) and CIFAR10 (model-c). \u2018*\u2019 indicates the number of times a layer is repeated. \u2018s\u2019 represent stride. Each convolutional layer is followed by ReLU activation. Batch-norm is used after each convolutional layer in model-c. Layers whose outputs are used by NRDM are highlighted in bold. 5.1.2 Network Architectures Classi\ufb01cation: We study eight models trained on the ImageNet dataset [18]. These models can be grouped into two categories. (a) Naturally trained: Five of these models are only trained on benign examples. These include Inceptionv3 (Inc-v3) [23], Inceptionv4 (Inc-v4), Inception Resnet v2 (IncRes-v2) [22] and Resnet v2152 (Res-152) [9] and VGG-19 [20]. (b) Adversarially trained: The other three, models including Adv-v3 [12], Inc-v3ens3 and IncRes-v2ens [25], are adversarially trained and made publicly available. The speci\ufb01c details about these models can be found in [12, 25]. Attacks are created for naturally trained models, while tested against all of them. For classi\ufb01cation on smaller datasets, we study three models each for MNIST and CIFAR10. Among these models, two are naturally trained and one is adversarially trained using saddle point optimization [16]. Adversarial examples are created for naturally trained models, named model-m and model-c for MNIST and CIFAR10, respectively (see Table 1). These examples are subsequently transferred to adversarially trained Madry\u2019s models [16] and naturally trained ResNet models, named res-m and res-m res-c conv2d(16, 3x3) conv2d(16, 3x3) 1\u2217rb ( conv2d(16, 3x3) conv2d(16, 3x3) 3\u2217rb ( conv2d(16, 3x3) conv2d(16, 3x3) 1\u2217rb ( conv2d(32, 3x3) conv2d(32, 3x3, s=2) 3\u2217rb ( conv2d(32, 3x3) conv2d(32, 3x3, s=2) 1\u2217rb ( conv2d(64, 3x3) conv2d(64, 3x3, s=2) 3\u2217rb ( conv2d(64, 3x3) conv2d(64, 3x3, s=2) softmax(10) avg-pool(8x8) softmax(10) Table 2: Architectures of naturally trained residual networks for MNIST (res-m) and CIFAR10 (res-c). \u2018\u2217\u2019 indicates the number of times a layer is repeated. \u2018s\u2019 and \u2018rb\u2019 represent stride and residual block respectively. Each convolutional layer is followed by a ReLU activation. Batchnorm is used after each convolutional layer in res-c. res-c for MNIST and CIFAR10 respectively (see Table 2). Object Detection: To demonstrate cross-task and cross-dataset transferability, we study naturally trained RetinaNet [13] performance against adversarial examples found by the NRDM approach (Algorithm 1) on the MS-COCO validation set. Segmentation: We evaluate the robustness of naturally trained SegNet-basic [3] against adversarial examples generated by the NRDM approach (Algorithm 1) on the CAMVID [4] test set. 5.1.3 Attack Parameters FGSM is a single-step attack. Its step size is set to 16. In the case of R-FGSM, we take a step of size \u03b1=16/3 in a random direction and then a gradient step of size 16\u2212\u03b1 to maximize model loss. The attack methods, I-FGSM, MI-FGSM and DIM, are run for ten iterations. The step size for these attacks is set to 1.6, as per standard practice. The momentum decay factor for MI-FGSM is set to one. This means that attack accumulates all the previous gradient information to perform the current update and is shown to have the best success rate [6]. For DIM, the transformation probability is set to 0.7. In the case of FFF [17], we train the adversarial noise for 10K iterations to maximize the response at the activation layers of VGG-16 [20]. For the NRDM (Algorithm 1), we used the VGG-16 [20] conv3-3 feature map as the representation loss. Since NRDM maximizes loss w.r.t a benign example, it does not suffer from the over\ufb01tting problem. We run NRDM for the maximum number of 100 iterations. The transferability of different attacks is compared against the number of iterations in Fig. 4. MIFGSM and DIM quickly reach to their full potential within ten iterations. The strength of I-FGSM strength decreases, 5 \fAccuracy Naturally Trained Adv. Trained Inc-v3 Inc-v4 Res-152 IncRes-v2 VGG-19 Adv-v3 Inc-v3ens3 IncRes-v2ens T-1 95.3 97.7 96.1 100.0 85.5 94.3 90.2 96.9 T-5 99.8 99.8 99.9 100.0 96.7 99.4 95.5 99.8 Table 3: Model accuracies are reported on original data set ImageNet-NIPS containing benign examples only. T-1: top-1 and T-5: top-5 accuracies. Best and second best performances are colorized. Naturally Trained Adv. Trained Attack Inc-v3 Inc-v4 Res-152 IncRes-v2 VGG-19 Adv-v3 Inc-v3ens3 IncRes-v2ens T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 Inc-v3 FGSM [7] 22.0\u2217 45.7\u2217 62.5 84.7 64.6 85.8 65.9 85.9 49.9 75.7 69.1 88.1 77.2 90.9 90.8 98.3 R-FGSM [25] 16.7\u2217 38.0\u2217 65.8 86.0 69.5 89.7 68.8 88.7 61.4 83.8 76.4 90.9 77.9 91.2 88.8 97.6 I-FGSM [8] 0.0\u2217 1.7\u2217 82.0 97.6 86.5 98.6 90.6 99.1 76.7 95.0 88.5 98.7 84.9 94.4 94.6 99.5 MI-FGSM [6] 0.0\u2217 1.5\u2217 47.1 78.8 47.1 84.5 52.5 81.9 47.3 76.7 71.6 89.8 73.8 90.7 88.3 98.0 TAP [27] 0.0\u2217 22.1 46.9 24.7 52.5 60.9 68.8 DIM [26] 0.2\u2217 1.3\u2217 27.8 63.1 42.1 75.2 34.6 65.4 40.2 71.4 65.2 87.9 68.3 89.6 86.3 97.5 Res-152 FGSM [7] 54.1 79.3 61.2 84.2 16.5\u2217 41.0\u2217 62.5 85.6 46.0 72.7 67.3 87.4 74.0 89.4 88.4 97.7 R-FGSM [25] 58.5 83.4 64.9 86.6 12.9\u2217 35.2\u2217 69.1 88.5 56.1 80.8 74.5 90.6 75.5 90.4 86.5 96.5 I-FGSM [8] 80.0 96.6 84.1 98.4 0.9\u2217 6.2\u2217 92.5 99.1 75.7 94.9 87.4 99.0 85.5 94.8 93.4 99.3 MI-FGSM [6] 43.5 76.8 49.9 79.2 0.9\u2217 5.1\u2217 54.8 82.4 46.8 76.0 72.6 90.7 71.1 90.1 86.0 97.5 TAP [27] 48.2 55.7 7.6\u2217 55.2 49.2 57.8 64.1 DIM [26] 20.1 51.2 22.0 54.6 0.6\u2217 4.2\u2217 24.6 57.3 33.3 62.6 53.5 82.5 55.2 83.1 74.4 94.1 IncRes-v2 FGSM [7] 61.7 83.8 69.6 87.6 68.4 89.6 50.1\u2217 73.9\u2217 52.3 76.5 72.0 89.6 79.0 91.6 90.0 97.7 R-FGSM [25] 66.6 87.0 71.8 89.4 73.5 91.5 46.1\u2217 71.3\u2217 62.9 84.1 75.5 91.2 79.3 91.5 87.4 97.3 I-FGSM [8] 62.8 88.4 68.3 91.9 77.2 94.8 1.1\u2217 2.6\u2217 71.4 91.7 85.6 97.5 83.8 95.6 89.8 98.4 MI-FGSM [6] 36.0 67.5 42.4 73.2 49.3 82.2 1.0\u2217 2.4\u2217 51.3 76.8 70.0 90.1 71.5 92.2 81.8 96.3 TAP [27] 25.9 33.2 53.5 4.8\u2217 60.5 79.1 87.8 DIM [26] 21.4 49.8 23.5 53.4 32.3 64.3 4.8\u2217 13.7\u2217 39.7 69.2 54.9 81.4 57.5 85.9 73.5 94.4 VGG16 FGSM [7] 30.1 56.0 34.0 58.0 36.6 65.2 42.2 66.1 9.1 27.9 48.8 72.6 53.5 79.5 72.8 91.1 R-FGSM [25] 41.5 67.9 45.1 72.5 49.2 78.4 54.9 77.7 12.9 35.8 63.9 86.2 63.5 85.3 77.1 93.0 I-FGSM [8] 69.4 93.0 75.3 94.5 79.5 95.7 87.2 97.9 18.3 56.1 82.2 97.5 80.9 93.7 91.5 99.1 MI-FGSM [6] 16.9 42.0 18.7 40.1 24.9 51.6 26.1 52.5 2.0 14.4 38.8 68.1 42.5 72.4 64.2 87.7 TAP [27] 23.9 28.1 23.9 32.3 38.8 41.9 63.8 DIM [26] 12.9 35.5 15.2 35.8 20.6 45.7 19.7 43.8 0.6 8.8 31.6 59.0 32.1 61.0 56.3 81.0 FFF [17] 61.7 80.7 60.8 78.7 72.8 90.1 76.1 90.1 44.0 68.0 79.6 93.1 83.1 93.1 92.8 98.5 NRDM 5.1 10.2 6.2 12.4 15.6 27.6 13.6 23.0 4.5 14.2 27.7 46.8 54.2 75.4 75.3 89.8 NRDM-DIM 4.9 10.5 5.7 12.0 16.0 28.6 12.7 22.6 5.0 14.0 28.7 45.7 52.9 73.8 74.0 89.5 Table 4: Model accuracies are reported under untargeted l\u221eadversarial attacks on ImageNet-NIPS with perturbation budget l\u221e\u226416 for pixel space [0-255]. T-1 and T-2 represent top-1 and top-5 accuracies, respectively. NRDM shows higher or competitive success rates for black-box models than FGSM [7], I-FGSM [8], MI-FGSM [6], TAP [27], DIM [26] and FFF [17]. NRDM-DIM combines input diversity as well as momentum with NRDM. \u2018\u2217\u2019 indicates the white-box attacks. Best and second best black-box attacks are colorized. while NRDM strength increases, with the number of attack iterations. 5.2. Input Transformations Different input transformations have been proposed to mitigate the adversarial effect but they can be easily broken in a white-box scenario. This is because an attacker can be adaptive and incorporate transformations into the adversary generation process. Even non-differentiable transformations can be by-passed by approximating them with an identity function [1]. However in a black-box scenario, the attacker does not have any knowledge of the transformation function along with the network architecture and its parameters. We test the strength of our adversarial attack against well studied transformations, including JPEG, total variation minimization (TVM) and median \ufb01ltering. We report our experimental results using the above-mentioned network architectures and input transformations in the following section. 6. Results Classi\ufb01cation: We report the performance of our attack against a number of CNN architectures on the ImageNetNIPS dataset in Table 4. The following insights can be drawn from our results. (1) In comparison to other stateof-the-art attacks, our approach consistently demonstrates a much higher transferability rate for naturally trained images. Speci\ufb01cally, NRDM attack have much higher trans6 \f5 10 15 20 25 30 35 40 45 50 100 Number of iterations 10 20 30 40 50 60 70 80 90 Accuracy I-FGSM  MI-FGSM  DIM  NRDM Figure 4: Accuracy of Inc-v3 for adversarial examples generated on VGG-16 by I-FGSM and MI-FGSM, DIM and NRDM. NRDM\u2019s strength increases with number of iterations, in contrast to MI-FGSM and DIM. Datasets \u2193 Attack \u2193 Naturally Trained Adv. Trained model-m res-m Madry-M MNIST FGSM 42.28\u2217 53.15 95.96 I-FGSM 40.66\u2217 51.04 96.64 MI-FGSM 40.66\u2217 48.19 95.96 NRDM 4.39\u2217 23.54\u2217 97.56 model-c res-c Madry-C CIFAR10 FGSM 5.47\u2217 24.19 85.54 I-FGSM 2.52\u2217 36.81 87.00 MI-FGSM 2.52\u2217 16.56 85.71 NRDM 11.92\u2217 23.98 86.99 Table 5: Model accuracies under untargeted l\u221eadversarial attacks on MNIST and CIFAR10 with perturbation budget l\u221e\u226476.5 and l\u221e\u22648, respectively, for pixel space [0255], as per standard practice [16]. NRDM shows higher or competitive success rates for black-box models compared to FGSM, I-FGSM and MI-FGSM. \u2018\u2217\u2019 indicates the whitebox attacks. Best and second best attacks are colorized. Dataset \u2193 Metric \u2193 Naturally Trained Adv. Trained model-m res-m Madry-M MNIST Accuracy 99.30 98.88 98.40 model-c res-c Madry-C CIFAR10 Accuracy 85.44 80.56 87.62 Table 6: Model accuracies on original test datasets for MNIST and CIFAR10 containing benign examples only. Best and second best performances are colorized. ferability on naturally trained models, bringing down top-1 accuracy of IncRes-v2 [22] from 100.0% (see Table 3) to 12.7% (see Table 4). (2) In comparison, MI-FGSM [6] and DIM [26] perform slightly better on adversarially trained ensemble models [25], with NRDM showing competitive success rate. This is because the MI-FGSM and DIM methods use decision boundary information while, NRDM is agnostic to decision-level information about the classi\ufb01er. Method No Attack NRDM l\u221e\u22648 l\u221e\u226416 No Defense 79.70 52.48 32.59 JPEG (quality=75) 77.25 51.76 32.44 JPEG (quality=50) 75.27 52.45 33.16 JPEG (quality=20) 68.82 53.08 35.54 TVM (weights=30) 73.70 55.54 34.21 TVM (weights=10) 70.38 59.52 34.57 MF (window=3) 75.65 49.18 30.52 Table 7: Segnet-Basic accuracies on CAMVID test set with and without input transformations against NRDM. Best and second best performances are colorized. Method No Attack NRDM l\u221e\u22648 l\u221e\u226416 No Defense 53.78 22.75 5.16 JPEG (quality=75) 49.57 20.73 4.7 JPEG (quality=50) 46.36 19.89 4.33 JPEG (quality=20) 40.04 19.13 4.58 TVM (weights=30) 47.06 27.63 6.36 TVM (weights=10) 42.79 32.21 9.56 MF (window=3) 43.48 19.59 5.05 Table 8: mAP (with IoU = 0.5) of RetinaNet is reported on the MS-COCO validation set with and without input transformations against NRDM. Best and second best performances are colorized. (3) We also test with adversarial examples found using different network architectures (i.e., Inc-v3, Res-152, IncRes-v2, VGG16). Overall, we conclude that the adversarial examples found in VGG-16 [20] space have very high transferability. Figure 3 shows a visual comparison of adversaries found by different attack algorithms. On small datasets (MNIST and CIFAR10), similar to other attacks, the NRDM becomes ineffective against adversarially trained Madry models [16] (see Tables 6 and 5) in blackbox settings. This shows that \ufb01nding better methods for adversarial training is a way forward to defend against these attacks. Input transformations can somewhat help to mitigate the adversarial effect in black-box settings (see Table 9). TVM is the most effective against all the attacks, while median \ufb01ltering perform better against DIM [26]. JPEG is the least effective against untargeted adversarial attacks. Segmentation: The NRDM attack created on CAMVID [4] in VGG-16 feature space is able to bring down the per pixel accuracy of Segnet-Basic by 47.11% within l\u221e\u226416 (see Table 7 and Fig 5). JPEG and TVM transformations are slightly effective but only at the cost of accuracy on benign examples. Object Detection: RetinaNet [13] collapses in the presence of adversaries found by NRDM on the MS-COCO validation set using the VGG-16 [20] feature space. Its mean average precision (mAP) with 0.5 intersection over union (IOU) drops from 53.78% to 5.16% under perturba7 \fNo Attack FGSM [7] R-FGSM [25] I-FGSM [8] MI-FGSM [6] DIM [26] NRDM T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 T-1 T-5 No Defense 95.3 99.8 30.1 56.0 41.5 67.9 69.4 93.0 16.9 42.0 12.9 35.5 5.1 10.2 JPEG (quality=75) 93.9 99.5 30.4 55.4 41.8 67.0 69.7 92.3 18.4 42.0 13.5 33.3 5.4 12.6 JPEG (quality=50) 91.3 99.3 31.0 55.4 40.5 65.3 68.7 91.8 18.1 42.1 13.1 34.4 6.5 12.8 JPEG (quality=20) 86.0 97.6 29.9 53.9 38.0 64.6 69.8 90.9 18.4 42.1 14.1 34.2 8.4 18.7 TVM (weights=30) 93.1 99.4 30.6 56.2 41.7 67.7 73.7 94.5 17.2 42.1 14.9 33.5 9.8 18.5 TVM (weights=10) 88.8 97.6 32.1 57.3 43.6 69.4 73.9 93.4 19.8 45.7 15.8 37.1 24.0 40.5 MF (window=3) 93.2 99.1 24.3 45.5 36.1 62.3 62.8 89.9 16.2 36.8 18.8 42.1 9.9 17.9 Table 9: Inc-v3 accuracy is reported with and without input transformations. Adversarial examples are generated for VGG-16 in white-box setting by FGSM, R-FGSM, I-FGSM, MI-FGSM, DIM and NRDM under perturbation budget l\u221e\u226416 and then transferred to Inc-v3. T-1 and T-2 represent top-1 and top-5 accuracies, respectively. Best and second best performances are colorized. (a) Original (b) Prediction for Original l\u221e\u226416 (c) Adversarial (d) Prediction for Adversarial Figure 5: Segnet-Basic output is shown for different images. (a) is the original image, while (b) shows predictions for the original image. (c) is the adversary found by NRDM algorithm 1, while (d) shows predictions for the adversarial image. The perturbation budget is written on the top of adversarial image. (a) Original l\u221e\u22648 (b) Adversarial (c) Original l\u221e\u226416 (d) Adversarial Figure 6: RetinaNet detection results are shown for different images. (a) and (c) show detection for the original images, while (b) and (d) show detection for adversaries found using NRDM algorithm 1. The perturbation budget is written on the top of each adversarial image. tion budget l\u221e\u226416 (see Table 8 and Fig 6). TVM is relatively more effective compared to other transforms against the NRDM attack. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1807.01216v2",
+                    "title": "Local Gradients Smoothing: Defense against localized adversarial attacks",
+                    "abstract": "Deep neural networks (DNNs) have shown vulnerability to adversarial attacks,\ni.e., carefully perturbed inputs designed to mislead the network at inference\ntime. Recently introduced localized attacks, Localized and Visible Adversarial\nNoise (LaVAN) and Adversarial patch, pose a new challenge to deep learning\nsecurity by adding adversarial noise only within a specific region without\naffecting the salient objects in an image. Driven by the observation that such\nattacks introduce concentrated high-frequency changes at a particular image\nlocation, we have developed an effective method to estimate noise location in\ngradient domain and transform those high activation regions caused by\nadversarial noise in image domain while having minimal effect on the salient\nobject that is important for correct classification. Our proposed Local\nGradients Smoothing (LGS) scheme achieves this by regularizing gradients in the\nestimated noisy region before feeding the image to DNN for inference. We have\nshown the effectiveness of our method in comparison to other defense methods\nincluding Digital Watermarking, JPEG compression, Total Variance Minimization\n(TVM) and Feature squeezing on ImageNet dataset. In addition, we systematically\nstudy the robustness of the proposed defense mechanism against Back Pass\nDifferentiable Approximation (BPDA), a state of the art attack recently\ndeveloped to break defenses that transform an input sample to minimize the\nadversarial effect. Compared to other defense mechanisms, LGS is by far the\nmost resistant to BPDA in localized adversarial attack setting.",
+                    "authors": "Muzammal Naseer, Salman H. Khan, Fatih Porikli",
+                    "published": "2018-07-03",
+                    "updated": "2018-11-19",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Deep neural network architectures achieve remarkable performance on critical applications of machine learning including sensitive areas such as face detection [16], malware detection [17] and autonomous driving [11]. However, the vulnerability of DNNs to adversarial examples limit their wide adoption in security critical applications [1]. It has been shown that adversarial examples can be created by minimally modifying the original input samples such that a DNN mis-classi\ufb01es them with high con\ufb01dence. DNNs are often criticized as black-box models; adversarial examples raise further concerns by highlighting blind spots of DNNs. At the same time, adversarial phenomena provide an opportunity to understand DNN\u2019s behavior to minor perturbations in visual inputs. Methods that generate adversarial examples either modify each image pixel by a small amount [24, 8, 14, 13] often imperceptible to human vision or few image pixels by a large visible amounts [20, 22, 4, 12, 7]. Pixel attack [22] changes few image pixels, but it requires small images (e.g., 32\u00d732) and does not provide control over noise location. Small noise patches were introduced by [20] in the form of glasses to cover human face to deceive face recognition systems. Similarly, Evtimov et al. [7] added noise patches as rectangular patterns on top of traf\ufb01c signs to cause misclassi\ufb01cation. Very recently, localized adversarial attacks, i.e., Adversarial patch [4] and LaVAN [12] have been introduced that can be optimized for triplets (misclassi\ufb01cation con\ufb01dence, target class, perturbed location). These practical attacks have demonstrated high strength and can easily bypass existing defense approaches. Therefore they present a signi\ufb01cant challenge for existing deep learning systems. Contributions: In this work, we study the behavior of localized adversarial attacks and propose an effective mechanism to defend against them (see Fig. 1). LaVAN and Adversarial patch add adversarial noise without affecting the original object in the image, and to some extent, they are complementary to each other. In an effort towards a strong defense against these attacks, this paper contributes as follows: \u2022 Motivated by the observation that localized adversarial attacks introduce high-frequency noise, we proarXiv:1807.01216v2  [cs.CV]  19 Nov 2018 \f(a) Impala (94%) (b) Ice Lolly (99%) (c) Impala (94%) (d) Squirrel Monkey (58%) (e) Toaster (91%) (f) Squirrel Monkey (57%) Figure 1: Inception v3 [23] con\ufb01dence scores are shown for example images. (a) and (d) represent benign examples from ImageNet [18], (b) and (e) are adversarial examples generated by LaVAN [12] and Adversarial patch [4] respectively, (c) and (f) show transformed adversarial images using our proposed LGS. As illustrated, LGS restores correct class con\ufb01dences. pose a transformation called Local Gradient Smoothing (LGS). LGS \ufb01rst estimates region of interest in an image with the highest probability of adversarial noise and then performs gradient smoothing in only those regions. \u2022 We show that by its design, LGS signi\ufb01cantly reduces gradient activity in the targeted attack region and thereby showing the most resistance to BPDA [2], an attack speci\ufb01cally designed to bypass transformation based defense mechanisms. \u2022 Our proposed defense outperforms other state-of-theart methods such as Digital watermarking, TVM, JPEG compression, and Feature squeezing in localized adversarial attacks setting [12, 4]. 2. Related Work Among the recent localized adversarial attacks, the focus of adversarial patch [4] is to create a scene independent physical-world attack that is agnostic to camera angles, lighting conditions and even the type of classi\ufb01er. The result is an image independent universal noise patch that can be printed and placed in the classi\ufb01er\u2019s \ufb01eld of view in a white box (when deep network model is known) or black box (when deep network model is unknown) setting. However, the size of the adversarial patch should be 10% of the image for the attack to be successful in about 90% cases [12]. This limitation was addressed by Karmoon et al. [12], who focused on creating localized attack covering as little as 2% of the image area instead of generating a universal noise patch. In both of these attacks [4, 12], there is no constraint on noise, and it can take any value within image domain, i.e., [0, 255] or [0, 1]. Defense mechanisms against adversarial attacks can be divided into two main categories. (a) Methods that modify DNN by using adversarial training [25] or gradient masking [15] and (b) techniques that modify input sample by using some smoothing function to reduce adversarial effect without changing the DNN [6, 5, 9, 26]. For example, JPEG compression was \ufb01rst presented as a defense by [6] and recently studied extensively by [5, 19]. [26] presented feature squeezing methods including bit depth reduction, median \ufb01ltering, Gaussian \ufb01ltering to detect and defend against adversarial attacks. Guo et al. [9] considered smoothing input samples by total variance minimization along with JPEG compression and image quilting to reduce the adversarial effect. Our work falls into the second category as we also transform the input sample to defend against localized adversarial attacks. However, as we will demonstrate through experiments, the proposed defense mechanism pro\fvides better defense against localized attacks compared to previous techniques. The paper is organized as follows: Section 3 discusses localized adversarial attacks, LaVAN and Adversarial patch in detail. Section 4 presents our defense approach (LGS) against these attacks. We discuss other related defense methods in Section 5.2. Section 5 demonstrates the effectiveness of the proposed method LGS in comparison to other defense methods against LaVAN and adversarial patch attacks. Section 5.3 discusses BPDA and resilience of different defense methods against it. Section 6 concludes the draft by discussing possible future directions. 3. Adversarial Attacks In this section, we provide a brief background to adversarial attacks and explain how LaVAN [12] and Adversarial patch [4] are different from traditional attacks. 3.1. Traditional Attacks The search for adversarial examples can be formulated as a constrained optimization problem. Given a discriminative classi\ufb01er F(y | x), an input sample x \u2208Rn, a target class \u00af y and a perturbation budget \u03f5, an attacker seeks to \ufb01nd a modi\ufb01ed input x\u2032 = x + \u03b4 \u2208Rn with adversarial noise \u03b4 to increase likelihood of the target class \u00af y by solving the following optimization problem: max x\u2032 F(y = \u00af y | x\u2032) subject to: \u2225x \u2212x\u2032\u2225p \u2264\u03f5 (1) This formulation produces well camou\ufb02aged adversarial examples but changes each pixel in the image. Defense methods such as JPEG compression [6, 5], Total variance minimization [9] and Feature squeezing [26] are effective against such attacks especially when the perturbation budget \u03f5 is not too high. 3.2. LaVAN LaVAN [12] differs from the formulation presented in Eq. 1 as it con\ufb01nes adversarial noise \u03b4 to a small region, usually away from the salient object in an image. It uses the following spatial mask to replace the small area with noise, as opposed to noise addition performed in traditional attacks: x\u2032 = (1 \u2212m) \u2299x + m \u2299\u03b4, s.t., m \u2208Rn and , (2) where \u2299is Hadamard product and \u03b4 represents adversarial noise. They also introduce a new objective function where at each iteration, optimization algorithm takes a step away from the source class and towards the target class simultaneously, as follows: max x\u2032 F(\u00af y | x\u2032) \u2212F(y | x\u2032) subject to: \u2225x \u2212x\u2032\u2225\u221e\u2264\u03f5, 0 \u2264\u03f5 \u22641, (3) where x\u2032 is given by Eq. 2. 3.3. Adversarial Patch Adversarial examples created using the methodology presented in Eq. 1 cannot be used in physical world attacks because adversarial noise loses its effect under different camera angles, rotations and lighting conditions. Athalye et al. [3] introduced an Expectation over Transformation (EoT) attack to create robust adversarial examples invariant to chosen set of transformations. Brown et al. [4] build upon Athalye\u2019s work and used EoT to create a scene independent robust noise patch con\ufb01ned to small region that can be printed and placed in the classi\ufb01er\u2019s \ufb01eld of view to cause misclassi\ufb01cation. To generate adversarial patch p\u2032, [4] proposed a patch operator A(p, x, l, t) for a given image x, patch p, location l and a set of transformation t. During optimization, patch operator A applies a set of transformations to the patch p and then projects it onto the image x at a location l to increase likelihood of target class \u00af y. p\u2032 = max p Ex\u223cX,t\u223cT,l\u223cL[F(\u00af y | A(p, x, l, t))] (4) where X represent training images, T represents distribution over transformations, and L is a distribution over locations in the image. 4. Defense: Local Gradients Smoothing Both of the above discussed attacks [12, 4] introduce high frequency noise concentrated at a particular image location and strength of such a noise becomes very prominent in image gradient domain. We propose that the effect of such adversarial noise can be reduced signi\ufb01cantly by suppressing high frequency regions without effecting the low frequency image areas that are important for classi\ufb01cation. An ef\ufb01cient way to achieve this is by projecting scaled normalized gradient magnitude map onto the image to directly suppress high activation regions. To this end, we \ufb01rst compute the magnitude of \ufb01rst-order local image gradients as follows: \u2225\u2207x(a, b) \u2225= s\u0012\u2202x \u2202a \u00132 + \u0012\u2202x \u2202b \u00132 , (5) where a, b denote the horizontal and vertical directions in the image plane. The range of the gradient magnitude calculated using the above equation is normalized for consistency across an image as follows: g(x) = \u2225\u2207x(a, b) \u2225\u2212\u2225\u2207x(a, b) \u2225min \u2225\u2207x(a, b) \u2225max \u2212\u2225\u2207x(a, b) \u2225min . (6) \f(a) (b) (c) (d) (e) (f) (g) (h) Figure 2: (a) and (e) show adversarial examples generated by LaVAN [12] and adversarial patch [4] respectively, (b) and (f) show normalized gradients magnitude before applying windowing operation to look for highest activation regions, (c) and (g) show concept of window search to estimate noise regions, (d) and (h) show normalized gradients magnitude after applying windowing operation. We refer to supplementary material for more examples. The normalized gradient g(x) is projected onto the original image to suppress noisy perturbations in the input data domain. This operation smooths out very high frequency image details. As demonstrated by our evaluations, these regions have high likelihood of being perturbed areas, but they do not provide signi\ufb01cant information for \ufb01nal classi\ufb01cation. The noise suppression is performed as follows: T (x) = x \u2299(1 \u2212\u03bb \u2217g(x)), (7) where \u03bb is the smoothing factor for LGS and \u03bb \u2217g(x) is clipped between 0 and 1. Applying this operation at a global image level, however, introduces image structural loss that causes a drop in classi\ufb01er\u2019s accuracy on benign examples. To minimize this effect, we design a block-wise approach where gradient intensity is evaluated within a local window. To this end, we \ufb01rst divide the gradient magnitude map into a total of K overlapping blocks of same size (\u03c4) and then \ufb01lter these blocks based on a threshold (\u03b3) to estimate highest activation regions which also have the highest likelihood of adversarial noise. This step can be represented as follows: g\u2032 h,w = W(g(x), h, w, \u03c4, o) \u2208R\u03c4, \u02c6 gh,w = ( g\u2032 h,w, if 1 |g\u2032 h,w| P i P j g\u2032 h,w(i, j) > \u03b3 0, otherwise. (8) where | \u00b7 | denotes the cardinality of each patch, o denotes the patch overlap, W(\u00b7) represent the windowing operation, h, w denote the vertical and horizontal components of the top left corner of the extracted window, respectively. We set the block size \u03c4 = 15\u00d715 with 5\u00d75 overlap and threshold is 0.1 in all of our experiments. The updated gradient blocks represented as \u02c6 gh,w are then collated to recreate the full gradient image: \u00af g = W\u22121({\u02c6 gh,w}K 1 ). Figure 2 shows the effect of windowing search on gradients magnitude maps. We further demonstrated LGS ef\ufb01ciency on challenging images in supplementary material. 5. Experiments 5.1. Protocol and Results Overview We used Inception v3 model [23] to experiment with various attack and defense mechanisms in all of our experiments. All attacks are carried out in white-box settings. We consider the validation set available with Imagenet-2012 dataset in our experiments. This set consists of a total of 50k images. We report top-1 accuracy of classi\ufb01er. Results are summarized in tables 1, 2 and 3. LaVAN [12] can be optimized for triplets (target, con\ufb01dence, location) but it is highly sensitive to noise location. Adversary loses its effect with even a small change to the pixel location. To reduce the computational burden and conduct experiments on a large scale, we randomly chose noise location along border areas of the image because they have the least probability to cover the salient object. We ran \fNo Attack 42x42 noise patch covering \u223c2% of image 52x52 noise patch covering \u223c3% of image 60x60 noise patch covering \u223c4% of image No Defense 75.61% 11.00% 2.79% 0.78% LGS [lambda=2.3] 71.05% 70.90% 69.84% 69.37% LGS [lambda=2.1] 71.50% 70.80% 69.54% 68.56% LGS [lambda=1.9] 71.84% 70.40% 68.84% 66.98% LGS [lambda=1.7] 72.30% 69.55% 67.32% 63.38% LGS [lambda=1.5] 72.72% 67.68% 64.13% 55.67% DW 52.77% 67.70% 66.19% 64.57% MF [window=3] 70.59% 63.90% 62.15% 59.81% GF [window=5] 61.75% 59.52% 57.68% 55.29% BF [window=5] 65.70% 61.53% 58.70% 55.59% JPEG [quality=80] 74.35% 18.14% 6.23% 2.06% JPEG [quality=60] 72.71% 25.69% 11.86% 4.85% JPEG [quality=40] 71.20% 37.10% 23.26% 12.73% JPEG [quality=30] 70.04% 45.00% 33.72% 22.04% JPEG [quality=20] 67.51% 52.84% 46.25% 37.19% JPEG [quality=10] 60.25% 53.10% 48.73% 43.59% TMV [weights=10] 70.21% 14.48% 4.64% 1.73% TMV [weights=20] 72.85% 13.24% 3.78% 1.17% TMV [weights=30] 73.85% 12.79% 3.53% 1.04% BR [depth=1] 39.85% 25.93% 15.14% 9.73% BR [depth=2] 64.61% 16.32% 6.15% 2.68% BR [depth=3] 72.83% 13.4% 3.89% 1.25% Table 1: Summary of Inception v3 performance against LaVAN attack on ImageNet validation set with and without defenses including local gradient smoothing (LGS), digital watermarking (DW), median \ufb01ltering (MF), Gaussian \ufb01ltering (GF), bilateral \ufb01ltering (BF), JPEG compression, total variance minimization (TVM) and bit-depth reduction (BR). Bold numbers represent the best accuracy of a certain defense against LAVAN attack. 1000 iterations of attack optimization per image. We terminate the optimization early if classi\ufb01er mis-classify with con\ufb01dence above than or equal to 99% or we let it run for at max 1000 iterations and attack is considered to be successful if the image label is changed to a random target (not equal to the true object class). Inceptionv3 model accepts 299x299 image as an input. Three adversarial noise masks with size 42x42 (\u223c2% of the image), 52x52 (\u223c3% of the image) and 60x60 (\u223c4% of the image) were applied. Table 1 presents summary of all the results. For the case of adversarial patch [4] attack, placing a patch of size 95x95 ( 10% of the image) randomly on all Imagenet validation set was not possible because it would cover most of salient objects details in an image. So we carefully created 1000 adversarial examples that model misclassi\ufb01ed as a toaster with a con\ufb01dence score at least 90%. We then applied all the defense techniques and reported results in Table 2. Figure 3 shows runtime of defense methods to process ImageNet [18] validation set. We used optimized python implementations. Speci\ufb01cally, we employed JPEG from Pillow, Total variance minimization (TVM), and Bilateral \ufb01ltering (BF) from scikit-image, Median \ufb01ltering (MF) and Gaussian \ufb01ltering (GF) from scipy, and LGS and Bit Depth Reduction (BR) are written in python 3.6 as well. All experiments were conducted on desktop windows computer equipped with Intel i7-7700k quad-core CPU clocked at 4.20GHz and 32GB RAM. Defense None LGS DW MF JPEG TVM BR Adversarial Patch 0% 90.5% 80% 49.10% 45% 1% 0% Table 2: Accuracy of Inception v3 against adversarial patch attack with and without defense. The size of adversarial noise is 95x95 covering \u223c10% of image. LGS is used with \u03bb = 2.3, DW in blind defense scenario, MF with window equal to 3, JPEG compression with quality equal to 30, TVM with weights equal to 10 and BR with depth 3. This hyperparameter choice was made for fair comparison such that the performance on benign examples from ImageNet is approximately the same (\ufb01rst column of Table 1). Results are reported for 1000 adversarial examples misclassi\ufb01ed as toaster with con\ufb01dence above than 90%. 5.2. Comparison with Related Defenses In this section, we report comparisons of our approach with other recent defense methods that transform the input sample to successfully reduce the adversarial effect. The compared methods include both global and local techniques. Note that our method processes image locally so it \f735 4172 1550 270 11640 170 4235 70 Time (in seconds) Defense Methods Defense Run Time Comparison  LGS DW MF GF BF JPEG TVM BR Figure 3: Computational cost comparison of defense methods to process 50k ImageNet validation images. Graph is shown in log scale for better visualization with actual processing times written on the top of each bar in seconds. has advantage over other defenses like JPEG, MF, TVM and BR that process image globally. First, we provide a brief description of the competing defenses which will allow us to elaborate further on the performance trends in Tables 1, 2 and 3. 5.2.1 Digital Watermarking Hayes et.al [10] presented two, non-blind and blind, defense strategies to tackle the challenge of localized attacks [12, 4]. Non-blind defense considers a scenario, where defender has the knowledge of adversarial mask location. This is unlikely scenario in the context of adversarial attacks because threat is over immediately, once the adversary provides the mask location. Localized attacks have the ability to change the attention of classi\ufb01er from the original object to adversarial mask. In their blind defense, authors [10] exploited the attention mechanism by \ufb01rst \ufb01nding the mask location using saliency map and then processing that area before inference. Using saliency map to detect adversarial mask location is the strength of this defense but at the same time its also the weakness of defense because on benign examples, saliency map will give the location of original object and hence processing original object will decrease the performance on clean examples. Authors [10] reported blind defense performance to protect VGG19 [21] on only 400 randomly selected images with 12% accuracy drop on clean images. We have tested this defense on imagenet validation set [18] (50k images). This method has the second best accuracy on adversarial examples after LGS but its accuracy on clean examples expectedly dropped by a large margin (22.8%). Tables 1, 2 and 3 summarizes the performance of digital watermarking [10]. 5.2.2 JPEG Compression [6, 5, 19] extensively studied JPEG compression to defend against adversarial attacks. This way high-frequency components are removed that are less important to human vision by using Discrete Cosine Transform (DCT). JPEG performs compression as follows: \u2022 Convert an RGB image Y CbCr color space, where Y and Cb, Cr represent luminance and chrominance respectively. \u2022 Down-sample the chrominance channels and apply DCT to 8 \u00d7 8 blocks for each channel. \u2022 Perform quantization of frequency amplitudes by dividing with a constant and rounding off to the nearest integer. As illustrated in Table 1, image quality decreases as the degree of compression increases which in turn decreases accuracy on benign examples. JPEG compression is not very effective against localized attacks, and its defending ability decreases a lot against BPDA. JPEG performance comparison is shown in Tables 1, 2 and 3 and Figure 4. 5.2.3 Feature Squeezing The main idea of feature squeezing [26] is to limit the explorable adversarial space by reducing resolution either by using bit depth reduction or smoothing \ufb01lters. We found that bit reduction is not effective against localized attacks, however smoothing \ufb01lter including Gaussian \ufb01lter, median \ufb01lter, and bilateral \ufb01lter reduces localized adversarial effect with reasonable accuracy drop on benign examples. Among smoothing \ufb01lters, median \ufb01lter outperforms Gaussian and bilateral \ufb01lters. Feature squeezing performance is shown in Tables 1, 2 and 3 and Figure 4. 5.2.4 Total Variance Minimization (TVM) Guo et al. [9] considered smoothing adversarial images using TVM along with JPEG compression and image quilting. TVM has the ability to measure small variations in the image, and hence it proved effective in removing small perturbations. As illustrated in Table 1, TVM becomes ineffective against large concentrated variations introduced by the localized attacks. Further comparisons are shown in Tables 2 and 3 and Figure 4. 5.3. Resilience to BPDA BPDA [2] is built on the intuition that transformed images by JPEG or TVM should look similar to original images, that is, T (x) \u2248x. BPDA approximate gradients for non-differentiable operators with combined forward propagation through operator and DNN while ignoring operator \f(a) Dragon\ufb02y (99%) (c) Cardoon (94%) (e) Cardoon (91%) (g) Cardoon (89%) (i) Dragon\ufb02y (70%) (k) Dragon\ufb02y (98%) (m) Dragon\ufb02y (99%) (b) Toaster (94%) (d) Sandpiper (89%) (f) Sandpiper (45%) (h) Sandpiper (55%) (j) Sandpiper (28%) (l) Toaster (90%) (n) Toaster (92%) Figure 4: Inception v3 con\ufb01dence score is shown on example images. (a,b) represent adversarial examples generated by LaVAN and adversarial patch respectively, (c,d) show transformed adversarial images using LGS with lambda equal to 2.3 respectively, (e,f) show transformed adversarial images using DW processing method respectively, (g,h) show transformed adversarial images using median \ufb01lter with window size 3 respectively, (i,j) show transformed adversarial images using JPEG with quality 30 respectively, (k,l) show transformed adversarial images using TVM with weights equal to 10 respectively, and (m,n) show transformed adversarial images using BR with depth 3. during the backward pass. This strategy allows BPDA to approximate true gradients and thus bypassing the defense. In the traditional attack setting like Projected Gradient Descent (PGD) [13], the explorable space available to BPDA is Rn because it can change each pixel in the image. In localized attack setting explorable space reduces to Rm<<n controlled by the mask size. LGS suppresses the highfrequency noise to near zero thereby signi\ufb01cantly reducing gradient activity in the estimated mask area and restricting BPDA to bypass defense. However, as it is the case with all defenses, increasing explorable space, i.e., distance limit in PGD attack [13] and mask size in the case of localized attack [12], protection ability of defense methods decreases. To test performance against BPDA in the localized setting, we randomly selected 1000 examples from Imagenet and attack is optimized against all defenses for the same target, location, mask size and number of iterations. Compared to other defenses methods, LGS signi\ufb01cantly reduces the explorable space for localized adversarial attacks within mask size equal to \u223c2% of the image as discussed in [12]. In the case of DW [10] defense, we tested BPDA against the proposed input processing given the mask location. Summary of attack success rate against defense methods is presented in Table 3. 6. Discussion and"
+                }
+            ],
+            "Salman Khan": [
+                {
+                    "url": "http://arxiv.org/abs/2312.07621v1",
+                    "title": "Spatiotemporal Event Graphs for Dynamic Scene Understanding",
+                    "abstract": "Dynamic scene understanding is the ability of a computer system to interpret\nand make sense of the visual information present in a video of a real-world\nscene. In this thesis, we present a series of frameworks for dynamic scene\nunderstanding starting from road event detection from an autonomous driving\nperspective to complex video activity detection, followed by continual learning\napproaches for the life-long learning of the models. Firstly, we introduce the\nROad event Awareness Dataset (ROAD) for Autonomous Driving, to our knowledge\nthe first of its kind. Due to the lack of datasets equipped with formally\nspecified logical requirements, we also introduce the ROad event Awareness\nDataset with logical Requirements (ROAD-R), the first publicly available\ndataset for autonomous driving with requirements expressed as logical\nconstraints, as a tool for driving neurosymbolic research in the area. Next, we\nextend event detection to holistic scene understanding by proposing two complex\nactivity detection methods. In the first method, we present a deformable,\nspatiotemporal scene graph approach, consisting of three main building blocks:\naction tube detection, a 3D deformable RoI pooling layer designed for learning\nthe flexible, deformable geometry of the constituent action tubes, and a scene\ngraph constructed by considering all parts as nodes and connecting them based\non different semantics. In a second approach evolving from the first, we\npropose a hybrid graph neural network that combines attention applied to a\ngraph encoding of the local (short-term) dynamic scene with a temporal graph\nmodelling the overall long-duration activity. Finally, the last part of the\nthesis is about presenting a new continual semi-supervised learning (CSSL)\nparadigm.",
+                    "authors": "Salman Khan",
+                    "published": "2023-12-11",
+                    "updated": "2023-12-11",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction 1 1.1 Video Action Recognition and Detection . . . . . . . . . . . . . . . 1 1.2 Event Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Road Event Detection . . . . . . . . . . . . . . . . . . . . 2 1.3 Temporal Action Localisation . . . . . . . . . . . . . . . . . . . . 2 1.4 Complex Activity Detection . . . . . . . . . . . . . . . . . . . . . 3 1.5 Scene Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.6 Continual Learning . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . 5 1.8 Organisation of the Thesis . . . . . . . . . . . . . . . . . . . . . . 6 1.9 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.10 List of Packages and Demos . . . . . . . . . . . . . . . . . . . . . 9 2 Related Work 11 2.1 Autonomous Driving Datasets . . . . . . . . . . . . . . . . . . . . 11 2.2 Temporal Action Localisation . . . . . . . . . . . . . . . . . . . . 12 2.3 Complex Activity Detection . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 One-Stage vs Two-Stage Approaches . . . . . . . . . . . . 13 2.4 Graph Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 xv \f2.4.1 GCN-Based Temporal Activity Localisation . . . . . . . . . 14 2.5 Continual Learning . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5.1 Continual Unsupervised Learning . . . . . . . . . . . . . . 15 2.5.2 Time-Varying Hidden Markov Models . . . . . . . . . . . . 15 2.6 Benchmark Datasets . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6.1 ROAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6.2 SARAS-ESAD . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6.3 Thumos-14 . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6.4 ActivityNet-1.3 . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6.5 50-Salad . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 Evaluation Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7.1 Classification Accuracy . . . . . . . . . . . . . . . . . . . 18 2.7.2 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7.3 Recall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7.4 F1-score . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7.5 Mean Average Precision . . . . . . . . . . . . . . . . . . . 19 2.7.6 Mean Absolute Error . . . . . . . . . . . . . . . . . . . . . 19 Part I : ROAD (The ROad event Awareness Dataset for Autonomous Driving) 21 3 The ROad event Awareness Dataset for Autonomous Driving 23 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 ROAD: a multi-label, multi-task dataset . . . . . . . . . . . 26 3.1.3 ROAD-R . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.4 Main Contributions . . . . . . . . . . . . . . . . . . . . . . 29 3.2 The Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 A Multi-Label Benchmark . . . . . . . . . . . . . . . . . . 31 3.2.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.3 Annotation Process . . . . . . . . . . . . . . . . . . . . . . 33 3.2.4 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.5 Quantitative Summary . . . . . . . . . . . . . . . . . . . . 35 3.3 Baseline and Challenge . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 3D-RetinaNet Architecture . . . . . . . . . . . . . . . . . . 36 3.3.2 Online Tube Generation via Agentness Score . . . . . . . . 37 3.3.3 The ROAD Challenge . . . . . . . . . . . . . . . . . . . . 38 3.4 ROAD-R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . 41 xvi \f3.5.1 Implementation Details . . . . . . . . . . . . . . . . . . . . 41 3.5.2 Experimental Results on ROAD . . . . . . . . . . . . . . . 42 3.5.3 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . 46 3.5.4 Challenge Results . . . . . . . . . . . . . . . . . . . . . . 46 3.5.5 Experimental Results of ROAD-R . . . . . . . . . . . . . . 49 3.6 Further Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.7 Summary of the Chapter . . . . . . . . . . . . . . . . . . . . . . . 54 Part II : Complex Activity Detection 55 4 Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.1 Action Tube Detection . . . . . . . . . . . . . . . . . . . . 62 4.2.2 3D Part-Based Deformable Layer . . . . . . . . . . . . . . 63 4.2.3 Graph Convolutional Network . . . . . . . . . . . . . . . . 63 4.2.4 Complete Framework . . . . . . . . . . . . . . . . . . . . . 65 4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.1 Datasets and Evaluation Metrics . . . . . . . . . . . . . . . 67 4.3.2 Action Tube Detection . . . . . . . . . . . . . . . . . . . . 69 4.3.3 Complex Activity Recognition . . . . . . . . . . . . . . . . 69 4.3.4 Temporal Activity Detection Comparison with State-of-theArt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.5 Ablation Studies . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4.1 Limitations and Future Work . . . . . . . . . . . . . . . . . 74 4.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5 A Hybrid Graph Network for Complex Activity Detection in Video 75 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2.1 Tube Detection and Feature Extraction . . . . . . . . . . . 81 5.2.2 Scene Graph Attention . . . . . . . . . . . . . . . . . . . . 82 xvii \f5.2.3 Temporal Graph Localisation . . . . . . . . . . . . . . . . 83 5.2.4 Loss Functions . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3.2 Implementation Details . . . . . . . . . . . . . . . . . . . . 86 5.3.3 Comparison with State-of-the-Art . . . . . . . . . . . . . . 87 5.3.4 Ablation Studies . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Part III : Continual Learning 95 6 Continual Semi-Supervised Learning 97 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.1.2 Overview of the CSSL Challenge . . . . . . . . . . . . . . 98 6.1.3 Overview of Hidden Markov Model as Latent Variable . . . 99 6.1.4 Main Contributions . . . . . . . . . . . . . . . . . . . . . . 100 6.2 Continual Semi-Supervised Learning Workshop and Challenge . . . 102 6.2.1 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2.2 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2.3 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2.5 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.2.6 Baselines . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.3 HMM Latent Variable Model for Unsupervised Data-Streams . . . . 109 6.3.1 Principles of HMM Latent Variable Model . . . . . . . . . 109 6.3.2 A Two-Stage Learning Process . . . . . . . . . . . . . . . . 110 6.3.3 Joint Distribution of the Data Stream . . . . . . . . . . . . 110 6.3.4 Discrete-State T-HMM for Continuous Classification . . . . 111 6.3.5 Random Sampling based Memory Replay . . . . . . . . . . 115 6.3.6 Alternative to HMM as Latent Variable Model . . . . . . . 116 6.4 Experimental Analysis of Challenge Baselines . . . . . . . . . . . . 116 6.4.1 Results on Activity Recognition . . . . . . . . . . . . . . . 117 6.4.2 Results on Crowd Counting . . . . . . . . . . . . . . . . . 118 6.5 Experimental Analysis of HMM latent Variable . . . . . . . . . . . 118 6.5.1 Datasets and Setups . . . . . . . . . . . . . . . . . . . . . . 118 6.5.2 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . 120 xviii \f6.6 Summary of the Chapter . . . . . . . . . . . . . . . . . . . . . . . 121 7 Conclusions and Future Research Directions 123 7.1 Summary of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 123 7.1.1 ROad event Awareness Dataset for Autonomous Driving . . 123 7.1.2 Complex Activity Detection . . . . . . . . . . . . . . . . . 124 7.1.3 Continual Semi-Supervised Learning . . . . . . . . . . . . 125 7.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . 126 7.2.1 Extension of ROAD . . . . . . . . . . . . . . . . . . . . . 126 7.2.2 Holistic Graph Approach for Complex Activity Detection and Prediction . . . . . . . . . . . . . . . . . . . . . . . . 126 7.2.3 Extension to Spatiotemporal Sentence Grounding . . . . . . 126 7.2.4 Continual Learning Extension to Regression . . . . . . . . 127 References 129 xix \fxx \fList of Figures 1.1 Our complex activity recognition framework workflow for two videos from the ROAD (autonomous driving) and SARAS (surgical robotics) datasets. The overall concept of detecting complex activities via a part-based deformation mechanism is the same in both cases. However, in ROAD background frames exist which do not contain any complex road activity. In contrast, the SARAS complex activities are actually the phases of a surgical procedure, which are contiguous without the need for a background label. . . . . . . . . . . . . . . 4 3.1 Use of labels in ROAD to describe typical road scenarios. (a) A green car is in front of the AV while changing lanes, as depicted by the arrow symbol. The associated event will then carry the following labels: in vehicle lane (location), moving left (action). Once the event is completed, the location label will change to: in outgoing lane. (b) Autonomous vehicle turning left from lane 6 into lane 4: lane 4 will be the outgoing lane as the traffic is moving in the same direction as the AV. However, if the AV turns right from lane 6 into lane 4 (a wrong turn), then lane 4 will become the incoming lane as the vehicle will be moving into the incoming traffic. The overall philosophy of ROAD is to use suitable combinations of multiple label types to fully describe a road situation, and allow a machine learning algorithm to learn from this information. . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Number of instances of each class of individual label-types, in logarithmic scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Proposed 3D-RetinaNet architecture for online video processing. . . 36 3.4 Success cases in which our model detects the actions and locations correctly, and only for those agents which are active. Ground truth bounding boxes and labels are shown in green, while the predictions of our model are shown in red. . . . . . . . . . . . . . . . . . . . . 47 xxi \f3.5 Some of the failure modes of our model, shown detecting inactive agents and/or assigning incorrect action and location labels. Ground truth bounding boxes and labels are shown in green, while the predictions of our model are shown in red. . . . . . . . . . . . . . . . 48 3.6 Examples of violations of {\u00acRedTL, \u00acGreenTL}. . . . . . . . . . . 51 3.7 ROAD-R and SOTA models. In the x-axis, there is the threshold \u03b8 \u2208[0.1, 0.9], step 0.1. . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1 Overall pipeline of our long-term complex activity detection framework. (i) The input video is first divided into snippets. (ii) Snippets are passed to an action tube detector module one by one. (iii) A partbased deformation module receives 3D VGG features and action tube locations and returns features for the salient (non-background) parts of each tube instance. (iv) A GCN module represents activity parts (action tubes) as nodes with the features generated by ROI pooling and builds edges with different semantics to construct a spatiotemporal graph representation. Finally (v), the graph representation features produced by GCN inference on the consolidated graph are used for temporal activity detection. . . . . . . . . . . . . . . . . . . . . 60 4.2 (a) AMTNet [1] generates micro-action tubes by predicting bounding boxes for the start and the end frame of a snippet. Predictions for a fixed number of intermediate frames are constructed by bilinear interpolation. (b) Complete action tubes are created by temporally linking micro action tubes by dynamic programming. . . . . . . . . 62 4.3 Our 3D deformable RoI pooling layer takes feature maps and action tube locations as input and arranges them into a fixed-size grid of components (here illustrated for size 3\u00d73). For each grid component an offset is generated and multiplied by the original tube feature to produce the final component features. . . . . . . . . . . . . . . . . 64 4.4 Our graph module takes as input the features generated for each tube by the ROI pooling layer T1,2,3...K and builds edges between them according to different semantics. ( order of appearance, spatial similarity, label). The overall graph is processed by a GCN to deliver a fixed-size graph representation. . . . . . . . . . . . . . . . . . . . . 65 4.5 Complex activity classification performance over ROAD dataset. . 70 4.6 Complex activity classification performance on SARAS-ESAD dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 xxii \f4.7 The performance of different components of our proposed method over ROAD and SARAS-ESAD dataset. The mAP at a threshold of 0.5 is reported for the test set of both datasets. . . . . . . . . . . . 72 4.8 The performance comparison of various edge types in our GCN modules. The mAP at a threshold of 0.5 is reported for the test set of both datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1 An overview of our Complex Activity Detection (CompAD) framework. The input video is divided into fixed-size snippets S1, . . . , SN (top); each snippet is then processed in three major steps (bottom). A) Firstly, scene objects (agents) are detected and tracked throughout the snippet to form agent tubes. 3D features are then extracted from all the cropped agent tubes (fai) as well as the local scene (fsi). B) Next, a fully-connected scene graph is constructed where agent nodes (in gray) are connected to each other and to the snippet node (in light blue). The local scene graph is processed using a graph attention network (GAT), resulting in intermediate scene features (f \u2032\u2032 si). C) Finally, all local scene features associated with individual snippets are temporally connected and processed using a global temporal graph to identify the boundaries of the activity using anchor proposals (shown in different colors). Each of the anchors represents a binary graph mask where snippets which belong to the activity receive label ybri = 1, 0 otherwise. . . . . . . . . . . . . . . . . . . . . . 76 5.2 Comparing TAL datasets (ActivityNet-1.3 and Thumos-14) with a CompAD dataset (ROAD) in terms of average activity and video durations and mean number of agents per snippet. . . . . . . . . . . . 78 5.3 Visualisation of our agent detection and tracking stage using a bird\u2019seye view of the spatiotemporal volume corresponding to a video segment of ActivityNet-1.3 dataset. The first half of the graph shows the plotting of agent tubes while the second half illustrates the scene graph of the detection agents. In this figure, we show examples of activities performed by multiple actors (agent tubes). . . . . . . . . 80 5.4 Visualisation of our agent detection and tracking stage using a bird\u2019seye view of the spatiotemporal volume corresponding to a video segment of Thumos-14 dataset. The first half of the graph shows the plotting of agent tubes while the second half illustrates the scene graph of the detection agents. Both of the examples show the activities performed by multiple agents (recognising sports activities). 82 xxiii \f5.5 Visualisation of our agent detection and tracking stage using a bird\u2019seye view of the spatiotemporal volume corresponding to a video segment of the ROAD dataset. The upper section shows an example of activity performed by only three agents. Below is the example of a night video where the activity is performed by multiple agents. The first half of the graph shows the plotting of agent tubes while the second half illustrates the scene graph of the detection agents. . . . 83 5.6 Our scene graph attention layer (Stage B of our approach, Fig. 5.1) takes the node features generated in the feature extraction Stage A as an input, and updates each node\u2019s features using node attention with respect to its all connected neighbours. Multi-head attention with H heads is applied to each node to further robustify the representation \u2013 averaging yields the final node features. To obtain a fixed-size overall representation for a specific local scene (snippet), the features of all its nodes are aggregated using a learnable weight matrix W2. . . . . 84 5.7 Class-wise average precision (%) with IoU threshold of 0.5 for ROAD dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.8 Class-wise average precision (%) with IoU threshold of 0.5 for Thumos14 dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.9 Performance comparison with the state-of-the-art over all the three datasets considered, as a function of average activity duration and average no. of agents per snippet. . . . . . . . . . . . . . . . . . . 90 5.10 Average mAP of variants of our method with: attended scene graph (\u2019Agents with GAT\u2019), scene graph (\u2019Agents without GAT\u2019) and \u2019Scene features only\u2019, over the all three datasets. . . . . . . . . . . . . . . . 91 5.11 The qualitative results of our proposed method for all three datasets. The green rectangles covering the snippets (local scenes) are the ground truth while the yellow boxes show the prediction of our model. 93 5.12 The qualitative results of our proposed method for all three datasets. The green rectangles covering the snippets (local scenes) are the ground truth while the yellow boxes show the prediction of our model. 94 6.1 Overview of the proposed latent variable pipeline. . . . . . . . . . . 100 xxiv \f6.2 Overall pipeline of our CAR baseline. 1) Firstly, a pre-trained EfficientNet model is fine-tuned over the supervised fold of the dataset (jointly over the 15 sequences). 2) The unlabeled validation and test folds are divided into subfolds. In each session video frames from each sub-fold are used in a self-training cycle in which a pseudo label PL with prediction probability score Ps is generated for each frame, but only the pseudolabels with Ps above a fixed threshold (namely, 0.4) are considered as ground truth for retraining the model. Frames with lower Ps are ignored. The model updated by self-training in session n is used as initial model in session n + 1. . . . . . . . . . 107 6.3 Time-varying HMM (dashed bars) vs. traditional HMM (colored bars).120 xxv \fxxvi \fList of Tables 2.1 Comparison of ROAD with similar datasets for perception in autonomous driving in terms of diversity of labels. The comparison is based on the number of classes portrayed and the availability of action annotations and tube tracks for both pedestrians and vehicles, as well as location information. Most competitor datasets do not provide action annotation for either pedestrians or vehicles. . . . . . . . 12 3.1 ROAD labels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 ROAD tasks and attributes. . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Constraint statistics. \u03a0n is the set of constraints r in \u03a0 with |r| = n, i.e., with n positive and negative labels. C = {\u00acA : A \u2208C}. Each row shows the number of rules r with |r| = n, and the average number of negative and positive labels in such rules. . . . . . . . . . 40 3.4 Splits of training, validation and test sets for the ROAD dataset with respect to weather conditions. The table shows the number of videos in each set or split. For splits, the first figure is the number of training videos, the second number that of validation videos. . . . . . . . . . 41 3.5 Frame-level results (mAP %) averaged across the three splits of ROAD. The considered models differ in terms of backbone network (2D, I3D, and Slowfast) and clip length (08 vs 32). The performance of YOLOv5 on agent detection is also reported. Detection threshold \u03b4 = 0.5. Both validation and test performance are reported for each entry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Video-level results (mAP %) averaged across the three ROAD splits. The models differ in terms of backbone network (2D, I3D and Slowfast) and test time clip length (08 vs 32). The performance of YOLOv5 on agent detection is also reported. Both validation and test performance are reported for each entry. . . . . . . . . . . . . . . . . . . 43 xxvii \f3.7 Number of videoand frame-level instances for each label (individual or composite), left. Corresponding frame-/video-level results (mAP@%) for each of the three ROAD splits (right). Val-n denotes the validation set for Split n. Results produced by an I3D backbone. 44 3.8 AV temporal action detection results (frame mAP%) averaged across all three splits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.9 Results (in video-mAP) of the winning entries to the ICCV 2021 ROAD challenge compared with the Slowfast and YOLOv5 baselines, at a detection threshold of 0.2. . . . . . . . . . . . . . . . . . 49 3.10 f-mAP@0.5 and f-mAP@0.75 for the current SOTA models. . . . . 52 4.1 List of ROAD complex activities with number of train and test snippets splits and brief description. . . . . . . . . . . . . . . . . . . . 67 4.2 SARAS video phases with number of train and test snippets and a brief description. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 Action tube detection performance on both the ROAD and SARASESAD datasets. Both Frame-mAP and Video-mAP at different IoU thresholds are reported for evaluation . . . . . . . . . . . . . . . . . 69 4.4 Comparative analysis of temporal activity localisation performance on ROAD and SARAS-ESAD, reporting mAP (%) at five different IoU thresholds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 ROAD activity localisation performance (mAP, %) for each complex road activity, at a standard IoU threshold of 0.5. . . . . . . . . . . . 71 4.6 SARAS-ESAD activity localisation performance (mAP, %) for each activity at a fixed IoU threshold of 0.5. . . . . . . . . . . . . . . . . 71 4.7 The performance comparison of different sequence lengths over both ROAD and SARAS-ESAD dataset, reporting mAP (%) at two different IoU thresholds. . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1 Comparing our approach with the state-of-the-art methods for CompAD available on the ROAD dataset. The mAP at the various standard thresholds is reported. Best results are in bold and second best underlined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Activity detection performance comparison with state-of-the-art methods on Thumos-14. mAP values at different IoU thresholds are reported for the test Thumos-14. Best results are in bold and second best underlined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 xxviii \f5.3 Activity detection performance comparison with state-of-the-art methods on ActivityNet-1.3. mAP values at different IoU thresholds are reported for validation sets of ActivityNet-1.3. Best results are in bold and second best underlined. . . . . . . . . . . . . . . . . . . 89 5.4 Effect of local scene graph topologies and feature aggregation strategies on the performance of our proposal. The average mAP for the relevant IoU thresholds is reported for each dataset. . . . . . . . . . 91 5.5 Performance of our method as a function of different snippet lengths over the three datasets. The average mAP for the relevant IoU thresholds is reported for each dataset. . . . . . . . . . . . . . . . . . . . 92 5.6 Performance of our method as a function of different temporal graph lengths over the three datasets. The average mAP for the relevant IoU thresholds is reported for each dataset. . . . . . . . . . . . . . . 92 6.1 Performance of the initial supervised model versus that of the incrementally updated models (separately on V and T or in combination), using three standard evaluation metrics. For each metric we report both class average (C) and weighted average (W). . . . . . . . . . . 117 6.2 Evaluation on the CCC validation (V ) and test (T) folds for different experimental setups. Here sup\u2212ft, upd\u2212V , and upd\u2212V +T refer to the supervised model fine-tuned on S, the model incrementally updated via self-training in sessions on the validation fold V , and the model incrementally updated on the combined validation V and test T folds, respectively. The average test MAE is reported for each model.117 6.3 Time-varying HMM (TV HMM) vs. traditional HMM (HMM) based on one-off training, where P, R, and F1 are precision, recall, and F1score, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.4 Time-varying HMM (TV HMM) vs. traditional HMM (HMM) based on continual retraining, where P, R, and F1 are precision, recall, and F1-score, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 120 6.5 Time-varying HMM (TV HMM) vs. traditional HMM (HMM) based on incremental update, where P, R, and F1 are precision, recall, and F1-score, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 120 xxix \fxxx \fNomenclature Acronyms C3D: Convolutional 3D Network I3D: Inflated Convolutional 3D Network mAP: mean Average Precision RGB: Red Green Blue, 3-Channel Camera output TAL: Temporal Action Localisation AUC: Area Under the Curve AV: Autonomous Vehicle BB: Bounding Box CompAD: Complex Activity Detection CL: Constrained Loss CO: Constrained Output CAR: Continual Activity Recognition CCC: Continual Crowd Counting CSSL: Continual Semi-Supervised Leanring CNN: Convolutional Neural Network EM: Expectation Maximisation FPN: Feature Pyramid Network fps: frames per second GAT: Graph Attention Network xxxi \fGCN: Graph Convolutional Network HMM: Hidden Markov Model HMC: Hierarchical Multi-label Classification IoU: Intersection over Union LSTM: Long Short-Term Memory recurrent neural network MAE: Mean Absolute Error MC: Multi-label Classification 1D: One Dimensional RNN: Recurrent Neural Network RE: ROAD Event SSL: Semi-Supervised Learning ESAD: the Endoscopic Surgeon Action Detection Dataset ROAD: The ROad Awareness Dataset for Autonomous Driving ROAD-R: The ROad Awareness Dataset with logical Requirements 3D: Three Dimensional (spatiotemporal for video) T-HMM: Time-varying Hidden Markov Model 2D: Two Dimensional (spatially for image) xxxii \fChapter 1 Introduction Scene understanding from videos has a significant role in the computer vision community due to its vast real-world applications such as autonomous driving [2, 3], surveillance [4], medical robotics [5] or team sports analysis [6]. The scene understanding problem can be addressed by a variety of techniques starting from simple video action recognition to complex activity detection. In this thesis, we address the problem in a comprehensive fashion, starting from individual event detection to holistic complex scene analysis (complex activity detection). 1.1 Video Action Recognition and Detection Video action recognition is the task of identifying and classifying human actions in video sequences. Video action detection, on the other hand, is the process of identifying and locating the actions of an individual within a video, in both space and time. In the current literature, a number of convolutional neural networks (CNNs) frameworks are used to learn spatial features from individual video frames, which are then fed to recurrent neural networks (RNNs) to model the temporal dynamics across frames [7]. Another approach is to use 3D CNNs, which can learn spatiotemporal features directly from the video data [8]. One of the key challenges in video action recognition and detection is the need to deal with the large variability in the appearance and motion of the actions across different videos. To address this, researchers often use techniques such as data augmentation and transfer learning to improve the robustness of the models [9,10]. Additionally, some researchers have proposed using attention mechanisms to allow the model to focus on the most relevant parts of the video when detecting and recognising actions [11]. Video action recognition and detection have applications to a wide range of areas such as surveillance, human-computer interaction and sports analytics, to name a few. 1 \f2 1.2 Event Detection Video event detection refers to the task of identifying and localising specific events or activities in a video. Such activities may be performed by a single agent (object) or by multiple agents at the same time. Examples include activities such as sports events [12], traffic incidents [13], disaster management [14, 15] or criminal activities. The most popular deep learning-based method for video event detection is the two-stream CNN, which uses two streams of CNNs, one for processing the RGB frames and the other for processing the optical flow frames [1]. Another popular method is the 3D CNN, which uses 3D convolutional kernels that can learn spatiotemporal features from the video. These features can then be used to detect events over time, making 3D CNNs useful for tasks such as road events for decision-making in autonomous driving. 1.2.1 Road Event Detection In this thesis, we define road events to be the collection of three labels (agent, action, and location) associated with a single \u201ctube\u201d (the spatiotemporal representation of a series of bounding boxes). The agent is an active road user (object); the event triple also contains the action(s) the latter performs (possibly more than one at the same time), and the location(s) in which this takes place (which may vary from the start to the end of the event itself), as seen from the point of the view of an autonomous vehicle. 1.3 Temporal Action Localisation The ability to recognise and localise activities temporally from untrimmed videos is a research problem which has been attracting significant attention, due in part to the rapid growth of online video generation. The current state-of-the-art in action detection is Temporal Action Localisation (TAL) whose approaches not only recognise the action label(s) present in a video, but also identify the start and end time of each action instance occurring. TAL can be instrumental in generating sports highlights [16], understanding road scenes in autonomous driving [17], video summarisation in surveillance videos [18] and video captioning [19]. A number of TAL methods [20\u201329] have recently been proposed. Whereas various new datasets have been recently proposed, the two most common relevant benchmarks remain ActivityNet 1.3 [30], and Thumos-14 [31]. State-of-the-art performance on Thumos-14 has improved in four years by some 19% [32]. \fChapter 1. Introduction 3 Almost all TAL approaches contemplate two major aspects: features/scene representation and temporal localisation. In the scene representation stage, snippets (continuous sequences of frames) are processed to understand the local scene in the video. Methods [27,28] typically employ pre-extracted features obtained using a sequential learning model (e.g., I3D [33]), often pre-trained on the Kinetics [34] dataset. Such features are then processed, e.g., via a temporal or semantic graph neural network, by applying appropriate feature encoding techniques or by generating temporal proposals, in an object detection style [35]. In the second step of TAL, actions need to be temporally localised. Existing methods vary in the way this is done, e.g. through temporal graphs [21, 25], boundary regression and proposals generation [20, 22, 28] or encoder-decoder approaches [32]. All such methods, however, are only applicable to shortor mid-duration actions that last for a few seconds (e.g., a person jumping or pitching a baseball). 1.4 Complex Activity Detection Complex activities, in opposition, can be defined as longer-term events, typically comprising a series of shorter actions. For example, an autonomous vehicle (AV) \u201cnegotiating a pedestrian crossing\u201d presents a complex activity first the AV drives along the road, then the traffic lights switches to red, the vehicle stops and several pedestrians cross the road. Eventually, the lights change back to green, and the AV drives off again. Whereas a number of studies have attempted to address the localisation of longer-term activities [36\u201338], these approaches mostly focus on activities performed by single actor. To the author\u2019s knowledge, [36] is the only existing study which attempts to model truly complex activities, performed by multiple actors over an extended period of time. The work, however, relies upon the availability of heavily-annotated datasets which include granular labels for the individual actions which make up a complex activity, and the corresponding frame-level bounding boxes. A general overview of a complex activity framework is visualised in Figure 1.1. 1.5 Scene Graphs A scene graph is a graph-based representation of a scene, where the nodes of the graph represent entities in the scene (such as objects, people, and locations) and the edges represent relationships between those entities (such as proximity, or interaction) [39]. Scene graphs are used in computer vision [40] and natural language processing [41] to represent the semantic structure of a scene and to facilitate tasks \f4 Figure 1.1: Our complex activity recognition framework workflow for two videos from the ROAD (autonomous driving) and SARAS (surgical robotics) datasets. The overall concept of detecting complex activities via a part-based deformation mechanism is the same in both cases. However, in ROAD background frames exist which do not contain any complex road activity. In contrast, the SARAS complex activities are actually the phases of a surgical procedure, which are contiguous without the need for a background label. \fChapter 1. Introduction 5 such as object detection, scene understanding, and natural language generation. Scene graphs are typically constructed by analysing images or videos, using techniques such as object detection, instance segmentation, and scene understanding [42]. The nodes of the graph are typically labeled with object categories, and the edges are labeled with the type of relationship between the entities. Due to the vast importance of scene graphs in scene understanding, in this thesis, we also adopt a scene graph representation for modelling complex activities. 1.6 Continual Learning Continual learning refers to the ability of a machine learning model to learn and adapt to new information over time, without forgetting previously acquired knowledge [43]. In other words, it allows a model to continuously improve its performance by learning from new data, even after it has been deployed in the real world. This is an important aspect of developing AI systems that can function in dynamic, realworld environments and adapt to changing conditions [44]. Continual learning is an active area of research in machine learning, with the goal of developing models that can learn and adapt throughout their lifetime. A typical example is that of an object detector which needs to be extended to include classes not originally in its list (e.g., \u2018donkey\u2019) while retaining its ability to correctly detect, say, a \u2018horse\u2019. The hidden assumption there is that we are quite satisfied with the model we have, and we merely wish to extend its capabilities to new settings and classes. In the last part of the thesis, we also proposed two continual learning paradigms in Chapter 6 for updating the machine learning model continually in a semi-supervised manner. 1.7 Contributions of the Thesis The main contributions of this thesis are the following: 1. A ROad event Awareness Dataset for Autonomous Driving (ROAD), the first of its kind, designed to support a paradigm shift toward the detection of semantically meaningful road events, and allow the testing of a range of tasks related to situation awareness for autonomous driving: agent and/or action detection, event detection, ego-action classification. 2. The ROad event Awareness Dataset with logical Requirements (ROAD-R), the first publicly available dataset whose commonsense requirements are expressed in full propositional logic, equipped with baseline models. \f6 3. A spatiotemporal scene graph generation and processing mechanism for complex activity detection able to cope with a variable number of parts while learning the overall semantics of an activity class. 4. An extension of 2D deformable RoI pooling to 3D for flexibly pooling features from the various components of the detected tubes, to create an overall representation for activity parts. 5. A hybrid graph network approach for general complex activity detection, comprising a local scene graph as well as a global temporal graph, capable of localising both long and short-term activities. 6. A feature extraction strategy for fine-grained temporal activity localisation without the need for additional annotation, consisting in detecting and tracking the relevant scene agents to construct tubes, followed by feature extraction from both the whole scene and each agent tube individually, as opposed to existing methods which only rely on snippet-level features. 7. A formal definition of the continual semi-supervised learning (CSSL) problem, and the associated training and testing protocols, with a suitable baseline. The first two benchmark datasets for the validation of semi-supervised continual learning approaches, one for classification (continual activity recognition) and one for regression (continual crowd counting), which we propose as the foundations for the first challenge in this new domain area. 8. A principled two-stage approach to the CSSL problem, based on a latent model formulation which reduces the task to continual supervised learning. 1.8 Organisation of the Thesis Firstly, we review all the relevant related work, including the benchmark datasets and evaluation metrics, in Chapter 2. Next, the main body of the thesis (Chapters 3 to 7) is divided into three parts; Part I, Part II, and Part III. Finally, the thesis is concluded in Chapter 7 by a summary and a discussion of possible future extensions. The theme of Part I 2.7.6 is ROAD (The ROad event Awareness Dataset for Autonomous Driving) which includes two chapters 3. In Chapter 3, we present the baseline ROAD dataset, which is designed to test an autonomous vehicle\u2019s ability to detect road events, defined as triplets composed by an active agent, the action(s) it performs, and the corresponding scene locations. We also introduce the ROad event Awareness Dataset with logical Requirements (ROAD-R) in Part I, the first publicly \fChapter 1. Introduction 7 available dataset for autonomous driving with requirements expressed as logical constraints, as a neurosymbolic extension to ROAD. Next, in Part II 3.7 we present the concept of Complex Activity Detection (CompAD), over two Chapters (Chapter 4 and Chapter 5). Chapter 4 presents a novel deformable, spatiotemporal scene graph approach for CompAD, consisting of three main building blocks: (i) action tube detection, (ii) the modelling of the deformable geometry of parts, and (iii) a graph convolutional network. The latter is further improved in Chapter 5 by addressing the CompAD problem using a hybrid graph neural network that combines attention applied to a graph encoding the local (short-term) dynamic scene with a temporal graph modelling the overall long-duration activity. Finally, Part III 5.4.2 is about continual learning and is composed of a single Chapter6. There, we present a new continual semi-supervised learning (CSSL) paradigm, which we proposed to the attention of the machine learning community via the IJCAI 2021 International Workshop on Continual Semi-Supervised Learning. In addition to the baseline made available for the challenge, we also propose to formulate the continual semi-supervised learning problem as a latent-variable one, in which the labels of a series of unlabelled, streaming data points play the role of the hidden states, while the instances play that of the observations. 1.9 Publications Part of thesis: the following is the list of publications that form the core of this PhD thesis. 1. Salman Khan, Izzeddin Teeti, Andrew Bradley, Mohamed Elhoseiny, and Fabio Cuzzolin, \u201cA Hybrid Graph Network for Complex Activity Detection in Video\u201d, Pre-print arXiv, (2023). 2. Gurkirt Singh, ..., Salman Khan, ..., and Fabio Cuzzolin, \u201cROAD: The ROad event Awareness Dataset for autonomous Driving\u201d, in IEEE Transactions on Pattern Analysis and Machine Intelligence (IEEE TPAMI) 45, pp. 1036-1054, January 2023. 3. E. Giunchiglia, Salman Khan, ..., F. Cuzzolin and T. Lukasiewicz, \u201cROAD-R: the Autonomous Driving Dataset for Learning with Requirements\u201d, Machine Learning, Springer, September 2022 (accepted with minor revision) 4. A. Shahbaz, Salman Khan, ... and F. Cuzzolin, \u201cInternational Workshop on Continual Semi-Supervised Learning: Introduction, Benchmarks and Baselines\u201d, in CSSL 2021, Lecture Notes in Artificial Intelligence Vol. 13418, pages 1-14, Springer, 2022 \f8 5. Salman Khan and Fabio Cuzzolin, \u201cSpatiotemporal Deformable Scene Graphs for Complex Activity Detection\u201d, in Proceedings of The 32nd British Machine Vision Conference (BMVC), 2021. 6. Eleonora Giunchiglia, Mihaela Catalina Stoian, Salman Khan, Fabio Cuzzolin and Thomas Lukasiewicz, \u201cROAD-R: The Autonomous Driving Dataset with Logical Requirements\u201d, in Proceedings of IJCAI 2022 Workshop on Artificial Intelligence for Autonomous Driving (AI4AD 2022) and IJCLR 2022. 7. Ajmal Shahbaz, Salman Khan, Mohammad Asiful Hossain, Vincenzo Lomonaco, Kevin Cannons, and Fabio Cuzzolin, International Workshop on Continual Semi-Supervised Learning: Introduction, Benchmarks and Baselines, in Proceedings of IJCAI Workshop, International Workshop on Continual Semi-Supervised Learning (2021). 8. Fabio Cuzzolin, Salman Khan, Ajmal Shahbaz \u201cContinual Semi-Supervised Learning for Classification via a Time-Varying Hidden Markov Model\u201d, Preprint arXiv 2023. Other works: Below is the list of other works which I published in the course of my PhD term, but are not directly connected to the topic of this thesis. 1. Muhammad, Khan, Hayat Ullah, Salman Khan, Mohammad Hijji, and Jaime Lloret. \u201dEfficient Fire Segmentation for Internet-of-Things-Assisted Intelligent Transportation Systems.\u201d IEEE Transactions on Intelligent Transportation Systems, September 2022. 2. Salman Khan, Khan Muhammad, Tanveer Hussain, Javier Del Ser, Fabio Cuzzolin, Siddhartha Bhattacharyya, Zahid Akhtar, and Victor Hugo C. de Albuquerque. \u201dDeepsmoke: Deep learning model for smoke detection and segmentation in outdoor environments.\u201d Expert Systems with Applications 182 (2021): 115125. 3. Teeti, Izzeddin, Salman Khan, Ajmal Shahbaz, Andrew Bradley, and Fabio Cuzzolin. \u201dVision-based Intention and Trajectory Prediction in Autonomous Vehicles: A Survey.\u201d Proceedings of IJCAI 2022 (Survey track), pages 56305637. 4. Teeti, Izzeddin, Valentina Musat, Salman Khan, Alexander Rast, Fabio Cuzzolin, and Andrew Bradley. \u201dVision in adverse weather: Augmentation using CycleGANs with various object detectors for robust perception in autonomous racing.\u201d ICML 2022 Workshop on Safe Learning for Autonomous Driving (SL4AD 2022). \fChapter 1. Introduction 9 5. Onyeogulu, Tochukwu, Amirul Islam, Salman Khan, Izzeddin Teeti, and Fabio Cuzzolin. \u201dSituation Awareness for Automated Surgical Check-listing in AIAssisted Operating Room.\u201d arXiv preprint arXiv:2209.05056 (2022). 6. Jin, Kaizhe, Adrian Rubio-Solis, Ravi Naik, Tochukwu Onyeogulu, Amirul Islam, Salman Khan, Izzeddin Teeti et al. \u201dIdentification of Cognitive Workload during Surgical Tasks with Multimodal Deep Learning.\u201d arXiv preprint arXiv:2209.06208 (2022). 1.10 List of Packages and Demos \u2022 All resources of our ROAD project (IEEE TPAMI 2022 [45] and The ROAD Workshop and Challenge: Event Detection for Situation Awareness in Autonomous Driving, ICCV 2021) are publicly available at: Workshop and Challenge link: https://sites.google.com/view/ roadchallangeiccv2021/ ROAD dataset: https://github.com/gurkirt/road-dataset 3D RetinaNet baseline: https://github.com/gurkirt/3D-RetinaNet Inference demo: https://www.youtube.com/watch?v=CmxPjHhiarA Challenge evalAI https://eval.ai/web/challenges/challenge-page/ 1059/overview. \u2022 The resources of our ROAD-R project [46] are publicly available at: code: https://github.com/EGiunchiglia/ROAD-R video: https://www.youtube.com/watch?v=_-Ll5d1VQXY. \u2022 The source code of our BMVC 2021 [17] is publicly available at: https:// github.com/salmank255/SDSG_Complex_Activity. \u2022 The source code of our CompAD paper Pre-print is publicly available at: https: //anonymous.4open.science/r/HG_CompAD-7697/README.md. \u2022 All resources of the continual learning project (Continual Semi-Supervised Learning Workshop and Challenge at IJCAI 2021 [47]) are publicly available at: Workshop link: https://sites.google.com/view/sscl-workshop-ijcai-2021/ Code: https://github.com/salmank255/cssl CAR challenge baseline: https://github.com/salmank255/IJCAI-2021-Continual \f10 CAR evalAI: https://eval.ai/web/challenges/challenge-page/ 984/overview CCC challenge baseline: https://github.com/Ajmal70/IJCAI_2021_ Continual_Crowd_Counting_Challenge CCC evalAI: https://eval.ai/web/challenges/challenge-page/ 986/overview. \fChapter 2 Related Work This chapter presents a literature review of the most relevant methods in the areas (discussed in Chapter 1) which most contribute to scene understanding. We start from a comparison of the existing major autonomous driving datasets (Section 2.1) with our ROAD dataset, which is presented in Chapter 3. Next, we provide a literature review of temporal action localisation, complex activity detection, graph convolutional networks, and continual learning in Sections 2.2, 2.3, 2.4, and 2.5, respectively. We also discuss the benchmark datasets used for the evaluation of our proposed methods in Section 2.6. Lastly, we introduce all the evaluation metrics (Section 2.7) used for the evaluation of the different tasks targeted in this thesis. 2.1 Autonomous Driving Datasets In recent years a multitude of AV datasets have been released, mostly focusing on object detection, scene segmentation, trajectory prediction, and action detection. We can categorise them into two main bins: (1) RGB without range data (single modality) which is relatively less time-consuming and expensive than building multimodal datasets including range data from LiDAR or radar and (2) RGB with range data (multimodal) that contain multiple types of data, such as lidar, camera, and radar data. To compare ROAD (proposed in Chapter 3) with autonomous driving perception datasets, we provide a comparative analysis of the recent state-of-the-art datasets in Table 2.1. As it can be noted in the table, the unique feature of ROAD is its diversity in terms of the types of actions and events portrayed, by all types of road agents in the scene. With 12 agent classes, 30 action classes and 15 location classes ROAD provides (through a combination of these three elements) a much more refined description of road scenes. 11 \f12 Table 2.1: Comparison of ROAD with similar datasets for perception in autonomous driving in terms of diversity of labels. The comparison is based on the number of classes portrayed and the availability of action annotations and tube tracks for both pedestrians and vehicles, as well as location information. Most competitor datasets do not provide action annotation for either pedestrians or vehicles. Dataset Class Num. Ann. frames Countries Loc. lab. Avg. obj./f Action Ann Tube Ann Ped. Veh. Ped. Veh. SYNTHIA [48] 13 13400 1 pixelwise ann. Cityscapes [49] 30 5000 2 pixel level sem. Waymo [50] 4 1,200,000 1 20 \u2713 \u2713 Apolloscape [51] 25 140,000 1 pixel level sem. \u2713 \u2713 PIE [52] 6 293,437 1 2.5 \u2713 \u2713 TITAN [53] 50 75,262 1 5.2 \u2713 \u2713 \u2713 \u2713 KITTI360 [54] 19 83,000 1 sem. ann. A*3D [55] 7 39,000 1 4.8 H3D [56] 8 27,721 1 16 \u2713 \u2713 Argoverse [57] 15 70,557 1 18 \u2713 \u2713 NuScense [2] 23 40,000 2 3D sem. seg. 20 \u2713 \u2713 DriveSeg [58] 12 5,000 1 sem. ann. Spatiotemporal action detection datasets UCF24 [59] 24 80175 1.5 \u2713 \u2713 AVA [60] 80 1,600,000 3 \u2713 \u2713 Multisports [61] 66 37701 2.5 \u2713 \u2713 ROAD [45] 43 122,000 1 \u2713 5 \u2713 \u2713 \u2713 \u2713 Ped. Pedestrian, Veh. Vehicle, ann. annotation, sem. seg. semantic segmentation 2.2 Temporal Action Localisation Significant progress has been made in recent years toward temporal action localisation (TAL) in untrimmed videos. The main goal of TAL is to remove the irrelevant background from the input videos and identify the start and end instants of the action of interest. Temporal localisation methods can be divided into supervised [62, 63] versus weakly/unsupervised ones [64\u201366]. While the former requires temporal annotations to learn a model able to identify the start and end of an action, in weaklysupervised settings the model is learned from partial annotation. In Part II of the thesis, our goal is the same as temporal activity localisation, with the difference that we wish to target complex, long duration events articulated into a possibly large number of parts (complex activity detection), whose internal structure cannot be leveraged by naive temporal segmentation methods. 2.3 Complex Activity Detection Most recent work on complex activity recognition concerns scalar sensors [67\u201369] or a combination of both scalar and vision sensors [70,71]. Recently, though, several vision-based complex activity recognition methods have been proposed [40, 63, 72\u2013 75] with the goal of understanding an overall scene by recognising and segmenting atomic actions. These methods can be further divided into (i) sliding windows ap\fChapter 2. Related Work 13 proaches [38, 76], in which an activity classifier is applied to each snippet, and (ii) boundaries analyses [77, 78], in which a model is trained to identify the start and end time of each action. Overall, current activity recognition methods are geared to recognise short-term activities via a combination of small atomic actions. Unlike existing approaches, the objective of our complex activity model (proposed in Chapter 4) is to understand long-term activities in dynamic scenes, such as the phases a surgical procedure is broken into, whose detection is crucial to inform the decision making of automated robotic assistants. 2.3.1 One-Stage vs Two-Stage Approaches Complex activity methods can been broadly divided into one-stage and two-stage approaches. The former [23, 79\u201381] detect actions/activities in a single shot, and can be easily trained in an end-to-end manner. For example, Wang et al. [82] detect actions using transformers which, unlike RNNs, do not suffer from nonparallelism and vanishing gradients. Most one-stage methods, however, only perform action classification, rather than spatiotemporal localisation. In contrast, Lin et al. [81] have recently proposed an anchor-free, one-stage light model which generates proposals, locates actions within them, and classifies them end-to-end. The latter group of methods [20, 22, 62, 83, 84], instead, consists of two stages, similarly to region-proposal object detectors. The first stage generates suitable proposals for predicting the start and end time of an activity, while the second stage extracts features and processes the proposals before passing them to both a classification head and a regression head (for temporal localisation). Some work, including [22], focus on the first stage to improve the quality of the proposals, while others focus on the processing or refining of the proposals. [84], for instance, uses an offthe-shelf method for proposal generation. The second stage consists of two networks, a \u2018disentanglement\u2019 network to separate the classification and regression representations, and a \u2018context aggregation\u2019 network to add them together. Such methods are not trainable end-to-end and limited to short-or mid-duration actions. In contrast, we propose a complex activity detection method in Chapter 5, which exhibits the advantages of both classes of methods, thanks to our hybrid graph approach capable of recognising and localising both shortand long-duration activities. 2.4 Graph Networks Recently, Graph Convolutional Networks (GCNs) have been widely used for action and activity detection and recognition, building on their success in different areas of computer vision such as point cloud segmentation [85, 86] and 3D object detec\f14 tion [87]. Relevant GCN approaches have been broadly focussing on either action recognition [88\u201390] or TAL [21,25]. In the former, videos are represented in different spatiotemporal formats such as 3D point clouds and time-space region graphs, and methods focus on recognising atomic actions only. In contrast, Zeng et al. [21] use GCN for temporal activity localisation by considering action proposals as nodes and a relation between two proposals as an edge. In opposition, in our model nodes are action tubes and their connections are based on an array of semantics. In another recent study, Xu et al. [25] generate graphs by considering temporal snippets as nodes and drawing connections between them based on temporal appearance and semantic similarity. Most graph-based activity detection methods [21, 25, 91] construct a graph for a whole video by taking snippets as nodes and their temporal linkage as edges, not paying much attention to the constituent atomic actions within each snippet, and are typically limited to shorter videos or memory dependent in case of larger videos due to adjacency matrices. For a large number of snippets (nodes) in the video required memory can be substantial. The adjacency matrix of the graph with n nodes (snippets) requires O(n2) memory, which can be impractical for large videos. In contrast, our proposed framework presented in Chapter 4 is designed to construct a scene graph for each snippet rather than the whole video which reflects the structure of a dynamic scene in terms of atomic action tubes (nodes) and the different types of relationships between them. 2.4.1 GCN-Based Temporal Activity Localisation GCNs have been extensively investigated for TAL [21, 25, 62, 92, 93]. GCN-based TAL methods can also be further divided into two-stage and one-stage methods. The former, once again, perform localisation after generating suitable proposals. For instance, in [21] two different types of boundary proposals are generated and then individually passed to the same graph, resulting in both an action label and a temporal boundary. One-stage GCN-based TAL methods, instead, solve the detection problem without proposals in one go by learning spatiotemporal features in an end-to-end manner. For examples, in [25] a graph is first generated by connecting the snippets both temporally and by virtue of meaningful semantics. This graph is then divided into subgraphs (anchors), where each anchor represents the activity in an untrimmed video. In contrast, Khan et al. [17] proposed a spatiotemporal scene graph-based long-term TAL method where each of the snippets is considered as a separate graph, which is heavily dependent on the particular actions present in the scene, and is only applicable to datasets providing (label and bounding box) annotations for each individual \fChapter 2. Related Work 15 action. In Chapter 5, we propose a new framework leveraging GCNs, by incorporating them in an overall hybrid graph capable of modelling both the local scenes, via a Graph Attention Network (GAT) [94], and the overall global activity via a temporal graph. GATs build on the transformer concept by applying attention to graphs, and were originally proposed in [94] for node classification. The idea is to update the representation of the current node with respect to its neighbours by applying attention to learn the importance of the various connections. In Chapter 5, GAT is used at the local scene level to learn the features of each node (active agent tube), to generate a more robust local scene representation. 2.5 Continual Learning Current continual learning methods can be categorised into three major families based on how the information of previous data is stored and used. Prior-focused methods [95] use a penalty term to regularise the parameters rather than a hard constraint. Parameter isolation methods [96] dedicate different parameters for different tasks to prevent interference. Finally, replay-based approaches store the examples\u2019 information in a \u2018replay buffer\u2019 or a generative model, which is used for rehearsal/retraining or to provide constraints on the current learning step [97]. 2.5.1 Continual Unsupervised Learning Continual unsupervised learning is a relatively unexplored territory due to the heavy dependency of existing works on the availability of ground truth information. Another challenge lies in the clear distinction of class boundaries for controlling model updates [98]. Clustering approaches that provide structure to the unlabelled stream are a popular theme of existing continual unsupervised works [98\u2013100]. A selfadaptive clustering approach [98] is proposed which does not rely on labelled data for model updates. The labelled data is initially leveraged to learn the cluster-toclass associations. Similarly, in [99] the pseudo labels for new data are obtained by a K-means global clustering approach, whereas the updated model without fully connected layers is employed as a feature extractor. 2.5.2 Time-Varying Hidden Markov Models A number of competing formulations exist for HMMs whose parameters depend on time. They have been employed recently for useful life prediction in engineering [101], analysing time-varying networks in EEG data [102], and incorporating \f16 periodic variability in animal movement [103]. Hidden semi-Markov models [104], in particular, have been proposed, which augment HMMs with time-varying parameters, so that the transition probability to a new state varies in accordance with the time spent in the current state. In opposition, the TVHMM model formulated in [105] encodes non-stationarity into a finite sequence of time varying transition densities and can simulate state duration of infinite length. A common approach [103, 106] is to model the transition probabilities as functions of a covariate process {zt}, which is observed in parallel to the main observation process [107]. As the covariates zt are simultaneously correlated with the hidden variables xt, it may be preferable to use lagged informative variables or artificial variables. In fact, in [108] a time-varying hidden Markov model with latent information was proposed in which an additional hidden variable (\u2018signal\u2019) with AR(1) dynamics drives the transition probabilities in the main hidden variable, xt. In [109], the authors observed that the states respond more to their own past values than any other variable. They thus proposed a model in which the state depends on the sign of an autoregressive latent variable, which induces time-varying probabilities in a non-Markovian process, as the probability of a certain state at time t depends on the lagged value of the latent variable and not the state at the previous time. 2.6 Benchmark Datasets In the literature, several datasets have been proposed to evaluate the performance of scene understanding from simple action detection to event detection and complex activity recognition. Here, in this chapter, we describe all the datasets used in this thesis. 2.6.1 ROAD ROAD (The ROad event Awareness Dataset for Autonomous Driving) [45] is a multilabeled dataset proposed for road agent, action, and location detection. The combination of these three labels is referred to in [45] as a \u2018road event\u2019. The ROAD dataset consists of a total of 22 videos with an average duration of 8 minutes, captured by the Oxford RobotCar [110] under diverse lighting and weather conditions. The dataset was further extended as a testbed for complex activity detection in [17]. Complex activities in the ROAD dataset belong to six different classes, including: negotiating an intersection, negotiating a pedestrian crossing, waiting in a queue, merging into the (ego) vehicle lane, sudden appearance (of other vehicles/agents), and (people) walking in the middle of the road. Activities can span up to two minutes and in\fChapter 2. Related Work 17 volve a large number of road agents. We introduce ROAD in Chapter 3 and evaluate our 3D-RetinaNet baseline in the same chapter. We also use ROAD extension for complex activity detection in Chapter 4 and Chapter 5 2.6.2 SARAS-ESAD ESAD (the Endoscopic Surgeon Action Detection Dataset) [111] is a benchmark devised for surgeon action detection in real-world endoscopic surgery videos. The goal of ESAD is contributing to increasing the potential and dependability of surgical assistant robots by realistically testing their awareness of the actions performed by a surgeon. In ESAD, surgeon actions are classed into 21 different categories and annotated with the help of professional surgeons. This dataset is used in Chapter 4 for complex activity detection. 2.6.3 Thumos-14 Thumos-14 [31] is one of the benchmark datasets for temporal action localisation. Thumos-14 contains 413 untrimmed temporally annotated videos categorised into 20 actions. Videos are characterised by a large variance in duration, from one second to 26 minutes. On average, each video contains 16 action instances. To compare our performance with the state-of-the-art, we also adopt the standard practice of using the validation set (200 videos) for training while evaluating our model on the test set (213 videos). We use this dataset for the evaluation of our complex activity framework in Chapter 5. 2.6.4 ActivityNet-1.3 ActivityNet-1.3 [30] is one of the largest action localisation datasets with around 20K untrimmed videos comprising 200 action categories. The videos are divided into training, validation, and testing folds according to a ratio of 2:1:1, respectively. The number of action instances per video is 1.65, which is quite low compared to Thumos-14. Following the previous art, we train our model on the training set and test it on the validation set. We use this dataset in Chapter 4 for the evaluation of our complex activity framework. 2.6.5 50-Salad 50-Salad [112] is an activity segmentation dataset with the aim of stimulate research on recognizing manipulative gestures. It consists of 50 videos of 25 cooks (actors) \f18 preparing salads, where each cook recorded two videos of themselves preparing salads by following different sequence of steps. The dataset contains two levels of annotations: \u201chigh-level\u201d labels only span three classes, whereas \u201clow-level\u201d annotation involves a total of 51 classes. 50-Salad is a multimodel dataset contains other modality besides RGB videos including RGB-D (depth information), 3-axis accelerometer devices attached to a knife, a mixing spoon, a small spoon, a peeler, a glass, an oil bottle, and a pepper dispenser. This dataset is used in Chapter 6 for the evaluation of our latent model. 2.7 Evaluation Metrics We also provide the standard evaluation metrics used for the evaluation of our methods proposed in this thesis. 2.7.1 Classification Accuracy Classification accuracy is a measure of how well a classification model is able to correctly predict the class labels of a given dataset. It is defined as the proportion of correctly classified instances to the total number of instances in the dataset. Classification accuracy is used in Chapter 4 for recognition of activities. 2.7.2 Precision Precision is a measure of the accuracy of a classification model\u2019s positive predictions. It is defined as the proportion of true positive predictions (correctly predicted positive instances) to the total number of positive predictions (true positives plus false positives). A high precision value means that the model has a low number of false positives, or that it is good at not classifying negative instances as positive. In other words, it means that the model is good at identifying only the relevant instances among the positive predictions. Precision is used in Chapter 4 and Chapter 6 for recognition of activities. 2.7.3 Recall Recall, also known as Sensitivity or the True Positive Rate (TPR), is a measure of a classification model\u2019s ability to correctly identify all relevant instances. It is defined as the proportion of true positive predictions (correctly predicted positive instances) to the total number of actual positive instances (true positives plus false negatives). \fChapter 2. Related Work 19 A high recall value means that the model has a low number of false negatives, or that it is good at identifying all relevant instances. In other words, it means that the model is good at detecting all the positive instances among all the actual positive instances. Recall is used in Chapter 4 and Chapter 6 for recognition of activities. 2.7.4 F1-score The F1 Score is a measure of a classification model\u2019s accuracy that combines precision and recall. It is defined as the harmonic mean of precision and recall. The F1 Score is a single value that represents the balance between precision and recall. A higher F1 Score indicates that the model has a better balance between precision and recall. It is particularly useful when the classes are imbalanced, or when the cost of false positives and false negatives is different. In these cases, it is possible to have a high accuracy score but a low F1 score if precision and recall are not in balance. F1 score is used in Chapter 4 and Chapter 6 for recognition of activities. 2.7.5 Mean Average Precision Mean Average Precision (mAP) is a common evaluation metric used in action detection, which is a subfield of computer vision that deals with recognizing and locating instances of specific actions within video sequences. MAP calculates the average precision across all classes of actions in a video dataset, by taking into account both the precision and recall of the action detection algorithm. Precision measures the proportion of true positive detections (correctly detected actions) to the total number of positive detections (both true positives and false positives). Recall measures the proportion of true positive detections to the total number of instances of the action that should have been detected. mAP is calculated by first computing the average precision for each action class, and then taking the mean of those values. A higher MAP value indicates a better performance of the action detection algorithm. This evaluation metric is used for two purposes; firstly for the spatiotemporal detection of events in Chapter 3. Secondly, used for the temporal evaluation of complex activities in Chapter 4 and Chapter 5. 2.7.6 Mean Absolute Error Mean Absolute Error (MAE) is a measure of the average magnitude of the errors in a set of predictions, without considering their direction. It measures the difference between the predicted values and the true values. \f20 MAE is a commonly used regression evaluation metric and it is relatively easy to understand and interpret. It is expressed in the same unit as the target variable and it gives an idea of the magnitude of the error, regardless of the direction of the error. MAE is used in Chapter 6 for the evaluation of continual crowd counting (regression problem). \fPart I : ROAD (The ROad event Awareness Dataset for Autonomous Driving) 21 \f22 \fChapter 3 The ROad event Awareness Dataset for Autonomous Driving 3.1 Introduction As anticipated, the first part of this thesis concerns individual event detection, specifically the detection of road events from an autonomous driving perspective. Injecting capabilities such as the understanding of dynamic road events and their evolution into autonomous vehicles has the potential to take situational awareness and decisionmaking closer to human-level performance. To this purpose, in this chapter, we introduce the ROad event Awareness Dataset (ROAD) for Autonomous Driving. ROAD is designed to test an autonomous vehicle\u2019s ability to detect road events. To the best of our knowledge, ROAD is the first multi-label multi-action autonomous driving dataset where each of the bounding box is associated with three labels (agent, action, and location). This unique labeling schema makes ROAD more demanding and challenging in comparison to its competitors. ROAD comprises videos originally from the Oxford RobotCar Dataset, annotated with bounding boxes showing the location in the image plane of each road event. We benchmark various detection tasks, proposing as a baseline a new incremental algorithm for online road event awareness termed 3D-RetinaNet. We also report the performance on the ROAD tasks of Slowfast and YOLOv5 detectors, as well as that of the winners of the ICCV2021 ROAD challenge, which highlights the challenges faced by situation awareness in autonomous driving. ROAD is designed to allow scholars to investigate exciting tasks such as complex (road) activity detection, future event anticipation and continual learning. Further down the line, we also introduce logical constraints to ROAD. The main intuition behind these logical constraints is that neural networks have proven to be very powerful at computer vision tasks. However, they often exhibit unexpected behaviours, violating known requirements and expressing background knowledge. 23 \f24 This calls for models (i) able to learn from the requirements, and (ii) be guaranteed to be compliant with the requirements themselves. Unfortunately, the development of such models is hampered by the lack of datasets equipped with formally specified requirements. Therefore, we introduce the ROad event Awareness Dataset with logical Requirements (ROAD-R), the first publicly available dataset for autonomous driving with requirements expressed as logical constraints. 3.1.1 Motivation The history of autonomous driving has witnessed significant milestones and advancements since the Defense Advanced Research Projects Agency (DARPA1) Grand Challenge 2004, which marked a pivotal moment in the field. The DARPA Grand Challenge was a groundbreaking competition organized by the U.S. Department of Defense\u2019s DARPA agency. Its aim was to accelerate the development of autonomous vehicle technology. The challenge required participating vehicles to navigate a designated route in a desert environment, relying solely on their autonomous capabilities. The main purpose of the Challenge was to push the boundaries of autonomous vehicle technology. The goal was to foster innovation and encourage advancements that could potentially benefit military applications, such as unmanned ground vehicles for reconnaissance or supply missions [113]. The DARPA Grand Challenge helped establish autonomous driving as a viable research area and laid the foundation for subsequent advancements in the field. In recent years, autonomous driving has emerged as a fast-growing research area. The race towards fully autonomous vehicles pushed many large companies, such as Google, Toyota, and Ford, to develop their own concept of robot-car [114\u2013116]. While self-driving cars are widely considered to be a major development and testing ground for the real-world application of artificial intelligence, major reasons for concern remain in terms of safety, ethics, cost, and reliability [117]. From a safety standpoint, in particular, smart cars need to robustly interpret the behaviour of the humans (drivers, pedestrians or cyclists) they share the environment with, in order to cope with their decisions. Situation awareness and the ability to understand the behaviour of other road users are thus crucial for the safe deployment of autonomous vehicles (AVs). Therefore, in this chapter, we only consider the context of vision-based autonomous driving [118] from video sequences captured by cameras mounted on the vehicle in a streaming, online fashion. While detector networks [119] are routinely trained to facilitate object and actor recognition in road scenes, this simply allows the vehicle to \u2019see\u2019 what is around it. The philosophy of this work is that robust self-driving capabilities require a deeper, 1https://www.darpa.mil/ \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 25 more human-like understanding of dynamic road environments (and of the evolving behaviour of other road users over time) in the form of semantically meaningful concepts, as a stepping stone for intention prediction and automated decision making. One advantage of this approach is that it allows the autonomous vehicle to focus on a much smaller amount of relevant information when learning how to make its decisions, in a way arguably closer to how decision-making takes place in humans. (a) (b) Figure 3.1: Use of labels in ROAD to describe typical road scenarios. (a) A green car is in front of the AV while changing lanes, as depicted by the arrow symbol. The associated event will then carry the following labels: in vehicle lane (location), moving left (action). Once the event is completed, the location label will change to: in outgoing lane. (b) Autonomous vehicle turning left from lane 6 into lane 4: lane 4 will be the outgoing lane as the traffic is moving in the same direction as the AV. However, if the AV turns right from lane 6 into lane 4 (a wrong turn), then lane 4 will become the incoming lane as the vehicle will be moving into the incoming traffic. The overall philosophy of ROAD is to use suitable combinations of multiple label types to fully describe a road situation, and allow a machine learning algorithm to learn from this information. On the opposite side of the spectrum lies end-to-end reinforcement learning. There, the behaviour of a human driver in response to road situations is used to train, in an imitation learning setting [120], an autonomous car to respond in a more \u2018human-like\u2019 manner to road scenarios. This, however, requires an astonishing amount of data from a myriad of road situations. For highway driving only, a relatively simple task when compared to city driving, Fridman et al. in [121] had to use a whole fleet of vehicles to collect 45 million frames. Perhaps more importantly, in this approach, the network learns a mapping from the scene to control inputs, without attempting to model the significant facts taking place in the scene or the reasoning of the agents therein. As discussed in [122], many authors [123, 124] have recently highlighted the insufficiency of models which directly map observations to actions [125], specifically in the scenario of the self-driving car. \f26 3.1.2 ROAD: a multi-label, multi-task dataset Concept. This work aims to propose a new framework for situation awareness and perception, departing from the disorganised collection of object detection, semantic segmentation or pedestrian intention tasks which is the focus of much current work. We propose to do so in a \u201cholistic\u201d, multi-label approach in which agents, actions and their locations are all ingredients in the fundamental concept of road event (RE). Road events are defined as triplets E = (Ag, Ac, Loc) composed by an active road agent Ag, the action(s) Ac it performs (possibly more than one at the same time), and the location(s) Loc2 in which this takes place (which may vary from the start to the end of the event itself), as seen from the point of the view of an autonomous vehicle. The complete list of all three label types is provided in Table 3.1. This takes the problem to a higher conceptual level, in which AVs are tested on their understanding of what is going on in a dynamic scene rather than their ability to describe what the scene looks like, putting them in a position to use that information to make decisions and a plot course of action. Modelling dynamic road scenes in terms of road events can also allow us to model the causal relationships between what happens; these causality links can then be exploited to predict further future consequences. To transfer this conceptual paradigm into practice, this chapter introduces ROAD, the first ROad event Awareness in Autonomous Driving Dataset, as an entirely new type of dataset designed to allow researchers in autonomous vehicles to test the situation awareness capabilities of their stacks in a manner impossible until now. Unlike all existing benchmarks, ROAD provides ground truth for the action performed by all road agents, not just humans. In this sense ROAD is unique in the richness and sophistication of its annotation, designed to support the proposed conceptual shift. We are confident this contribution will be very useful moving forward for both the autonomous driving and the computer vision community. However, at the same time, it is also worth mentioning that the quantisation of the action and activities in the context of autonomous driving with several limitation and challenges including computational complexity, robustness to uncertainty, and control and decision-making. Features. ROAD is built upon (a fraction of) the Oxford RobotCar Dataset [126], by carefully annotating 22 carefully selected, relatively long-duration videos. Road events are represented as \u2019tubes\u2019, i.e., time series of frame-wise bounding box detections. ROAD is a dataset of significant size, most notably in terms of the richness and complexity of its annotation rather than the raw number of video frames. A total of 122K video frames are labelled for a total of 560K detection bounding boxes in 2Note: The label location Loc in our case does not refer to the spatiotemporal coordinates (x, y) of the bounding box. \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 27 turn associated with 1.7M unique individual labels, broken down into 560K agent labels, 640K action labels and 499K location labels. The dataset was designed according to the following principles. \u2022 A multi-label benchmark: each road event is composed by the label of the (moving) agent responsible, the label(s) of the type of action(s) being performed, and labels describing where the action is located. \u2022 Each event can be assigned multiple instances of the same label type whenever relevant (e.g., an RE can be an instance of both moving away and turning left). \u2022 The labelling is done from the point of view of the AV: the final goal is for the autonomous vehicle to use this information to make the appropriate decisions. \u2022 The meta-data is intended to contain all the information required to fully describe a road scenario: an illustration of this concept is given in Figure 3.1. After closing one\u2019s eyes, the set of labels associated with the current video frame should be sufficient to recreate the road situation in one\u2019s head (or, equivalently, sufficient for the AV to be able to make a decision). In an effort to take action detection into the real world, ROAD moves away from human body actions almost entirely, to consider (besides pedestrian behaviour) actions performed by humans as drivers of various types of vehicles, shifting the paradigm from actions performed by human bodies to events caused by agents. As shown in our experiments, ROAD is more challenging than current action detection benchmarks due to the complexity of road events happening in real, nonchoreographed driving conditions, the number of active agents present and the variety of weather conditions encompassed. Tasks. ROAD allows one to validate manifold tasks associated with situation awareness for self-driving, each associated with a label type (agent, action, location) or a combination thereof: 1. Agent detection, the output is in the form of \u2019agent tubes\u2019 collecting the BBs associated with an active road agent in consecutive frames. 2. Action detection, the output is in the form of \u2019action tubes\u2019 (series of bounding box (BB) detections with an attached action label). 3. Location detection, the output is in the form of \u2019agent-location\u2019 tubes collecting the BBs associated with an agent and its location in consecutive frames. 4. Agent-action detection, the detector is trained on pairs of agent-action labels. The output is again in the form of a tube. \f28 5. Road event detection, as well as the AV\u2019s decisions make use of all the three types of information provided by ROAD, detecting RE triplets is crucial for autonomous driving applications. 6. Temporal AV actions detection, the output of this task is the class of action the AV performs in each video frame, from a list suitably compiled. For each task, one can assess both frame-level detection, which outputs independently for each video frame the bounding box(es) of the instances there present and the relevant class labels, and video-level detection, which consists in regressing the whole series of temporally-linked bounding boxes (i.e., in current terminology, a \u2019tube\u2019) associated with an instance, together with the relevant class label. In this chapter, we conduct tests on both. All tasks come with both the necessary annotation and a shared baseline, which is described in Section 3.3. Table 3.1: ROAD labels. Agents Actions Locations Pedestrian Move away AV lane Car Move towards Outgoing lane Cyclist Move Outgoing cycle lane Motorbike Brake Incoming lane Medium vehicle Stop Incoming cycle lane Large vehicle Indicating left Pavement Bus Indicating right Left pavement Emergency vehicle Hazards lights on Right pavement AV traffic light Turn left Junction Other traffic light Turn right Crossing location Overtake Bus stop Wait to cross Parking Cross from left Cross from right Crossing Push object Red traffic light Amber traffic light Green traffic light 3.1.3 ROAD-R In this chapter, we also generalize hierarchical multi-label classification (HMC) problems by introducing multi-label classification problems with (full) propositional logic requirements. Thus, given a multi-label classification problem with labels A, B, and \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 29 C, we can, for example, write the requirement: (\u00acA \u2227B) \u2228C, stating that for each data point in the dataset either the label C is predicted, or B but not A are predicted. Then, we present the ROad event Awareness Dataset with logical Requirements (ROAD-R), the first publicly available dataset for autonomous driving with requirements expressed as logical constraints. ROAD-R extends the ROAD dataset by manually annotated ROAD-R with 243 constraints, each verified to hold for each bounding box. A typical constraint is thus \u201ca traffic light cannot be red and green at the same time\u201d, while there are no constraints like \u201cpedestrians should cross at crossings\u201d, which should always be satisfied in theory, but which might not be in real-world scenarios. Given ROAD-R, we considered 6 current state-of-the-art (SOTA) models, and we showed that they are not able to learn the requirements just from the data points, as more than 90% of the times, they produce predictions that violate the constraints. 3.1.4 Main Contributions The major contributions of this work are thus the following. \u2022 A conceptual shift in situation awareness centred on a formal definition of the notion of road event, as a triplet composed by a road agent, the action(s) it performs and the location(s) of the event, seen from the point of view of the AV. \u2022 A new ROad event Awareness Dataset for Autonomous Driving (ROAD), the first of its kind, designed to support this paradigm shift and allow the testing of a range of tasks related to situation awareness for autonomous driving: agent and/or action detection, event detection, ego-action classification. \u2022 A multi-label classification problem with propositional logic requirements. \u2022 Introduction of ROAD-R, which is the first publicly available dataset whose requirements are expressed in full propositional logic. \u2022 Considered 6 SOTA models and show that on ROAD-R, they produce predictions violating the requirements more than \u223c90% of the times, Instrumental to the introduction of ROAD as the benchmark of choice for semantic situation awareness, we propose a robust baseline for online action/agent/event detection (termed 3D-RetinaNet) which combines state-of-the-art single-stage object \f30 detector technology with an online tube construction method [127], with the aim of linking detections over time to create event tubes [128, 129]. Results for two additional baselines based on a Slowfast detector architecture [130] and YOLOv53 (for agent detection only) are also reported and critically assessed. Related publications: The work of this chapter appeared in IEEE transactions on pattern analysis and machine intelligence (IEEE TPAMI 2022) [45] and The ROAD Workshop and Challenge: Event Detection for Situation Awareness in Autonomous Driving at ICCV 20214. The dissertation\u2019s author was the co-author of the TPAMI paper and co-organiser of the workshop and challenge. The main contributions of the dissertation\u2019s author to this project are: adding a new backbone (SlowFast) to the 3D RetinaNet architecture (baseline for ROAD), implementing a YOLOv5 for agents detection, and setting up the evaluation site for the challenge. All the resources of this project are publically available and are accessed from ROAD dataset5, 3D RetinaNet baseline6, Inference demo7, and Challenge evalAI8. The work of ROAD-R appeared in IJCAI 2022 Workshop on Artificial Intelligence for Autonomous Driving (AI4AD 2022)9 and IJCLR 2022 the International Joint Conference on Learning and Reasoning [46]. This work received the best paper award at IJCAI 2022 AI4AD and won the best student paper award at IJCLR 2022. The dissertation\u2019s author was the co-author in both papers. The main contribution of the dissertation\u2019s author to this project is providing the vision baselines for on ROAD dataset for logical constraint. All the resources of this project are publically available and are accessed from code10, video11. Outline: The remainder of the chapter is organised as follows. Section 3.2 presents our ROAD dataset in full detail, including: its multi-label nature (Sec. 3.2.1), data collection (Sec. 3.2.2), annotation (Sec. 3.2.3), the tasks it is designed to validate (Sec. 3.2.4), and a quantitative summary (Sec. 3.2.5). Section 3.3 presents an overview of the proposed 3D-RetinaNet baseline, and recalls the ROAD challenge organised by some of us at ICCV 2021 to disseminate this new approach to situation awareness within the autonomous driving and computer vision communities, using ROAD as the benchmark. Section 3.4 describes ROAD-R. Experiments are described 3https://github.com/ultralytics/yolov5. 4https://sites.google.com/view/roadchallangeiccv2021/ 5https://github.com/gurkirt/road-dataset 6https://github.com/gurkirt/3D-RetinaNet 7https://www.youtube.com/watch?v=CmxPjHhiarA 8https://eval.ai/web/challenges/challenge-page/1059/overview 9https://learn-to-race.org/workshop-ai4ad-ijcai2022/ 10https://github.com/EGiunchiglia/ROAD-R 11https://www.youtube.com/watch?v=_-Ll5d1VQXY \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 31 in Section 3.5, where a number of ablation studies are reported and critically analysed in detail, together with the results of the ROAD challenge\u2019s top participants. Section 3.6 outlines additional exciting tasks the dataset can be used as a benchmark for in the near future, such as future event anticipation, decision making and machine theory of mind [122]. Conclusions and future work are outlined in Section 3.7. 3.2 The Dataset 3.2.1 A Multi-Label Benchmark The ROAD dataset is specially designed from the perspective of self-driving cars, and thus includes actions performed not just by humans but by all road agents in specific locations, to form road events (REs). REs are annotated by drawing a bounding box around each active road agent present in the scene, and linking these bounding boxes over time to form \u2019tubes\u2019. As explained, to this purpose three different types of labels are introduced, namely: (i) the category of road agent involved (e.g. Pedestrian, Car, Bus, Cyclist); (ii) the type of action being performed by the agent (e.g. Moving away, Moving towards, Crossing and so on), and (iii) the location of the road user relative the autonomous vehicle perceiving the scene (e.g. In vehicle lane, On right pavement, In incoming lane). In addition, ROAD labels the actions performed by the vehicle itself. Multiple agents might be present at any given time, and each of them may perform multiple actions simultaneously (e.g. a Car may be Indicating right while Turning right). Each agent is always associated with at least one action label. The complete list of all three labels types are given in Table 3.1. Agent labels. Within a road scene, the objects or people able to perform actions which can influence the decision made by the autonomous vehicle are termed agents. We only annotate active agents (i.e., a parked vehicle or a bike or a person visible to the AV but located away from the road are not considered to be \u2019active\u2019 agents). Three types of agent are considered to be of interest, in the sense defined above, to the autonomous vehicle: people, vehicles and traffic lights. For simplicity, the AV itself is considered just like another agent: this is done by labelling the vehicle\u2019s bonnet. People are further subdivided into two sub-classes: pedestrians and cyclists. The vehicle category is subdivided into six sub\u2013classes: car, small\u2013size motorised vehicle, medium\u2013size motorised vehicle, large\u2013size motorised vehicle, bus, motorbike, emergency vehicle. Finally, the \u2018traffic lights\u2019 category is divided into two sub\u2013classes: Vehicle traffic light (if they apply to the AV) and Other traffic light (if they apply to other road users). Only one agent label can be assigned to each active agent present in the scene at any given time. Action labels. Each agent can perform one or more actions at any given time \f32 instant. For example, a traffic light can only carry out a single action: it can be either red, amber, green or \u2018black\u2019. A car, instead, can be associated with two action labels simultaneously, e.g., Turning right and Indicating right. Although some road agents are inherently multitasking, some action combinations can be suitably described by a single label: for example, pushing an object (e.g. a pushchair or a trolley-bag) while walking can be simply labelled as Pushing object. The latter was our choice. AV own actions. Each video frame is also labelled with the action label associated with what the AV is doing. To this end, a bounding box is drawn on the bonnet of the AV. The AV can be assigned one of the following seven action labels: AV-move, AV-stop, AV-turn-left, AV-turn-right, AV-overtake, AV-move-left and AV-move-right. Location labels. Agent location is crucial for deciding what action the AV should take next. As the final, long-term objective of this project is to assist autonomous decision making, we propose to label the location of each agent from the perspective of the autonomous vehicle. For example, a pedestrian can be found on the right or the left pavement, in the vehicle\u2019s own lane, while crossing or at a bus stop. The same applies to other agents and vehicles as well. There is no location label for the traffic lights as they are not movable objects, but agents of a static nature and well-defined location. To understand this concept, Fig. 3.1 illustrates two scenarios in which the location of the other vehicles sharing the road is depicted from the point of view of the AV. Traffic light is the only agent type missing location labels, all the other agent classes are associated with at least one location label. 3.2.2 Data Collection ROAD is composed of 22 videos from the publicly available Oxford Robot Car Dataset [126] (OxRD) released in 2017 by the Oxford Robotics Institute12, covering diverse road scenes under various weather conditions. The OxRD dataset, collected from the narrow streets of the historic city of Oxford, was selected because it presents challenging scenarios for an autonomous vehicle due to the diversity and density of various road users and road events. Note, however, that our labelling process (described below) is not limited to OxRD. In principle, other autonomous vehicle datasets (e.g. [131, 132]) may be labelled in the same manner to further enrich the ROAD benchmark, we plan to do exactly so in the near future. Video selection. Within OxRD, videos were selected with the objective of ensuring diversity in terms of weather conditions, times of the day and types of scenes recorded. Specifically, the 22 videos have been recorded both during the day (in strong sunshine, rain or overcast conditions, sometimes with snow present on the 12http://robotcar-dataset.robots.ox.ac.uk/ \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 33 surface) and at night. Preprocessing. First, the original sets of video frames were downloaded and demosaiced, in order to convert them to red, green, and blue (RGB) image sequences. Then, they were encoded into proper video sequences using ffmpeg13 at the rate of 12 frames per second (fps). Although the original frame rate in the considered frame sequences varies from 11 fps to 16 fps, we uniformised it to keep the annotation process consistent. 3.2.3 Annotation Process Annotation tool. Annotating tens of thousands of frames rich in content is a very intensive process; therefore, a tool is required which can make this process both fast and intuitive. For this work, we adopted Microsoft\u2019s VoTT14. The most useful feature of this annotation tool is that it can copy annotations (bounding boxes and their labels) from one frame to the next, while maintaining a unique identification for each box, so that boxes across frames are automatically linked together. Moreover, VoTT also allows for multiple labels, thus lending itself well to ROAD\u2019s multi-label annotation concept. Annotation protocol. All salient objects and actors within the frame were labelled, with the exception of inactive participants (mostly parked cars) and objects/actors at large distances from the ego vehicle, as the latter were judged to be irrelevant to the AV\u2019s decision-making. This can be seen in the attached 30-minute video15 portraying ground truth and predictions. As a result, pedestrians, cyclists, and traffic lights were always labelled. Vehicles, on the other hand, were only labelled when active (i.e., moving, indicating, being stopped at lights or stopping with hazard lights on the side of the road). As mentioned, only parked vehicles were not considered active (as they do not arguably influence the AV\u2019s decision-making) and were thus not labelled. Event label generation. Using the annotations manually generated for actions and agents in the multi-label scenario as discussed above it is possible to generate event-level labels about agents, e.g. Pedestrian / Moving towards the AV On right pavement or Cyclist / Overtaking / In vehicle lane. Any combinations of location, action and agent labels are admissible. If location labels are ignored, the resulting event labels become location-invariant. Namely, given an agent tube and the continuous temporal sequence of action labels attached to its constituent bounding box detections, we can generate action tubes by looking for changes in the action label series associated with each agent tube. For instance, a Car appearing in a video might be first Moving away before Turning left. 13https://www.ffmpeg.org/ 14https://github.com/Microsoft/VoTT/ 15https://www.youtube.com/watch?v=CmxPjHhiarA. \f34 The agent tube for the car will then be formed by two contiguous agent-action tubes: a first tube with label pair Car / Moving away and a second one with pair Car / Turning left. Table 3.2: ROAD tasks and attributes. Task type Problem type Output Multiple labels Active agent Detection Box&Tube No Action Detection Box&Tube Yes Location Detection Box&Tube Yes Event Detection Box&Tube Yes AV-action Temporal detection Start/End No 3.2.4 Tasks ROAD is designed as a sandbox for validating the six tasks relevant to situation awareness in autonomous driving outlined in Sec. 3.1.2. Five of these tasks are detection tasks, while the last one is a frame-level action recognition task sometimes referred to as \u2019temporal action segmentation\u2019 [133], Table 3.2 shows the main attributes of these tasks. All detection tasks are evaluated both at frame-level and at video(tube-)level. Frame-level detection refers to the problem of identifying in each video frame the bounding box(es) of the instances there present, together with the relevant class labels. Video-level detection consists in regressing a whole series of temporally-linked bounding boxes (i.e., in current terminology, a \u2019tube\u2019) together with the relevant class label. In our case, the bounding boxes will mark a specific active agent in the road scene. The labels may issue (depending on the specific task) either from one of the individual label types described above (i.e., agent, action or location) or from one of the meaningful combinations described in 3.2.3 (i.e., event). Below we list all the tasks for which we currently provide a baseline, with a short description. 1. Active agent detection (or agent detection) aims at localising an active agent using a bounding box (frame-level) or a tube (video-level) and assigning a class label to it. 2. Action detection seeks to localise an active agent occupied in performing a specific action from the list of action classes. 3. In agent location detection (or location detection) a label from the relevant list of locations (as seen from the AV) is sought and attached to the relevant bounding box or tube. \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 35 Figure 3.2: Number of instances of each class of individual label-types, in logarithmic scale. 4. Road event detection (or event detection) consists in assigning to each box or tube a triplet of class labels. 5. Autonomous vehicle temporal action segmentation is a frame-level action classification task in which each video frame is assigned a label from the list of possible AV own actions. We refer to this task as \u2019AV-action segmentation\u2019, similarly to [133]. 3.2.5 Quantitative Summary Overall, 122K frames extracted from 22 videos were labelled, in terms of both AV own actions (attached to the entire frame) and bounding boxes with attached one or more labels of each of the three types: agent, action, location. In total, ROAD includes 560K bounding boxes with 1.7M instances of individual labels. The latter figure can be broken down into 560K instances of agent labels, 640K instances of action labels, and 499K instances of location labels. Based on the manually assigned individual labels, we could identify 454K instances of triplets (event labels). The number of instances for each individual class from the three lists is shown in Fig. 3.2 (frame-level, in orange). The 560K bounding boxes make up 7, 029, 9, 815, 8, 040, and 8, 394 tubes for the label types agent, action, location, and event, respectively. Figure 3.2 also shows the number of tube instances for each class of individual label types as number of video-level instances (in blue). The figure also highlights the presence of an imbalance in the dataset distribution among the classes. Specifically, certain classes exhibit a significant disparity in sample sizes, for example, \u201dcar\u201d has a considerably larger number of samples compared to \u201demergency vehicle.\u201d \f36 backbone CNN class+box subnets TxWxH x256 TxWxH x256 Conv 1x3x3 Conv 1x3x3 Conv 3x3x3 Conv 3x3x3 class+box subnets class subnet box subnet class+box subnets TxWxH xCxA TxWxH x4xA 3xTxWxH input sequence of frames spatial avg pool Tx1x1x256 Conv 3x1x1 TxCa a C number of all classes A number of anchors C  number of AV action classes Figure 3.3: Proposed 3D-RetinaNet architecture for online video processing. 3.3 Baseline and Challenge Inspired by the success of recent 3D CNN architectures [33] for video recognition and of feature-pyramid networks (FPN) [134] with focal loss [135], we propose a simple yet effective 3D feature pyramid network (3D-FPN) with focal loss as a baseline method for ROAD\u2019s detection tasks. We call this architecture 3D-RetinaNet. 3.3.1 3D-RetinaNet Architecture The data flow of 3D-RetinaNet is shown in Figure 3.3. The input is a sequence of T video frames. As in classical FPNs [134], the initial block of 3D-RetinaNet consists of a backbone network outputting a series of forward feature pyramid maps, and of lateral layers producing the final feature pyramid composed by T feature maps. The second block is composed by two sub-networks which process these features maps to produce both bounding boxes (4 coordinates) and C classification scores for each anchor location (over A possible locations). In the case of ROAD, the integer C is the sum of the numbers of agent, action, location, and agent-action-location (event) classes, plus one reserved for an agentness score. The extra class agentness is used to describe the presence or absence of an active agent. As in FPN [134], we adopt ResNet50 [136] as the backbone network. 2D versus 3D backbones. In our experiments we show results obtained using three different backbones: frame-based ResNet50 (2D), inflated 3D (I3D) [33] and Slowfast [130], in the manner also explained in [130,137]. Choosing a 2D backbone makes the detector completely online [127], with a delay of a single frame. Choosing an I3D or a Slowfast backbone, instead, causes a 4-frame delay at detection time. Note that, as Slowfast and I3D networks makes use of a max-pool layer with stride \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 37 2, the initial feature pyramid in the second case contains T/2 feature maps. Nevertheless, in this case we can simply linearly upscale the output to T feature maps. AV action prediction heads. In order for the method to also address the prediction of the AV\u2019s own actions (e.g. whether the AV is stopping, moving, turning left etc.), we branch out the last feature map of the pyramid (see Fig. 3.3, bottom) and apply spatial average pooling, followed by a temporal convolution layer. The output is a score for each of the Ca classes of AV actions, for each of the T input frames. Loss function. As for the choice of the loss function, we adopt a binary crossentropy-based focal loss [135]. We chose a binary cross entropy because our dataset is multi-label in nature for which we only train a single binary cross entropy function. The rationale behind selecting a focal-type loss function stems from the anticipation that it could potentially help the neural network in addressing the challenge of long tail distribution of data (see Figure 3.2), where some classes have significantly more samples than others. The focal-type loss accomplishes this by giving more weight to samples from minority classes during training, enabling the model to better learn their distinctive characteristics. 3.3.2 Online Tube Generation via Agentness Score The autonomous driving scenario requires any suitable method for agent, action or event tube generation to work in an online fashion, by incrementally updating the existing tubes as soon as a new video frame is captured. For this reason, this work adopts a recent algorithm proposed by Singh et al. [127], which incrementally builds action tubes in an online fashion and at real-time speed. To be best of our knowledge, [127] was the first online multiple action detection approach to appear in the literature, and was later adopted by almost all subsequent works [138\u2013140] on action tube detection. Linking of detections. We now briefly review the tube-linking method of Singh et al. [127], and show how it can be adapted to build agent tubes based on an \u2019agentness\u2019 score, rather than build a tube separately for each class as proposed in the original paper. This makes the whole detection process faster, since the total number of classes is much larger than in the original work [127]. The proposed 3D-RetinaNet is used to regress and classify detection boxes in each video frame potentially containing an active agent of interest. Subsequently, detections whose score is lower than 0.025 are removed and non-maximal suppression is applied based on the agentness score. At video start, each detection initialises an agentness tube. From that moment on, at any time instance t the highest scoring tubes in terms of mean agentness score up to t \u22121 are linked to the detections with the highest agentness score in frame t which display an Intersection-over-Union (IoU) overlap with the latest detection in \f38 the tube above a minimum threshold \u03bb. The chosen detection is then removed from the pool of frame-t detections. This continues until the tubes are either assigned or not assigned a detection from current frame. Remaining detections at time t are used to initiate new tubes. A tube is terminated after no suitable detection is found for n consecutive frames. As the linking process takes place, each tube carries scores for all the classes of interest for the task at hand (e.g., action detection rather than event detection), as produced by the classification subnet of 3D-RetinaNet. We can then label each agentness tube using the k classes that show the highest mean score over the duration of the tube. 3.3.3 The ROAD Challenge To introduce the concept of road event, our new approach to situation awareness and the ROAD dataset to the computer vision and AV communities, some of us have organised in October 2021 the workshop \u201dThe ROAD challenge: Event Detection for Situation Awareness in Autonomous Driving\u201d16. For the challenge, we selected (among the tasks described in Sec. 3.2.4) only three tasks: agent detection, action detection and event detection, which we identified as the most relevant to autonomous driving. As standard in action detection, evaluation was done in terms of video mean average precision (video-mAP). 3D-RetinaNet was proposed as the baseline for all three tasks. Challenge participants had 18 videos available for training and validation. The remaining 4 videos were to be used to test the final performance of their model. This split was applied to all the three challenges (split 3 of the ROAD evaluation protocol, see Section 3.5.2). The challenge opened for registration on April 1 2021, with the training and validation folds released on April 30, the test fold released on July 20 and the deadline for submission of results set to September 25. For each stage and each Task the maximum number of submissions was capped at 50, with an additional constraint of 5 submissions per day. The workshop, co-located with ICCV 2021, took place on October 16 2021. In the validation phase we had between three and five teams submit between 15 and 17 entries to each of three challenges. In the test phase, which took place after the summer, we noticed a much higher participation with 138 submissions from 9 teams to the agent challenge, 98 submissions from 8 teams to the action challenge, and 93 submission from 6 teams to the event detection challenge. The methods proposed by the winners of each challenge are briefly recalled in Section 3.5.4. 16https://sites.google.com/view/roadchallangeiccv2021/. \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 39 Benchmark maintenance. After the conclusion of the ROAD @ ICCV 2021 workshop, the challenge has been re-activated to allow for submissions indefinitely. The ROAD benchmark will be maintained by withholding the test set from the public on the eval.ai platform17, where teams can submit their predictions for evaluation. Training and validation sets can be downloaded from https://github. com/gurkirt/road-dataset. 3.4 ROAD-R ROAD-R extends the ROAD dataset by introducing a set \u03a0 of 243 constraints that specify the space of admissible outputs. In order to improve the usability of our dataset, we write the constraints in a way that allows us to easily express \u03a0 as a single formula in conjunctive normal form (CNF). The above can be done without any loss in generality, as any propositional formula can be expressed in CNF, and is important because many solvers expects formulas in CNF as input. Thus, each requirement in \u03a0 has form: l1 \u2228l2 \u2228\u00b7 \u00b7 \u00b7 \u2228ln , (3.1) where n \u22651, and each li is either a negative label \u00acA or a positive label A. The requirements have been manually specified following three steps: 1. an initial set of constraints \u03a01 was manually created, 2. a subset \u03a02 \u2282\u03a01 was retained by eliminating all those constraints that were entailed by the others, 3. the final subset \u03a0 \u2282\u03a02 was retained by keeping only those requirements that were always satisfied by the ground-truth labels of the entire ROAD-R dataset. Finally, redundancy in the constraints has been automatically checked with RELSAT18. Note that our process of gathering and further selecting the logical requirements follows more closely the software engineering paradigm rather than the machine learning view. To this end, we ensured that the constraints were consistent with the provided labels from the ROAD dataset in the sense that they were acting as strict conditions to be absolutely satisfied by the ground-truth labels, as emphasized in the third step of the annotation pipeline above. Table 3.3 gives a high-level description of the properties of the set \u03a0 of constraints. Notice that, with a slight abuse of notation, in the tables we use a set based notation for the requirements. Each requirement of form (3.1) thus becomes {l1, l2, . . . , ln}. 17https://eval.ai/web/challenges/challenge-page/1059/overview 18https://github.com/roberto-bayardo/relsat/ \f40 Table 3.3: Constraint statistics. \u03a0n is the set of constraints r in \u03a0 with |r| = n, i.e., with n positive and negative labels. C = {\u00acA : A \u2208C}. Each row shows the number of rules r with |r| = n, and the average number of negative and positive labels in such rules. n |\u03a0n| avgr\u2208\u03a0n(|r \u2229C|) avgr\u2208\u03a0n(|r \u2229C|) 2 215 1.995 0.005 3 5 1 2 7 1 1 6 8 6 1 7 9 6 1 8 10 1 0 10 12 1 1 11 14 1 0 14 15 7 1 14 Total 243 1.87 0.96 Such notation allows us to express the properties of the requirements in a more succinct way. In addition to the information in the tables, we report that of the 243 constraints, there are two in which all the labels are positive (expressing that there must be at least one agent and that every agent but traffic lights has at least one location), and 214 in which all the labels are negative (expressing mutual exclusion between two labels). All the constraints with more than two labels have at most one negative label, as they express a one-to-many relation between actions and agents (like in our dataset \u201cif something is crossing, then it is a pedestrian or a cyclist\u201d). Constraints like \u201cpedestrians should cross at crossings\u201d, which might not be satisfied in practice, are not included. Additionally embedding such logical constraints would require, e.g., using modal operators and, while it would be an interesting study to see the impact on the model\u2019s predictions when adding more complex layers to the expressivity of our logic, we opted for using a simpler logic in this first instance. This also provides more transparency to the wider research community, as the full propositional logic covers a vast range of applications that do not require extra logical operators. Notice that the 243 requirements restrict the number of admissible prediction to 4985868 \u223c5 \u00d7 106, thus ruling out (241 \u22124985868) \u223c1012 non-admissible predictions.19 In principle, the set of admissible predictions can be further reduced by adding other constraints. Indeed, the 243 requirements are not guaranteed to be complete from every possible point of view: as standard in the software development cycle, the requirement specification process deeply involves the stakeholders of the system (see, e.g., [141]). For example, we decided not to include constraints like \u201cit is not possible to both move towards and move away\u201d, which were not satisfied by all 19The number of admissible predictions has been computed with relsat: https://github.com/robertobayardo/relsat/. \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 41 Table 3.4: Splits of training, validation and test sets for the ROAD dataset with respect to weather conditions. The table shows the number of videos in each set or split. For splits, the first figure is the number of training videos, the second number that of validation videos. Condition sunny overcast snow night Training and validation 7 7 1 3 Split-1 7/0 4/3 1/0 3/0 Split-2 7/0 7/0 1/0 0/3 Split-3 4/3 7/0 1/0 3/0 Testing 1 1 1 1 the data points because of errors in the ground truth labels. In these cases, we decided to dismiss the constraint in order to maintain (i) consistency between the knowledge provided by the constraints and by the data points, and (ii) backward compatibility. As an additional point, we underline that, even though the annotation of the requirements introduces some overhead in the annotation process, it is also the case that the effort of manually writing 243 constraints (i) is negligible when compared to the effort of manually annotating the 22 videos, and (ii) can improve such last process, e.g., allowing to prevent errors in the annotation of the data points. 3.5 Experimental Validation In this section we present results on the various tasks the ROAD dataset is designed to benchmark (see Sec. 3.2.4). 3.5.1 Implementation Details The results are evaluated in terms of both frame-level bounding box detection and tube detection. In the first case, the evaluation measure of choice is frame mean average precision (f-mAP). We set the Intersection over Union (IoU) detection threshold to 0.5 (signifying a 50% overlap between the predicted and true bounding box). For the second set of results, we use video mean average precision (video-mAP), as information on how the ground-truth BBs are temporally connected is available. These evaluation metrics are standard in action detection [127,138,142\u2013144]. We also evaluate actions performed by AV, as described in 3.2.1. Since this is a temporal segmentation problem, we adopt the mean average precision metric computed at frame-level, as standard on the Charades [133] dataset. We use sequences of T = 8 frames as input to 3D-RetinaNet. Input image size is set to 512 \u00d7 682. This choice of T is the result of GPU memory constraints; however, at test time, we unroll our convolutional 3D-RetinaNet for sequences of 32 frames, showing that it can be deployed in a streaming fashion. We initialise the \f42 backbone network with weights pretrained on Kinetics [34]. For training we use an SGD optimiser with step learning rate. The initial learning rate is set to 0.01 and drops by a factor of 10 after 18 and 25 epochs, up to an overall 30 epochs. The parameters of the tube-building algorithm (Sec. 3.3.2) are set by cross validation. For ROAD we obtain \u03bb = 0.5 and k = 4. 3.5.2 Experimental Results on ROAD Three splits: modelling weather variability For the benchmarking of the ROAD tasks, we divided the dataset into two sets. The first set contains 18 videos for training and validation purposes, while the second set contains 4 videos for testing, equally representing the four types of weather conditions encountered. The group of training and validation videos is further subdivided into three different ways (\u2019splits\u2019). In each split, 15 videos are selected for training and 3 for validation. Details on the number of videos for each set and split are shown in Table 3.4. All 3 validation videos for Split-1 are overcast; 4 overcast videos are also present in the training set. As such, Split-1 is designed to assess the effect of different overcast conditions. Split-2 has all 3 night videos in the validation subset, and none in the training set. It is thus designed to test model robustness to day/night variations. Finally, Split-3 contains 4 training and 3 validation videos for sunny weather: it is thus designed to evaluate the effect of different sunny conditions, as camera glare can be an issue when the vehicle is turning or facing the sun directly. Note that there is no split to simulate a bias towards snowy conditions, as the dataset only contains one video of that kind. The test set (bottom row) is more uniform, as it contains one video from each environmental condition. Results on the various tasks Results are reported for the tasks discussed in Section 3.2.4. Frame-level results across the five detection tasks are summarised in Table 3.5 using the frame-mAP (f-mAp) metric, for a detection threshold of \u03b4 = 0.5. The reported figures are averaged across the three splits described above, in order to assess the overall robustness of the detectors to domain variations. Performance within each split is evaluated on both the corresponding validation subset and test set. Each row in the Table shows the result of a particular combination of backbone network (2D, I3D, or Slowfast) and test-time sequence length (in a number of frames, 8 and 32). Frame-level results vary between 16.8% (events) and 65.4% (agentness) for I3D, and between 23.9% and 69.2% for Slowfast. \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 43 Table 3.5: Frame-level results (mAP %) averaged across the three splits of ROAD. The considered models differ in terms of backbone network (2D, I3D, and Slowfast) and clip length (08 vs 32). The performance of YOLOv5 on agent detection is also reported. Detection threshold \u03b4 = 0.5. Both validation and test performance are reported for each entry. Model Agentness Agents Actions Locations Events 2D-08 51.8/63.4 30.9/39.5 15.9/22.0 23.2/30.8 10.6/12.8 2D-32 52.4/64.2 31.5/39.8 16.3/22.6 23.6/31.4 10.8/13.0 I3D-08 52.3/65.1 32.2/39.5 19.3/25.4 24.5/34.9 12.3/16.5 I3D-32 52.7/65.4 32.3/39.2 19.7/25.9 24.7/35.3 12.6/16.8 Slowfast-08 68.8/69.2 41.9/47.5 26.9/31.1 34.6/37.3 18.1/23.7 Slowfast-32 69.3/68.7 42.6/43.7 27.3/31.7 34.8/36.4 18.0/23.9 YOLOv5 57.9/56.9 Table 3.6: Video-level results (mAP %) averaged across the three ROAD splits. The models differ in terms of backbone network (2D, I3D and Slowfast) and test time clip length (08 vs 32). The performance of YOLOv5 on agent detection is also reported. Both validation and test performance are reported for each entry. Model Agents Actions Locations Events Detection threshold \u03b4 = 0.2 2D-08 22.2/25.1 10.3/13.9 18.2/24.8 12.8/14.7 2D-32 22.6/25.0 11.2/14.5 18.5/25.9 13.0/15.3 I3D-08 23.2/26.5 14.1/15.8 20.8/25.8 14.9/17.4 I3D-32 24.4/26.9 14.3/17.5 21.3/27.1 15.9/17.9 Slowfast-08 24.1/29.0 16.0/20.5 28.3/33.0 18.9/22.4 Slowfast-32 24.2/28.6 16.0/19.55 29.0/29.7 19.1/22.5 YOLOv5 38.8/43.3 Detection threshold \u03b4 = 0.5 2D-08 8.9/7.5 2.3/3.0 5.2/6.1 5.1/5.3 2D-32 8.3/8.0 2.7/3.3 5.6/7.1 5.0/5.7 I3D-08 9.2/9.6 4.0/4.3 5.8/6.9 4.6/5.4 I3D-32 9.7/10.2 4.0/4.6 6.4/7.7 4.8/6.1 Slowfast-08 7.1/8.9 3.9/4.7 7.1/11.0 6.5/6.6 Slowfast-32 8.3/9.8 3.7/4.4 8.4/10.0 5.3/7.3 YOLOv5 18.7/13.9 Video-level results are reported in terms of video-mAP in Table 3.6. As for the frame-level results, tube detection performance (see Sec. 3.3.2) is averaged across the three splits. One can appreciate the similarities between frameand video-level results, which follow a similar trend albeit at a much lower absolute level. Again, results are reported for different backbone networks and sequence lengths. Streaming deployment. Increasing test sequence length from 8 to 32 does not much impact performance. This indicates that, even though the network is trained on 8-frame clips, being fully convolutional (including the heads in the temporal direction), it can be easily unrolled to process longer sequences at test time, making it easy to deploy in a streaming fashion. Being deployable in an incremental fashion is a must for autonomous driving applications; this is a quality that other tubelet\f44 Table 3.7: Number of videoand frame-level instances for each label (individual or composite), left. Corresponding frame-/video-level results (mAP@%) for each of the three ROAD splits (right). Val-n denotes the validation set for Split n. Results produced by an I3D backbone. Number of instance Frame-mAP@0.5/Video-mAP@0.2 Train subset #Boxes/#Tubes Train-1 Train-2 Train-3 Eval subset All Val-1 Val-2 Val-3 Test Val-1 Test Val-2 Test Val-3 Test Agent 559142/7029 60103/781 79119/761 83750/809 82465/1138 44.5/30.1 34.0/25.7 17.2/16.0 40.9/27.4 35.3/27.1 42.6/27.5 Action 639740/9815 69523/1054 89142/1065 95760/1111 94669/1548 26.2/17.0 26.6/17.4 11.7/11.4 25.3/17.3 21.2/14.6 25.7/17.9 Location 498566/8040 56594/851 67116/864 77084/914 70473/1295 34.9/28.6 35.2/26.4 13.7/12.1 33.9/26.3 25.4/23.2 36.7/28.6 Event 453626/8394 43569/883 65965/963 72152/967 64545/1301 17.7/18.6 15.9/15.8 6.4/11.8 16.4/18.9 13.7/17.2 18.1/18.9 Number of instances Frame-AP AV-action 122154/490 17929/67 18001/56 16700/85 20374/82 57.9 45.7 33.5 43.6 43.7 48.2 based online action detection methods [138\u2013140] fail to exhibit, as they can only be deployed in a sliding window fashion. Interestingly, the latest work on streaming object detection [145] proposes an approach that integrates latency and accuracy into a single metric for real-time online perception, termed \u2018streaming accuracy\u2019. We will consider adopting this metric in the future evolution of ROAD. Impact of the backbone. Broadly speaking, the Slowfast [130] and I3D [33] versions of the backbone perform as expected, much better than the 2D version. A Slowfast backbone can particularly help with tasks that require the system to \u2018understand\u2019 movement, e.g. when detecting actions and road events, at least at 0.2 IoU. Under more stringent localisation requirements (\u03b4 = 0.5), it is interesting to notice how Slowfast\u2019s advantage is quite limited, with the I3D version often outperforming it. This is due to the fact that the 3D-RetinaNet baseline model\u2019s choice of fixed-length input that may not fully leverage the strengths of the Slowfast backbone, which excels in scenarios requiring motion information and temporal context, such as video analysis. Therefore, any critique of the model should consider this limitation and its potential impact on capturing temporal dependencies and motion-related features. Finally, even stronger backbones using transformers [146,147] can be plugged in our 3D-RetinaNet baseline as backbone. Level of task challenge. The overall results on event detection (last column in both Table 3.5 and Table 3.6) are encouraging, but they remain in the low 20s at best, showing how challenging situation awareness is in road scenarios. Comparison across tasks. The headline figures are not really comparable since, as we know, the number of classes per task varies. More importantly, within-class variability is often lower for composite labels. For example, the score for Indicating right is really low, whereas Car / Indicating-right has much better performance. This is because the within-class variability of the pair Car / Indicating-right is much lower than that of Indicating right, which puts together instances of differently-looking types of vehicles (e.g. buses, cars and vans) all indicating right. Interestingly, results on agents are comparable among the four baseline models (especially for f-mAP and \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 45 v-mAP at 0.2, see Tables 3.5 and 3.6). YOLOv5 for Agent detection. For completeness, we also trained YOLOv520 for the detection of active agents. The results are shown in the last row of both Table 3.5 and Table 3.6. Keeping is mind that YOLOv5 is trained only on single input frames, it shows a remarkable improvement over the other baseline methods for active agent detection. We believe that is because YOLOv5 is better at the regression part of the detection problem \u2013 namely, Slowfast has a recall of 71% compared to the 94% of YOLOv5, so that Slowfast has a 10% lower mAP for active agent detection. We leave the combination of YOLOv5 for bounding box proposal generation and Slowfast for proposal classification as a promising future extension, which could lead to a general improvement across all tasks. Validation vs test results. Results on the test set are, on average, superior to those on the validation set. This is because the test set includes data from all weather/visibility conditions (see Table 3.4), whereas for each split the validation set only contains videos from a single weather condition. E.g., in Split 2 all validation videos are nighttime ones. Results under different weather conditions Table 3.7 shows, instead, the results obtained under the three different splits we created on the basis of the weather/environmental conditions of the ROAD videos, discussed in Section 3.5.2 and summarised in Table 3.4. Note that the total number of instances (boxes for frame-level results or tubes for video-level ones) of the five detection tasks is comparable for all the three splits. We can see how Split-2 (for which all three validation videos are taken at night and no nighttime videos are used for training, see Table 3.4) has the lowest validation results, as seen in Table 3.7 (Train-2, Val-2). When the network trained on Split-2\u2019s training data is evaluated on the (common) test set, instead, its performance is similar to that of the networks trained on the other splits (see Test columns). Split-1 has three overcast videos in the validation set, but also four overcast videos in the training set. The resulting network has the best performance across the three validation splits. Also, under overcast conditions one does not have the typical problems with nighttime vision, nor glares issues as in sunny days, as it has sunny videos in both train and validation sets. These results seem to attest a certain robustness of the baseline to weather variations, for no matter the choice of the validation set used to train the network parameters (represented by the three splits), the performance on test data (as long as the latter fairly represents a spectrum of weather conditions) is rather stable. 20https://github.com/ultralytics/yolov5 \f46 Results of AV temporal action detection Table 3.8 shows the results of using 3D-RetinaNet to temporally detect AV-action classes, averaged across all three splits on both validation and test set. As we can see, the results for classes AV-move and AV-stop are very good, we think because these two classes are predominately present in the dataset. The performance of the \u2019turning\u2019 classes is reasonable, but the results for the bottom three classes are really disappointing. We believe this is mainly due the fact that the dataset is very heavily biased (in terms of number of instances) towards the other classes. As we do intend to further expand this dataset in the future by including more and more videos, we hope the class imbalance issue can be mitigated over time. A measure of performance weighing mAP using the number of instances per class could be considered, but this is not quite standard in the action detection literature. At the same time, ROAD provides an opportunity for testing methods designed to address class imbalance. 3.5.3 Qualitative Results Finally, we provide some qualitative results of our baseline model in terms of success and failure modes. Cases in which the baseline work accurately are illustrated in Figure 3.4, where the model is shown to detect only those agents which are active (i.e., are performing some actions) and ignore all the inactive agents (namely, parked vehicles). Agent prediction is very stable across all the examples, whereas action and location prediction show some weakness in some case: for instance, the nighttime example in the second row of the second column, where both the cars in front are moving away in the outgoing lane but our method fails to label their location correctly. In contrast, the failure modes illustrated in Figure 3.5 are cases in which the model fails to assign to agents the correct label, and also detects agents which are not active (e.g. often parked cars, see the white vehicle in the top row, first column or the red vehicle in the third row, first column). 3.5.4 Challenge Results Table 3.9 compares the results of the top teams participating in our ROAD @ ICCV 2021 challenge with those of the Slowfast and YOLOv5 baselines, at a tube detection threshold of 0.2. The challenge server remains open at https://eval.ai/web/ challenges/challenge-page/1059/overview, where one can consult the latest entries. Agent detection. The agent detection challenge was won by a team formed by Chenghui Li, Yi Cheng, Shuhan Wang, Zhongjian Huang, Fang Liu of Xidian Uni\fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 47 Figure 3.4: Success cases in which our model detects the actions and locations correctly, and only for those agents which are active. Ground truth bounding boxes and labels are shown in green, while the predictions of our model are shown in red. versity, with an entry using YOLOv5 with post-processing. In their approach, agents are linked by evaluating their similarity between frames and grouping them into a tube. Discontinuous tubes are completed through frame filling, using motion information. Also, the authors note that YOLOv5 generates some incorrect bounding boxes, scattered in different frames, and take advantage of this by filtering out the shorter tubes. As shown in Table 3.9, the postprocessing applied by the winning entry significantly outperforms our off-the-shelf implementation of YOLOv5 on agent detection. Action detection. The action detection challenge was won by Lijun Yu, Yijun \f48 Figure 3.5: Some of the failure modes of our model, shown detecting inactive agents and/or assigning incorrect action and location labels. Ground truth bounding boxes and labels are shown in green, while the predictions of our model are shown in red. Qian, Xiwen Chen, Wenhe Liu and Alexander G. Hauptmann of team CMU-INF, with an entry called \u201cArgusRoad: Road Activity Detection with Connectionist Spatiotemporal Proposals\u201d, based on their Argus++ framework for real-time activity recognition in extended videos in the NIST ActEV (Activities in Extended Video ActEV) challenge21. The had to adapt their system to be run on ROAD, e.g. to construct tube proposals rather than frame-level proposals. The approach is a rather complex cascade of object tracking, proposal generation, activity recognition and temporal localisation stages [148]. Results show a significant (5%) improvement 21https://actev.nist.gov/. \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 49 Table 3.8: AV temporal action detection results (frame mAP%) averaged across all three splits. No instances Frame-mAP@0.5 Model I3D 2D Eval subset All Val Test Val Test Av-move 81196/233 92.0 96.6 83.0 87.8 Av-stop 31801/108 92.2 98.5 65.3 68.4 Av-turn-right 3826/50 46.1 63.0 35.0 57.7 Av-turn-left 3787/56 69.0 59.8 55.1 42.9 Av-overtake 599/12 4.9 1.1 2.7 2.5 Av-move-left 537/15 0.5 0.8 0.5 0.5 Av-move-right 408/16 10.5 0.6 4.0 2.0 Total/Mean 122154/490 45.0 45.8 35.1 37.4 Table 3.9: Results (in video-mAP) of the winning entries to the ICCV 2021 ROAD challenge compared with the Slowfast and YOLOv5 baselines, at a detection threshold of 0.2. Task Top team Slowfast YOLOv5 Winners Agent detection Xidian 29.0 43.3 52.4 Action detection CMU-INF 20.5 25.6 Event detection IFLY 22.4 24.7 over the Slowfast baseline, which is close to state-of-the-art in action detection, but still at a relatively low level (25.6%) Event detection. The event detection challenge was won by team IFLY (Yujie Hou and Fengyan Wang, from the University of Science and Technology of China and IFLYTEK). The entry consisted in a number of amendments to the 3D-RetinaNet baseline, namely: bounding box interpolation, tuning of the optimiser, ensemble feature extraction with RCN, GRU and LSTM units, together with some data augmentation. Results show an improvement of above 2% over Slowfast, which suggests event better performance could be achieved by applying the ensemble technique to the latter. 3.5.5 Experimental Results of ROAD-R As a first step, we ran 6 SOTA temporal feature learning architectures as part of a 3D-RetinaNet model [149] (with a 2D-ConvNet backbone made of Resnet50 [136]) for event detection and evaluated to which extent constraints are violated. We considered: 1. 2D-ConvNet (C2D) [150]: a Resnet50-based architecture with an additional temporal dimension for learning features from videos. The extension from 2D to 3D is done by adding a pooling layer over time to combine the spatial \f50 features. 2. Inflated 3D-ConvNet (I3D) [33]: a sequential learning architecture extendable to any SOTA image classification model (2D-ConvNet based), able to learn continuous spatio-temporal features from the sequence of frames. 3. Recurrent Convolutional Network (RCN) [8]: a 3D-ConvNet model that relies on recurrence for learning the spatio-temporal features at each network level. During the feature extraction phase, RCNs exploit both 2D convolutions across the spatial domain and 1D convolutions across the temporal domain. 4. Random Connectivity Long Short-Term Memory (RCLSTM) [151]: an updated version of LSTM in which the neurons are connected in a stochastic manner, rather than fully connected. In our case, the LSTM cell is used as a bottleneck in Resnet50 for learning the features sequentially. 5. Random Connectivity Gated Recurrent Unit (RCGRU) [151]: an alternative version of RCLSTM where the GRU cell is used instead of the LSTM one. GRU makes the process more efficient with fewer parameters than the LSTM. 6. SlowFast [130]: a 3D-CNN architecture that contains both slow and fast pathways for extracting the sequential features. A Slow pathway computes the spatial semantics at low frame rate while a Fast pathway processes high frame rate for capturing the motion features. Both of the pathways are fused in a single architecture by lateral connections. We trained 3D-RetinaNet22 using the same hyperparameter settings for all the models: (i) batch size equal to 4, (ii) sequence length equal to 8, and (iii) image input size equal to 512 \u00d7 682. All the models were initialized with the Kinetics pre-trained weights. An SGD optimizer [152] with step learning rate was used. The initial learning rate was set to 0.0041 for all the models except SlowFast, for which it was set to 0.0021 due to the diverse nature of slow and fast pathways. All the models were trained for 30 epochs and the learning rate was made to drop by a factor of 10 after 18 and 25 epochs. The machine used for the experiments has 64 CPUs (2.2 GHz each) and 4 Titan RTX GPUs having 24 GB of RAM each. To measure the models\u2019 performance, we used the frame mean average precision (f-mAP), which is the standard metric used for action detection (see, e.g., [153,154]), with IoU threshold equal to 0.5 and 0.75, indicated as f-mAP@0.5 and f-mAP@0.75, respectively. The results for the SOTA models at IoU threshold 0.5 and 0.75 are reported in Table 3.10, column \u201cSOTA\u201d. 22https://github.com/gurkirt/3D-RetinaNets. \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 51 (a) I3D (b) C2D (c) RCGRU (d) RCLSTM (e) RCN (f) SlowFast Figure 3.6: Examples of violations of {\u00acRedTL, \u00acGreenTL}. To measure the extent to which each system violates the constraints, we used the following metrics: \u2022 the percentage of non-admissible predictions, \u2022 the average number of violations committed per prediction, and \u2022 the percentage of constraints violated at least once, while varying the threshold \u03b8 from 0.1 to 0.9 with step 0.1. The results are in Fig. 5.9, where (to improve readability) we do not plot the values corresponding to \u03b8 = 0.0 and \u03b8 = 1.0. For \u03b8 = 0.0 (resp., \u03b8 = 1.0), all the predictions are positive (resp., negative), and thus the corresponding values are (in order) 100%, 214, and 214/243 (resp., 100%, 2, and 2/243). 3.6 Further Extensions By design, ROAD is an open project which we expect to evolve and grow over time. \f52 (a) Percentage of predictions violating at least one constraint. (b) Average number of violations committed per prediction. (c) Percentage of constraints violated at least once. Figure 3.7: ROAD-R and SOTA models. In the x-axis, there is the threshold \u03b8 \u2208[0.1, 0.9], step 0.1. Table 3.10: f-mAP@0.5 and f-mAP@0.75 for the current SOTA models. IoU threshold Model 0.5 0.75 C2D 27.57 12.66 I3D 30.12 16.29 RCGRU 30.78 15.98 RCLSTM 30.49 16.64 RCN 29.64 15.82 SlowFast 28.79 14.40 Extension to other datasets and environments. In the near future, we will work towards completing the multi-label annotation process for a larger number of frames coming from videos spanning an even wider range of road conditions. Further down the line, we plan to extend the benchmark to other cities, countries and sensor configurations, to slowly grow towards an even more robust, \u2019in the wild\u2019 setting. In particular, we will initially target the Pedestrian Intention Dataset (PIE, [52]) and Waymo [155]. The latter one comes with spatiotemporal tube annotation for pedestrian and vehicles, much facilitating the extension of ROAD-like event annotation there. Event anticipation/intent prediction. ROAD is an oven-ready playground for action and event anticipation algorithms, a topic of growing interest in the vision community [156, 157], as it already provides the kind of annotation that allows researchers to test predictions of both future event labels and future event locations, both spatial and temporal. Anticipating the future behaviour of other road agents is crucial to empower the AV to react timely and appropriately. The output of this Task should be in the form of one or more future tubes, with the scores of the associated class labels and the future bounding box locations in the image plane [158]. We will shortly propose a baseline method for this Task, but we encourage researchers in the area to start engaging with the dataset from now. Autonomous decision making. In accordance with our overall philosophy, we will design and share a baseline for AV decision making from intermediate semantic \fChapter 3. The ROad event Awareness Dataset for Autonomous Driving 53 representations. The output of this Task should be the decision made by the AV in response to a road situation [159], represented as a collection of events as defined in this chapter. As the action performed by the AV at any given time is part of the annotation, the necessary meta-data is already there. Although we did provide a simple temporal segmentation baseline for this task seen as a classification problem, we intend in the near future to propose a baseline from a decision making point of view, making use of the intermediate semantic representations produced by the detectors. Machine theory of mind [160] refers to the attempt to provide machines with (limited) ability to guess the reasoning process of other intelligent agents they share the environment with. Building on our efforts in this area [122], we will work with teams of psychologists and neuroscientists to provide annotations in terms of mental states and reasoning processes for the road agents present in ROAD. Note that theory of mind models can also be validated in terms of how close the predictions of agent behaviour they are capable of generating are to their actual observed behaviour. Assuming that the output of a theory of mind model is intention (which is observable and annotated) the same baseline as for event anticipation can be employed. Continual event detection. ROAD\u2019s conceptual setting is intrinsically incremental, one in which the autonomous vehicle keeps learning from the data it observes, in particular by updating the models used to estimate the intermediate semantic representations. The videos forming the dataset are particularly suitable, as they last 8 minutes each, providing a long string of events and data to learn from. To this end, we plan to set a protocol for the continual learning of event classifiers and detectors and propose ROAD as the first continual learning benchmark in this area [161]. Alternatives to ROAD-R. Throughout our ROAD-R, we used logical constraints to satisfy the requirements and avoid the prediction of impossible scenarios. The main intuition behind using logical constraints instead of probabilistic priors is that probabilistic priors do not give any guarantee that the constraints will be satisfied. In ROAD-R, the constraints we employed predominantly pertained to impossible scenarios. This means that the majority of our constraints are rooted in expressing physical impossibilities. This distinction between impossible and improbable scenarios is pivotal when considering alternative probabilistic methods. In the future, we are aiming to consider alternative probabilistic methods that can effectively handle both impossible and improbable scenarios. This reflection can enhance the robustness and applicability of our research findings, ensuring that they are not limited solely to situations of physical impossibility but also account for highly improbable yet impactful events. By doing so, we aim to provide a more comprehensive and versatile framework for decision-makers and stakeholders who rely on our research to navigate uncertain real-world situations. \f54 3.7 Summary of the Chapter This chapter proposed a strategy for situation awareness in autonomous driving based on the notion of road events, and contributed a new ROad event Awareness Dataset for Autonomous Driving (ROAD) as a benchmark for this area of research. The dataset, built on top of videos captured as part of the Oxford RobotCar dataset [126], has unique features in the field. Its rich annotation follows a multi-label philosophy in which road agents (including the AV), their locations and the action(s) they perform are all labelled, and road events can be obtained by simply composing labels of the three types. The dataset contains 22 videos with 122K annotated video frames, for a total of 560K detection bounding boxes associated with 1.7M individual labels. Baseline tests were conducted on ROAD using a new 3D-RetinaNet architecture, as well as a Slowfast backbone and a YOLOv5 model (for agent detection). Both frame\u2013mAP and video\u2013mAP were evaluated. Our preliminary results highlight the challenging nature of ROAD, with the Slowfast baseline achieving a video-mAP on the three main tasks comprised between 20% and 30%, at low localisation precision (20% overlap). YOLOv5, however, was able to achieve significantly better performance. These findings were reinforced by the results of the ROAD @ ICCV 2021 challenge, and support the need for an even broader analysis, while highlighting the significant challenges specific to situation awareness in road scenarios. Our dataset is extensible to a number of challenging tasks associated with situation awareness in autonomous driving, such as event prediction, trajectory prediction, continual learning and machine theory of mind, and we pledge to further enrich it in the near future by extending ROAD-like annotation to major datasets such as PIE and Waymo. We also proposed a new dataset for the task of learning with requirement, called ROAD-R. We showed that SOTA models most of the time violate the requirements, and how it is possible to exploit the requirements to create models that are compliant with (i.e., strictly satisfy) the requirements while improving their performance. \fPart II : Complex Activity Detection 55 \f56 \fChapter 4 Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 4.1 Introduction In the second part of the thesis, we are taking one step ahead of ROAD event detection presented in the previous chapter (Chapter 3) and address a novel problem of Complex Activity Detection. In this chapter, long-term complex activity recognition and localisation framework is presented and can be crucial for decision-making in autonomous systems such as smart cars and surgical robots. Here we address the problem via a novel deformable, spatiotemporal scene graph approach, consisting of three main building blocks: (i) action tube detection, (ii) the modelling of the deformable geometry of parts, and (iii) a graph convolutional network. Firstly, action tubes are detected in a series of snippets. Next, a new 3D deformable RoI pooling layer is designed for learning the flexible, deformable geometry of the constituent action tubes. Finally, a scene graph is constructed by considering all parts as nodes and connecting them based on different semantics such as order of appearance, sharing the same action label and feature similarity. We also contribute fresh temporal complex activity annotation for the recently released ROAD autonomous driving and SARAS-ESAD surgical action datasets and show the adaptability of our framework to different domains. Our method is shown to significantly outperform graph-based competitors on both augmented datasets. 57 \f58 4.1.1 Motivation Complex activity recognition is attracting much attention in the computer vision research community due to its significance for a variety of real-world applications, such as autonomous driving [2, 3], surveillance [4], medical robotics [5] or team sports analysis [6]. In autonomous driving, for instance, it is extremely important that the vehicle understands dynamic road scenes, in order, e.g., to accurately predict the intention of pedestrians and forecast their trajectories to inform appropriate decisions. In surveillance, group activities rather than actions performed by individuals need to be monitored. Robotic assistant surgeons need to understand what the main surgeon is doing throughout a complex surgical procedure composed of many short-term actions and events [162], in order to suitably assist them. Prior to the advent of deep learning techniques, the study of atomic and complex activities was a vibrant research area that encompassed a range of methodologies [163,164]. The fundamental works from the 1990s and the early 2000s focused on understanding and modeling these activities through various approaches such as grammars [165], Bayesian networks [166], and graph-based recognition [167]. Researchers explored the decomposition of complex actions into fundamental atomic actions, which were basic and indivisible components of more intricate behaviors. These foundational works laid the groundwork for understanding how activities are composed and structured, providing insights into the underlying patterns and dependencies. By analyzing the relationships between atomic actions and their compositions, researchers were able to formulate models that captured the essence of human actions and interactions. This chapter revisits the concepts of atomic and complex activities within the framework of the present era, building upon the legacy of past research while also addressing the novel challenges and opportunities introduced by deep learning techniques. Recent methods for action or activity recognition and localisation can be broadly divided into two categories; single atomic action [25,168\u2013170] and multiple atomic action recognition/localisation [40, 63, 72\u201375]. The former methods only focus on identifying the start and end of an action performed in a short video portraying a single instance, leveraging datasets such as UCF-101 [171] or Charades [172]. The latter set of approaches consider videos which contain a number of atomic actions or multiple repetitions of the same action. Methods in this category do address complex activity recognition, as their aim is to understand an overall, dynamic scene by detecting and identifying its constituent components. Datasets used for complex activity detection are Epic-kitchens [173], THUMOS14 [31] or ActivityNet v1.3 [30]. Both classes of methods are geared towards merely recognising and localising short term action or activities that lasts for only a few frames or seconds. \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 59 4.1.2 Objective Unlike all existing methods, in this chapter, we present a framework capable of recognising complex, long-term activities, validated in the fields of autonomous driving and surgical robotics but of general applicability and extendable to other domains. More precisely, by \u2018complex activity\u2019 we mean an ensemble of \u2018atomic\u2019 actions, each performed by an individual agent present in the environment, which extends over a period of time and which collectively has a meaning. These activities can be contiguous, occurring sequentially without any gap or background (in the case of surgical robotics). Alternatively, these activities may occur at different positions within the video, with background segments interspersed between them (in the case of autonomous driving). For instance, in the autonomous driving context, one can define the complex activity Negotiating intersection as composed of the following atomic actions: an autonomous vehicle (AV) moves along its lane; the traffic light regulating the vehicle\u2019s lane turns red, while the light turns green for a traversing road; a number of other vehicles pass through the intersection from both sides; the AV\u2019s light turns green again; the AV resumes moving and crosses the intersection. The proposed pipeline (Fig. 4.1) is divided into three parts: (i) action tube detection, (ii) part-based feature extraction and learning via 3D deformable RoI pooling, and (iii) a graph generation strategy to process a variable number of parts and their connections, aimed at learning the overall semantics of a dynamic scene representing a complex activity. Action tube detection is a necessary pre-processing step, aimed at spatially and temporally locating the atomic actions present [127,128,158,174\u2013177]. Note that the tube detector needs to ensure a fixed-size representation for each activity part (atomic action). Here, in particular, we adopt AMTNet [1], as the latter describes action tubes of any duration using a fixed number of bounding box detections. 4.1.3 Main Contributions Our contribution is twofold. Firstly, our novel 3D deformable RoI pooling layer, inspired by standard deformable and modulated RoI pooling [178, 179], is not only designed to work with 3D data but is also capable of learning feature representations for tubes of variable spatiotemporal shape. Secondly, an original Graph Convolutional Network (GCN) module constructs a graph by considering individual tubes as nodes and connecting them via edges encoding diverse semantics, namely: appearance order, sharing of the same action label, and spatial feature similarity. The spatiotemporal scene graph so constructed is then processed by a stack of graph convolutional layers resulting in graph representation features, which are used to train a classifier for recognising complex activities, followed by a localisation stage which \f60 Figure 4.1: Overall pipeline of our long-term complex activity detection framework. (i) The input video is first divided into snippets. (ii) Snippets are passed to an action tube detector module one by one. (iii) A part-based deformation module receives 3D VGG features and action tube locations and returns features for the salient (non-background) parts of each tube instance. (iv) A GCN module represents activity parts (action tubes) as nodes with the features generated by ROI pooling and builds edges with different semantics to construct a spatiotemporal graph representation. Finally (v), the graph representation features produced by GCN inference on the consolidated graph are used for temporal activity detection. uses a sliding window approach. The framework is evaluated using two real-world datasets springing from completely different domains: ROAD [180] for situation awareness in autonomous driving and SARAS-ESAD [181] for surgical action detection, both providing videolevel annotation in the form of (atomic) action tubes. In this work we augment these datasets with suitable annotation on the start and end time of each instance of complex activity (road activities in ROAD vs surgical phases in SARAS-ESAD). The main contributions of this work are therefore: \u2022 A novel framework for long-term complex activity recognition and localisation. \u2022 An original deformable 3D RoI pooling approach for flexibly pooling features from the various components of the detected tubes, to create an overall representation for activity parts. \u2022 A spatiotemporal scene graph generation and processing mechanism able to cope with a variable number of parts while learning the overall semantics of an activity class. \u2022 Augmented annotation for two newly-released datasets aimed at making them suitable benchmarks for future work on complex activity detection. \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 61 Our results clearly indicate that the detection task (both at atomic and complex activity level) is extremely challenging on the real-world data which forms these newly annotated benchmarks, when compared to existing academic datasets. We hope this will stimulate further original thinking to address these challenges. Our method is shown to clearly outperform two recent state of the art graph-based competitors [21,25] on both augmented datasets. Related publications: The work presented in this chapter is published in BMVC 2021 [17]. The dissertation author was the primary investigator in [17]. Our code is also available online1. Outline: This chapter is organised as follows. Firstly, we start by presenting the proposed methodology of our spatiotemporal deformable scene graphs framework in Section 4.2. The proposed methodology is divided into building blocks including Action Tube Detection, 3d Part-Bsed Deformable Layer, and Graph Convolutional Network described in Section 4.2.1, Section 4.2.2, and Section 4.2.3, respectively. Next, the detailed experimental evaluation of our proposed method is reported in Section 4.3. Finally, a summary with limitations of the proposed method is given in Section 4.4. 4.2 Proposed Method Whereas modern complex activity recognition and localisation methods focus on single action or collections of short-term atomic actions, the issue of detecting complex activities articulated into a number atomic actions and spanning longer intervals of time is still rather unexplored. Crucial to the identification of complex video activities is the modelling of the relations among the constituent actions. Deformable models that represented activities as graphs of atomic parts provide such a framework. The existing part-based methods are investigated for action detection and prediction tasks, but their potential for the recognition of long-term activities has not been explored so far. In this chapter, we propose to achieve this via a combination of deformable pooling of features and a spatiotemporal graph representation that employs multiple semantics. The workflow of our approach is illustrated in Figure 4.1. The input video is divided into fixed-sized snippets which are fed to an action tube detector, followed by a novel 3D deformable RoI pooling stage which computes parts features, which are later processed by GCN for activity detection. 1https://github.com/salmank255/SDSG_Complex_Activity \f62 4.2.1 Action Tube Detection To provide a fixed-size representation for the instances of atomic actions composing a complex activity, we adopt AMTnet [1]. AMTnet is a two-stream online action tube detector that uses both RGB and optical flow information (although here we only use the RGB stream). The main rationale for using AMTnet is that it generates tubes in an incremental manner while preserving a fixed-size representation. Architectural Details. AMTNet uses VGG-16 [182] as baseline CNN feature extractor. The last two fully-connected layers of VGG-16 are replaced by two convolutional layers, and add four extra convolutional layers at the end. AMTNet takes sequence of RGB frames as an input with a fixed temporal interval \u25b3between consecutive frames, i.e., {ft, ft+\u25b3}. The input to AMTNet is in the format [BS \u00d7 Sq \u00d7 D \u00d7 H \u00d7 W], where BS is the training batch size, Sq is the sequence length (in this case a pair), D is the dimensionality (equal to 3 as we are dealing with RGB frames), while H and W are the height and width of each frame (300 \u00d7 300 in our case). As typical in action detection, AMTNet uses both a classification and a regression layer for recognition and detection, respectively, with the goal of predicting action \u2018microtubes\u2019 defined by pairs of consecutive detections. The method predicts bounding boxes for a pair of frames separated by fixed gap \u25b3, while the bounding boxes for intermediate frames are generated by interpolation. In this work, atomic action instances are represented as 3 micro-tubes with \u25b3= 3 for an overall tube length of L = 12 frames, aligned with our snippet length. Complete action tubes are incrementally generated by AMTNet by temporally linking the micro-tubes predicted by the network. (see Figure 4.2). Figure 4.2: (a) AMTNet [1] generates micro-action tubes by predicting bounding boxes for the start and the end frame of a snippet. Predictions for a fixed number of intermediate frames are constructed by bilinear interpolation. (b) Complete action tubes are created by temporally linking micro action tubes by dynamic programming. \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 63 4.2.2 3D Part-Based Deformable Layer The feature extractor in our framework is a novel 3D deformable RoI pooling layer which encodes the spatiotemporal geometry of the action tubes which correspond to the activity parts. This is an extension of the existing standard deformable RoI pooling layer [178] that has the ability to extract and learn features from an action tube rather than a 2D bounding box. The rationale behind using the 3D deformable RoI pooling layer is that it allow us to learn at training time how the geometric shape of atomic action tubes (as regions of the video considered as a spatiotemporal volume) varies across instances of the same class. Intuitively, the shape of the bounding box detections forming a tube (and therefore the shape of the tube itself) will vary with e.g. the viewpoint, as well as the particular style with which the action is performed by a certain agent (for instance, a cyclist can turn right making a narrower as opposed to a wider turn). The principle of our 3D deformable RoI pooling operation is shown in Figure 4.3. Like the classical deformable RoI pooling layer, our module also includes standard RoI pooling (used in all region proposal-based object detection methods), a fully connected layer, and offsets which encode the amount of geometric deformation. Firstly, standard RoI pooling is applied to the provided feature map X and bounding boxes locations forming an action tube (L \u00d7 [x,y,w,h]), by subdividing the tube into a pooled feature map grid of fixed-size in both the spatial and the temporal dimensions: L \u00d7 k \u00d7 k. Here L = 12 is the fixed action tube length, while k is a free parameter which determines the \u2018bin size\u2019, i.e., the number of grid locations detections are divided into in each spatial dimension (see Figure 4.3 again). Next, for each bin in the grid, normalised offsets (representing the degree of deformation of the grid components of each action tube) are generated for these feature maps using a fully-connected layer, which are then transformed using an element-wise product with the original RoI\u2019s width and height. Offsets are also multiplied by a scalar value to modulate their magnitude (empirically set to 0.1), making them invariant to the different possible sizes of the RoI. In our framework, this layer takes the VGG features extracted by AMTNet and each detected action tube separately as an input and returns an overall feature map which encodes both the appearance and the shape (through the above offsets) of each atomic action. 4.2.3 Graph Convolutional Network As our purpose is to achieve a comprehensive understanding of the dynamic scene which comprises a complex activity, we propose to use a graph convolutional network to model and exploit the relations between the constituent action tubes. Unlike the tree structure of classical part-based models (which requires to fix the number of \f64 Figure 4.3: Our 3D deformable RoI pooling layer takes feature maps and action tube locations as input and arranges them into a fixed-size grid of components (here illustrated for size 3 \u00d7 3). For each grid component an offset is generated and multiplied by the original tube feature to produce the final component features. parts [183]), (spatiotemporal) graphs allow us to flexibly describe a complex activity composed by a variable number of actions (nodes) of different type, and to encode the different semantic relationships between them. The functioning of our GCN module is illustrated in Figure 4.4. Graph Construction. When constructing the activity graphs, the input video is subdivided into consecutive, non-overlapping, fixed-length snippets. For each snippet, a separate graph is built with a variable number of nodes corresponding to the number of detected activity parts (action tubes) within the snippet. The initial representation of the nodes is provided by their RoI features. We consider three different types of connections: (i) the order of appearance (from left to right) of each action tube (bearing in mind that in autonomous driving, for instance, road activities tend to follow a specific order, e.g., pedestrian crossing the road followed by vehicles engaging an intersection); (ii) the spatial similarity of the node features inspired from the edge convolution proposed in [85], which dynamically constructs edges between the nodes according to their feature distances. The distance threshold in our case is fixed, but can be a learnable parameter by updating via an edge convolution [85]; (iii) node type, meant as the sharing the same action label, as this provides very relevant information for the determination of the activity class. As a result three spatiotemporal scene graphs are constructed having the same nodes but with different edges. While the second and third graph are undirected, the appearance order graph is a directed one. However, when merging the three graphs an undirected version of the \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 65 Figure 4.4: Our graph module takes as input the features generated for each tube by the ROI pooling layer T1,2,3...K and builds edges between them according to different semantics. ( order of appearance, spatial similarity, label). The overall graph is processed by a GCN to deliver a fixed-size graph representation. order graph is used. These graphs are then combined by taking a union of all edges to create a single homogeneous graph representing the overall scene. Namely, two nodes are connected in the merged graph iff they are connected in at least one of the three graphs. Graph Convolution and Representation. Given the final graph, global graph embedding is applied to extract the context of each snippet portraying a complex activity. In our GCN approach we apply a stack of three graph convolutional layers followed by a graph readout layer. The latter encapsulates the final graph representation by taking the mean of the hidden convolutional representation, resulting in a fixed-sized feature representation which is invariant to the number of nodes and edges. 4.2.4 Complete Framework The complete framework is the concatenation of the aforementioned three modules. Firstly, we divide the video V into N Snippets S1,2,3,...,N, with each Snippet Si consisting of a fixed (M) number of frames: Si = F 1,2,3,...,M. Each snippet is passed to the action tube detection module AT which returns K action tubes each composed by L = 12 bounding boxes with labels BA and intermediate VGG features XA represented as BA M\u00d7K\u00d75, XA M\u00d764\u00d7300\u00d7300 \u2208AT. Action tube locations and features are then passed to our 3D deformable RoI pooling layer DRoI which returns a fixedsized (i.e., 7 \u00d7 7) grid of components whose dimensionality is equal to the number (64) of convolutional layers: XDRoI K\u00d764\u00d7M\u00d77\u00d77. These features are then fed to our \f66 Algorithm 4.1 Complex Activity Detection 1 Input: Input Video V divided into N number of Snippets S consists of fixed M number of frames F 1,2,3,...,M 2 Initialization: Load pretrained action tube detector ATD on desire dataset, import our 3D deformable RoI pooling DRoI and sparsity Sp modules 3 for i \u22081,2,3,..., N do 4 Extract features and detect action tubes BA M\u00d7K\u00d75, XB M\u00d764\u00d7300\u00d7300 \u2190 ATD(Si) 5 XDRoI K\u00d764\u00d7M\u00d77\u00d77 \u2190DRoI(BA M\u00d7K\u00d75, XB M\u00d764\u00d7300\u00d7300) 6 XSp K\u00d732\u00d7M\u00d77\u00d77 \u2190Sp(XDRoI K\u00d764\u00d7M\u00d77\u00d77) 7 XFc1 K\u00d7M\u00d7512 \u2190Fc1(XSp K\u00d732\u00d7M\u00d77\u00d77) 8 XFc2 2048 \u2190Fc2(XFc1 K\u00d7M\u00d7512) 9 CAi \u2190Softmax(XFc2 2048) 10 if CAi\u22121 == CAi then 11 Same activity else 12 if CAi\u22121 == CAi+1 then 13 same activity else 14 end of activity 15 start new activity end end end 16 Output: Start and end frames with activity labels GCN module G, where a graph with K nodes and E edges is processed to yield a fixed-sized feature representation XG 2048 \u2208G. Finally, the latter features are fed to a Softmax classifier to classify the snippet into their respective activity category. For localisation we use a sliding window approach in which snippets are sequentially classified, using a dual verification mechanism. Namely, as our purpose is to recognise and detect long-term activities, if there appears to be a random false positive or false negative between two snippets belonging to the same class we simply ignore it and consider it as having the same activity label. The detection algorithm is described in detail in Algorithm 4.1. Implementation. Before training our overall architecture, we separately train AMTNet for action tube detection over both datasets. Note that we had to design from scratch suitable data loaders for the two datasets, as the format of the annotation there is completely different from that of the original datasets AMTNet was validated upon. As mentioned, our 3D RoI pooling layer includes a temporal dimension to learn and extract tube parts of a tube rather than of a 2D object. In our experiments we also convert a more recent version of deformable RoI pooling called \u2019modulated\u2019 deformable RoI pooling to the 3D case. In the GCN module we construct graphs \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 67 Table 4.1: List of ROAD complex activities with number of train and test snippets splits and brief description. No Complex activity Train snippets Test snippets Scenario/Description 1 Negotiating intersection 523 414 Autonomous vehicle (AV) stops at intersection/junction, waits for other vehicles to pass, then resumes. 2 Negotiating pedestrian crossing 381 25 AV Stops at pedestrian crossing signal, waits for pedestrian(s) and cycles to cross, then resumes. 3 Waiting in a queue 570 447 AV stops and waits in a queue for a signal (either pedestrian crossing or junction traffic light). 4 Merging into vehicle lane 68 3 AV stops and waits for bus/car to merge into the vehicle lane from the opposite side of the road. 5 Sudden appearance 3 3 A pedestrian/cycle/vehicle suddenly appears in front of the vehicle, coming from a side street. 6 Walking in the middle of road 13 19 A pedestrian walking in the middle of the road in front of the AV, causing the AV to slow down or stop. online for each snippet during training using a PyTorch data loader [184]. For the design of the GCN architecture we use the Deep Graph Library (DGL) [185] with a PyTorch back-end, which supports the processing of graphs of various length in a single mini-batch. Overall, architecture is implemented using the PyTorch [184] deep learning library with OpenCV and Scikit-learn. For training we used a machine equipped with 4 Nvidia GTX 1080 GPUs with 12GB VRAM each. 4.3 Experiments 4.3.1 Datasets and Evaluation Metrics In this chapter, we used two datasets for evaluating our approach, both already annotated at video level for action tubes detection. ROAD [180]: ROAD (the ROad event Awareness Dataset for autonomous driving) is annotated for road action and event detection. The complete detail of ROAD dataset is given in Section 2.6. In this work, we augmented the annotation of the ROAD dataset for complex road activity detection. We used a total of 19 videos with an average duration of 8 minutes each, 12 of which were selected for training and the remaining 7 for testing. We temporally annotated the ROAD videos by specifying the start and end frame for six different classes of complex road activities we inferred from video inspection. For example, a \u2018Negotiating intersection\u2019 activity class can be defined which is made up of the following \u2018atomic\u2019 events: Autonomous Vehicle (AV)-move + Vehicle traffic \f68 Table 4.2: SARAS video phases with number of train and test snippets and a brief description. No Complex activity/ Phase # Train snippets Test snippets Scenario/Description 1 Phase#1 (Bladder mobilization) 1808 999 Tools only used for mobilization 2 Phase#2 (AFP Dissection) 6263 4455 Anytime only Anterior Prostatic Fat (AFP) is removed 3 Phase#3 (Bladder neck transection) 8353 2419 Whenever the scissor cut the neck, until all the neck is transected 4 Phase#4 (NVP dissection) 361 46 Located just under the seminal vescicles. 5 Phase#5 ( Exposure of seminal vescicles) 2858 800 White bags surrounding the bladder neck. Comprises fat removal. 6 Phase#6 (Urethral division) 2553 1685 Start even when removing fat on the urethra. 7 Phase#7 (Prostate liberation) 1092 2903 After urethral division. It is not AFP. Stops only when completely free. 8 Phase#8 (Vesicourethral anastomosis) 4430 4690 Every time there is string and needle in the scene. Stops when urethra is linked to bladder light / Green + AV-stop + Vehicle(s) / Stopped / At junction+ AV-move. Activity class statistics are listed and described in Table 4.1. SARAS-ESAD [181]: ESAD (the Endoscopic Surgeon Action Detection Dataset) is a benchmark devised for surgeon action detection in real-world endoscopic surgery videos described in Section 2.6. In this work, we took a step forward and annotated ESAD in terms of complex activities corresponding to the different phases of the surgical procedure portrayed by the videos (namely, radical prostatectomy). For example, Phase # 3 corresponds to \u2018Bladder neck transection\u2019, in which a scissor cuts the neck of the bladder until it is transected. Phases and their statistics are again reported in Table 4.2. The complex activities (surgical phases) in SARAS were defined by professional surgeons experts in radical prostatectomy. As standard in the surgical context, such phases are consecutive without the need for any background activity class. For more details please see [181]. Evaluation Metrics: For the evaluation of action tube detection performance we used the standard frame/video mean Average Precision (mAP) at different IoU thresholds \u03b4 (namely, 0.2, 0.3, 0.5, 0.75) on both datasets. Complex activity recognition was evaluated using classification accuracy, precision, recall and F-score. For complex activity localisation we used the standard protocol mAP over the temporal dimension used by all relevant methods. \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 69 Table 4.3: Action tube detection performance on both the ROAD and SARAS-ESAD datasets. Both Frame-mAP and Video-mAP at different IoU thresholds are reported for evaluation ROAD SARAS-ESAD Methods / IoU threshold \u03b4 0.2 0.3 0.5 0.75 0.2 0.3 0.5 0.75 Singh et al. [180] (frame-mAP) 25.9 Singh et al. [180] (video-mAP) 17.5 4.6 Bawa et al. [181] (frame-mAP) 24.3 12.2 AMTNet (frame-mAP) 22.3 18.1 15.4 11.0 30.4 24.6 18.7 7.9 AMTNet (video-mAP) 11.6 7.9 3.8 13.7 10.1 8.8 5.4 4.3.2 Action Tube Detection A detailed comparative analysis of AMTNet over different action detection datasets can be found in the original paper [1]. Here we briefly report the performance of AMTNet on our two datasets of choice, as AMTNet was never tested there. Table 4.3 reports both frame-mAP and video-mAP results, and compares AMTNet with the proposed baselines for the two datasets: the ROAD baseline (termed 3D RetinaNet [180]), and the ESAD baseline [181], a vanilla implementation of RetinaNet (only providing frame-level results). AMTNet performed better than [181] on SARASESAD, while being inferior to [180] on ROAD. Remember that the main rationale for using AMTNet is that it can provide a fixed-size representation for the tubes (as required by our framework), motivating us to compromise on accuracy. 4.3.3 Complex Activity Recognition Next, we provide a detailed analysis of complex activity classification using our approach on both the ROAD and SARAS-ESAD datasets. The performance for each class in both datasets is illustrated in Fig. 4.5 and Fig. 4.6 using all metrics. It is apparent that the ROAD dataset is characterised by significant fluctuations in class-wise performance, with higher recognition accuracy for activities that appear more often, e.g. \u2018waiting in a queue\u2019, as opposed to infrequent ones (e.g. \u2018sudden appearance\u2019). In SARAS-ESAD each activity class does contain enough samples for good training, while the diversity in phases still poses a challenge. In fact, performance on this dataset is a function of the complexity of the phases and the visual similarity of the constituent atomic actions, not just the amount training data. For example, Phase 5 and 6 both include \u2018fat removal\u2019 actions, complicating their differentiation. \f70 Figure 4.5: Complex activity classification performance over ROAD dataset. Table 4.4: Comparative analysis of temporal activity localisation performance on ROAD and SARAS-ESAD, reporting mAP (%) at five different IoU thresholds. ROAD SARAS-ESAD Methods / IoU threshold \u03b4 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 P-GCN [21] 60.0 56.7 53.9 50.5 46.4 57.9 55.6 53.4 49.0 45.1 G-TAD [25] 62.1 59.8 55.6 52.2 48.7 59.1 56.7 54.5 49.8 46.9 Ours 77.3 74.6 71.2 66.7 59.4 62.9 59.6 58.2 55.3 51.5 4.3.4 Temporal Activity Detection Comparison with State-ofthe-Art To evaluate the performance of our complex activity detection approach we reimplemented two state-of-the-art activity localisation methods \u2013 P-GCN [21] and GTAD [25], as neither ROAD nor ESAD were ever used for complex activity detection. The major changes we made during re-implementation are: (i) data loading (as both P-GCN and G-TAD were designed to be trained and tested on pre-extracted features), and (ii) replacing the regression part with a sliding window approach for the SARAS-ESAD dataset, as the latter lacks a background class. ROAD. For activity detection in ROAD we used an additional \u2018background\u2019 class, which indicates either no action or presence of action(s) without any solid indication. Whenever no action tube was detected we would use the entire frame as RoI for our parts deformation module to understand the overall scene. Temporal activity detection performance on ROAD, measured via mAP at five different IoU thresholds, is reported in Table 4.4 for both our approach and the two competitors. \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 71 Figure 4.6: Complex activity classification performance on SARAS-ESAD dataset. Table 4.5: ROAD activity localisation performance (mAP, %) for each complex road activity, at a standard IoU threshold of 0.5. Method / Activities Negotiating intersection Negotiating pedestrian crossing Waiting in queue Merging into vehicle lane Sudden appearance Walking middle of road PGCN [21] 44.3 53.8 74.4 50.1 21.7 34.1 GTAD [25] 47.8 57.3 70.6 55.2 24.3 37.1 Ours 51.2 72.3 89.8 84.1 17.8 41.3 Class-wise results for each complex activity at a standard IoU threshold of 0.5 are reported in Table 4.5. SARAS-ESAD. Temporal activity detection on this dataset much relates to activity recognition, as surgical phases are contiguous. Both the average mAP of the methods at five IoU thresholds and the class-wise performance for each activity (phase) at a standard IoU threshold of 0.5 are reported in Table 4.4 and 4.6, respectively. From the results it is clear how our method outperforms the chosen state-of-the-art methods by a reasonable margin. Table 4.6: SARAS-ESAD activity localisation performance (mAP, %) for each activity at a fixed IoU threshold of 0.5. Method / Activities Phase#1 Phase#2 Phase#3 Phase#4 Phase#5 Phase#6 Phase#7 Phase#8 P-GCN [21] 56.7 43.2 52.3 59.1 33.8 59.4 14.8 41.2 G-TAD [25] 51.1 46.6 57.2 63.8 29.4 62.2 19.3 45.7 Ours 57.5 54.1 69.3 60.2 31.1 71.3 16.5 52.4 \f72 4.3.5 Ablation Studies We ran a number of ablation studies on both datasets to understand the influence of the various components of our model. Effect of GCN. First of all, we demonstrated the significance of our GCN by conducting three different types of experiments; i) Snippet features from the entire scene without engaging in action tube detection or utilising GCN.. ii) Action features without GCN where we first detect the action tubes and then extract the deformable features from the tubes. It is worth noting that, in this particular instance, we constrained the number of action tubes to a fixed limit of 10. iii) Action features with GCN, our proposed framework, incorporating action features and integrating the GCN. The clear difference among these three types of setups can be observed in Fig. 4.7. Snippet features only Actions features without GCN Actions features with GCN Figure 4.7: The performance of different components of our proposed method over ROAD and SARAS-ESAD dataset. The mAP at a threshold of 0.5 is reported for the test set of both datasets. Effect of Edges. Next, we ablated the effect of various types of edges employed in our GCN for graph construction. This ablation analysis includes all three types of edges used for graph construction (explained in Section 4.2.3) individually and combined. The comprehensive performance for each edge type over both ROAD and SARAS-ESAD datasets is presented in Fig. 4.8. \fChapter 4. Spatiotemporal Deformable Scene Graphs for Complex Activity Detection 73 Figure 4.8: The performance comparison of various edge types in our GCN modules. The mAP at a threshold of 0.5 is reported for the test set of both datasets. Effect of Sequence Length. Lastly, we showed how snippet duration, which represents the time span of the local scene impacts our model. We conducted experiments with four different sizes (12, 18, 24, and 30), as given in Table 4.7. Notably, when evaluating our model on ROAD, the most favorable results were achieved with a sequence length of 18. This choice is attributed to the datasets\u2019 characteristics, as the activities within them tend to be of average duration. Conversely, when evaluating on SARAS-ESAD, we observed superior performance with a sequence length of 24, aligning with the predominantly the much longer duration of activities in this dataset. Table 4.7: The performance comparison of different sequence lengths over both ROAD and SARAS-ESAD dataset, reporting mAP (%) at two different IoU thresholds. ROAD SARAS-ESAD Sequence length / IoU threshold \u03b4 0.1 0.5 0.1 0.5 12 77.3 59.4 62.9 51.5 18 79.7 60.8 64.6 52.8 24 74.1 55.3 67.2 54.5 30 70.6 52.2 66.1 53.7 \f74 4.4 Summary of the chapter 4.4.1 Limitations and Future Work The main limitation of this work is that it relies on action tube detection. From our results, the existing tube detectors are not reliable enough to perform well over challenging real-world datasets such as those we adopted here. Clearly, if the tube detector misses an important atomic action this will affect the overall activity detection performance. Nevertheless our results show that, even when using a suboptimal detector, our approach is capable of significantly outperforming state of the art methods on our new benchmarks. Detection is challenging on SARAS and ROAD because of their real-world nature: surgical images are indistinct, road scenes come with incredible variations. These benchmarks show how even the best detectors suffer a huge drop in performance when moving from \u2018academic\u2019 benchmarks to real ones. We hope the realism of these two extremely challenging datasets will stimulate real progress and new original thinking in the field. In the future our primary target will be the design of a more accurate action tube detector with the ability to perform better in challenging scenarios such as those portrayed in ROAD or SARAS-ESAD. We will also explore the end-to-end training of the entire model in all its three components. Further down the line, we will update our S/T scene graph approach to properly model the heterogenous nature of the graph [186], and extend it to a more complete representation of complex dynamic events in which nodes (rather than correspond all to action tubes) may be associated with any relevant elements of a dynamic scene, such as objects, agents, actions, locations and their attributes (e.g. red, fast, drivable, etc). 4.4.2 Conclusion In this chapter, we presented a spatiotemporal complex activity detection framework which leverages both part deformation and a heterogenous graph representation. Our approach is based on three building blocks; action tube detection, part-based deformable 3D RoI pooling for feature extraction and a GCN module which processes the variable number of detected action tubes to model the overall semantics of a complex activity. In an additional contribution, we temporally annotated two recently released benchmark datasets (ROAD and ESAD) in terms of long-term complex activities. Both datasets come with video-level action tube annotation, making them suitable benchmarks for future work in this area. We thoroughly evaluated our method, showing the effectiveness of our 3D part-based deformable model approach for the detection of complex activities. \fChapter 5 A Hybrid Graph Network for Complex Activity Detection in Video 5.1 Introduction Following the limitations of our previous work (relies on action tube detection) explained in Chapter 4. In this Chapter, we propose a new Complex Activity Detection (CompAD) paradigm that can be applicable to a variety of computer vision domains, including autonomous driving and sports analytics without additional action annotation. CompAD can be seen as the advanced version of Temporal Action Localisation (TAL), which aims to recognise what actions take place in a video and identify their start and end time. While TAL mostly focuses on short-term actions, CompAD extends the analysis to long-term activities, and does so by modelling a complex video activity\u2019s internal structure. In this chapter, we address the CompAD problem using a hybrid graph neural network that combines attention applied to a graph encoding the local (short term) dynamic scene with a temporal graph modelling the overall longduration activity. Our contribution is threefold. i) Firstly, we propose a novel feature extraction technique which, for each video snippet, generates spatiotemporal \u2018tubes\u2019 for the active elements (\u2018agents\u2019) in the (local) scene by detecting individual objects, tracking them and then extracting 3D features from all the agent tubes as well as the overall scene. ii) Next, we construct a local scene graph where each node (representing either an agent tube or the scene) is connected to all other nodes. Attention is then applied to this graph to obtain an overall representation of the local dynamic scene. iii) Finally, all local scene graph representations are temporally connected to each other via a temporal graph, with the aim of estimating the complex activity class together with its start and end time. The proposed framework outperforms all previous state-of-the-art methods on three benchmark datasets: Thumos-14, ActivityNet-1.3, and the more recent ROAD. 75 \f76 5.1.1 Motivation The ability to detect and recognise activities from untrimmed videos is a research problem which is attracting significant attention, due in part to the rapid growth of online video generation in domains including sports [6], autonomous driving [2], medical robotics [187] and surveillance [188]. The current state-of-the-art in action detection is Temporal Action Localisation (TAL) whose approaches not only recognise the action label(s) present in a video, but also identify the start and end time of each action instance occurring. TAL can be instrumental in generating sports highlights [16, 61], understanding road scenes in autonomous driving [17], video summarisation in surveillance videos [18] and video captioning [19, 189]. A number of TAL methods [20\u201329] have recently been proposed, competing to achieve state-ofthe-art performance on selected benchmark datasets. Whereas various new datasets have been recently proposed, the two most common relevant benchmarks remain ActivityNet 1.3 [30], and Thumos-14 [31]. State-of-the-art performance on Thumos-14 has improved in four years by some 19% [32]. Figure 5.1: An overview of our Complex Activity Detection (CompAD) framework. The input video is divided into fixed-size snippets S1, . . . , SN (top); each snippet is then processed in three major steps (bottom). A) Firstly, scene objects (agents) are detected and tracked throughout the snippet to form agent tubes. 3D features are then extracted from all the cropped agent tubes (fai) as well as the local scene (fsi). B) Next, a fully-connected scene graph is constructed where agent nodes (in gray) are connected to each other and to the snippet node (in light blue). The local scene graph is processed using a graph attention network (GAT), resulting in intermediate scene features (f\u2032\u2032 si). C) Finally, all local scene features associated with individual snippets are temporally connected and processed using a global temporal graph to identify the boundaries of the activity using anchor proposals (shown in different colors). Each of the anchors represents a binary graph mask where snippets which belong to the activity receive label ybri = 1, 0 otherwise. \fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 77 Almost all TAL approaches contemplate two major aspects: features/scene representation and temporal localisation. In the scene representation stage, snippets (continuous sequences of frames) are processed to understand the local scene in the video. Methods [26\u201329] typically employ pre-extracted features obtained using a sequential learning model (e.g., I3D [33]), often pre-trained on the Kinetics [34] dataset. Such features are then processed, e.g., via a temporal or semantic graph neural network, by applying appropriate feature encoding techniques or by generating temporal proposals, in an object detection style [35]. In the second step of TAL, actions need to be temporally localised. Existing methods vary in the way this is done, e.g. through temporal graphs [21, 25], boundary regression and proposals generation [20, 22, 28] or encoder-decoder approaches [32]. All such methods, however, are only applicable to shortor mid-duration actions that last for a few seconds (e.g., a person jumping or pitching a baseball). As recently pointed out in e.g. [36,37], in real-world applications a challenge is posed by complex activities, longer-term events comprising a series of elementary actions, often performed by multiple agents. For example, an Autonomous Vehicle (AV) negotiating a pedestrian crossing is engaged in a complex activity: First it drives along the road, then the traffic lights change to red, the vehicle stops and several pedestrians cross the road. Eventually, the lights turn green again and the AV drives off. Theoretically, TAL methods can be employed to temporally segment complex activities, in practice such approaches are only employed to detect shortor midduration actions lasting a few seconds at most (e.g., a person jumping or pitching a baseball). The activities contemplated by the most common datasets are of this nature. Fig. 5.2 compares two standard TAL benchmarks with the recently released ROAD dataset, explicitly designed for complex activity detection. Activities in the ROAD dataset last longer than those in ActivityNet or Thumos, with twice as many agents per snippet, making them more complex in nature. In this paper, we argue that standard TAL approaches are ill equipped to detect complex activities, as they fail to model both the global temporal structure of the activity and its fine-grained structure, in terms of the agents and elementary actions involved and their relations. We may thus define (strong) Complex Activity Detection (CompAD) as the task of recovering, given an input video, the temporal extent of the activities there present, as well as their inner structure in terms of the agents or elementary actions involved. A weaker CompAD is one in which the only expected output is the temporal segmentation, with the internal structure of the activity estimated as a means for achieving segmentation with no annotation available. While a small number of studies have attempted to detect complex, long-duration \f78 activities [36\u201338], to our knowledge [17] is the only existing study which attempts to tackle CompAD as defined above. The work, however, relies upon the availability of heavily-annotated datasets which include granular labels for the individual actions which make up a complex activity, and the corresponding frame-level bounding boxes. This is a serious limitation, for it prevents [17] from being usable for pure temporal segmentation and compared with prior art there. Further, [17] focuses only on the graph representation of snippets, neglecting the long-term modelling of complex activities. Figure 5.2: Comparing TAL datasets (ActivityNet-1.3 and Thumos-14) with a CompAD dataset (ROAD) in terms of average activity and video durations and mean number of agents per snippet. 5.1.2 Objectives This work aims to push the boundaries of temporal action localisation towards fullblown Complex Activity Detection (CompAD), intended as the ability to detect longterm, complex activities, leveraging datasets providing temporal segmentation annotation only. We do so by modelling and leveraging a complex video activity\u2019s internal structure, but without resorting to any additional fine-grained annotation concerning individual actions. Our proposed framework (Fig. 5.1) is composed of three stages: A) feature extraction; B) a scene graph attention network designed to learn the importance of each active object (\u2018agent\u2019) within the local dynamic scene; and C) a temporal graph of attended scene graphs for the localisation of complex activities of arbitrary duration. Our feature extraction scheme (A) differs from what typically done in prior art where spatiotemporal features are extracted from whole snippets only. In contrast, we first detect the relevant active objects in the scene and track them within each snippet to build for each of them an agent tube (a series of related bounding boxes). A pre-trained tracker is used to allow the method to be deployable to any datasets with only temporal annotation (no bounding box annotation for the scene agents is required), while enabling a finer-grained description of the internal structure of a \fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 79 complex activity. Agent tubes are resized to a standard spatial size to create \u2018mini videos\u2019 from which features can be extracted. A pre-trained 3D feature extraction model is then used to encode both the \u2018cropped\u2019 tubes and the whole snippet. To suitably represent the local dynamic scene within each snippet, a local scene graph (B) is constructed using three different topologies including a star graph, star graph with same agent label, and a fully-connected (the effects of all three topologies are explained in Section 5.3.4 Ablation study). The local scene graph is then processed using a scene graph attention (SGAT) network for extracting the overall scene representation. The rationale behind using the SGAT is its ability to compute the importance of each node in the context of its neighbors, thus modelling the structure of complex scenes. Finally (C), the learned local scene graphs are connected to each other by constructing a temporal graph, with the aim of recognising the activity label and identifying its temporal boundaries (start and end time) in a class-agnostic manner. Each node in the temporal graph learns two things; differentiating between the background and the activity region and recognising the activity label. Such a strategy enables us to model and detect both short and long-duration activities. 5.1.3 Contributions Our main contributions are, therefore: \u2022 A hybrid graph network approach for general complex activity detection, comprising a local scene graph as well as a global temporal graph, capable of localising both long and short-term activities. \u2022 A feature extraction strategy for fine-grained temporal activity localisation without the need for additional annotation, consisting in detecting and tracking the relevant scene agents to construct tubes, followed by feature extraction from both the whole scene and each agent tube individually, as opposed to existing methods which only rely on snippet-level features. \u2022 Comprehensive experiments showing how our approach leveraging weak CompAD outperforms the most recent TAL state-of-the-art across the board on both Thumos-14 and the recent ROad event Awareness Dataset (ROAD) [17,45], which portrays long-duration road activities involving multiple road agents over sometimes several minutes, while being extremely competitive on the standard ActivityNet 1.3 benchmark, showing a clear dominance on classical TAL approaches as the duration and the complexity of the activities increase. \f80 Related publications: The work presented in this chapter is a preprint ArXiv. The dissertation author is the primary investigator of this research work. Our code is also available online1. Outline: The remainder of the Chapter is organised as follows. The proposed methodology of our framework is presented in Section 5.2 while the main building blocks including tube detection and features extraction in Section 5.2.1, scene graph attention in 5.2.2, temporal graph localisation in 5.2.3, and loss functions in 5.2.4, respectively. Further, the detailed experimental evaluation of the proposed method is reported in Section 5.3. Finally, a summary with limitations of the proposed method is given in Section 5.4. Figure 5.3: Visualisation of our agent detection and tracking stage using a bird\u2019s-eye view of the spatiotemporal volume corresponding to a video segment of ActivityNet-1.3 dataset. The first half of the graph shows the plotting of agent tubes while the second half illustrates the scene graph of the detection agents. In this figure, we show examples of activities performed by multiple actors (agent tubes). 5.2 Proposed Method The proposed framework, outlined in the Introduction, is illustrated in Fig. 5.1. An input untrimmed video V is divided into N snippets S = S1, ..., Si, Si+1, ..., SN (each snippet is a pre-defined constant length of consecutive frames). Each snippet is then passed to the tube detection and feature extraction module which returns a feature vector for both the snippet fsi and the individual agent tubes fa1 i , fa2 i ,...,fan i , where n 1https://anonymous.4open.science/r/HG_CompAD-7697/README.md \fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 81 is the number of agents present in snippet i. These features are then forwarded to the local scene graph attention layer for learning the attention of each agent in the context of its neighbours. This returns an aggregate feature representation for the whole scene (f \u2032\u2032 si). These aggregate local scene features for all the snippets, f \u2032\u2032 s1,f \u2032\u2032 s2,...,f \u2032\u2032 sN, are then connected using a global temporal graph for the generation of the activity class label \u02c6 ysi and activity boundary labels \u02c6 ybri using anchor proposals in a classagnostic manner. 5.2.1 Tube Detection and Feature Extraction As mentioned, one of the contributions of this chapter is a new strategy for feature extraction and representation which consists in analysing the finer structure of the local dynamic scene, rather extracting features from whole snippets only. Objects Detection and Tracking. We first detect scene objects in each frame of the snippet using an object detector pre-trained on the COCO dataset [190], which comprises of 80 different types of objects. However, we select a relevant list of object types (termed agents) which depends on the dataset (the lists of classes selected for each dataset is given in Section 5.3.2 \u2013 Implementation Details). The detected agents are then tracked using a pre-trained tracker throughout the snippet in order to construct agent tubes. The latter are of variable length, depending on the agent itself and its role in each snippet. A pictorial illustration of agent tubes in an example video segment from all three datasets, showing both a sample frame, with overlaid the detection bounding boxes, and a bird\u2019s-eye view of the agent tubes and a local scene graph for the snippet it belongs to. ActivityNet-1.3 is one of the largest temporal action localisation datasets which includes activities performed by both individual and multiple objects (agents). Therefore, we illustrate the example of multi-agent activities in Fig. 5.3. Thumos-14 covers different types of sports actions performed by single or mostly multiple agents. Two sample actions; baseball pitch and soccer penalty are visualised in Fig. 5.4. In the figure, it can be seen that our model construct the seen graph of almost all agents despite the low-resolution videos. ROAD. The activities in the road dataset are mostly performed by a large number of agents, however, there are a few cases where the number of detected agents is less in number as shown in Fig. 5.5 (above). On the other hand, in Fig. 5.5 (below) we also show an example of a night video where the activity is performed by a large number of agents moving very promptly. Feature Extraction. Next, all the detected agent tubes are brought to a standard size and passed to the pre-trained 3D CNN model along with the whole snippet for spatiotemporal feature extraction. The adopted 3D CNN model allows variable \f82 Figure 5.4: Visualisation of our agent detection and tracking stage using a bird\u2019s-eye view of the spatiotemporal volume corresponding to a video segment of Thumos-14 dataset. The first half of the graph shows the plotting of agent tubes while the second half illustrates the scene graph of the detection agents. Both of the examples show the activities performed by multiple agents (recognising sports activities). length inputs, and outputs a fixed-sized feature vector for each of the agent tubes and the snippet. 5.2.2 Scene Graph Attention In a second stage, a scene graph representation is used to describe the scene in terms of features extracted from both the overall snippet and the agent tubes. Graph Generation. The scene graph is constructed using three different topologies: a star graph connecting each agent node to the scene node, a star topology with also links between agents sharing a label, and a fully-connected one. The influence of topology is shown in Sec. 5.3.4. Graph Attention. As mentioned, our scene graph attention layer is inspired by the GAT concept [94], originally designed for node classification. While we follow a similar attention mechanism, here we amend the attention layer in order to extract aggregate features from all the nodes, to be passed in turn to our localisation layer in the third stage. The workflow of our scene graph attention layer is shown in Fig. 5.6. All input node features are linearly transformed using a weight matrix W1, followed by selfattention to find the importance of each node with respect to its connected neighbours. An activation function LeakyReLU is applied to the features for nonlinearity, resulting in final output features for all of the nodes. These are then normalised using \fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 83 Figure 5.5: Visualisation of our agent detection and tracking stage using a bird\u2019s-eye view of the spatiotemporal volume corresponding to a video segment of the ROAD dataset. The upper section shows an example of activity performed by only three agents. Below is the example of a night video where the activity is performed by multiple agents. The first half of the graph shows the plotting of agent tubes while the second half illustrates the scene graph of the detection agents. a softmax function to make each node representation (f \u2032 siaj i) comparable with that of all the nodes connected to it (f \u2032 si). The self-attention process is further improved by applying a multi-head attention strategy inspired by transformers [94]. Attention is applied to the node features individually. For each node, the average over the H heads is computed, resulting in an attended feature vector for each node. Finally, to get a fixed-size representation for the whole scene, the output features of all the nodes are aggregated using another learnable weight matrix W2, which outputs the final feature representation f \u2032\u2032 si for the whole (local) scene. 5.2.3 Temporal Graph Localisation In Stage C of our framework, activity recognition and localisation are performed using a GCN. The final features from all the local scenes (f \u2032\u2032 s1, ..., f \u2032\u2032 si,f \u2032\u2032 si+1,...,f \u2032\u2032 sN), as outputted by the scene graph attention layer, are temporally connected to build a \f84 Figure 5.6: Our scene graph attention layer (Stage B of our approach, Fig. 5.1) takes the node features generated in the feature extraction Stage A as an input, and updates each node\u2019s features using node attention with respect to its all connected neighbours. Multihead attention with H heads is applied to each node to further robustify the representation \u2013 averaging yields the final node features. To obtain a fixed-size overall representation for a specific local scene (snippet), the features of all its nodes are aggregated using a learnable weight matrix W2. global temporal graph (see Fig. 5.1, C). Our GCN network for processing the global temporal graph is divided into two parts. The first part consists of three 1D convolutional layers designed to learn the temporal appearance of all the local scenes with boundaries. Each convolutional layer is followed by a sigmoid activation for non-linearity. The second part generates the anchor proposals of the temporally learned features via pre-defined anchors, where each of the anchors acts as a binary mask over the whole graph. Overall, the GCN module provides two different outputs. i) Activity classification: the list of predicted activity labels for all the snippets in the video (\u02c6 ys1, ..., \u02c6 ysi, ..., \u02c6 ysN) is produced, where the dimensionality of the output vector \u02c6 y is equal to the number of classes. ii) Activity localisation: activities are localised using binary masked class-agnostic anchor proposals. The Intersection over Union (IoU) measure between each anchor and the ground truth (true temporal extension of the activity) is computed, and the anchors with maximum IoU are selected to train the model for localising the boundaries of any activity, regardless of its activity label. The final output of our temporal graph is a one-hot binary vector (\u02c6 ybr1, ..., \u02c6 ybri, ..., \u02c6 ybrN) for each series of snippets (video), where \u02c6 ybri = 1 iff snippet Si belongs to the activity, = 0 when the snippet does not belong to it. 5.2.4 Loss Functions Our problem is multi-objective, as we aim at not only recognising the label of the activity taking place but also finding its boundary (start and end time). Therefore, we combined two losses one for learning the class label (classification) and the other for finding the start and end time (regression). The details of both losses are as follows: \fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 85 Given the ground truth ys = {ys1, . . . , ysi, . . . , ysN} and predicted \u02c6 ys = {\u02c6 ys1, . . . , \u02c6 ysi, . . . , \u02c6 ysN} activity labels, and the ground truth ybr = {ybr1, . . . , ybri, . . . , ybrN} and predicted \u02c6 ybr = {\u02c6 ybr1, . . . , \u02c6 ybri, . . . , \u02c6 ybrN} temporal extent labels, our overall loss function is formulated as: L = \u03bb \u00b7 LAct + LBr. (5.1) The second component denotes the binary cross entropy (BCE) loss LBr = {l1, ..., li, ..., lN}T, where li = \u2212wi h ybri \u00b7 log \u02c6 ybri + (1 \u2212ybri) \u00b7 log(1 \u2212\u02c6 ybri) i , driving the recognition of the activity\u2019s temporal extent in class-agnostic manner. LAct denotes, instead, the BCEWithLogitsLoss [191], defined as LAct = {l1,c, . . . , li,c, . . . , lN,c}T, where li,c = \u2212wi,c h pc \u00b7 ysi,c \u00b7 log \u03c3(\u02c6 ysi,c) +(1 \u2212ysi,c \u00b7 log(1 \u2212\u03c3(\u02c6 ysi,c)) i . (5.2) Here i is the index of the sample in the batch, c is the class number, pc is the weight of a positive sample for class c, and \u03c3 is the sigmoid function. This loss component is used to predict the activity labels {ysi}, and combines classical binary cross entropy with a sigmoid layer [191]. This combination is proven to be numerically stronger (as it leverages the log-sum-exp trick) than using a plain sigmoid separately followed by a BCE loss. Finally, \u03bb is a weight term, which we set to the number of selected anchor proposals (128 in our case). 5.3 Experiments 5.3.1 Datasets To evaluate the performance of our proposed method, we used three benchmark datasets including ROAD, Thumoas-14, and ActivityNet-1.3. Firstly, ROAD is used to compare our method with state-of-the-art CompAD methods. Next, Thumos-14 and ActivityNet-1.3 are used to show the effectiveness of our method over temporal action localisation methods. The complete detail of all these three datasets can be found in Section 2.6. \f86 5.3.2 Implementation Details Evaluation metrics. In our experiments, mean Average Precision (mAP) was used as an evaluation metric (explained in Section 2.7), using different IoU thresholds for the different datasets. According to the official protocols for the various benchmarks, the following lists of temporal IoU thresholds were selected: {0.1, 0.2, 0.3, 0.4, 0.5} for ROAD, {0.3, 0.4, 0.5, 0.6, 0.7} for Thumos-14 and {0.5, 0.7, 0.95} for ActivityNet1.3. Feature extraction. Firstly, the agent tubes are constructed by detecting scene objects using a YOLOv5 detector [192] pre-trained on the COCO dataset. Detections are then tracked throughout a snippet using DeepSort [193]. Then, features are extracted using an I3D network pre-trained on the Kinetics dataset [34], from both the entire snippet and each cropped agent tube individually. The object categories were reduced to six for the ROAD dataset to only cover the agents actually present in the road scenes portrayed there. As the other two datasets (ActivityNet-1.3 and Thumos-14) are general purpose, in their case we retained all the 80 classes present in the COCO dataset. Scene Graph Attention. The local scene graph was generated by producing a list of tuples [(0, 1), (0, 2), ...], where the first index relates to the source node and the second number indexes the target node. The reason for preferring this structure over an adjacency matrix was to limit memory usage. For node attention learning, we initialised our architecture using the weights of the GAT model [94] pre-trained on the PPI dataset [194] and applied 4 attention layers with {4, 4, num of classes, and num of classes} heads, respectively. The number of classes num of classes is equal to 201, 21, and 7 in ActivityNet-1.3, Thumos-14 and ROAD, respectively. The Temporal Graph is a stack of three 1D convolution layers on the final representation of the temporally connected local scenes. The size of the input to the first convolutional layer is the number N of local scenes (snippets), multiplied by the number of heads in the last attention layer. The length of our temporal graph is fixed to N. Videos with number of snippets less than or equal to N are passed directly to the temporal graph; longer videos are split into multiple chunks containing N snippets each. The output is a one-hot vector of activity labels of size N and a collection of 128 proposals (binary graph masks) also of length N. Loss Functions. Our problem is multi-objective, as we aim at not only recognising the label of the activity taking place but also finding its boundary (start and end time). Our overall loss function is thus the weighted sum of BCEWithLogitsLoss [191] (for activity classification) and standard binary cross entropy (for tem\fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 87 poral localisation). Full details can be found in the Section 5.2.4. 5.3.3 Comparison with State-of-the-Art ROAD. To validate our CompAD approach, we first compared ourselves with the prior art available for the ROAD dataset and re-implemented the current state-ofthe-art TAL methods; TallFormer [37] and ActionFormer [195] there (see Table 5.1). The standard IoU thresholds for mAP computation on ROAD range from 0.1 to 0.5. The detailed class-wise average precision of our methods is visualised in Fig. 5.7. Table 5.1: Comparing our approach with the state-of-the-art methods for CompAD available on the ROAD dataset. The mAP at the various standard thresholds is reported. Best results are in bold and second best underlined. ROAD Methods 0.1 0.2 0.3 0.4 0.5 Avg P-GCN [21] 60.0 56.7 53.9 50.5 46.4 53.5 G-TAD [25] 62.1 59.8 55.6 52.2 48.7 55.6 STDSG [17] 77.3 74.6 71.2 66.7 59.4 69.8 TallFormer [37] 78.4 74.9 70.3 63.8 57.1 68.9 ActionFormer [195] 76.5 73.7 72.6 64.4 58.2 69.0 Ours 82.1 77.4 73.3 69.5 62.9 73.0 Figure 5.7: Class-wise average precision (%) with IoU threshold of 0.5 for ROAD dataset. Thumos-14. Our hybrid graph approach is compared with the state of the art on Thumos-14 in Table 5.2. We followed the standard evaluation metric there: mAP with IoU thresholds ranging from 0.3 to 0.7 and their average. Competitor \f88 methods were published in top vision conferences in 2018-2022. The recent ActionFormer [195] approach achieves the highest average mAP 66.8. The detailed class-wise average precision of our method is visualised in Fig. 5.7. Table 5.2: Activity detection performance comparison with state-of-the-art methods on Thumos-14. mAP values at different IoU thresholds are reported for the test Thumos-14. Best results are in bold and second best underlined. Thumos-14 Methods Conference 0.3 0.4 0.5 0.6 0.7 Average BSN [20] ECCV\u201918 53.5 45.0 36.9 28.4 20.0 36.8 P-GCN [21] ICCV\u201919 63.6 57.8 49.1 \u2014 \u2014 \u2014 BMN [22] ICCV\u201919 56.0 47.4 38.8 29.7 20.5 38.5 GTAN [23] CVPR\u201919 57.8 47.2 38.8 \u2014 \u2014 \u2014 MGG [24] CVPR\u201919 53.9 46.8 37.4 29.5 21.3 37.8 G-TAD [25] CVPR\u201920 54.5 47.6 40.2 30.8 23.4 39.3 BU-MR [196] ECCV\u201920 53.9 50.7 45.4 38.0 28.5 43.3 BC-GNN [62] ECCV\u201920 57.1 49.1 40.4 31.2 23.1 40.2 BSN++ [197] AAAI\u201921 59.9 49.5 41.3 31.9 22.8 41.1 TCANet [198] CVPR\u201921 60.6 53.2 44.6 36.8 26.7 44.4 AFSD [81] CVPR\u201921 67.3 62.4 55.5 43.7 31.1 52.0 MUSES [199] CVPR\u201921 68.9 64.0 56.9 46.3 31.0 53.4 VSGN [200] ICCV\u201921 66.7 60.4 52.4 41.0 30.4 50.2 ContextLoc [201] ICCV\u201921 68.3 63.8 54.3 41.8 26.2 50.9 RTD-Net [202] ICCV\u201921 68.3 62.3 51.9 38.8 23.7 49.0 CPN [28] WACV\u201922 68.2 62.1 54.1 41.5 28.0 50.7 E2E-TAD [27] CVPR\u201922 69.4 64.3 56.0 46.4 34.9 54.2 RefactorNet [26] CVPR\u201922 70.7 65.4 58.6 47.0 32.1 54.8 RCL [203] CVPR\u201922 70.1 62.3 52.9 42.7 30.7 51.7 LDCLR [32] AAAI\u201922 72.1 65.9 57.0 44.2 28.5 53.5 TallFormer [37] ECCV\u201922 76.1 \u2014 63.2 \u2014 34.5 59.2 STALE [204] ECCV\u201922 68.9 64.1 57.1 46.7 31.2 52.9 ActionFormer [195] ECCV\u201922 82.1 77.8 71.0 59.4 43.9 66.8 Ours \u2014 78.2 69.5 62.7 50.1 36.9 59.8 ActivityNet-1.3. Table 5.3 also compares our proposal with the state-of-the-art methods on ActivityNet-1.3. The mAP IoU thresholds used for comparison are, according to protocol, 0.5, 0.75 and 0.95. Our approach performs the best on average and for IoU = 0.5 and 0.95, while coming a very close second for IoU = 0.75 with RefactorNet [26], proving our hypothesis that modelling the fine-grain structure of activity is conducive to better performance even for short-term actions. To sum up, our proposal clearly outperforms all previous approaches over the ROAD dataset, showing the potential of this approach for modelling and detecting long, complex activities. Further, our approach outperforms all competitors over ActivityNet and achieves a reasonable performance (second best) on Thumos-14, \fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 89 Figure 5.8: Class-wise average precision (%) with IoU threshold of 0.5 for Thumos-14 dataset. Table 5.3: Activity detection performance comparison with state-of-the-art methods on ActivityNet-1.3. mAP values at different IoU thresholds are reported for validation sets of ActivityNet-1.3. Best results are in bold and second best underlined. ActivityNet-1.3 Methods Conference 0.5 0.75 0.95 Average BSN [20] ECCV\u201918 46.45 29.96 8.02 30.03 P-GCN [21] ICCV\u201919 48.26 33.16 3.27 31.11 GTAN [23] CVPR\u201919 52.61 34.14 8.91 34.31 G-TAD [25] CVPR\u201920 50.36 34.60 9.02 34.09 BU-MR [196] ECCV\u201920 43.47 33.91 9.21 30.12 BC-GNN [62] ECCV\u201920 50.6 34.8 9.4 34.3 BSN++ [197] AAAI\u201921 51.27 35.70 8.33 34.88 TCANet [198] CVPR\u201921 51.91 34.92 7.46 34.43 AFSD [81] CVPR\u201921 52.4 35.3 6.5 34.4 MUSES [199] CVPR\u201921 50.02 34.97 6.57 33.99 VSGN [200] ICCV\u201921 52.4 36.0 8.4 35.1 ContextLoc [201] ICCV\u201921 56.01 35.19 3.55 34.23 RTD-Net [202] ICCV\u201921 47.2 30.7 8.6 30.8 E2E-TAD [27] CVPR\u201922 50.47 35.99 10.83 35.10 RefactorNet [26] CVPR\u201922 56.6 40.7 7.4 38.6 RCL [203] CVPR\u201922 55.15 39.02 8.27 37.65 LDCLR [32] AAAI\u201922 58.14 36.30 6.16 35.24 TallFormer [37] ECCV\u201922 54.1 36.2 7.9 35.6 STALE [204] ECCV\u201922 56.5 36.7 9.5 36.4 ActionFormer [195] ECCV\u201922 53.5 36.2 8.2 35.6 Ours \u2014 60.6 40.3 11.1 39.3 \f90 which is characterised by shorter, simpler activities. This proves that modelling the fine-grain structure of activities is conducive to better performance, even for simpler events. Fig. 5.9 illustrates how our method increasingly outperforms the prior art as the duration and complexity of the activities increases. Figure 5.9: Performance comparison with the state-of-the-art over all the three datasets considered, as a function of average activity duration and average no. of agents per snippet. 5.3.4 Ablation Studies We ran a number of ablation studies to understand the influence of the various components. Effect of Agent Nodes. Firstly, we showed the advantage of using a scene graph modelling and connecting single agents compared to only using whole scene features. Namely, we removed the local scene graph from our pipeline (Fig. 5.1, stage B) and passed the 3D features of the whole scene as a node to the global temporal graph. The significant performance drop can be clearly observed in Fig. 5.10 over all three datasets. Effect of Edges and Aggregation. Next, we constructed our local scene graph using three different types of edge connections. Firstly, a fully-connected scene graph where each node is connected to every other node. Secondly, a star structure where all of the agent nodes are connected to the scene node only. Lastly, a star structure with additional connections between agent nodes sharing the same label. We also validated two different techniques for extracting the final representation from the local scene graph, which we named \u2018Aggregated\u2019 and \u2018Scene\u2019. In the former, the feature representation is extracted by aggregating those of all the attended nodes. In the latter, only the feature vector related to the scene node (after attention) is retained. The effect of the possible combinations of graph topologies and aggregation strategies is shown in Table 5.4 \u2013 a fully-connected scene graph with Aggregated \fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 91 Figure 5.10: Average mAP of variants of our method with: attended scene graph (\u2019Agents with GAT\u2019), scene graph (\u2019Agents without GAT\u2019) and \u2019Scene features only\u2019, over the all three datasets. Table 5.4: Effect of local scene graph topologies and feature aggregation strategies on the performance of our proposal. The average mAP for the relevant IoU thresholds is reported for each dataset. Methods Thumos-14 ActivityNet-1.3 ROAD Fully connected (Aggregated) 59.8 37.4 73.0 Fully connected (Scene) 49.2 31.4 62.7 Star structure (Aggregated) 51.2 39.3 62.3 Star structure (Scene) 44.8 36.6 57.9 Star + same label (Aggregated) 52.2 35.3 64.9 Star + same label (Scene) 41.9 35.3 59.2 features performs the best in two cases over three, while the star topology performed best for ActivityNet. Effect of Sequence Length. To explore the effect on our model of snippet duration (the temporal extent of the local dynamic scene), we performed experiments with four different sizes (12, 18, 24, and 30), reported in Table 5.5. On Thumos-14 and ROAD the top scores were obtained by selecting a sequence length of 24, due to the nature of the activities present in these datasets, which last longer. On the other hand, on ActivityNet-1.3 we achieved the best performance using a sequence length of 18, as most activities there are shorter in duration. Effect of Temporal Graph Length. In addition to the temporal span of the local scene graph, we also ablated the effect of the length of the temporal graph. By this, we mean the number of local scene nodes used in the temporal graph for recognition and localisation. Four different temporal lengths (128, 256, 512, 1024) are com\f92 Table 5.5: Performance of our method as a function of different snippet lengths over the three datasets. The average mAP for the relevant IoU thresholds is reported for each dataset. Snippet length Thumos-14 ActivityNet-1.3 ROAD 12 52.7 31.3 56.9 18 59.8 37.6 62.6 24 54.3 39.3 73.0 30 49.7 34.5 70.0 Table 5.6: Performance of our method as a function of different temporal graph lengths over the three datasets. The average mAP for the relevant IoU thresholds is reported for each dataset. Temporal length Thumos-14 ActivityNet-1.3 ROAD 128 59.8 29.6 48.3 256 52.9 32.7 58.5 512 50.3 39.3 69.2 1,024 46.8 34.1 73.0 pared in Table 5.6. The best performance is achieved on ActivityNet-1.3 using a shorter temporal length, due to the average duration of the videos there and its portraying shorter-term activities. In contrast, on Thumos-14 and ROAD our approach performed the best using longer temporal graphs, as activities there are typically of longer duration. Qualitative Results. To help the reader visualise the output of our proposed method, we show some qualitative detection results on all the three datasets in Fig. 5.11 and Fig. 5.12. The figure shows one sample per dataset, and portrays a series of local scenes (snippets), skipping some for visualisation purposes, with superimposed the ground truth (in green) and the prediction of our model (in red). For example, for ActivityNet-1.3 an instance of the \u2018Layup drill in basketball\u2019 class is shown in which the activity starts with snippet 7 and ends with snippet 42. Our model predicts the activity to start from snippet 6 and end with snippet 43. 5.4 Summary of the chapter 5.4.1 Future Work In the future we intend to progress further from incremental inference to incremental training, by learning to constructing activity graphs in an incremental manner, thus opening the way to applications such as future activity anticipation [205] and pedestrian intent prediction [206], while overcoming the need for a fixed-size snippet rep\fChapter 5. A Hybrid Graph Network for Complex Activity Detection in Video 93 Figure 5.11: The qualitative results of our proposed method for all three datasets. The green rectangles covering the snippets (local scenes) are the ground truth while the yellow boxes show the prediction of our model. resentation. A further exciting line of research is the modelling of the uncertainty associated with complex scenes, in either the Bayesian [207] or the full epistemic setting [208,209]. 5.4.2 Conclusion Whereas most existing work focuses on short-term actions, this chapter explicitly addressed the problem of detecting longer-term, complex activities using a novel hybrid graph neural network-based framework which combines both scene graph attention and a temporal graph to model activities of arbitrary duration. Our proposed framework is divided into three main building blocks: agent tube detection and feature extraction; a local scene graph construction with attention; and a temporal graph for recognising the class label and localising each activity instance. We tested our method on three benchmark datasets, showing the effectiveness of our method in detecting both short-term and long-term activities, thanks to its ability to model their finer-grained structure without the need for extra annotation. \f94 Figure 5.12: The qualitative results of our proposed method for all three datasets. The green rectangles covering the snippets (local scenes) are the ground truth while the yellow boxes show the prediction of our model. \fPart III : Continual Learning 95 \f96 \fChapter 6 Continual Semi-Supervised Learning 6.1 Introduction The last part of the thesis is about continual learning, while interesting works have been recently directed at continual learning from streaming data in a fully supervised setting, especially focusing on avoiding catastrophic forgetting, continual learning in a semi-supervised setting remains a wide-open research question. Thus in this chapter, we present a new continual semi-supervised learning (CSSL) paradigm, proposed to the attention of the machine learning community via the IJCAI 2021 International Workshop on Continual Semi-Supervised Learning (CSSL@IJCAI)1, with the aim of raising the field\u2019s awareness about this problem and mobilising its effort in this direction. After a formal definition of continual semi-supervised learning and the appropriate training and testing protocols, the chapter introduces two new benchmarks specifically designed to assess CSSL on two important computer vision tasks: activity recognition and crowd counting. We describe the Continual Activity Recognition (CAR) and Continual Crowd Counting (CCC) challenges built upon those benchmarks, the baseline models proposed for the challenges, and describe a simple CSSL baseline which consists in applying batch self-training in temporal sessions, for a limited number of rounds. The results show that learning from unlabelled data streams is extremely challenging, and stimulate the search for methods that can encode the dynamics of the data stream. In addition to the challenge baseline, we also propose to formulate the continual semi-supervised learning problem as a latent-variable one, in which the labels of a series of unlabelled, streaming datapoints play the role of the hidden states, while the instances play that of the observations. In any realistic setting the process generating instances and labels, modelled by a probability distribution, will vary in time. The aim is for the model to be updated in such a way as to adapt to a (slowly) shifting 1https://sites.google.com/view/sscl-workshop-ijcai-2021/ 97 \f98 target domain, without artificial subdivisions into tasks. To accommodate for this, latent variable temporal graphical models whose parameters also vary in time need to be considered. In particular, we thoroughly analyse the proposed framework in a classification setting, for the case in which the data stream is modelled by a timevarying hidden Markov model. 6.1.1 Motivation The continual learning problem has been recently the object of much attention in the machine learning community, with a focus on preventing the model updated in the light of new data from \u2018catastrophic forgetting\u2019 its initial abilities. A typical example is that of an object detector which needs to be extended to include classes not originally in its list (e.g., \u2018donkey\u2019) while retaining its ability to correctly detect, say, a \u2018horse\u2019. The unspoken assumption there is that we are quite satisfied with the model we have, and merely wish to extend its capabilities to new settings and classes. This way of posing the continual learning problem, however, is in rather stark contrast with widespread real-world situations in which an initial model is trained using limited data, only for it to then be deployed without any additional supervision. Think of a person detector used for surveillance purposes on a busy street. Even after having been trained extensively on public datasets, experience shows that its performance in its target setting will suboptimal. In this scenario, the goal is for the model to be incrementally updated using the new (unlabelled) data, in order to adapt to its target domain. In such settings, the goal is quite the opposite of that of the classical scenario described above: the initial model is usually poor and not a good match for the target domain. To complicate things further, the target domain itself may change over time, both periodically (e.g., night/day cycles) and in asynchronous, discrete steps (e.g., when a new bank opens within the camera\u2019s field of view or a new building is erected with a new entrance). 6.1.2 Overview of the CSSL Challenge In the first part of this chapter, we formalise this problem as one of continual semisupervised learning, in which an initial training batch of labelled data points is available and can be used to train an initial model, but then the model is incrementally updated exploiting the information provided by a time series of unlabelled data points, each of which is generated by a data generating process (modelled by a probability distribution) which varies with time. Our formulation of CSSL is similar to online domain adaptation (ODA). Both CSSL and ODA involve adaptation to changing \fChapter 6. Continual Semi-Supervised Learning 99 data distributions, they have distinct focuses and characteristics. The main difference between ODA and our CSSL is that ODA primarily deals with adapting a model to changes in the data distribution across different domains (tasks). This is particularly relevant in applications where the target domain is dynamic and constantly evolving [210\u2013212]. CSSL, on the other hand, focuses on the scenario where a model incrementally learns from both labeled and unlabeled data over time. The key challenge is to continually adapt to new data without catastrophic forgetting of previously learned knowledge. While this may involve adapting to some degree of changing data distributions, the primary emphasis is on incremental learning and avoiding model degradation over time. In CSSL, we do not assume any artificial subdivision into \u2018tasks\u2019, but allow the data-generating distribution to be an arbitrary function of time. While batch semi-supervised learning (SSL) has seen renewed interest in recent times, thanks to relevant work in the rising field of unsupervised domain adaptation [213] and various approaches based on classical SSL self-training [214,215], where the dominant trend is based on adversarial approaches to reduce the discrepancy between source and target domain features [213]. A panoply of deep self-training approaches have been proposed, inspired by classical SSL self-training [214], which alternate between generating pseudo-labels for the unlabelled data-points and retraining a deep classifier [215]. In opposition, continual SSL is still a rather unexplored field. The reason is that, while in the supervised case it is clear what information the streaming data points carry, in the semi-supervised case it is far from obvious what relevant information carried by the streaming instance should drive model update. 6.1.3 Overview of Hidden Markov Model as Latent Variable In response to the CSSL challenge, we propose to model the stream of instances and ground truth values as a (temporal) latent variable model, in which the instances play the role of the observations, and the targets that of the hidden variables. To accommodate for data distributions that shift in time, this latent variable model needs to be time-variant, i.e., its parameters must be a function of time. In this chapter we investigate, in particular, the case in which the latent variable model belongs to the class of time-varying hidden Markov models (T-HMMs) [105, 108], although deep generative models can also be envisaged. We focus on classification tasks, applied to the continual activity recognition problem setting defined in 6.1.2. Our approach is two-step. For each time instant, t, in the first stage, the parameters of the latent model of the data stream are updated incrementally, together with the sequence of state (label) estimates up to the present time instant. In the case of T-HMMs, the parameters are: i) the time-varying matrix of transition probabilities \f100 Feature Extractor  (Convolu\u0001onal Neural Network) Memory  Replay Latent Model Hidden Markov Model (Discrete/Time-varying) Session wise Latent Model Supervised data stream One-o\ufb00  training Unsupervised par\u0001\u0001oned stream Con\u0001nual  Retraining Incremental Update Figure 6.1: Overview of the proposed latent variable pipeline. between labels and ii) the time-varying parameters of the emission probability densities (typically, Gaussian) for generating observations given the current state (label). Such a T-HMM latent variable model can be suitably initialised using the information provided by the initial, supervised training set. Based on the obtained sequence of most likely (pseudo-)label estimates2, a suitable continual supervised learning approach (in our case, memory replay with random sampling [97]) can be applied to incrementally update the model. Figure 6.1 shows the overall pipeline of the proposed framework. 6.1.4 Main Contributions Summarising, the main contributions of this chapter are: 1. A formal definition of the continual semi-supervised learning problem, and the associated training and testing protocols. 2. The first two benchmark datasets for the validation of semi-supervised continual learning approaches, one for classification (continual activity recognition, CAR) and one for regression (continual crowd counting, CCC), which we propose as the foundations of the first challenges in this domain. 3. Results produced by a simple strategy in which self-training is applied in sessions to competitive baseline models on both benchmarks, as the foundations for the above challenges. 4. A principled two-stage approach to the problem based on a latent model formulation reducing the task to continual supervised learning. 2In fact, an HMM outputs a series of probability vectors P(yt|x1, \u00b7 \u00b7 \u00b7 , xt), where yt is the current label and x\u03c4 denotes the observation at time \u03c4. See Conclusions. \fChapter 6. Continual Semi-Supervised Learning 101 5. An efficient solution based on a time-varying hidden Markov models, able to model streaming data issued from a time-shifting target data distribution. 6. State-of-the-art results produced by our approach on the recently released Continual Activity Recognition (CAR) dataset, as well as the 50Salads dataset adapted to continual learning thanks to a new, original protocol. Related publications: The work of this chapter appeared in International Workshop on Continual Semi-Supervised Learning [47], Continual Semi-Supervised Learning Workshop and Challenge at IJCAI 20213, and under preparation IEEE transactions on pattern analysis and machine intelligence (IEEE TPAMI). The dissertation\u2019s author was co-organiser of the workshop and challenge and co-author of the IEEE TPAMI paper. The main contributions of the dissertation\u2019s author to this project are: leading the effort of continual activity recognition (continual semi-supervised classification), HMM latent baseline implementation, and CAR baseline for challenge. All the resources of this project are publically available and are accessed from Code4, CAR challenge baseline5, CAR evalAI6,CCC challenge baseline7, and CCC evalAI8. Outline: The structure of the chapter is as follows. Firstly, we explained Continual Semi-Supervised Learning Workshop and Challenge in Section 6.2 including learning (Section 6.2.1) and testing (Section 6.2.2) protocols. The CSSL workshop and challenge also include benchmarks (Section 6.2.3), challenges (Section 6.2.4), and baselines (Sections 6.2.6. Secondly, we then explained HMM latent variable model in Section 6.3. The latent variable model is then divided into principles (Section 6.3.1), learning process (Section 6.3.2), distribution of data stream (Section 6.3.3, discrete-state T-HMM for continuous classification (Section 6.3.4), and memory replay (Section 6.3.5). Next, the experimental results of CSSL workshop and challenge and latent variable are given in Section 6.4 and Section 6.5, respectively. Finally, in Section 6.6 we provide the summary of the chapter. 3https://sites.google.com/view/sscl-workshop-ijcai-2021/ 4https://github.com/salmank255/cssl 5https://github.com/salmank255/IJCAI-2021-Continual-Activity-Recognition-Challenge 6https://eval.ai/web/challenges/challenge-page/984/overview 7https://github.com/Ajmal70/IJCAI_2021_Continual_Crowd_Counting_ Challenge 8https://eval.ai/web/challenges/challenge-page/986/overview \f102 6.2 Continual Semi-Supervised Learning Workshop and Challenge We begin by formalising the continual semi-supervised learning problem, and by defining the appropriate training and testing protocols for this new learning setting. 6.2.1 Learning The continual semi-supervised learning problem can be formulated as follows. Given an initial batch of supervised training examples, T0 = {(xj, yj), j = 1, . . . , N}, and a stream of unsupervised examples, Tt = {x1, . . . , xt, . . .}, we wish to learn a model f(x|\u03b8) mapping instances x \u2208X to target values y \u2208Y, depending on a vector \u03b8 \u2208\u0398 of parameters. When Y is discrete (i.e., a list of labels) the problem is one of continual semisupervised classification; when Y is continuous we talk about continual semi-supervised regression. In both cases we wish the model to be updated incrementally as the string of unsupervised samples come in: namely, based on the current instance xt at time t, model \u03b8t\u22121 is mapped to a new model \u03b8t. 6.2.2 Testing How should such a series of models be evaluated? In continual supervised learning (see e.g. [97]), a series of ground-truth target values is available and can be exploited to update the model. To avoid overfitting, the series of models outputted by a continual supervised learner, f(x|\u03b81), . . . , f(x|\u03b8t), . . . cannot be tested on the same data it was trained upon. Thus, basically all continual (supervised) learning papers set aside from the start a set of instances upon which the incremental models are tested (e.g., in the Core50 dataset the last three sessions are designated as test fold [216]). In principle, however, as each model f(x|\u03b8t) is estimated at time t it should also be tested on data available at time t, possibly coming from a parallel (test) data stream. An idealised such scenario is one in which a person detector is continually trained on data coming from one surveillance camera, say in a shopping mall, and the resulting model is deployed (tested) in real time on all other cameras in the same mall. \fChapter 6. Continual Semi-Supervised Learning 103 In our continual semi-supervised setting, model update cannot make use of any ground truth target values. If the latter are somehow available, but not shown to the learner for training purposes, the performance of the learner can in fact be evaluated on the stream of true target values by testing at each time t model f(x|\u03b8t) on the data pair (xt, yt). A reasonable choice for a performance measure is the average loss of the series of models on the contemporary data pair: X t=1,...,T l(f(xt|\u03b8t), yt). (6.1) In our experiments, a standard 0/1 loss, which translates into classical accuracy, is used for classification tasks. 6.2.3 Benchmarks To empirically validate CSSL approaches in a computer vision setting we created two benchmark datasets, one designed to test continual classification (CAR), and one for continual regression (CCC). Continual activity recognition (CAR) dataset The MEVA dataset To allow the validation of continual semi-supervised learning approaches in a realistic classification task we created a new continual activity recognition (CAR) dataset derived from the very recently released MEVA (Multiview Extended Video with Activities) dataset [217]. MEVA is part of the EctEV (Activities in Extended Video) challenge9. As of December 2019, 328 hours of ground-camera data and 4.2 hours of Unmanned Arial Vehicle video had been released, broken down into 4304 video clips, each 5 minutes long. These videos were captured at 19 sites (e.g. School, Bus station, Hospital) of the Muscatatuck Urban Training Center (MUTC), using a team of over 100 actors performing in various scenarios. There are annotations for 22.1 hours (266 fiveminute-long video clips) of data. The original annotations are available on GitLab10. Each video frame is annotated in terms of 37 different activity classes relevant to video surveillance (e.g. person opens facility door, person reads document, vehicle picks up person). Each activity is annotated in terms of class labels and bounding boxes around the activity of interest. Whenever activities relate to objects or other persons (e.g., in person loads vehicle the person usually puts an object into 9https://actev.nist.gov/ 10https://gitlab.kitware.com/meva/meva-data-repo/tree/master/ annotation/DIVA-phase-2/MEVA/ \f104 the vehicle\u2019s trunk; in person talks to person a number of people listen to the person speaking), these object(s) or people are also identified by a bounding box, to allow human-object interaction analyses. The CAR dataset The original MEVA dataset comes with a number of issues, from our standpoint: (i) multiple activity classes can take place simultaneously, whereas in our formulation (at least for now) only one label can be emitted at any given time instant; (ii) the quality of the original annotation is uneven, with entire instances of activities missing. As our tests are about classification, we can neglect the bounding box information. For these reasons we set about creating our own continual activity recognition (CAR) dataset by selecting 45 video clips from MEVA and generating from scratch a modified set of annotations spanning a reduced set of 8 activity classes (e.g. person enters scene through structure, person exits vehicle, vehicle starts). Those 8 classes have been suitably selected from the original 37 to ensure that activity instances from different classes do not temporally overlap, so that we can assign a single label to each frame. Each instance of activity is annotated with the related start and end frame. Frames that do not contain any relevant activity label are assigned to a \u2018background\u2019 class. The goal is to classify the activity label of each input video frame. For some MEVA sites, contiguous annotated videos exist with no gap between the end of the first video and the start of the second video. For some sites, \u2018almost\u2019 contiguous videos separated by short (5 or 15 min) gaps are available. Finally, videos from a same site separated by hours or days exist. Accordingly, our CAR benchmark is composed of 15 sequences, broken down into three groups: 1. Five 15-minute-long sequences from sites G326, G331, G341, G420, and G638 formed by three original videos which are contiguous. 2. Five 15-minute-long sequences from sites G329, G341, G420, G421, G638 formed by three videos separated by a short gap (5-20 minutes). 3. Five 15-minute-long sequences from sites G420, G421, G424, G506, and G638 formed by three original videos separated by a long gap (hours or days). Each of these three evaluation settings is intended to simulate a different mix of continuous and discrete domain dynamics. The CAR dataset including annotation and scripts is available on GitHub11. 11https://github.com/salmank255/IJCAI-2021-Continual-Activity-Recognition-Challen \fChapter 6. Continual Semi-Supervised Learning 105 The continual crowd counting (CCC) dataset Crowd counting is the problem of, given a video frame, counting the number of people present in the frame. While intrinsically a classification problem, crowd counting can be posed as a regression problem by manually providing for each training frame a density map [218]. To date, crowd counting is mostly considered an image-based task, performed on single video frames. Few attempts have been made to extend the problem to the video domain [219], [220], [221]. To the best of our knowledge, continual crowd counting has never been posed as a problem, not even in the fully supervised context \u2013 thus, there are no standard benchmarks one can adopt in this domain. For this reason we set about assembling the first benchmark dataset for continual crowd counting (CCC). Our CCC dataset is composed by 3 sequences, taken from existing crowd counting datasets: 1. A single 2,000 frame sequence originally from the Mall dataset12 [222]. 2. A single 2,000-frame sequence originally from the UCSD dataset13 [223]. 3. A 750-frame sequence from the Fudan-ShanghaiTech (FDST) dataset14, composed by 5 clips, 150 frames long, portraying a same scene [224]. The ground truth for the CCC sequences (in the form of a density map for each frame) was generated by us for all three datasets following a standard annotation protocol15. The CCC dataset complete with scripts for download and generating the baseline results is available on GitHub16. 6.2.4 Challenges CAR challenge From our problem definition (Sec. 6.2), once a model is fine-tuned on the supervised portion of a data stream it is then both incrementally updated using the unlabelled portion of the same data stream and tested there, using the provided ground truth. Incremental training and testing are supposed to happen independently for each sequence, as the aim is to simulate real-world scenarios in which a smart device with continual learning capability can only learn from its own data stream. However, as CAR sequences do not contain instances of all 9 activities, the initial supervised training is run there on the union of the supervised folds for each sequence. 12https://www.kaggle.com/c/counting-people-in-a-mall 13http://www.svcl.ucsd.edu/projects/peoplecnt/ 14https://drive.google.com/drive/folders/19c2X529VTNjl3YL1EYweBg60G70G2D-w 15https://github.com/svishwa/crowdcount-mcnn 16https://github.com/Ajmal70/IJCAI_2021_Continual_Crowd_Counting_ Challenge \f106 Split. Each data stream (sequence) in our benchmark(s) is divided into a supervised fold (S), a validation fold (V ) and a test fold (T), the last two unsupervised. Specifically, in CAR given a sequence the video frames from the first five minutes (5 x 60 x 25 = 7,500 samples) are selected to form the initial supervised training set S. The second clip (another 5 minutes) is provided as validation fold to tune the CSSL strategy. The remaining clip (5 minutes) is used as test fold for testing the performance of the (incrementally updated) classifier. Evaluation. Performance is evaluated as the average performance of the incrementally updated classifier over the test fold for all the 15 sequences. More specifically, we evaluate the average accuracy (percentage of correctly classified frames) over the test folds of the 15 sequences. Remember that, however, in our setting each test frame is classified by the current model available at time t for that specific sequence. This distinguishes our evaluation setting from classical ones in which all samples at test time are processed by the same model, whose accuracy is then assessed. CCC challenge Unlike the CAR challenge, in the crowd counting case (as this is a regression problem) each sequence is treated completely independently. Split. For the CCC challenge we distinguish two cases. For the 2,000-frame sequences from either the UCSD or the Mall dataset, S is formed by the first 400 images, V by the following 800 images, and T by the remaining 800 images. For the 750-frame sequence from the FDST dataset, S is the set of the first 150 images, V the set of the following 300 images, and T the set of remaining 300 images. Evaluation. For continual crowd counting, MAE (Mean Absolute Error) is adopted (as standard in the field) to measure performance. MAE is calculated using predicted and ground truth density maps in a regression setting. 6.2.5 Tasks For our IJCAI challenge we therefore set four different validation experiments (tasks). 1. In the first task (CAR-absolute) the goal is to achieve the best average performance across all the (test folds of the) 15 sequences in the CAR dataset. The choice of the baseline action recognition model is left to the participants. 2. In the second task (CAR-incremental) the goal is to achieve the best performance differential between the model as updated through the chosen CSSL strategy and the original model fine-tuned over the supervised fold. We thus evaluate the difference between the average performance of the incrementally \fChapter 6. Continual Semi-Supervised Learning 107 Figure 6.2: Overall pipeline of our CAR baseline. 1) Firstly, a pre-trained EfficientNet model is fine-tuned over the supervised fold of the dataset (jointly over the 15 sequences). 2) The unlabeled validation and test folds are divided into subfolds. In each session video frames from each sub-fold are used in a self-training cycle in which a pseudo label PL with prediction probability score Ps is generated for each frame, but only the pseudolabels with Ps above a fixed threshold (namely, 0.4) are considered as ground truth for retraining the model. Frames with lower Ps are ignored. The model updated by self-training in session n is used as initial model in session n + 1. updated model on the test fold and the average performance of initial model, also on the test fold. The baseline recognition model is set by us (see Baselines). 3. Task CCC-absolute seeks the best average performance over the test fold of the 3 sequences of the CCC dataset. The choice of the baseline crowd counting model is left to the participants to encourage them to push its performance to the limit. 4. Finally, task CCC-incremental seeks the best performance differential between the initial and the updated model over the test fold, averaged across the three sequences. The baseline crowd counting model is set. 6.2.6 Baselines To provide a baseline for the above tasks, and assess the degree of difficulty of the challenges, we decided to adopt a simple strategy which consists of classical (batch) semi-supervised self-training performed in a series of sessions, for both the activity and the crowd counting experiments. Baselines are provided regarding the initial action recognition model to be used in the CAR tests, the base crowd counter to be used in the CCC tests, and the semi-supervised incremental learning process itself. Baseline activity recognition model The baseline model is the recent EfficientNet network (model EfficientNet-B5) [225], pre-trained on the large-scale ImageNet dataset over 1000 classes. For our tests we initialised the model using those weights and changed the number of classes to 9 \f108 activities (see Section 6.2.3). Detailed information about its implementation, along with pre-trained models, can be found on Github17. The model is easily downloadable using the Python command \u201cpip\u201d (pip install efficientnet-pytorch). Baseline crowd counter For the baseline crowd counting model, we selected the Multi-Column Convolutional Neural Network (MCNN) [226], whose (unofficial) implementation is publicly available18 and uses PyTorch. MCNN was considered state-of-the-art when released in 2016, but is still commonly used as a standard baseline in more recent works, due to its competitive performance on public datasets (e.g., [227, 228]. MCNN made significant contributions to crowd counting by proposing a network architecture betterequipped to deal with differing head sizes due to image resolution, perspective effects, distances between the camera and the people within the scene, etc. In a nutshell, this robustness to scale was achieved via a network composed of three parallel CNN columns, each of which used filters with different receptive field sizes, allowing each column to specialize for a particular scale of human head. Pre-trained MCNN models are available for both the ShanghaiTech A and the ShanghaiTech B datasets. For our tests, as well as the Challenge, we chose to adopt the ShanghaiTechB pre-trained model. Baseline incremental learning strategy As mentioned, our baseline for incremental learning from unlabelled data stream is instead based on a vanilla self-training approach [229]. For each sequence, the unlabelled data stream (without distinction between validation and test folds) is partitioned into a number of sub-folds. Each sub-fold spans 1 minute in the CAR challenges, so that each unlabelled sequence is split into 10 sub-folds. Sub-folds span 100 frames in the CCC challenges, so that the UCSD and MALL sequences comprise 16 sub-folds whereas the FDST sequence contains only 6 sub-folds. Starting with the model initially fine-tuned on the supervised portion of the data stream, self-training is successively applied in a batch fashion in sessions, one session for each sub-fold, for a single epoch (as we noticed that using multiple epochs would lead to results degradation). Self-training requires to select a confidence threshold above which predictions are selected as pseudolabels. In our tests we set a confidence threshold of 0.4. The predictions generated by the model obtained after one round of self-training upon a sub-fold are stored as baseline predictions for the current subfold. The model updated after each self-training session is used as initial model for 17https://github.com/lukemelas/EfficientNet-PyTorch 18https://github.com/svishwa/crowdcount-mcnn \fChapter 6. Continual Semi-Supervised Learning 109 the following session. 6.3 HMM Latent Variable Model for Unsupervised DataStreams In the second part of the chapter, the problem is defined as the same as in Section 6.2 including Learning and Testing. However, in this part, we only consider the continual semi-supervised classification. 6.3.1 Principles of HMM Latent Variable Model In this chapter, we wish to devise an approach to the continual semi-supervised learning problem adhering to the following principles. 1. Continual semi-supervised learning, as defined in Section 6.2, should reduce to continual supervised learning once one or more hypotheses are made about the target values of the unsupervised stream. 2. At each time step t, a constant number, M, of such hypotheses should be examined. 3. The admissible target assignments for the unlabelled data stream up to time t should be sampled according to their likelihood to be correct, assuming this information is available. 4. For each target assignment hypothesis under consideration, the model should be updated incrementally. 5. To realistically model real-world applications, a budget should be enforced on the total number of (past) example instances one has access to at any given time step, t. 6. Assuming ground truth target values y1, . . . , yt, . . . are available, the validation of the learner should take place simultaneously with the training process, by measuring at each time instant, t, the loss of the current model, f(x|\u03b8t), on the current data pair, (xt, yt), and taking the average over the entire data stream. Principle 3) assumes that a likelihood model for the data stream of targets is maintained at each time step, t, so that we can sample from it. In the classification case (when Y is finite), a likelihood model for the admissible label assignments over \f110 the (current) sequence of instances, x1, . . . , xt, is a finite probability distribution,19 PY (y1, . . . , yt|\u0398t Y ) over Yt . = Y\u00d7\u00b7 \u00b7 \u00b7\u00d7Y (t times), conditional on a vector of parameters \u0398t Y = {\u03b81 Y , . . . , \u03b8t Y }. A similar likelihood model can be defined for continual regression problems. 6.3.2 A Two-Stage Learning Process In this chapter, we propose a general approach to continual semi-supervised learning (Section 6.2) which adheres to the principles stated in Section 6.3.1. The approach is incremental, i.e., the \u2018best\u2019 model, f(x|\u03b8), mapping instances, x \u2208X, to target values, y \u2208Y, is updated at each time step, t, based on the new evidence, xt. Namely, at each time step t: 1. The most likely sequence(s) of target values y1, . . . , yt are estimated based on the new piece of evidence (unlabelled instance) xt. 2. Based on the current estimate of the most likely labellings, a continual supervised learning task is posed and a memory replay-based approach is used to update the model. More specifically, in this work we propose to estimate the most likely target assignments for the current data-stream (Step 1) via a latent variable model of the data stream of pairs (xt, yt). This is discussed in Section 6.3. Although in principle any such model can be employed, including deep generative models, we elaborate in particular on the case in which time-varying hidden Markov models are used to model the data and the way the data-generating distribution changes over time for a continual classification task. While the pseudo-labels estimated by the latent variable model are used to incrementally update the model at training time, at testing time the performance of the series of models (and, as a consequence, of the overall learner) can be measured versus the series of ground-truth labels, assuming those are available, as dictated by Principle 6. 6.3.3 Joint Distribution of the Data Stream We assume that the unsupervised data stream, up to the current time instant t, can be modelled by the joint distribution p1:t D ((x1, y1), . . . , (xt, yt)|\u03b81 D, . . . , \u03b8t D), (6.2) 19In this chapter we use P (uppercase) to denote finite probability distributions, and p (lowercase) to denote probability densities. \fChapter 6. Continual Semi-Supervised Learning 111 under the additional principle that: 7. The data-generating probability distribution pt D = p((xt, yt)|\u03b8t D), from which the pair (xt, yt) is assumed to be sampled, can be a function of time. Any likelihood model for the target assignments (Principle 3) should thus be derived from the joint, time-varying probability distribution (6.2), possibly under sensible simplifying assumptions. Principle 7) allows the target distribution to vary over time, reflecting material, realistic real-world conditions. This principle makes our approach more general than those which assume the existence, next to the source domain, of a single target domain, as in classical domain adaptation [230]. It is also more general than \u2018taskbased\u2019 approaches to continual learning [231], in which the data are assumed to be generated in a series of discrete tasks, each associated with its own data generating process, whose boundaries typically are given. 6.3.4 Discrete-State T-HMM for Continuous Classification The data stream of pairs is composed by two parallel series x1, . . . , xt, . . . and y1, . . . , yt, . . ., of which only the sequence of instances is observed, while the (unknown) target values form a series of hidden variables. This amounts to representing the unsupervised data stream using a latent variable model in which the targets y1, . . . , yt, . . . play the role of the hidden variables. In this chapter, we consider hidden Markov models (HMMs) [232] as one specific instance of a latent variable model that is applicable to the posed problem setup. In the continual classification case, states are discrete and correspond to the labels. To cater to Principle 7), we seek to learn a time-varying HMM from the data stream which can model a data distribution that shifts with time. Model formulation In a discrete-state, continuous observation time-varying hidden Markov model [105, 108,233] the joint distribution (6.2) assumes the form: p1:t D ((x1, y1), . . . , (xt, yt)|\u03b81 D, . . . , \u03b8t D) = P(y1|\u03b81 D)p(x1|y1, \u03b81 D) \u00b7 t Y \u03c4=2 P(y\u03c4|y\u03c4\u22121, \u03b8\u03c4\u22121 D )p(x\u03c4|y\u03c4, \u03b8\u03c4 D). (6.3) The model parameters \u03b8\u03c4 D = {\u03b80, \u03b8\u03c4 T, \u03b8\u03c4 \u0393} are time-varying, to model the fact that, according to Principle 7), the (target) data distribution varies in general with time. These parameters describe20: (i) the (discrete) probability of the initial state, P(y1|\u03b80), 20The notation used for HMM parameters follows that in [232]. \f112 (ii) the discrete, time-dependent transition probability for the hidden variables (in this case, the labels), A\u03c4 . = P \u03c4(y\u03c4+1|y\u03c4) = P(y\u03c4+1|y\u03c4, \u03b8\u03c4 T), where \u03c4 \u2208{1, . . . , t} denotes any time instant in the current data-stream, and (iii) the conditional density of the (continuous) observations (instances), given the state (label), \u0393\u03c4(x\u03c4|y\u03c4) = p(x\u03c4|y\u03c4, \u03b8\u03c4 \u0393). Under this model, the structure of the time-dependent data probability distribution is pt D((xt, yt)|\u03b8t D) = p(xt|yt, \u03b8t \u0393) \u00b7 P(yt), (6.4) where P(yt) = P(yt|yt\u22121, \u03b8t\u22121 T ) \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 P(y2|y1, \u03b81 T) \u00b7 P(y1|\u03b80). (6.5) In other words, the probability distribution generating the data at time t is parameterised by: the parameters, \u03b8t \u0393, that determine the distribution of the instances given the label at time t, those that determine the series of transitions 1 \u21922 \u2192\u00b7 \u00b7 \u00b7 \u2192t of the Markov chain of labels, and the probability of the initial label. This formulation constrains the way the target domain distribution p((x, y)|\u03b8D) can vary in time, while still allowing for a rather general process model. Learning the Parameters of a Discrete-State T-HMM Modelling time dependency with covariates. While the above formulation is completely general, the mainstream approach to T-HMM is to model the dependency of the parameters on time (and the transition probabilities, in particular) using a multinomial logistic model [234] of the kind A\u03c4(i, j) = P(y\u03c4 = i|y\u03c4\u22121 = j) . = exp(g(z\u03c4, i, j)) 1 + PN\u22121 h=1 exp(g(z\u03c4, i, h)) , (6.6) where N is the number of states, i, j \u2208{1, . . . , N}, and g(z\u03c4, i, j) is a function of a covariate process {z\u03c4, \u03c4 = 1, . . .}, which is also observable. As each entry of the transition matrix is a function of the covariate variables, their values drive the dynamics of the transition probabilities. The transition matrix can then be expressed as A\u03c4 = P(y\u03c4+1|y\u03c4, \u03b8\u03c4 T) = P(y\u03c4+1|y\u03c4, \u03b8L, z\u03c4), where \u03b8L is the vector of parameters of the logistic model (6.6) (in particular, of the function g), and z\u03c4 is the covariate vector at time \u03c4. Covariates on the initial probability can also be accounted for, in the form of a single vector, z0. Covariates as functions of time. The shape of the function, g, is a design choice. A common option is g(z\u03c4, i, j) = \u03d5ij + z\u03c4\u03d1ij, where \u03d5ij and \u03d1ij are unknown coef\fChapter 6. Continual Semi-Supervised Learning 113 ficients, which are unique to the transition between state i and state j. The resulting transition matrix has |\u03b8L| = N \u00b7 (N \u22121) parameters, twice those of a traditional, time-invariant transition matrix. In many practical applications, time and date stamps can act as effective covariates. For instance, in surveillance settings, the time stamp can induce a suitable periodic behaviour in a system, e.g. to detect people entering a building. The date stamp can induce annual cycles, or facilitate the gradual shift of the parameter values over time. As shown in [103], a range of transition models can be designed by plugging in a different function of time (intended as the main covariate quantity) as the argument g(z\u03c4, i, j) of the exponential in (6.6). Parameter estimation via Expectation Maximisation. Just as in traditional HMMs, the Expectation-Maximisation (EM) algorithm [235] can be used to recursively estimate the parameters of a T-HMM in the logistic dependency from covariates formulation, as well as to estimate the most likely series of states [236], given one or more training sequences of observations: {(xm 1 , . . . , xm Tm), m = 1, . . . , M}. In the T-HMM case, the parameter estimation algorithm takes as input, in addition to each training sequence of observations x1, . . . , xT, a parallel sequence of covariate vectors, z1, . . . , zT, and outputs both the parameters \u03b8L of the logistic model (6.6) and those of the emission probabilities. EM is well-known to be prone to converge to local, as opposed to global, minima of the log-likelihood of the data. In our experiments, EM is run multiple times on the same sequence in order to extract \u2018consensus\u2019 parameters. Initialisation via the supervised fold. While the initial probabilities of the hidden variables (labels) can be estimated via EM with covariates in the same framework explained above for the transition matrix, for our problem it makes more sense to estimate them using the information provided by the initial, labelled training set T0, sampled from a \u2018source\u2019 data distribution p((x, y)|\u03b80 D). In fact, from T0 we can extract the empirical marginal of the labels, \u02c6 P(y|\u03b80 D), by counting and normalising the occurrences of the labels in the initial training data, and set: P(y1|\u03b80) = \u02c6 P(y|\u03b80 D). (6.7) Covariates in the emission probabilities. To the best of our knowledge, the extension of the dependence on covariates to the emission probabilities p(x\u03c4|y\u03c4, \u03b8\u03c4 \u0393) has not been discussed in the literature. Nevertheless, this appears to be conceptually straighforward, for instance by plugging a logistic dependency of the mean vector \u00b5\u03c4 and covariance matrix \u03a3\u03c4 in the classical multivariate Gaussian approach from a sequence of covariates (e.g., time or functions thereof). We will address this extension in future work. \f114 Discrete state HMM vs. continuous state space The choice of using labels as the state space in HMMs is often a strong and meaningful solution for specific tasks. For example, in our case, the primary objective is to train the model on the sequences that are already available and are inherently discrete in nature (train set). We are not expecting a change in the number of states as each state represents a specific class and these classes will remain the same during the training and prediction. In contrast, using a continuous state space related to physics or perception may introduce complexity that is unnecessary for tasks focused on discrete events. This complexity can hinder model interpretability and complicate the mapping between states and meaningful labels. The model with continue state space would be useful in other continual learning settings such as class-incremental learning, where the objective is to train the model of new classes. Protocols for training the latent model In the incremental learning setting described in Section 6.3, the desired model (in this case, the classifier) is incrementally updated at each time instant t using the unlabelled observation xt, and tested on the ground-truth output yt. In our latent-model approach to the problem, this incremental updating exploits the series of pseudooutputs (labels) generated by the latent model. Classical (T-)HMM training, however, assumes that model parameters are learned in a batch way from one or more training sequences of observations. The question thus arises of what training sequences the latent model should be learned from, whether or not in fact the latent model should be learned from the single unsupervised data stream at disposal, and at what time instants in the data-stream T-HMM learning should take place. We can identify a number of options. 1. Separate (batch) training. The option which is closest to the traditional setting is to identify the latent variable model once and for all from one or more training sequences of instances gathered from the site of interest, and set aside for this purpose. This obviously assumes that such sequences are available, which might not be the case. Alternatively, if only one sequence is available for both training and testing, one can learn the latent model from the labelled section T0 of the data stream and never update it, but simply use it to generate pseudolabels for the unlabelled section Tt. 2. Continuous (batch) training. The second option is to learn the latent model from the test unsupervised data stream itself (as the only one available), but to learn a new version of the T-HMM as each new instance becomes available. Given a new instance xt: \fChapter 6. Continual Semi-Supervised Learning 115 (a) We use the current sequence x1, . . . , xt to re-estimate the parameters {\u03b8L, \u03b8\u0393} of the T-HMM. (b) We estimate the probability P(y1, . . . , yt) of all possible labellings for the current sequence under the newly learnt model. This protocol fits well our philosophy of continually learning and revising our view of the world as new observations are gathered. However, it is very computationally expensive and possibly unfeasible for deployment on edge devices such as surveillance cameras. 3. Continual (batch) training. An intermediate, less expensive option is to update the latent model at regular intervals, rather than for each new unlabelled sample (hence, continually, rather than continuously). In this case: (a) The latent model is first initialised using the string of instances (which are temporally ordered) in the supervised training set T0 (without making use of the label information). (b) The latent model is then re-trained in sessions, each corresponding to a section of the unlabelled data stream. This approach requires us to suitably partition the data stream Tt into sessions. The limit case is that in which a single learning session is used, after which the latent model is not updated further. We will elaborate on this option in our Experiments. Sampling labellings as a state estimation problem Once the parameters \u03b8L, \u03b8\u0393 of the latent T-HMM model have been learned from the training set of the unsupervised data streams, given a new unsupervised data stream x1, . . . , xT computing its most likely labelling \u02c6 y1, . . . , \u02c6 yT reduces to estimating the most likely sequence of hidden states. In fact, a sequence of discrete probability distributions P(yt|x1, . . . , xt; z1, . . . , zt; \u03b80, \u03b8L, \u03b8\u0393) over the hidden variables can be obtained by simply \u2018running\u2019 the model in a forward pass. 6.3.5 Random Sampling based Memory Replay After obtaining the estimated pseudo-labels from session-wise continual retraining, the continual semi-supervised learning problem is simplified to continual supervised learning. A random sampling [97] based memory replay method is employed to incrementally update the model. An initial memory buffer of size M is comprised \f116 of 1000 samples randomly selected from the supervised training set, T0, and the first session, T1, of the suitably partitioned data stream. The model is then incrementally updated at the end of each session via a randomly selected memory buffer. At the end of every session, the memory buffer is filled with new samples. Then, the memory buffer is updated to contain 1000 samples that are obtained through random selection. This incremental updating process continues until the last session of the partitioned unlabelled stream is reached. Details on how the data stream is partitioned can be found in Section 6.5.1. The effectiveness of random sampling-based memory replay along with one-off training and continual retraining is detailed in Section 6.5.2. 6.3.6 Alternative to HMM as Latent Variable Model In our method, we used HMMs as latent variable modeling for pseudo-label generation, however, there are alternative models that you can consider. These alternatives may offer different strengths and may be more suitable for specific tasks or data characteristics. Some of the alternatives are as follows: \u2022 Variational Autoencoders (VAEs): In CSSL, you can train a VAE on the labeled data and then generate pseudo-labels for unlabeled data by mapping them to the learned latent space. \u2022 Gaussian Mixture Models (GMMs): GMMs are probabilistic models that assume data points are generated from a mixture of Gaussian distributions. They can be used for clustering and semi-supervised learning. In this context, GMMs can help cluster data points and assign pseudo-labels based on cluster assignments. \u2022 Recurrent Neural Networks (RNNs) and Long Short-Term Memory networks (LSTMs): RNNs and LSTMs are neural network architectures capable of modeling sequential data. They can be used to capture temporal dependencies in data and generate pseudo-labels based on the learned representations. 6.4 Experimental Analysis of Challenge Baselines We ran experiments on both continual classification and continual regression using two new benchmarks of our own creation, which formed the basis of an IJCAI 2021 workshop and challenge21. The following description of the experimental setups mimics very closely the guidelines provided for the IJCAI challenge. 21https://sites.google.com/view/sscl-workshop-ijcai-2021/ \fChapter 6. Continual Semi-Supervised Learning 117 Table 6.1: Performance of the initial supervised model versus that of the incrementally updated models (separately on V and T or in combination), using three standard evaluation metrics. For each metric we report both class average (C) and weighted average (W). Validation Fold Test Fold Methods / Evaluation metrics Precision Recall F1-Score Precision Recall F1-Score C W C W C W C W C W C W sup \u2212ft \u2212union 0.20 0.74 0.25 0.68 0.16 0.70 0.16 0.82 0.15 0.77 0.14 0.79 upd \u2212V / upd \u2212T 0.18 0.73 0.19 0.73 0.14 0.72 0.14 0.81 0.15 0.81 0.14 0.80 upd \u2212V + T 0.17 0.73 0.19 0.73 0.14 0.72 0.14 0.80 0.14 0.79 0.13 0.79 Table 6.2: Evaluation on the CCC validation (V ) and test (T) folds for different experimental setups. Here sup\u2212ft, upd\u2212V , and upd\u2212V +T refer to the supervised model fine-tuned on S, the model incrementally updated via self-training in sessions on the validation fold V , and the model incrementally updated on the combined validation V and test T folds, respectively. The average test MAE is reported for each model. Validation Fold V Test Fold T FDST UCSD MALL FDST UCSD MALL sup \u2212ft 5.17 6.36 6.65 9.59 8.34 13.56 upd \u2212V 6.05 7.58 7.40 8.00 7.45 16.38 upd \u2212V + T 8.32 8.28 9.36 9.16 8.93 9.46 6.4.1 Results on Activity Recognition Three types of experiments were performed to measure: (i) the performance of the baseline model after fine-tuning on the union of the supervised folds for the 15 CAR sequences (sup \u2212ft \u2212union), without any continual learning; (ii) the performance obtained by incrementally updating the model (in sessions) using the unlabelled validation (upd \u2212V ) or test (upd \u2212T) data streams, considered separately; (iii) the performance of self-training over validation and training folds considered as a single contiguous data stream, again in a session-wise manner (upd \u2212V + T). The results are shown in Table 6.1. The standard classification evaluation metrics precision, recall, and F1-score were used for evaluation. For each evaluation metric we computed both the class average (obtained by computing the score for each class and taking the average without considering the number of training samples in each class, i.e., F1class1 + F1class2 \u00b7 \u00b7 \u00b7 + F1class9 ) and the weighted average (which uses the number of samples Wi in each class i, i.e., F1class1 \u2217W1 + F1class2 \u2217W2 \u00b7 \u00b7 \u00b7 + F1class9 \u2217W9 ). It can be noted that on the validation fold continual updating does improve performance to some extent, especially under Recall and F1 score. Improvements in Recall are visible on the test fold as well. All in all the baseline appears able to extract information from the unlabelled data stream. \f118 6.4.2 Results on Crowd Counting Table 6.2 shows a quantitative analysis of the performance of the fine-tuned supervised model (sup) and two incrementally updated models (upd) on the validation and the test split, respectively, using the mean absolute error (MAE) metric. The experiments were performed under three different experimental settings: (i) the initial model fine tuned on the supervised folds (sup\u2212ft), (ii) a model incrementally updated on the validation fold (upd\u2212V ), and (iii) a model incrementally updated on both the validation and test folds considered as a single data stream (upd\u2212V +T). In Table 6.2 all three models are tested on both the V and T portions of the data streams. The initial model was trained for 100 epochs on the supervised fold (S), whereas the incremental models were self-trained for 5 epochs in each session (we also tried 2 epochs with no significant change observed). When compared with classical papers in the crowd counting literature, the MAE values in Table 6.2 are noticeable higher. Concretely, a recent work that incorporates optical flow to improve crowd counting achieved MAEs of 1.76, 0.97, and 1.78 on the FDST, UCSD, and Mall dataset, respectively [219]. Also, the original MCNN implementation yielded MAEs of 3.77 and 1.07 on FDST and UCSD (performance was not reported on Mall) [224, 226]. The higher MAEs reported in our work are expected, due to the significantly different and more challenging training protocol (i.e., batch training in standard crowd counting work vs. continual learning here). For example, standard papers in crowd counting following the typical evaluation protocol employ 800 training images for the Mall and UCSD datasets; whereas, in our problem setting only 400 images are used for supervised training. Previous batch results should be seen as upper bounds to our incremental task, whereas ours should be considered as new baselines for a new problem setting. While model updating does not seem to be effective (on average) on the validation streams, the effect is quite significant on the test fold (right side of the Table), with important performance gains. 6.5 Experimental Analysis of HMM latent Variable 6.5.1 Datasets and Setups Continual activity recognition (CAR) dataset. The CAR dataset (explained in Section 6.2.3) is used also for evaluating the latent variable and comparison with the previous challenge baseline. According to the protocol of the CAR dataset, each video sequence is divided into a supervised fold (S), corresponding to the first 5 minutes, a validation fold (V), corresponding to the sec\fChapter 6. Continual Semi-Supervised Learning 119 Table 6.3: Time-varying HMM (TV HMM) vs. traditional HMM (HMM) based on one-off training, where P, R, and F1 are precision, recall, and F1-score, respectively. HMM TV HMM P R F1 P R F1 CAR 0.86 0.18 0.24 0.83 0.37 0.49 50Salad low timesplit cooksplit 0.22 0.05 0.08 0.226 0.052 0.077 0.076 0.023 0.018 0.076 0.023 0.018 50Salad high timesplit cooksplit 0.39 0.23 0.27 0.4 0.24 0.26 0.407 0.236 0.256 0.405 0.238 0.26 ond 5-minute video, and a test fold (T), corresponding to the last 5-minute video. The last two folds are unsupervised. In our experiments, for each sequence, the unlabelled data stream (i.e. the last 10 minutes, without distinction between validation and test folds) is partitioned into a number of sub-folds. Each sub-fold spans 1 minute in the CAR dataset, so that each unlabelled sequence is split into 10 sub-folds. 50 Salads activity detection dataset. We also validate our method using the 50 Salads dataset, which consists of 50 videos by 25 cooks (actors) preparing salads, where each cook recorded two videos of themselves preparing salads by following different sequence of steps. The dataset contains two levels of annotations: \u201chigh-level\u201d labels only span three classes, whereas \u201clowlevel\u201d annotation involves a total of 51 classes. We added a background class to both settings, yielding a total number of classes of 4 and 52, respectively. To enforce a similar protocol to that of the CAR dataset, we introduced two splits to the 50 Salad dataset for training and unsupervised continual updates. The splits are defined as follows: 1. Split by time selects the first video for each cook as the training set and the second as the validation set. In this set-up, we do not have test set, so we only update our model continually over the validation set. 2. Split by cook divides the set of 25 cooks into 9, 8, and 8 for train, validation, and test sets, respectively. For evaluation, we perform 10-fold random split cross-validation and report the averaged results. Session sub-folds span 1.28 minutes on average for the 50 Salads dataset, as the duration of each video is slightly longer than in the CAR dataset. \f120 Table 6.4: Time-varying HMM (TV HMM) vs. traditional HMM (HMM) based on continual retraining, where P, R, and F1 are precision, recall, and F1-score, respectively. HMM TV HMM P R F1 P R F1 CAR 0.81 0.28 0.39 0.81 0.40 0.52 50Salad low timesplit cooksplit 0.07 0.02 0.03 0.07 0.028 0.035 0.058 0.020 0.025 0.059 0.018 0.023 50Salad high timesplit cooksplit 0.14 0.26 0.17 0.14 0.36 0.20 0.402 0.257 0.288 0.407 0.237 0.261 Table 6.5: Time-varying HMM (TV HMM) vs. traditional HMM (HMM) based on incremental update, where P, R, and F1 are precision, recall, and F1-score, respectively. Traditional HMM Time-varying HMM P R F1 P R F1 CAR 0.79 0.34 0.46 0.80 0.34 0.46 50Salad low timesplit cooksplit 0.25 0.15 0.19 0.309 0.179 0.214 0.155 0.031 0.044 0.144 0.029 0.04 50Salad high timesplit cooksplit 0.34 0.21 0.24 0.42 0.25 0.27 0.440 0.274 0.306 0.409 0.259 0.278 0 10 20 30 40 50 60 70 80 90 100 Time Cook Time Cook Time Cook Time Cook Time Cook Time Cook CAR 50Salad Low 50Salad High CAR 50Salad Low 50Salad High CAR 50Salad Low 50Salad High One-o\ufb00 Con\u0002nual Retraining Memory Replay Precision Recall F1-Score Time-Varing HMM Figure 6.3: Time-varying HMM (dashed bars) vs. traditional HMM (colored bars). 6.5.2 Quantitative Analysis The performance of Time-varying HMM (T-HMM) and traditional HMM (HMM) was assessed using three different experimental setups, namely: single, one-off training; continual retraining; and random sampling-based memory replay. In the first two settings, only the accuracy of the pseudo-labels generated by the latent model was assessed, whereas the third setting evaluated the whole framework in which the image classifier is incrementally updating using the pseudo-labels generated in sessions by the latent model, also incrementally updated. The quantitative comparison was performed using standard precision (P), recall (R) and F1-score (F1) performance \fChapter 6. Continual Semi-Supervised Learning 121 metrics. The quantitative analysis from the Table 6.3,6.4 and 6.5 are further summarized in the Figure 6.3. T-HMM vs. HMM It can be seen from Table 6.3, 6.4 and 6.5 that the T-HMM significantly performs better in terms of the F1 and R. T-HMM is better overall in all the three experimental setups. There were big gains in terms of R and F1 in the case of CAR dataset. The overall performance of T-HMM was quite low in case of 50Salad low as it comprises of relatively higher number of classes (51) as described in 6.5.1. Continual retraining vs. One-off training It can be seen that the continual retraining improves the performance of not only T-HMM but also in the case of the HMM (Table 6.4). The HMM performance improves significantly in terms of F1 (+15%). Similarly, R also gained (+10%). The T-HMM significantly performs better than the HMM (+12% gain in R and F1 for CAR dataset). However, there was not a significant gain in terms of 50Salad dataset either low or high classes. The performance of T-HMM and HMM were close at some instances. Incremental Update vs. Continual retraining The general trend of improvement can be seen in the case of incremental update via a random sampling memory replay as well. The incremental update helped improve the overall recall metric leading to better F1 score. In general, the overall improved F1 score can be found from +5-15% (for CAR dataset). While a similar trend of not much gain was observed for the 50Salad dataset. 6.6 Summary of the Chapter In this chapter, we present CSSL workshop and challenge and a latent variable model. In the CSSL workshop and challenge, approach to the recently defined continual semi-supervised learning problem, in which the labels of the unsupervised data stream are estimated via a time-varying hidden Markov model. We formulated the continual semi-supervised learning problem and proposed suitable training and testing protocols for it. To encourage future work in the area we created the first two benchmark datasets, as the foundations of our IJCAI 2021 challenge. We proposed a simple strategy based on batch self-training a baseline model in sessions. The results show that, in both the activity recognition and the crowd counting challenge, the baseline appears in fact to be able to extract information from the unlabelled data stream. Nevertheless, the need for a more sophisticated approach leveraging the dynamics of the data stream is clear. Next, we proposed a latent variable model approach to the defined continual semisupervised learning problem, in which the labels of the unsupervised data stream are estimated via a time-varying hidden Markov model. Once the most likely label \f122 assignment is estimated, the problem reduces to a continual supervised learning one, and can be tackled by, for instance, random sampling memory replay. Employing a time-varying hidden Markov model as the latent model allows us to track the evolution of the target domain over time, under realistic assumptions, and therefore achieve better performance. The latent model was evaluated in three different experimental setups on two public datasets adapted to continual learning settings. The proposed solution showed a trend of better performance (a gain of 5-15% in F1 score) in all the experimental settings. The model of the label dynamics we proposed makes some weak assumptions on the series of labels: in particular, that they can be described by a Markov chain and that emission probabilities can be described as multivariate Gaussian distributions. Our experiments support the view that these assumptions are fairly realistic \u2013 nevertheless, more general approaches should be explored. In the future, we plan to investigate the use of different logistic models of the dependency of the transition probabilities of the T-HMM from time, and extend the proposed latent variable approach to continual semi-supervised regression. \fChapter 7 Conclusions and Future Research Directions In this thesis, we have presented a variety of novel methods for scene understanding divided into three parts. Part I 2.7.6: The ROad event awareness dataset for autonomous driving (Chapter 3), focuses on our novel approach to road event detection and the associated datasets and baselines. Part II 3.7 discusses complex activity detection (Chapter 4, Chapter 5) and the spatio-temporal modelling of the scene for holistic scene understanding. Finally, Part III 5.4.2 proposes a new paradigm for continual learning (Chapter 6) allowing us to update models incrementally by leveraging streaming unsupervised data. To conclude the thesis, this chapter is divided into two sections. First, the methods proposed in this thesis are summarised in Section 7.1. Secondly, we illustrated possible research directions for future work in Section 7.2. 7.1 Summary of the Thesis 7.1.1 ROad event Awareness Dataset for Autonomous Driving We proposed a strategy for situation awareness in autonomous driving based on the notion of road events, and contributed a new ROad event Awareness Dataset for Autonomous Driving (ROAD) as a benchmark for this area of research. The dataset, built on top of videos captured as part of the Oxford RobotCar dataset [126], has unique features in the field. Its rich annotation follows a multi-label philosophy in which road agents (including the AV), their locations, and the action(s) they perform are all labelled, and road events can be obtained by simply composing labels of the three types. The dataset contains 22 videos with 122K annotated video frames, for a total of 560K detection bounding boxes associated with 1.7M individual labels. 123 \f124 Baseline tests were conducted on ROAD using a new 3D-RetinaNet architecture, as well as a SlowFast backbone and a YOLOv5 model (for agent detection). Both frame\u2013mAP and video\u2013mAP were evaluated. Our preliminary results highlight the challenging nature of ROAD, with the SlowFast baseline achieving a video-mAP on the three main tasks comprised between 20% and 30%, at low localisation precision (20% overlap). YOLOv5, however, was able to achieve significantly better performance. These findings were reinforced by the results of the ROAD @ ICCV 2021 challenge, and support the need for an even broader analysis, while highlighting the significant challenges specific to situation awareness in road scenarios. In another study we proposed a new learning framework, termed \u201clearning with requirements\u201d, and a new dataset for this task, called ROAD-R (the ROad event Awareness Dataset with logical Requirements), derived from ROAD. We showed that SOTA models most of the time violate the requirements, and how it is possible to exploit the requirements to create models that are compliant with (i.e., strictly satisfy) the requirements while improving their performance. We envision that requirement specification will become a standard step in the development of machine learning models, to guarantee their safety, as it is in any software development process. In this sense, ROAD-R may be followed by many other datasets with formally specified requirements. 7.1.2 Complex Activity Detection Moving up in the scene understanding reasoning chain, we presented a spatiotemporal complex activity detection framework that leverages both part deformation and a heterogenous graph representation. Our approach is based on three building blocks; action tube detection, part-based deformable 3D RoI pooling for feature extraction, and a GCN module that processes the variable number of detected action tubes to model the overall semantics of a complex activity. In an additional contribution, we temporally annotated two recently released benchmark datasets (ROAD and ESAD) in terms of long-term complex activities. Both datasets come with video-level action tube annotation, making them suitable benchmarks for future work in this area. We thoroughly evaluated our method, showing the effectiveness of our 3D part-based deformable model approach for the detection of complex activities. This complex activity model works well with datasets endowed with spatiotemporal action annotations. This, however, is a serious limitation for the method cannot be applied to datasets without such ground truth. To overcome this limitation, in another study we proposed a novel hybrid graph neural network-based framework that combines both scene graph attention and a temporal graph to model activities of arbitrary duration. Our proposed framework is divided into three main building \fChapter 7. Conclusions and Future Research Directions 125 blocks: agent tube detection and feature extraction; a local scene graph construction with attention; and a temporal graph for recognising the class label and localising each activity instance. We tested our method on three benchmark datasets, showing the effectiveness of our method in detecting both short-term and long-term activities, thanks to its ability to model their finer-grained structure without the need for extra annotation. 7.1.3 Continual Semi-Supervised Learning Lastly, in the last part of this thesis, we formulated an original \u201ccontinual semisupervised learning\u201d problem and proposed suitable training and testing protocols for it. To encourage future work in the area we created the first two benchmark datasets, as the foundations of our CSSL @ IJCAI 2021 challenge. We proposed a simple strategy based on batch self-training a baseline model in sessions. The results show that, in both the activity recognition and the crowd counting challenge, the baseline appears in fact to be able to extract information from the unlabelled data stream. Nevertheless, the need for a more sophisticated approach leveraging the dynamics of the data stream is clear. To move in this direction we then formulated a latent variable model approach to the continual semi-supervised learning problem, in which the labels of the unsupervised data stream are estimated via a time-varying hidden Markov model. Once the most likely label assignment is estimated, the problem reduces to a continual supervised learning one and can be tackled by, for instance, random sampling memory replay. Employing a time-varying hidden Markov model as the latent model allows us to track the evolution of the target domain over time, under realistic assumptions, and therefore achieve better performance. The latent model was evaluated in three different experimental setups on two public datasets adapted to continual learning settings. The proposed solution showed a trend of better performance (a gain of 5-15% in F1 score) in all the experimental settings. The model of the label dynamics we proposed makes some weak assumptions on the series of labels: in particular, that they can be described by a Markov chain and that emission probabilities can be described as multivariate Gaussian distributions. Our experiments support the view that these assumptions are fairly realistic \u2013 nevertheless, more general approaches should be explored. \f126 7.2 Future Research Directions 7.2.1 Extension of ROAD Our ROAD dataset is extensible to a number of challenging tasks associated with situation awareness in autonomous driving, such as event prediction, trajectory prediction, continual learning, and machine theory of mind. We are currently actively working to scale up our benchmark by at least one order of magnitude by extending ROAD-like annotation to major autonomous driving datasets such as PIE and WAYMO. We also intend to propose to organise a follow-up Workshop (possibly a series of events) at the next suitable upcoming major computer vision (e.g. ICCV, CVPR) and robotics (e.g. ICRA, IROS) conferences, in order to start mobilising both communities towards the further development of the benchmark and associated array of tasks and baselines. An additional workshop on neurosymbolic learning in the real world is being considered as well. 7.2.2 Holistic Graph Approach for Complex Activity Detection and Prediction In the future we intend to progress further from incremental inference to incremental training, by learning to construct activity graphs in an incremental manner, thus opening the way to applications such as future activity anticipation [205] and pedestrian intent prediction [206], while overcoming the need for a fixed-size snippet representation. A further exciting line of research is the modelling of the uncertainty associated with complex scenes, in either the Bayesian [207] or the full epistemic setting [208,209]. 7.2.3 Extension to Spatiotemporal Sentence Grounding We also aim to expand our activity modelling to include spatiotemporal sentence grounding. More in detail, we plan to integrate our vision models for complex activity detection with language models, in order to localise the specific moment in video based on an input language query. The ultimate goal of our spatiotemporal sentence grounding will be to localise the object tubes in untrimmed videos, where the object may only exist in a very small segment of the video with the help of an input query sentence. \fChapter 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2310.17493v2",
+                    "title": "A Hybrid Graph Network for Complex Activity Detection in Video",
+                    "abstract": "Interpretation and understanding of video presents a challenging computer\nvision task in numerous fields - e.g. autonomous driving and sports analytics.\nExisting approaches to interpreting the actions taking place within a video\nclip are based upon Temporal Action Localisation (TAL), which typically\nidentifies short-term actions. The emerging field of Complex Activity Detection\n(CompAD) extends this analysis to long-term activities, with a deeper\nunderstanding obtained by modelling the internal structure of a complex\nactivity taking place within the video. We address the CompAD problem using a\nhybrid graph neural network which combines attention applied to a graph\nencoding the local (short-term) dynamic scene with a temporal graph modelling\nthe overall long-duration activity. Our approach is as follows: i) Firstly, we\npropose a novel feature extraction technique which, for each video snippet,\ngenerates spatiotemporal `tubes' for the active elements (`agents') in the\n(local) scene by detecting individual objects, tracking them and then\nextracting 3D features from all the agent tubes as well as the overall scene.\nii) Next, we construct a local scene graph where each node (representing either\nan agent tube or the scene) is connected to all other nodes. Attention is then\napplied to this graph to obtain an overall representation of the local dynamic\nscene. iii) Finally, all local scene graph representations are interconnected\nvia a temporal graph, to estimate the complex activity class together with its\nstart and end time. The proposed framework outperforms all previous\nstate-of-the-art methods on all three datasets including ActivityNet-1.3,\nThumos-14, and ROAD.",
+                    "authors": "Salman Khan, Izzeddin Teeti, Andrew Bradley, Mohamed Elhoseiny, Fabio Cuzzolin",
+                    "published": "2023-10-26",
+                    "updated": "2023-10-30",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Detecting and recognising activities in untrimmed videos is a challenging research problem, with applications to, e.g., sports [14], autonomous driving [9], medical robotics [53] and surveillance [58]. Temporal Action Localisation (TAL) approaches not only recognise the action label(s) present in a video, but can also identify the start and end time of each activity instance, enabling the generation of sports highlights [25,60], the understanding of road scenes in autonomous driving [21], the video summarisation of surveillance videos [54] and video captioning [23,38]. A number of TAL methods [3, 13, 27, 28, 31, 34, 35, 52, 55, 59] have recently been proposed, competing to achieve state-of-theart performance [12,61] on accepted benchmarks. Whereas various new datasets have been recently proposed, the two most common relevant benchmarks remain ActivityNet 1.3 [7], and Thumos-14 [16]. State-of-the-art performance on Thumos-14 has improved in four years by some 19% [64]. Almost all TAL approaches comprise a features/scene representation stage and a temporal localisation stage. In the former, snippets (continuous sequences of frames) are processed to understand the local scene in the video. Methods [3, 13, 31, 52] typically employ pre-extracted features obtained using a sequential learning model (e.g., I3D [10]), often pre-trained on the Kinetics [18] dataset. Features are then processed, e.g., via a temporal or semantic graph neural network, by applying appropriate encoding techniques or by generating temporal proposals, in an object detection style [11]. In the second stage, TAL approaches temporally localise activities in various ways, e.g. via temporal graphs [55,59], boundary regression and proposals generation [13,27,28] or encoder-decoder methods [64]. As recently pointed out in e.g. [12, 20], in real-world applications a challenge is posed by complex activities, longer-term events comprising a series of elementary actions, often performed by multiple agents. For example, an Autonomous Vehicle (AV) negotiating a pedestrian crossing is engaged in a complex activity: First it drives along the road, then the traffic lights change to red, the vehicle stops and several pedestrians cross the road. Eventually, the lights turn green again and the AV drives off. Theoretically, TAL methods can be employed to temporally segment complex activities, in practice such approaches are only employed to detect shortor mid-duration actions lasting a few seconds at most (e.g., a person jumping or pitching a baseball). The activities contemplated by the most common datasets are of this nature. Fig. 2 compares two standard TAL benchmarks with the recently released ROAD dataset, explicitly designed for complex activity detection. Activities in the ROAD dataset last longer 1 arXiv:2310.17493v2  [cs.CV]  30 Oct 2023 \fFigure 1. An overview of our Complex Activity Detection (CompAD) framework. The input video is divided into fixed-size snippets S1, . . . , SN (top); each snippet is then processed in three major steps (bottom). A) Firstly, scene objects (agents) are detected and tracked throughout the snippet to form agent tubes. 3D features are then extracted from all the cropped agent tubes (fai) as well as the local scene (fsi). B) Next, a local scene graph is constructed where agent nodes (in gray) are connected to each other and to the snippet node (in light blue). The local scene graph is processed using a graph attention network (GAT), resulting in intermediate scene features (f \u2032\u2032 si). C) Finally, all local scene features associated with individual snippets are temporally connected and processed using a global temporal graph to identify the boundaries of the activity using anchor proposals (shown in different colors). than those in ActivityNet or Thumos, with twice as many agents per snippet, making them more complex in nature. In this paper we argue that standard TAL approaches are ill equipped to detect complex activities, as they fail to model both the global temporal structure of the activity and its fine-grained structure, in terms of the agents and elementary actions involved and their relations. We may thus define (strong) Complex Activity Detection (CompAD) as the task of recovering, given an input video, the temporal extent of the activities there present, as well as their inner structure in terms of the agents or elementary actions involved. A weaker CompAD is one in which the only expected output is the temporal segmentation, with the internal structure of the activity estimated as a means for achieving segmentation with no annotation available. While a small number of studies have attempted to detect complex, long-duration activities [12,20,43], to our knowledge [21] is the only existing study which attempts to tackle CompAD as defined above. The work, however, relies upon the availability of heavily-annotated datasets which include granular labels for the individual actions which make up a complex activity, and the corresponding frame-level bounding boxes. This is a serious limitation, for it prevents [21] from being usable for pure temporal segmentation and compared with prior art there. Further, [21] focuses only on the graph representation of snippets, neglecting the long-term modelling of complex activities. Figure 2. Comparing TAL datasets (ActivityNet-1.3 and Thumos14) with a CompAD dataset (ROAD) in terms of average activity and video durations and mean number of agents per snippet. Objectives. This work aims to push the boundaries of temporal action localisation to tackle (weak) CompAD, leveraging datasets providing temporal segmentation annotation only. We do so by modelling and leveraging a complex video activity\u2019s internal structure, but without resorting to any additional fine-grained annotation concerning individual actions. Nevertheless, our proposal is easily generalisable to strong CompAD whenever individual action/agent annotation is available, in an end-to-end trainable setting. Our proposed framework (Fig. 1) is composed of three stages: A) feature extraction; B) a scene graph attention network designed to learn the importance of each active object (\u2018agent\u2019) within the local dynamic scene; and C) a temporal 2 \fgraph of attended scene graphs for the localisation of complex activities of arbitrary duration. Our feature extraction scheme (A) differs from what typically done in prior art where spatiotemporal features are extracted from whole snippets only. In contrast, we first detect the relevant active objects (agents) in the scene and track them within each snippet to build for each an agent tube (a series of related bounding boxes). We use a pretrained tracker to allow the method to be deployable to any datasets with only temporal annotation (no bounding box annotation for the scene agents is required), while leveraging a fine-grained description of the internal structure of a complex activity in the form of a graph of agents. A pretrained 3D feature extraction model is used to extract features from both the agent tubes and the whole snippet. To represent the local dynamic scene within each snippet, a local scene graph (B) is constructed using three different topologies (Sec. 3 .2). The local scene graph is then processed using a scene graph attention (SGAT) network to extract an overall scene representation, because of its ability to compute the importance of each node in the context of its neighbors, thus modelling the structure of complex scenes. Finally (C), the learned local scene graphs are connected to each other by constructing a temporal graph, with the aim of recognising the activity label and identifying its temporal boundaries (start and end time) in a class-agnostic manner. Our main Contributions are, therefore: \u2022 An original hybrid graph network approach for general complex activity detection, comprising a local scene graph as well as a global temporal graph, capable of localising both complex and shorter-term activities and able to perform both weak and strong CompAD, depending on what annotation is available. \u2013 A scene graph attention network for learning the importance of each agent in the context of a (local) dynamic scene. \u2013 A temporal graph of activated scene graphs for the detection of the start and end of an activity of arbitrary duration. \u2022 Comprehensive experiments showing how our approach leveraging weak CompAD outperforms the most recent TAL state-of-the-art across the board on ActivityNet-1.3, Thumos-14 and the recent ROad event Awareness Dataset (ROAD) [21,44], which portrays long-duration road activities involving multiple road agents over sometimes several minutes, showing clear dominance on classical TAL approaches. 2 . Related Work Recently, a series of works on activity detection have been proposed including spatiotemporal methods and graph-based methods, with recent advances in Graph Attention showing promise in many applications, including trajectory prediction for autonomous driving [22, 47], social recommendation [45] and 3D object tracking [6]. The state-of-the-art approaches to activity detection and graph approaches are summarised below. 2 .1. One-stage vs two-stage approaches TAL methods can been broadly divided into one-stage and two-stage approaches. The former [5,15,26,35] detect actions/activities in a single shot, and can be easily trained in an end-to-end manner. For example, Wang et al. [50] detect actions using transformers which, unlike RNNs, do not suffer from nonparallelism and vanishing gradients. Most one-stage methods, however, only perform action classification, rather than spatiotemporal localisation. In contrast, Lin et al. [26] have recently proposed an anchor-free, onestage light model which generates proposals, locates actions within them, and classifies them end-to-end. The latter group of methods [2, 27, 28, 51, 65], instead, consist of two stages, similarly to region-proposal object detectors. The first stage generates suitable proposals for predicting the start and end time of an activity, while the second stage extracts features and processes the proposals before passing them to both a classification head and a regression head (for temporal localisation). Some works, including [27], focus on the first stage to improve the quality of the proposals, while others focus on the processing or refining of the proposals. [65], for instance, uses an offthe-shelf method for proposal generation. The second stage consists of two networks, a \u2018disentanglement\u2019 network to separate the classification and regression representations, and a \u2018context aggregation\u2019 network to add them together. Such methods are not trainable end-to-end and limited to short-or mid-duration actions. In contrast, this paper proposes a CompAD method which exhibits the advantages of both classes of methods, thanks to our hybrid graph approach capable of recognising and localising both shortand long-duration activities. 2 .2. Graph Convolutional Networks approaches Graph Convolutional Networks (GCNs) have been extensively investigated for TAL [2,40,55,56,59]. GCN-based TAL methods can also be further divided into two-stage and one-stage methods. The former, once again, perform localisation after generating suitable proposals. E.g., in [59] two different types of boundary proposals are generated and then individually passed to the same graph, resulting in both an action label and a temporal boundary. One-stage GCN-based TAL methods, instead, solve the detection problem without proposals in one go by learning spatiotemporal features in an end-to-end manner. For examples, in [55] a graph is first generated by connecting the snippets both temporally and by virtue of meaningful semantics. This graph is then divided into sub-graphs (anchors), where each anchor represents the activity in an 3 \funtrimmed video. In contrast, [21] proposed a spatiotemporal scene graph-based long-term TAL method where each of the snippets is considered as a separate graph, which is heavily dependent on the particular actions present in the scene, and is only applicable to datasets providing (label and bounding box) annotations for each individual action. This study proposes leveraging GCNs, by incorporating them in an overall hybrid graph capable of modelling both the local scenes, via a Graph Attention Network (GAT) [48], and the overall global activity via a temporal graph. GATs build on the transformer concept by applying attention to graphs, and were originally proposed in [48] for node classification. The idea is to update the representation of the current node with respect to its neighbours by applying attention to learn the importance of the various connections. In this paper, GAT is used at the local scene level to learn the features of each node (active agent tube), to generate a more robust local scene representation. 3 . Proposed Methodology The proposed framework is illustrated in Fig. 1. An input untrimmed video V is divided into N snippets S = S1, ..., Si, Si+1, ..., SN (each snippet is a pre-defined constant length of consecutive frames). Each snippet is then passed to the tube detection and feature extraction module which returns a feature vectors for both the snippet fsi and the individual agent tubes fa1 i , fa2 i ,...,fan i , where n is the number of agents present in snippet i. These features are then forwarded to the local scene graph attention layer for learning the attention of each agent in the context of its neighbours. This returns an aggregate feature representation for the whole scene (f \u2032\u2032 si). These aggregate local scene features for all the snippets, f \u2032\u2032 s1,f \u2032\u2032 s2,...,f \u2032\u2032 sN , are then connected using a global temporal graph for the generation of the activity class label \u02c6 ysi and activity boundary labels \u02c6 ybri using anchor proposals in a class-agnostic manner. 3 .1. Tube Detection and Feature Extraction As mentioned, one of the contributions of this paper is a new strategy for feature extraction and representation which consists in analysing the finer structure of the local dynamic scene, rather extracting features from whole snippets only. Objects Detection and Tracking. We first detect scene objects in each frame of the snippet using an object detector pre-trained on the COCO dataset [29], which comprises of 80 different types of objects. However, we select a relevant list of object types (agents) which depends on the dataset (the lists of classes selected for each dataset is given in Section 4 .2 \u2013 Implementation Details). The detected agents are then tracked using a pre-trained tracker throughout the snippet in order to construct agent tubes. The latter are of variable length, depending on the agent itself and its role in each snippet. Fig. 3 pictorially illustrates agent tubes in an example video segment from the ROAD dataset, showing Figure 3. Visualisation of our agent detection and tracking stage using a bird\u2019s-eye view of the spatiotemporal volume corresponding to a video segment of the ROAD dataset. The upper section shows a random frame from the segment (with bounding boxes). Below, the detected agent tubes are plotted in space and time together with different possible local scene graph representations. The agent tubes we extract from the scene are of variable sizes. The way scene object motion affects the spatial and temporal extent of the tubes can be appreciated. Additional scenarios with visualisation are illustrated in the Supplementary material. both a sample frame, with overlaid the detection bounding boxes, and a bird\u2019s-eye view of the agent tubes and local scene graph for the snippet it belongs to. Feature Extraction. Next, all the detected agent tubes are brought to a standard size and passed to the pre-trained 3D CNN model along with the whole snippet for spatiotemporal feature extraction. The adopted 3D CNN model allows variable length inputs, and outputs a fixed-sized feature vector for each of the agent tubes and the snippet. 3 .2. Scene Graph Attention In the second stage, a scene graph representation is used to describe the scene in terms of features extracted from both the overall snippet and the agent tubes. Graph Generation. The scene graph is constructed using three different topologies: a star graph connecting each agent node to the scene node, a star topology with also links between agents sharing a label, and a fully-connected one (Fig. 3). The influence of topology is shown in Sec. 4 .4. Graph Attention. As mentioned, our scene graph attention layer is inspired by the GAT concept [48], originally 4 \fFigure 4. Our scene graph attention layer (Stage B of our approach, Fig. 1) takes the node features generated in the feature extraction Stage A as an input, and updates each node\u2019s features using node attention with respect to its all connected neighbours. Multi-head attention with H heads is applied to each node to further robustify the representation \u2013 averaging yields the final node features. To obtain a fixed-size overall representation for a specific local scene (snippet), the features of all its nodes are aggregated using a learnable weight matrix W2. designed for node classification. While we follow a similar attention mechanism, here we amend the attention layer in order to extract aggregate features from all the nodes, to be passed in turn to our localisation layer in the third stage. The workflow of our scene graph attention layer is shown in Fig. 4. All input node features are linearly transformed using a weight matrix W1, followed by self-attention to find the importance of each node with respect to its connected neighbours. An activation function LeakyReLU is applied to the features for nonlinearity, resulting in final output features for all of the nodes. These are then normalised using a softmax function to make each node representation (f \u2032 siaj i ) comparable with that of all the nodes connected to it (f \u2032 si). The self-attention process is further improved by applying a multi-head attention strategy inspired by transformers [48]. Attention is applied to the node features individually. For each node, the average over the H heads is computed, resulting in an attended feature vector for each node. Finally, to get a fixed-size representation for the whole scene, the output features of all the nodes are aggregated using another learnable weight matrix W2, which outputs the final feature representation f \u2032\u2032 si for the whole (local) scene. 3 .3. Temporal Graph Localisation In Stage C of our framework, activity recognition and localisation are performed using a GCN. The final features from all the local scenes (f \u2032\u2032 s1, ..., f \u2032\u2032 si,f \u2032\u2032 si+1,...,f \u2032\u2032 sN ), as outputted by the scene graph attention layer, are temporally connected to build a global temporal graph (see Fig. 1, C). Our GCN network for processing the global temporal graph is divided into two parts. The first part comprises three 1D convolutional layers designed to learn the temporal appearance of all the local scenes with boundaries, each followed by a sigmoid activation for non-linearity. The second part generates the anchor proposals of the temporally learned features via pre-defined anchors, where each of the anchors acts as a binary mask over the whole graph. Overall, the GCN module provides two different outputs. i) Activity classification: the list of predicted activity labels for all the snippets in the video (\u02c6 ys1, ..., \u02c6 ysi, ..., \u02c6 ysN ) is produced, where the dimensionality of the output vector \u02c6 y is equal to the number of classes. ii) Activity localisation: activities are localised using binary masked class-agnostic anchor proposals. The Intersection over Union (IoU) measure between each anchor and the ground truth (true temporal extension of the activity) is computed, and the anchors with maximum IoU are selected to train the model for localising the boundaries of any activity, regardless of its activity label. The final output of our temporal graph is a one-hot binary vector (\u02c6 ybr1, ..., \u02c6 ybri, ..., \u02c6 ybrN ) for each series of snippets (video), where \u02c6 ybri = 1 iff snippet Si belongs to the activity, = 0 when the snippet does not belong to it. 4 . Experimental Evaluation 4 .1. Datasets ROAD (The ROad event Awareness Dataset for Autonomous Driving) [44] is a multi-labeled dataset proposed for road agent, action, and location detection. The combination of these three labels is referred to in [44] as a \u2018road event\u2019. The ROAD dataset consists of a total of 22 videos with an average duration of 8 minutes, captured by the Oxford RobotCar [36] under diverse lighting and weather conditions. The dataset was further extended as a testbed for CompAD in [21]. Complex activities in the ROAD dataset belong to six different classes, including: negotiating an intersection, negotiating a pedestrian crossing, waiting in a queue, merging into the (ego) vehicle lane, sudden appearance (of other vehicles/agents), and (people) walking in the middle of the road. Activities can span up to two minutes and involve a large number of road agents. Thumos-14 [16] is a benchmark datasets for TAL. It contains 413 untrimmed temporally annotated videos categorised into 20 actions. Videos are characterised by a large variance in duration, from one second to 26 minutes. On average, each video contains 16 action instances. To compare our performance with the state-of-the-art, we adopt standard practice of using the validation set (200 videos) for training while evaluating our model on the test set (213 videos). 5 \fActivityNet-1.3 [7] is one of the largest action localisation datasets with around 20K untrimmed videos comprising 200 action categories. The videos are divided into training, validation, and testing folds according to a ratio of 2:1:1, respectively. The number of action instances per video is 1.65, which is quite low compared to Thumos-14. Following the previous art, we train our model on the training set and test it on the validation set. 4 .2. Implementation Details Evaluation metrics. In our experiments, mean Average Precision (mAP) was used as an evaluation metric, using different IoU thresholds for the different datasets. According to the official protocols for the various benchmarks, the following lists of temporal IoU thresholds were selected: {0.1, 0.2, 0.3, 0.4, 0.5} for ROAD, {0.3, 0.4, 0.5, 0.6, 0.7} for Thumos-14 and {0.5, 0.7, 0.95} for ActivityNet-1.3. Feature extraction. Firstly, the agent tubes are constructed by detecting scene objects using a YOLOv5 detector [17] pre-trained on the COCO dataset. Detections are then tracked throughout a snippet using DeepSort [4]. Then, features are extracted using an I3D network pre-trained on the Kinetics dataset [18], from both the entire snippet and each cropped agent tube individually. The object categories were reduced to six for the ROAD dataset to only cover the agents actually present in the road scenes portrayed there. As the other two datasets (ActivityNet-1.3 and Thumos-14) are general purpose, in their case we retained all the 80 classes present in the COCO dataset. Scene Graph Attention. The local scene graph was generated by producing a list of tuples [(0, 1), (0, 2), ...], where the first index relates to the source node and the second number indexes the target node. The reason for preferring this structure over an adjacency matrix was to limit memory usage. For node attention learning, we initialised our architecture using the weights of the GAT model [48] pre-trained on the PPI dataset [66] and applied 4 attention layers with {4, 4, num of classes, and num of classes} heads, respectively. The number of classes is equal to 201, 21, and 7 in ActivityNet-1.3, Thumos-14 and ROAD, respectively. The Temporal Graph is a stack of three 1D convolution layers on the final representation of the temporally connected local scenes. The size of the input to the first convolutional layer is the number N of local scenes (snippets), multiplied by the number of heads in the last attention layer. The length of our temporal graph is fixed to N. Videos with number of snippets less than or equal to N are passed directly to the temporal graph; longer videos are split into multiple chunks containing N snippets each. The output is a one-hot vector of activity labels of size N and a collection of 128 proposals (binary graph masks) also of length N. Loss Functions. Our problem is multi-objective, as we aim at not only recognising the label of the activity taking place but also finding its boundary (start and end time). Our Table 1. Comparing our approach with the state-of-the-art methods for CompAD available on the ROAD dataset. The mAP at the various standard thresholds is reported. Best results are in bold and second best underlined. ROAD Methods 0.1 0.2 0.3 0.4 0.5 Avg P-GCN [59] 60.0 56.7 53.9 50.5 46.4 53.5 G-TAD [55] 62.1 59.8 55.6 52.2 48.7 55.6 STDSG [21] 77.3 74.6 71.2 66.7 59.4 69.8 TallFormer [12] 78.4 74.9 70.3 63.8 57.1 68.9 ActionFormer [61] 76.5 73.7 72.6 64.4 58.2 69.0 Ours 82.1 77.4 73.3 69.5 62.9 73.0 overall loss function is thus the weighted sum of BCEWithLogitsLoss [1] (for activity classification) and standard binary cross entropy (for temporal localisation). Full details can be found in the Supplementary material. 4 .3. Comparison with State-of-the-Art ROAD. To validate our CompAD approach, we first compared ourselves with the prior art available for the ROAD dataset and re-implemented the current state-of-theart TAL methods; TallFormer [12] and ActionFormer [61] there (see Table 1). The standard IoU thresholds for mAP computation on ROAD range from 0.1 to 0.5. Thumos-14. Our hybrid graph approach is compared with the state of the art on Thumos-14 in Table 2. We followed the standard evaluation metric there: mAP with IoU thresholds ranging from 0.3 to 0.7 and their average. Competitor methods were published in top computer vision venus in 2018-2023. These methods are grouped into two categories whether the optical flow (OF) modality is used or not especially at inference time. The recent TriDet [42] approach achieves the highest average mAP 69.3 by using both OF and RGB modalities while our method outperforms all RGB-based models. Figure 5. Performance comparison with the state-of-the-art over all the three datasets considered, as a function of average activity duration and average no. of agents per snippet. ActivityNet-1.3. Table 2 also compares our proposal with the state-of-the-art methods on ActivityNet-1.3. The mAP IoU thresholds used for comparison are 0.5, 0.75, and 0.95. Our approach achieves the best mAP for all IoU 6 \fTable 2. Activity detection performance comparison on Thumos-14 and ActivityNet-1.3. mAP values (%) at different IoU thresholds are reported for the test and validation sets of Thumos-14 and ActivityNet-1.3, respectively. The models are grouped by whether the model relies on optical flow (OF) or not. Best results are in bold and second best underlined. Thumos-14 ActivityNet-1.3 Methods Venue OF 0.3 0.4 0.5 0.6 0.7 Average 0.5 0.75 0.95 Average BSN [28] ECCV\u201918 \u2713 53.5 45.0 36.9 28.4 20.0 36.8 46.4 30.0 8.0 30.0 P-GCN [59] ICCV\u201919 \u2713 63.6 57.8 49.1 \u2014 \u2014 \u2014 48.3 33.2 3.3 31.1 BMN [27] ICCV\u201919 \u2713 56.0 47.4 38.8 29.7 20.5 38.5 50.1 34.8 8.3 33.8 G-TAD [55] CVPR\u201920 \u2713 54.5 47.6 40.2 30.8 23.4 39.3 50.4 34.6 9.0 34.1 BC-GNN [2] ECCV\u201920 \u2713 57.1 49.1 40.4 31.2 23.1 40.2 50.6 34.8 9.4 34.3 BSN++ [46] AAAI\u201921 \u2713 59.9 49.5 41.3 31.9 22.8 41.1 51.3 35.7 8.3 34.9 MUSES [32] CVPR\u201921 \u2713 68.9 64.0 56.9 46.3 31.0 53.4 50.0 35.0 6.6 34.0 ContextLoc [63] ICCV\u201921 \u2713 68.3 63.8 54.3 41.8 26.2 50.9 56.0 35.2 3.5 34.2 CPN [13] WACV\u201922 \u2713 68.2 62.1 54.1 41.5 28.0 50.7 \u2014 \u2014 \u2014 \u2014 RefactorNet [52] CVPR\u201922 \u2713 70.7 65.4 58.6 47.0 32.1 54.8 56.6 40.7 7.4 38.6 RCL [49] CVPR\u201922 \u2713 70.1 62.3 52.9 42.7 30.7 51.7 55.1 39.0 8.3 37.6 LDCLR [64] AAAI\u201922 \u2713 72.1 65.9 57.0 44.2 28.5 53.5 58.1 36.3 6.2 35.2 ActionFormer [61] ECCV\u201922 \u2713 82.1 77.8 71.0 59.4 43.9 66.8 53.5 36.2 8.2 35.6 Re2TAL [62] CVPR\u201923 \u2713 77.4 72.6 64.9 53.7 39.0 61.5 55.3 37.9 9.0 37.0 TriDet [42] CVPR\u201923 \u2713 83.6 80.1 72.9 62.4 47.4 69.3 54.7 38.0 8.4 36.8 GTAN [35] CVPR\u201919 \u2717 57.8 47.2 38.8 \u2014 \u2014 \u2014 52.6 34.1 8.9 34.3 TadTR [33] TIP\u201922 \u2717 59.6 54.5 47.0 37.8 26.5 45.1 49.6 35.2 9.9 34.3 E2E-TAD [31] CVPR\u201922 \u2717 69.4 64.3 56.0 46.4 34.9 54.2 50.8 36.0 10.8 35.1 ActionFormer [61] ECCV\u201922 \u2717 69.8 66.0 58.7 48.3 34.6 55.5 53.2 35.1 8.0 34.9 TAGS [39] ECCV\u201922 \u2717 59.8 57.2 50.7 42.6 29.1 47.9 54.4 34.9 8.7 34.9 TallFormer [12] ECCV\u201922 \u2717 76.1 \u2014 63.2 \u2014 34.5 59.2 54.1 36.2 7.9 35.5 DL-Net [57] ICASSP\u201923 \u2717 61.3 55.8 47.7 37.6 26.4 \u2013 50.3 35.0 9.3 34.3 DCMD [24] CVPR\u201923 \u2717 70.5 65.8 59.2 50.1 38.2 56.8 53.7 35.9 8.6 35.6 Ours \u2014 \u2717 78.2 69.5 62.7 50.1 36.9 59.8 60.6 40.3 11.1 39.3 thresholds and even outperforms all the methods including the one using both OF and RGB modalities. To sum up, our proposal clearly outperforms all previous approaches over the ROAD dataset, showing the potential of this approach for modelling and detecting long, complex activities. Further, our approach outperforms all competitors over Thumos-14 and ActivityNet. The comparison of methods with respect to the complexity of the datasets is illustrated in Fig. 5 shows how increasingly outperforms the prior art as the duration and complexity of the activities increases. 4 .4. Ablation Studies Effect of Agent Nodes. Firstly, we showed the advantage of modelling the scene as a graph of agents, compared to simply using whole scene features. Namely, we removed the local scene graph from our pipeline (Fig. 1, stage B) and passed the 3D features of the whole scene as a node to the global temporal graph. The significant performance drop can be clearly observed in Fig. 6 over all three datasets. Effect of Edges and Aggregation. Next, we studied the influence of different types of edge connections in the local scene graph: (i) a fully-connected scene graph; (ii) a star structure where each agent node is connected to the scene Figure 6. Average mAP of variants of our method with: attended scene graph (\u2019Agents with GAT\u2019), scene graph (\u2019Agents without GAT\u2019) and \u2019Scene features only\u2019, over the all three datasets. node only; (iii) a star structure with additional connections between agent nodes sharing the same label. We also validated two different techniques for extracting the final representation from the local scene graph, named \u2018Aggregated\u2019 and \u2018Scene\u2019. In the former, the feature representation is extracted by aggregating those of all the attended nodes. In the latter, only the feature vector related to the scene node (after attention) is retained. Table 3 shows the effect of all possible combinations of graph topologies and aggregation strategies. A fully-connected scene graph 7 \fwith Aggregated features performs best in two cases over three, while the star topology best suits ActivityNet. Table 3. Effect of local scene graph topologies and feature aggregation strategies on the performance of our proposal. Thumos-14 Act.Net-1.3 ROAD Topology Aggr. Scene Aggr. Scene Aggr. Scene Fully 59.8 49.2 37.4 31.4 73.0 62.7 Star 51.2 44.8 39.3 36.6 62.3 57.9 Star + 52.2 41.9 35.3 32.7 64.9 59.2 Effect of Sequence Length. To explore the effect on our model of snippet duration (the temporal extent of the local dynamic scene), we performed experiments with four different sizes (12, 18, 24, and 30), reported in Table 4. On ActivityNet-1.3 and ROAD, the top scores were obtained by selecting a sequence length of 24, due to the nature of the activities present in these datasets, which last longer. On the other hand, on Thumos-14 we achieved the best performance using a sequence length of 18, as most activities there are shorter in duration. Table 4. Average mAP over all IoU thresholds of our method as a function of different snippet lengths for the three datasets. Snippet length Thumos-14 ActivityNet-1.3 ROAD 12 52.7 31.3 56.9 18 59.8 37.6 62.6 24 54.3 39.3 73.0 30 49.7 34.5 70.0 Effect of Temporal Graph Length. We also ablated the effect of the length of the temporal graph, i.e., the number of local scene nodes composing the global graph (Table 5). Four different graph lengths (128, 256, 512, 1024) are compared. The best performance is achieved on Thumos-14 using a smaller number of scene nodes, due to the average duration of the videos there and its portraying shorter-term activities. In contrast, on ROAD and ActivityNet-1.3 our approach performed the best using longer temporal graphs. Qualitative Results. To help the reader visualise the output of our proposed method, we show some qualitative detection results on all three datasets in Fig. 7. The figure shows one sample per dataset, and portrays a series of local scenes (snippets), skipping some for visualisation purposes, Table 5. Average mAP over all IoU thresholds of our method as a function of different temporal graph lengths for the three datasets. Temporal length Thumos-14 ActivityNet-1.3 ROAD 128 59.8 29.6 48.3 256 52.9 32.7 58.5 512 50.3 39.3 69.2 1,024 46.8 34.1 73.0 Figure 7. Qualitative results of our method on all 3 datasets. The green rectangles spanning the snippets (local scenes) are the ground truth; the red boxes denote our model\u2019s predictions. with superimposed the ground truth (in green) and the prediction of our model (in red). For example, for ActivityNet1.3 an instance of the \u2018Layup drill in basketball\u2019 class is shown in which the activity starts with snippet 7 and ends with snippet 42. Our model predicts the activity to start from snippet 6 and end with snippet 43. 5 ."
+                }
+            ],
+            "Muhammad Ferjad Naeem": [
+                {
+                    "url": "http://arxiv.org/abs/2212.02291v1",
+                    "title": "I2MVFormer: Large Language Model Generated Multi-View Document Supervision for Zero-Shot Image Classification",
+                    "abstract": "Recent works have shown that unstructured text (documents) from online\nsources can serve as useful auxiliary information for zero-shot image\nclassification. However, these methods require access to a high-quality source\nlike Wikipedia and are limited to a single source of information. Large\nLanguage Models (LLM) trained on web-scale text show impressive abilities to\nrepurpose their learned knowledge for a multitude of tasks. In this work, we\nprovide a novel perspective on using an LLM to provide text supervision for a\nzero-shot image classification model. The LLM is provided with a few text\ndescriptions from different annotators as examples. The LLM is conditioned on\nthese examples to generate multiple text descriptions for each class(referred\nto as views). Our proposed model, I2MVFormer, learns multi-view semantic\nembeddings for zero-shot image classification with these class views. We show\nthat each text view of a class provides complementary information allowing a\nmodel to learn a highly discriminative class embedding. Moreover, we show that\nI2MVFormer is better at consuming the multi-view text supervision from LLM\ncompared to baseline models. I2MVFormer establishes a new state-of-the-art on\nthree public benchmark datasets for zero-shot image classification with\nunsupervised semantic embeddings.",
+                    "authors": "Muhammad Ferjad Naeem, Muhammad Gul Zain Ali Khan, Yongqin Xian, Muhammad Zeshan Afzal, Didier Stricker, Luc Van Gool, Federico Tombari",
+                    "published": "2022-12-05",
+                    "updated": "2022-12-05",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction In Zero-Shot Learning (ZSL), we task an image classi\ufb01cation model trained on a set of seen classes to generalize to a disjoint set of unseen classes using shared auxiliary information. While there has been great progress made in the \ufb01eld, most works treat the auxiliary information to be \ufb01xed to a set of human-labeled attributes [16, 37, 48, 53]. While powerful, these attributes are hard to annotate and expensive to scale [46,58]. Unsupervised alternatives to attributes rely on pretrained word embeddings which provide limited 1First and second author contributed equally. Describe Mallard? Describe House Wren? Laysan Albatross wings are sooty black and they\u00a0mainly feed on squid and small fish ...\u00a0\u00a0 House wren's underpart color ranges from brown to pure white and are mostly found in North America ... Mallard's beak is yellow in color and they have glossy bottle-green head ... Describe Blue Jay? Large Languge Model (LLM) Views of class Blue Jay Blue Jays have white chest and underparts. They can be found in most of the eastern and central United States. Blue Jays feed on seeds, soft fruits and small vertebrates. Their bill, eyes and legs are all blacks. Blue Jays are noisy bold and aggressive but are slow fliers. They have blue hue and crown of feathers on head. Describe Laysan Albatross? Query Class K-shot examples Keys Figure 1. Different annotators focus on different attributes when describing a class. Large Language Models prompted with each of these annotations as k-shot examples can reveal complementary information about a class for zero-shot image classi\ufb01cation. We refer to multiple LLM-generated descriptions as views of a class. information about a class. Recent works [7,20,35,39] show that text documents from internet sources like Wikipedia can provide great auxiliary information for ZSL. Since these web documents describe a queried class in detail, they provide more information for the ZSL model compared to word embeddings. However, these methods only rely on a single source of text documents like Wikipedia, which might not suf\ufb01ciently represent all classes a model is faced with. Multiple sources of text documents of a class can provide complementary information for the ZSL model. For example, in the case of birds, one source might focus more on the patterns of the feather, while another source might better describe the belly and the face of the bird. However, \ufb01nding multiple good sources of text documents for each class requires additional annotation effort. Large Language Models (LLM) [5, 12, 63] trained on web-scale text have shown impressive abilities of using their learned information to solve a multitude of tasks. These models can be conditioned with a k-shot prompt to generalize to a wide set of applications [5, 31, 60] using knowledge from multiple sources they were trained on. In 1 arXiv:2212.02291v1  [cs.CV]  5 Dec 2022 \fthis work, we aim to generate multiple text descriptions of a class, that we recall as \u201cviews\u201d hereinafter, with an LLM using a k-shot prompting strategy. We show that the LLM can act as a mixture of annotators conditioned on different annotation styles to generate complementary information about a class. Moreover, we propose a novel model, I2MVFormer, which utilizes our memory-ef\ufb01cient summary modules to extract discriminative information from each view of a class with the aim of learning a multi-view class embedding. Our contributions in this work are as follows. 1) We provide the \ufb01rst study into using an LLM to generate auxiliary information for zero-shot image classi\ufb01cation. Moreover, we propose a prompting strategy to extract multiple descriptions from an LLM that reveal complementary information about a class. 2) We propose I2MVFormer, a novel transformer-based model for zero-shot image classi\ufb01cation which exploits multiple complementary sources of text supervision to learn a class embedding. I2MVFormer utilizes our Single-View Summary (SVSummary) module to extract rich discriminative information from each class view. This information is utilized by our Multi-View Summary (MVSummary) module to represent a class-level set of tokens from multiple views. The multi-view tokens are aligned with the image to maximize global and local compatibility between the images and the multiple views. 3) Our I2MVFormer achieves signi\ufb01cant performance gains to establish a new state-of-the-art (SOTA) in unsupervised class embeddings in ZSL on three public benchmarks AWA2 [22], CUB [48] and FLO [36]. 2. Related Work Zero-shot learning aims to learn a model that can generalize beyond the seen classes it was trained on. This is accomplished by using side information that is shared with a set of disjoint unseen classes. Towards this, several methods learn a compatibility function between the image feature and a class embedding representing the auxiliary information [9, 19, 28, 41, 51, 62]. These methods often suffer from bias against unseen classes. To address this, another family of methods additionally learns the distribution of the features of images using a generative model [4, 21, 43, 53, 67, 68]. Approaches in this category focus on learning a class conditional generator that can generate features of unseen classes [4, 21] or exploit semantic information about a class to generate features of unseen classes directly [44, 67]. Once learned on seen classes, these models generate features of unseen classes using its auxiliary information to tackle the bias issue. Other works focus on learning improved visual-semantic embeddings [8,17,26,62] and training better image feature extractors [54, 59, 69]. However, all these methods assume that the set of auxiliary information is \ufb01xed to human-labeled attributes [16,33,37,48,53]. However, labeling attributes is expensive and hard to scale on large datasets as it requires expert annotators [46,48,58]. Unsupervised semantic embedding aims to learn the semantic embedding of seen and unseen classes using side information that does not require human intervention. The most in\ufb02uential works in this direction use word embeddings from a pretrained model to encode semantic similarities [30, 38, 45, 56] and re\ufb01ne them using knowledge graphs [6, 18, 29, 34, 49]. VGSE [55] learns a class embedding by using image patches and class embedding vectors. Several works have explored leveraging text documents from sources like Wikipedia to learn class embeddings since they contain rich information about a class. The literature in this direction exploits pretrained language models [1, 7, 14, 15, 20, 39, 67] in addition to noise minimization through a prede\ufb01ned vocabulary [39] or part detection network [15, 67]. However, these works treat the embedding of the document with a pretrained model as \ufb01xed. Recently I2DFormer [35] propose a transformer-based model that learns a class embedding from raw text. Unlike zeroshot transfer models like CLIP [40], which only maximizes the global compatibility between an image and text embedding, I2DFormer maximizes both the global and the local compatibility of the text features against the image. However, I2DFormer relies on expensive local attention between each image patch and document token, which does not scale well to large text sources. Large Language Models (LLM) like GPT-3 [5], OPT [63] and PaLM [12] are trained on very large web-scale datasets. Once trained, these models have impressive abilities towards zero-shot and few-shot inference on a multitude of tasks such as Open Question and Answering [66], generating code [11], text summarization [32], etc. These models rely on a k-shot prompt de\ufb01ning the problem they have to solve along with 0 or k examples. Once prompted for the target class, they use their stored knowledge from web-scale training to generate text for the target class. Recently some works have tried to pair these models with vision models using generated text [60] or adding vision as a modality [2]. Other works explore prompting vision-language models [25, 31, 50] for continual learning, image caption generation, and action understanding. However, no work has yet explored leveraging an LLM to generate auxiliary information for zero-shot image classi\ufb01cation. 3. Generating multiple text views with an LLM We de\ufb01ne a text description of a class as its \u201cview\u201d. This is analogous to a document used in previous work [35]. A single view from a source like Wikipedia as used in previous works [1,7,35,39], while powerful, can present knowledge gaps for less frequent classes. These classes might be described better in other, more specialized databases on 2 \fText Transformer SVSummary Transformer MVSummary I2MVLocalSearch CLS Cross Attention Linear Keys Dot product Large Language Model (LLM) Mean Attention AttPool CLS CLS CLS CLS CLS CLS CLS CLS CLS CLS CLS CLS LLM Text View Wiki Article Blue Jays are native to eastern North America. They measure 22\u201330 cm from bill to tail, weigh 70\u2013100 g and have a U-shaped collar around neck ... I2MVGlobal MLP ADD A person wants to recognize {\u2019type\u2019} in images. They come across {class name} and search online for facts about {class name}. They think the following description of {class name} is a good description Template K-Shot prompt Query Class Prompting LLM for Multiple Text Views Blue Jays Blue Jays are passerine birds in the family Corvidae, native to North America. They are known to raid the nests of other birds and even eat the young in the nests ... LLM Text View LLM Text View They have a long tails and a crest on their head. Blue Jays are bright blue with tails barred with black and have white throats ... House Wren's underparts ... Mallard's bill is yellow ... Laysan Albatross have Sooty wing  color ...  Blue Jays are medium-sized songbirds with bright blue bodies. They feed on nuts, seeds, insects, eggs, and even small vertebrates ... Image Transformer Figure 2. I2MVFormer, our novel Transformer based model uses LLM-generated document supervision conditioned on multiple annotators and noisy Wiki documents as multiple views of a class to learn a zero-shot model. Our SVSummary module generates a single view level CLS representation and local summary. The CLS summaries over multiple views are used to align the global features against the image. Our MVSummary module uses the summary tokens of each view to generate a multi-view summary of the class. This multi-view summary is aligned with the per-patch feature of the image in our I2MVLocalSearch module. Together the two modules learn a highly discriminative multi-view class embedding. the internet. We propose to use LLMs to generate multiple views of a class, exploiting the property of these models of storing the knowledge from multiple internet sources. In this section, we describe our novel prompting method to automatically generate multi-view text descriptions of object classes using a pretrained LLM. While LLMs have impressive zero-shot abilities, practically, they require signi\ufb01cant prompt engineering to get good zero-shot output, as noted in several works [23\u201325, 27,50,64,65]. Since LLMs are few-shot learners [5], given k-shot examples, they can generate output for any class they are prompted for. We exploit this few-shot ability of LLMs to generate multiple text descriptions of each class representing multiple views. Given a set of annotated examples of how to describe a few classes of a dataset in natural language, the LLM can be prompted to mimic the labeling style to generate text supervision for all classes in a dataset. We require f + 1 annotated examples to generate f views from multiple inferences of the LLM. The extra example is reserved for replacement whenever the query class is present in the k-shot examples. We observed that without this, the LLM is susceptible to producing the exact class description as the one in the examples. To curate f + 1 examples without requiring signi\ufb01cant annotation cost, we rely on the Wiki articles released by [35]. These articles are \ufb01ltered only to contain sections of Wikipedia labeled to contain visual information. This is done as sources like Wikipedia contain a lot of noise in the form of non-visual information. We notice that different wiki articles denote different annotation styles as the content is sourced from a pool of volunteers. In our work, we use f = 3 as the number of views generated by LLM for each class. We randomly select four classes from each dataset and curate their wiki articles from [35] to only focus on the visually relevant details. This allows us to obtain a relatively rich source of auxiliary information representing multiple sources without drastically increasing annotation effort. We now want the LLM to use these as examples of class descriptions to generate similar examples over all classes in the dataset. We append each labeled example with the following prompt. \u201cA person wants to recognize {\u2019type\u2019} in images. They come across {class name} and search online for facts about {class name}. They think the following description of {class name} is a good description.\u201d In the given template, we use type as \u201canimals\u201d, \u201cbirds\u201d and \u201c\ufb02owers\u201d for AWA2 [52], CUB [48] and FLO [36], respectively, moreover {class name} de\ufb01nes the name of the labelled class. The target class is then entered into the template and fed as the input to the LLM with the k-shot examples. The LLM generates a description for this target class conditioned on the labeling style of our k-shot examples. We use k = 2 for most of our experiments, i.e., each view is generated conditioned on 2 labeled examples. These 2 shots are a combination of the 3 + 1 labeled examples for the 3 views of each class. We later show in supplementary that the LLM is fairly robust to the choice of prompt given the same k-shot example. 4. I2MVFormer Most methods in zero-shot literature either rely on human-labeled attributes [16, 33, 37, 48, 53] or pretrained word embeddings [6, 18, 29, 34, 45, 49, 56] as auxiliary information. Recently I2DFormer [35] has shown that text can be a powerful substitute without requiring signi\ufb01cant labeling effort. Multiple descriptions, referred to as views in this work, of a class can provide complementary infor3 \fmation of each class to the model and potentially lead to better performance. Our proposed model Image to MultiView Transformer (I2MVFormer) is designed to exploit the knowledge in multiple views of each class to learn a zero-shot model. I2MVFormer consists of two streams of transformers to process the visual and the textual data as shown in Figure 2. I2MVFormer aligns the global information available in each view with the image along with the \ufb01ne-grained information. Moreover, I2MVFormer reduces the memory footprint of I2DFormer while signi\ufb01cantly outperforming it on three public benchmarks. Notations. We represent the classes in the training set as Ys and the unseen classes only available at test time as Yu. Let T = {(x, y, v)|x \u2208X s, y \u2208Ys, v \u2208Vs} be our training set where x denotes an RGB image from the training images X s, y is its label belonging to the seen classes Ys, v = {d1, d2....dq} is the set of documents representing q views of the class and Vs the set of views of the seen classes. At test time an additional set Vu is made available as the views of novel classes where V = Vs + Vu and Vs \u2229Vu = \u2205. The ZSL task requires the model to predict a class from the set of unseen classes Yu, and the Generalized ZSL (GZSL) requires the model to predict a class over both seen and unseen classes Y = Ys + Yu. 4.1. Image Transformer On the image side, our model learns F, an image transformer as an embedding function. Given an image x \u2208 RH\u00d7W \u00d7C, we reshape it into a sequence of \ufb02attened 2D patches xp \u2208RN\u00d7(P 2C), where (H, W) is the size of an input image with C as the RGB channels, (P, P) is the size of each image patch, and N = HW/P 2 is the resultant number of patches. Moreover, we append a CLS token to xp as the input to the image transformer to learn a global image representation. Inspired by recent advances in multimodal learning [61], we use a pretrained frozen image transformer [13] followed by a learnable MLP layer which maps the features to a joint image-text embedding space with dimentionality r. F outputs ICLS(x) \u2208Rr as the global image feature and Ip(x) \u2208RN\u00d7r as the patch-wise image embedding for the input image. 4.2. SVSummary: Extracting class level summary from each view. In our multi-view setup, each training class y is associated with v, the set of documents representing multiple views of the class. The model is now faced with an increased amount of text compared to existing works that rely on a single view. This makes existing solutions [35] that rely on exhaustive attention on all document tokens against the image computationally very expensive. Our SingleViewSummary (SVSummary) module is a text transformer that aims to represent the most discriminative information available in each text description (view) of a class into a \ufb01xed set of Summary tokens. It is important to note that we de\ufb01ne a summary as the most important features described in the text rather than its classical de\ufb01nition of a human consumable shortened description of the text. Given v = {d1, d2, ..., dn}, the set of documents representing views of class y, we pre-tokenize each document and represent it by a pretrained word embedding model similar to [35]. We learn a shallow MLP on top to improve upon the pretrained word embedding model. The output of this yields dt \u2208RM\u00d7r for each document dt in the view set where M is the length of the document and r is the feature dimension. We introduce Single-View Summary tokens Ssv \u2208RT \u00d7r as a set of T learnable tokens. These tokens are introduced to specialize them for the downstream image compatibility task instead of using all M tokens of a view. Given T < M, this results in a reduced constant memory cost of the later cross-modal alignment independent of the length of each view. Ssv is appended to each view and passed through a learnable text transformer with several Transformer encoding blocks. On the output side, we take the token representations corresponding to Ssv for every single view dt to get its summary Ssv(dt). For v, the set of views of a class, this step yields \u02c6 v = {Ssv(d1), Ssv(d2), ..., Ssv(dq)} \u2208Rq\u00d7T \u00d7r as the learnable summary for each view of a class. The \ufb01rst summary token of each view is specialized as the CLS token, which contains the global feature of this view for global alignment in the next step. 4.3. I2MVGlobal: Learning global compatibility between image and multiple views I2MVGlobal aims to align the global feature of an image x with the ground truth class y using the view set v. To accomplish this, we use the CLS token of each view in Figure 2 and take a mean over the view dimension q to get a global feature across all views de\ufb01ned as VCLS(v) \u2208Rr. We align this feature with the global image-level feature ICLS from the image encoding function F. To accomplish this, we de\ufb01ne a scoring function sCLS that measures the compatibility between ICLS and VCLS as a dot product: sCLS(x, v) = ICLS(x) \u00b7 VCLS(v). (1) The learning objective aims to assign high values to correct pairs and low scores otherwise. For a particular training instance (x, y, v), we minimize the following cross-entropy loss over Vs the set of views for the seen classes. LCLS = \u2212log \u0012 exp sCLS(x, v) P v\u2032\u2208Vs exp sCLS(x, v\u2032) \u0013 (2) 4 \f4.4. MVSummary: Generating multi-view summary from all views Previous work in ZSL [35] has shown that only aligning the global feature of an image to a text description can lead to over-\ufb01tting to seen classes. The existing solution to mitigate this relies on expensive cross-attention between all image and document tokens which becomes increasingly expensive as increased text is available for each class. Our SVSummary module allows this at reduced memory cost as it generates a \ufb01xed number of summary tokens irrespective of the length of a view. We concatenate the summary tokens of each view (except the CLS token, which is used in I2MVGlobal) in the view dimension q to de\ufb01ne \u02c6 vlocal \u2208Rq(T \u22121)\u00d7r, the summary of all views. This set will grow linearly, given increased views of each class, and can become expensive to align against the local features of an image. We mitigate this through our Multi-View Summary Transformer (MVSummary) module, which introduces a set of multi-view summary tokens Smv \u2208RT \u00d7r with T learnable tokens. These tokens are aimed at summarizing the local summaries of all views into one class-level multi-view summary. We concatenate Smv with \u02c6 vlocal and use it as input to a learnable Transformer encoder block. The output feature representation corresponding to tokens in Smv represent the multiview local summary vmv \u2208RT \u00d7r for a given class. 4.5. I2MVLocalSearch: Fine-grained alignment between image and summary Our I2MVLocalSearch aims to align the patch-wise image features Ip(x) \u2208RN\u00d7r with the multi-view summary tokens of each class vmv. The core idea is that each summary token specializes in an aspect of a class de\ufb01ned by the multiple views of the class. Aligning an image to these encourage the model to focus on local features that are important for \ufb01ne-grained classi\ufb01cation. We de\ufb01ne this as a query search problem where Q = Ip(x)Wq de\ufb01nes the visual query set, and K = vmv(v)Wk, V = vmv(v)Wv de\ufb01ne the keys to compare with and values to combine with. Wq, Wk and Wv are learnable linear mappings \u2208Rr\u00d7r. The \ufb01rst step involves computing an attention matrix between Q and K and using it to calculate a new multi-view representation of image patches Imvpatch = softmax( QKT \u221ar )V \u2208RN\u00d7r. We want to align this representation with the ground truth class y. We de\ufb01ne a learnable attention pooling to get an image-level feature from these patch representations. We use a learnable Image summary token SIm \u2208Rr as the query Qmv. The keys Kmv and values Vmv are de\ufb01ned by passing Imvpatch through a linear layer. We compute attention between the query and key and use these values to compute intermediate representation I\u2032 mv \u2208Rr. This is passed through a learnable two-layer MLP and a skip connection to get the multi-view image feature Imv \u2208Rr as: I\u2032mv(x, v) = softmax(QmvKT mv \u221ar )Vmv Imv(x, v) = I\u2032 mv(x, v) + MLP(I\u2032 mv(x, v)) (3) The multi-view image feature is used to predict a local alignment score slocal by a learnable linear layer J \u2208R1\u00d7r. This score is optimized with a cross-entropy loss Llocal, slocal(x, v) = J(Imv) LLocal = \u2212log( exp slocal(x, v) P v\u2032\u2208Vs exp slocal(x, v\u2032)) (4) To summarize, SVSummary allows each document view a chance to describe the class. The MVSummary uses these to generate a multi-view class summary which is subsequently used by our I2MVLocalSearch. This multi-view summary is aligned with the patch-level image features to promote \ufb01ne-grained feature learning. 4.6. Inference. Given an input image x, a prediction \u02c6 y corresponds to the view set that yields the highest compatibility score among unseen classes for ZSL and among both seen and unseen classes for GZSL: \u02c6 y = arg max v\u2032\u2208V sCLS(x, v\u2032). (5) 5. Experiments We conduct extensive experimentation on three popular ZSL datasets Animal with Attributes 2 (AWA2) [22], Caltech-UCSD Birds (CUB) [48] and Oxford Flowers (FLO) [36] using the evaluation protocol and data splits proposed by Xian et al. [53]. We do not use any humanlabeled attributes similar to other works in unsupervised class embeddings. In the following, we discuss implementation details, detailed experiments and their conclusions. Implementation Details. We use PaLM540B [12] as the Large Language Model (LLM) for our main experiments prompted with two shots per view at a temperature value of 0.9. We use 3 LLM generated views in addition to the wiki articles released by [35] for our main experiments. The SVSummary is implemented as a two-block deep text transformer. The MVSummary transformer is implemented with a similar con\ufb01guration to SVSummary. T the number of summary tokens is set as 64 for CUB and 128 for AWA and FLO. We use GloVe [38] as the initial token representation similar to [35]. We use the VIT/B16 checkpoint trained on ImageNet1K as the pretrained Image Transformer to be consistent with previous work. The patch projection and 5 \fZero-Shot Learning Generalized Zero-Shot Learning Model Auxiliary Information AWA2 CUB FLO AWA2 CUB FLO T1 T1 T1 u s H u s H u s H GloVe [38] CLSN 52.1 20.4 21.6 42.1 75.3 54.0 16.2 43.6 23.6 14.4 88.3 24.8 GloVe [38] Wiki 61.6 29.0 25.8 49.5 78.1 60.6 23.8 62.6 34.5 14.7 91.0 25.3 LongFormer [3] Wiki 44.2 22.6 8.8 41.6 81.8 55.2 19.9 41.0 26.8 8.8 89.8 16.0 MPNet [47] Wiki 61.8 25.8 26.3 58.0 76.4 66.0 20.6 44.3 28.2 22.2 96.7 36.1 TF-IDF [42] Wiki 46.4 39.9 34.0 29.6 87.6 44.2 29.0 52.1 37.3 28.9 94.8 44.3 VGSE [55] IMG + CLSN 69.6 37.1 56.9 82.8 67.4 27.6 70.6 39.7 I2DFormer [35] Wiki 76.4 45.4 40.0 66.8 76.8 71.5 35.3 57.6 43.8 35.8 91.9 51.5 3-LLM (ours) 69.7 46.0 41.9 65.2 80.4 72.0 36.6 59.5 45.3 37.4 94.2 53.5 3-LLM + Wiki (ours) 77.3 47.0 43.0 68.6 77.4 72.7 38.5 59.3 46.7 40.4 80.1 53.8 I2MVFormer (ours) Wiki 73.6 42.1 41.3 66.6 82.9 73.8 32.4 63.1 42.8 34.9 96.1 51.2 3-LLM (ours) 76.4 47.8 44.4 72.7 81.3 76.8 40.1 58.0 47.4 41.1 91.1 56.6 3-LLM + Wiki (ours) 79.6 51.1 46.2 75.7 79.6 77.6 42.5 59.9 49.7 41.6 91.0 57.1 Table 1. Comparing our I2MVFormer with baseline. Our I2MVFormer signi\ufb01cantly improves on the baselines to set a new SOTA for unsupervised class embeddings. We report top-1 accuracy (T1) on unseen classes for ZSL, and seen/unseen (s/u) classes and their harmonic mean (H) for GZSL. We see that the 3-LLM generated views provide complementary information to the wiki articles and signi\ufb01cantly improve the performance. Moreover, we see that I2MVFormer is better at consuming multi-view knowledge compared I2DFormer. Finally, we see that I2MVFormer with LLM supervision alone can outperform I2DFormer with Wiki article indicating that the LLM alone can generate rich class descriptions. Best results within a method are underlined. Best results overall are in bold. MLP in SVSummary are two layers with ReLU and LayerNorm. For GZSL, we apply calibrated stacking [10] to calibrate the activations of unseen classes on a held-out set. We use the Adam optimizer with a learning rate of 1e\u22123 and the model converges in \u224824 hours. LCLS and LLocal are combined with weights ablated on the validation set. Detailed training details and examples of LLM-generated views are available in the supplementary. Our experimentation framework is implemented in PyTorch and the model can be trained on a single A100 40GB GPU. For the previous SOTA, I2DFormer, we concatenate the text in different views as it is designed for a single view (document) of a class. Performance of pretrained/ classical embedding baselines like GloVe [38], TF-IDF [42], VGSE [55] etc. are taken from [35] with their training setup. We report the top1 per-class mean accuracy in ZSL. In GZSL, we report the top-1 per-class mean accuracy on seen (s) and unseen (u) classes separately along with their harmonic mean (H). 5.1. Comparing with State-of-the-Art. We compare our results with state-of-the-art in unsupervised class embeddings in Table 1 and show that LLMgenerated multiple views can signi\ufb01cantly improve the performance in ZSL. Moreover, we show that I2MVFormer signi\ufb01cantly outperforms all the previous methods to set a new state-of-the-art in unsupervised class embeddings on all three datasets. Our detailed observations are as follows. LLM documents vs Wiki documents. We observe that Wiki articles from [35] and the 3-LLM generated views provide complementary information to consistently improve the ZSL and GZSL performance on all datasets across all metrics. This validates our hypothesis that ZSL models can bene\ufb01t from multiple perspectives of a class and the LLM is able to generate them without signi\ufb01cant annotation effort. Compared to previous SOTA results of I2DFormer with Wiki, I2MVFormer achieves an absolute improvement of 3.2% on AWA, 5.7% on CUB and 6.2% on FLO in ZSL. Similar improvements are seen in the GZSL setting, where we see consistent improvements. We observe that I2MVFormer is better at consuming multi-view knowledge compared to the previous SOTA I2DFormer validating our hypothesis that per-view processing of text allows for extracting richer information from each view. Since our model is speci\ufb01cally developed for multi-view documents, we see that it is on par with I2DFormer across a single view of the Wiki article but achieves signi\ufb01cant improvement once multiple views are introduced. Finally, we see that I2MVFormer with only LLM generated views surpasses I2DFormer with Wiki documents on the three datasets indicating that LLM alone can generate highly discriminative class descriptions for zero-shot image classi\ufb01cation. This con\ufb01rms our hypothesis that LLM with targeted prompting can provide multiple highly discriminative views of a class. Learning per-view summary vs text concatenation. I2MVFormer processes each document in the multiple views independently before generating class level VCLS 6 \fComponents AWA CUB FLO LCLS LLocal SVS MVS T1 T1 T1 a) \u2713 73.6 45.6 38.9 b) \u2713 \u2713 74.1 48.5 39.1 c) \u2713 \u2713 \u2713 57.7 32.5 24.2 d) \u2713 \u2713 \u2713 78.4 49.0 43.2 e) \u2713 \u2713 \u2713 \u2713 79.6 51.1 46.2 Table 2. Ablating over I2MVFormer, we con\ufb01rm the importance of each component of our model. We observe that LCLS and LLocal are complementary to each other. Moreover, SVSummary (SVS) and MVSummary(MVS) reduce the complexity of crossmodal attention while improving performance. and local multi-view summary tokens vmv. This is in contrast to I2DFormer, which concatenates all views into a single text sequence for global and local alignment. We see that the I2MVFormer strategy remains superior as the model can \ufb01rst extract highly discriminative facts from each view and later combine them together in the learned CLS and local summary tokens. When all views are concatenated, the large text sequence can contain repeated information. Moreover, learning local alignment on such a large sequence works less optimally than allowing the model to \ufb01rst extract a set of highly discriminative local summary tokens. We see that I2MVFormer consistently outperforms I2DFormer at LLM across 3 and LLM+Wiki supervision across 4 views. These improvements are in addition to the reduced memory cost as I2MVFormer requires half the GPU memory compared to I2DFormer across 4 views. 5.2. Ablation over I2MVFormer. We ablate over the various components of our model in Table 2 using LLM+Wiki views. Rows a) and b) only optimize for the global feature between the image and text using LCLS. Row a) optimizes for the VCLS generated by the concatenation of all views, while Row b) introduces our SVSummary module. We see that learning a per-view summary while reducing the cost of attention in the text, also offers a performance improvement. Row c) only optimizes the \ufb01ne-grained alignment between image patches and multiview summary tokens using LLocal. We see that this alone performs worse than the global head as \ufb01ne-grained alignment is a hard problem to optimize as also noted in previous works [35, 57]. Row d) and e) optimize for both the LCLS and LLocal. We observe that the two losses are complementary and result in a signi\ufb01cant improvement in performance as the model aligns the image and text modality with global as well as local features. Row d) uses a concatenation of per-view summary tokens while Row e) uses our MVSummary module to \ufb01rst learn a set of tokens representing a multi-view summary of the class for learning local alignment. We con\ufb01rm that MVSummary improves the perforZero-Shot Learning Generalized Zero-Shot Learning Shots AWA2 FLO AWA2 FLO T1 T1 u s H u s H a) 0 shot 73.0 40.7 66.6 79.1 72.3 38.0 85.7 52.7 b) 1 shot unique 74.2 42.1 68.8 82.8 75.1 39.8 89.9 55.2 c) 2 shots repeated 73.1 43.1 67.8 79.9 73.4 39.7 90.1 55.1 d) 2 shots unique 76.4 44.4 72.7 81.3 76.8 41.1 91.1 56.6 Table 3. Ablating over different prompting strategies, we observe that k-shot prompting works better than 0 shot prompting resulting in richer class descriptions. Moreover, unique k-shot examples serve better at generating multiple views than repeated kshot examples for each view. mance of the model while reducing its memory complexity in cross-modal attention. 5.3. Ablation on generating views from LLM. In this section, we study how to generate good text supervision from an LLM for zero-shot image classi\ufb01cation. Unless mentioned, we do not use the Wiki article as a view to only study the impact of the LLM. In\ufb02uence of LLM k-shot prompting on performance. We study the impact of k-shot prompting in Table 3 on generating 3 views per class. We observe that even in zeroshot prompting in row a), the views generated by the LLM allow for a very competitive model further validating that LLM can serve as a search engine for generating class supervision in ZSL. We observe that with 1-shot prompting in row b), we see an improvement over zero-shot as the LLM is now aware of what sort of information we require per class. For our 2-shot prompting, we test repeatedly querying the model three times with the same 2-shot example in row c) vs providing a unique combination of the 4 examples in row d). We observe that while repeating the same 2-shot example generates competitive views, these provide limited additional information. We see the best performance in row d) where we prompt the model with unique 2-shot examples. This allows the model to combine two labeling styles to generate a combined perspective that contains more information as evidenced by the improved accuracy numbers. We expect a further increase in performance if the LLM has access to more unique k-shot examples. However, this would require labeling 10 k-shot examples which are already 20% of the classes for AWA. Since we are interested in learning semantic embeddings with minimal supervision, we leave studying this for future works. Impact of Multiple views on performance. We study the impact of introducing multiple views as supervision for I2MVFormer in Table 4. We observe that increased views generated with each k-shot example consistently improve the performance. The best performance is achieved by the three views and LLM for a total of 4 perspectives about each class available to the model. This further validates our hy7 \fZero-Shot Learning Generalized Zero-Shot Learning Views from LLM AWA2 FLO AWA2 FLO T1 T1 u s H u s H 1 71.6 39.0 67.5 75.2 71.2 34.6 88.0 49.6 2 74.8 43.6 70.5 80.2 75.0 37.7 91.0 53.3 3 76.4 44.4 72.7 81.3 76.8 41.1 91.1 56.6 3 + Wiki 79.6 46.2 75.7 79.6 77.6 41.6 91.0 57.1 4 76.6 44.5 72.9 81.2 76.8 40.5 89.6 55.8 Table 4. Ablating over number of views, we observe that each view provides a useful source of information for I2MVFormer and improves the model performance. pothesis that increased views representing different annotator biases can improve the zero-shot performance. The introduction of LLM as an annotator enables it without requiring actual human annotators as the LLM has stored the knowledge available online and can use it to mimic the annotators available in the k-shot example. Comparing the performance of 2 views in Table 4 with using 3 views across the repeated 2-shot example in row c) of Table 3, we observe that the model generally bene\ufb01ts more from increased perspectives used in the k-shot example of each view than increased views without new perspective in the k-shot example. Moreover, we observe that the fourth view from LLM which repeats the k-shot example performs worse than using the Wiki article as the fourth view. This hints that the knowledge retrieval ability of an LLM is impacted by the information available in its k-shot examples. We expect there to be a potential further increase in performance if the LLM has access to more unique k-shot examples but this again comes at increased labeling cost. 0.2 0.4 0.6 0.8 Temperature 68 70 72 74 76 ZSL Accuracy 1-Shot 2-Shot (a) AWA 0.2 0.4 0.6 0.8 Temperature 38 39 40 41 42 43 44 ZSL Accuracy 1-Shot 2-Shot (b) FLO Figure 3. Ablation over Temperature values. We observe a performance increase with the temperature value of the LLM. However, at 2-shots, the model is fairly robust to changing temperature and achieves impressive performance at all temperature values. Impact of temperature value of LLM on performance. We study the impact of the temperature value of the LLM in Figure 3. The temperature value of the LLM controls the distribution it explores. A high value means the language model will sample from further away from the mean and introduce more stochasticity. We observe from Figure 3 that the performance of I2MVFormer improves with the temperature value as the LLM can represent more diverse text views of the class with its effect most profound in Zero-Shot Learning Generalized Zero-Shot Learning LLM AWA2 FLO AWA2 FLO T1 T1 u s H u s H PaLM 62B 74.0 38.6 66.1 82.3 73.3 37.1 70.3 48.6 GPT3 175B 74.2 44.2 68.8 81.0 74.2 40.4 83.4 54.5 PaLM 540B 76.4 44.4 72.7 81.3 76.8 41.1 91.1 56.6 Table 5. Ablating over different LLM, we observe that our prompting strategy can be used with different LLMs for generating powerful supervision for zero-shot image classi\ufb01cation. one-shot prompting. However, this temperature value can end up being yet another hyperparameter to ablate over. We notice that in two-shot prompting, this temperature value has a smaller impact on the performance of I2MVFormer as the 2 examples provide better conditioning for each view to constrict the model in what kind of information we require for each class. We see a smaller change in performance across different temperature values with the best performance achieved at a temperature of 0.9 on both datasets. Impact of size and family of LLM on performance. We study the impact of different LLM in Table 5 and observe that the largest model PaLM540B provides the best auxiliary information leading to the best performance. Moreover, we observe that the smaller 60B version of PaLM achieves very promising performance indicating that while increased parameters do bring more performance, the \u201csmaller\u201d LLM can still achieve impressive results. As we compare the results of PaLM with GPT3, we observe that the larger PaLM540B model outperforms GPT3 for our 2-shot setup as also noted in the original PaLM manuscript [12]. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2209.10304v1",
+                    "title": "I2DFormer: Learning Image to Document Attention for Zero-Shot Image Classification",
+                    "abstract": "Despite the tremendous progress in zero-shot learning(ZSL), the majority of\nexisting methods still rely on human-annotated attributes, which are difficult\nto annotate and scale. An unsupervised alternative is to represent each class\nusing the word embedding associated with its semantic class name. However, word\nembeddings extracted from pre-trained language models do not necessarily\ncapture visual similarities, resulting in poor zero-shot performance. In this\nwork, we argue that online textual documents, e.g., Wikipedia, contain rich\nvisual descriptions about object classes, therefore can be used as powerful\nunsupervised side information for ZSL. To this end, we propose I2DFormer, a\nnovel transformer-based ZSL framework that jointly learns to encode images and\ndocuments by aligning both modalities in a shared embedding space. In order to\ndistill discriminative visual words from noisy documents, we introduce a new\ncross-modal attention module that learns fine-grained interactions between\nimage patches and document words. Consequently, our I2DFormer not only learns\nhighly discriminative document embeddings that capture visual similarities but\nalso gains the ability to localize visually relevant words in image regions.\nQuantitatively, we demonstrate that our I2DFormer significantly outperforms\nprevious unsupervised semantic embeddings under both zero-shot and generalized\nzero-shot learning settings on three public datasets. Qualitatively, we show\nthat our method leads to highly interpretable results where document words can\nbe grounded in the image regions.",
+                    "authors": "Muhammad Ferjad Naeem, Yongqin Xian, Luc Van Gool, Federico Tombari",
+                    "published": "2022-09-21",
+                    "updated": "2022-09-21",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction \u201cWhat does a tiger look like? It is a \ufb01erce animal that looks like a scary, big cat with stripes.\" Tigers are not native to Japan, yet when the travelers coming from China described them in relation to native animals, it inspired a range of historic paintings depicting tigers in Japan. Humans possess an impressive ability to imagine and identify unseen objects from pure language descriptions. In computer vision, the ability to predict unseen classes is called zero-shot learning, which can be achieved by transferring knowledge from seen classes using auxiliary side information (or semantic embeddings) e.g., attributes [22], word embeddings [17], etc. Although remarkable progress has been made, most of prior works [22, 1, 57, 71, 31, 10] rely on human annotated attributes as the side information. While attributes are appealing, they are often costly to annotate [48, 65, 51] and scale to large datasets. Towards unsupervised semantic embeddings [46, 17, 2], word embeddings can be easily obtained from pre-trained language models [35]. Yet, they often do not re\ufb02ect \ufb01ne-grained visual similarities, thus limiting the performance [2]. The goal of this work is to learn visually aligned unsupervised semantic embeddings from online textual documents for zero-shot image classi\ufb01cation. With the advent of the Internet, the collective knowledge of humans about the world has been distilled into online encyclopedias like Wikipedia. These encyclopedias present a rich source of \ufb01ne-grained auxiliary information for a model. While the entries (referred to as documents) may describe an object class with rich visual details, they 36th Conference on Neural Information Processing Systems (NeurIPS 2022). arXiv:2209.10304v1  [cs.CV]  21 Sep 2022 \ftend to contain a lot of noise. For example, an entry for \u2018horse\u2019 can de\ufb01ne its appearance as well as interesting historic events it participated in. While the former is helpful for a visual model, the latter might introduce noise making it challenging to fully exploit this knowledge. In this work, we propose a novel model Image to Document Transformer (I2DFormer) that learns to align image and document pairs with their global representations as well as with tokenwise representations, i.e., image patches and document words. As a result, without any imagelevel language supervision, our model is able to develop an understanding of different parts of an animal, its habitat, etc, leading to a more discriminative semantic embedding. We summarise our contributions as: (1) We propose a novel transformer based framework for ZSL with noisy documents. (2) Our novel Image to Document Attention (I2D Attention) module learns to identify visually discriminative properties in a document leading to a more discriminative semantic embedding. (3) Our model I2DFormer consistently improves the SOTA in unsupervised semantic embeddings on three challenging datasets, i.e., AWA2, CUB and FLO. Moreover, we qualitatively demonstrate that our model learns highly interpretable results. (4) We show that the learned document embedding can be used with any existing ZSL model to signi\ufb01cantly improve its performance. To the best of our knowledge, I2DFormer is the \ufb01rst method to learn an attention-based embedding from noisy documents for ZSL without relying on any pretrained part localization model or attribute vocabulary. 2 Related Works Zero-shot Learning aims to generalize a model trained on seen classes onto a disjoint set of unseen classes using shared auxiliary information available for both sets [22]. Several methods in this direction learn a compatibility function between the image and the class embedding space [42, 30, 8, 27, 1, 68, 56, 26]. Another competing line of work uses generative models like GANs to learn the feature space of seen and unseen classes [58, 59, 70, 69, 21, 44]. A complementary line of work focuses on learning improved visual-semantic embeddings [24, 68, 18, 7] and training better image encoders [66, 71, 61]. Semantic embeddings are a crucial building block for all of these methods. However, despite its importance, it is a less studied topic. Human labeled attributes [57, 34, 51, 16, 29] have become the de-facto semantic embedding for most methods. However, they are hard and expensive to scale as they require human experts [48, 65, 51]. Learning semantic embeddings with minimal supervision aims to use cheap to obtain side information to learn a semantic embedding with minimal label information. Several works have explored using text corpora as an alternative source of semantic embeddings. Some approaches include using word embeddings from pretrained language models [63, 35, 47, 28] and knowledge graphs [52, 19, 5, 30, 27] to encode semantic similarities. Another line of work aims to directly learn semantic embeddings from documents containing information about classes. Earlier works in this direction used TF-IDF [43] to directly embed the document in a joint image space [14]. Successive works have focused on reducing the noise in the document by using prede\ufb01ned attribute vocabulary [3], learning better weights for TF-IDF embeddings [37] or complementing these embeddings with a part detection network [15, 69]. Recent works have incorporated Transformer based language models to directly embed a document to a semantic embedding [20, 6]. However, all these works either learn the semantic embedding against the global image representation or use a pretrained part detector for the human-labeled attributes to \ufb01lter the relevant details. VGSE [62] instead proposes to directly learn semantic embeddings from images of seen classes and extrapolate them to the unseen classes by measuring their class name similarities. Our model, I2DFormer instead uses both the knowledge in text documents and the images of seen classes to learn a semantic embedding and ZSL model. Learning cross-modal attention between image and text to ground text in images without region level supervision has been a long-studied problem in visual question answering, image captioning, etc. [11, 12, 40, 41]. Methods in this line of work learn a mapping between the region level features from an image and its caption. More recently, Transformers [50] have made a breakthrough in this \ufb01eld with models like ViLBERT [25] and FILIP [41] that learn a cross-modal attention to learn cross modal embeddings. They show that the grounding of text in the image naturally emerges as a by-product [60]. However, these works rely on having access to image-level text which is expensive to obtain. Our model instead addresses the much more challenging problem of learning a cross-modal embedding and attention from images and their class-level text document. 2 \fDocument Document  feature Image  feature colors on belly feathers song is \u2026Green Voiletear  has shining green  and bronzy colors  on the belly and  feathers. Their song  is sharp\u2026 \u2026Tropical Kindbird has a yellow  bottom, whitish head, and gray green  feathers. The bird was formally  described in 1819\u2026. ..Cape May Warbler has yellowish  underparts, and characteristic tiger  strips below the head. The bird sings  from high perches\u2026 Image  Document Image Transformer Document  Transformer CLS CLS Image Patch features Word Features Linear Wk Linear Wq Linear Wv the and Their sharp 0.0 1.0 0.2 0.2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.3 0.1 0.2 0.2 0.2 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0` 0.0 0.0 0.0 Attention Word to Patch Attention Cross Modal  Mixer Global  pooling Linear H P1 P2 P3 PN P4 P5 P6 P7 P8 Glove token  embedding I2D Attention I2D Global Patch to Word attention \u2715 . \u2715 Image Transformer Document Transformer Q K V green  belly forest violet around A Dot product softmax pool Image Matrix multiplication linear I2D Attention I2D Global Image patch embedding Document word embedding Figure 1: I2DFormer, our novel Transformer based model, uses noisy documents as auxiliary information to learn a zero-shot model. The \ufb01rst part of the model, I2D Global, learns to encode images and noisy documents to a shared embedding space. In order to distill discriminative local information from the document, we propose a novel I2D Attention module that learns \ufb01ne-grained interactions between image patches and document words. Together, the two modules learn a highly discriminative document semantic embedding I2DEmb. 3 Image to Document Transformer (I2DFormer) The vast majority of existing ZSL works utilize either human-annotated attributes or word embeddings as auxiliary information. In this work, we instead utilize the textual collection of encyclopedia (wiki) entries of classes as side information given the wealth of free document collections describing object classes available on the internet. To achieve that, we propose I2DFormer (Figure 1), a pure-transformer ZSL framework that learns to align image and document pairs with their global representations and with token-wise representations i.e., image patches and document words. Notations. We de\ufb01ne the classes that are included in the training set as seen classes Ys, and the classes that are excluded from training as unseen classes Yu. Let T = {(x, y, d)|x \u2208X s, y \u2208Ys, d \u2208Ds} be our training set where x denotes an RGB image from the training images X s, y is its label belonging to the seen classes Ys, d is a document e.g., Wikipedia article, containing textual descriptions of the object class y, and Ds is a collection of documents describing seen classes. At test time, another collection of documents Du describing the unseen classes Yu will be made available to the model. This simulates an internet query to fetch extra information about an unseen class. Those documents will be used as the side information to connect seen and unseen classes. The task of ZSL is to make a prediction among only unseen classes, while GZSL needs to predict both seen and unseen classes. 3.1 I2D Global: Learning joint Image-Document Embeddings with Transformer Our model is a dual-stream transformer architecture that learns, respectively, an embedding function F, an image transformer [13], for images, and G, a document transformer [50], for text documents. The \ufb01rst part of our model learns a global compatibility between the Image and the Document by our Image to Document(I2D) Global module. On the image side, given an input image x \u2208RH\u00d7W \u00d7C, we reshape it into a sequence of \ufb02attened 2D patches xp \u2208RN\u00d7(P 2\u00d7C), where (H, W) is the size of an input image with C as the RGB channels, (P, P) is the size of each image patch, and N = HW/P 2 is the resultant number of patches. Moreover, we append a CLS token to xp as the input to the image transformer to learn a global image representation. Inspired by LiT [67], we use a pretrained frozen image transformer [13]. This is followed by a learnable feature projection layer that maps the image embeddings to a joint image-document embedding space with dimensionality r. The image encoder F outputs fCLS(x) \u2208Rr as the global image feature and fp(x) \u2208RN\u00d7r as the patch-wise image embedding for the input image where r is the feature dimension. On the document side, given a document d consisting of M words, we get its token-wise input feature representation with a pretrained word embedding model. Note that we use words and tokens interchangeably as we use GloVe word features as tokens [35]. Since each document consists of 3 \fa typically long sequence of words, we further pass this feature representation through a learnable MLP as a token projection layer to reduce the feature dimension and the memory footprint, yielding dt \u2208RM\u00d7r, where r is the feature dimension as the output of the token projection layer. Our learnable document transformer consists of transformer encoder blocks with multi-head attention. We append a CLS \u2208Rr token to this sequence and pass it through the document transformer to get gCLS(d) \u2208Rr as the global document embedding and gt(d) \u2208RM\u00d7r as the word-wise text embedding for the input document. We later refer to the learned gCLS(d) as a document embedding (semantic embedding) I2DEmb that can be used by any ZSL method. We de\ufb01ne a scoring function s : X \u00d7 D \u2192R that measures the similarity of any image x and document d pair. The scoring function computes the dot product between global image embedding fCLS(x) and document embedding gCLS(d) , formulated as s(x, d) = fCLS(x) \u00b7 gCLS(d). (1) The learning objective is to make the scoring function assign high scores to correct image and document pairs and low scores to incorrect ones. Therefore, for a particular training instance (x, y, d), and Ds the collection of documents belonging to seen classes, we minimize the following cross-entropy loss, LCLS = \u2212log( exp s(x, d) P d\u2032\u2208Ds exp s(x, d\u2032)) (2) 3.2 I2D Attention: Learning Image Patch to Document Word Attention Our I2D Global module essentially aligns image-document pairs using their global representations. While this paradigm has been popularized by in\ufb02uential works like CLIP [38], it relies on a large amount of image-text pairs to learn all discriminative local features and represent them in the output of CLS token. However, we are dealing with a more challenging problem where the number of training images is small (a few thousand) and there is only one document associated with each class. Aligning two modalities at a global level will be prone to over\ufb01tting and hard to generalize to unseen classes at test time. Moreover, our documents are directly collected from the Internet and therefore are noisy e.g., a large portion of the words are irrelevant to visual appearance. To address these challenges, we propose I2D Attention, a novel cross-modality attention module, to learn \ufb01ne-grained interaction between image patches and document words, capturing local features de\ufb01ned in the document such as body parts of an animal, their habitat in the form of image background, etc. We argue that learning these local mappings allows a model to generalize beyond the seen classes. Our I2D Attention module takes as inputs the patch-wise embeddings fp(x) \u2208RN\u00d7r of the image and the token-wise embeddings gt(d) \u2208RM\u00d7r of the document. We task the model with searching for the visually-relevant words in the documents using image patches as the queries. More speci\ufb01cally, we de\ufb01ne Q = fp(x)Wq as the image queries, K = gt(d)Wk as the text keys to compare with, and V = gt(d)Wv as the text values to mix with after the search, where Wq, Wk and Wv are learnable linear transformations, all in size r\u00d7r. The I2D Attention module estimates the cross-modal attention A(x, d) \u2208RN\u00d7M by computing a dot product between every image patch and word pair followed by a softmax, A(x, d) = softmax(QKT \u221ar ) (3) This attention matrix is used to compute new feature representations fpa(x, d) \u2208RN\u00d7r for all image patches as linear combinations of rows of the value matrix V i.e., fpa(x, d) = A(x, d)V . Intuitively, this operation recomputes the image patch embeddings using the token-wise embeddings of relevant words in a document. To obtain the image-level embedding, we apply global pooling on the patch dimension N of the new patch embeddings fpa(x, d), yielding \u02c6 fpa(x, d) \u2208R1\u00d7r. Afterwards, we compute the local alignment score between an image-document pair by applying a simple linear layer, slocal(x, d) = H( \u02c6 fpa) (4) where H \u2208Rr\u00d71 is a learnable linear layer. Given a particular training example (x, y, d), we optimize the following cross-entropy loss, Llocal = \u2212log( exp slocal(x, d) P d\u2032\u2208Ds exp slocal(x, d\u2032)) (5) 4 \fWe do not use any skip connection similar to previous cross-modal attention blocks like ViLBERT [25] as we want the attention weighted embedding to directly give us a linearly separable representation. Our I2D Attention Module searches for relevant patch features in the document for each training class y \u2208Ys and learns to associate visual concepts with the noisy text in the document. The classi\ufb01cation loss Llocal maximizes the contribution of discriminative words in the document and minimizes the contribution of irrelevant details about a class. Furthermore, calculating cross-entropy over the full seen set ensures that the model is aware of similar attributes between \ufb01ne-grained classes and can pick additional cues to separate such classes. I2D Attention introduces minimal learnable parameters with only 4 additional linear layers and rather forces F and G to minimize irrelevant details in the tokenwise fp(x) and gt(d) as well as the global fCLS(x) and gCLS(d) embeddings. This is in contrast to architectures like ViLBERT [25] where several self-attention layers are stacked on top of the cross-modal module to further learn the output embedding with self-attention. We later show in our experiments that this can hurt the performance in our data constrained zero-shot learning setup. Although documents have been explored before in ZSL, prior work uses \ufb01xed document embeddings that are encoded with TF-IDF [14, 69, 15] or extracted from a pretrained language model [6, 20]. In contrast, our document embeddings are aided by the attention module to identify important details and thus are also assisted by this additional visual information. 3.3 Inference Given an input image x, we search for the document \u02c6 d that yields the highest compatibility score, \u02c6 d = arg max d\u2032\u2208D s(x, d\u2032). (6) The search space includes only documents of unseen classes in zero-shot learning i.e., D = Du, and all classes in generalized zero-shot learning (GZSL) i.e., D = Ds \u222aDu. The \ufb01nal prediction is simply the class label associated with the document \u02c6 d. For GZSL, we apply calibrated stacking [9] to calibrate the activations of unseen classes on a held-out set to reduce the bias towards seen classes. We only use the output of the global prediction as it is computationally cheaper and has distilled the knowledge of patch-to-token interactions while training. The attention between image patch and document words is computed as the explainability of the model\u2019s decision when required. 4 Experiments We conduct extensive experiments on Animals with Attributes2 (AWA2) [57], Caltech-UCSD Birds (CUB) [51] and Oxford Flowers (FLO) [32], which are widely used datasets in ZSL. We follow the evaluation protocol and data splits proposed by Xian et al. [57]. Since the main focus of this work is to learn unsupervised semantic embeddings, we do not use any human-annotated attributes. In the following, we \ufb01rst describe how documents are collected and implementation details. Then, we quantitatively compare against SOTA unsupervised semantic embeddings methods and ZSL methods. Finally, we show quantitative results to demonstrate the interpretability of our method. Collecting documents. We use online sources for documents that can be queried with minimal human supervision. These sources contain useful knowledge about each class but might have a lot of noise as irrelevant textual details. For AWA2, we use A-Z Animals [53], an animal encyclopedia. For CUB, we use AllAboutBirds [54], a bird-watching encyclopedia. For FLO, we use a collection of gardening blogs and Wikipedia [55] to collect documents for these classes. However, we found documents for \ufb02owers to be less focused on the patterns of petals and pistils and rather more focused on the general description of the plant and its taxonomic biological classi\ufb01cation. FLO is therefore a challenging dataset to generalize from document-based embeddings. We adopt a simple \ufb01ltering step on these collected articles similar to [20]. We look at the documents for 10% of classes of each dataset and identify sections that contain relevant information about the class. The rest of the documents are \ufb01ltered to only contain these sections. The average size of a document is \u2248400 words. To put this into perspective, models like CLIP [38, 64] use image captions of at max 64 tokens [38, 36]. The long length of the documents presents an additional challenge. Document examples and their links are included in the supplementary. The collected documents will be released after the review process. Training Details. We implement our model in PyTorch and train on an Nvidia A100 GPU. We use the VIT/B16 checkpoint trained on ImageNet 1k by [13] as the pretrained Image Transformer. The 5 \fZero-Shot Learning Generalized Zero-Shot Learning Semantic AWA2 CUB FLO AWA2 CUB FLO Embedding Source T1 T1 T1 u s H u s H u s H GloVe[35] CLSN 52.1 20.4 21.6 42.1 75.3 54.0 16.2 43.6 23.6 14.4 88.3 24.8 GloVe[35] DOC 61.6 29.0 25.8 49.5 78.1 60.6 23.8 62.6 34.5 14.7 91.0 25.3 LongFormer[4] DOC 44.2 22.6 8.8 41.6 81.8 55.2 19.9 41.0 26.8 8.8 89.8 16.0 MPNet[49] DOC 61.8 25.8 26.3 58.0 76.4 66.0 20.6 44.3 28.2 22.2 96.7 36.1 TF-IDF[43] DOC 46.4 39.9 34.0 29.6 87.6 44.2 29.0 52.1 37.3 28.9 94.8 44.3 VGSE[62] IMG + CLSN 69.6 37.1 56.9 82.8 67.4 27.6 70.6 39.7 I2DFormer(Ours) IMG + DOC 76.4 45.4 40.0 66.8 76.8 71.5 35.3 57.6 43.8 35.8 91.9 51.5 Table 1: Comparing our I2DFormer with unsupervised semantic embedding methods using the same image feature and method (our I2D Global module). In ZSL, we report top-1 accuracy (T1) on unseen classes, in GZSL on seen/unseen (s/u) classes and their harmonic mean (H). We consider semantic embeddings that are either directly extracted (with a pretrained language model) or learned from different sources including classnames (CLSN), document (DOC), a combination of image and classnames (IMG+CLSN), and a combination of image and document (IMG+DOC). Our I2DFormer signi\ufb01cantly improves on the baselines to set a new SOTA for unsupervised class embeddings. Zero-Shot Learning Generalized Zero-Shot Learning Semantic AWA2 CUB FLO AWA2 CUB FLO Type ZSL Model Embeddings T1 T1 T1 u s H u s H u s H Discriminative SJE[2] GloVe 56.6 27.1 13.1 41.3 83.4 55.3 14.4 51.6 22.5 4.6 93.2 8.7 VGSE 70.1 31.6 49.9 84.8 62.8 23.1 57.5 33.0 I2DEmb(Ours) 72.6 38.2 33.4 55.8 82.6 66.6 25.0 56.2 34.6 18.5 87.1 30.5 APN[61] GloVe 73.8 20.7 15.2 57.6 84.6 68.5 19.6 32.6 24.5 12.8 39.4 19.3 VGSE 74.0 34.3 65.0 72.4 68.5 23.2 52.9 32.1 I2DEmb(Ours) 74.5 40.6 35.4 65.5 76.9 70.7 30.0 49.9 37.5 32.0 85.3 46.5 Generative GAZSL[69] GloVe 63.7 37.5 20.9 22.2 90.8 35.6 5.93 36.2 10.2 8.38 97.3 15.4 VGSE 74.7 35.7 29.5 93.8 44.9 10.5 51.8 10.5 I2DEmb(Ours) 83.1 42.9 34.2 56.8 94.7 71.0 15.9 50.4 24.1 28.8 90.1 43.7 f-VAEGAN-D2[59] GloVe 70.7 31.8 32.1 65.7 69.5 67.6 23.9 55.7 33.5 25.0 99.0 39.9 VGSE 75.0 40.7 70.8 79.0 74.7 32.7 57.5 41.7 I2DEmb(Ours) 85.1 41.9 36.9 73.2 81.7 77.2 33.4 57.3 42.2 30.0 97.3 45.8 Disc. I2DFormer(Ours) I2DEmb (Ours) 76.4 45.4 40.0 66.8 76.8 71.5 35.3 57.6 43.8 35.8 91.9 51.5 Table 2: Comparing I2DFormer with baseline ZSL methods, under various unsupervised semantic embeddings we see that our model and embeddings I2DEmb set a new SOTA. In ZSL, we report top-1 accuracy (T1) on unseen classes, in GZSL on seen/unseen (s/u) classes and their harmonic mean (H). Best embedding results within a method are underlined. Best results overall are bolded. image patch projection and token projection layers are implemented as a shallow MLP. Maxpool or Meanpool are chosen as global pooling by ablation. The model is trained with Adam optimizer with a learning rate of 1e\u22123 and takes \u224824 hours to converge. LCLS and Llocal relative weights are chosen by ablation. More details are available in the supplementary. For baseline methods, we use the CLS features from the same VIT/B16 checkpoint with author\u2019s implementations. We ablate these methods over multiple hyperparameters to report the best run. For VGSE, we use the semantic embeddings released by the original authors (not available for FLO). 4.1 Comparison with SOTA unsupervised semantic embeddings In this section, we compare with existing unsupervised semantic embeddings where they are obtained without using human supervision using the same ZSL method (our I2D global module). Compared semantic embeddings. For GloVe (classname) [35], we simply extract GloVe vectors of class names. This method has been adopted by many prior ZSL methods [33, 45, 17, 2, 30] due to its simplicity. For GloVe (Document) [35], we average over the feature vectors of each word in the document. LongFormer [4] is a text transformer model trained for documents and outputs a CLS embedding given a document. MPNet[49] is the current SOTA Sentence Transformer model[39] trained to optimize embeddings for natural language classi\ufb01cation tasks. Since the original model is 6 \ftrained for short sequences, we average over the individual sentence embeddings similar to [20, 6]. TF-IDF [43] stands for Term Frequency-Inverse Document Frequency, which has been used by some prior ZSL methods [14, 23]. VGSE [62] learns the semantic embeddings from image patches and word embeddings of class names. Since these embedding models generate one embedding for the whole document, we replace the Document Transformer with an equally deep MLP. Results. From Table 1, we observe that our method I2DFormer consistently outperforms all semantic embedding methods in both ZSL and GZSL. Compared to GloVe (Document) [35], which also serves as an input to our method (without the average over words), the learned embedding of our model achieves an impressive 76.5% accuracy vs 61.6% on AWA2, 45.4 % vs 29.0 % on CUB and 40.0% vs 25.8% on FLO with a relative 1.25\u00d7, 1.5\u00d7 and 1.6\u00d7 improvement each. This shows that our learned document embedding assisted by our I2D attention module signi\ufb01cantly improves the zero-shot performance. We see that these improvements are also consistent in GZSL where we see a signi\ufb01cant improvement in the HM. Similar results are observed for other pretrained language semantic embeddings Longformer, MPNet and TF-IDF [4, 49, 43]. Since the original embedding models for these baselines were only trained on language data, the generated semantic embedding is unlikely to capture the most visually discriminative features described in the document. Our model however is able to learn a more informed semantic embedding thanks to supervision from the images of the seen classes. Comparing rows 1 and 2, we see that the use of documents over classnames leads to a major improvement as documents capture better class similarities. Finally, compared to VGSE [62], the current SOTA unsupervised semantic embedding, we observe that our model again substantially outperforms it. While both VGSE and our model exploit patch-wise similarities in images of different classes to learn a class embedding, our model is additionally able to complement this embedding with localized information available from the documents thanks to our I2D Attention. 4.2 Comparing with SOTA ZSL methods In this section, we compare our full model with existing SOTA zero-shot models across baseline embeddings and our learned document embedding. For a fair comparison, we evaluate those methods with the same VIT/B16 image features. We show in Table 2 that our method I2DFormer, and our learned document embeddings I2DEmb achieve SOTA performance. Compared to baselines, our model I2DFormer or our learned embedding I2DEmb consistently outperform all baseline ZSL methods and embeddings to establish a new SOTA. I2DFormer achieves SOTA ZSL performance on CUB and FLO, the \ufb01ne-grained datasets. On CUB, I2DFormer achieve an impressive 45.4% compared to the closest 42.9% of GAZSL that also uses our I2DEmb. On FLO, I2DFormer achieves 40.0% compared to the closest 36.9% of f-VAEGAN-D2 that again uses our I2DEmb. In GZSL, on CUB, I2DFormer achieves 43.8% HM compared to the closest 42.2% of f-VAEGAN-D2 (I2DEmb). On FLO, I2DFormer achieves an impressive 51.5% HM compared to the closest 45.8% of f-VAEGAN-D2 (I2DEmb). We would like to emphasize that our model is outperforming both the generative baselines in GZSL on these two datasets. Generative models have previously been shown to be the most competitive baselines in these datasets. However, since I2DFormer learn a \ufb01ne-grained attention between the image patches and the words in the article, it is able to outperform these baselines with this extra knowledge without requiring feature generation. On AWA2, a coarse classi\ufb01cation dataset, we see that I2DFormer achieves SOTA performance among the Discriminative baselines. However, the best performance is achieved by the Generative baseline f-VAEGAN-D2 using our I2DEmb on this dataset. f-VAEGAN-D2 with I2DEmb achieves the best ZSL accuracy of a remarkable 85.1% vs. the closest 83.1% achieved by GAZSL(I2DEmb), with I2DFormer being the third. In GZSL, f-VAEGAN-D2 with I2DEmb achieves SOTA with an impressive HM of 77.2% followed by 74.7% of the same method with VGSE embeddings. I2DFormer is a second in this setting surpassing the remaining 3 ZSL methods across all embeddings. However, these baselines are only able to outperform I2DFormer with our learned I2DEmb. 4.3 Ablation study What kind of Patch to Word Attention is required in ZSL? We study the importance of learning patch to word attention for Document based embeddings in Table 3a. We see that while only training I2DGlobal can learn a competitive ZSL model, it signi\ufb01cantly improves and achieves SOTA performance with the introduction of our novel I2D Attention module in I2DFormer. We see a 7 \fModel AWA2 CUB FLO I2DGlobal 69.4 37.2 37.2 I2DGlobal + FILIP[64] 67.3 35.7 38.3 ViLBERT[25] 75.0 29.9 21.3 I2DFormer 76.4 45.4 40.0 (a) Input Embedding AWA2 CUB FLO LongFormer[4] 51.5 34.3 21.8 MPNet[49] 65.5 36.1 28.3 Word2Vec[28] 74.0 43.8 37.8 GloVe[35] 76.4 45.4 40.0 (b) Table 3: a) Ablation over I2DFormer. The proposed I2DGlobal module greatly bene\ufb01ts from the addition of I2D Attention to achieve SOTA performance. Comparing against FILIP and VilBERT cross-modal attention, we see that I2D Attention achieves SOTA. b) Ablating over input embeddings for our Document Transformer we see that older models like Word2Vec and GloVe serve as better input representation than modern Transformer-based language models. relative 14%, 16%, and 8% improvement over I2DGlobal. This validates our hypothesis that the patch to word attention distills its knowledge to the global I2DEmb, improving its performance. In the same table, we also ablate over 2 competing cross-modal attention modules. FILIP [64] is a recent method that proposes to associate each image patch to its most attended word. We see that this hurts the performance when using noisy Documents. ViLBERT [25] proposes a cross-modal attention module which is paired with a self-attention block [50] to learn an image embedding. We see that while this improves the performance over I2DGlobal on AWA2, it leads to worse performance on our \ufb01ne-grained datasets CUB and FLO potentially due to the bigger model requiring more training data. Our I2D Attention outperforms both these baselines and achieves SOTA performance. What kind of input text representation works best for I2DFormer? We ablate over several pretrained word/ token representations to be used as an input to our Document Transformer in Table 3b and note that GloVe [35] achieves the best result. We observe that the two Transformer based language models LongFormer [4] and MPNet [49] perform much worse than older baselines Word2Vec [28] and GloVe [35]. We conjecture that this is due to the Document Transformer having limited text data for the seen classes while training. Transformer-based models generate different word features for the same word with self-attention [50]. Documents of unseen classes use the same and additional vocabulary in new sentences causing a distribution shift in their input representation. 4.4 Qualitative Results Document Transformer attention for I2DEmb. We look at the learned attention over documents of unseen classes in Figure 2a and plot the top 8 most attended words across the Document Transformer attention heads for I2DEmb. On AWA2, we see that class name is complemented with human-like labelled attributes for these classes such as the color of the animal, type of the feet, and habitat etc. For the \ufb01ne-grained datasets, CUB and FLO, we see that for similar classes like the two warblers, the model learns similar attributes like \u201cruby-crowned\u201d as well as discriminating \u201ctiger stripes\u201d vs \u201cchestnut patterns\u201d. We con\ufb01rm our hypothesis that a learned document embedding will focus on discriminating properties of the class from the noisy document. Visualizing Image to Document attention as the row of the attention matrix in Figure 2b we see that the top 3 words contributing to the fp for these patch are visually grounded in the image. I2D Attention is able to develop this localization in the image for unseen classes from the noisy documents without any patch, word associated ground truth. We further see that the class name can repeatedly be the most important cue for the model in multiple sentences of the document for some patches. Visualizing Document word to Image attention as the column of the attention matrix in Figure 3, we see the impressive localization ability of I2DFormer for the top attended words in I2DEmb. We see that the model is able to localize the unseen classes horse and giraffe in the image despite never observing them while training. The discriminating properties like the hoofed legs are also localized in the image. For CUB, we see that between the two very similar images of two unseen classes, the model identi\ufb01es the yellow bottom as an important property from the two different documents of the ground truth class. However, the model is further able to identify the discriminative tiger stripes of the Cape May Warbler to differentiate it from the Tropical Kingbird which has gray-green feathers leading to correct classi\ufb01cation. Finally, on FLO, the localization ability of our model remains consistent where the Peruvian lily is identi\ufb01ed by localizing it as a Lily and identifying its 8 \fClassname Top attended words for I2DEmb AWA2 Blue Whale polar, whale, enormous, water, greyish, massive, blue,migrating Sheep grass, sleeker, wooly, stocky, rams, horns, sheep, hooves Seal frigid, fur, hind, cetaceans, pinnipeds, seal, ocean Giraffe markings, hoofed, giraffe, enormous, mammals, reddish, woodlands, leaves CUB Green Violetear bronzy, bolivia, nicaragua, green chest, canopy, colibri, shining, glittering Tropical Kingbird gray-headed, kingbird, green, venezuela, whitish, \ufb02ycatcher, gray-green feathers, plumage Cape May Warbler yellowish, breast, short-trailed, green, tiger stripes, olive, ruby-crowned, cape Chestnutsided Warbler chestnut, markings, wingbars, crown, green, ruby-crowned, warbler, oak FLO Pink Primrose buttercup, ranunculus, stigmas, wild\ufb02ower, primrose, four-petaled, evening blooms, amapola, Globe Thistle weed, daisy, wooly, thistles, \ufb02orets, sharp toothed, wrinkled, asteraceae Peruvian Lily lily, tuber, stripped, curving petals, \ufb02ecked, alstroemeria, streaked, resupinate Tiger lily lily, tiger, bulblets, capsules, lilium, tigrinum, pollinated, england (a) Base, Vegetation,  Leaves Eyeing, Coniferous,  Head Understory, Warbler,  Necklaced Giraffe, Woodland,  Mammal Giraffe, Giraffe,  Giraffe Legs, Woodland,  approaching Asteraceae, Flower,  Floret Marigold,  Flowerhead, Floret Daisy, Capitulum,  Asteracea (b) Figure 2: a) Top attended words for I2DEmb for unseen classes in the Document Transformer consist of discriminative properties available in the document. b) Visualizing Image patch to Word attention, we see that the top 3 most important words are visually grounded in the image. Legs Horse Horse Hoofed Giraffe Giraffe Leaves White Globe Thistle Sharp Toothed Soft Wrinkled Cape May Warbler Underpart Yellowish Tiger Stripes Tropical Kingbird Yellow bottom Whiteish Gray-green feathers Peruvian lily Lily Stripped Curving Petals Figure 3: Visualizing Word to Image Attention in the most attended document words for I2DEmb, we see that our model I2DFormer has learned to localize them in the image without any paired patch-word supervision. This learned attention differentiates the two similar birds in the second row by identifying and localizing tiger stripes and gray-green as discriminative properties. stripped and curved petals. Similarly for Globe Thistle, the model is able to differentiate the sharp teeth, soft and wrinkled parts of the \ufb02ower. The prevalence of these words as top attended words in the document transformer and their impressive localization veri\ufb01es our hypothesis that the attention module distills its knowledge to the CLS head. A model that does not learn patch to word attention can miss these properties if they are not deemed important among the seen classes. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2102.01987v3",
+                    "title": "Learning Graph Embeddings for Compositional Zero-shot Learning",
+                    "abstract": "In compositional zero-shot learning, the goal is to recognize unseen\ncompositions (e.g. old dog) of observed visual primitives states (e.g. old,\ncute) and objects (e.g. car, dog) in the training set. This is challenging\nbecause the same state can for example alter the visual appearance of a dog\ndrastically differently from a car. As a solution, we propose a novel graph\nformulation called Compositional Graph Embedding (CGE) that learns image\nfeatures, compositional classifiers, and latent representations of visual\nprimitives in an end-to-end manner. The key to our approach is exploiting the\ndependency between states, objects, and their compositions within a graph\nstructure to enforce the relevant knowledge transfer from seen to unseen\ncompositions. By learning a joint compatibility that encodes semantics between\nconcepts, our model allows for generalization to unseen compositions without\nrelying on an external knowledge base like WordNet. We show that in the\nchallenging generalized compositional zero-shot setting our CGE significantly\noutperforms the state of the art on MIT-States and UT-Zappos. We also propose a\nnew benchmark for this task based on the recent GQA dataset. Code is available\nat: https://github.com/ExplainableML/czsl",
+                    "authors": "Muhammad Ferjad Naeem, Yongqin Xian, Federico Tombari, Zeynep Akata",
+                    "published": "2021-02-03",
+                    "updated": "2021-05-03",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction A \u201cblack swan\u201d was ironically used as a metaphor in the 16th century for an unlikely event because the western world had only seen white swans. Yet when the European settlers observed a black swan for the \ufb01rst time in Australia in 1697, they immediately knew what it was. This is because humans posses the ability to compose their knowledge of known entities to generalize to novel concepts. Since visual concepts follow a long tailed distribution [44, 49], it is not possible to gather supervision for all concepts. Therefore, recognizing shared and discriminative properties of objects and reasoning about their various states has evolved as an essential part of human intelligence. Once familiar with the semantic meaning of these concepts, we can recognize unseen compositions of them without any supervision. While there is a certain degree old cute cat car dog old  dog cute  dog old  car cute  cat Visual Primitives Visual Primitives (States and Objects) cute old cat car dog Compositional Space cute  cat cute  dog old  car old  dog old dog cute cat cute dog old car Images ? Figure 1: We aim to build a classi\ufb01er for a novel state of a known object (e.g. old dog) given the knowledge of the shared primitives state and object in the training set. of compositionality in modern vision systems, e.g. feature sharing, most models are not compositional in the classi\ufb01er space and treat every class as an independent entity requiring training for any new concept. In this work, we study the state-object compositionality problem also known as Compositional Zero-Shot Learning (CZSL)[34]. The goal is to learn the compositionality of observed objects and their states as visual primitives to generalize to novel compositions of them as shown in \ufb01gure 1. Some notable existing works in this \ufb01eld include learning a transformation network on top of individual classi\ufb01ers [34], treating states as linear transformations of object vectors [35], learning modular networks conditioned on compositional classes [40] and learning object embeddings that are symmetric under different states [28]. However, these works treat each state-object composition independently, ignoring the rich dependency structure of different states, objects and their compositions. For example, learning the composition old dog is not only dependent on the state old and object dog, but also can be supported by other compositions like cute dog, old car, etc. We argue that such dependency structure provides a strong regularization which allows the network to better generalize to novel compositions. We therefore propose to exploit this dependency relationship by constructing a compositional graph to learn embeddings that are globally consistent. Our contributions are as follows: (1) We introduce a novel graph formulation named Compositional Graph Embedding (CGE) to model the dependency relationship of visual primitives and compositional classes. This graph can arXiv:2102.01987v3  [cs.CV]  3 May 2021 \fbe created independently of an external knowledge base like WordNet [32]. (2) Observing that visual primitives are dependent on each other and their compositional classes (\ufb01gure 1), we propose a multimodal compatibility learning framework that learns to embed related states, objects and their compositions close to each other and far away from the unrelated ones. (3) We propose a new benchmark called CGQA for the task of CZSL. This dataset is curated from the recent GQA[15] dataset with diverse compositional classes and clean annotations compared to datasets used in the community. (4) Our model signi\ufb01cantly improves the state of the art on all the metrics on MIT-States, UT-Zappos and CGQA datasets. 2. Related work Compositionality can loosely be de\ufb01ned as the ability to decompose an observation into its primitives. These primitives can then be used for complex reasoning. One of the earliest attempts in computer vision in this direction can be traced to Hoffman [14] and Biederman [4] who theorized that visual systems can mimic compositionality by decomposing objects to their parts. Compositionality at a fundamental level is already included in modern vision systems. Convolutional Neural Networks (CNN) have been shown to exploit compositionality by learning a hierarchy of features[58, 25]. Transfer learning[6, 8, 10, 38] and few-shot learning[12, 41, 30] exploit the compositionality of pretrained features to generalize to data constraint environments. Visual scene understanding[18, 9, 17, 29] aims to understand the compositionality of concepts in a scene. Nevertheless, these approaches still requires collecting data for new classes. Zero-Shot Learning aims at recognizing novel classes that are not observed during training [24]. This is accomplished by using side information that describes novel classes e.g. attributes [24], text descriptions [42] or word embeddings [45]. Some notable approaches include learning a compatibility function between image and class embeddings [1, 59] and learning to generate image features for novel classes [53, 60]. Graph convolutional networks (GCN) [21, 47, 19] have shown to be promising for zero-shot learning. Wang et al. [47] propose to directly regress the classi\ufb01er weights of novel classes with a GCN operated on an external knowledge graph (WordNet [32]). Kampffmeyer et al.[19] improve this formulation by introducing a dense graph to learn a shallow GCN as a remedy for the laplacian smoothing problem [27]. Graph Convolutional Networks are a special type of neural networks that exploit the dependency structure of data (nodes) de\ufb01ned in a graph. Current methods [21] are limited by the network depth due to over smoothing at deeper layers of the network. The extreme case of this can cause all nodes to converge to the same value [27]. Several works have tried to remedy this by dense skip connections [54, 26], randomly dropping edges [43] and applying a linear combination of neighbor features [50, 23, 22]. A recent work in this direction from Chen et al.[33] combines residual connections with identity mapping. Compositional zero-shot learning stands at the intersection of compositionality and zero-shot learning and focuses on state and object relations. We aim to learn the compositionality of objects and their states from the training set and are tasked with generalizing to unseen combination of these primitives. Approaches in this direction can be divided into two groups. The \ufb01rst group is directly inspired by [14, 4]. Some notable methods including learning a transformation upon individual classi\ufb01ers of states and objects [34], modeling each state as a linear transformation of objects [35], learning a hierarchical decomposition and composition of visual primitives[55] and modeling objects to be symmetric under attribute transformations[28]. An alternate line of works argues that compositionality requires learning a joint compatibility function with respect to the image, the state and the object[2, 40, 48]. This is achieved by learning a modular networks conditioned on each composition [40, 48] that can be \u201crewired\u201d for a new compositions. Finally a recent work from Atzmon et al. [2] argue that achieving generalization in CZSL requires learning the causality of visual transformation through a causal graph where the latent representation of primitives are independent of each other. Our proposed method lies at the intersection of several discussed approaches. We learn a joint compatibility function similar to [2, 40, 48] and utilize a GCN similar to [47, 19]. However, our approach exploits the dependency structure between states, objects and compositons which has been overlooked by previous CZSL approaches [2, 40, 48]. Instead of using a prede\ufb01ned knowledge graph like WordNet [32] to regress pretrained classi\ufb01ers of the seen classes [47, 19], we propose a novel way to build a compositional graph and learn classi\ufb01ers for all classes in an end-to-end manner. In contrast to Atzmon et al.[2] we explicitly promote the dependency between all primitives and their compositions in our graph. This allows us to learn embeddings that are consistent with the whole graph. Finally, unlike all existing methods [34, 35, 2, 40, 48, 55], we do not rely on a \ufb01xed image feature extractor and train our pipeline end-to-end. 3. Approach We consider the image classi\ufb01cation task where each image is associated with a label that is composed of a state (e.g. cute) and an object (e.g. dog). The goal of compositional zero-shot learning (CZSL) [34] is to recognize the compositional labels that are not observed during \fNovel Input Graph Output Graph GCN cute old Images cat car dog cute cat old car cute dog old dog ? cute old cat car cute cat old car cute dog old dog Input Graph Output Graph dog GCN Compositional Space old dog cute cat cute dog old car Compositional Space Images old dog cute cat cute dog old car ? cute old cat car cute cat old car cute dog old dog dog ? cute old cat car cute cat old car cute dog old dog dog cute  cat cute  dog old  car old  dog Figure 2: Compositional Graph Embed (CGE) learns a globally consistent joint embedding space between image features and classes of seen and unseen compositions from a graph. In our novel graph formulation, nodes are connected if a dependency exists in form of a compositional label e.g. old, car and old car. We backpropagate the classi\ufb01cation loss through the seen compositional nodes to the GCN G and the feature extractor F. Hence, the representation of e.g. the dog is compatible with its different states and the representation of old dog aggregates the knowledge from old, dog, cute dog, old car etc. training. This is particularly challenging as the states significantly change the visual appearance of an object hindering the performance of the classi\ufb01ers. We propose a novel formulation to the problem, namely Compositional Graph Embedding (CGE), which constructs a compositional graph and adopts a graph convolutional network to learn the dependency structure between labels. An overview of our approach is shown in Figure 2. It builds on the compatibility learning framework that learns a classagnostic scoring function between an image and a compositional label. The input image is encoded with an image feature extractor F, while the classi\ufb01er weights for the compositional label are learned by a composition function G. The key insight of our approach is that leveraging the dependency relationship between states, objects and their compositions is bene\ufb01cial for recognizing unseen compositions. 3.1. Compatibility Learning Framework for CZSL Task formulation. We formalize the CZSL task as follows. Let T = {(x, y)|x \u2208X, y \u2208Ys} where T stands for the training set, x denotes an image in the RGB image space X and y is its label belonging to one of the seen labels Ys. Each label is a tuple y = (s, o) of a state s \u2208S and an object o \u2208O with S and O being the set of states and objects respectively. The task of CZSL is to predict a set of novel labels Yn that consists of novel compositions of states S and objects O i.e., Ys \u2229Yn = \u2205. Following [40, 52], we study this problem in the generalized compositional zeroshot setting where the test set includes images from both seen and novel compositional labels Y = Ys \u222aYn. Compatibility function. Learning state and object classi\ufb01ers separately is prone to over\ufb01t to labels observed during training because states and objects are not independent e.g. the appearance of the state sliced varies signi\ufb01cantly with the object (e.g. apple or bread). Therefore, we chose to model them jointly by learning a compatibility function f : X \u00d7 S \u00d7 O \u2212 \u2192R that captures the compatibility score between an image, a state and an object. Given a speci\ufb01c input image x, we predict its label y = (s, o) by searching the state and object composition that yields the highest compatibility score: f(x, s, o) = F(x; W) \u00b7 G(s, o; \u0398) (1) where F(x; W) \u2208Rd is the image feature extracted from a pretrained feature extractor, G(s, o; \u0398) \u2208Rd is a function that outputs the label embedding of the state-object pair (s, o), (W,\u0398) are respectively the learnable parameters of F and G, and (\u00b7) is the dot product. The compatibility function assigns high scores to the correct triplets i.e., image x and its label (s, o), and low scores to the incorrect ones. The label embedding can be also interpreted as the classi\ufb01er weights for the label (s, o) and we use the two terms interchangeably. Our compatibility learning framework is closely related to [34, 40]. LabelEmbed [34] parameterizes the compositional embedding function with a multi-layer perceptron and computes the compositions from the word embeddings (e.g. word2vec [31]) of states and objects, while TMN [40] adopts a modular network as the image feature extractor and a gating network as the compositional embedding function. We argue that there exists a complex dependency structure between states, objects and their compositions and learning this dependency structure is crucial. To this end, we propose to integrate the compositional embedding function G as a graph convolutional neural network (GCN) which adds an inductive bias to the inherent structure between states, objects, and their combination de\ufb01ned by our compositional graph introduced next. \f3.2. Compositional Graph Embedding (CGE) We propose the Compositional Graph Embedding (CGE) framework integrating the Graph Convolutional Networks (GCN) [21] to the compositional embedding function G(s, o) that learns the label embedding for each compositional label y = (s, o) \u2208Y in an end to end manner. The GCN network exploits the dependency structure in a prede\ufb01ned compositional graph from states, objects and their compositions (including both seen and unseen labels). In the following, we \ufb01rst de\ufb01ne the compositional graph, then introduce the node features and \ufb01nally explain how to learn a GCN for the CZSL task. Compositional graph. Our graph consists of K = |S| + |O| + |Y| nodes that represent states S, objects O and compositional labels Y. Two nodes are connected if they are related. The key insight of our graph is that each compositional label y = (s, o) \u2208Y de\ufb01nes a dependency relationship between the state s, object o and their composition y. To this end, we build the edges of the graph by connecting (s, o), (s, y) and (o, y) for every y = (s, o) \u2208Y. In addition, each node is also connected to itself. Note that the edges in our graph are unweighted and undirected, leading to a symmetric adjacency matrix L \u2208RK\u00d7K where element Lij = 1 if there is a connection between nodes i and j otherwise Lij = 0. Despite its simplicity, we \ufb01nd that our compositional graph provides the accurate dependency structure to recognize unseen compositional labels. Node features. GCN [21, 33] operates on node features in a neighborhood de\ufb01ned by the graph. Therefore, after obtaining the compositional graph, we need to represent each node with a proper feature embedding. We chose to use the word embeddings [31, 5] pretrained on a large text corpus e.g. Wikipedia, because they capture rich semantic similarities among words i.e., dog is closer to cat than to car in the word embedding space. Speci\ufb01cally, every state or object node in the compositional graph is represented by the word embedding associated to its corresponding state or object name. We compute the node features of the the compositional label (e.g. cute dog) by averaging the word embeddings of the corresponding state (e.g. cute) and object (e.g. dog) names. As indicated in [31], by adding word embeddings we achieve compositionality in the semantic space. We represent the input node features with a matrix E \u2208RK\u00d7P where K is the total number of nodes and each row denotes the P-dim feature of a graph node. Graph convolutional network for CZSL. GCN [21] is an ef\ufb01cient multi-layer network to learn new feature representation of nodes for a downstream task that are consistent with the graph structure. Here, we apply the GCN to tackle the CZSL task by directly predicting the compositional label embeddings. The input of our GCN consists of the compositional graph, represented by the adjencency matrix L \u2208RK\u00d7K and the node feature matrix E \u2208RK\u00d7P . Speci\ufb01cally, each GCN layer computes the hidden representation of each node by convolving over neighbor nodes using a simple propagation rule [21] also known as a spectral convolution, H(l+1) = \u03c3(D\u22121LH(l)\u0398(l)) (2) where \u03c3 represents the non-linearity activation function ReLU, H(l) \u2208RK\u00d7U denotes the hidden representations in the lth layer with H(0) = E and \u0398 \u2208RU\u00d7V is the trainable weight matrix with V learnable \ufb01lters operating over U features of H(l). D \u2208RK\u00d7K is a diagonal node degree matrix which normalizes rows in L to preserve the scale of the feature vectors. By stacking multiple such layers, the GCN propagates the information through the graph to obtain better node embeddings for both the seen and unseen compositional labels. For example, our GCN allows an unseen compositional label e.g. old dog to aggregate information from its neighbor nodes e.g. old, dog, cute dog, and old car that are observed (see Figure 2). Objective. As the objective of the GCN is to predict the classi\ufb01er weights of the compositional labels, the node embedding of the output layer in the GCN has the same dimentionality as the image feature F(x). This indicates that our compositional embedding function becomes G(s, o) = H(N) y where H(N) is the output node embedding matrix and H(N) y denotes the row corresponding to the compositional label y = (s, o). We then optimize the following crossentropy loss to jointly learn the image feature extractor and GCN in an end-to-end manner, min W,\u0398 1 |T | |T | X i=1 \u2212log( exp f(xi, si, oi) P j\u2208Ys exp f(xi, sj, oj)) (3) where f is the compatibility function de\ufb01ned in Equation 1, y = (si, oi) is the ground truth label of image xi, label y\u2032 = (sj, oj) denotes any seen compositional class, W and \u0398 are the learnable parameters of the feature extractor and the GCN respectively. Intuitively, the cross-entropy loss enables the compatibility function to assign the high scores for correct input triplets. Inference. At test time, given an input image x, we derive a prediction by searching the compositional label that yields the highest compatibility score, arg max y=(s,o)\u2208Y f(x, s, o). (4) It is worth noting that our model works in the challenging generalized CZSL setting [40], where both seen and unseen compositional classes (i.e. Y = Ys \u222aYn) are predicted. Discussion. To the best of our knowledge, our Compositional Graph Embedding (CGE) is the \ufb01rst end-to-end learning method that jointly optimizes the feature extractor F \fTraining Validation Test Dataset s o sp i sp up i sp up i MIT-States[16] 115 245 1262 30k 300 300 10k 400 400 13k UT-Zappos[56] 16 12 83 23k 15 15 3k 18 18 3k C-GQA (Ours) 453 870 6963 26k 1173 1368 7k 1022 1047 5k Table 1: Dataset statistics for CZSL: We use three datasets to benchmark our method against the baselines. C-GQA (ours): our proposed dataset splits from Stanford GQA dataset [15]. (s: # states, o: # objects, sp: # seen compositions, up: # unseen compositions, i: # images) and the compositional embedding function G for the task of compositional zero-shot learning. Compared to prior CZSL works [40, 28, 34, 35] our CGE does not over\ufb01t while optimizing the CNN backbone of F (see supplementary) as it is regularized by the compositional graph that de\ufb01nes the dependency relationship between classes making the end-to-end training bene\ufb01cial. Compared to previous GCN work [47, 19] that utilizes GCNs to regress the \ufb01xed classi\ufb01er weights to learn classi\ufb01ers of novel classes, we directly use image information to learn classi\ufb01ers for both seen and novel classes. Moreover, while [47, 19] rely on a known knowledge graph like WordNet[32] describing the relation of novel classes to seen classes, our CGE cannot rely on existing knowledge graphs like WordNet[32] because they do not cover compositional labels. Therefore, we propose to construct the graph by exploiting the dependency relationship de\ufb01ned in the compositional classes. We \ufb01nd that propagating information from seen to unseen labels through this graph is crucial for boosting the CZSL performance. 4. Experiments After introducing our experimental setup, we compare our results with the state of the art, ablate over our design choices and present some qualitative results. Datasets. We perform our experiments on three datasets (see detailed statistics in Table 1). MIT-States[16] consists of natural objects in different states collected using an older search engine with limited human annotation leading to signi\ufb01cant label noise [2]. UT-Zappos[56, 57] consists of images of a shoes catalogue which is arguably not entirely compositional as states like Faux leather vs Leather are material differences not always observable as visual transformations. We use the GCZSL splits from [40]. To address the limitations of these two datasets,we propose a split built on top of Stanford GQA dataset [15] originally proposed for VQA and name it Compositional GQA (C-GQA) dataset (see supplementary for the details). CGQA contains over 9.5k compositional labels making it the most extensive dataset for CZSL. With cleaner labels and a larger label space, our hope is that this dataset will inspire further research on the topic. Figure 4 shows some samples from the three datasets. Metrics. As the models in zero-shot learning problems are trained only on seen Ys labels (compositions), there is an inherent bias against the unseen Yn labels. As pointed out by [7, 40], the model thus needs to be calibrated by adding a scalar bias to the activations of the novel compositions to \ufb01nd the best operating point and evaluate the generalized CZSL performance [40] for a more realistic setting. We adopt the evaluation protocol of [40] and report the Area Under the Curve (AUC) (in %) between the accuracy on seen and unseen compositions at different operating points with respect to the bias. The best unseen accuracy is calculated when the bias term is large leading to predicting only the unseen labels, also known as zero-shot performance. In addition, the best seen (base class) performance is calculated when the bias term is negative leading to predicting only the seen labels. As a balance between the two, we also report the best harmonic mean. To emphasize that this is different from the traditional zero-shot learning evaluation, we add the term \u201cbest\u201d in our results. Finally, we report the state and object accuracy on the novel labels to show the improvement in classifying the visual primitives. We emphasize that the dataset splits we propose for C-GQA and use from [40] for MIT-States and UT-Zappos do no not violate the zero-shot assumption as results are ablated on the validation set. Some works in CZSL use older splits that lack a validation set and thus use indirect full label supervision[52] by ablating over the test set. We therefore advice future works to rely on the new splits. Training details. To be consistent with the state of the art, we use a ResNet18 [13] backbone pretrained on ImageNet as the image feature extractor F. For a fair comparison with the models that use a \ufb01xed feature extractor, we introduce a simpli\ufb01cation of our method named CGEff. We learn a 3 layer fully-connected (FC) network with ReLU[36], LayerNorm[3] and Dropout[46] while keeping the feature extractor \ufb01xed for this baseline. We use a shallow 2-layer GCN with a hidden dimension of 4096 as G (detailed ablation on this is presented in section 4.2). On MIT-States, we initialize our word embeddings with a concatenation of pretrained fasttext[5] and word2vec models[31] similar to [51]. On UT-Zappos and C-GQA, we initialize the word embeddings with word2vec(ablation reported in supplementary). We use Adam Optimizer[20] with a learning rate of 5e\u22126 for F and a learning rate of 5e\u22125 for G. We implement our method in PyTorch[37] and train on a Nvidia V100 GPU. For state-of-the-art comparisons, we use the authors\u2019 implementations where available. The code for our method and the new dataset C-GQA will be released upon acceptance. \fMIT-States UT-Zap50K C-GQA Method AUC Best AUC Best AUC Best Val Test HM Seen Unseen s o Val Test HM Seen Unseen s o Val Test HM Seen Unseen s o AttOp[35] 2.5 1.6 9.9 14.3 17.4 21.1 23.6 21.5 25.9 40.8 59.8 54.2 38.9 69.6 0.9 0.3 2.9 11.8 3.9 8.3 12.5 LE+[34] 3.0 2.0 10.7 15.0 20.1 23.5 26.3 26.4 25.7 41.0 53.0 61.9 41.2 69.2 1.2 0.6 5.3 16.1 5.0 7.4 15.6 TMN[40] 3.5 2.9 13.0 20.2 20.1 23.3 26.5 36.8 29.3 45.0 58.7 60.0 40.8 69.9 2.2 1.1 7.7 21.6 6.3 9.7 20.5 SymNet[28] 4.3 3.0 16.1 24.4 25.2 26.3 28.3 25.9 23.9 39.2 53.3 57.9 40.5 71.2 3.3 1.8 9.8 25.2 9.2 14.5 20.2 CGEff (ours) 6.8 5.1 17.2 28.7 25.3 27.9 32.0 38.7 26.4 41.2 56.8 63.6 45.0 73.9 3.6 2.5 11.9 27.5 11.7 12.7 26.9 CGE (ours) 8.6 6.5 21.4 32.8 28.0 30.1 34.7 43.2 33.5 60.5 64.5 71.5 48.7 76.2 5.0 3.6 14.5 31.4 14.0 15.2 30.4 Table 2: Comparison with the state of the art: We compare our Compositional Graph Embed (CGE) with the state of the art on Validation and Test AUC (in %); best unseen, seen and harmomic mean (HM) accuracies (in %) as well as state (s) and object (o) prediction accuracies (in %) on widely used MIT-States and UT-Zappos datasets as well as our proposed C-GQA dataset. 4.1. Comparing with the State of the Art We compare our results with the state of the art in Table 2 and show that our Compositional Graph Embed(CGE) outperforms all previous methods by a large margin and establishes a new state of the art for Compositional Zero-shot Learning. Our detailed observations are as follows. Generalized CZSL performance. Our framework demonstrates robustness against the label noise on MIT-States noted previously in [2]. For the generalized CZSL task, our CGE achieves a test AUC of 6.5% which is an improvement of over 2\u00d7 compared to the last best 3.0% from SymNet. Similarly, as our method does not only improve results on seen labels but also unseen labels, it signi\ufb01cantly boosts the state of the art harmonic mean, i.e. 16.1% to 21.4%. When it comes to state and object prediction accuracy, we observe an improvement from 26.3% to 30.1% for states and 28.3% to 34.7% for objects. Although our results signi\ufb01cantly improve the state of the art on all metrics, the state and object accuracies are quite low, partially due to the label noise for this dataset. Similar observations are con\ufb01rmed on UT-Zappos, where we achieve a signi\ufb01cant improvement on the state of the art with an AUC of 33.5% compared to 29.3% from TMN. An interesting observation is that SymNet, i.e. the state of the art on MIT States, with an AUC of 23.9% does not achieve the best performance in the generalized CZSL setting on UT Zappos. We conjecture that this is because the state labels in this dataset are not entirely representing visual transformations, something this method was designed to exploit. In this dataset, our fully compositional model improves the best harmonic mean wrt the state of the art signi\ufb01cantly (45.0% with TMN vs 60.5% ours). Note that, this is due to a signi\ufb01cant accuracy boost achieved on unseen compositions (60.0% vs 71.5%). Finally on the proposed splits of the GQA dataset [15], i.e. C-GQA dataset, we achieve a test AUC of 3.6% outperforming the closest baseline by a 2\u00d7. Note that, since C-GQA has a compositional space of over 9.3k concepts, it is signi\ufb01cantly harder than MIT-States and UT-Zappos while being truly compositional and containing cleaner labels. The state and the object accuracies of our method are 15.2% and 30.4%, i.e. signi\ufb01cantly higher than the state of the art. However these results also indicate the general dif\ufb01culty of the task. Similarly, our best seen and best unseen accuracies (31.4% and 14.0%) indicate a large room for improvement on this dataset, which may encourage further research with our C-GQA dataset on the CZSL task. We also make an interesting observation on all three datasets. While SymNet uses an object classi\ufb01er that is trained independently from the compositional pipeline, our method consistently outperforms it on object accuracy. We conjecture that this is because a compositional network sensitive to the information about the states is also a better object classi\ufb01er, since it disentangles what it means to be an object from the state it is observed in, preventing biases to properties like textures [11]. This insight can be an avenue for future improvement in object classi\ufb01cation. Impact of feature representations. To quantify the improvement of our graph formulation on the same feature representations as the state of the art, we also present results of our CGE with a \ufb01xed feature extractor (Resnet18), i.e. denoted by CGEff, in Table 2. We see that this version of our model also consistently outperforms the state of the art by a large margin on MIT-States and C-GQA while matching the performance on UT-Zappos. Especially on MITStates, the improvement over the state of the art is remarkable, i.e. 5.1% test AUC vs 3.0% test AUC with SymNet. In summary, this shows that our method bene\ufb01ts from both the knowledge propagation in the compositional graph and from learning better image representations. For a fair comparison, we also allowed the previous base\fConnections in Graph AUC Best HM a) Direct Word Embedding 5.9 19.4 b) (s,y) and (o,y), no self-loop on y 7.6 18.6 c) (s,y) and (o,y) 8.1 22.7 d) CGE: (s,y), (o,y) and (s,o) 8.6 23.3 e) Extra WordNet hierarchy on O 7.9 22.0 Table 3: Ablation over the graph connections validates the structure of our proposed graph on the validation set of MIT-States dataset. We start from directly using the word embeddings as classi\ufb01er weights to learning a globally consistent embedding from a GCN as the classi\ufb01er weights (s: states, o: objects, y: compositional labels). lines to train end-to-end with F. However, this results in a signi\ufb01cant performance drop indicating they are unable to jointly learn the feature extractor against the RGB space. To address this limitation, some works[53, 48] have proposed to use a generative network to learn the distribution of image features in zero-shot problems. We, on the other hand, don\u2019t need to rely on an expensive generative network and jointly learn the image representations and the compositional classi\ufb01ers in an end-to-end manner. 4.2. Ablation study In this section we ablate our CGE model with respect to the graph connections, the graph depth and graph convolution variants. Graph connections. We perform an ablation study with respect to the various connections in our compositional graph on the validation set of MIT-States and report results in Table 3. In the Direct Word Embedding variant, i.e. row (a) our label embedding function G is an average operation of state and object word embeddings. We see that, directly using word embedding of compositional labels as the classi\ufb01er weights leads to an AUC of 5.9. In row (b) we represent a graph with connections between states (s) to compositional labels (y) and objects to compositional labels (y) but remove the self connection for the compositional label. In this case, the \ufb01nal representation of compositional labels from the GCN only combines the hidden representations of states and objects leading to an AUC of 7.6. Row (c) represents the graph that has self connections from each compositional label in addition to the connections between states and compositional labels as well as objects and compositional labels as in row (b). We see that this variant achieves an AUC of 8.1 indicating that the hidden representation of compositional classes is bene\ufb01cial. Row (d) is our \ufb01nal model where we additionally incorporate the connections between states and objects in a pair to model the dependency between them. We observe 2 4 6 8 Number of Layers 7 8 9 AUC GCN Type GCN GCNII Figure 3: Graph convolution and depth: We compare the spectral convolution GCN[21] with the recent GCNII[33] that aims to address the over smoothing issue at increasing depth. We perform the comparison at various depths of the GCN network on the validation set of MIT-States. that learning a representation that is consistent with states, objects and the compositional labels increases the AUC from 8.1 to 8.6 validating our choice of connections in the graph. We again emphasize that in the absence of an existing knowledge graph for compositional relations, our simple but well designed graph structure is able to capture the dependencies between various concepts. While our \ufb01nal CGE does not employ an external knowledge graph, we can utilize an existing graph like WordNet [32] to get the hierarchy of the object classes similar to some baselines in zero-shot learning [47, 19]. Row (e) represents a model that exploits object hierarchy in addition to our compositional graph discussed earlier. This leads to additional 418 nodes to model the parent child relation of the objects. We see that this results in a slight performance drop with an AUC of 7.9 because this graph may not be compatible with the compositional relations. Graph architecture. We ablate over the architecture of the graph at various depths from 2-8 layers to quantify the degree of knowledge propagation needed to achieve best performance. From Figure 3 we observe that a shallow architecture at 2 layers achieves the best AUC of 8.6 outperforming the deeper con\ufb01guration. This is an established problem for the spectral graph convolution and is caused by laplacian smoothing across deeper layers[27]. To study if we are limited by a shallow representation, we use a more recent formulation of graph convolution named GCNII[33]. This method introduces a few key improvements like skip connections that remedy the laplacian smoothing problem. We see that while GCNII suffers less from the smoothing problem and maintains performance at deeper architectures, It only achieves an AUC of 7.2 for the best performing model. Since our graph is exploiting close relations between the states, objects and the compositions introduced by the dense connections for visual primitives, we are not held back by the shallow architecture. We advice future works to explore richer graphs that can facilitate deeper models. \fSmall Elephant Small Elephant Tiny Elephant Young Elephant Crushed Granite Crushed Granite Crushed Rock Crushed Stone Wet Cat Tiny Animal Wet Cat Young Cat Thawed Meat Damp Basement Wet Basement Wet Foam Rusty Gate Rusty Gate Weathered Steel Rusty Fence Satin Sandals Satin Sandals Haircalf Heels Satin Heels Synthetic Midcalf Synthetic Midcalf Leather Kneehigh Synthetic Kneehigh Canvas Loafers Canvas Loafers Canvas Flats Cotton Flats Suede Midcalf Synthetic Ankle Suede Midcalf Sheepskin Ankle Leather Slippers Canvas Flats Leather Flats Canvas Loafers Baseball Field Baseball Field Brown Field Large Field White Bus Gray Bus White Bus Silver Bus Stripped Person Standing Person Orange Jacket Blue Shirt Red Coat Red Coat Red Jacket Maroon Jacket Covered Hill Covered Hill White Mountain Large Snow MIT-States UT-Zappos C-GQA (a) Retrieving compositional labels Old Car Broken Car Synthetic Ankle-Boot Leather Ankle-Boot Concrete Walkway Asphalt Walkway Mossy pond Puffy Pizza Seen Unseen Original Dataset Cross Dataset MIT-States UT-Zappos C-GQA MIT-States MIT-States C-GQA C-GQA Dataset Retrieval Cross Dataset Retrieval (b) Retrieving images Figure 4: Qualitative results. Left: We show the top-3 predictions of our model for some examples. We observe from the \ufb01rst four columns that all the predictions of the model are meaningful, but the model is only incentivized when it matches the label. The task of CZSL is a multi label one and future datasets need to account for this. The last column shows some examples of suboptimal labels and wrong predictions. Right: We show good candidates for retrival on all three dataset and then we perform cross-dataset retrieval for a unseen composition across C-GQA and MIT-States. 4.3. Qualitative results We show some qualitative results for the novel compositions with their top-3 predictions in Figure 4 (left). The \ufb01rst three columns present some instances where the top prediction matches the label. For MIT-States and C-GQA, we notice that the remaining two answers of the model contain information visually present in the image but not in the label highlighting a limitation of current CZSL benchmarks. Different groups of states like color, age, material etc. can represent different information for the same object and are thus not mutually exclusive. For example in column 4, row 1 the image of the cat consist of a size, surface texture and age all present in the label space of the dataset and the output of the model. However the label for this image only contains its surface texture. This puts an upper limit on compositional class accuracy dependent on the number of groups associated with an object in the label space. Speci\ufb01cally, column 4 of Figure 4 (left) counts as a wrong prediction but all of the top 3 predictions represent correct visual information for MIT-States and C-GQA. Unless the model learns the annotator bias, it can not achieve a perfect accuracy. Finally in column 5, we show some instances of sub-optimal and wrong labels. Speci\ufb01cally, the image in row 1 is entirely missing the thawed meat represented in the label, the image in row 2 can not suf\ufb01ciently communicate the material information while the label in row 3 does not contain the dominant information in the image. In Figure 4 (right) we \ufb01rst show image retrieval results from seen and unseen compositions. We can see that for all three datasets our method returns the correct top images for the query. We then perform cross-dataset retrieval between MIT-States and C-GQA for an unseen composition. We show a representative image from the original dataset and the top-3 retrievals from the cross dataset. While the datasets have a distribution shift between them, we see that retrievals are still meaningful. On MIT-States 2/3 retrieved images match a Mossy pond while the 3rd image is a grass \ufb01eld confused with the query. Similar trend is observed for the model trained on C-GQA for retrieval of a puffy pizza. The model confuses the top retrieval with a casserole followed by two images of pizzas. Nevertheless, the cross dataset retrieval shows promise towards further generalization for future works. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2002.09797v2",
+                    "title": "Reliable Fidelity and Diversity Metrics for Generative Models",
+                    "abstract": "Devising indicative evaluation metrics for the image generation task remains\nan open problem. The most widely used metric for measuring the similarity\nbetween real and generated images has been the Fr\\'echet Inception Distance\n(FID) score. Because it does not differentiate the fidelity and diversity\naspects of the generated images, recent papers have introduced variants of\nprecision and recall metrics to diagnose those properties separately. In this\npaper, we show that even the latest version of the precision and recall metrics\nare not reliable yet. For example, they fail to detect the match between two\nidentical distributions, they are not robust against outliers, and the\nevaluation hyperparameters are selected arbitrarily. We propose density and\ncoverage metrics that solve the above issues. We analytically and\nexperimentally show that density and coverage provide more interpretable and\nreliable signals for practitioners than the existing metrics. Code:\nhttps://github.com/clovaai/generative-evaluation-prdc.",
+                    "authors": "Muhammad Ferjad Naeem, Seong Joon Oh, Youngjung Uh, Yunjey Choi, Jaejun Yoo",
+                    "published": "2020-02-23",
+                    "updated": "2020-06-28",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.LG",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Assessing a generative model is dif\ufb01cult. Unlike the evaluation of discriminative models P(T|X) that is often easily done by measuring the prediction performances on a few labelled samples (Xi, Ti), generative models P(X) are assessed by measuring the discrepancy between the real {Xi} and generated (fake) {Yj} sets of high-dimensional data points. Adding to the complexity, there are more than one way of measuring distances between two distributions each with its own pros and cons. In fact, even human judgement based measures like Mean Opinion Scores (MOS) are not ideal, as practitioners have diverse opinions on what the \u201cideal\u201d generative model is (Borji, 2019). *Equal contribution 1Clova AI Research, Naver Corp. 2Technische Universit\u00a8 at M\u00a8 unchen, Germany 3 \u00b4 Ecole polytechnique f\u00b4 ed\u00b4 erale de Lausanne (EPFL), Switzerland. Correspondence to: Jaejun Yoo <jaejun.yoo88@gmail.com>. Proceedings of the 37 th International Conference on Machine Learning, Vienna, Austria, PMLR 119, 2020. Copyright 2020 by the author(s). Nonetheless, there must be some measurement of the quality of generative models for the progress of science. Several quantitative metrics have been proposed, albeit with their own set of trade-offs. For example, Fr\u00b4 echet Inception Distance (FID) score (Heusel et al., 2017), the most popular metric in image generation tasks, has empirically exhibited good agreements with human perceptual scores. However, FID summarises the comparison of two distributions into a single number, failing to separate two important aspects of the quality of generative models: \ufb01delity and diversity (Sajjadi et al., 2018). Fidelity refers to the degree to which the generated samples resemble the real ones. Diversity, on the other hand, measures whether the generated samples cover the full variability of the real samples. Recent papers (Sajjadi et al., 2018; Simon et al., 2019; Kynk\u00a8 a\u00a8 anniemi et al., 2019) have introduced precision and recall metrics as measures of \ufb01delity and diversity, respectively. Though precision and recall metrics have introduced the important perspectives in generative model evaluation, we show that they are not ready yet for practical use. We argue that necessary conditions for useful evaluation metrics are: (1) ability to detect identical real and fake distributions, (2) robustness to outlier samples, (3) responsiveness to mode dropping, and (4) the ease of hyperparameter selection in the evaluation algorithms. Unfortunately, even the most recent version of the precision and recall metrics (Kynk\u00a8 a\u00a8 anniemi et al., 2019) fail to meet the requirements. To address the practical concerns, we propose the density and coverage metrics. By introducing a simple yet carefully designed manifold estimation procedure, we not only make the \ufb01delity-diversity metrics empirically reliable but also theoretically analysable. We test our metric on generative adversarial networks, one of the most successful generative models in recent years. We then study the embedding algorithms for evaluating image generation algorithms. Embedding is an inevitable ingredient due to the high-dimensionality of images and the lack of semantics in the RGB space. Despite the importance, the embedding pipeline has been relatively less studied in the existing literature; evaluations of generated images mostly rely on the features from an ImageNet pretrained model (Salimans et al., 2016; Heusel et al., 2017; Sajjadi et al., 2018; Simon et al., 2019; Kynk\u00a8 a\u00a8 anniemi et al., arXiv:2002.09797v2  [cs.CV]  28 Jun 2020 \fReliable Fidelity and Diversity Metrics for Generative Models 2019; Deng et al., 2009). This sometimes limits the fair evaluation and provides a false sense of improvement, since the pre-trained models inevitably include the dataset bias (Torralba & Efros, 2011; Geirhos et al., 2019). We show that such pre-trained embeddings often exhibit unexpected behaviours as the target distribution moves away from the natural image domain. To exclude the dataset bias, we consider using randomly initialised CNN feature extractors (Ulyanov et al., 2018). We compare the evaluation metrics on MNIST and sound generation tasks using the random embeddings. We observe that random embeddings provide more macroscopic views on the distributional discrepancies. In particular, random embeddings provide more sensible evaluation results when the target data distribution is signi\ufb01cantly different from ImageNet statistics (e.g. MNIST and spectrograms). 2. Backgrounds Given a real distribution P(X) and a generative model Q(Y ), we assume that we can sample {Xi} and {Yj}, respectively. We need an algorithm to assess how likely the sets of samples are arising from the same distribution. When the involved distribution families are tractable and the full density functions can be easily estimated, statistical testing methods or distributional distance measures (e.g. Kullback-Leibler Divergence or Expected Likelihood) are viable. However, when the data P(X) are complex and high-dimensional (e.g. natural images), it becomes dif\ufb01cult to apply such measures naively (Theis et al., 2016). Because of the dif\ufb01culty, the evaluation of samples from generative models is still an actively researched topic. In this section, we provide an overview of existing approaches. We describe the most widely-used evaluation pipeline for image generative models (\u00a72.1), and then introduce the prior works on \ufb01delity and diversity measures (\u00a72.2). For a more extensive survey, see (Borji, 2019). 2.1. Evaluation pipeline It is dif\ufb01cult to conduct statistical analyses over complex and high-dimensional data X in their raw form (e.g. images). Thus, evaluation metrics for image generative models largely follow the following stages: (1) embed real and fake data ({Xi} and {Yj}) into a Euclidean space RD through a non-linear mapping f like CNN feature extractors, (2) construct real and fake distributions over RD with the embedded samples {f(Xi)} and {f(Yj)}, and (3) quantify the discrepancy between the two distributions. We describe each stage in the following paragraphs. Embeddings. It is often dif\ufb01cult to de\ufb01ne a sensible metric over the input space. For example, \u21132 distance over the image pixels \u2225Xi \u2212Yj\u22252 is misleading because two perceptually identical images may have great \u21132 distances (one-pixel translation) (Theis et al., 2016). To overcome this dif\ufb01culty, researchers have introduced ImageNet pre-trained CNN feature extractors as the embedding function f in many generative model evaluation metrics (Salimans et al., 2016; Heusel et al., 2017; Sajjadi et al., 2018; Kynk\u00a8 a\u00a8 anniemi et al., 2019) based on the reasoning that the \u21132 distance in the feature space \u2225f(Xi) \u2212f(Yj)\u22252 provide sensible proxies for the human perceptual metric (Zhang et al., 2018). Since we always use embedded samples for computing metrics, we write Xi and Yj for f(Xi) and f(Yj), respectively. In this work, we also adopt ImageNet pre-trained CNNs for the embedding, but we also criticise their use when the data distribution is distinct from the ImageNet distribution (\u00a74.2). We suggest randomly-initialised CNN feature extractors as an alternative in such cases (Ulyanov et al., 2018; Zhang et al., 2018). We show that for MNIST digits or sound spectrograms with large domain gaps from ImageNet, random embeddings provide more sensible evaluation measures. Building and comparing distributions. Given embedded samples {Xi} and {Yj}, many metrics conduct some form of (non-)parametric statistical estimation. Parzen window estimates (Bengio et al., 2013) approximate the likelihoods of the fake samples {Yj} by estimating the density P(X) with Gaussian kernels around the real samples {Xi}. On the parametric side, Inception scores (IS) (Salimans et al., 2016) estimate the multinomial distribution P(T|Yj) over the 1000 ImageNet classes for each sample image Yj and compares it against the estimated marginalised distribution P(T) with the KL divergence. Fr\u00b4 echet Inception Distance (FID) (Heusel et al., 2017) estimates the mean \u00b5 and covariance \u03a3 for X and Y assuming that they are multivariate Gaussians. The distance between the two Gaussians is computed by the Fr\u00b4 echet distance (Dowson & Landau, 1982), also known as the Wasserstein-2 distance (Vaserstein, 1969). FID has been reported to generally match with human judgements (Heusel et al., 2017); it has been the most popular metric for image generative models in the last couple of years (Borji, 2019). 2.2. Fidelity and diversity While single-value metrics like IS and FID have led interesting advances in the \ufb01eld by ranking generative models, they are not ideal for diagnostic purposes. One of the most important aspects of the generative models is the trade-off between \ufb01delity (how realistic each input is) and diversity (how well fake samples capture the variations in real samples). We introduce variants of the two-value metrics (precision and recall) that capture the two characteristics separately. Precision and recall. (Sajjadi et al., 2018) have reported the pathological case where two generative models have similar FID scores, while their qualitative \ufb01delity and diver\fReliable Fidelity and Diversity Metrics for Generative Models Fake samples Real samples Precision = = 100% 1 + 1 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1 Density = 2 + 1 + 1 + 1 + 1 2 + 2 + 2 + 2 + 2 = 60% k = 2 Real manifold Real outlier (a) Precision versus density. Fake manifold k = 1 Recall = = 100% 5 5 Coverage = = 20% 1 5 Real manifold k = 1 (b) Recall versus coverage. Figure 1. Overview of metrics. Two example scenarios for illustrating the advantage of density over precision and coverage over recall. Note that for recall versus coverage \ufb01gure, the real and fake samples are identical across left and right. sity results are different. For a better diagnosis of generative models, (Sajjadi et al., 2018) have thus proposed the precision and recall metrics based on the estimated supports of the real and fake distributions. Precision is de\ufb01ned as the portion of Q(Y ) that can be generated by P(X); recall is symmetrically de\ufb01ned as the portion of P(X) that can be generated by Q(Y ). While conceptually useful, they have multiple practical drawbacks. It assumes that the embedding space is uniformly dense, relies on the initialisation-sensitive k-means algorithm for support estimation, and produces an in\ufb01nite number of values as the metric. Improved precision and recall. (Kynk\u00a8 a\u00a8 anniemi et al., 2019) have proposed the improved precision and recall (P&R) that address the above drawbacks. The probability density functions are estimated via k-nearest neighbour distances, overcoming the uniform-density assumption and the reliance on the k-means. Our proposed metrics are based on P&R; we explain the full details of P&R here. P&R \ufb01rst constructs the \u201cmanifold\u201d for P(X) and Q(Y ) separately, the object is nearly identical to the probabilistic density function except that it does not sum to 1. Precision then measures the expected likelihood of fake samples against the real manifold and recall measures the expected likelihood of real samples against the fake manifold: precision := 1 M M X j=1 1Yj\u2208manifold(X1,\u00b7\u00b7\u00b7 ,XN) (1) recall := 1 N N X i=1 1Xi\u2208manifold(Y1,\u00b7\u00b7\u00b7 ,YM) (2) where N and M are the number of real and fake samples. 1(\u00b7) is the indicator function. Manifolds are de\ufb01ned manifold(X1, \u00b7 \u00b7 \u00b7 , XN) := N [ i=1 B(Xi, NNDk(Xi)) (3) where B(x, r) is the sphere in RD around x with radius r. NNDk(Xi) denotes the distance from Xi to the kth nearest neighbour among {Xi} excluding itself. Example computation of P&R is shown in Figure 1. There are similarities between the precision above and the Parzen window estimate (Bengio et al., 2013). If the manifolds are formed by superposition instead of union in Equation 3 and the spheres B(\u00b7, \u00b7) are replaced with Gaussians of \ufb01xed variances, then the manifold estimation coincides with the kernel density estimation. In this case, Equation 1 computes the expected likelihood of fake samples. 3. Density and Coverage We propose novel performance measures density and coverage (D&C) as practically usable measures that successfully remedy the problems with precision and recall. 3.1. Problems with improved precision and recall Practicality of the improved precision and recall (P&R) is still compromised due to their vulnerability to outliers and computational inef\ufb01ciency. Building the nearest neighbour manifolds (Equation 3) must be performed carefully because the spheres around each sample are not normalised according to the their radii or the relative density of samples in the neighbourhood. Consequently, the nearest neighbour manifolds generally overestimate the true manifold around the outliers, leading to undesired effects in practice. We explain the drawbacks at the conceptual level here; they will be quanti\ufb01ed in \u00a74. Precision. We \ufb01rst show a pathological case for precision in Figure 1a. Because of the real outlier sample, the manifold is overestimated. Generating many fake samples around the real outlier is enough to increase the precision measure. Recall. The nearest neighbour manifold is built upon the \fReliable Fidelity and Diversity Metrics for Generative Models fake samples in Equation 2. Since models tend to generate many unrealistic yet diverse samples, the fake manifold is often an overestimation of the actual fake distribution. The pathological example is shown in Figure 1b. While the fake samples are far from the modes in real samples, the recall measure gains points for real samples contained in the overestimated fake manifold. Another problem with relying on fake manifolds for the recall computation is that the manifold must be computed per model. For example, to generate the recall-vs-iteration curve for training diagnosis, the k-nearest neighbours for all fake samples must be computed (O(kM log M)) for every data point. 3.2. Density and coverage We remedy the issues with P&R above and propose new metrics: density and coverage (D&C). Density. Density improves upon the precision metric by \ufb01xing the overestimation of the manifold around real outliers. Precision counts the binary decision of whether the fake data Yj contained in any neighbourhood sphere {B(Xi, NNDk(Xi))}i of real samples (Equation 1). Density, instead, counts how many real-sample neighbourhood spheres contain Yj (Equation 4). The manifold is now formed by the superposition of the neighbourhood spheres {B(Xi, NNDk(Xi))}i, and a form of expected likelihood of fake samples is measured. In this sense, the density measure is at the midway between the precision metric and the Parzen window estimate. Density is de\ufb01ned as density := 1 kM M X j=1 N X i=1 1Yj\u2208B(Xi,NNDk(Xi)) (4) where k is for the k-nearest neighbourhoods. Through this modi\ufb01cation, density rewards samples in regions where real samples are densely packed, relaxing the vulnerability to outliers. For example, the problem of overestimating precision (100%) in Figure 1a is resolved using the density measure (60%). Note that unlike precision, density is not upper bounded by 1; it may be greater than 1 depending on the density of reals around the fakes. Coverage. Diversity, intuitively, shall be measured by the ratio of real samples that are covered by the fake samples. Coverage improves upon the recall metric to better quantify this by building the nearest neighbour manifolds around the real samples, instead of the fake samples, as they have less outliers. Moreover, the manifold can only be computed per dataset, instead of per model, reducing the heavy nearest neighbour computations in recall. Coverage is de\ufb01ned as coverage := 1 N N X i=1 1\u2203j s.t. Yj\u2208B(Xi,NNDk(Xi)). (5) It measures the fraction of real samples whose neighbourhoods contain at least one fake sample. Coverage is bounded between 0 and 1. 3.3. Analytic behaviour of density & coverage The simplest sanity check for an evaluation metric is whether the metric attains the best value when the intended criteria are met. For generative models, we examine if D&C attain 100% performances when the real and fake distribution are identical (P d = Q). We show that, unlike P&R, D&C yield an analytic expression for the expected values E[density] and E[coverage] for identical real and fake, and the values approach 100% as the sample sizes (N, M) and number of neighbourhoods k increase. This analysis further leads to a systematic algorithm for selecting the hyperparameters (k, N, M). Lack of analytic results for P&R. From Equation 1, the expected precision for identical real and fake is E[precision] = P[X0 \u2208manifold(X1, \u00b7 \u00b7 \u00b7 , XN)] (6) = P \" X0 \u2208 N [ i=1 B(Xi, NNDk(Xi)) # (7) = P \u0002 \u222aN i=1Ak i \u0003 (8) where Ak i is the event {\u2225X0 \u2212Xi\u2225< NNDk(Xi)}. Since (Ak i )i are not independent with complex dependence structures, a simple expression for Equation 8 does not exist. Same observation holds for E[recall]. Analytic derivations for D&C. We derive the expected values of D&C under the identical real and fake. Lemma 1. E[density] = 1. Proof. The expected density boils down to E[density] = 1 k N X i=1 P(Bk N) = N k P(Bk N) where Bk N is the event where \u2225Y \u2212X1\u2225is at most kth smallest among the random variables S := {\u2225X1 \u2212 X2\u2225, \u00b7 \u00b7 \u00b7 , \u2225X1 \u2212XN\u2225}. Since the random variables S\u222a{\u2225Y \u2212X1\u2225} are identical and independently distributed, any particular ranking of them are equally likely with probability 1/N!. Since the number of rankings of N values with one of them at a particular rank is (N \u22121)!, we have P(Bk N) = k \u00b7 (N \u22121)! \u00b7 1 N! = k N . \u25a0 Lemma 2. E[coverage] = 1 \u2212 (N \u22121) \u00b7 \u00b7 \u00b7 (N \u2212k) (M + N \u22121) \u00b7 \u00b7 \u00b7 (M + N \u2212k). (9) Moreover, as M = N \u2192\u221e, E[coverage] \u21921 \u2212 1 2k . \fReliable Fidelity and Diversity Metrics for Generative Models Precision Density Gaussian FFHQ Gaussian FFHQ Analytic 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.00 0.25 0.50 0.75 1.00 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.00 0.25 0.50 0.75 1.00 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.00 0.25 0.50 0.75 1.00 Recall Coverage Gaussian FFHQ Gaussian FFHQ Analytic 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 8 9 10 k 32000 16000 8000 4000 2000 1000 500 Samples 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2. Metrics under identical real and fake distributions. Results with Gaussian and FFHQ data are shown. For density and coverage, the expected values derived in \u00a73.3 are plotted. In each case, the values are shown with respect to the varying number of samples N = M (assume the same number of real and fake samples) and the nearest neighbours k. Proof. The expected coverage is estimated as E[coverage] = 1 N N X i=1 P(\u2203j s.t. Yj \u2208B(Xi, NNDk(Xi)) | {z } (\u22c6) ) = 1 \u2212P(\u2200j, Yj / \u2208B(X1, NNDk(X1)) | {z } (\u22c6\u22c6) ) using the fact that the events (\u22c6) is symmetrical with respect to i. The event (\u22c6\u22c6) can be re-written as min j \u2225Yj \u2212X1\u2225\u2265\u2225X1 \u2212X\u03b2\u2225for at most k \u03b2\u2019s. We write Z\u03b2 := \u2225X1 \u2212X\u03b2\u2225for \u03b2 \u2208{2, \u00b7 \u00b7 \u00b7 , N} and f Zj := \u2225X1 \u2212Yj\u2225for j \u2208{1, \u00b7 \u00b7 \u00b7 , M}. Then, the set {Z\u03b2}N \u03b2=2 \u222a{f Zj}M j=1 of N + M \u22121 random variables is independent and identically distributed. The probability of (\u22c6\u22c6) can be equivalently described as: Assume there are M + N \u22121 non-negative real numbers Z distributed according to iid \u223cP. Colour M of them red uniformly at random and colour the rest N \u22121 blue. What is the chance that the k \u22121 smallest among Z are all coloured blue? Since any assignment of red and blue colours is equally likely, we compute the probability by counting the ratio of possible colour assignments where k smallest elements are coloured blue. The formula is written as \u0000M+N\u2212k\u22121 M \u0001 \u0000M+N\u22121 M \u0001 = (N \u22121) \u00b7 \u00b7 \u00b7 (N \u2212k) (M + N \u22121) \u00b7 \u00b7 \u00b7 (M + N \u2212k). \u25a0 Note that the expected values do not depend upon the distribution type or the dimensionality D of the data. 3.4. Hyperparameter selection Given the analytic expression for the expected D&C for identical real and fake distributions, we can systematically choose k, M, and N. We set the aim of hyperparameter selection as: E[coverage] > 1\u2212\u03f5 for identical real and fake. Since E[coverage] does not depend upon the exact distribution type or dimensionality of data (Equation 9), the hyperparameters chosen as above will be effective across various data types. We verify the consistency of D&C across data types in Figure 2. It contains plots of the P&R and D&C values for identical real and fake distributions from (1) 64-dimensional multivariate standard Gaussians, (2) 4096dimensional ImageNet pre-trained VGG embeddings of the FFHQ face images, and (3) analytic estimations in \u00a73.3. While P&R exhibits signi\ufb01cantly different estimated values across the Gaussians and the FFHQ embeddings, D&C metrics agree on all three types of estimates, con\ufb01rming the independence on data type. This conceptual advantage leads to a con\ufb01dent choice of evaluation hyperparameters that are effective across data types. In practice, we choose the hyperparameters to achieve E[coverage] > 0.95. For the sake of symmetry, we \ufb01rst set M = N. We then set M = N = 10 000 to ensure a good approximation E[coverage] \u22481 \u2212 1 2k , while keeping the computational cost tractable. k = 5 is then suf\ufb01cient to ensure E[coverage] \u22480.969 > 0.95. The exact principle behind the choices of those values for P&R (k = 3, M = N = 50 000) is unknown. 4. Experiments We empirically assess the proposed density and coverage (D&C) metrics and compare against the improved precision and recall (P&R) (Kynk\u00a8 a\u00a8 anniemi et al., 2019). Evaluating \fReliable Fidelity and Diversity Metrics for Generative Models Fake Real Fake outliers Real outliers \u00b5 = \u22121 \u00b5 = 0 \u00b5 = +1 \u22121.0 \u22120.5 0.0 0.5 1.0 \u00b5 0.0 0.5 1.0 precision \u22121.0 \u22120.5 0.0 0.5 1.0 \u00b5 0.0 0.5 1.0 recall No-outlier case One outlier in real One outlier in fake \u22121.0 \u22120.5 0.0 0.5 1.0 \u00b5 0.0 0.5 1.0 density \u22121.0 \u22120.5 0.0 0.5 1.0 \u00b5 0.0 0.5 1.0 coverage Figure 3. Robustness to outliers for toy data. Behaviour of the four metrics when the real distribution is \ufb01xed X \u223cN(0, I) and fake distribution Y \u223cN(\u00b5, I) shifts with \u00b5 \u2208[\u22121, 1]. We further consider two outlier scenarios where a sample at x = +3 is added either to the set of real or fake samples. Real distribution Fake distributions Sequential dropping Simultaneous dropping 0.00 0.25 0.50 0.75 1.00 Concentration on the \ufb01rst mode 0.0 0.5 1.0 precision 0.00 0.25 0.50 0.75 1.00 Concentration on the \ufb01rst mode 0.0 0.5 1.0 recall 0.00 0.25 0.50 0.75 1.00 Concentration on the \ufb01rst mode 0.0 0.5 1.0 density 0.00 0.25 0.50 0.75 1.00 Concentration on the \ufb01rst mode 0.0 0.5 1.0 coverage Figure 4. Sensitivity to mode dropping for toy data. Assuming the mixture-of-Gaussian real distribution X, we simulate two paths in which the perfect fake distribution Y drops to a single mode. Under the sequential dropping, Y drops a mode one at a time; under the simultaneous dropping, Y gradually decreases samples in all but one mode. Corresponding behaviours of the four metrics are shown. evaluation metrics is dif\ufb01cult as the ground truth metric values are often unavailable (which is why the evaluation metrics are proposed in the \ufb01rst place). We carefully select sanity checks with toy and real-world data where the desired behaviours of evaluation metrics are clearly de\ufb01ned. At the end of the section, we study the embedding pipeline (\u00a72.1) and advocate the use of randomly initialised embedding networks under certain scenarios. 4.1. Empirical results on density and coverage We build several toy and real-world data scenarios to examine the behaviour of the four evaluation metrics: P&R and D&C. We \ufb01rst show results on toy data where the desired behaviours of the \ufb01delity and diversity metrics are wellde\ufb01ned (\u00a74.1.1). We then move on to diverse real-world data cases to show that the observations extend to complex data distributions (\u00a74.1.2). 4.1.1. SANITY CHECKS WITH TOY DATA We assume Gaussian or mixture-of-Gaussian distributions for the real X and fake Y in R64 (D = 64). We simulate largely two scenarios: (1) Y moves away from X (Figure 3) and (2) Y gradually fails to cover the modes in X (Figure 4). We discuss each result in detail below. Translating Y away from X. We set X \u223cN(0, I) and Y \u223cN(\u00b51, I) in R64 where 1 is the vector of ones and I is the identity matrix. We study how the metrics change as \u00b5 varies in [\u22121, 1]. In Figure 3, without any outlier, this setting leads to the decreasing values of P&R and D&C as \u00b5 moves away from 0, the desired behaviour for all metrics. However, P&R show a pathological behaviour when the distributions match (\u00b5 = 0): their values are far below 1 (0.68 precision and 0.67 recall). D&C, on the other hand, achieve values close to 1 (1.06 density and 0.97 coverage). D&C detect the distributional match better than P&R. Translating with real or fake outliers. We repeat the previous experiment with exactly one outlier at 1 \u2208R64 in either real or fake samples (Figure 3). Robust metrics must not be affected by one outlier sample. However, P&R are vulnerable to this one sample; precision increases as \u00b5 grows above 0.5 and recall increases as \u00b5 decreases below \u22120.4. The behaviour is attributed to the overestimation \fReliable Fidelity and Diversity Metrics for Generative Models of the manifold on the region enclosed by the inliers and the outlier (\u00a73.1). This susceptibility is a serious issue in practice because outliers are common in realistic data and generated outputs. On the other hand, the D&C measures for the outlier cases largely coincide with the no-outlier case. D&C are sturdy measures. Mode dropping. We assume that the real distribution X is a mixture of Gaussians in R64 with ten modes. We simulate the fake distribution Y initially as identical to X, and gradually drop all but one mode. See Figure 4 for an illustration. We consider two ways the modes are dropped. (1) Each mode is dropped sequentially and (2) weights on all but one mode are decreased simultaneously. Under the sequential mode dropping, both recall and coverage gradually drop, capturing the decrease in diversity. However, under the simultaneous dropping, recall cannot capture the decrease in diversity until the concentration on the \ufb01rst mode reaches 90% and shows a sudden drop when the fake distribution becomes unimodal. Coverage, on the other hand, decreases gradually even under the simultaneous dropping. It reliably captures the decrease in diversity in this case. 4.1.2. SANITY CHECKS WITH REAL-WORLD DATA Having veri\ufb01ed the metrics on toy Gaussians, we assess the metrics on real-world images. As in the toy experiments, we focus on the behaviour of metrics under corner cases including outliers and mode dropping. We further examine the behaviour with respect to the latent truncation threshold \u03c8 (Karras et al., 2019). As the embedding network, we use the fc2 layer features of the ImageNet pre-trained VGG16 (Kynk\u00a8 a\u00a8 anniemi et al., 2019). Outliers. Before studying the impact of outliers on the evaluation metrics, we introduce our criterion for outlier detection. Motivated by (Kynk\u00a8 a\u00a8 anniemi et al., 2019), we use the distance to the kth nearest neighbour among the fake samples. According to this criterion, we split the fake samples into 10 : 1 for inliers and outliers. We experiment with fake images from StyleGAN on CelebA (Liu et al., 2015) and LSUN-bedroom (Yu et al., 2015). Example images of inliers and outliers are shown in Figure 5. We observe that the outliers have a more distortions and atypical semantics. We examine the behaviour of recall and coverage as the outliers are gradually added to the pool of fake samples. In Figure 5, we plot recall and coverage relative to their values when there are only inliers. As outliers are added, recall increases more than 11% and 15% on CelebA and LSUN bedroom, respectively, demonstrating its vulnerability to outliers. Coverage, on the other hand, is stable: less than 2% increase with extra outliers. Mode dropping. As in the toy experiments, we study mode dropping on the MNIST digit images. We treat the ten CelebA LSUN-Bedroom 0.0 2.5 5.0 7.5 10.0 ratio of outliers to inliers (%) 1.00 1.02 1.04 1.06 1.08 1.10 values relative to the 0% outlier case recall coverage 0.0 2.5 5.0 7.5 10.0 ratio of outliers to inliers (%) 1.000 1.025 1.050 1.075 1.100 1.125 1.150 values relative to the 0% outlier case recall coverage Inliers Inliers Worst outliers Worst outliers Figure 5. Recall versus coverage against the amount of outliers. Example inliers and the worst outliers are shown below. 0.2 0.4 0.6 0.8 1.0 Ratio of single-mode samples 0.00 0.25 0.50 0.75 1.00 Density Precision (a) Precision vs density 0.2 0.4 0.6 0.8 1.0 Ratio of single-mode samples 0.00 0.25 0.50 0.75 1.00 Coverage Recall (b) Recall vs coverage Figure 6. Metrics under mode dropping on MNIST. Behaviour of the metrics under varying degrees of the mode dropping towards a single class on MNIST with R64. The real distribution has the uniform class distribution while the fake distribution has an increasing ratio of the class \u201c0\u201d. classes as modes and simulate the scenario where a generative model gradually favours a particular mode (class \u201c0\u201d) over the others. We use the real data (MNIST images) with decreasing number of classes covered as our fake samples (sequential dropping). The results are shown in Figure 6. \fReliable Fidelity and Diversity Metrics for Generative Models 0.0 0.2 0.4 0.6 0.8 1.0 \u03c8 0.0 0.5 1.0 Precision T4096 R4096 R64 0.0 0.2 0.4 0.6 0.8 1.0 \u03c8 0.0 0.5 1.0 Recall 0.0 0.2 0.4 0.6 0.8 1.0 \u03c8 0 2 4 Density 0.0 0.2 0.4 0.6 0.8 1.0 \u03c8 0.0 0.5 1.0 Coverage \u03c8 = 0.2 \u03c8 = 0.4 \u03c8 = 0.6 \u03c8 = 0.8 \u03c8 = 1.0 Figure 7. Behaviour of metrics with \u03c8. The latent truncation threshold \u03c8 is applied over over StyleGAN generated images on FFHQ. Qualitative examples are given. While recall is unable to detect the decrease in overall diversity until the ratio of class \u201c0\u201d is over 90%, coverage exhibits a gradual drop. Coverage is superior to recall at detecting mode dropping. Resolving \ufb01delity and diversity. The main motivation behind two-value metrics like P&R is the diagnosis involving the \ufb01delity and diversity of generated images. We validate whether D&C successfully achieve this. We perform the truncation analysis on StyleGAN (Karras et al., 2019) on FFHQ. The truncation technique is used in generative models to arti\ufb01cially manipulate the learned distributions by thresholding the latent noise Z with \u03c8 \u22650 (Brock et al., 2018). In general, for greater \u03c8, data \ufb01delity decreases and diversity increases, and vice versa. This stipulates the desired behaviours of the P&R and D&C metrics. The results are shown in Figure 7. With increasing \u03c8, precision and density decrease, while recall and coverage increase. Note that density varies more than precision as \u03c8 increases, leading to \ufb01ner-grained diagnosis for the \ufb01delity. Relation to negative log likelihood metrics. For variational auto-encoders (VAE, (Kingma & Welling, 2013)) and their variants, negative log likelihood (NLL) has been a popular metric for quantifying the ability of the trained model to represent and replicate the data distribution. While NLL provides a single view of the generative model, the D&C metrics provide further diagnostic information. For example, we have trained a convolutional VAE with MNIST. The mean NLL is computed on the reconstructed test data; the D&C Real MNIST Real spectrogram DCGAN WaveGAN MNIST Sound Methods T4096 R4096 R64 T4096 R4096 R64 Precision 0.07 0.60 0.72 0.64 0.71 0.78 Recall 0.08 0.38 0.58 0.03 0.46 0.54 Density 0.02 0.31 0.53 0.27 0.46 0.62 Coverage 0.02 0.56 0.70 0.02 0.47 0.59 Table 1. Failure of trained embeddings. Evaluation results of generated samples beyond natural images using T4096, R4096, and R64. Corresponding real and fake qualitative samples shown. are computed over randomly-initialised VGG features of dimension 64 (\u00a74.2). To simulate the \ufb01delity-diversity tradeoff for VAEs, we consider truncating the latent variable z at magnitude 1. The NLL for vanilla \u2212log p(xtest|z) versus truncated latent \u2212log p(xtest|z, \u2225z\u2225\u22641) is 41.4 and 40.2, not providing meaningful granularity on \ufb01delity-diversity trade-off. For the same setup, D&C provide further details: density has not changed much (0.208\u21920.205), while coverage drops signi\ufb01cantly (0.411\u21920.291) as the result of truncation. Latent variable truncation for our VAE model leads to a drop in diversity, while not improving the \ufb01delity. 4.2. Random Embeddings Image embedding is an important component in generative model evaluation (\u00a72.1). Yet, this ingredient is relatively less studied in existing literature. In this section, we explore the limitations of the widely-used ImageNet embeddings when the target data is distinct from the ImageNet samples. In such cases, we propose random embeddings as alternatives. In the following experiments, we write T4096 for the fc2 features of VGG16 pre-trained on ImageNet, R4096 for the fc2 features of a randomly initialised VGG16, and R64 for the fc2\u2032 features of a randomly initialised VGG16 where the prime indicates the replacement of the 4096 dimensions with 64 dimensions. We experiment on MNIST and a sound dataset, Speech Commands Zero Through Nine (SC09). 4.2.1. METRICS WITH R64 ON NATURAL IMAGES We study the behaviour of metrics for different truncation thresholds (\u03c8) on StyleGAN generated FFHQ images (Fig\fReliable Fidelity and Diversity Metrics for Generative Models ure 7). We con\ufb01rm that the metrics on R64 closely follow the trend with T4096. P&R and D&C over R64 can be used in ImageNet-like images to capture the \ufb01delity and diversity aspects of model outputs. 4.2.2. METRICS WITH R64 BEYOND NATURAL IMAGES We consider the scenario where the target distribution is signi\ufb01cantly different from the ImageNet statistics. We use DCGAN generated images (Radford et al., 2016) on MNIST (LeCun et al., 1998) and WaveGAN generated spectrograms (Donahue et al., 2019) on Speech Commands Zero Through Nine (SC09) dataset. The qualitative examples and corresponding metrics are reported in Table 1. Compared to the high quality of generated MNIST samples, the metrics on T4096 are generally low (e.g. 0.047 density), while the metrics on R64 report reasonably high scores (e.g. 0.491 density). For sound data, likewise, the metrics on T4096 do not faithfully represent the general \ufb01delity and diversity of sound samples. The real and fake sound samples are provided at http://bit.ly/38DIMAA and http://bit.ly/2HAm8NB. The samples consist of human voice samples of digit words \u201czero\u201d to \u201cnine\u201d. The fake samples indeed lack enough \ufb01delity yet, but they do cover diverse number classes. Thus, the recall (0.029) and coverage (0.020) values under T4096 are severe underestimations of the actual diversity. Under the R64, the recall and coverage values are in the more sensible range: 0.572 and 0.653, respectively. When the target data domain signi\ufb01cantly differ from the embedding training domain, R64 may be a more reasonable choice. 5."
+                }
+            ],
+            "Munawar Hayat": [
+                {
+                    "url": "http://arxiv.org/abs/1901.07711v1",
+                    "title": "Max-margin Class Imbalanced Learning with Gaussian Affinity",
+                    "abstract": "Real-world object classes appear in imbalanced ratios. This poses a\nsignificant challenge for classifiers which get biased towards frequent\nclasses. We hypothesize that improving the generalization capability of a\nclassifier should improve learning on imbalanced datasets. Here, we introduce\nthe first hybrid loss function that jointly performs classification and\nclustering in a single formulation. Our approach is based on an `affinity\nmeasure' in Euclidean space that leads to the following benefits: (1) direct\nenforcement of maximum margin constraints on classification boundaries, (2) a\ntractable way to ensure uniformly spaced and equidistant cluster centers, (3)\nflexibility to learn multiple class prototypes to support diversity and\ndiscriminability in feature space. Our extensive experiments demonstrate the\nsignificant performance improvements on visual classification and verification\ntasks on multiple imbalanced datasets. The proposed loss can easily be plugged\nin any deep architecture as a differentiable block and demonstrates robustness\nagainst different levels of data imbalance and corrupted labels.",
+                    "authors": "Munawar Hayat, Salman Khan, Waqas Zamir, Jianbing Shen, Ling Shao",
+                    "published": "2019-01-23",
+                    "updated": "2019-01-23",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Deep neural networks are data hungry in nature and require large amounts of data for successful training. For imbalanced datasets, where several (potentially important) classes have a scarce representation, the learned models are biased towards highly abundant classes. This is because the scarce classes have less representations during training which results in a mismatch between the joint distribution model for training p(x, y) and test sets p(x\u2032, y\u2032). This leads to lower recall rates for rare classes, which are otherwise critically desirable in numerous scenarios. As an example, a malignant lesion is rare compared to benign ones, but should not be miss-classi\ufb01ed. The soft-max loss is a popular choice for conventional recognition tasks. However, through extensive experiments, we show that it is less suitable to handle mismatch between \ud835\udf0e \ud835\udc30\ud835\udc68 \ud835\udc30\ud835\udc69 \ud835\udc30\ud835\udc6a \ud835\udc51\ud835\udc35,\ud835\udc36 \ud835\udc51\ud835\udc34,\ud835\udc35 \ud835\udc51\ud835\udc34,\ud835\udc36 \u03bb Figure 1. Af\ufb01nity Loss integrates classi\ufb01cation and clustering in a single objective. It\u2019s \ufb02exible formulation in Euclidean space allows enforcing margin between classes, control over learned clusters, number of class-prototypes and the distance between classprototypes. Such max-margin learning greatly helps in overcoming class imbalance by learning balanced classi\ufb01cation regions and generalizable class boundaries. train and test distributions. This is partly due to no direct enforcement of margins in the classi\ufb01cation space and the lack of a principled approach to control intra-class variations and inter-class separation. Here, we propose that max-margin learning can improve generalization which can help mitigate classi\ufb01er bias towards more frequent classes by learning balanced representations for all classes. Remarkably, some recent efforts focus on introducing max-margin constraints within the soft-max loss function [10, 33, 32]. Since soft-max loss computes similarities in the angular domain (vector dot-product or cosine similarity), direct enforcement of angular margins is ill-posed and existing works either involve approximations or make restricting assumptions (e.g., points lying on a hypersphere). In this paper, we propose a novel loss formulation that enhances generalization by jointly reducing intra-class variations and maximizing inter-class distances. A notable difference from the previous works is the automatic learning of 1 arXiv:1901.07711v1  [cs.CV]  23 Jan 2019 \fclass representative prototypes in the Euclidean space with inherent \ufb02exibility to enforce certain geometric constraints on the learned prototypes. This is in contrast to soft-max loss where more abundant classes tend to occupy additional space in the projected feature space and rare classes get a skewed representation. The proposed objective is named the \u2018Af\ufb01nity loss function\u2019 as it is based on a Gaussian similarity metric de\ufb01ned in terms of Bergman divergence. The proposed loss formulation learns to map input images to a highly discriminative Euclidean space where the distance with class representative prototypes provides a direct similarity measure for each class. The class prototypes are key points in the embedding space around which feature points are clustered [43]. The af\ufb01nity loss function promotes the classi\ufb01er to have a simpler, balanced and more generalizable inductive bias during training. The proposed loss function thus provides the following advantages: \u2022 An inherent mechanism to jointly cluster and classify feature vectors in the Euclidean space. \u2022 A tractable way to ensure uniformly spaced and equidistant class prototypes (when embedding dimension d and prototype number n are related as: n < d + 1). \u2022 Along-with uniformly spaced prototypes, our formulation ensures that the clusters formed around the prototypes are uniformly shaped (in terms of second order moments). \u2022 The resulting classi\ufb01er shows robustness against different levels of label noises and imbalances amongst classes. The proposed loss function is a differentiable module which is applicable to different network architectures, and complements the commonly deployed regularization techniques including dropout, weight decay and momentum. Through extensive evaluations on a number of datasets, we demonstrate that it achieves a highly balanced and generalizable classi\ufb01er, leading to signi\ufb01cant improvements over previous techniques. 2. Related Work Class-imbalanced Learning: Imbalanced datasets exhibit complex characteristics and learning from such data requires designing new techniques and paradigms. The existing class imbalance approaches can be divided into two main categories, 1) data-level, and 2) algorithm-level approaches. The data-level schemes modify the distribution of data e.g., by oversampling the minority classes [41, 7, 14, 15, 21] or undersampling the majority classes [25, 3]. Such approaches are usually susceptible to redundancy and over-\ufb01tting (for over-sampling) and critical information loss (for under-sampling). In comparison, the algorithm level approaches improve the classi\ufb01er itself e.g., through cost-sensitive learning. Such methods incorporate prior knowledge about classes based upon their signi\ufb01cance or representation in the training data [26, 38, 23]. These methods have been applied to different classi\ufb01ers including SVMs [48], decision trees [61] and boosting [49]. Some works further explore ensemble of cost-sensitive classi\ufb01ers to tackle imbalance [19, 24]. A major challenge associated with these cost-sensitive methods is that the class-speci\ufb01c costs are only de\ufb01ned at the beginning, and they lack mechanisms to dynamically update the costs during the course of training. Deep Imbalanced Learning: Some recent attempts have been made to learn deep models from imbalanced data [20, 23, 5, 52, 36]. For example, the method in [20] \ufb01rst learns to under sample the training data using a neural network, followed by Synthetic Minority Oversampling TEchnique (SMOTE) based technique to re-balance the data. Deep models are trained to directly optimize the imbalanced classi\ufb01cation accuracy in [52, 36]. Wang et. al. [53] propose a meta learning approach to progressively transfer the model parameters from majority towards less-frequent classes. Some works [23, 5] train cost sensitive deep networks, which alternatively optimize class costs and network weights. Continually determining class costs while training a deep model is still an open and challenging research problem, and makes optimization intractable in learning from large scale datasets [18]. Joint Loss Formulation: Popular loss functions used for classi\ufb01cation in deep networks include hinge loss, softmax loss, Euclidean loss and contrastive loss [22]. A triplet loss could simultaneously perform recognition and clustering, however its training is prohibitive due to huge number of triplet combinations on large-scale datasets [40]. Since these loss functions are limited in their capability to achieve discriminability in feature space, recent literature explores the combination of multiple loss functions. To this end, [44] showed that the combination of soft-max and contrastive losses concurrently enforce intra-class compactness and inter-class separability. On a similar line, [54] proposed \u2018center loss\u2019 that uses separate objectives for classi\ufb01cation and clustering. Max-margin Learning: Margin-maximizing learning objectives have been traditionally used in machine learning. Hinge loss in Support vector machines is one of the pioneering max-margin learning framework [16]. Some recent works aim to integrate max-margin learning with crossentropy loss function. Among these, Large-margin softmax [33] enforces inter-class separability directly on the dot-product similarity while SphereFace [32] and ArcFace [10] enforce multiplicative and additive angular margins on the hypersphere manifold, respectively. The hypersphere \fassumption for feature space makes the resulting loss less generalizable to applications other than face recognition. Furthermore, enforcing margin based separation in angular domain is an ill-posed problem and either requires approximations or assumptions (e.g., unit sphere) [12]. THis paper proposes a new \ufb02exible loss function which simultaneously performs clustering and classi\ufb01cation, and enables direct enforcement of the max-margin constraints. We describe the proposed loss formulation next. 3. Max-margin Framework We propose a hybrid multi-task formulation to perform learning on imbalanced datasets. The proposed formulation combines classi\ufb01cation and clustering in a single objective that minimizes intra-class variations while simultaneously achieving maximal inter-class separation. We \ufb01rst explain why traditional Soft-max Loss (SL) is unsuitable for large-margin learning and then introduce our novel objective function. 3.1. Soft-max Loss Given an input-output pair {xi, yi}, a deep neural network transforms input to a feature space representation fi using a function F parameterized by \u03b8 i.e., f = F(x; \u03b8). The soft-max loss can then compute the discrepancy between prediction and ground-truth in the label space as follows: Lsm = 1 N X i \u2212log \u0010 exp(wT yifi) P j exp(wT j fi) \u0011 , (1) where i \u2208[1, N], j \u2208[1, C], N and C are number of training examples and classes respectively. It is worth noting that we have included the last fully connected layer in the de\ufb01nition of soft-max loss which will be useful in further analysis. Also, for the sake of brevity, we do not mention unit biases in Eq. 1. Although soft-max loss is one of the most popular choices for multi-class classi\ufb01cation, in the following discussion, we argue that it is not suitable for class imbalanced learning due to several limitations. Limitations of SL: The loss function in Eq. 1 computes inner vector product \u27e8w, f\u27e9which measures the projection of feature representation on to each of the class vectors wj. The goal is to perfectly align fi with the correct class vector wyi such that the data likelihood is maximized. Due to the reliance of the oft-max loss on vector dot-product, it has the following limitations: \u2022 No inherent mechanism to ensure max margin constraints. Computation of inter-class margin for softmax loss is intractable [12]. Large margin constraints promote better generalization in imbalanced distributions and robustness against input perturbations [9]. \u2022 The learned projection vectors are not necessarily equi-spaced in the classi\ufb01cation space. That is, ideally the angle between closest projection vectors should be equal (e.g., 2\u03c0 k in 2D where k is the number of classes). However, in practice the projection vectors for majority classes occupy more angular space compared with minority classes. This has been visualized in Fig. 2 on imbalanced MNIST dataset, and leads to poor generalization to test samples. \u2022 The length \u2225wj \u22252 of the learnt projection vectors for different classes is not necessarily the same. It has been shown in the literature that the minority class projection vectors are weaker (i.e., with less magnitude) compared with the majority classes [33]. Costsensitive learning which arti\ufb01cially augments the magnitude of the minority class projection vectors has been shown to be effective for imbalance learning [23]. Unsuitability of SL for Imbalanced Learning: We attribute the above limitations to not directly enforcing the max-margin constraints on the classi\ufb01cation boundaries. Consider the de\ufb01nition of soft-max loss (Eq. 1) in terms of dot-products wT f, we can simplify the expression as follows: Li sm \u221d X j\u0338=yi exp(wT j fi \u2212wT yifi) (2) The decision boundary for a class pair {j, k} is given by the case where wT j F(x) = wT k F(x), i.e., the class boundaries are shared between the pair of classes. Further, minimization of Li sm requires wT j F(x) > wT k F(x) : k \u0338= j for correct class assignment to x. This is a \u2018relative constraint\u2019 and therefore the soft-max loss Lsm does not necessarily: (a) reduces intra-class variations, (b) enforces a margin between each class pair. To address these issues, we propose our new loss function next. 3.2. Max-margin Learning with Hybrid Objective Euclidean space similarity measure: Instead of computing similarities with class prototypes using vector dotproduct, we propose to measure class similarities for an input feature in the Euclidean space using a Gaussian similarity measure in terms of Bergman divergence (squared \u21132 distance): d(fi, wj) = exp \u0010 \u2212\u2225fi \u2212wj \u22252 \u03c3 \u0011 , (3) where, \u03c3 denotes a weighting parameter. This provides us: (a) the \ufb02exibility to directly enforce margin maximizing constraints, (b) have equi-spaced classi\ufb01cation boundaries for multiple classes, (c) control the variance of learned \f(a) Softmax (b) Clustering+Softmax (c) Affinity Loss Minority Majority Figure 2. 2D feature space projections in terms of penultimate layer activations. The model is trained on imbalanced MNIST data (by retaining only 10% of the samples for digits 0-4) using different losses: (a) soft-max loss learns \ufb02oral petals in angular space, note that the minority class feature vectors are weaker (shorter in length) and occupy less angular space. (b) center loss reduces intra-class variations by performing clustering. However, the minority class vectors tend to be congested near the center and are confused amongst each other (c) the proposed af\ufb01nity loss learns equi-spaced clusters of uniform shapes for both majority and minority classes. clusters and therefore enhancing intra-class compactness, (d) the freedom to use standard distance measures in Euclidean domain to measure similarity and most importantly (e) simultaneous classi\ufb01cation and clustering in a single objective function. Proposition 1. The similarity function d(a, b) is a valid similarity metric for any real-valued inputs. Proof. The real-valued similarity function d(a, b) will de\ufb01ne a valid similarity metric if it satis\ufb01es the following conditions [30]: \u2022 Non-negativity: d(a, b) \u22650 \u2022 Symmetry: d(a, b) = d(b, a) \u2022 Equivalence: d(a, a) = d(b, b) = d(a, b) iff a = b \u2022 Self-similarity: d(a, a) \u2265d(a, b) \u2022 Triangular similarity: d(a, b) + d(b, c) \u2264d(a, c) + d(b, b) Since, all above conditions are true for d(\u00b7), therefore, it is a valid similarity metric. Relation between Dot-product and Gaussian Similarity: The proposed Gaussian similarity measure is related to the dot-product as follows: d(fi, wj) = exp \u0010 \u2212\u2225fi \u22252 + \u2225wj \u22252 \u22122\u27e8wj, f\u27e9 \u03c3 \u0011 , (4) \u27e8wj, f\u27e9= \u03c3 log d(fi, wj)+ \u2225fi \u22252 + \u2225wj \u22252 2 (5) Intuitively, the above relation implies the dependence of soft-max loss on the scale/magnitude of feature vectors and class prototypes. It leads to two conclusios: (1) It can be seen that d(fi, wj) is bounded between [0, 1] since \u2225fi \u22252 + \u2225wj \u22252\u22652\u27e8wj, f\u27e9, while \u27e8wj, f\u27e9can have large magnitudes. (2) The Gaussian measure can be considered as an inverse chord distance when magnitudes of vectors are normalized to be equal. The dot product in that case is directly proportional to the Gaussian similarity and both similarity measures will behave similarly if no additional constraints are included in our proposed similarity measure. However, the main \ufb02exiblity with our formulation is the explicit introduction of margin constraints, which we introduce next. Enforcing margin between classes: Note that some variants of soft-max loss introduce angle based margin constraints [32, 10], however, the margins in angular domain are computationally expensive and implemented only as approximations due to intractability. Our formulation allows a more direct margin penalty in the loss function. The proposed max margin loss function based on Eq. 3 is given by, Lmm = X j max \u00000, \u03bb + d(fi, wj) \u2212d(fi, wyi) \u0001 : j \u0338= yi, (6) where d(fi, wj) is the similarity of the sample with its true class, d(fi, wyi) is its similarity with other classes, and \u03bb is the enforced margin. Uniform classi\ufb01cation regions: The soft-max loss does not ensure uniform classi\ufb01cation regions for all classes. As a result, undersampled minority classes get a shrinked representation in the feature space compared to more frequent classes. To ensure equi-distant weight vectors, we propose \fSkin Lesion Dataset (DIL) Number of Images 50% Data 50% Data Faces Dataset (VGG2) 50% Data Figure 3. Data Imbalance due to long-tail distribution. to apply a regularization on the learned class weights. This regularizer is termed as a \u2018diversity regularizer\u2019 as it enforces all class centers (w) to be uniformly spread out in the feature space. The diversity regularizer is formally de\ufb01ned as follows: R(w) = E[(\u2225wj \u2212wk \u22252 \u2212\u00b5)2], s.t. j < k, (7) \u00b5 = 2 C2 \u2212C X j<k \u2225wj \u2212wk \u22252 (8) where \u00b5 is the mean distance between all class prototypes. Multi-centered learning: For challenging classi\ufb01cation problems, the feature space may be partitioned such that all samples belonging to the same class are not co-located in a single region. Therefore, clustering all same class samples with a single prototype (class center) will not be optimal in such cases. To resolve this limitation, we introduce a novel multi-centered learning paradigm based on our max-margin framework. Instead of learning a single projection vector wj for each class, the proposed framework enables learning multiple projection vectors {wt}j per-class. Speci\ufb01cally, we can learn m projection vectors per class, where similarity of a feature vector fi with a class j is given by: d(fi, wj) = max \b exp \u0000\u2212\u2225fi \u2212wj,t \u22252 \u03c3 \u0001\t , t = [1, m] (9) Max-margin loss is then de\ufb01ned similar to Eq. 6 above. The overall loss function therefore becomes: L = Lmm + R(w). (10) The diversity regularizer for the multi-center case is enforced on the similarity between all m \u2217C prototypes. 4. Experiments To demonstrate the effectiveness of the proposed af\ufb01nity loss, we perform experiments on datasets which exhibit natural imbalance. These include Dermo\ufb01t Image Library (DIL) for skin lesion classi\ufb01cation and large scale image datasets for facial veri\ufb01cation. We further extensively evaluate various components of the proposed approach by systematically generating imbalance and introducing different levels of label noise. Through these empirical evaluations, we provide an evidence of the robustness of the proposed method against different data imbalance levels and noisy training labels. A brief description about the evaluated datasets is presented next. 4.1. Datasets Skin Melanoma Dataset (DIL): Edinburgh Dermo\ufb01t Image Library (DIL) contains 1300 images belonging to 10 skin lesion categories including melanomas, seborrhoeic keratosis and basal cell carcinomas. The images are based upon diagnosis from dermatologists and dermatopathologists. The number of images vary amongst categories (between 24 and 331, mean 130, median 83), and show significant imbalance, with 50% of all images belonging to only top two classes (Fig. 3). Similar to [2], we perform two experiments, considering \ufb01ve and ten class splits respectively, and report results for 3-fold cross validation. Face Recognition: Datasets used to train large scale face recognition models have natural imbalance. This is because the data is web-crawled, and images for some identities are easily available in abundance compared with others. For unconstrained face recognition, we train our model on VGG2 [4], which is a large scale dataset with inherent class imbalance. We evaluate the trained network on four different datasets. These include two popular widely used benchmarks i.e., Labelled Faces in the Wild (LFW) [27] and YouTube Faces (YTF) [55]. We further evaluate on Celebrities in Frontal Pro\ufb01le (CFP) [39] and Age Database (AgeDB) [35]. VGG2: facial image dataset [4] contains 3.31 million images belonging to 8, 631 identities. The number of samples for each subject exhibit imbalance and vary from 80 to 843 with a mean of 362. The data is collected from the Internet and has real-life variations in the form of ethnicites, head poses, illumination changes and age groups. LFW: Labelled Faces in the Wild (LFW) [27] contains 13, 233 static images of 5749 individuals collected over the Internet in real-life situations. We follow the standard evaluation protocol \u2018unrestricted with labeled outside data\u2019 [27] and test on 6000 pairs for face veri\ufb01cation. YTF: YouTube Faces (YTF) [55] has 3425 videos belonging to 195 different subjects. The length of video sequences varies between 48 and 6070 frames, with an average of 181.3 frames. We follow the standard evaluation protocol for face veri\ufb01cation on 5000 video pairs. CFP: contains 10 frontal and 4 pro\ufb01le view images for 500 different identities [39]. Two evaluation protocols are used based upon the type of images in the gallery and probe: frontal-frontal (FF) and frontal-pro\ufb01le (FP). Each protocol has 10 runs, each with 700 face pairs (350 same and 350 different). AgeDB: has 12, 240 images acquired in-the-wild for 440 \fsubjects [35]. Along with variations across expression deformations, head poses and illumination conditions, a distinct feature of this dataset is the diversity across ages of the subjects, which ranges between 3 and 101 years, with an average of 49 years. Test evaluation protocol has four groups with different age gaps (5, 10, 20 and 30 years). Each group contains ten splits, each having 600 face image pairs (300 same, 300 different). We use the most challenging split with 30 years gap. Imbalanced MNIST: Standard MNIST has 70, 000 handwritten images of digits (0-9), 60, 000 of these images are used for training (600/class) and the remaining 10,000 for testing (100/class). For this paper, we perform experiments on the standard evaluation split, as well as by systematically creating imbalance in the training set. For this, we reduce the even and odd digit samples to 10% and 25%. We further perform ablative study (Sec. 4.6) by gradually introducing different imbalance ratios amongst classes and noise levels in the training labels. 4.2. Experimental Settings For experiments on DIL dataset, ResNet-18 backbone is used in combination with the proposed af\ufb01nity loss. For training the model to learn features for face veri\ufb01cation tasks, we deploy Squeeze and Excitation (SE) networks [17] with ResNet-50 backbone and af\ufb01nity loss. The face images are cropped and re-sized to 112 \u00d7 112 using multi-task cascaded Convolution Neural Network (CNN) [59]. The model is trained using random horizontal \ufb02ips as data augmentation. The features extracted after the global pooling layer are then used for face veri\ufb01cation evaluations on different datasets. The experiments on MNIST are performed on a simple network with four hidden layers having three convolution layers (32, 64 and 128 \ufb01lters of 5 \u00d7 5), one fully connected layer (128 neurons), and an output layer. The model is trained with Stochastic Gradient Descent (SGD) optimizer with momentum and learning rate decay. For ablative study in Sec. 4.6, we only change the output soft-max layer with the proposed Af\ufb01nity loss layer and keep rest of the architecture \ufb01xed. 4.3. Results and Analysis Table. 2 present our experimental results on DIL dataset. In Exp#1, we report average performance for 3 fold cross validation on \ufb01ve classes (Actinic Keratosis, Basal Cell Carcinoma, Melanocytic Nevus, Squamous Cell Carcinoma and Seborrhoeic Keratosis). Compared with existing state of the art [23], we achieve an absolute gain of 10.9% on Exp#1. For Exp#2 on DIL dataset, all 10 classes are considered. Evaluations on 3 fold cross validation in Table 2 show a signi\ufb01cant performance improvement of 7.7% for Exp#2. Confusion matrix analysis for class-wise accuracy comparison in Fig 4 shows that the performance boost is Methods (using stand. split) Performances Deeply Supervised Nets [29] 99.6% Generalized Pooling Func. [28] 99.7% Maxout NIN [6] 99.8% Imbalanced (\u2193) CoSen CNN [23] Af\ufb01nity Loss Stand. split 99.3% 99.6% 10% of odd digits 98.6% 99.3% 10% of even digits 98.4% 99.3% 25% of odd digits 98.9% 99.4% 25% of even digits 98.5% 99.5% Table 1. Evaluations on Imbalanced MNIST Database. Methods Performances (using stand. split) Exp#1 (5-classes) Exp#2 (10-classes) Hierarchical-KNN [2] 74.3 \u00b1 2.5% 68.8 \u00b1 2.0% Hierarchical-Bayes [1] 69.6 \u00b1 0.4% 63.1 \u00b1 0.6% Flat-KNN [2] 69.8 \u00b1 1.6% 64.0 \u00b1 1.3% CoSen CNN [23] 80.2 \u00b1 2.5% 72.6 \u00b1 1.6% Af\ufb01nity Loss 91.1 \u00b1 1.7% 80.3 \u00b1 2.1% Table 2. Evaluation on DIL Database. more pronounced for minority classes with lower representations. We attribute this to the capability of the proposed method to simultaneously optimize within class compactness by performing feature space clustering, and enhance inter-class separability by enforcing max-margin constraints. Our method achieves competitive performance on LFW and YTF datasets in Table 3. The performances on LFW and YTF are already saturated with many recent methods surpassing human-level results. The top performing methods on these datasets have been trained on much larger models with signi\ufb01cantly more data and model parameters. Further evaluations on other facial recognition benchmarks achieve veri\ufb01cation accuracies of 95.9%, 99.5% and 96.0% on AgeDB30, CFP-FF and CFP-FP datasets respectively. These results prove the effectiveness of the proposed approach for large scale imbalanced learning. It is worth noting that our proposed Af\ufb01nity loss does not require additional compute and memory and is easily scalable to larger datasets. This is in contrast to some of the existing loss formulations (such as triplet loss [40] and contrastive loss [13]) which do enhance feature space discriminability, but suffer scalability to large data due to substantial possible combinations of training pairs, triplets or quintuplets. 4.4. Generalization To test the generalization of the proposed method for different imbalance levels, we gradually reduce the training set by changing the representation of the minority class samples on MNIST data. Speci\ufb01cally, we gradually alter the majority to minority class ratios (up-to 1 : 0.025) by randomly dropping samples of the \ufb01rst \ufb01ve digits (0\u22124). Under these settings, we therefore have signi\ufb01cantly lower repre\f(a) CosSen CNN [23] (b) Af\ufb01nity Loss Figure 4. Confusion matrices for Exp#1 on DIL dataset. Methods #Models Train Data LFW YTF DeepFace [46] 3 4M 97.35 91.4 FaceNet [40] 1 200M 99.63 95.4 Web-scale [47] 4 4.5M 98.37 VGG Face [37] 1 2.6M 98.95 97.3 DeepID2+ [45] 25 0.3M 99.47 93.2 Baidu [31] 1 1.3M 99.13 Center Face [54] 1 0.7M 99.28 94.9 Marginal Loss [11] 1 4M 99.48 95.98 Noisy Softmax [8] 1 Ext. WebFace 99.18 94.88 Range Loss [60] 1 1.5M 99.52 93.7 Augmentation [34] 1 WebFace 98.06 Center Invariant Loss [56] 1 WebFace 99.12 93.88 Feature transfer [58] 1 4.8M 99.37 Softmax+Contrastive [44] 1 WebFace 98.78 93.5 Triplet Loss [40] 1 WebFace 98.7 93.4 Large Margin Softmax [33] 1 WebFace 99.10 94.0 Center Loss [54] 1 WebFace 99.05 94.4 SphereFace [32] 1 WebFace 99.42 95.0 CosFace [51] 1 WebFace 99.33 96.1 LMLE [18] 1 WebFace 99.51 95.8 Af\ufb01nity Loss 1 VGG2 99.65 97.3 Table 3. Face Veri\ufb01cation Performance on LFW and YTF datasets. Figure 5. The effect of label noise on the soft-max and af\ufb01nity loss functions. sentation for half of the classes. The experimental results in terms of error rates against fraction of retained minority class samples are shown in Fig. 6. We also repeat these experiments for standard soft-max loss. The comparison in Fig. 6 demonstrates a consistently superior performance of the proposed loss function across all settings. The effect on achieved performance is more noticeable for larger imbalance levels between majority and minority classes. The proposed Af\ufb01nity loss enhances inter-class separability irreFigure 6. Robustness analysis against different imbalance levels (fraction of retained minority class samples) spective of the class frequencies by enforcing margin maximization constraints. Soft-max loss does not have inherent margin learning characteristics. Further, compared with soft-max loss, where intra-class variations can vary across classes depending upon their representative samples, af\ufb01nity loss learns uniformed sized clusters. As visualized in Fig. 2, feature space within class disparities are observed for soft-max loss with minority classes occupying compact regions compared with their majority counterparts. In comparison, our proposed loss formulation is \ufb02exible, and allows learnt class prototypes to be equi-spaced and form uniformly shaped clusters. This reduces bias towards the less frequent cases and enhances the overall generalization capabilities, thus yielding a more discriminatively learnt feature space and an improved performance. 4.5. Robustness against Noisy Labels For many real-world applications, the acquired data has noisy labels, and generalization of the learning methods against label noise is highly desirable [42, 50, 19]. To check the robustness of our proposed approach against noisy labels in the training data, we randomly \ufb02ip the classes of MNIST training samples. The fraction of the miss-labelled samples is gradually increased from 10% to 50% with an increment of 10%. In order to avoid over-\ufb01tting on the noisy data, we deploy early stopping [57], and \ufb01nish training when the performance on a held-out cross validation set starts to degrade. For comparison, we repeat all experiments using standard soft-max loss. The experimental results in Fig. 5 show that that the proposed Af\ufb01nity loss performs better across the entire range of different noise levels. Although, the performance for both soft-max and af\ufb01nity losses degrades with increasing noise factors, the proposed af\ufb01nity loss shows more robustness, specially for larger noise ratios, with comparatively less performance degradation. The multi-centered learning in our loss provides \ufb02exibility to the noisy samples to associate themselves with class prototypes which are different from the non-noisy and clean samples. \fFigure 7. Effect of changing parameter \u03c3 that controls the spread of clusters. Results on imbalanced MNIST show that increasing the cluster variance above a certain point results in overlapped clusters and higher error rate. Figure 8. Performance for different number of clusters per class. 4.6. Ablation Number of Cluster Centers: A unique aspect of the proposed af\ufb01nity loss is its multi-centered learning which provides us the \ufb02exibility to have multiple class prototypes for each class. Here, we perform experiments on the imbalanced MNIST dataset (10% representation for \ufb01rst \ufb01ve digits), by gradually changing the number of representative prototypes m per class from 1 to 20. The experimental results in terms of error rates vs prototypes m in Fig. 8 show that the best performance is achieved for m \u22655. Fewer prototypes per class (m \u22645) yield relatively poor performance. The proposed method performs consistently when prototypes are increased beyond 5. Such multi-centered learning supports diversity in input samples. It is speci\ufb01cally helpful in scenarios with complex data distributions where large differences are observed amongst samples of the same class. Such diverse samples might not necessarily cluster around a single region, and could form multiple clusters by virtue of the proposed multi-centered learning mechanism. Furthermore, our experiments in Sec. 4.5 show that by providing \ufb02exible class prototypes, multi-centered learning proves an effective and robust scheme against noisy samples. Cluster Spread \u03c3: The parameter \u03c3 in Eq. 3 determines the cluster spread and helps achieve uniform intra-class variations. Our 2D visualization of the learnt features in Fig. 2 demonstrate that the clusters for each class are uniformly sized for both the majority and minority classes. This is in contrast to the traditional soft-max loss, where shrinked feature space regions are observed for the minority classes. For our proposed loss formulation, the size of the cluster is directly related with the value of parameter \u03c3, with larger \u03c3 Distance d Similarity Performance ||a \u2212b||1 exp \u2212(d/\u03c3) 99.3 ||a \u2212b||2 exp \u2212(d/\u03c3) 99.3 ||a \u2212b||1 1/(1 + d) \u2212 ||a \u2212b||2 1/(1 + d) 99.1 Table 4. Evaluation with different combinations of distance and similarity measures. indicating larger variance for a cluster. We perform experiment on imbalanced MNIST dataset for different values of the the parameter \u03c3 = {.1, .5, 1, 5, 1e1, 2e1, 5e1}. The results in Fig. 7 show that the optimal performance is achieved for values of \u03c3 between 5 and 20. Very high values of \u03c3 results in larger cluster spreads causing overlaps and confusion amongst classes and lower classi\ufb01cation performance. Distance and Similarity Metrics: Our original af\ufb01nity loss formulation in Eq. 3 \ufb01rst computes the squared \u21132 distance between the feature f and class prototype w, which is then converted to a similarity measure using the Gaussian metric. In this experiment, we evaluate different combinations of distance and similarity metrics. \u21131 and \u21132 metrics are used to compute distances, whereas Gaussian and inverse distance (de\ufb01ned by 1 1+x) are the two similarity measures. We perform these experiments on imbalanced MNIST data (by retaining 10% samples for \ufb01rst \ufb01ve digits). Table. 4 shows our evaluation results. The proposed scheme works well with all combinations except for \u21131 distance and Gaussian similarity, where it fails to converge. The best performance is achieved for Gaussian similarity in combination with squared \u21132 distance. 5."
+                }
+            ],
+            "Salman H. Khan": [
+                {
+                    "url": "http://arxiv.org/abs/1906.03650v1",
+                    "title": "Unsupervised Primitive Discovery for Improved 3D Generative Modeling",
+                    "abstract": "3D shape generation is a challenging problem due to the high-dimensional\noutput space and complex part configurations of real-world objects. As a\nresult, existing algorithms experience difficulties in accurate generative\nmodeling of 3D shapes. Here, we propose a novel factorized generative model for\n3D shape generation that sequentially transitions from coarse to fine scale\nshape generation. To this end, we introduce an unsupervised primitive discovery\nalgorithm based on a higher-order conditional random field model. Using the\nprimitive parts for shapes as attributes, a parameterized 3D representation is\nmodeled in the first stage. This representation is further refined in the next\nstage by adding fine scale details to shape. Our results demonstrate improved\nrepresentation ability of the generative model and better quality samples of\nnewly generated 3D shapes. Further, our primitive generation approach can\naccurately parse common objects into a simplified representation.",
+                    "authors": "Salman H. Khan, Yulan Guo, Munawar Hayat, Nick Barnes",
+                    "published": "2019-06-09",
+                    "updated": "2019-06-09",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction \u2018The objects seen could be constructed out of parts with which we are familiar.\u2019 L.G. Roberts Computer vision in its early days saw the emergence of parts-based representations for object representation and scene understanding [23]. As early as 1963, Roberts [27] presented an approach to represent objects using a set of 3D polyhedral shapes. Subsequently, Guzman [10] introduced a collection of parts that appear in generic line drawings and demonstrated how they can be used to recognize 2D curved shapes. The generalized cylinders based representation to describe curved objects of Binford [3] was a signi\ufb01cant breakthrough. It was developed further, including a pioneering contribution by Biederman, who introduced a set of basic primitives (termed as \u2018geons\u2019 meaning geometrical ions) and linked it with the object recognition in human cognitive system [2]. Figure 1: Compared to traditional 3D generative modeling approaches (top) that directly generate 3D shape, our approach (bottom) transitions incrementally from a simple primitive based representation towards a complete 3D shape. Such a hierarchical approach provides better control and interpretability for generative networks. Furthermore, a major novelty of this work is an unsupervised primitive discovery approach that underpins the proposed generative pipeline. Very recently, early research towards automatic discovery of shape primitives using deep networks have been reported in the literature. Tulsiani et al. [39] proposed a CNN model to predict the size and transformation parameters of primitives that were assembled together to represent generic 3D shapes. Their main draw-back is the inability to jointly represent different object categories using a single model. This requires a class-speci\ufb01c CNN training procedure, that is both time-consuming and dif\ufb01cult to scale to a large number of categories. Zou et al. [47] proposed a generative model based on RNNs to recover a 3D shape de\ufb01ned by primitives from an input depth image. Their model, however, requires primitive-level shape labellings for training, requires an accurate depth map as input and works only for a set of three related classes (i.e., chair, table and night stand). In this work, we propose to incorporate a generic primitive based representation in the 3D generative modeling 1 arXiv:1906.03650v1  [cs.CV]  9 Jun 2019 \fprocess to enhance the scalability of learned models. Our \ufb01rst major contribution is the automatic primitive discovery in 3D shapes. Such a shape representation can provide several key bene\ufb01ts such as: (a) It factorizes the 3D generation process into a set of simpler steps, that de\ufb01nes a natural topdown \ufb02ow in the existing bottom up generation pipelines. (b) It offers a highly compact representation compared to volumetric representations such as a voxel or a TSDF. (c) Shape primitives provide a level of abstraction in the generation process, that makes it easy to understand and manipulate the output from generative models. (d) A global representation of a shape encoded by a few primitives allows a better intuition about the object parts, their physical properties (e.g., stability and solidness) and their mutual relationships (e.g., support and contact) [11]. (e) Such a shape description provides invariance to pose changes by explicitly estimating object size and transformation using our proposed primitive Generative Adversarial Network (GAN), the network separates viewpoint changes from the actual shape changes. In a nutshell, we introduce the principal of modularity in the existing generative pipelines. Our main contributions are summarized below: \u2022 A factorized generative model that improves 3D generation by introducing a simpler auxiliary task focused on learning primitive representation. \u2022 A fully unsupervised approach based on a high-order Conditional Random Field (CRF) model to jointly optimize shape abstractions over closely related sub-sets of 3D models. Our model considers appearance, stability and physical properties of the primitives and their mutual relationships such as overlap and co-occurrence. \u2022 The proposed model is jointly trained on all object categories and avoids expensive category speci\ufb01c training procedures adopted by earlier approaches. The proposed approach can be used to incorporate intermediate levels of user input and can render more sophisticated outputs on top of that. From another perspective, it can be used to analyze the intermediate part-based representations learned by GAN that provides better interpretability and transparent generation process. 2. Related Work 3D Generative Models: Wu et al. [41] were the \ufb01rst to extend the 2D GAN framework [8] to generate 3D shapes. They demonstrated that the representations learned by the discriminator are generalizable and outperform other unsupervised classi\ufb01cation methods. Another similar approach was proposed in [38] that used a Wasserstein loss [1] for 3D GAN. However, [38, 41] do not address primitive based shape modeling for a hierarchical shape generation pipeline. Notably, some recent efforts in 2D image generation built a hierarchy of stacked GANs to generated stage-wise outputs [13, 40, 44]. Huang et al. [13] used a combination of encoder, generator and discriminator blocks to perform joint top-down and bottom-up information exchange for improved image generation. However, they operate on learned feature representations and do not enhance model interpretability. Besides, a common limitation of above mentioned methods is the lack of control over the latent representations and resulting dif\ufb01culties in generating data with desired attributes. Primitive Discovery: Cuboids have been extensively used in the previous literature to represent objects, parts and scene structural elements due to their simple form [32, 18, 15, 23]. The identi\ufb01cation of recurring parts and objects has also been studied under the problems of cosegmentation and unsupervised learning [28, 34, 29]. In 3D shapes, some efforts aim at parts discovery and modeling their mutual arrangements in large-scale shapes datasets [45, 6]. Recently, Tulsiani et al. [39] proposed a deep learning based approach to describe a shape with a combination of cuboid primitives. Their approach requires learning a separate model for each set of shapes belonging to the same category. Therefore, their model is not fully unsupervised and dif\ufb01cult to scale to a large number of object categories. In this work, we address these limitations and further propose a factorized generative model for improved shape generation. Model Based 3D Reconstruction: The pioneering work of Roberts [27] lead to several efforts in recovering 3D layout of a scene from a single image. However, the 3D reconstruction from a single image is still an unsolved problem. Given the success of deep networks, recent approaches have proposed several incarnations of these models for 3D reconstruction. Izadinia et al. [14] generated 3D CAD models from a single indoor scene by detecting objects class and its pose using deep CNNs, followed by synthesizing scenes using CAD models from the ShapeNet library [4]. However, in contrast to these works, we do not have the prior knowledge about a speci\ufb01ed set of primitives, rather we aim to automatically learn the shared parts across 3D shapes. 3. Primitive Discovery in 3D Shapes In the \ufb01rst stage, we automatically discover 3D primitives from generic object shapes. Our goal is to learn common recurring primitives in 3D shapes in an unsupervised manner. We introduce a higher-order CRF model that incorporates several physical and volumetric properties of primitives to identify a consistent shape description. We propose a multiview primitive discovery approach that discretizes the 3D space without losing much shape information and allows a computationally ef\ufb01cient alternative to direct 3D primitive \ufb01tting. Furthermore, since our objective is to discover shared primitives among various models, direct cuboid \ufb01tting in the original 3D space leads to more instance speci\ufb01c and less category generalizable primitives. Our CRF model is \fFigure 2: An overview of the proposed approach. Our model consists of a Primitive GAN that generates a parsimonious representation that is used by the 3D VAE GAN in the next stage to recover a complete 3D shape. explained next. 3.1. Proposed CRF Model Our goal is to automatically discover 3D primitives to represent generic 3D shapes without any supervision. For this purpose, we design a CRF model that allows ef\ufb01cient inference and adequately incorporates rich relationships between primitives and complete shapes. Suppose, we have a dataset D = {x1 . . . xM} consisting of M 3D shapes. For each shape xm, assume a candidate set of box proposals generated via bottom-up grouping (see Sec. 3.3), denoted as B = {b1 . . . bN}, where N is the total number of box proposals. The segmented regions obtained by grouping are denoted by R = {r1 . . . rR}. We also use a set of binary variables V = {v1 . . . vN} and S = {s1 . . . sR}, where each vi and sr is associated with a box proposal and a segmented region, respectively. The variable vi denotes whether a cuboid is selected as a representative primitive or not. We develop a CRF model to encapsulate the relationships between primitives both locally as well as globally. The Gibbs energy formulation of the CRF is given by: E(V|D) = X i \u03c8u(vi) + X i<j \u03c8p(vi, vj) + X \u03c8h(V, T ), where, \u03c8u, \u03c8p and \u03c8h denote the unary, pairwise and higherorder potentials respectively and T represents the set of primitives from similar shapes. Next, we elaborate on each of the three potentials. 3.1.1 Unary Potential The unary potential for each primitive candidate denotes its likelihood for a valid simpli\ufb01ed representation of the 3D shape. This potential encodes physical and geometric properties of each box. We explain the individual cost terms within the unary potential below. Frontal convex hull Convex  part Concave  part (\ud835\udc63\ud835\udc53) (\ud835\udc4e\ud835\udc53) Face area \ud835\udc65\ud835\udc58 \ud835\udc51(\ud835\udc65\ud835\udc58, Conv(\ud835\udc4b)) Extended  cuboid (a) Compactness (b) Convexity (c) Cuboid support Figure 3: Visual illustration of costs. Volumetric occupancy: This cost (coc i ) estimates the empty volume within the ith primitive. It is de\ufb01ned as coc i = noc i /nt i, where noc i and nt i are the number of empty and total voxels respectively. Shape uniformity: This cost (csu i ) measures the uniformity of the shape along the primitive sides that were used to propose the candidate primitive. It is calculated by taking the average entropy of the surface normal direction distribution for the relevant initial segmented regions. Primitive compactness: The cost (cpc i ) estimates how tightly a 3D shape is enclosed by the primitive. It is calculated using average ratio between the empty surface area on each face and the actual face area (af): cpc i = P f\u2208F af \u2212vf af , where F is the set of visible faces of primitive (Fig. 3). Support cost: A valid primitive is likely to be supported by nearby shape parts. This cost calculates nearby support by considering a 5% enlarged box and taking the ratio: csc i = nsc i nex i \u2212nsc i , where nex i and nsc i denote the number of voxels in the extended and original box primitive respectively. Shape convexity: This cost determines the degree to which a shape-part is convex. For the regions associated with each primitive proposal, we \ufb01rst obtain a 3D frontal convex hull that only covers the visible 3D points from a single view. We then obtain the mean of distances between the 3D points and the frontal convex hull. It is given by: cco i = P \u2200views P k d(xk,Conv(X)) N , where, xk \u2208X, Conv(X) is the frontal convex hull for X, d denotes shortest distance between xk and Conv(X) (see Fig. 3). A large value of \fshape convexity cost (cco i ) denotes that the shape is concave, while a small value denotes a convex shape. As convex shapes are more common in indoor scenes, a soft cost based on convexity is helpful. Shape symmetry: For each primitive, we measure the cost (css i ) denoting re\ufb02ective symmetry of its enclosed 3D shape. For this purpose, we perform SVD decomposition to calculate three principal axis and measure the average overlap between the original points and their re\ufb02ected versions. This overlap is measured as the distance between the neighboring point\u2019s position and normal direction [16]. Given the three principal orthogonal directions X = {a, b, c} of maximum variation and the corresponding Eigenvalues denoted by \u03c0x, the following relation is used to measure symmetry: css i = 1 P x \u03c0x X x \u03c0x lx \u0010 X j \u2225pP i x j \u2212p P i x\u2032 nj \u2225+ (1 \u2212qP i x j \u00b7 q P i x\u2032 nj ) \u0011 , where, x\u2208X, j\u2208|P i|, P i denotes the point cloud of ith primitive, Px\u2032 denotes the \ufb02ipped point cloud along x direction, pj and qj denote the jth point and its normal respectively, lx denotes the length of primitive along the x direction and nj is the nearest neighbor of jth point in Px\u2032. The individual costs listed above are fused together to obtain the per primitive unary cost as follows: \u03c8u(vi) = \u27e8\u00b5u, w \u25e6ci\u27e9, where ci = [coc i , csu i , cpc i , csc i , cco i , css i ]. (1) Here \u27e8\u00b7, \u00b7\u27e9, \u25e6denote inner and Hadamard products, \u00b5u is the cost weight vector and w is the normalizing vector calculated on the validation set to obtain mutually comparable costs. 3.1.2 Pairwise and High-order Potentials Primitive Overlap: The pairwise potential considers the intersection relationships between primitive pairs. Since valid primitives do not signi\ufb01cantly overlap each other, the goal is to penalize a con\ufb01guration that violates this physical constraint. This cost cpw is measured as an intersection between the two cuboids normalized by the volume of the smaller cuboid: \u03c8p(vi, vj) = \u00b5pwcpwvivj, where \u00b5pw > 0 is the weighting parameter. In practice, we introduce an auxiliary boolean variable yij to linearize the pairwise intersection cost by replacing vi, vj in the above cost. Primitive Parsimony: Motivated by the minimum description length principle, we aim to obtain a parsimonious representation of 3D shapes. In other words, we discourage using additional primatives if a small number is adequate to represent an object. A penalty on the number of active primitives is therefore introduced as a higher-order potential, \u03b8par h (V) = \u00b5par PN i=1 vi, s.t., \u00b5par > 0, where \u00b5par is the weight of the potential. Coverage Potential: The minimization of costs de\ufb01ned above will lead to a null primitive assignment. An important requisite is to obtain a representation that maximally covers the 3D shape [39]. This constraint is formulated as maximizing the surface area enclosed by primitives: \u03b8cov h (V, S) = \u00b5cov X k ccov k sk, s.t., \u00b5cov < 0, sk \u2264 X i:rk\u2208bi vi. (2) Here, \u00b5cov k denotes weight and the cost ccov k is set equal to the area of the segmented region rk. Co-occurrence Potential: We assume a set T of matched primitives for all vertices vi \u2208V. Each element ti = {\u02c6 v1 . . . \u02c6 vJ} \u2208T comprises of boolean variables \u02c6 vj for all J primitives identi\ufb01ed in similar shapes that are matched to primitive \u2018i\u2019. The co-occurrence potential is de\ufb01ned as: \u03b8coc h (V, T ) = \u00b5coc X ij ccoc ij uij, s.t., uij = vi\u02c6 vj, \u00b5coc < 0, vi \u2264 X j \u02c6 vj (3) The variables uij and \u00b5coc denote the auxiliary boolean variable and the weight respectively. The cost ccoc ij is de\ufb01ned as the Intersection over Union (IoU) measure between vi and \u02c6 vj. We next describe the procedure used to \ufb01nd the set T for each 3D shape. First, for each 3D volumetric object, a set of similar shapes is found via k-nearest neighbors in the feature space. The feature mapping is performed by obtaining a single 2D rendered image and feeding it forward through an off-theshelf deep network [37] pre-trained on the ImageNet dataset. Afterwards, we form a complete bipartite graph G = {N, E} with nodes N and edges E. Assume that the capacity of each edge e connecting nodes p and q is denoted by wp,q = \u2212ce. The cost ce is de\ufb01ned by the IoU calculated for each edge in the bipartite graph. A canonical representation is obtained for 3D shapes by aligning their principal axes and matching spatial dimensions before primitive IoU calculation. The goal is to calculate maximum weight matching M between the disjoint partitions P and Q de\ufb01ned over the mth shape and its nearest neighbors. As a result, primitives within the 3D shape will have best matches that will preferably cooccur in similar 3D shapes. This problem can be formulated as an Integer Program (IP), but its solution is NP hard. To this end, we alternatively solve the following primal-dual linear relaxations of the original IP: Primal: min X p,q wp,q yp,q, s.t. X p\u2208P yp,q = 1, X q\u2208Q yp,q = 1, yp,q \u22650, p \u2208P, q \u2208Q. (4) \fSince the relaxed LP does not guarantee an optimal solution, we also construct a dual to the original LP where both are solved alternatively to \ufb01nd the optimal matching. The following lower-bound is maximized in the dual formulation: Dual: max X p\u2208P zp + X q\u2208Q zq s.t. zp + zq \u2264wp,q, (p, q) \u2208E (5) The algorithm runs in several iterations maintaining a feasible solution to the dual problem, and tries to \ufb01nd a feasible solution to the primal problem that satis\ufb01es complementary slackness i.e., a perfect matching M with only tight edges [31]. Note that if the matching is not perfect, the exposed nodes in the graph do not have corresponding co-occurrence constraints during the \ufb01nal optimization. 3.2. Model Inference For a given 3D shape dataset D, the proposed CRF model represents each shape xm with a set of primitive shapes. The CRF inference is formulated as a Mixed Integer Linear Program (MILP): V\u2217= argmin V E(V|D) s.t. vi = {0, 1}, yi,j \u22650, yij \u2264vi, yij \u2264vj, yij \u2265vi + vj \u22121, sk \u2264 X i:rk\u2208bi vi, sk \u22641, \u00b5par > 0, \u00b5par > 0, \u00b5par > 0, \u00b5cov < 0, \u00b5coc < 0, uij \u22650, uij \u2264vi, uij \u2264\u02c6 vj, uij \u2265vi + \u02c6 vj \u22121, vi \u2264 X j \u02c6 vj \u2200i, \u2200i, j, \u2200k (6) We use branch and bound algorithm [21] to ef\ufb01ciently solve the MILP based inference procedure. 3.3. Primitive Proposal Generation Here, we describe our proposed multi-view approach to generate primitive candidates. Given a polygon mesh of a 3D CAD model, we obtain rendered depth views of the model from six equi-spaced virtual viewpoints around the object. The virtual camera viewpoints were divided into two groups, one looking horizontally at the center of the upright object and the second camera viewpoint was chosen at an upward elevation of 150 such that the camera points towards the volume. These viewpoints were alternatively applied to get six rendered depth images that were subsequently used to obtain bottom-up polyhedron proposals. The rendered views provide sparse incomplete point clouds of the 3D shape that are mapped in the same frame of reference using a projective transformation. As an initial step, we generate a set of 3D box proposals via bottom-up grouping. First, a normal image is calculated based on the 3D sparse point cloud for each view. Next, rough surface segmentations are obtained by clustering the 3D points that are co-located, have similar appearance and whose normals point in the same direction. Spurious segmented regions are removed by dropping regions with a small number of 3D points. We then calculate all closely lying region pairs, that can potentially form the two visible surfaces of a bounding box enclosing a part of the 3D shape. For each pair, a bounding box is tightly \ufb01t to generate a candidate primitive. 4. Generative Modeling for Shape Generation The primitives discovered in an unsupervised manner allow us to factorize the shape generation process into two stages. The \ufb01rst GAN learns to generate novel primitive con\ufb01gurations that represent 3D shapes. The second GAN builds on this initial representation and \ufb01lls in local details to generate a complete 3D shape. A Variational Auto-encoder (VAE) connects the two generative models. The overall pipeline therefore transitions from simple shape parametrization to more complex 3D shape generation. By introducing a simpler auxiliary task in the generative modeling, we achieve three key advantages: (a) In contrast to existing 3D generative models that are separately trained for each object category, our model is jointly trained on all shape classes, (b) It provides better interpretability of generator\u2019s latent space and can incorporate user input to generate desired shapes, (c) The learned model achieves better 3D generation results and demonstrates highly discriminative features. We explain the generative modeling pipeline below. 100-d noise vector 256-d FC Layer Reshape (128,1,2) Batch Norm Leaky ReLU ReLU Dropout Upsampling (2,2) Convolution (128,3,3) Upsampling (2,2) Convolution (64,3,3) Upsampling (2,2) Convolution (1,3,3) Tanh Generator Input Convolution (32,3,3) Convolution (64,3,3) Discriminator Convolution (128,3,3) Convolution (256,2,2) Parametric Encoding Parametric  Encoding Convolution (64,4,4,4) Convolution (128,4,4,4) Encoder Convolution (256,4,4,4) Convolution (512,4,4,4) 2048-d FC Layer Reshape (256,2,2,2) Conv Transpose (128,4,4,4) Conv Transpose (64,4,4,4) Conv Transpose (32,4,4,4) Conv Transpose (16,4,4,4) Generator \ud835\udf07 \ud835\udf0e 500-d FC 500-d FC 256,2,2,2 128,4,4,4 64,8,8,8 Figure 4: Primitive GAN Architecture. 4.1. Primitive GAN Network Architecture: The primitive GAN consists of a generator and a discriminator network (Fig. 4) [17]. The training process is framed as a game between the two competing networks. The generator maps a random sample (e.g., from a Gaussian distribution) to the original data space. The discriminator operates in the data space and predicts whether an input sample is real or fake. The interesting aspect of our design is that an arbitrary number of primitives, t \u2208[1, N] (N = 6 in our case), are predicted for each 3D shape. This \ufb02exibility is crucial because different object types are rep\fFigure 5: 3D Shape GAN Architecture. It \ufb01rst uses a VAE to encode the parametric representation of primitives and then learns to generate complete 3D shapes with an adversarial objective. Output tensors are shown with dotted lines. 100-d  256Resha Dropout Upsam Convolu Upsam Convolu Upsam Convol Tanh Generator Input Convol Convol Discriminator Convolu Convolu Parametric Encoding Parametric  Encoding Convolution (64,4,4,4) Convolution (128,4,4,4) Encoder Convolution (256,4,4,4) Convolution (512,4,4,4) 2048-d FC Layer Reshape (256,2,2,2) Conv Transpose (128,4,4,4) Conv Transpose (64,4,4,4) Conv Transpose (32,4,4,4) Conv Transpose (16,4,4,4) Reshape (1,32,32,32) Generator Convolution (32,4,4,4,4) Convolution (64,4,4,4) Discriminator Convolution (128,4,4,4) Convolution (256,4,4,4) 3D Shape \ud835\udf07 \ud835\udf0e Batch Norm Leaky ReLU ReLU Dropout Tanh 500-d FC 500-d FC FC Layer 256,2,2,2 128,4,4,4 64,8,8,8 32,16,16,16 resented by different number of part primitives. N can be set higher at the cost of slower inference. Each primitive is encoded by a shape parameter set \u03b8s \u2208R15 including the box dimensions (i.e., height, width, depth), translation (along x, y and z axes) and the rotation matrix (with nine parameters). An additional parameter \u03b8l is included to denote the likelihood whether the primitive will be selected in the overall shape or not. This likelihood is used as a parameter to obtain a sample from Bernoulli distribution that denotes the existence of a primitive [39]. In this way, a 3D shape is encoded as a signi\ufb01cantly lower dimensional parametric representation. Loss Function: To allow a stable training of GAN, we used an improved form of Wasserstein GAN [1]. The WGAN exhibits better convergence behavior by employing Wasserstein distance as an objective. It enforces the discriminator model to remain within the space of 1-Lipschitz functions by weight clipping that can lead to sub-optimal convergence behavior. Instead of weight clipping, we used the gradient penalty introduced in [9] to restrict the norm of the gradients of the discriminator\u2019s output with respect to its input. The game between D and G is formulated as the following min-max objective function: min G max D E x\u223cPr [D(x)] \u2212 E \u02dc x\u223cPg [D(\u02dc x)]\u2212 \u03bb E \u02c6 x\u223cP\u02c6 x [(\u2225\u2207\u02c6 xD(\u02c6 x)\u22252 \u22121)2], (7) where Pr is the real data distribution, Pg is the generator distribution modeled as \u02dc x = G(z) such that z is a random sample from a \ufb01xed distribution and P\u02c6 x is the distribution de\ufb01ned with uniformly sampled points between the pairs of samples belonging to Pr and Pg. 4.2. 3D Shape VAE-GAN Network Architecture: The generative model for 3D shape generation consists of a combination of a variational auto-encoder and an adversarial network (Fig. 5). The complete architecture consists of three blocks, an encoder, a generator and a discriminator. The parametric shape representation is \ufb01rst converted to a coarse 3D shape in the form of a voxelized grid. The encoder maps this representation to the parameters of a variational distribution by applying a series of 3D convolutional and down-sampling operations. The generator then operates on this a random sample from this parameterized distribution and generates a new 3D shape to deceive discriminator, while the discriminator is trained to correctly categorize the real and fake 3D shapes. Remarkably, in contrast to primitive GAN, the shape GAN consists of 3D operations to accurately model the data distribution of 3D shapes. Loss Function: The loss function has the same form as for the case of primitive GAN, however, a regularization is applied on the input latent representation of generator to match it to a \ufb01xed known distribution (a unit Gaussian). This constraint is formulated as minimizing the KullbackLeibler (KL) divergence between the Gaussian and encoded distribution as follows: Lvae = KL(N(\u00b5, \u03c3)||N(0, I)). (8) The reparametrization trick proposed in [19] is used to perform back-propagation through the stochastic sampling from the distribution N(\u00b5, \u03c3). 5. Experiments Primitive Discovery: We evaluate the primitive detection accuracy on ModelNet10 dataset and report results in Table 1. Speci\ufb01cally, we convert the shapes and the 3D primitive representations to a 50x50x50 voxelized output. Evaluation is performed by accounting for the matched voxel predictions for both outputs. We obtain a high recall rate of 83% that con\ufb01rms the correct enclosure of shape parts by primitives. In contrast, a lower precision is obtained because shape parts are often hollow, that give rise to unmatched empty voxels. In our case, recall is a more accurate measure to asses the quality of primitives. Example results for primitive generation are shown in Fig. 7. \fFigure 6: Qualitative results for 3D shape generation. Left: Generated shapes from our model. Right: Comparisons with [42], [41] and [38] respectively from bottom to top. Wu et. al [2016] Wu et. al [2015] Smith et. al [2017] Evaluation Measure Performance Recall 83.0% Precision 19.9% Accuracy 60.8% F-measure 0.321 Table 1: Results for the primitive generation approach on ModelNet. A high recall shows that the predicted primitives generally tightly enclose the original 3D shape. Supervised Unsupervised Methods Accuracy Methods Accuracy PointNet (CVPR\u201917) [24] 86.2% T-L Net (ECCV\u201916) [7] 74.4% OctNet (CVPR\u201917) [26] 83.8% 3D-GAN (NIPS\u201916) [41] 83.3% Vol-CNN (Arxiv\u201919) [25] 86.5% Vconv-DAE (ECCV\u201916)[33] 75.5% EC-CNNs (CVPR\u201917)[35] 83.2% 3D-DescripNet (CVPR\u201918) [43] 83.8% Kd-Net (ICCV\u201917) [20] 88.5% 3D-GAN (Ours) 84.5% SO-Net (CVPR\u201918) [22] 90.8% Primitive GAN (Ours) 86.4% Table 2: Classi\ufb01cation performance on the ModelNet40 dataset. Type Method Accuracy Supervised 3D ShapeNets (CVPR\u201915) [42] 93.5% EC-CNNs (CVPR\u201917) [35] 90.0% Kd-Net (ICCV\u201917) [20] 93.5% LightNet (3DOR\u201917) [46] 93.4% SO-Net (CVPR\u201918) [22] 95.5% Unsupervised Light Field Descriptor (CGF\u201903) [5] 79.9% Vconv-DAE (ECCV\u201916) [33] 80.5% 3D-GAN (NIPS\u201916) [41] 91.0% 3D-DescripNet (CVPR\u201918) [43] 92.4% 3D-WINN (AAAI\u201919) [12] 91.9% 3D-GAN (Ours) 91.2% Primitive GAN (Ours) 92.2% Table 3: Classi\ufb01cation performance on the ModelNet10. Unsupervised Shape Classi\ufb01cation: To illustrate the improved performance of proposed generative model, we evaluate the representations learned by our discriminator (convergence plot is shown in Fig. 8). A typical way of evaluating representations learned without supervision is to use them as features for classi\ufb01cation. Note that the Primitive GAN model is only trained in an unsupervised manner on the ModelNet10 dataset, but tested on both ModelNet10 and ModelNet40 datasets. We extract intermediate feature layers from the discriminator, concatenate them and train a single layer neural network classi\ufb01er. The classi\ufb01cation results are shown in Tables 2 and 3. Our method beats all other unsupervised techniques by a fair margin of 1.9% Table 4: Inception scores for 3D shape generation. Best and Secondbest scores are shown in color. Method IS 3D-ShapeNet (CVPR\u201915) [42] 4.13\u00b10.19 3D-VAE (ICLR\u201915) [19] 11.02\u00b10.42 3D-GAN (NIPS\u201916) [41] 8.66\u00b10.45 3D-DescripNet (CVPR\u201918) [43] 11.77\u00b10.42 3D-WINN (AAAI\u201919) [12] 8.81\u00b10.18 Ours (Primitive GAN) 11.52\u00b10.33 on ModelNet40 dataset. On ModelNet10, we achieve a competitive performance as compared to the state-of-theart [43]. Note that some unsupervised methods have used class speci\ufb01c models, extra datasets (such as ShapeNet) and higher feature dimensions compared to ours. The proposed method also compares well with recent best performing fully supervised methods on both datasets. These approaches employ other tricks e.g., EC-CNNs [36] performs voting over 12 views of each test model at test time. Inception Scores: To quantitatively evaluate the generated 3D shapes, we report Inception Score (IS) in Table 4. The IS characterizes generated objects based on two distinct criterion: the quality of 3D outputs and their diversity [30]. The quality of generated outputs is measured by the conditional probability p(y|x), where y is the output label and x is the input shape. The diversity of a sample is computed by the marginal distribution R z p(y|x = G(z))dz. The KL-divergence between the two gives the Inception score: IS = exp(E[KL(p(y|x)||p(y))]). Notably, using a single model for shape generation, our model achieves the second best IS score on ModelNet10 which denotes the diversity and the quality of generates shapes. Primitive Based Reconstruction: In order to evaluate \fFigure 7: Automatically discovered primitive representations of 3D shapes in an unsupervised manner. Example results are shown for common indoor objects such as chair, table, desk, bathtub, sofa, monitor, toilet and nightstand. Our approach learns to represent common shapes in a parsimonious form that is consistent for examples belonging to the same category. Figure 8: Discriminator loss during 3D GAN training on the ModelNet dataset. Method Bed Bookcase Chair Desk Sofa Table Overall AlexNet-fc8\u2020 [7] 29.5 17.3 20.4 19.7 38.8 16.0 23.6 AlexNet-conv4\u2020 [7] 38.2 26.6 31.4 26.6 69.3 19.1 35.2 T-L Network\u2020 [7] 56.3 30.2 32.9 25.8 71.7 23.3 40.0 3D-VAE-GAN [41] 49.1 31.9 42.6 34.8 79.8 33.1 45.2 VAE-IWGAN [38] 65.7 44.2 49.3 50.6 68.0 52.2 55.0 Primitive GAN\u2217 68.4 52.2 47.5 56.9 77.1 60.0 60.4 Table 5: Reconstruction results for voxel prediction on IKEA dataset (AP is reported). \u2020Accuracies are reported from [41]. \u2217Primitive GAN uses primitive representations obtained from 3D shapes and therefore has more supervision relative to compared methods that propose shapes from 2D images. the reconstruction performance of the proposed GAN model, we test our approach on the IKEA dataset (Table 5). Previous works e.g., [41] aim to reconstruct a 3D model from a single color image. However, in our case, the VAE-GAN model is trained on parametric inputs representing a set of basic primitives instead of image inputs. Therefore, we \ufb01rst run our proposed primitive discovery algorithm on the IKEA dataset to estimate primitive representations and afterwards use these to reconstruct full shapes. Note that this dataset consists of 759 images with 1039 object crops and corresponding models that belong to six objects classes namely bed, bookcase, chair, desk, sofa, and table. Since the dataset shapes are at 20x20x20, we downscale the original network output to lower resolution for evaluation. Ablation study on cuboid detection: It is important to note that the proposed CRF formulation is an integrated framework where several potentials are optimized jointly. For example, useful primitives that are shared across similar shapes cannot be detected if we exclude any of the co-occurrence, coverage, parsimony or overlap potentials. Intuitively, one can understand that potentials like shape coverage and parsimony have opposite goals and they balance each other to get an optimal representation. However, we do run an ablation study with different types of unary potentials whose results are provided in Table 6 below. We also include a case where only unary costs are used to pick up the top four (average primitive number in dataset) primitives. We note that all potentials contribute to \ufb01nal performance and excluding one or some of them leads to lower recall rates. Table 6: Ablation study on ModelNet10 for unsupervised primitive detection. The unary cost itself is insuf\ufb01cient to generate a good primitive representation. The best result is achieved with our full model. Method Recall w/o Volumetric occupancy 72.8 w/o Shape uniformity 81.5 w/o Primitive compactness 77.2 w/o Support cost 82.9 w/o Shape symmetry 80.0 Unary only (top 4 boxes) 51.6 Full model 83.0 6."
+                }
+            ],
+            "Izzeddin Teeti": [
+                {
+                    "url": "http://arxiv.org/abs/2308.04589v2",
+                    "title": "Temporal DINO: A Self-supervised Video Strategy to Enhance Action Prediction",
+                    "abstract": "The emerging field of action prediction plays a vital role in various\ncomputer vision applications such as autonomous driving, activity analysis and\nhuman-computer interaction. Despite significant advancements, accurately\npredicting future actions remains a challenging problem due to high\ndimensionality, complex dynamics and uncertainties inherent in video data.\nTraditional supervised approaches require large amounts of labelled data, which\nis expensive and time-consuming to obtain. This paper introduces a novel\nself-supervised video strategy for enhancing action prediction inspired by DINO\n(self-distillation with no labels). The Temporal-DINO approach employs two\nmodels; a 'student' processing past frames; and a 'teacher' processing both\npast and future frames, enabling a broader temporal context. During training,\nthe teacher guides the student to learn future context by only observing past\nframes. The strategy is evaluated on ROAD dataset for the action prediction\ndownstream task using 3D-ResNet, Transformer, and LSTM architectures. The\nexperimental results showcase significant improvements in prediction\nperformance across these architectures, with our method achieving an average\nenhancement of 9.9% Precision Points (PP), highlighting its effectiveness in\nenhancing the backbones' capabilities of capturing long-term dependencies.\nFurthermore, our approach demonstrates efficiency regarding the pretraining\ndataset size and the number of epochs required. This method overcomes\nlimitations present in other approaches, including considering various backbone\narchitectures, addressing multiple prediction horizons, reducing reliance on\nhand-crafted augmentations, and streamlining the pretraining process into a\nsingle stage. These findings highlight the potential of our approach in diverse\nvideo-based tasks such as activity recognition, motion planning, and scene\nunderstanding.",
+                    "authors": "Izzeddin Teeti, Rongali Sai Bhargav, Vivek Singh, Andrew Bradley, Biplab Banerjee, Fabio Cuzzolin",
+                    "published": "2023-08-08",
+                    "updated": "2023-08-20",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Computer vision techniques have advanced to the point at which they are able to outperform humans at certain object recognition tasks [55]. However, for many computer vision applications, a higher-level understanding of the scene is required. For example, achieving human-level performance in autonomous vehicles remains a formidable challenge [47]. One of the key reasons for this gap is the inherent difficulty in understanding what may happen next. Thus there is a growing recognition of the importance of prediction. Prediction plays a crucial role in enhancing the decision-making process of autonomous systems by anticipating the future behaviour of dynamic elements in the environment, e.g. other vehicles, pedestrians, and cyclists thus ensuring safer operation. Moreover, prediction also facilitates the development of high-level understanding within autonomous systems, enabling more nuanced and contextually appropriate planning, leading to smoother interactions with other agents on the road [30]. However, prediction poses its own set of challenges, encompassing spatial, temporal, social, and stochastic dimensions [50]. Modelling these dimensions requires complex models, such as [43, 45, 61, 37], which require significant amounts of data often scarce and costly to gather and annotate. To address this, leveraging the abundance of unlabelled data through self-supervised methods offers an enticing opportunity to enhance performance with minimal impact upon resources. While existing self-supervised prediction methods, including [54, 63, 27], have shown promise, they have limitations that hinder their effectiveness. Firstly, their predictive capability is limited to a very short-term horizon (typically one frame ahead) which is impractical for autonomous driving scenarios requiring longer-term predictions. Secondly, these methods often involve a two-stage process [54], which is computationally expensive and time-consuming. Finally, they are typically designed for a specific architecture, lacking the ability to generalize across different architectures. In this paper, we present a novel one-stage selfsupervised representation learning strategy specifically dearXiv:2308.04589v2  [cs.CV]  20 Aug 2023 \fTemporal  Downsampling EMA SGD Pretraining Fine-tunning Past Future t 2 \u2026 t+1 t+tPred t+ 2 \u2026 1 3 1 t+1 t+tPred \u2026 \u2026 FPD  Loss Figure 1: Overview of the proposed Temporal DINO. The student model processes the past frames (x1:t), while the teacher processes both the past and future frames (x1:t+tP red) . A Future-past Distillation loss is applied to their representations (S\u03b8 and T\u03d5) to guide the student to capture the future temporal context from the teacher. signed for videos. Our proposed approach draws inspiration from the image-based self-supervised DINO [7] model and extends its application to the temporal dimension. Leveraging a student-teacher framework, our method guides the student model to focus on the most informative (temporal) features, enabling accurate predictions of future events. The student model learns to attend to the relevant cues in the past and current moments, extracting valuable information that aids in forecasting forthcoming actions. Our approach addresses the aforementioned limitations by significantly extending the prediction range beyond one frame, enabling more practical and effective autonomous driving applications. Moreover, our proposed method eliminates the need for a two-stage process, reducing computational complexity and saving valuable time. Finally, our approach is not limited to a single architecture but is a wrapper strategy that can be used to improve prediction performance across various architectures, enhancing the generalisability and applicability of the method. The contributions of this work include: \u2022 A novel one-stage self-supervised representation learning strategy for videos, addressing limitations in prediction range, computational complexity, and architectural generalisability. \u2022 Proof-of-concept evaluation of the approach in both autonomous driving, and a more general task with various deep-learning architectures (including 3DCNN, Transformer, and LSTM), showcasing its effectiveness on real-world challenging data, and versatility to use on other models and domains. \u2022 Identification of the optimal model architecture and loss function for capturing long-range dependencies in video data, shedding light on the most effective design choices for robust action prediction in autonomous driving scenarios. In the following sections, we provide a comprehensive overview of related work in self-supervised learning for images and videos, and action prediction (Section 2), followed by a detailed description of our proposed method (Section 3). We then present the experimental setup, including datasets, evaluation metrics, and training details (Section 4), and discuss the results and analysis of our experiments (Section 5). Finally, we provide a thorough discussion of our findings, and highlight potential applications and future research directions (Section 6). 2. Related Work 2.1. Image-based Self-supervised Learning Two types of self-supervised strategies are commonly employed in computer vision: pretext methods and contrastive learning. The former involves utilizing specific internal properties or tasks of the input data to learn useful representations. For instance, some models learn the contextual information of the entire scene by analyzing small image patches [10]. Other approaches focus on tasks like re-colourization of grayscale images [64], predicting transformations between different views of the same image [14], or determining the order of patches extracted from an image [38, 35]. With the introduction of Vision Transformer (ViT) [11], patch-based self-supervised methods have gained prominence in the literature [21, 3, 2]. Particularly, masked auto-encoders (MAE) [21] have emerged as a preferred mechanism for model pretraining due to their superior performance on downstream tasks. Contrastive learning methods, on the other hand, focus on maximizing the dissimilarity between features extracted \ffrom different samples to encourage discriminative representations [12, 5, 58]. Some discriminative self-supervised algorithms leverage instance-level discrimination to create distinct feature representations for different examples, leading to robust feature learning [22, 9, 17, 7]. Other approaches draw inspiration from clustering mechanisms [6, 1]. The DINO method [7], for example, employs selfdistillation that trains a student model on local crops and a teacher model on the entire image, then find the loss between both representations. The teacher will push the student to learn global representations by seeing only the local ones. Whilst these techniques offer significant potential on images, their performance is limited on video-related tasks, such as action prediction. Thus, there is a need for advancements in video self-supervised learning methods. 2.2. Video-based Self-supervised Learning Video-based self-supervised learning strategies, akin to their image counterparts, encompass both pretext and contrastive approaches [46]. However, videos introduce an additional dimension, namely the temporal dimension. The inherent order of frames within a video sequence offers intrinsic properties that can be leveraged to develop effective self-supervised learning mechanisms. Despite this potential, the research in this specific domain remains relatively limited [51, 4, 36, 57]. To address this gap, recent works have explored different strategies in video-based self-supervised learning. For instance, [20] employed complementary information from RGB and optical flow streams to co-train their model using contrastive loss. [18] utilized a 3D-CNN architecture with a video transformer and trained the model using contrastive loss on positive pairs of video sequences. CVRL [42] employed a discriminative learning approach, utilizing two augmented views of the same video as positive examples and views of other videos as negatives. Another method, VideoMoCo [39], extended the MoCo [22] framework to videos by randomly dropping frames from the video and learning the same representation for each random input. In line with these advancements, our proposed model introduces a novel approach where the teacher model incorporates a longer temporal sequence than the student model. This design choice aims to provide the student with a wider temporal context, encompassing future frames that the student has not yet observed. By leveraging this extended temporal context within the student-teacher framework, our model aims to enhance the student\u2019s ability to learn representations that capture long-term dependencies and improve its performance in video-based self-supervised learning. 2.3. Supervised Action Prediction Solving the action prediction task requires modelling the different dimensions of the problem, including the spatial, temporal, stochastic, and social dimensions, since the driving environment is dynamic, uncertain, and multi-agent. To model the temporal dimension, [13, 29] used 3D-CNN, [43, 16] Recurrent Neural Networks, while [61] used Transformers. Regarding the stochastic dimension, GANs [28], and CVAE models [16, 61] were utilised. To model the multi-agent aspect of the problem, different types of Graph Neural Networks of different connectivity, sparsity and homogeneity were used [15, 45, 16], semantic segmentation cues [44, 34] and social pooling [40]. However, all of those methods are supervised; they require a huge amount of labelled data which is time-consuming and expensive. 2.4. Self-supervised Action Prediction In recent studies, limited approaches have been explored in the field of self-supervised action prediction. Zatsarynna et al. [63] employed the contrastive loss of InfoNCE [19] to encourage proximity between temporally adjacent video clips in the embedding space. To preserve the order of the clips, they complemented the InfoNCE loss with an order loss in the form of cross-entropy. Their proposed model focused on using 3D-CNN as the backbone architecture, and its evaluation was limited to this specific backbone without examining the performance on other architectures. Another approach by Kochakarn et al. [27] utilized graph contrastive learning with the SimCLR loss function [9] to learn more informative embeddings for the prediction task. They incorporated an attention mechanism to achieve explainable action prediction, albeit limited to predicting only the next one frame. In contrast, our proposed method extends the prediction horizon to include the next 3, 6, and 12 frames, offering a more comprehensive temporal context. Furthermore, their approach specifically focused on graphs, while our method is designed to work with 3DCNN, Transformers, and LSTMs, providing broader applicability. It is worth noting that both of these aforementioned approaches rely heavily on the use of contrastive loss, which requires careful crafting of augmentations. This dependency on specific augmentations can limit the generalisability and robustness of the learned representations. In contrast, Tran et al. [54] adopted a knowledge distillation approach, where a teacher model trained on recognition tasks transfers its knowledge to a prediction model. However, this method follows a two-stage process involving training a teacher model for recognition and then distilling the knowledge into a student model for prediction. This two-stage approach introduces additional time and computational costs, which may impact the scalability and practicality of the method, particularly in real-time or resourceconstrained scenarios. Furthermore, their evaluation was focused on the I3D architecture [8] without exploring the performance on other architectures. \fThese related works provide valuable insights into selfsupervised action prediction approaches. However, they have certain limitations in terms of the backbone architectures considered, the prediction horizons addressed, the reliance on specific augmentations, and the two-stage training process. In contrast, our one-stage proposed method aims to overcome these limitations by leveraging a different loss formulation and considering multiple backbone architectures while extending the prediction horizon, thereby contributing to the advancement of self-supervised action prediction techniques. 3. Methodology 3.1. Representation Learning for Action Prediction In this study, our objective is to learn a mapping function f in an unsupervised manner, which takes an unlabelled and untrimmed video clip consisting of t \u2208R+ frames as input. The goal is to map the 4D input clip x1:t \u2208RT \u00d7C\u00d7H\u00d7W , to a feature vector g(x1:t) \u2208Rd, such that the learned features effectively transfer to the downstream task of action prediction. To achieve this, we draw inspiration from the DINO approach and adopt a student-teacher setup, as depicted in Figure 1. The student network S\u03b8 processes only the past frames x1:t during both training and inference, without access to future frames. On the other hand, the teacher network T\u03d5 processes both the past and future frames x1:t+tP red, where tP red denotes the length of the future sequence. To ensure consistency in architecture, we downsample the sequence processed by the teacher network to match the number of frames processed by the student. The downsampling is performed by determining a sampling frequency, which is calculated as (t+tP red)/t. For instance, if t = 12 frames and tP red = 12 frames, the student processes the past (12) frames, while the teacher processes (24 frames) with a step of 2, resulting in 12 frames. Despite processing sequences of the same sequence length, the teacher network has access to a wider temporal context. 3.2. Future-past Distillation Loss During training, we introduce a knowledge distillation loss between the final embeddings of the student and teacher networks. This loss guides the student to distil knowledge about the future from the teacher, despite not having direct access to future frames. The aim is to teach the student to focus on the most relevant features from the past frames that contribute to predicting the future. In contrast to DINO, our Future-Past Distillation (FPD) loss is defined in the Cosine Similarity form, instead of using cross-entropy. This formulation is motivated by the findings of our ablation analysis, which indicate that the Cosine-based FPD loss yields improved performance on downstream tasks. The learning objective for the pretraining stage of future-past distillation is expressed in Equation 1. The student network parameters \u03b8 are updated using backpropagation optimized by stochastic gradient descent (SGD), while the teacher network parameters \u03d5 are updated using an Exponential Moving Average (EMA) based on the student network, with a scheduled momentum variable (m) as shown in Equation 2. \u03b8\u2217, \u03d5\u2217= arg min \u03b8,\u03d5 LF P D \u0010 S\u03b8(x1:t), T\u03d5(x1:t+tP red) \u0011 (1) \u03d5i+1 = mi \u00d7 \u03d5i + (1 \u2212mi) \u00d7 \u03b8i (2) 3.3. Downstream Task Definition The objective of pretraining is to enhance the performance of the model in the downstream task of predicting driver\u2019s actions. Given a past (observed) clip of length t \u2208R+, the task is to predict the Ego-vehicle (driver) action in each frame in the next tP red \u2208R+ frames. Building upon the optimal pretrained student model S\u03b8\u2217, the prediction model f will use the student model as a backbone and add a classification head on top of it. Subsequently, it performs further optimization (fine-tuning) on either both the backbone and the head parameters or solely the latter (we conducted experiments on both scenarios) for the prediction task. The objective is to map the learned features S\u03b8\u2217(X1:t) to the future action labels yt+1:t+tP red, formally, f\u03c8 : S\u03b8\u2217(X1:t) \u2192yt+1:t+tP red. Given the nature of a classification task, cross-entropy (CE) loss is utilized to guide the optimization process and refine the learning objective for the downstream task, as illustrated in Equation 3. \u03b8\u2217\u2217, \u03c8\u2217= arg min \u03b8\u2217,\u03c8 LCE \u0012 f\u03c8 \u0010 S\u03b8\u2217(x1:t) \u0011 , yt+1:t+tP red \u0013 (3) 4. Experiments 4.1. Datasets We adhered to the convention in self-supervised learning, utilizing two distinct datasets for distinct purposes: a larger dataset for pretext task pretraining and a smaller dataset for downstream task fine-tuning. Specifically, we employed the Kinetics-400 dataset [25] and the ROad event Awareness Dataset (ROAD) [48]. Kinetics-400 It is designed for action recognition and comprises over 240,000 videos. On average, each video spans 10 seconds and is assigned a single label from a pool of 400 possible action classes. The dataset\u2019s substantial video collection has facilitated its adoption in numerous \fvideo self-supervised methods [51, 39]. For our purposes, we disregard the label information as it will not be used during pretraining. ROAD The ROad event Awareness Dataset (ROAD) [48] is built on a fraction of Oxford RobotCar Dataset [33], and it is extended with multi-label annotations for action recognition, localisation, and prediction tasks within the context of autonomous driving. It comprises 22 videos from an egocentric view as shown in Figure 2, each with an 8-minute duration and a frame rate of 12 frames per second (fps). Importantly, it contains labels indicating the actions performed by the ego vehicle (driver). Notably, the dataset encompasses seven distinct ego-vehicle actions: Move, Stop, Turn Left, Turn Right, Overtake, Move Left, and Move Right. These labels serve as training data for fine-tuning our models to predict the driver\u2019s actions in future frames during the prediction task. 4.2. Models We conducted a series of experiments utilizing four diverse deep-learning architectures. R3D [52] This architecture employs a 3D convolutional neural network (3D-CNN) backbone for processing video data. It is based on the ResNet-18 [23] architecture, which has proven to be successful in image recognition tasks. Swin [31] This architecture utilizes a Transformer-based 3D backbone for video processing. It is based on the Transformer architecture, originally introduced in the context of image recognition as Vision Transformer (ViT) [11]. ResNet-LSTM This architecture combines a convolutional neural network (CNN)-based 2D backbone using the ResNet-50 architecture for image processing, along with a Long Short-Term Memory (LSTM) layer for capturing temporal dependencies. ViT-LSTM Same as the previous but replacing the ResNet with a Transfomer-based 2D backbone. The first two models use spatio-temporal backbones, which extract spatial and temporal features simultaneously, while the last two use a spatial backbone and connect it using a temporal one. These models exhibit differences in their depth and working mechanisms, offering a diverse range of approaches to our experiments. By leveraging these varied architectures, we can comprehensively investigate our proposed representation learning strategy, examine their performance, and compare their effectiveness according to the experiments outlined in the subsequent section. 4.3. Experimental Protocol In accordance with the conventions of self-supervised learning experiments, we evaluated the performance of our models on the downstream task of action prediction under three distinct protocols: 1) Full-Supervised: This protocol represents the results obtained from the model without employing the pretraining strategy. The model was trained solely on the labelled data of the prediction task. 2) Linear Probing: In this protocol, the proposed strategy was applied, and the model was fine-tuned by solely updating the parameters of the prediction head. The backbone of the model was kept frozen during this fine-tuning process. 3) Fine-tuning: This protocol involved applying the proposed strategy and performing full fine-tuning of the entire network, including both the backbone and the prediction head. All parameters of the model were updated during the fine-tuning process. By examining the model\u2019s performance across these three protocols, we can gain insights into the effectiveness and impact of the pretraining strategy on action prediction. 4.4. Implementation Details We used Pytorch framework [41] to implement the models, and the Precision (P) metric was used to evaluate their performance on the action prediction task. All experiments were conducted on NVIDIA A30 graphics cards. Below are the details of the pretraining and fine-tuning. Pretraining For pretraining, we utilized either the full dataset of Kinetics-400 or the training split of ROAD (depending on the experiment). The pretraining was optimised for 1000 (50) epochs using SGD optimiser with a learning rate of 0.005 (0.001) and a batch size of 64 (32) for Kinetics-400 and ROAD, respectively. Additionally, a cosine scheduler was employed for updating the momentum variable of the exponential moving average (EMA). Concerning the models, the LSTM architecture had a hidden dimension of 512, and the small structure of ViT (ViT-s) was used. Fine-tuning Fine-tuning was performed on the ROAD dataset, with a data split of 60% for training, 20% for validation, and 20% for testing. Cross-entropy loss was employed as the objective function for fine-tuning, and it was optimised for 10 epochs using the SGD optimizer with a learning rate of 0.001, and a batch size of 32. 4.5. Ablations In this section, we present different ablations to study the effects of different architectural components on the model \f(a) ROAD dataset (b) UCF101 dataset Figure 2: Sample images from ROAD and UCF101 datasets. performance. Specifically, we investigate the effects of variations in input sequence temporal length, self-supervised objective, backbone selection, and the strategy\u2019s performance on another downstream task (action recognition). Detailed explanations of each ablation study are provided below. Backbone and Temporal Length The choice of model backbone plays a critical role in the model\u2019s ability to learn effective features. Similarly, the length of the input sequence contributes to the model\u2019s capability to capture temporal dynamics and visual changes within the scene. Longer sequence lengths generally provide more visual data, but they may also introduce challenges in modelling long-term dependencies and potentially lead to performance degradation, as shown in Table 1. In this specific ablation study, We conducted experiments with the four backbones mentioned in Section 4.2, and we varied the temporal depth by using sequence lengths of 3, 6, and 12 frames for each backbone. We trained these models using the three experimental protocols outlined in Section 4.3. The prediction performance of the four models with different input lengths under the three protocols is summarized in Table 1. Loss Function Existing literature demonstrates that the choice of learning objective in self-supervised algorithms significantly influences the performance of models on downstream tasks [46]. In our experiments, we employed three widely used loss functions: Cross-entropy, Cosine Similarity, and Mean Squared Error (MSE) loss. The evaluation of these loss functions was performed on R3D and Swin backbones, and the corresponding results are summarized in Table 2. Action Recognition In addition, we performed supplementary experiments to assess the impact of our proposed strategy on a different video-based downstream task, namely Action Recognition. For this purpose, the models underwent pretraining on the Kinetics-400 dataset and subsequent fine-tuning on the UCF101 dataset [49], shown in Figure 2. Table 4 presents the state-of-the-art (SOTA) performance for action recognition. 5. Results Observing Table 1, it is evident that our proposed strategy yielded notable enhancements in the prediction performance of all backbone architectures, albeit to varying extents. Notably, the impact of the strategy is particularly pronounced on the 2D-based backbones, which initially exhibited comparatively lower results in the fullysupervised setup. This observation aligns with expectations since these backbones lack inherent spatio-temporal modelling capabilities. However, when trained with TDINO, the ResNet-LSTM and ViT-LSTM backbones witnessed substantial improvements, with average increases of 22.5 and 10.1 in terms of precision points (PP), respectively. Additionally, the video backbones, R3D and Swin, experienced PP gains of 5.2 and 1.8, respectively. Table 3 shows the comparison of our strategy to other SOTA methods, showing that it surpasses the performance of both supervised and self-supervised approaches on action prediction on ROAD. Within the same table, the column \u2018Supervised\u2019 indicates a fully supervised method, the opposite denotes a self-supervised approach, and \u2018Frozen\u2019 means the finetuning process exclusively updates the classification head while leaving the pre-trained backbone untouched, the opposite indicates that the entire model was updated during the fine-tunning. \fTable 1: The precision of different backbones with varying input lengths under the three protocols mentioned in Section 4.3. Backbone Pretrained on Interval (frames) Linear Probe Fine-tunning Supervised Improvement R3D ROAD 3 36.6 77.7 74.1 3.6 6 29.7 69.2 64.9 4.3 12 36.2 53.3 45.7 7.6 Swin ROAD 3 84.7 87.2 86.4 0.8 6 75.8 82.6 81.7 0.9 12 60.1 60.9 57.1 3.8 ResNet+LSTM Kinetics-400 3 77.6 84.6 62.9 21.7 6 70.3 81.8 58.3 23.5 12 58.3 76.7 54.3 22.4 ViT+LSTM Kinetics-400 3 73.5 77.7 69.3 8.4 6 70.2 76.7 65.5 11.2 12 64.21 66.2 55.5 10.7 Table 2: Evaluation of three common loss functions on R3D and Swin backbones. Backbone Loss Linear Probe Fine-tunning Supervised Improvment R3D MSE 32.0 37.3 45.7 -8.4 Cosine 36.2 53.3 45.7 7.6 Cross-entropy 30.9 50.4 45.7 4.7 Swin MSE 57.2 57.4 57.1 0.3 Cosine 60.1 60.9 57.1 3.8 Cross-entropy 49.6 58.4 57.1 1.3 To have a better understanding of the generalisation capability of the proposed model, we compare the performance of T-DINO with SOTA methods on another task, human action recognition (on UCF101), summarised in Table 4. The results highlight the effectiveness of the enhanced temporal modelling offered by T-DINO when applied to the R3D backbone, Furthermore, examining the results for varying input sequence intervals across each backbone, it becomes apparent that greater improvements are observed at longer input sequences. This suggests that pretraining with T-DINO equips the backbones with enhanced abilities to capture and model long-term dependencies in the data. Notably, the Swin-transformer-based models demonstrate higher accuracy, attributed to the superior representation capabilities offered by Transformers. We observed that transformerbased spatial feature extractor (ViT) combined with LSTMbased temporal sequence modelling results in a huge improvement in the downstream task. Analyzing the results obtained from the longest input configuration consisting of 12 frames, it is evident that models pretrained on the larger dataset, Kinetics-400, exhibit superior performance compared to those pretrained on the ROAD dataset. Moreover, the models that incorporate separate modelling of spatial and temporal relationships, such as ViT+LSTM and ResNet+LSTM, outperform the models that jointly model these relationships, namely R3D and Swin, by a margin of 16.6 and 5.7 percentage points, respectively. In regard to the selection of the most optimal loss function, the findings presented in Table 2 indicate that T-DINO pretrained using the Cosine Similarity loss surpasses the performance of models trained with MSE or Cross-entropy loss. Of significant importance, self-supervised models typically require extensive datasets and a high number of pretraining epochs to achieve satisfactory generalization on downstream tasks. However, our proposed strategy, pretrained on the ROAD dataset, exhibits a relatively comparable level of performance to models pretrained on the larger Kinetics-400 dataset. Notably, the ROAD dataset possessed a substantially smaller size and was pretrained with a lower number of epochs. These findings demonstrate T-DINO\u2019s resource and time efficiency in achieving desirable results. \fTable 3: Comparison of the proposed methods with SOTA on ROAD for the action prediction task when the input length is 12 frames Method Pretrained on Backbone Supervised Frozen Precision R3D [53] Road R3D \u2713 \u2717 45.7 Swin [32] Road Swin \u2713 \u2717 57.1 Resnet+LSTM [62] Road ResNet50 \u2713 \u2717 54.3 ViT+LSTM Road ViT \u2713 \u2717 55.5 VideoMAE [51] Kinetics ViT \u2717 \u2717 75.1 T-DINO (ours) Road R3D \u2717 \u2713 53.3 T-DINO (ours) Kinetics ResNet+LSTM \u2717 \u2713 58.3 T-DINO (ours) Kinetics ResNet+LSTM \u2717 \u2717 76.7 T-DINO (ours) Kinetics ViT+LSTM \u2717 \u2713 64.2 T-DINO (ours) Kinetics ViT+LSTM \u2717 \u2717 66.2 Table 4: Comparison of the proposed methods with SOTA on UCF101 for action recognition task. Method Year Pretrained on Arch. Supervised Frozen Acc. ClipOrder [59] 2019 UCF101 R3D \u2717 \u2717 72.40 3D ST-puzzle [26] 2019 Kinetics C3D \u2717 \u2717 65.80 Wang et al. [56] 2019 UCF101 C3D \u2717 \u2717 61.20 PRP [60] 2020 Kinetics R3D \u2717 \u2717 72.10 SpeedNet [4] 2020 Kinetics S3D-G \u2717 \u2717 81.10 CSJ [24] 2021 K+UCF101 R(2+3)D \u2717 \u2717 79.50 VideoMoCo [39] 2021 Kinetics R(2+1)D \u2717 \u2717 78.7 CACL [18] 2022 UCF101 R(2+1)D \u2717 \u2717 82.5 VideoMAE [51] 2022 Kinetics ViT \u2717 \u2717 96.1 T-DINO (ours) 2023 Kinetics ResNet+LSTM \u2717 \u2717 80.26 T-DINO (ours) 2023 Kinetics R3D \u2717 \u2717 83.9 T-DINO (ours) 2023 Kinetics ViT+LSTM \u2717 \u2717 85.86 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2201.03246v3",
+                    "title": "Vision in adverse weather: Augmentation using CycleGANs with various object detectors for robust perception in autonomous racing",
+                    "abstract": "In an autonomous driving system, perception - identification of features and\nobjects from the environment - is crucial. In autonomous racing, high speeds\nand small margins demand rapid and accurate detection systems. During the race,\nthe weather can change abruptly, causing significant degradation in perception,\nresulting in ineffective manoeuvres. In order to improve detection in adverse\nweather, deep-learning-based models typically require extensive datasets\ncaptured in such conditions - the collection of which is a tedious, laborious,\nand costly process. However, recent developments in CycleGAN architectures\nallow the synthesis of highly realistic scenes in multiple weather conditions.\nTo this end, we introduce an approach of using synthesised adverse condition\ndatasets in autonomous racing (generated using CycleGAN) to improve the\nperformance of four out of five state-of-the-art detectors by an average of\n42.7 and 4.4 mAP percentage points in the presence of night-time conditions and\ndroplets, respectively. Furthermore, we present a comparative analysis of five\nobject detectors - identifying the optimal pairing of detector and training\ndata for use during autonomous racing in challenging conditions.",
+                    "authors": "Izzeddin Teeti, Valentina Musat, Salman Khan, Alexander Rast, Fabio Cuzzolin, Andrew Bradley",
+                    "published": "2022-01-10",
+                    "updated": "2023-01-02",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Autonomous driving is a current focus of many governments, startups, already-established corporations, and research labs [7]. Many studies aim to create an SAE level 4 or level 5 autonomous vehicle (AV) [17], able to operate in unstructured environments and adverse conditions. From the AV\u2019s perspective, an adverse condition is any condition with which the detection model has never experienced during training. Of particular interest are the effects of weather and lighting effects that change the appearance of objects, such as droplets (which can occlude objects and blur images), or night driving. Research in AVs has stimulated the development of autonomous racing, where cars designed for racing compete \ufb01ercely on racetracks, potentially under varied weather conditions which may be subject to rapid changes. The close, competitive nature of racing ampli\ufb01es the effects of any performance gains (or weaknesses) in the AV control system in a safe, controlled test environment [9]. Autonomous racing is therefore an ideal application to test the extreme operating limits of AVs, accelerating the development process of AV algorithms including object detection in adverse conditions. Two main approaches exist to AV control systems. The \ufb01rst is an \u2018end-to-end\u2019 AI-based approach that takes visual inputs and outputs control commands [2]. The end-to-end nature of such systems, however, makes them black boxes: they lack explainability and are dif\ufb01cult to debug. Furthermore, they require an extensive amount of labelled training data; however, gathering such data is time-consuming and expensive. The second approach breaks down the AV control system into a pipeline of interacting subsystems including perception, decision making, and control [1] enhancing explainability, debugging, and a higher con\ufb01dence in the ability of the autonomous vehicle to handle a variety of road scenarios. Both approaches, however, are vulnerable to adverse conditions, as the latter distorts the visual input upon which they depend. AVs see (perceive) the world through visual inputs, acquired by sensors such as (optical) cameras and LiDAR both of which are susceptible to adverse conditions including snow, rain, fog, and night [28]. Rain droplets can accumulate *Autonomous Driving & Intelligent Transport Group, Oxford Brookes University \u2020Oxford Robotics Institute, University of Oxford \u2021Visual Arti\ufb01cial Intelligence Laboratory, Oxford Brookes University \u00a7for email correspondence: abradley@brookes.ac.uk 1 arXiv:2201.03246v3  [cs.CV]  2 Jan 2023 \fon camera lenses, occluding objects and agents, or distorting their shapes. If a LiDAR beam intersects a rain droplet at a short distance from the transmitter, the raindrop can re\ufb02ect enough of the beam back to the receiver to cause the raindrop to be detected as an object [12]. Night driving is particularly challenging for optical sensors; the level of re\ufb02ected light from target objects or agents to the camera sensor is low, causing dim or barely recognisable objects. Perception tasks, including object detection, semantic segmentation, action recognition, and trajectory prediction are hampered by poor visual inputs caused by adverse weather, resulting in missed and incorrect detections [28] of critical importance for AVs to safely \u2019see\u2019 other road users. Errors in detections will propagate through downstream tasks including path planning and control, causing inaccurate and unreliable manoeuvres and compromising the safety of other road users. In the particular application of AV racing, a small error in perception may result in a considerable deviation from the optimal path [21] and a loss of time or even penalties or accidents. In both on-road and racing applications, the AV industry has a critical and urgent need to mitigate or deal with the signi\ufb01cant degradation of perception caused by adverse weather conditions [39, 13]. Since both end-to-end and independent-subsystems approaches typically use machine learning models to understand the visual inputs, one way to improve them in adverse weather is to retrain those models on more data that contains adverse weather conditions. However, obtaining this data is time-consuming, expensive, and at the mercy of the weather. It also introduces a very costly and laborious process of (manually) labelling the collected data for different tasks like detection or segmentation. To overcome this challenge, researchers have suggested synthesising adverse weather datasets using simulators [52] or simple augmentation algorithms [13]. However, the applicability of these approaches for enhancement of object detection in a real-world setting is questionable, due to the lack of visual realism in the imagery. A recent promising approach is harnessing CycleGAN to generate more realistic weather augmentations using style transfer. A small number of studies have explored the use of CycleGAN to improve perception performance in adverse weather in the context of on-road autonomous driving tasks [33, 47, 29], but as yet, no studies have attempted to exploit this potential in the context of autonomous racing. Furthermore, various object detection models have been proposed for real-time use in AV applications [19, 20, 38, 44], however few studies have compared different detectors for use in autonomous driving applications [15] let alone with a detailed comparison of the detection performance and latencies introduced. This study represents the \ufb01rst attempt to harness CycleGAN-based weather appearance transfer to generate realistic training data for multiple adverse weather conditions in autonomous racing, and explores the performance of \ufb01ve leading object detectors for real-time object detection in adverse conditions. Thus, the contributions of this study are: 1. An approach to using individual synthesised adverse weather conditions to improve detection performance under multiple real weather conditions in autonomous racing. 2. Performance improvements for four state-of-the-art object detectors under multiple real adverse weather conditions by training them on synthesised adverse weather datasets. 3. Benchmark results for \ufb01ve different object detectors, a comparative analysis of their performance in terms of accuracy and speed in both sunny and adverse conditions, and the identi\ufb01cation of the optimal detector for the speci\ufb01c task of object detection for autonomous racing cars. 2 Related work 2.1 Object detection in Autonomous Racing Traf\ufb01c cones are used to regulate traf\ufb01c and to mark racing circuit layout, therefore detecting them is crucial for autonomous driving and autonomous racing. The earliest cones detection methods proposed simple colour-based techniques using \ufb01lters and thresholds to \ufb01nd cones [18]. Generally, these techniques are quite unreliable and are characterised by many false positives [16], because they ignore a fundamentally distinctive feature of cones: their shape. Template matching methods, which account for both colour and shape of cones, were also attempted for cone detection [51, 53]. Although they show accuracy improvements, their results are poor when cones are slightly deformed, partially occluded, or appear at multiple scales. Feature-based detectors have also been explored by [45], which used Histogram of Oriented Gradients (HOG) [10] to detect cones. In the current decade, Convolutional Neural Networks (CNNs), have proven to be highly accurate for many perception \ftasks, such as recognition, localisation, segmentation, and, most importantly, detection [22]. CNNs are characterised by multiple feature extraction stages that can automatically learn representations from the data, thus, trained on the right dataset, CNN-based models can detect objects even when occluded, deformed, or rescaled. For cone detection in a racing context, papers such as [35] employed a colour-based detector, whereas [45] used an HOG representation. Many systems, however, including [11, 42, 9], use different versions of the well-known YOLO detector [36] because of its speed, accuracy, and reliability. A variety of object detectors were used in AV racing, however, there is no clear study that makes comparative analysis between these models and therefore an investigation into the ideal detector, considering detection accuracy and speed, would be a useful contribution to the \ufb01eld. Despite being the most accurate and most commonly used class of detectors, however, CNN-based models display a signi\ufb01cantly degraded performance in adverse weather conditions [13]. While this can be mitigated by training the model on datasets that contain adverse weather, the collecting and labelling of such datasets is expensive, laborious, and at the mercy of the prevailing weather conditions. An alternative solution is generating such data using simulators, physics models, or style transfer techniques like CycleGAN. However, augmentation affects the performance of different detectors in different ways. This paper makes a comparative analysis between different object detectors in terms of accuracy, speed, and their behaviour against augmentation. 2.2 Synthetic weather generation Numerous approaches were proposed to generate synthetic adverse weather images. Simple approaches have utilised open-source frameworks like OpenGL [34], and image augmentation libraries including [13], however, they lack visual realism. More sophisticated methods were proposed to generate more realistic adverse weather images. They harness physics models to model the dynamics of raindrops [8], as well as, the behaviour of the light going through them [4]. Despite their improved realism, such models demand in-depth knowledge of many controlling parameters, which, in turn, are empirically determined. Due to the dif\ufb01culty in generating synthetic data using simple augmentations and physics engines (low realism, high number of free parameters), the autonomous driving research has seen an increase in the use of GANs and its variants to address the issue of data scarcity in adverse weather conditions. As collecting data in harsh conditions and providing ground truth for it is an expensive and time-consuming task, these approaches have been employed to generate datasets in various conditions based on existing datasets and their corresponding ground truth. The \ufb01rst to explore this avenue for autonomous driving was the work of [32], which generates new images with snow and various light conditions for the purpose of improving feature matching. Other applications focus on improving vehicle detection during nighttime [24], object detection and semantic segmentation [31]. Other articles have used CycleGAN in the context of autonomous driving tasks [33, 47, 29]. The latter is the closest to our work; using CycleGAN to create fake adverse conditions for road-going AVs in general rather than racing AVs. While the augmentations demonstrated an improvement in perception performance in adverse weather, only one detector was used, and no consideration was given to how different detectors may perform when augmented training data is used, or their suitability for real-time use on-board a vehicle. 2.3 Adverse weather in autonomous racing Adverse weather affects both the vehicle dynamics (due to, e.g., ice, water puddles or soil on the road surface) and the perception stack of the autonomous vehicle (due to both detection occlusions and changes in the road surface). Different methods have been proposed to address the problem of adverse weather from a perception point of view. [52] who improved an end-to-end model by tackling weather appearance with the use of the existing Airsim simulator [39] to generate various time-of-day appearances. The method is scalable, however, it generates data only in day time, and the generated images are not realistic, thus the data may still be susceptible to differences between simulation and real world. Although such models might perform well on simulation images, they may fail on images coming from real-world unconstrained environments. [13] who employed existing simple open-source augmentation libraries to generate adverse weather datasets, but likewise potentially suffers from similar issues. Other methods including [33, 46] used GANs to improve tasks downstream from perception in urban and rural autonomous driving. To the best of our knowledge, this study represents the \ufb01rst attempt to harness CycleGAN-based weather appearance transfer to generate realistic datasets for multiple adverse weather conditions, in order to improve object detection in real adverse conditions in the context of autonomous racing. \f(a) Real sunny weather (b) Real night (c) Real droplet Figure 1: Real datasets 3 Data 3.1 Real-world dataset For the object detection task, we used three real datasets of various conditions (Figure 1) that were captured and annotated from an autonomous racing vehicle\u2019s vision sensor. The \ufb01rst dataset consists of more than 10, 125 cones captured in sunny conditions; the second (1, 548 cones) is captured under low light conditions after sunset; the \ufb01nal dataset (1, 197 cones) was captured in conditions where adherent droplets on lens completely occlude and distort some of the cones (Figure 1c, in contrast to the \ufb01gures shown in [11], which show no drastic changes in the cones\u2019 appearance). We used the \ufb01rst dataset to train and test the \ufb01ve object detectors (70% training, 30% testing), and kept the last two datasets to test the detectors against out-of-sample adverse illumination and rainy conditions. We exclusively focus on two object classes: blue and yellow cones. 3.2 Synthetic dataset Since obtaining data with ground truth in adverse conditions is a timeand effort-intensive task, we employ a CycleGAN architecture [54] (Figure 2) to generate synthetic night and rainy (droplet) conditions with the datasets that we already have. A CycleGAN is an image-to-image translation model that learns to map images from domain A to domain B while preserving the structure of the image. It is a variant of the original GAN model, one in which two generators are trained simultaneously (one to learn the mapping from A \u2192B and the other the mapping from B \u2192A), while enforcing a cycle-consistency [54]. Cycle-consistency is enforced by training each GAN on the output of the other: the second generator receives as input the output of the \ufb01rst generator (image in domain B) and is trained to reconstruct the original image (image in domain A), by applying a reconstruction loss between the real image in domain A and its reconstruction GBA(GAB(A)) = GBA( \u02c6 B) = \u02dc A \u2248A. Similarly, the \ufb01rst generator receives as input the output of the second generator (image in domain A), and is trained to reconstruct the original image (image in domain B), by applying a reconstruction loss between the real image in domain B and its reconstruction GAB(GBA(B)) = GAB( \u02c6 A) = \u02dc B \u2248B. In order to generate synthesised images that contain night or droplets conditions, we trained off-the-shelf CycleGAN implementation [54] on two (non-racing) datasets from a mixture of resources. The \ufb01rst CycleGAN was trained on frames from a YouTube video captured at night, in London\u2019s streets [41] (Figure 3b), while the second CycleGAN was trained on in-house acquired images that contain droplets (Figure 3c), both were trained at a resolution of 512 \u00d7 512. Then, we used them to transfer the night or droplet styles to the sunny images, ending up with 911 \u2018fake\u2019 night images (Figure 3d) and 800 \u2018fake\u2019 droplet images (Figure 3e), both having the same ground truth labels as the original sunny images. Since these fake conditions are meant for training the detectors, we split them into 70% training set (637 fake night and 560 fake droplet cone images) and 30% validation set (274 fake night and 240 fake droplet cone images). We used the training set to train the detectors and the validation set to measure their performance while training. Training the detectors with fake adverse conditions then testing them on real adverse conditions allows us to verify that fake generated images are a good proxy for real adverse conditions images. \fFigure 2: CycleGAN architecture 3.3 Qualitative analysis of the generated datasets FID measure is the 2-Wasserstein distance [30] between two multidimensional Gaussian distributions in the feature space: N(\u00b5s, \u03a3s) the distribution of some neural network features of the synthesized images generated by the GAN and N(\u00b5r, \u03a3r) the distribution of the same neural network features from the real images used to train the GAN. Mathematically, FID = ||\u00b5r \u2212\u00b5s||2 2 + trace(\u03a3r + \u03a3s \u22122(\u03a3r\u03a3s)1/2) [30]. FID measure is 83.56 and 138.39 for night and droplet conditions, respectively. The lower the value of FID measure, the closer the distribution of the generated dataset to the that for the real dataset, thus, the more similar is the generated dataset to the real dataset. Therefore, the results suggest that our arti\ufb01cially generated night is more similar to a real night than our generated droplet images are to real images with droplets. 4 Object Detectors We decided to test our concept on \ufb01ve state-of-the-art object detection models, not only to validate our concept, but also to run a comparative analysis between these detectors and identify the optimal one, which is crucial information for autonomous racing research. The following \ufb01ve object detectors were chosen due to their high speed, high accuracy, or because they can be ef\ufb01ciently deployed for autonomous driving applications, YOLOv3-tiny has fewer convolution layers than YOLOv3 [37]. With fewer layers, the tiny version is claimed to be the fastest detection model [50], making it ideal for high-speed applications like autonomous racing. Counterintuitively, this high speed does not signi\ufb01cantly compromise accuracy for our problem which only contemplates two classes (blue and yellow cones). Scaled YOLOv4 [48] was chosen because it scales the resolution, width, and depth (number of layers) of the original YOLOv4 [5], making it more ef\ufb01cient and robust. Furthermore, their authors claim it delivers the highest accuracy on the Microsoft COCO (Common Objects In Context) dataset [26]. A modi\ufb01ed, effective and real-time version of YOLOv4 was implemented in [6] for object detection in autonomous driving. YOLOv5s [20] is the small version of YOLOv5. It differs from YOLOv4 by being computationally more ef\ufb01cient in terms of storage space and training time. In terms of accuracy and inference speed, however, there is no quantitative evidence on which is better. Consequently, YOLOv5s was chosen to be compared with other detectors in our autonomous racing domain. [14] adopted YOLOv5 medium for real-time detection in autonomous driving, their system achieved a balance between accuracy and latency as it scored the second place at the Streaming Perception Challenge (Workshop on Autonomous Driving at CVPR 2021), while running in real-time. Additionally, [3] implemented a modi\ufb01ed version of YOLOv5 called YOLO-Z to detect small objects in the context of autonomous racing. Faster-RCNN [38] is a well-known Region Proposal Network (RPN)-based object detection paradigm. The model was initially proposed with a VGG [40] backbone; here we used a Feature Pyramid Network (FPN) [25] backbone because of its superior performance [49]. Faster-RCNN was extended to stereo inputs in [23] to build a 3D object detector for autonomous cars that achieved state-of-the-art results. Ef\ufb01cientDet [44] applies weighted bi-directional FPN and compound scaling to the Ef\ufb01cientNet classi\ufb01er [43], result\fing in a highly computationally ef\ufb01cient detector. Given the principal goal of detecting cones in real-time, we selected Ef\ufb01cientDet-D0 which is the lightest and most ef\ufb01cient among the 9 Ef\ufb01cientDet models. 5 Experiments 5.1 Experimental Method The general approach of the experiments is to train the \ufb01ve detectors on sunny weather to then test them on real adverse conditions. After that, we retrain them on fake adverse conditions then test them again on real adverse conditions. The goal is to record the difference in performance between models trained on (real) sunny condition data versus models trained on (fake) adverse conditions. If the latter proved to be better, then we would have demonstrated that our (synthesised) datasets are an effective proxy for (real) adverse condition data in the domain of autonomous racing. Furthermore, experiments aim to analysis the response of detectors to augmentation and identify the detector with best response. Additionally, the inference time of detectors is recorded to identify their suitability to real-time operation. Our experiments were divided into four stages. Firstly, all \ufb01ve detectors were trained on sunny weather data (see Table 1, \ufb01rst row). This set the baseline against which the rest of the models were compared. Secondly, the detectors were trained on Sunny and Fake Night (see Table 1, second row, and Figure 3d). Thirdly, they were trained on Sunny and Fake Droplet data (third row and Figure 3e). Finally, they were trained on Sunny, Fake Night and Fake Droplet data (forth row). In each stage the trained detectors were tested on Sunny, Real Night, and Real Droplet data (see Section 3.1). In total, 5 detectors were trained on 4 sets of data and tested on 3 sets of data, yielding a total of 60 tests (5 \u00d7 4 \u00d7 3). The results are reported in Table 1, while a performance comparison across stages is shown in Figure 4. (a) Real sunny (original image) (b) Sample used to train CycleGAN to generate night (c) Sample used to train CycleGAN to generate droplets (d) Generated night (e) Generated droplets Figure 3: a) An original image in sunny weather, which is the input of the CycleGAN. b, c) Examples of images used to train the CycleGAN to generate night and droplet conditions. d, e) the generated images based on the input image (a) with the style of the training images (b) and (c). \f5.2 Experimental Setup In all our experiments the training was conducted on a GPU machine equipped with 4 Nvidia GTX 1080 GPUs with 12GB VRAM each. To perform fair comparisons between all \ufb01ve object detectors we \ufb01xed the value of several hyperparameters, including number of epochs (to 300), batch size (to 64), and input image size (to 512 \u00d7 512 pixels). For testing, we used an in-car PC mounted on a racing car equipped with a single Nvidia GTX 1050 TI GPU, with a batch size of 1 and an image size of 512 \u00d7 512 pixels. Performance was measured using the standard mean Average Precision (mAP), with a constant Intersection over Union (IoU) threshold of 0.5 (Table 1). In addition, we measured the computational ef\ufb01ciency by recording for each detector the \ufb02oating point operations per second (FLOPS) and the inference time in frames per seconds (fps) (last two rows of Table 1). Data splits Detection model mAP Training set Test set YOLOv3 SYOLOv4 YOLOv5 FRCNN Ef\ufb01cientDet Sunny Sunny 0.89 0.93 0.95 0.97 0.45 Real Night 0.43 0.21 0.22 0.23 0.24 Real Droplet 0.53 0.54 0.53 0.57 0.43 Sunny + Fake Night Sunny 0.89 0.89 0.97 0.96 0.47 Real Night 0.62 0.39 0.73 0.67 0.55 Real Droplet 0.57 0.55 0.58 0.60 0.53 Sunny + Fake Droplet Sunny 0.89 0.88 0.96 0.95 0.36 Real Night 0.57 0.33 0.25 0.30 0.26 Real Droplet 0.54 0.51 0.55 0.62 0.36 Sunny + Fake Night + Fake Droplet Sunny 0.89 0.86 0.97 0.97 0.37 Real Night 0.68 0.66 0.74 0.71 0.25 Real Droplet 0.58 0.55 0.61 0.60 0.37 Speed (fps) \u2191 74.63 8.77 13.68 3.01 26.53 FLOPS (B) \u2193 12.9 109 16.4 180 2.5 Table 1: Detailed comparative analysis of mAP for all \ufb01ve object detectors (tested on Sunny, Real Night, and Real Droplet) when trained on four different training sets. The aim of that is to show the difference of detectors\u2019 performance on real adverse conditions when trained on fake adverse condition against that when trained on sunny condition only. mAP is measured at a standard IoU threshold of 0.5. The plus (+) shows the concatenation of real and augmented data for training. The best model of each trainingtesting stage (each row). Frame rate (in fps) and FLOPS (in billions) comparison of all \ufb01ve object detectors, tested using an AV\u2019s on-board PC with GTX 1050 TI. 6 Results 6.1 Impact of synthetic adverse conditions Detector performance (mAP) over the 60 experiments discussed in Section 5.1 is reported in Table 1. Figure 4 shows the difference in object detection performance of the models trained on synthetic adverse conditions dataset against that of their baseline versions (trained only on Sunny weather). Figure 4a compares the models trained upon Fake Night to their corresponding baseline (1st and 2nd rows in Table 1). It is clear that, after training the detectors on Fake Night, their performance on Real Night improves by an average of 32 percentage points (pp). Perhaps surprisingly, training on Fake Night not only improved the detectors\u2019 performance on real night, but also on Real Droplet by an average of 4.5 pp. A possible reason for this is that by introducing Fake Night in the training dataset, the models were trained to infer more diverse images that are out-of-sample of the original Sunny Weather training data. Figure 4b compares the models trained upon Fake Droplet to their respective baseline (1st and 3rd rows in Table 1). When trained on Fake Droplet, YOLOv3, YOLOV5 and Faster-RCNN show improvements on Real Droplet, and just like the previous stage, they also show improvements on the other real adverse condition (Real Night). Counterintuitively, training on Fake Droplets slightly reduced performance in Real Droplet conditions and improved detections in Real Night using YOLOv4 and Ef\ufb01cientDet. Finally, Figure 4c shows the difference between The \u2018monolithic\u2019 versions of the models (those trained on Sunny, \fFake Night and Fake Droplet) and their related baseline (1st and 4th rows in Table 1). The \u2018monolithic\u2019 versions exhibit slightly better performance on real adverse conditions than those trained on a single fake adverse condition. When tested on Real Night, the monolithic models (excluding Ef\ufb01cientDet) show a mean performance improvement of 42.7 pp, while the models trained only on Fake Night show a mean increase of 32.7 pp. In Real Droplet conditions, the monolithic models show a mean increase of 4.4 pp, while the models trained only on Fake Droplet show a slight degradation of 1.2 pp. Figure 5 show the qualitative results of YOLOv5 night detections before and after being trained on synthetic night. The average improvements on all conditions for night is higher than that for droplets. This is due to the baseline models having lower performance on Fake Night than on Fake Droplets, so there was more room for improvement in the former. Furthermore, this agrees with the fact that the synthetic Fake Night data is more visually similar to the Real Night data than the Fake Droplet data is to Real Droplet data (as the Fake Night dataset has a lower FID than the Fake Droplet data (Section 3.3)). This intuitively might be due to the fact that night images, by nature, are captured with lower re\ufb02ective light than images taken during day-time, resulting in images that have lower number of colours and details than images captured during the day. This, in turn, makes it easier for generative models like CycleGAN to generate them because, again, they have fewer colours and details than day-time images. +0.2 -4.0 +1.4 -1.0 +1.7 +0.8 -5.0 +0.2 -2.0 -8.8 +0.4 -7.0 +1.3 0 -8.4 +18.9 +18.0 +50.2 +43.8 +31.0 +14.8 +12.0 +3.1 +7.2 +1.1 +25.6 +45.0 +52.1 +48.1 +0.5 +3.8 +1.0 +4.7 +3.0 +10.2 +1.7 -3.0 +1.1 +5.0 -7.0 +5.3 +1.3 +7.6 +3.5 -5.5 0 10 20 30 40 50 60 70 80 90 100 Yolov3 SYolov4 Yolov5 FRCNN E\ufb03ctDet Yolov3 SYolov4 Yolov5 FRCNN E\ufb03ctDet Yolov3 SYolov4 Yolov5 FRCNN E\ufb03ctDet a) Trained upon (Sunny + Fake Night) b) Trained upon (Sunny + Fake Droplet) c) Trained upon (Sunny + Fake Night + Fake Droplet) mAP (%) Sunny (Test set) Real Night (Test set) Real Droplet (Test set) Baseline models Figure 4: Dashed bars show the baseline performance (mAP) of 5 models trained on Sunny and tested on Sunny, Real Night, and Real Droplet (1st row of Table 1). The coloured continuous bars and coloured numbers show the improvement (+ve value) or degradation (-ve value) of performance caused by retraining the models on a) Sunny+Fake Night. b) Sunny+Fake Droplet, c) Sunny+Fake Night+Fake Droplet, (2nd, 3rd, and 4th rows of Table 1). (a) Before training on synthetic night and synthetic droplet (mAP=0.22) (b) After training on synthetic night and synthetic droplet (mAP=0.74) Figure 5: Impact of augmentation on enhancing night detections using YOLOv5 6.2 Detector behaviour using synthetic adverse conditions When models are trained on augmented (Fake) datasets, their performance difference on Real Sunny varies. the performance of YOLOv3 and YOLOV5 experienced marginal (negligible) improvement or degradation. However, ScaledYOLOv4 and Ef\ufb01cientDet performance was slightly degraded. The different detection models use a varied architecture for the neck structure (which is placed between the head and backbone to aggregate as much information extracted by the backbone as possible before it is fed to the head) YOLOv3 and YOLOV5 use a Path Aggregation Network (PANet) [27] as a neck and they exhibit a slight performance enhancement, while ScaledYOLOv4, FRCNN, and Ef\ufb01cientDet use FPN [25] and they generally show a degradation \fof performance. The results suggest that detectors with an FPN neck are more susceptible to augmentations degrading good weather performance than those using PANet. Among the \ufb01ve detectors, YOLOv5 is the most robust in all weather conditions when trained on augmented data, it reported an improvement of 1.3, 52.1, and 7.6 pp on real sunny, real night, and real droplet respectively. On the other hand, Ef\ufb01cientDet performed the worst after augmentations, exhibiting a performance degradation of -8.4 and -5.5 pp for Sunny and Real Droplet and an improvement of +0.5 on Real Night. 6.3 Object Detectors\u2019 Performance Looking at Table 1, it is observed that, in the absence of adverse weather training data (i.e., when only the sunny weather training dataset is available), Faster-RCNN is the most accurate model for sunny and rainy weather. However, YOLOv3-tiny is arguably the optimal model for this application, as object detection performance is only marginally reduced from that of Faster-RCNN in sunny and rainy conditions, but exhibits double the mAP in night-time tests while simultaneously being 25 times faster and 18 times more computationally ef\ufb01cient than Faster-RCNN. In contrast, when synthetic adverse condition training data is available, YOLOv5 and FRCNN appeared to be the best choices in terms of accuracy and robustness in all weather conditions. After training on synthetic adverse conditions (fake night and fake droplet), they scored the highest mAP across the board when tested on sunny weather, real night, and real droplet. YOLOv5 is also capable of achieving these results while running at a high frame rate of almost 14 fps. In terms of raw computational ef\ufb01ciency alone (last row of Table 1), Ef\ufb01cientDet is the best model among the \ufb01ve detectors. It has the lowest number of \ufb02oating point operations per second, which makes it ideal for low computational capacity hardware. Although Ef\ufb01cientDet has (5 times) lower FLOPS than YOLOv3, the latter is (2.8 times) faster than the former. This is because the term FLOPS describes the number of operations per second not the total number of operations. In other words, Ef\ufb01cientDet might have more operations in total than YOLOv3, but it divides and executes them over a longer time span (thus slower), ending up with a small number of operations per second. Having a lower FLOPs means the model is hardware-friendly and can be installed on relatively light processors including the popular NVIDIA Jetson. However, Ef\ufb01cientDet performance is decreased in almost all weather conditions with the use of augmented training data suggesting it is a poor choice when using augmented training data in adverse conditions. Furthermore, Table 1 in terms of inference speed, YOLOv3 is the fastest model by a clear margin triple the speed of Ef\ufb01cientDet, and an order of magnitude faster than SYOLOv4 and Faster-RCNN. Interestingly, the number of frames it can process per second is more than the summation of frames processed by all the other models combined. While running at this high speed, YOLOv3 achieved a similar performance to that of YOLOv5. Therefore, in case a lower powered processor hardware is to be used or a high frame rate is critical, YOLOv3 is the optimal choice. It is worth mentioning that the frame rates in Table 1 were measured while running the detectors on the AV racing PC mentioned in section 5.2 which (at the time of writing) is 4 years old. State-of-the-art on-board hardware will further improve the performance of all detectors. 7"
+                }
+            ],
+            "Mohamed Elhoseiny": [
+                {
+                    "url": "http://arxiv.org/abs/1904.01109v6",
+                    "title": "Creativity Inspired Zero-Shot Learning",
+                    "abstract": "Zero-shot learning (ZSL) aims at understanding unseen categories with no\ntraining examples from class-level descriptions. To improve the discriminative\npower of zero-shot learning, we model the visual learning process of unseen\ncategories with inspiration from the psychology of human creativity for\nproducing novel art. We relate ZSL to human creativity by observing that\nzero-shot learning is about recognizing the unseen and creativity is about\ncreating a likable unseen. We introduce a learning signal inspired by\ncreativity literature that explores the unseen space with hallucinated\nclass-descriptions and encourages careful deviation of their visual feature\ngenerations from seen classes while allowing knowledge transfer from seen to\nunseen classes. Empirically, we show consistent improvement over the state of\nthe art of several percents on the largest available benchmarks on the\nchallenging task or generalized ZSL from a noisy text that we focus on, using\nthe CUB and NABirds datasets. We also show the advantage of our approach on\nAttribute-based ZSL on three additional datasets (AwA2, aPY, and SUN). Code is\navailable.",
+                    "authors": "Mohamed Elhoseiny, Mohamed Elfeki",
+                    "published": "2019-04-01",
+                    "updated": "2019-12-03",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction With hundreds of thousands of object categories in the real world and countless undiscovered species, it becomes unfeasible to maintain hundreds of examples per class to fuel the training needs of most existing recognition systems. Zipf\u2019s law, named after George Zipf (1902\u22121950), suggests that for the vast majority of the world-scale classes, only a few examples are available for training, validated earlier in language (e.g., [64, 65]) and later in vision (e.g., [44]). This problem becomes even more severe when we target recognition at the \ufb01ne-grained level. For example, there exists tens of thousands of bird and \ufb02ower species, but the largest available benchmarks have only a few hundred classes motivating a lot of research on classifying instances of unseen classes, known as Zero-Shot Learning (ZSL). Figure 1: Generalizing the learning of zero-shot models requires a deviation from seen classes to accommodate recognizing unseen classes. We carefully model a learning signal that inductively encourages deviation of unseen classes from seen classes, yet not pushed far that the generation fall in the negative hedonic unrealistic range on the right and loses knowledge transfer from seen classes. Interestingly, this curve is similar to the famous Wundt Curve in the Human Creativity literature (Martindale, 1990) [34]. People have a great capability to identify unseen visual classes from text descriptions like \u201cThe crested auklet is subspecies of birds with dark-gray bodies tails and wings and orange-yellow bill. It is known for its forehead crests, made of black forward-curving feathers.\u201d; see Fig 1 (bottom). We may imagine the appearance of \u201ccrested auklet\u201d in different ways yet all are correct and may collectively help us understand it better. This imagination notion been modeled in recent ZSL approaches (e.g., [19, 30, 20, 63, 58]) successfully adopting deep generative models to synthesize visual examples of an unseen object given its semantic description. After training, the model generates imaginary data for each unseen class transforming ZSL into a standard classi\ufb01cation task with the generated data. However, these generative ZSL methods do not guarantee the discrimination between seen and unseen classes since the generations are not motivated with a learning signal to deviate from seen classes. For example, \u201cParakeet Auklet\u201d as a seen class in Fig 1 (left) has a visual text description [55] that signi\ufb01cantly overlaps with \u201cCrested Auklet\u201d description, yet one can identify \u201cCrested Auklet\u201d\u2019s 1 arXiv:1904.01109v6  [cs.CV]  3 Dec 2019 \funique \u201c\u2019black forward-curving feathers\u201d against \u201cParakeet Auklet\u201d from text. The core of our work is to address the question of how to produce discriminative generations of unseen visual classes from text descriptions by explicitly learning to deviate from seen classes while allowing transfer to unseen classes. Let\u2019s imagine the space of conditional visual generations from class descriptions on an intensity map where light regions implies seen and darker regions implies unseen. These class descriptions are represented in a shared space between the unseen (dark) and the seen (light) classes, and hence the transfer is expected. In existing methods, this transfer signal is formulated by encouraging the generator to produce quality examples conditioned only on the descriptions of the seen classes ( light regions only). In this inductive zero-shot learning, class descriptions of unseen classes are not available during training and hence can not used as a learning signal to explicitly encourage the discrimination across unseen and seen classes. Explicitly modeling an inductive and discriminative learning signal from the dark unseen space is at the heart of our work. Creativity Inspiration to Zero-shot Learning. We propose to extend generative zero-shot learning with a discriminative learning signal inspired from the psychology of human creativity. Colin Marindale [34] proposes a psychological theory to explain the perception of human creativity. The de\ufb01nition relates likability of an art piece to novelty by \u201cthe principle of least effort\u201d. The aesthetic appeal of an art work \ufb01rst increases when it deviates from existing work till some point, then decreases when the deviation goes too far. This means that it gets dif\ufb01cult to connect this art to what we are familiar with, and hence deems it hard to understand and hence appreciate. This principle can be visualized by the Wundt Curve where the X axis represents novelty and Y axis represents likability like an inverted U-shape; similar to the curve in Fig 1. We relate the Wundt curve behavior in producing creative art to a desirable generalized ZSL mode that has a better capability to distinguish the \u201ccrested auklet\u201d unseen class from the \u201cparakeet auklet\u201d seen class given how similar they are as mentioned before; see Fig 1. A generative ZSL model that cannot deviate generations of unseen classes from instances of seen classes is expected to underperform in generalized zero-shot recognition due to confusion; see Fig 1(left). As the deviation capability increases, the performance is expected to get better but similarly would decrease when the deviation goes too far producing unrealistic generation and reducing the needed knowledge transfer from seen classes; see Fig 1(middle and right). Our key question is how to properly formulate deviation from generating features similar to existing classes while balancing the desirable transfer learning signal. Contributions. 1) We propose a zero-shot learning approach that explicitly models generating unseen classes by learning to carefully deviate from seen classes. We examine a parametrized entropy measure to facilitate learning how to deviate from seen classes. Our approach is inspired from the psychology of human creativity; and thus we name it Creativity Inspired Zero-shot Learning (CIZSL). 2) Our creativity inspired loss is unsupervised and orthogonal to any Generative ZSL approach. Thus it can be integrated with any GZSL while adding no extra parameters nor requiring any additional labels. 3) By means of extensive experiments on seven benchmarks encompassing Wikipedia-based and attribute-based descriptions, our approach consistently outperformed stateof-the-art methods on zero-shot recognition, zero-shot retrieval, and generalized zero-shot learning using several evaluation metrics. 2. Related Work Early Zero-Shot Learning(ZSL) Approaches A key idea to facilitate zero-shot learning is \ufb01nding a common semantic representation that both seen and unseen classes can share. Attributes and text descriptions are shown to be effective shared semantic representations that allow transferring knowledge from seen to unseen classes. Lampert et al. [27] proposed a Direct Attribute Prediction (DAP) model that assumed independence of attributes and estimated the posterior of the test class by combining the attribute prediction probabilities. A parallelly developed, yet similar model was developed by Farhadi et al. [13]. Visual-Semantic Embedding ZSL. Relaxing the unrealistic independence assumption, Akata et al. [2] proposed an Attribute Label Embedding(ALE) approach that models zero-shot learning as a linear joint visual-semantic embedding. In principal, this model is similar to prior existing approaches that learn a mapping function from visual space to semantic space [61, 48]. This has been also investigated in the opposite direction [61, 48] as well as jointly learning a function for each space that map to a common space [59, 29, 3, 43, 50, 12, 1, 32, 31, 51]. Generative ZSL Approaches The notion of generating arti\ufb01cial examples has been recently proposed to model zeroshot learning reducing it to a conventional classi\ufb01cation problem [19, 30, 20, 63]. Earlier approaches assumed a Gaussian distribution prior for visual space to every class and the probability densities for unseen classes are modeled as a linear combination of seen class distributions [19]. Long et al. [30] instead proposed a one-to-one mapping approach where synthesized examples are restricted. Recently, Zhu et al. [63], Xian et al. [58], and Verma et al.[26] relaxed this assumption and built on top of generative adversarial networks (GANs) [17, 40] to generate examples from unseen class descriptions. Different from ACGAN [37], Zhu et al. added a visual pivot regularizer (VPG) encourages generations of each class to be close to the average of its corresponding real features. \fSemantic Representations in ZSL (e.g., Attributes, Description). ZSL requires by de\ufb01nition additional information (e.g., semantic description of unseen classes) to enable their recognition. A considerable progress has been made in studying attribute representation [27, 28, 2, 15, 61, 59, 29, 3, 43, 1]. Attributes are a collection of semantic characteristics that are \ufb01lled to uniquely describe unseen classes. Another ZSL trend is to use online textual descriptions [11, 12, 39, 41, 29]. Textual descriptions can be easily extracted from online sources like Wikipedia with a minimal overhead, avoiding the need to de\ufb01ne hundreds of attributes and \ufb01lling them for each class/image. Elhoseiny et al. [11] proposed an early approach for Wikipediabased zero-shot learning that combines domain transfer and regression to predict visual classi\ufb01ers from a TF-IDF textual representation [45]. Qiao et al. [39] proposed suppress the noise in the Wikipedia articles by encouraging sparsity of the neural weights to the text terms. Recently, part-based zero-shot learning model [12] was proposed with a capability to connect text terms to its relevant parts of objects without part-text annotations. More recently, Zhu et al. [63] showed that suppressing the non-visual information is possible by the predictive power of the their model to synthesize visual features from the noisy Wikipedia text. Our work also focus on the challenging task of recognizing objects based on Wikipedia articles and is also a generative model. Unlike existing, we explicitly model the careful deviation of unseen class generations from seen classes . Visual Creativity. Computational Creativity studies building machines that generate original items with realistic and aesthetic characteristics [33, 35, 7]. Although GANs [17, 40, 22] are a powerful generative model, yet it is not explicitly trained to create novel content beyond the training data. For instance, a GAN model trained on art works might generate the \u201cMona Lisa\u201d again, but would not produce a novel content that it did not see. It is not different for some existing style transfer work [16, 8] since there is no incentive in these models to generate a new content. More recent work adopts computational creativity literature to create novel art and fashion designs [9, 46]. Inspired by [34], Elgammalet al. [9] adapted GANs to generate unconditional creative content (paintings) by encouraging the model to deviate from existing painting styles. Fashion is a 2.5 trillion dollar industry and has an impact in our everyday life, this motivated [46] to develop a model that can for example create an unseen fashion shape \u201cpants to extended arm sleeves\u201d. The key idea behind these models is to add an additional novelty loss that encourage the model to explore the creative space of image generation. 3. Background GANs [17, 40] train the generator G, with parameters \u03b8G, to produce samples that the Discriminator D believe they are real. On the other hand, the Discriminator D, with parameters \u03b8D, is trained to classify samples from the real distribution pdata as real (1), and samples produced by the generator as fake (0); see Eq 2. min \u03b8G LG = min \u03b8G X zi\u2208Rn log(1 \u2212D(G(zi))) (1) min \u03b8D LD = min \u03b8D X xi\u2208D,zi\u2208Rn \u2212log D(xi) \u2212log(1 \u2212D(G(zi)))) (2) where zi is a noise vector sampled from prior distribution pz and x is a real sample from the data distribution pdata. In order to learn to deviate from seen painting styles or fashion shapes, [9, 46] proposed an additional head for the discriminator D that predicts the class of an image (painting style or shape class). During training, the Discriminator D is trained to predict the class of the real data through its additional head, apart from the original real/fake loss. The generator G is then trained to generate examples that are not only classi\ufb01ed as real but more importantly are encouraged to be hard to classify using the additional discriminator head. More concretely, LG = LG real/fake + \u03bbLG creativity (3) The common objective between [9] and [46] is to produce novel generations with high entropy distribution over existing classes but they are different in the loss function. In [9], LG creativity is de\ufb01ned as the binary cross entropy (BCE) over each painting style produced by the discriminator additional head and the uniform distribution (i.e., 1 K , K is the number of classes). Hence, this loss is a summation of BCE losses over all the classes. In contrast, Sbai et al. [46] adopted the Multiclass Cross Entropy (MCE) between the distribution over existing classes and the uniform distribution. To our knowledge, creative generation has not been explored before conditioned on text and to also facilitate recognizing unseen classe, two key differences to our work. Relating computational creativity to zero-shot learning is one of the novel aspects in our work by encouraging the deviation of generative models from seen classes. However, proper design of the learning signal is critical to (1) hallucinate class text-descriptions whose visual generations can help the careful deviation, (2) allow discriminative generation while allowing transfer between seen and unseen classes to facilitate zero-shot learning. 4. Proposed Approach Problem De\ufb01nition. We start by de\ufb01ning the zero-shot learning setting. We denote the semantic representations of unseen classes and seen classes as tu i = \u03c6(T u k ) \u2208T and ts i \u2208T respectively, where T is the semantic space (e.g., features \u03c6(\u00b7) of a Wikipedia article T u k ). Let\u2019s denote the seen data as Ds = {(xs i, ys i , ts i)}N s i=1, where N s is the number of training(seen) image examples, where xs i \u2208X denotes the visual features of the ith image in the visual space \fFigure 2: Generator G is trained to carefully deviate from seen to unseen classes without synthesizing unrealstic images. Top part: G is provided with a hallucinated text th and trained to trick discriminator to believe it is real, yet it encourages to deviate learning from seen classes by maximizing entropy over seen classes given th. Bottom part: G is provided with text of a seen class ts and is trained to trick discriminator to believe it is real with a corresponding class label(low-entropy). X, ys i is the corresponding category label. We denote the number of unique seen class labels as Ks. We denote the set of seen and unseen class labels as S and U, where the aforementioned ys i \u2208S. Note that the seen and the unseen classes are disjointed, i.e., S \u2229U = \u2205. For unseen classes, we are given their semantic representations, one per class, {tu i }Ku i=1, where Ku is the number of unseen classes. The zero-shot learning (ZSL) task is to predict the label yu \u2208U of an unseen class visual example xu \u2208X. In the more challenging Generalized ZSL (GZSL), the aim is to predict y \u2208U\u222aS given x that may belong to seen or unseen classes. Approach Overview. Fig. 2 shows an overview of our Creativity Inspired Zero-Shot Learning model(CIZSL). Our method builds on top of GANs [17] while conditioning on semantic representation from raw Wikipedia text describing unseen classes. We denote the generator as G: RZ \u00d7 RT \u03b8G \u2212 \u2212 \u2192RX and the discriminator as D : RX \u03b8D \u2212 \u2212 \u2192 {0, 1}\u00d7Lcls, where \u03b8G and \u03b8D are parameters of the generator and the discriminator as respectively, Lcls is the set of seen class labels (i.e., S = {1 \u00b7 \u00b7 \u00b7 Ks}). For the Generator G and as in [58], the text representation is then concatenated with a random vector z \u2208RZ sampled from Gaussian distribution N(0, 1); see Fig. 2. In the architecture of [63], the encoded text tk is \ufb01rst fed to a fully connected layer to reduce the dimensionality and to suppress the noise before concatenation with z. In our work, the discriminator D is trained not only to predict real for images from the training images and fake for generated ones, but also to identify the category of the input image. We denote the real/fake probability produced by D for an input image as Dr(\u00b7), and the classi\ufb01cation score of a seen class k \u2208S given the image as Ds,k(\u00b7). Hence, the features are generated from the encoded text description tk, as follows \u02dc xk \u2190G(tk, z). The discriminator then has two heads. The \ufb01rst head is an FC layer that for binary real/fake classi\ufb01cation. The second head is a Ks-way classi\ufb01er over the seen classes. Once our generator is trained, it is then used to hallucinate fake generations for unseen classes, where conventional classi\ufb01er could be trained as we detail later in Sec 4.3. The generator G is the key imagination component that we aim to train to generalize to unseen classes guided by signals from the discriminator D. In Sec 4.1, we detail the de\ufb01nition of our Creativity Inspired Zero-shot Signal to augment and improve the learning capability of the Generator G. In Sec 4.2, we show how our proposed loss can be easily integrated into adversarial generative training. 4.1. Creativity Inspired Zero-Shot Loss (CIZSL) We explicitly explore the unseen/creative space of the generator G with a hallucinated text (th \u223cph text). We de\ufb01ne ph text, as a probability distribution over hallucinated text description that is likely to be unseen and hard negatives to seen classes. To sample th \u223cph text, we \ufb01rst pick two seen text features at random ts a, ts b \u2208S. Then we sample th by interpolating between them as th = \u03b1ts a + (1 \u2212\u03b1)ts b (4) where \u03b1 is uniformally sampled between 0.2 and 0.8. We discard \u03b1 values close to 0 or 1 to avoid sampling a text feature very close to a seen one. We also tried different ways to sample \u03b1 which modi\ufb01es ph text like \ufb01xed \u03b1 = 0.5 or \u03b1 \u223cN(\u00b5 = 0.5, \u03c3 = 0.5/3) but we found uniformally sampling from 0.2 to 0.8 is simple yet effective; see ablations at Appendix E. We de\ufb01ne our creativity inspired zeroshot loss LC G based on G(th, z) as follows LC G = \u2212Ez\u223cpz,th\u223cps text[Dr(G(th, z))] +\u03bbEz\u223cpz,th\u223cph text[Le({Ds,k(G(th, z))}k=1\u2192Ks)] (5) We encourage G(th, z) to be real (\ufb01rst term) yet hard to classify to any of the seen classes (second term) and hence achieve more discrimination against seen classes; see Fig. 2 \f(top). More concretely, the \ufb01rst term encourage the generations given th \u223cph text to trick the discriminator to believe it is real (i.e., maximize Dr(G(th, z)). This loss encourages the generated examples to stay realistic while deviating from seen classes. In the second term, we quantify the dif\ufb01culty of classi\ufb01cation by maximizing an entropy function Le that we de\ufb01ne later in this section. Minimizing LC G connects to the principal of least effort by Martindale et.al. 1990, where exaggerated novelty would decrease the transferability from seen classes (see visualized in Fig. 1). Promoting the aforementioned high entropy distribution incents discriminative generation. However, it does not disable knowledge transfer from seen classes since the unseen generations are encouraged to be an entropic combination of seen classes. We did not model deviation from seen classes as an additional class with label Ks + 1 that we always classify G(th, z) to, since this reduces the knowledge transfer from seen classes as we demonstrate in our results. De\ufb01nition of Le : Le is de\ufb01ned over the seen classes\u2019 probabilities, produced by the second discriminator head {Ds,k(\u00b7)}k=1\u2192Ks (i.e., the softmax output over the seen classes). We tried different entropy maximization losses. They are based on minimizing the divergence between the softmax distribution produced by the discriminator given the hallucinated text features and the uniform distribution. Concretely, the divergence, also known as relative entropy, is minimized between {Ds,k(G(th, z))}k=1\u2192Ks)) and { 1 Ks }k=1\u2192Ks; see Eq 6. Note that similar losses has been studied in the context of creative visual generation of art and fashion(e.g., [9, 46]). However, the focus there was mainly unconditional generation and there was no need to hallucinate the input text th to the generator, which is necessary in our case; see Sec 3. In contrast, our work also relates two different modalities (i.e., Wikipedia text and images). LKL e = Ks X k=1 1 Ks Ds,k(G(th, z)) LSM e (\u03b3, \u03b2) = 1 \u03b2 \u22121 \" Ks X k=1 (Ds,k(G(th, z)1\u2212\u03b3( 1 Ks ) \u03b3 ) 1\u2212\u03b2 1\u2212\u03b3 \u22121 # (6) Several divergence/entropy measures has been proposed in the information theory literature [42, 52, 23, 4, 21]. We adopted two divergence losses, the well-known KullbackLeibler(KL) divergence in LKL e and the two-parameter Sharma-Mittal(SM) [21] divergence in LSM e which is relatively less known; see Eq 6. It was shown in [4], that other divergence measures are special case of Sharma-Mittal(SM) divergence by setting its two parameters \u03b3 and \u03b2. It is equivalent to R\u00b4 enyi [42] when \u03b2 \u21921 (single-parameter), Tsallis divergence [52] when \u03b3 = \u03b2 (single-parameter), Bhattacharyya divergence when\u03b2 \u21920.5 and \u03b3 \u21920.5, and KL divergence when \u03b2 \u21921 and \u03b3 \u21921 (no-parameter). So, when we implement SM loss, we can also minimize any of the aforementioned special-case measures; see details in Appendix B. Note that we also learn \u03b3 and \u03b2 when we train our model with SM loss. 4.2. Integrating CIZSL in Adversarial Training The integration of our approach is simple that LC G de\ufb01ned in Eq 5 is just added to the generator loss; see Eq 7. Similar to existing methods, when the generator G is provided with text describing a seen class ts, its is trained to trick the discriminator to believe it is real and to predict the corresponding class label (low-entropy for ts versus higentropy for th); see Fig 2(bottom). Note that the remaining terms, that we detail here for concreteness of our method, are simlar to existing generative ZSL approaches [58, 63] Generator Loss The generator loss is an addition of four terms, de\ufb01ned as follows LG = LC G \u2212Ez\u223cpz,(ts,ys)\u223cps text[Dr(G(ts, z))+ Ks X k=1 ys klog(Ds,k(G(ts, z)))] + 1 Ks Ks X k=1 ||Ez\u223cpz[G(tk, z)] \u2212Ex\u223cpk data[x]||2 (7) The \ufb01rst term is our creativity inspired zero-shot loss LC G, described in Sec 4.1. Note that seen class text descriptions {tk}k=1\u2192Ks are encouraged to predict a low entropy distribution since loss is minimized when the corresponding class is predicted with a high probability. Hence, the second term tricks the generator to classify visual generations from seen text ts as real. The third term encourages the generator to be capable of generating visual features conditioned on a given seen text. The fourth term is an additional visual pivot regularizer that we adopted from [63], which encourages the centers of the generated (fake) examples for each class k (i.e., with G(tk, z)) to be close to the centers of real ones from sampled from pk data for the same class k. Similar to existing methods, the loss for the discriminator is de\ufb01ned as: LD =Ez\u223cpz,(ts,ys)\u223cps text[Dr(G(ts, z))] \u2212Ex\u223cpdata[Dr(x)] + LLip \u22121 2Ex,y\u223cpdata[ Ks X k=1 yklog(Ds,k(x))] \u22121 2Ez\u223cpz,(ts,ys)\u223cps text[ Ks X k=1 ys klog(Ds,k(G(ts, z)))] (8) where y is a one-hot vector encoding of the seen class label for the sampled image x, ts and ys are features of a text description and the corresponding on-hot label sampled from seen classes ps text. The \ufb01rst two terms approximate Wasserstein distance of the distribution of real features and fake features. The third term is the gradient penalty to enforce \fthe Lipschitz constraint: LLip = (|| \u25bd\u02dc x Dr(\u02dc x)||2 \u22121)2, where \u02dc x is the linear interpolation of the real feature x and the fake feature \u02c6 x; see [18]. The last two terms are classi\ufb01cation losses of the seen real features and fake features from text descriptions of seen category labels. Training. We construct two minibatches for training the generator G, one from seen class ts and from the halllucinated text th to minimize LG (Eq. 7) and in particular LC G (Eq. 5). The generator is optimized to fool the discriminator into believing the generated features as real either from hallucinated text th or the seen text ts. In the mean time, we maximize their entropy over the seen classes if the generated features comes from hallucinated text th \u223cph text or to the corresponding class if from a real text ts. Training the discriminator is similar to existing works; see in Appendix C a detailed algorithm and code to show how G and D are alternatively trained with an Adam optimizer. Note that when Le has parameters like \u03b3 and \u03b2 for SharmaMittal(SM) divergence ( Eq 6), that we also learn. 4.3. Zero-Shot Recognition Test After training, the visual features of unseen classes can be synthesized by the generator conditioned on a given unseen text description tu, as xu = G(tu, z). We can generate an arbitrary number of generated visual features by sampling different z for the same text tu. With this synthesized data of unseen classes, the zero-shot recognition becomes a conventional classi\ufb01cation problem. We used nearest neighbor prediction, which we found simple and effective. 5. Experiments We investigate the performance of our approach on two class-level semantic settings: textual and attribute descriptions. Since the textual based ZSL is a harder problem, we used it to run an ablation study for zero-shot retrieval and generalized ZSL. Then, we conducted experiments for both settings to validate the generality of our work. Cross-Validation The weight \u03bb of our loss in Eq 5 is a hyperparameter that we found easy to tune on all of our experiments. We start by splitting the data into training and validation split with nearly 80-20% ratio for all settings. Training and validation classes are selected randomly prior to the training. Then, we compute validation performance when training the model on the 80% split every 100 iterations out of 3000 iterations. We investigate a wide range of values for \u03bb, and the value that scores highest validation performance is selected to be used at the inference time. Finally, we combine training and validation data and evaluate the performance on testing data. Zer-Shot Performance Metrics. We use two metrics widely used in in evaluating ZSL recognition performance: Standard Zero-shot recognition with the Top-1 unseen class accuracy and Seen-Unseen Generalized Zero-shot performance with Area under Seen-Unseen curve [6]. The Top-1 accuracy is the average percentage of images from unseen classes classifying correctly to one of unseen class labels. However, this might be incomplete measure since it is more realistic at inference time to encounter also seen classes. Therefore, We also report a generalized zero-shot recognition metric with respect to the seen-unseen curve, proposed by Chao et al. [6]. This metric classi\ufb01es images of both seen S and unseen classes U at test time. Then, the performance of a ZSL model is assessed by classifying these images to the label space that covers both seen classes and unseen labels T = S \u222aU. A balancing parameter is used sample seen and unseen class test accuracy-pair. This pair is plotted as the (x, y) co-ordinate to form the Seen-Unseen Curve(SUC). We follow [63] in using the Area Under SUC to evaluate the generalization capability of class-level text zero-shot recognition, and the haromnic mean of SUC for attribute-based zero-shot recognition. In our model, we use the trained GAN to synthesize the visual features for both training and testing classes. 5.1. Wikipedia based ZSL Results (4 benchmarks) Text Representation. Textual features for each class are extracted from corresponding raw Wikipedia articles collected by [11, 12]. We used Term Frequency-Inverse Document Frequency (TF-IDF) [45] feature vector of dimensionality 7551 for CUB and 13217 for NAB. Visual Representation. We use features of the part-based FC layer in VPDE-net [61]. The image are fed forward to the VPDE-net after resizing to 224 \u00d7 224, and the feature activation for each detected part is extracted which is of 512 dimensionality. The dimensionalities of visual features for CUB and NAB are 3583 and 3072 respectively. There are six semantic parts shared in CUB and NAB:\u201chead\u201d, \u201cback\u201d, \u201cbelly\u201d, \u201cbreast\u201d, \u201cleg\u201d, \u201cwing\u201d, \u201ctail\u201d. Additionally, CUB has an extra part which is \u201cleg\u201d which makes its feature representation 512D longer compared to NAB (3583 vs 3072). More details in the Appendix F. Datasets. We use two common \ufb01ne-grained recognition datasets for textual descriptions: Caltech UCSD Birds2011 (CUB) [54] and North America Birds (NAB) [53]. CUB dataset contains 200 classes of bird species and their Wikipedia textual description constituting a total of 11,788 images. Compared to CUB, NAB is a larger dataset of birds, containing a 1011 classes and 48,562 images. Splits. For both datasets, there are two schemes to split the classes into training/testing (in total four benchmarks): Super-Category-Shared (SCS) or easy split and SuperCategory-Exclusive Splitting (SCE) or hard split, proposed in [12]. Those splits represents the similarity of the seen to unseen classes, such that the former represents a higher similarity than the latter. For SCS (easy), unseen classes are deliberately picked such that for every unseen class, there is \fMetric Top-1 Accuracy (%) Seen-Unseen AUC (%) Dataset CUB NAB CUB NAB Split-Mode Easy Hard Easy Hard Easy Hard Easy Hard CIZSL SM-Entropy (ours \ufb01nal) 44.6 14.4 36.5 9.3 39.2 11.9 24.5 6.4 CIZSL SM-Entropy (replace 2nd term in Eq 5 by Classifying th as new class) 43.2 11.31 35.6 8.5 38.3 9.5 21.6 5.6 CIZSL SM-Entropy (minus 1st term in Eq 5) 43.4 10.1 35.2 8.3 35.0 8.2 20.1 5.4 CIZSL SM-Entropy: (minus 2nd term in Eq 5) 41.7 11.2 33.4 8.1 33.3 10.1 21.3 5.1 CIZSL Bachatera-Entropy (\u03b3 = 0.5, \u03b2 = 0.5) 44.1 13.7 35.9 8.9 38.9 10.3 24.3 6.2 CIZSL Renyi-Entropy (\u03b2 \u21921) 44.1 13.3 35.8 8.8 38.6 10.3 23.7 6.3 CIZSL KL-Entropy (\u03b3 \u21921, \u03b2 \u21921) 44.5 14.2 36.3 8.9 38.9 11.6 24.3 6.2 CIZSL Tsallis-Entropy (\u03b2 = \u03b3) 44.1 13.8 36.7 8.9 38.9 11.3 24.5 6.3 CIZSL SM-Entropy: (minus 1st and 2nd terms in Eq 5)= GAZSL [63] 43.7 10.3 35.6 8.6 35.4 8.7 20.4 5.8 Table 1: Ablation Study using Zero-Shot recognition on CUB & NAB datasets with two split settings each. CIZSL is GAZSL [63]+ our loss Figure 3: Seen Unseen Curve for Parakeet Auklet (Seen, y-axis) vs Crested Auklet (Unseen, xaxis) for GAZSL[57] and GAZSL[57]+ CIZSL. at least one seen class with the same super-category. Hence, the relevance between seen and unseen classes is very high, deeming the zero-shot recognition and retrieval problems relatively easier. On the other end of the spectrum, SCE (hard) scheme, the unseen classes do not share the supercategories with the seen classes. Hence, there is lower similarity between the seen and unseen classes making the problem harder to solve. Note that the easy split is more common in literature since it is more Natural yet the deliberately designed hard-split shows the progress when the super category is not seen that we also may expect. Ablation Study (Table 5). Our loss is composed of two terms shown that encourage the careful deviation in Eq 5. The \ufb01rst term encourages that the generated visual features from the hallucinated text th to deceive the discriminator believing it is real, which restricts synthesized visual features to be realistic. The second term maximizes the entropy using a deviation measure. In our work, ShamaMittal(SM) entropy parameters \u03b3 and \u03b2 are learnt and hence adapt the corresponding data and split mode to a matching divergence function, leading to the best results especially in the generalized SUAUC metric; see \ufb01rst row in Table 5. We \ufb01rst investigate the effect of deviating the hallucinated text by classifying it t a new class Ks + 1, where Ks is the number of the seen classes. We found the performance is signi\ufb01cantly worse since the loss would signi\ufb01cantly increase indecencies against seen classes and hence reduces seen knowledge transfer to unseen classes; see row 2 in Table 5. When we remove the \ufb01rst term (realistic constraints), the performance degrades especially under the generalized Seen-Unseen AUC metric because generated visual features became unrealistic; see row 3 in Table 5 (e.g., 39.2% to 35.0% AUC drop for CUB Easy and 11.9%-8.2% drop for CUB Hard). Alternatively, when we remove the second term (entropy), we also observe a signi\ufb01cant drop in performance showing that both losses are complementary to each other; see row 4 in Table 5 (e.g., 39.2% to 33.5% AUC drop for CUB Easy and 11.9%-10.1% drop for CUB Hard). In our ablation, applying our approach without both terms (our loss) is equivalent to [63], shown is the last row in Table 5 as one of the least performing baselines. Note that our loss is applicable to other generative ZSL methods as we show (a) CUB with SCS (easy) split (b) CUB with SCE (hard) split (c) NAB with SCS (easy) split (d) NAB with SCE (hard) split Figure 4: Seen-Unseen accuracy Curve with two splits: SCS(easy) and SCE(hard). Ours indicates GAZSL+ CIZSL in our state-of-the-art comparisons later in this section. We also compare different entropy measures to encourage the deviation from the seen classes: Kullback-Leibler (KL), R\u00b4 enyi [42], Tsallis [52], Bhattacharyya [23]; see rows 5-8 in Table 5. All these divergences measure are special cases of the two parameter (\u03b3 , \u03b2) Sharma-Mittal(SM) [21] divergence that we implemented. For instance, Renyi [42] and Tsallis [52] on the other hand only learns one parameter and achieves comparable yet lower performance. Bhattacharyya [23] and KL have no learnable parameters an achieves lower performance compared to SM. Zero-Shot Recognition and Generality on [58] and [63]. Fig 5 shows the key advantage of our CIZSL loss, doubling the capability of [57] from 0.13 AUC to 0.27 AUC to distinguish between two very similar birds: Parakeet Auklet (Seen class) and Crested Auklet (unseen class), in 200-way classi\ufb01cation; see Appendix A for details. Table 2 shows state-of-the-art comparison on CUB and NAB datasets for both their SCS(easy) and SCE(hard) splits (total of four benchmarks). Our method shows a signi\ufb01cant advantage \fMetric Top-1 Accuracy (%) Seen-Unseen AUC (%) Dataset CUB NAB CUB NAB Split-Mode Easy Hard Easy Hard Easy Hard Easy Hard WAC-Linear [11] 27.0 5.0 \u2013 \u2013 23.9 4.9 23.5 \u2013 WAC-Kernel [10] 33.5 7.7 11.4 6.0 14.7 4.4 9.3 2.3 ESZSL [43] 28.5 7.4 24.3 6.3 18.5 4.5 9.2 2.9 ZSLNS [39] 29.1 7.3 24.5 6.8 14.7 4.4 9.3 2.3 SynCfast [5] 28.0 8.6 18.4 3.8 13.1 4.0 2.7 3.5 ZSLPP [12] 37.2 9.7 30.3 8.1 30.4 6.1 12.6 3.5 FeatGen [58] 43.9 9.8 36.2 8.7 34.1 7.4 21.3 5.6 FeatGen[58]+ CIZSL 44.2+0.3 12.1 +2.3 36.3 +0.1 9.8 +1.1 37.4 +2.7 9.8+2.4 24.7+3.4 6.2 +0.6 GAZSL [63] 43.7 10.3 35.6 8.6 35.4 8.7 20.4 5.8 GAZSL [63] + CIZSL 44.6 +0.9 14.4 +4.1 36.6 +1.0 9.3 +0.7 39.2+3.8 11.9+3.2 24.5+4.1 6.4+0.6 Table 2: Zero-Shot Recognition on class-level textual description from CUB and NAB datasets with two-split setting. Top-1 Accuracy(%) Seen-Unseen H AwA2 aPY SUN AwA2 aPY SUN DAP [28] 46.1 33.8 39.9 \u2013 9.0 7.2 SSE [62] 61.0 34.0 51.5 14.8 0.4 4.0 SJE [3] 61.9 35.2 53.7 14.4 6.9 19.8 LATEM [56] 55.8 35.2 55.3 20.0 0.2 19.5 ESZSL [43] 58.6 38.3 54.5 11.0 4.6 15.8 ALE [2] 62.5 39.7 58.1 23.9 8.7 26.3 CONSE [36] 44.5 26.9 38.8 1.0 \u2013 11.6 SYNC [5] 46.6 23.9 56.3 18.0 13.3 13.4 SAE [25] 54.1 8.3 40.3 2.2 0.9 11.8 DEM [61] 67.1 35.0 61.9 25.1 19.4 25.6 DEVISE [15] 59.7 39.8 56.5 27.8 9.2 20.9 GAZSL [63] 58.9 41.1 61.3 15.4 24.0 26.7 GAZSL [63] + CIZSL 67.8 +8.9 42.1 +1.0 63.7 +2.4 24.6+9.2 25.7 +1.7 27.8 +1.1 FeatGen [58] 54.3 42.6 60.8 17.6 21.4 24.9 FeatGen [58] + CIZSL 60.1+5.8 43.8+1.2 59.4\u22120.6 19.1+1.5 24.0+2.6 26.5+1.6 cycle-(U)WGAN [14] 56.2 44.6 60.3 19.2 23.6 24.4 cycle-(U)WGAN [14] + CIZSL 63.6+7.4 45.1 +0.5 64.2 +3.9 23.9+4.7 26.2 +2.6 27.6+3.2 Table 3: Zero-Shot Recognition on class-level attributes of AwA2, aPY and SUN datasets. CUB NAB 25% 50% 100% 25% 50% 100% ESZSL [43] 27.9 27.3 22.7 28.9 27.8 20.9 ZSLNS [39] 29.2 29.5 23.9 28.8 27.3 22.1 ZSLPP [12] 42.3 42.0 36.3 36.9 35.7 31.3 GAZSL [63] 49.7 48.3 40.3 41.6 37.8 31.0 GAZSL [63]+ CIZSL 50.3+0.6 48.9 +0.6 46.2+5.9 41.0\u22120.6 40.2+2.4 34.2+3.2 Table 4: Zero-Shot Retrieval using mean Average Precision(mAP) (%) on CUB and NAB with SCS(easy) splits. compared to the state of the art especially in generalized Seen-Unseen AUC metric ranging from 1.0-4.5% improvement. Fig 4 visualizes Seen-Unseen curves for our four benchmarks CUB (east and hard splits) and NABirds (easy and hard splits) where ours has a signi\ufb01cant advantage compared to state-of-the-art on recognizing unseen classes; see our area under SU curve gain in Fig 4 against the runner-up GAZSL. The average relative SU-AUC improvement on the easy splits is 15.4% and 23.56% on the hard split. Meaning, the advantage of our loss becomes more clear as splits get harder, showing a better capability of discriminative knowledge transfer. We show the generality of our method by embedding it with another feature generation method, FeatGen [58], causing a consistent improvement. All the methods are using same text and visual representation. Zero-Shot Retrieval. We investigate our model\u2019s performance for zero-shot retrieval task given the Wikipedia article of the class using mean Average Precision (mAP), the common retrieval metric. In table 4, we report the performance of different settings: retrieving 25%, 50%, 100% of the images at each class. We follow [63] to obtain the visual center of unseen classes by generating 60 examples for the given text then computing the average. Thus, given the visual center, the aim is to retrieve images based on the nearest neighbor strategy in the visual features space. Our model is the best performing and improves the MAP (100%) over the runner-up ( [63]) by 14.64% and 9.61% on CUB and NAB respectively. Even when the model fails to retrieve the exact unseen class, it tends to retrieve visually similar images; see qualitative examples in Appendix D. 5.2. Attribute-based Zero-Shot Learning Datasets. Although it is not our focus, we also investigate the performance of our model\u2019s zero-shot recognition ability using different semantic representation. We follow the GBU setting [57], where images are described by their attributes instead of textual describtion deeming the problem to be relatively easier than textual-description zeroshot learning. We evaluated our approach on the following datasets: Animals with Attributes (AwA2) [27], aPascal/aYahoo objects(aPY) [13] and the SUN scene attributes dataset [38]. They consist of images covering a variety of categories in different scopes: animals, objects and scenes respectively. AwA contains attribute-labelled classes but aPY and SUN datasets have their attribute signature calculated as the average of the instances belonging to each class. Zero-Shot Recognition. On AwA2, APY, and SUN datasets, we show in Table 3 that our CIZSL loss improves three generative zero-shot learning models including GAZSL [63], FeatGen [58], and cycle-(U)WGAN [14]. The table also shows our comparison to the state-ofthe-art where we mostly obtain a superior performance. Even when obtaining a slightly lower score than state-ofthe-art on AWA2, our loss adds a 9.2% Seen-Unseen H absolute improvement to the non-creative GAZSL [63]. We also evaluated our loss on CUB-T1(Attributes) benchmark [57], where the Seen-Unseen H for GAZSL [63]] and GAZSL [63]+CIZSL are 55.8 and 57.4, respectively. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1709.01148v1",
+                    "title": "Link the head to the \"beak\": Zero Shot Learning from Noisy Text Description at Part Precision",
+                    "abstract": "In this paper, we study learning visual classifiers from unstructured text\ndescriptions at part precision with no training images. We propose a learning\nframework that is able to connect text terms to its relevant parts and suppress\nconnections to non-visual text terms without any part-text annotations. For\ninstance, this learning process enables terms like \"beak\" to be sparsely linked\nto the visual representation of parts like head, while reduces the effect of\nnon-visual terms like \"migrate\" on classifier prediction. Images are encoded by\na part-based CNN that detect bird parts and learn part-specific representation.\nPart-based visual classifiers are predicted from text descriptions of unseen\nvisual classifiers to facilitate classification without training images (also\nknown as zero-shot recognition). We performed our experiments on CUBirds 2011\ndataset and improves the state-of-the-art text-based zero-shot recognition\nresults from 34.7\\% to 43.6\\%. We also created large scale benchmarks on North\nAmerican Bird Images augmented with text descriptions, where we also show that\nour approach outperforms existing methods. Our code, data, and models are\npublically available.",
+                    "authors": "Mohamed Elhoseiny, Yizhe Zhu, Han Zhang, Ahmed Elgammal",
+                    "published": "2017-09-04",
+                    "updated": "2017-09-04",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Recognizing visual categories only from the class description is an appealing characteristic of human learning and generalization, which is desirable to be modeled for better machine intelligence. This problem is known as \u201czeroshot\u201d learning/classi\ufb01cation. In practice, this is motivated by the lack of annotated training data for most object categories and especially at the \ufb01ne-grained level, which has been observed by several researches (e.g., [40, 52]). For instance, there exist tens of thousands of bird categories among which images are available for only few-hundred-categories in existing datasets (< 5%) [48]. Some bird categories are scarce in the real-world\u2013 it is very hard to \ufb01nd the \u201cCrested ibis\u201d around us and even in a zoo. Earlier zero-shot recognition methods rely on describing * Both authors contributed equally to this work Figure 1. People can learn from text descriptions at part-level visual classes by a set of semantically meaningful properties known as attributes [16, 24]. The underlying principle behind the success of attributes on zero-shot learning is that they are modeled as an intermediate layer between class labels and images, which enable transfer of shared concepts/attributes from seen classes to unseen classes. More recent attribute methods improve the information transfer across classes by joint embedding of images and attributes [4, 49, 10, 5]. While attributes can semantically describe classes with human interpretability without any images, they typically require domain experts to be de\ufb01ned. It is also necessary to collect hundreds of these attribute annotations for each of the seen and unseen classes which is discouraging. Towards reducing the gap between machine and human intelligence on this task, recent methods [14, 26, 6, 36] explored zero-shot learning from online text descriptions, which in turn avoids the burden of heavy attributes annotations for each class. What makes this setting very challenging is that these descriptions comes in the form of noisy encyclopedia articles that include not only visual descriptions about the visual appearance but also discussion about the category\u2019s behavior, breading, immigration, etc. Our work aims at designing an interpretable model in this direction. Prior works [36, 39, 26, 6] use a wholistic feature representation for both the object and the text description (e.g., term frequency vector is common for the bird text description and a visual feature vector for the whole object). Contributions In our work, we propose an effective model that can relate text information of visual categories to images with part-based regularization. Fig. 1 illustrates the text-part connectivity capability that we aim to model in our work, where birds are recognized from text description by relating text terms to parts in the image (e.g, relating the 1 arXiv:1709.01148v1  [cs.CV]  4 Sep 2017 \fbill to the head of the bird). Note that this task is unlike existing visual grounding tasks (e.g., [35, 18]), which requires object-(text phrase) annotations during training and has been mainly studied at the object level/not at part-level. Our method is able to quell the noise in the text descriptions by eliminating irrelevant text information without requiring part-text correspondence annotation or part annotations at test time. Our model is composed of two networks, \u201cVisual Part Detector / Encoder network\u201d (VPDE-net) and \u201cPart Zero-Shot Classi\ufb01er prediction network\u201d (PZSC-net). The VPDE-net is fed with bird images, detects the bird parts, and learns CNN feature representation for every part. The PZSC-net predicts part-based zero-shot classi\ufb01er from the noisy text description of bird classes, which is executed on the part-CNN representation produced by the VPDE-net. Besides evaluating on the CUB dataset [48], we also set up new zero-shot benchmarks by extending the NABirds dataset [43] with a corresponding unstructured text article extracted from Wikipedia and AllaboutBirds website [2]. This is \ufb01ve times bigger than the largest existing benchmark for text-based zero shot learning. 2. Related Work Attribute-based methods: Besides manually speci\ufb01ed attributes (e.g., [25, 16, 24, 33]), several researchers have explored various attribute applications and attempted to automatically discover these attributes [9, 37, 29, 38]. Recent approaches model attributes in a continuous space (e.g., [4, 21]). The main idea of these approaches is to learn a transformation matrix W that correlates attributes to images \u2013we name these methods transformation-based approaches. Other zero-shot approaches used graph/hyper-graphs built on attributes and class labels (e.g., [17, 20]). In contrast to graph/hyper-graph based approaches, transformation-based approaches have recently shown better performance and are meanwhile simpler and more ef\ufb01cient on \ufb01ne-grained recognition (e.g., [39, 6, 5]). Text-based methods: More relevant to this paper is the research direction exploring using text articles from the web to predict zero-shot visual classi\ufb01ers. Elhoseiny et al. [14] proposed an approach to that combines domain transfer and regression to predict visual classi\ufb01ers from a TF-IDF textual representation. Bo et al. [26] adopted deep neural networks to predict convolutional classi\ufb01ers, leading to a noticeable improvement on zero-shot classi\ufb01cation. Very recently, Qiao et al. [36] revisited the importance of regularization on zeroshot learning. They show that attribute-based formulation like [39] achieves competitive zero-shot performance when applied to text by just replacing the attribute representation with textual feature vectors. They further demonstrated that the noise in the text descriptions could be suppressed by encouraging group sparsity on the connections to the textual terms. Similar to transformation-based approaches, most of these text-based methods (e.g., [14, 36, 39]) are based also on learning transformations that relates images to text in a common space. In our view, most of the recent progress has been achieved by better visual representations using deep neural networks (e.g., [26]) and/or better regularization to suppress noise in texts (e.g. [36, 39]). In our work, we build on top the existing methods and demonstrate that zeroshot recognition could be signi\ufb01cantly improved by partbased regularization in contrast to the whole image in the aforementioned approaches. It is important to mention that in [3], Akata et al. studied zero-shot learning with multiple cues and they used bird parts. There are two key differences to our work. (1) In [3], multiple sources from WordNet [31] and word embeddings [30, 34] are used in addition to text terms, while we only uses text terms. (2) They used annotations of 19 bird parts for training, however, at test time the method is not able to locate these parts and hence require the part test annotations to relate to their multiple cues. In our work, we demonstrated signi\ufb01cantly better performance using only text terms and with no part annotation needed at test time. Moreover, at training time, only annotations of 7 parts are needed instead of 19 that are easier to collect. Other language& vision methods: In other tasks like image-captioning (e.g., [22, 47, 15]),VQA (e.g., [7]), and image-sentence similarity (e.g., [23, 45]), better performance has been demonstrated with better image and language representations. The text annotations in the typical datasets for these methods are carefully collected at the image-level by crowdsourcing services (e.g., 5 captions per sentences in MS-COCO [28] or Flick30K datasets [50]). In contrast to these settings, the text descriptions in our work come at the category level (e.g., one text description for \u201cCardinal\u201d class). Hence, there is much less text in our setting and meanwhile the text is much noisier as we described earlier. In our experiments, we set up an image-sentence similarity baseline to study the performance of the representations in methods when applied to very noisy text as in our setting with only the small portion of the text is related visually. 3. Proposed Approach Connecting unstructured text into bird parts requires language and a visual representations that facilitates mutual transfer at the part level from text to images and vice versa. We also aim at a formulation that does not require text-topart labeling at training time nor it does require oracle part annotations at test time (e.g., [3]). Fig. 2 shows an overview of our learning framework. Our approach starts by a simple raw text representation involving term frequencies; see Sec 3.1. The text representation is then fed into a dimensionality reduction step followed by multi-part transformation to predict a visual classi\ufb01er at the visual part level. The predicted classi\ufb01er is applied on the part-based feature representations that are learnt through a deep Convolutional Neural Network (CNN). In the following subsections, we describe the text and visual part encoders, then de\ufb01ne our \fproblem and the proposed approach on top of these encoders. 3.1. Text Encoder Similar to [14, 26], text articles are \ufb01rst tokenized into words and the stop words are removed. Then, a simple Term Frequency-Inverse Document Frequency(TF-IDF) feature vector is extracted [41]. We denote the TF-IDF representation of a text article t by t \u2208RdT , where dT is the number of terms in the TF-IDF text representation. 3.2. Visual Parts CNN Detector/Encoder (VPDE) Detecting semantic parts facilitates modeling a representation that can be related to unstructured text terms at the part-level. It was shown in [51] that bird parts can be detected at precision of 93.40% vs 74.0% with earlier methods [27]. We adopt fast-RCNN framework [19] with VGG16 architecture [42] to detect seven small bird-parts using the small-part proposal method proposed in [51]. The seven parts in order are (1) head, (2) back, (3) belly, (4) breast, (5) leg, (6) wing and (7) tail; see Fig. 2. We denote the input image to the visual part encoder as x. First, the image x is processed through VGG16 convolutional layers. The proposed regions by [51] on x are then ROI pooled with a 3\u00d73 grid. Then, they are then passed through an 8-way classi\ufb01er (7 parts + background) and a bounding box regressor. Each part p is assigned to the region with the highest con\ufb01dence of part p if that con\ufb01dence is greater than a threshold (i.e. 1/7). If the highest con\ufb01dence of part p is less than the threshold, part p is considered as missing. The detected part regions are then passed to the visual encoder sub-network, which ROI(3 \u00d7 3) pools these regions and eventually encode each part into a 512 dimensional learning representation. When a part is missing, a region of all zeros is passed to the encoder-sub-network. We denote these part-learning representations of a bird image x as x(1), x(2), \u00b7 \u00b7 \u00b7 , x(P ); see the \ufb02ow from x to the part representation in Fig. 2 (top-part starting from the blue arrow at the top-left). We will detail later how the Visual Part Detector/Encoder (VPDE) network is trained. We denote the dimensionality of the part features as dP , where x(p) \u2208RdP \u2200p and dP = 512 in our work. 3.3. Problem De\ufb01nition During training, the information comes from images and text descriptions of K seen classes. We denote the learning representations of the detected parts of N training examples as {X (p) \u2208RdP \u00d7N}, p = 1 : P, where P is the number of parts. We denote the text representation of K seen classes as T \u2208RdT \u00d7K. We de\ufb01ne Y \u2208{0, 1}N\u00d7K as the label matrix of each example in one-hot representation (i.e., each row in Y is a vector of zeros except at the corresponding class label index). At test time, the text features are given for \u02c6 K classes, where we need to assign the right label among them to each test image. Formally, the label assignment of an image x is de\ufb01ned as k\u2217= arg max k P X p=1 z(p)(tk) T \u00b7 x(p), k = 1 : \u02c6 K (1) where {x(1), x(2), \u00b7 \u00b7 \u00b7 , x(P )} is the part learned representation of image x, tk is the text representation of class k, and z(p)(t) is a function that takes a text representation t and predicts a visual classi\ufb01er weights for part p. In our work, we aim at jointly learning and regularizing z(p)(\u00b7), \u2200p \u22081 : P to encourage text terms to correlate with sparse set of parts. 3.4. Part Zero-Shot Classi\ufb01er Prediction (PZSC) Part visual classi\ufb01er prediction functions are de\ufb01ned as z(p)(t) = tTWt TWp x, \u2200p \u22081 : P (2) where Wt \u2208Rd\u00d7dT is a dimensionality reduction matrix, which projects the text representation t \u2208RdT into a latent space, Wp x \u2208Rd\u00d7dP for each part p then regress the projected text representation into a classi\ufb01er for part p; see Fig. 2 (bottom part starting from the blue arrow at the bottom-left). Hence, z(p)(t) \u2200p are mainly controlled by Wp x and Wt since t is the input. We will elaborate next on how Wt and Wp x \u2200p are trained jointly. 3.5. Model Optimization and Training An interesting research direction regularizes zero-shot learning by introducing different structures to the learning parameters (e.g., [39, 36]). In [39]. Minimizing the variance of the projections from image to attribute space and vice versa is the key to improving attribute-based zero-shot prediction. In [36], Qiao et al. used l2,1 sparsity regularization, proposed in [32], to encourage sparsity on the text terms, and showed its capability to suppress noisy text terms and improve zero-shot classi\ufb01cation from text. We got inspired by these regularization techniques to train our our framework in Fig. 2 with the following cost function: min W1 x,\u00b7\u00b7\u00b7 ,WP x ,Wt ||( P X p=1 X(p)TWp x T)WtT \u2212Y||2 F + \u03bb1 P X p=1 ||Wp x TWtT||2 F + \u03bb2 P X p=1 ||Wp x TWt||2,1 (3) where || \u00b7 ||F is the Frobenius norm. The \ufb01rst term in Eq. 3 encourages that for every image xj, PP p=1 z(p)(tk) T\u00b7x(p) j = PP p=1(tT kWt TWp x)T\u00b7x(p) j to be equal to 1 if k is the ground truth class, 0 if other classes. This enables z(p)(t) to predict part classi\ufb01ers for an arbitrary text t (i.e. high (\u21921) for the right class, low (\u21920) for others). The second term bounds the variance of the functions {z(p)(t) = tTWt TWp x \u2200p}. More importantly, the third term imposes structure on Wt and {Wp x \u2200p}, to encourage connecting every text term with sparse set of parts (i.e., every text term attends to as few parts as possible ). The third term PP p=1 ||Wp x TWt||2,1 is de\ufb01ned \fPart  Proposals  RoI pooling FC layers and Regressor Outputs: a) Part Bbox regressor b)  Softmax ( part score) Semantic Part detections Shared Conv layers 1. Head 2. Back 3. Belly 4. Breast 5. Leg 6. Wing 7. Tail      Semantic  part ROI  pooling FC layers  (softmax loss  over seen  classes) The Parakeet Auklet is a small (23 cm) auk with a short  orange-bill that is upturned to  give the bird its curious fixed  expression. The bird's plumage  is dark-above and white-below with a single white plume projecting back from the eye Wt 7-Part Zero-Shot Classifier prediction  512 512 Part FC layers ( one for each part separately) The Parakeet Auklet is a small  (23 cm) auk with a short  orange-bill that is upturned to  give the bird its curious fixed  expression. The bird's plumage  is dark-above and white-below  with a single white plume  projecting back from the eye 512 512 512 512 512 x t x(1) x(2) x(3) x(4) x(5) x(6) x(7) Detection  sub-network Encoder sub-network t From VGG16 arch Visual Parts Detector/ Encoder TFIDF z(3)(t) z(7)(t) z(1)(t) z(2)(t) z(4)(t) z(5)(t) z(6)(t) dX= 512 Figure 2. Our approach (best seen in color): On the bottom is the core of our approach where the input is a pure text description and produces classi\ufb01er through a dimensionality reduction transformation Wt following by part projections W p x , p = 1 : P, where P is the number of parts. The produced P classi\ufb01ers are then applied on the part learning representation produced through detected parts from the top visual CNN. RoI refers to Region of Interest Pooling [19].FC refers to Fully connected layers. VGG conv layers refer to the \ufb01rst \ufb01ve convolutional layers in VGGNet-16 [42] as PP p=1 PdT i=1 ||Wp x Twi t||2, wi t is the ith column in Wt matrix that corresponds to the ith text term, Wp x Twi t \u2208RdX are the weights that connect the pth part to ith text term. Hence, the third term encourages group sparsity over the parameter groups that connect every text term i to every part p (i.e. Wp x Twi t), which encourages terms to be connected to parts sparsely. Optimization: The parameters of our model include part detection sub-network parameters and part representation sub-network parameters for Visual Part Detector/Encoder (VPDE) network, and {Wp x, p = 1 : P}, Wt for the part zero-shot classi\ufb01er predictor (PZSC) network. The VPDE network is trained by alternate optimization over the detector and the representation sub-networks with the training images. The detector sub-network is optimized through softmax loss over 8 outputs (7 parts and background) and bounding box regression to predict the \ufb01nal box for each detected part. The representation sub-network is optimized over by softmax loss over the seen/training classes. The convolutional layers are shared between the detection and representation subnetworks (VGG16 conv layers in our work); see Fig. 2(toppart) and supplementary for architecture details. After training VPDE network, we solve the objective function in Eq. 3 to train the Part Zero-Shot Classi\ufb01er predictor. The cost function in Eq. 3 is convex if optimized for either Wt or {Wp x, p = 1 : P} individually but not convex on both. Hence, we solve Eq. 3 by an alternate optimization, where we \ufb01x Wt and solve for {Wp x, p = 1 : P}, then \ufb01x {Wp x, p = 1 : P} and solve for Wt. Solving for Wt: Following the ef\ufb01cient l2,1 group sparsity optimization method in [32], the solution to this subproblem could be ef\ufb01ciently achieved by sequentially solving to following problem until convergence. min Wt,{Dp l ,\u2200p} ||( P X p=1 X(p)TWp x T)WtT \u2212Y||2 F + \u03bb1 P X p=1 ||Wp x TWtT||2 F + \u03bb2 P X p=1 Tr (Wp x TWtDp l Wt TWp x) (4) where Dp l is a diagonal matrix with the i-th diagonal element is 1/(2||Wp x(wi z)(l\u22121)||2)2 at the the l-th iteration, where (wi z)(l\u22121) is the i-th column of Wt solution at iteration l\u22121. \fAlgorithm 1: Alternate Optimization to solve Eq. 3 Input :T, Y, X(1), \u00b7 \u00b7 \u00b7 X(p) Output :Wt, W1 x, \u00b7 \u00b7 \u00b7 , WP x 1 Initialize Wt and W1 x, \u00b7 \u00b7 \u00b7 , WP x with standard Gaussian distribution. 2 Initialize Wt turn = false 3 for l=1 \u00b7 \u00b7 \u00b7 L do 4 Update D(p) l \u2200p 5 if (Wt turn = true) then 6 Find Wt with Eq. 4 by Quassi-Newton BFGS ; 7 else 8 Find {Wp x} with Eq. 5 by Quassi-Newton BFGS ; 9 Wt turn = not Wt turn 10 if Converges then 11 Break 12 end 13 end We realized that it is hard to \ufb01nd a closed-form solution to Eq. 4 or even reduce it to the Sylvester Equation [8]. Hence, we solve Eq. 4 by Quasi-Newton with Limited Memory BFGS Updating (i.e., gradient-based optimization). The derived gradients for Eq. 4 sub-problem are attached in the supplementary materials. Solving for Wp x : In this step, we solve the following sub-problem. min {Dp l ,Wp x,\u2200p} ||( P X p=1 X(p)TWp x T)WtT \u2212Y||2 F + \u03bb1 P X p=1 ||Wp x TWtT||2 F + \u03bb2 P X p=1 Tr (Wp x TWtDp l WtTWp x) (5) where Dp l is a diagonal matrix with the i-th diagonal element is 1/(2||(Wp x)(l\u22121)wi z||2)2 at the l-th iteration, where (Wp x)(l\u22121) is the solution of Wp x for part p at iteration l \u22121. Similar to Eq. 4, we solve Eq. 5 by Quasi-Newton with BFGS Updating. The derived gradients for Eq. 5 sub-problem are attached in the supplementary materials. Algorithm 1 shows the overall optimization process that solves Wt and W1 x, \u00b7 \u00b7 \u00b7 , WP x jointly. 4. Experiments 4.1. Experiment setting Datasets: We compare the proposed method with stateof-the-art approaches on two datasets: CUB2011 [48] and NABirds [44]. Both are bird datasets for \ufb01ne-grained classi\ufb01cation. Important parts of the bird in each image are annotated with locations by experts. CUB2011 dataset contains 200 categories of bird species with a total of 11,788 images. Compared with CUB2011, NABirds is a larger dataset of birds with 1011 classes and 48562 images. It constructs a hierarchy of bird classes, including 555 leaf nodes and 456 parent nodes, starting from the root class \u201cbird\u201d. Only leaf nodes are associated with images, and the images for parent class can be collected by merging all images of its children nodes. In practice, we found some pairs of classes merely differ in gender. For example, the parent node \u201cAmerican Kestrel\u201d are divided to \u201cAmerican Kestrel (Female, immature)\u201d and \u201cAmerican Kestrel (Adult male)\u201d. Since we cannot \ufb01nd the Wikipedia articles for this subtle division of classes, we merged such pairs of classes to their parent. After such processing, we \ufb01nally have 404 classes, each one is associated with a set of images, as well as the class description from Wikipedia. We collected the raw textual sources from English-language Wikipedia-v01.02.2016. We manually veri\ufb01ed all the articles and augmented classes with limited descriptions from the all-about-birds website [2]. We plan to release this data and the NABird benchmarks that we set up. Two split setting: To split the dataset to training/testing set, we have designed two kinds of splitting schemes, in terms of how close the seen classes are to the unseen classes: Super-Category-Shared splitting (SCS), Super-CategoryExclusive splitting(SCE). In the dataset, some classes often are the further division of one category. For example, both \u201cBlack footed Albatross\u201d and \u201cLaysan Albatross\u201d belong to the category \u201cAlbatross\u201d in CUB2011, and both \u201cCooper\u2019s Hawk\u201d and \u201cHarris\u2019s Hawk\u201d are under the category \u201cHawks\u201d in NABirds. For SCS, unseen classes are deliberately picked in the condition that there exists seen classes with the same Super-Category. In this scheme, the relevance between seen classes and unseen classes is very high. On the contrary, in SCE, all classes under the same category as unseen classes would either belong to the seen or the unseen classes. For instance, if \u201cBlack Footed Albatross\u201d is an unseen class then all other albatrosses are unseen classes as well and so no albatrosses are seen during training. It is not hard to see that the relevance between seen and unseen classes is minimal in the SCE-split. Intuitively, SCE-split is much harder compared to SCS-split. These strategies for zero-shot splits were used on CUBirds dataset in the literature but in different works and were not compared to each other. For SCS-split on CUB2011, we use the same splitting to [3, 36], where 150 classes for training and 50 classes for testing. For SCE-split on CUB2011, we use the same splitting to [14], where the \ufb01rst 80% classes are considered as seen classes and used for training. To design these two splitting schemes in the NABirds, we \ufb01rst check the class hierarchy. There exist 22 children nodes under the root category (bird) in the hierarchy. We found that the number of descendants under the 22nd children (Perching Birds) are much greater than the average descendants of the remaining 21 classes (205 vs.10). To eliminate this imbalance, we further divide this category to its children. With the combination of 29 children of this category and \fother 21 children of the root, we ended up with 50 super categories (21+29). For SCS-split, we randomly pick 20% of descendant classes under each super categories as unseen classes. For SCE-split, we randomly pick 20% of super categories and consider all their-descendant classes as unseen are considered the seen classes. For both splits, there are totally 323 training (seen) classes and 81 testing (unseen) classes, respectively. For ease of presentation, we sometimes refer to the SCS-split as the easy-split and to SCE-split as the hard-split. Textual Representation: We extract the text representation according to the scheme described in Section 3.1. The dimensionality of TF-IDF feature for CUB2011 and NABirds are 11083 and 13585, respectively. Image representation: As described in Section 3.2, the part regions are \ufb01rst detected and then passed to the VPDE network. 512-dimensional feature vector is extracted for each semantic part. For CUB2011 dataset, we only use seven semantic parts to train the VPDE network; illustrated in Fig. 2. For NABird dataset, we used only six visual parts with the \u201cleg\u201d part removed, since there is no annotations for the \u201cleg\u201d part in the NABirds dataset. 4.2. Performance evaluation Baselines and Competing Methods: The performance of our approach is compared to six state-of-the-art algorithms: SJE [6], MCZSL [3], ZSLNS [36], ESZSL [39], WAC [14]. The source code of ESZSL and ZSLNS are available online, and we get the code of WAC [14, 13] from its author. For MCZSL and SJE, since their source codes are not available, we directly copy the highest scores for non-attribute settings reported in [3, 6]. Image-sentence baseline [46]: Additionally, we used a state of the art Model [46] for image-sentence similarity by breaking down each text document into sentences and considering it as a positive sentence for all images in the corresponding class. Then we measure the similarities between an image to class by averaging its similarity to all sentences in that class. Images were encoded using VGGNet [42] and sentences were encoded by an RNN with GRU activations [12]. The purpose of this experiment is to study how RNN representation of the sentences perform in our setting with noisy text descriptions. We \ufb01rst compare our approach with MCZSL, which is among the best performing state-of-art methods. Both our approach and MCZSL utilizes part annotations provided by the CUB2011 datasets.However, in contrast to MCZSL, which directly uses part annotations to extract image feature in the test phase, our approach is merely based on the detected semantic parts during both training and testing. Less accurate detection of semantic parts will surely degrade the accuracy for the \ufb01nal zero-shot classi\ufb01cation. In order to make a fair comparison with MCZSL, we also report our result using the ground-truth annotations of semantic parts at methods Accuracy MCZSL [3](BoW) 26.0 MCZSL [3](word2vec) 32.1 MCZSL [3](Comb) 34.7 Ours-DET 37.2 Ours-ATN 43.6 Table 1. Performance comparison with the accuracy (%) on CUB2011 Dataset. In [3], the approach is evaluated with different textual representation: BoW, word2vec, and their combination. test-time. The results of our approach based on the detected parts and ground-truth parts are denoted by \u201cOurs-DET\u201d and \u201cOurs-ATN\u201d, respectively. In Table 1, we compared to the same benchmark reported in [3], which is the SCS-split on CUBirds 2011 dataset. The results show that our performance is 9% better than [3] (43.6% vs 34.7%) although we only used a simple TF-IDF text representation compared to multiple cues used in MCZSL like text, WordNet and word2vec. Note also that the 34.7% achieved by [3] used 19 part annotations during training and testing (the whole image, head, body, full object, and 15 part locations annotated), while we only used 7 parts to achieve the 43.6%. Table 1 also shows that our method still perform 2.5% better even when using the detected parts at test time (37.2% Ours-DET vs 34.7% MCSZSL using ground truth annotations). In all the following experiments, we only used our approach with the detected parts (i.e. \u201cOurs-DET\u201d). Zero-shot Top-1 Accuracy. For standard zero-shot image classi\ufb01cation, we calculate the mean Top-1 accuracy obtained on unseen classes. We performed comprehensive experiments on both SCS-(easy) and SCS-(hard) splits on both CUBirds and NABirds. Note that some of these methods were applied on attributes prediction (e.g., ZSLNS [36], SynC [10], ESZSL [39] ) or image-sentence similarity (e.g.,Order Embedding [46]). We used the publicly available code of these methods and other text-based methods like (ZSLNS [36], WAC [14], WAC-kernel [13]) to apply them on our setting. Note that the conventional split setting for zero-shot learning is Super-Category Shared splitting (i.e. SCS-(easy) split). We think evaluating the performance on both the SCS-(easy) and the SCE-(hard) splits are complementary and hence we report the performance on both of them. In Table 2, we show the comparisons between our method to all the baselines on the CUB2011 easy and hard benchmarks, where method outperforms all the baselines by a noticeable margin on both the easy and the hard benchmarks. Note that the image-sentence similarity baseline (i.e. Order Embedding [46]) is among the least-performing methods. We think the reason is the level of noise which is addressed by the other methods by regularizing the text information at the term level, while the representation unit in [46] is the whole sentence. Similarly, Table 3 shows the results on NABirds easy and hard benchmarks, where the performance of our approach is also superior over the competing methods. It is worth mentioning that the WAC-method is not \fscalable since the its training parameters depend on the number of image-class pair. We trained it for 6 days on 64GB RAM machine and report the results of the latest snapshot in Table 3. methods SCS(Easy) SCE(Hard) WAC-Linear [14] 27.0 5.0 WAC-Kernel [13] 33.5 7.7 ESZSL [39] 28.5 7.4 SJE [6] 29.9 \u2013 ZSLNS [36] 29.1 7.3 SynCfast [10] 28.0 8.6 SynCOV O [10] 12.5 5.9 Order Embedding [46] 17.3 5.9 Ours-DET 37.2 9.7 Table 2. Top-1 accuracy (%) on CUB2011 Dataset in two different split settings. Note that some of these methods are attribute-based methods but applicable in our setting by replacing attribute vectors with text features. methods SCS(Easy) SCE(Hard) WAC-Kernel [13] 11.4 6.0 ESZSL [39] 24.3 6.3 ZSLNS [36] 24.5 6.8 SynCfast [10] 18.4 3.8 Ours-DET 30.3 8.1 Table 3. Top-1 accuracy (%) on NABird Dataset splits. Generalized Zero-Shot Learning Performance. The conventional zero-shot learning that we discussed earlier classi\ufb01es test examples into unseen classes without considering the seen classes in test phase. Because the seen classes are often the most common, it is hardly realistic to assume that we will never encounter them during the test phase [11]. To get rid of such an assumption, Chao et al. [11] recently proposed a more general metric for generalized zero-shot learning (GZSL). We here brie\ufb02y review how it generally measures the capability of recognizing not only unseen data, but also seen data. Let S, U denote the label spaces of seen classes, unseen classes; T = S \u222aU, the joint label space. AU\u2192T and AS\u2192T are the accuracies of classifying seen data and unseen data into joint label space. The labels are computed using the Eq. 6: y = arg max c\u2208T f(x) \u2212\u03bbI[c \u2208S] (6) where I[.] \u2208{0, 1} indicates whether c is a seen class and \u03bb is the penalty factor. x is set to seen data or unseen data to calculate AU\u2192T and AS\u2192T , respectively. As \u03bb increases or decreases, data are encouraged to be classi\ufb01ed to unseen classes or seen classes, respectively. In the cases where \u03bb is extremely large or small, all data will assigned with unseen class label or seen class label, respectively. Therefore, we can generate a series of pairs of classi\ufb01cation accuracies (\u27e8AU\u2192T , AS\u2192T \u27e9) by tuning values of \u03bb. Considering these pairs as points with AU\u2192T as x-axis and AS\u2192T as y-axis, we can draw the Seen-Unseen accuracy Curve(SUC). The Area Under SUC (AUSUC), as a widely-used measure of curves, can well assess the performance of an classi\ufb01er in balance of the con\ufb02icting AU\u2192T and AS\u2192T ) measurements. The Seen-Unseen accuracy Curve of our method and other state-of-the-art approaches are shown in Fig. 3. The performance of our work is superior over all other methods in term of the AUSUC score. Although WAC linear apparently achieves a high performance on seen classes, its poor performance of classifying unseen classes indicates that it doesn\u2019t learn much knowledge that can be effectively transferred to unseen classes. On the contrary, ZSLNS has a relatively good accuracy AU\u2192T , but its lower AS\u2192T compared with other methods indicates that the success of unseen classes\u2019 classi\ufb01cation may come from the overweighted regularizers. Our method remarkably outperforms other methods in term of both the classi\ufb01cation of unseen classes, and also achieves a relative high accuracy in recognition of seen classes. The curves in Fig. 3 demonstrate our method\u2019s capability of balancing the classi\ufb01cation of unseen classes and seen classes (0.304 AUSUC for Ours-DET compared to 0.239 for the best performing baseline). We also demonstrated the effectiveness of our performance on NABirds dataset in Fig. 4 (0.126 AUSUC for Ours-DET compared to 0.093 for the best performing baseline). In addition to these GZSL results on SCS-splits, we also report the Seen/Unseen curves on the SCE-splits in the supplementary due to space. Model Analysis and Qualitative Examples. We also analyzed the the connections between the terms and parts in the learnt parameters, which is Wp x Twi t for the connection between term i and part p on CUBirds dataset (SCS-split). Fig. 6 shows the l2 norm of Wp x Twi t for each part separately and only on the top 30 terms for each part sorted by \u2225Wp x Twi t\u22252. Fig. 8 shows the percentage of overlap between these terms for every pair of parts, which shows that every part focus on its relevant concepts yet there is still a shared portion that includes shared concepts like color and texture. In Fig. 6, we show the the summation of these connections for every part and compare between \u201cOurs-DET\u201d and \u201cOurs-ATN\u201d to analyze the effect of detecting the parts versus using part annotations. We observe that more concepts/terms are discovered and connected to head for \u201cOursATN\u201d, while more concepts are learnt for \u201cbreast\u201d for \u201cOursDET\u201d. This is also consistent with the Top-1 accuracy if each part is individually used for recognition; see the Top-1 Acc for each part separately in Fig. 6 (right). This observation shows if we have a perfect detector, head will be one of the most important part to be connected to terms which is intuitive. We also observed the same conclusion on both SCS and SCE splits on NABirds and SCE on CUBirds; see additional analysis \ufb01gures for these splits in the supplementary. We further demonstrate these part-to-term connectivity by some qualitative examples in Fig. 7. For each bird, the top related term for each part is printed based on ranking \f0 0.1 0.2 0.3 0.4 AU!T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 AS!T ESZSL:          0.185 ZSLNS:          0.147 WAC_kernal: 0.225 WAC_linear:  0.239 SynC_Fast:    0.131 SynC_RegV:   0.017 Ours:             0.304 Figure 3. CUBirds Seen-Unseen accuracy Curve (SCE split) 0 0.05 0.1 0.15 0.2 0.25 0.3 AU!T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 AS!T ESZSL:          0.092 ZSLNS:          0.070 WAC_kernal: 0.041 SynC_Fast:    0.026 Ours:             0.130 Figure 4. NABirds Seen-Unseen accuracy Curve (SCS split) Head Back Belly Breast Leg Wing Tail part name 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Ours-ATN Ours-DET \u2211 ||\ud835\udc4a $ %\ud835\udc64' (||) *+ (,Figure 5. Connection to Text Terms 0 10 20 30 40 50 60 70 80 90 100 0 1 10 -3 0 10 20 30 40 50 60 70 80 90 100 0 1 10 -3 0 10 20 30 40 50 60 70 80 90 100 0 1 10 -3 0 10 20 30 40 50 60 70 80 90 100 0 1 10 -3 0 10 20 30 40 50 60 70 80 90 100 0 1 10 -3 0 10 20 30 40 50 60 70 80 90 100 0 1 10 -3 0 10 20 30 40 50 60 70 80 90 100 0 1 10 -3 ||\ud835\udc4a # $\ud835\udc64& '||( Head 5.73% Back 19.5% Belly 17.01% Leg 22.16% Breast 25.67% Tail 14.55% Wing 18.07% Figure 6. Connection to Text Terms (CU Birds dataset\u2013SCS Split with with 37.2% Top1-Acc). On the right, Top1-Acc is shown per part Head: billed    (0.0075)    Back: declined(0.0092)   Belly: marsh (0.0029)  Breast: white (0.0108)   Leg: marshes (0.0008) Wing: marsh (0.0088) Tail: brown (0.0265)   Head: birds     (3.655e-08) Back: boring    (0.0089)    Belly: backed    (0.0084) Breast: recently (0.0054)  Leg:  toed         (0.0026)   Wing: black  (2.957e-07)  Tail: peninsula (0.0038)  Head: shore (0.0037) Back: prey     (0.0080) Belly: gray     (0.0038) Breast: black (0.0052) Leg: yellow    (0.0072) Wing: chicks (0.0056) Tail: white      (0.0189) Figure 7. Part-to-Term connectivity (From left to right: \u201cLeast Tern\u201d, \u201cMarsh Wren\u201d, \u201cThree-toed Woodpecker\u201d from CUBirds-SCS split) 0 Tail Wing 0.5 Leg Breast Belly Tail 1 Back Wing Leg Breast Head Belly Back Head Figure 8. Top-30 terms Overlap between every two parts(CUBirds-SCS) the terms by x(p)Wp x Twi zti k, where ti k is the ith dimension of the text representation of the predicted class k (i.e., only the text terms that exist in the text description of class k are considered). The \ufb01gure shows the capability of our method to ground concepts to its location in the image. In the right example, like \u201ctoes\u201d is strongly connected to leg\u2013 the connection strength is shown between parenthesis. In the middle example, \u201cbilled\u201d concept is connected to head, \u201cwhite\u201d is connected to the breast, and \u201cbrown\u201d is connected to the tail. In the left example, \u201cyellow\u201d is connected to leg. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1604.00466v3",
+                    "title": "Automatic Annotation of Structured Facts in Images",
+                    "abstract": "Motivated by the application of fact-level image understanding, we present an\nautomatic method for data collection of structured visual facts from images\nwith captions. Example structured facts include attributed objects (e.g.,\n<flower, red>), actions (e.g., <baby, smile>), interactions (e.g., <man,\nwalking, dog>), and positional information (e.g., <vase, on, table>). The\ncollected annotations are in the form of fact-image pairs (e.g.,<man, walking,\ndog> and an image region containing this fact). With a language approach, the\nproposed method is able to collect hundreds of thousands of visual fact\nannotations with accuracy of 83% according to human judgment. Our method\nautomatically collected more than 380,000 visual fact annotations and more than\n110,000 unique visual facts from images with captions and localized them in\nimages in less than one day of processing time on standard CPU platforms.",
+                    "authors": "Mohamed Elhoseiny, Scott Cohen, Walter Chang, Brian Price, Ahmed Elgammal",
+                    "published": "2016-04-02",
+                    "updated": "2016-04-08",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction People generally acquire visual knowledge by exposure to both visual facts and to semantic or language-based representations of these facts, e.g., by seeing an image of \u201ca person petting dog\u201d and observing this visual fact associated with its language representation . In this work, we focus on methods for collecting structured facts that we de\ufb01ne as structures that provide attributes about an object, and/or the actions and interactions this object may have with other objects. We introduce the idea of automatically collecting annotations for second order visual facts and third order visual facts where second order facts <S,P> are attributed objects (e.g., <S: car, P: red>) and singleframe actions (e.g., <S: person, P: jumping>), and third order facts specify interactions (i.e., <boy, petting, dog>). This structure is helpful for designing machine learning algorithms that learn deeper image semantics from caption data and allow us to model the relationships between facts. In order to enable such a setting, we need to collect these structured fact annotations in the form of (language view, visual view) pairs (e.g., <baby, sitting on, chair> as the language view and an image with this fact as a visual view) to train models. (Chen et al., 2013) showed that visual concepts, from a prede\ufb01ned ontology, can be learned by querying the web about these concepts using image-web search engines. More recently, (Divvala et al., 2014) presented an approach to learn concepts related to a particular object by querying the web with Google-N-gram data that has the concept name. There are three limitations to these approaches. (1) It is dif\ufb01cult to de\ufb01ne the space of visual knowledge and then search for it. It is further restricting to de\ufb01ne it based on a prede\ufb01ned ontology such as (Chen et al., 2013) or a particular object such as (Divvala et al., 2014). (2) Using image search is not reliable to collect data for concepts with few images on the web. These methods assume that the top retrieved examples by imageweb search are positive examples and that there are images available that are annotated with the searched concept. (3) These concepts/facts are not structured and hence annotations lacks information like \u201cjumping\u201d is the action part in <person, jumping >, or \u201cman\u2019 and \u201chorse\u201d are interacting in <person, riding, horse >. This structure is important for deeper understanding of visual data, which is one of the main motivations of this work. The problems in the prior work motivate us to propose a method to automatically annotate structured facts by processing image caption data since arXiv:1604.00466v3  [cs.CL]  8 Apr 2016 \fFigure 1: Structured Fact Automatic Annotation facts in image captions are highly likely to be located in the associated images. We show that a large quantity of high quality structured visual facts could be extracted from caption datasets using natural language processing methods. Caption writing is free-form and an easier task for crowd-sourcing workers than labeling secondand third-order tasks, and such free-form descriptions are readily available in existing image caption datasets. We focused on collecting facts from the MS COCO image caption dataset (Lin et al., 2014) and the newly collected Flickr30K entities (Plummer et al., 2015). We automatically collected more than 380,000 structured fact annotations in high quality from both the 120,000 MS COCO scenes and 30,000 Flickr30K scenes. The main contribution of this paper is an accurate, automatic, and ef\ufb01cient method for extraction of structured fact visual annotations from image-caption datasets, as illustrated in Fig. 1. Our approach (1) extracts facts from captions associated with images and then (2) localizes the extracted facts in the image. For fact extraction from captions, We propose a new method called SedonaNLP for fact extraction to \ufb01ll gaps in existing fact extraction from sentence methods like Clausie (Del Corro and Gemulla, 2013). SedonaNLP produces more facts than Clausie, especially <subject,attribute> facts, and thus enables collecting more visual annotations than using Clausie alone. The \ufb01nal set of automatic annotations are the set of successfully localized facts in the associated images. We show that these facts are extracted with more than 80% accuracy according to human judgment. 2 Motivation Our goal by proposing this automatic method is to generate language&vision annotations at the factlevel to help study language&vision for the sake of structured understanding of visual facts. Existing systems already work on relating captions directly to the whole image such as (Karpathy et al., 2014; Kiros et al., 2015; Vinyals et al., 2015; Xu et al., 2015; Mao et al., 2015; Antol et al., 2015; Malinowski et al., 2015; Ren et al., 2015). This gives rise to a key question about our work: why it is useful to collect such a large quantity of structured facts compared to caption-level systems? We illustrate the difference between captionlevel learning fact-level learning that motivates this work by the example in Fig 1. Caption-level learning systems correlate captions like those on top of Fig. 1(top-left) to the whole image that includes all objects. Structured Fact-level learning systems are instead fed with localized annotations for each fact extracted form the image caption; see in Fig. 1(right), Fig. 6, and 7 in Sec. 6. Fact level annotations are less confusing training data than sentences because they provide more precise information for both the language and the visual views. (1) From the language view, the annotations we generate is precise to list a particular fact (e.g., <bicycle,parked between, parking posts>). (2) From the visual view, it provide the bounding box of this fact; see Fig 1. (3) A third unique part about our annotations is the structure: e.g., <bicycle,parked between, parking posts> instead of \u201ca bicycle parked between parking posts\u201d. Our collected data has been used to develop methods that learn hundreds of thousands of image facts, as we introduced and studied in (Mohamed Elhoseiny, 2016). The results shows that fact-level learning is superior compared to caption-level learning like (Kiros et al., 2015), as shown in Table 4 in (Mohamed Elhoseiny, 2016) (16.39% accuracy versus 3.48% for (Kiros et al., 2015)). It further shows the value of the associated structure in the (16.39% accuracy versus 8.1%) in Table 4(Mohamed Elhoseiny, 2016)). Similar results also shown on a smaller scale in Table 3 in (Mohamed Elhoseiny, 2016). \f3 Approach Overview We propose a two step automatic annotation of structured facts: (i) Extraction of structured fact from captions, and (ii) Localization of these facts in images. First, the captions associated with the given image are analyzed to extract sets of clauses that are considered as candidate <S,P>, and <S,P,O> facts. Captions can provide a tremendous amount of information to image understanding systems. However, developing NLP systems to accurately and completely extract structured knowledge from free-form text is an open problem. We extract structured facts using two methods: Clausie (Del Corro and Gemulla, 2013) and Sedona( detailed later in Sec 4); also see Fig 1. We found Clausie (Del Corro and Gemulla, 2013) missed many visual facts in the captions which motivated us to develop Sedona to \ufb01ll this gap as detailed in Sec. 4. Second, we localize these facts within the image (see Fig. 1). The successfully located facts in the images are saved as fact-image annotations that could be used to train visual perception models to learn attributed objects, actions, and interactions. We managed to collect 380.409 highquality secondand third-order fact annotations (146,515 from Flickr30K Entities, 157,122 from the MS COCO training set, and 76,772 from the MS COCO validation set). We present statistics of the automatically collected facts in the Experiments section. Note that the process of localizing facts in an image is constrained by information in the dataset. For MS COCO, the dataset contains object annotations for about 80 different objects as provided by the training and validation sets. Although this provides abstract information about objects in each image (e.g., \u201cperson\u201d), it is usually mentioned in different ways in the caption. For the \u201cperson\u201d object, \u201cman\u201d, \u201cgirl\u201d, \u201ckid\u201d, or \u201cchild\u201d could instead appear in the caption. In order to locate secondand third-order facts in images, we started by de\ufb01ning visual entities. For the MS COCO dataset (Lin et al., 2014), we de\ufb01ne a visual entity as any noun that is either (1) one of the MS COCO dataset objects, (2) a noun in the WordNet ontology (Miller, 1995; Leacock and Chodorow, 1998) that is an immediate or indirect hyponym of one of the MS COCO objects (since WordNet is searchable by a sense and not a word, we perform word sense disambiguation on the sentences using a state-of-the-art method (Zhong and Ng, 2010)), or (3) one of scenes the SUN dataset (Xiao et al., 2010) (e.g., a \u201crestaurant\u201d). We expect visual entities to appear either in the S or the O part (if exists) of a candidate fact. This allows us to then localize facts for images in the MS COCO dataset. Given a candidate third-order fact, we \ufb01rst try to assign each S and O to one of the visual entities. If S and O elements are not visual entities, then the fact is ignored. Otherwise, the facts are processed by several heuristics, detailed in Sec 5. For instance, our method takes into account that grounding the plural \u201dmen\u201d in the fact <S:men, P: chasing, O: soccer ball > may require the union of multiple \u201dman\u201d bounding boxes. In the Flickr30K Entities dataset (Plummer et al., 2015), the bounding box annotations are presented as phrase labels for sentences (for each phrase in a caption that refers to an entity in the scene). A visual entity is considered to be a phrase with a bounding box annotation or one of the SUN scenes. Several heuristics were developed and applied to collect these fact annotations, e.g. grounding a fact about a scene to the entire image; detailed in Sec 5. 4 Fact Extraction from Captions We extract facts from captions using Clausie (Del Corro and Gemulla, 2013) and our proposed SedonaNLP system. In contrast to Clausie, we address several challenging linguistic issues by evolving our NLP pipeline to: 1) correct many common spelling and punctuation mistakes, 2) resolve word sense ambiguity within clauses, and 3) learn a common spatial preposition lexicon (e.g., \u201cnext to\u201d, \u201con top of\u201d, \u201cin front of\u201d) that consists of over 110 such terms, as well as a lexicon of over two dozen collection phrase adjectives (e.g., \u201dgroup of\u201d, \u201dbunch of\u201d, \u201dcrowd of\u201d, \u201dherd of\u201d). For our purpose, these strategies allowed us to extract more interesting structured facts that Clausie fails at which include (1) more discrimination between single versus plural terms, (2) extracting positional facts (e.g., next to). Additionally, SedonaNLP produces attribute facts that we denote as <S, A>; see Fig 4. Similar to some existing systems OpenNLP (Baldridge, 2014) and ClearNLP (Choi, 2014), the SedonaNLP platform also performs many common NLP \fFigure 3: Accumulative Percentage of SP and SPO facts in COCO 2014 captions as number of verbs increases tasks: e.g., sentence segmentation, tokenization, part-of-speech tagging, named entity extraction, chunking, dependency and constituency-based parsing, and coreference resolution. SedonaNLP itself employs both open-source components such as NLTK and WordNet, as well as internallydeveloped annotation algorithms for POS and clause tagging. These tasks are used to create more advanced functions such as structured fact annotation of images via semantic triple extraction. In our work, we found SedonaNLP and Clausie to be complementary for producing a set of candidate facts for possible localization in the image that resulted in successful annotations. Varying degrees of success have been achieved in extracting and representing structured triples from sentences using <subject, predicate, object> triples. For instance, (Rusu et al., 2007) describe a basic set of methods based on traversing the parse graphs generated by various commonly available parsers. Larger scale text mining methods for learning structured facts for question answering have been developed in the IBM Watson PRISMATIC framework (Fan et al., 2010). While parsers such as CoreNLP (Manning et al., 2014) are available to generate comprehensive dependency graphs, these have historically required signi\ufb01cant processing time for each sentence or have traded accuracy for performance. In contrast, SedonaNLP currently employs a shallow dependency parsing method that runs in some cases 8-9X faster than earlier cited methods running on identical hardware. We choose a shallow approach with high, medium, and low con\ufb01dence cutoffs after observing that roughly 80% Figure 4: Examples of caption processing and <S,P,O> and <S,P> structured fact extractions. of all captions consisted of 0 or 1 Verb expressions (VX); see Fig. 3 for MSCOCO dataset (Lin et al., 2014). The top 500 image caption syntactic patterns we observed can be found on our supplemental materials. These syntactic patterns are used to learn rules for automatic extraction for not only <S,P,O>, but also <S,P>, and <S,A>, where <S,P>, are subject-action facts and <S,A> are subject-attribute facts. Pattern examples and statistics for MS COCO are shown in Fig. 5. In SedonaNLP, structured fact extraction was accomplished by learning a subset of abstract syntactic patterns consisting of basic noun, verb, and preposition expressions by analyzing 1.6M caption examples provided by the MS COCO, Flickr30K, and Stony Brook University Im2Text caption datasets. Our approach mirrors existing known art with the addition of internallydeveloped POS and clause tagging accuracy improvements through the use of heuristics listed \fFigure 2: SedonaNLP Pipeline for Structured Fact Extraction from Captions Figure 5: Examples of the top observed Noun (NX), Verb (VX), and Preposition (IN) Syntactic patterns. below to reduce higher occurrence errors due to systematic parsing errors: (i) Mapping past participles to adjectives (e.g., stained glass), (ii) Denesting existential facts (e.g., this is a picture of a cat watching a tv.), (iii) Identifying auxiliary verbs (e.g., do verb forms). In Fig. 4, we show an example of extracted <S,P,O> structured facts useful for image annotation for a small sample of MS COCO captions. Our initial experiments empirically con\ufb01rmed the \ufb01ndings of IBM Watson PRISMATIC researchers who indicated big complex parse trees tend to have more wrong parses. By limiting a frame to be only a small subset of a complex parse tree, we reduce the chance of error parse in each frame (Fan et al., 2010). In practice, we observed many correctly extracted structured facts for the more complex sentences (i.e., sentences with multiple VX verb expressions and multiple spatial prepositional expressions) \u2013 these facts contained useful information that could have been used in our joint learning model but were conservatively \ufb01ltered to help ensure the overall accuracy of the facts being presented to our system. As improvements are made to semantic triple extraction and con\ufb01dence evaluation systems, we see potential in several areas to exploit more structured facts and to \ufb01lter less information. Our full <S,P,O> triple and related tuple extractions for MS COCO and Flickr30K datasets are available in the supplemental material. 5 Locating facts in the Image In this section, we present details about the second step of our automatic annotation process introduced in Sec. 3. After the candidate facts are extracted from the sentences, we end up with a set Fs = {fi l }, i = 1 : Ns for statement s, where Ns is the number of extracted candidate fact fi l , \u2200i from the statement s using either Clausie (Del Corro and Gemulla, 2013) or Sedona-3.0. The localization step is further divided into two steps. The mapping step maps nouns in the facts to candidate boxes in the image. The grounding step processes each fact associated with the candidate boxes and outputs a \ufb01nal bounding box if localization is successful. The two steps are detailed in the following subsections. 5.1 Mapping The mapping step starts with a pre-processing step that \ufb01lters out a non-useful subset of Fs and produces a more useful set F\u2217 s that we try to locate/ground in the image. We perform this step by performing word sense disambiguation using the state-of-the-art method (Zhong and Ng, 2010). The word sense disambiguation method provides \feach word in the statement with a word sense in the wordNet ontology (Leacock and Chodorow, 1998). It also assigns for each word a part of speech tag. Hence, for each extracted candidate fact in Fs we can verify if it follows the expected part of speech according to (Zhong and Ng, 2010). For instance, all S should be nouns, all P should be either verbs or adjectives, and O should be nouns. This results in a \ufb01ltered set of facts F\u2217 s. Then, each S is associated with a set of candidate boxes in the image for secondand third-order facts and each O associated with a set or candidate boxes in the image for third-order facts only. Since entities in MSCOCO dataset and Flickr30K are annotated differently, we present how the candidate boxes are determined in each of these datasets. MS COCO Mapping: Mapping to candidate boxes for MS COCO reduces to assigning the S for second-order and third-order facts, and S and O for third-order facts. Either S or O is assigned to one of the MSCOCO objects or SUN scenes classes. Given the word sense of the given part (S or O), we check if the given sense is a descendant of MSCOCO objects senses in the wordNet ontology. If it is, the given part (S or O) is associated with the set of candidate bounding boxes that belongs to the given object (e.g., all boxes that contain the \u201cperson\u201d MSCOCO object is under the \u201cperson\u201d wordnet node like \u201cman\u201d, \u2019girl\u2019, etc). If the given part (S or O) is not an MSCOCO object or one of its descendants under wordNet, we further check if the given part is one of the SUN dataset scenes. If this condition holds, the given part is associated with a bounding box of the whole image. Flickr30K Mapping: In contrast to MSCOCO dataset, the bounding box annotation comes for each entity in each statement in Flickr30K dataset. Hence, we compute the candidate bounding box annotations for each candidate fact by searching the entities in the same statement from which the clause is extracted. Candidate boxes are those that have the same name. Similarly, this process assigns S for second-order facts and assigns S and O for secondand third-order facts. Having \ufb01nished the mapping process, whether for MSCOCO or Flickr30K, each candidate fact fi l \u2208F\u2217 s, is associated with candidate boxes depending on its type as follows. <S,P> : Each fi l \u2208F\u2217 s of second-order type is associated with one set of bounding boxes bi S, which are the candidate boxes for the S part. bi O could be assumed to be always an empty set for second-order facts. <S,P,O> : Each fi l \u2208F\u2217 s of third-order type is associated with two sets of bounding boxes bi S and bi S as candidate boxes for the S and P parts, respectively. 5.2 Grounding The grounding process is the process of associating each fi l \u2208F\u2217 s with an image fv by assigning fl to a bounding box in the given MS COCO image scene given the bi S and bi O candidate boxes. The grounding process is relatively different for the two dataset due to the difference of the entity annotations. Grounding: MS COCO dataset (Training and Validation sets) In the MS COCO dataset, one challenging aspect is that the S or O can be singular, plural, or referring to the scene. This means that one S could map to multiple boxes in the image. For example, \u201cpeople\u201d maps to multiple boxes of \u201cperson\u201d. Furthermore, this case could exist for both the S and the O. In cases where either S or O is plural, the bounding box assigned is the union of all candidate bounding boxes in bi S. The grounding then proceeds as follows. <S,P> facts: (1) If the computed bi S = \u2205for the given fi l , then fi l fails to ground and is discarded. (2) If S singular, fi v is the image region that with the largest candidate bounding box in bi S. (3) If S is plural, fi v is the image region that with union of the candidate bounding boxes in bi S. <S,P, O> facts: (1) If bi S = \u2205and bi O = \u2205, fi l fails to ground and is ignored. (2) If bi S \u0338= \u2205and bi O \u0338= \u2205, then bounding boxes are assigned to S and O such that the distance between them is minimized (though if S or O is plural, the assigned bounding box is the union of all bounding boxes for bi S or bi O respectively), and the grounding is assigned the union of the bounding boxes assigned to S and O. (3) If either bi S = \u2205or bi O = \u2205, then a bounding box is assigned to the present object (the largest bounding box if singular, or the union of all bounding boxes if plural). If the area of this region compared to the area of the whole scene is greater than a threshold th = 0.3, then the fi v is associated to the whole image of the scene. Otherwise, \fTable 1: Human Subject Evaluation by MTurk workers % Dataset (responses) Q1 Q2 Q3 yes no Yes No a b c d e f g MSCOCO train 2014 (4198) 89.06 10.94 87.86 12.14 64.58 12.64 3.51 5.10 0.86 1.57 11.73 MSCOCO val 2014 (3296) 91.73 8.27 91.01 8.99 66.11 14.81 3.64 4.92 1.00 0.70 8.83 Flickr30K Entities2015 (3296) 88.94 11.06 88.19 11.81 70.12 11.31 3.09 2.79 0.82 0.39 11.46 Total 89.84 10.16 88.93 11.07 66.74 12.90 3.42 4.34 0.89 0.95 10.76 Table 2: Human Subject Evaluation by Volunteers % (This is another set of annotations different from those evaluated by MTurkers) Volunteers Q1 Q2 Q3 yes No Yes No a b c d e f g MSCOCO train 2014 (400) 90.75 9.25 91.25 8.75 73.5 8.25 2.75 6.75 0.5 0.5 7.75 MSCOCO val 2014 (90) 97.77 2.3 94.44 8.75 84.44 8.88 3.33 1.11 0 0 2.22 Flickr30K Entities 2015 (510) 78.24 21.76 73.73 26.27 64.00 4.3 1.7 1.7 0.7 1.18 26.45 fi l fails to ground and is ignored. Grounding: Flickr30K dataset The main difference in Flickr30K is that for each entity phrase in a sentence, there is a box in the image. This means there is no need to have cases for single and plural. Since in this case, the word \u201cmen\u201d in the sentence will be associated with the set of boxes referred to by \u201cmen\u201d in the sentences. We union these boxes for plural words as one candidate box for \u201cmen\u201d We can also use the information that the object box has to refer to a word that is after the subject word, since subject usually occurs earlier in the sentence compared to object. We union these boxes for plural words. <S,P> facts: If the computed bi S = \u2205for the given fi l , then fi l fails to ground and is discarded. Otherwise, the fact is assigned to the largest candidate box in if there are multiple boxes. <S,P, O> facts: <S,P, O> facts are handled very similar to MSCOCO dataset with two main differences. a) The candidate boxes are computed as described for the case of Flickr30K dataset. b) All cases are handled as single case, since even plural words are assigned one box based on the nature of the annotations in this dataset. 6 Experiments 6.1 Human Subject Evaluation We propose three questions to evaluate each annotation: (Q1) Is the extracted fact correct (Yes/No)? The purpose of this question is to evaluate errors captured by the \ufb01rst step, which extracts facts by Sedona or Clausie. (Q2) Is the fact located in the image (Yes/No)? In some cases, there might be a fact mentioned in the caption that does not exist in the image and is mistakenly considered as an annotation. (Q3) How accurate is the box assigned to a given fact (a to g)? a (about right), b (a bit big), c (a bit small), d (too small), e (too big), f (totally wrong box), g (fact does not exist or other). Our instructions on these questions to the participants can be found in this anonymous url (Eval, 2016). We evaluate these three questions for the facts that were successfully assigned a box in the image, because the main purpose of this evaluation is to measure the usability of the collected annotations as training data for our model. We created an Amazon Mechanical Turk form to ask these three questions. So far, we collected a total of 10,786 evaluation responses, which are an evaluation of 3,595 (fv, fl) pairs (3 responses/ pair). Table 2 shows the evaluation results, which indicate that the data is useful for training, since\u224883.1% of them are correct facts with boxes that are either about right, or a bit big or small (a,b,c). We further some evaluation responses that we collected from volunteer researchers in Table 2 showing similar results. Fig. 6 shows some successful qualitative results that include four extracted structured facts from MS COCO dataset (e.g., <person, using, phone>, <person, standing>, etc). Fig 7 also show a negative example where there is a wrong fact among the extracted facts (i.e., <house, ski>). The main reason for this failure case is that \u201chow\u201d is mistyped as \u201chouse\u201d; see Fig 7. The supplementary materials includes all the captions of these examples and also additional qualitative examples. 6.2 Hardness Evaluation of the collected data In order to study how the method behave in both easy and hard examples. This section present statistics of the successfully extracted facts and re\flate it to the hardness of the extraction of these facts. We start by de\ufb01ning hardness of an extracted fact in our case and its dependency on the fact type. Our method collect both secondand third-order facts. We refer to candidate subjects as all instances of the entity in the image that match the subject type of either a second-order fact <S,P> or a third-order fact <S,P,O>. We refer to candidate objects as all instances in the image that match the object type of a third-order fact <S,P,O>. The selection of the candidate subjects and candidate objects is a part of our method that we detailed in Sec 5. We de\ufb01ne the hardness for second order facts by the number of candidate subjects and the hardness of third order facts by the number of candidate subjects multiplied by the number of candidate objects. In Fig 16 and 17, the Y axis is the number of facts for each bin. The X axis shows the bins that correspond to hardness that we de\ufb01ned for both Figure 6: Se6eral Facts successfully extracted by our method from two MS COCO scenes Figure 7: An example where one of the extracted facts are not correct due to a spelling mistake second and third order fats. Figure 16 shows a histogram of the dif\ufb01culties for all Mturk evaluated examples including both the successful and the failure cases. Figure 17 shows a similar histogram but for but for subset of facts veri\ufb01ed by the Turkers with Q3 as (about right). The \ufb01gures show that the method is able to handle dif\ufb01culty cases even with more than 150 possibilities for grounding. We show these results broken out for MSCOCO and Flickr30K Entities datasets and for each fact types in the supplementary materials. Figure 8: (All MTurk Data) Hardness histogram after candidate box selection using our method Figure 9: (MTurk Data with Q3=about right)Hardness histogram after our candidate box selection 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1512.00818v2",
+                    "title": "Zero-Shot Event Detection by Multimodal Distributional Semantic Embedding of Videos",
+                    "abstract": "We propose a new zero-shot Event Detection method by Multi-modal\nDistributional Semantic embedding of videos. Our model embeds object and action\nconcepts as well as other available modalities from videos into a\ndistributional semantic space. To our knowledge, this is the first Zero-Shot\nevent detection model that is built on top of distributional semantics and\nextends it in the following directions: (a) semantic embedding of multimodal\ninformation in videos (with focus on the visual modalities), (b) automatically\ndetermining relevance of concepts/attributes to a free text query, which could\nbe useful for other applications, and (c) retrieving videos by free text event\nquery (e.g., \"changing a vehicle tire\") based on their content. We embed videos\ninto a distributional semantic space and then measure the similarity between\nvideos and the event query in a free text form. We validated our method on the\nlarge TRECVID MED (Multimedia Event Detection) challenge. Using only the event\ntitle as a query, our method outperformed the state-of-the-art that uses big\ndescriptions from 12.6% to 13.5% with MAP metric and 0.73 to 0.83 with ROC-AUC\nmetric. It is also an order of magnitude faster.",
+                    "authors": "Mohamed Elhoseiny, Jingen Liu, Hui Cheng, Harpreet Sawhney, Ahmed Elgammal",
+                    "published": "2015-12-02",
+                    "updated": "2015-12-16",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Every minute, hundreds of hours of video are uploaded to video archival site such as YouTube (Google2014 ). Developing methods to automatically understand the events captured in this large volume of videos is necessary and meanwhile challenging. One of the important tasks in this direction is event detection in videos. The main objective of this task is to determine the relevance of a video to an event based on the video content (e.g., feeding an animal, birthday party; see Fig. 1). The cues of an event in a video could include visual objects, scene, actions, detected speech (by Automated Speech Recognition(ASR)), detected text (by Optical Character Recognition (OCR)), and audio concepts (e.g. music and water concepts). Search and retrieval of videos for arbitrary events using only free-style text and unseen text in particular has been a dream in computational video and multi-media understanding. This is referred as \u201czero-shot event detection\u201d, because there is no positive exemplar videos to train a detector. Due to the proliferation of videos, especially consumer-generated Copyright c \u20dd2016, Association for the Advancement of Arti\ufb01cial Intelligence (www.aaai.org). All rights reserved. (a) Grooming an Animal (1) \u201cbrushing dog\u201d, weight = 0.67 (2) \u201ccombing dog\u201d, weight = 0.66 (3) \u201cclipping nails\u201d, weight = 0.52 (b) Birthday Party (1) \u201ccutting cake\u201d, weight = 0.72 (2) \u201cblowing candles\u201d, weight = 0.65 (3) \u201copening presents\u201d, weight = 0.59 Figure 1: Top relevant Concepts from a pre-de\ufb01ned multimedia concept repository and their automatically-assigned weights as a part of our Event Detection method videos (e.g., YouTube), zero-shot search and retrieval of videos has become an increasingly important problem. Several research works have been proposed to facilitate performing the zero-shot learning task by establishing an intermediate semantic layer between events or generally categories (i.e., concepts or attributes) and the low-level representation of a multimedia content from the visual perspective. (Lampert, Nickisch, and Harmeling 2009) and (Farhadi et al. 2009) were the two \ufb01rst to use attribute learning representation for the zero-shot setting for object recognition in still images. Attributes were similarly adopted for recognizing human actions (Liu, Kuipers, and Savarese 2011); attributes are generalized and denoted by concepts in this context. Later, (Liu et al. 2013) proposed Concept Based Event Retrieval (CBER) for videos InTheWild. Even though these methods facilitate zero-shot event detection, they only capture the visual modality and more importantly they assume that the relevant concepts for a query event are manually de\ufb01ned. This manual de\ufb01nition of concepts, also known as semantic query editing, is a tedious task and may be biased due to the limitation of human knowledge. Instead, we aim at automatically generating relevant concepts by leveraging information from distributional semantics. Recently, several systems were proposed for zero-shot event detection methods (Wu et al. 2014; Jiang et al. 2014b; 2014a; Chen et al. 2014; Habibian, Mensink, and Snoek arXiv:1512.00818v2  [cs.CV]  16 Dec 2015 \f2014). These approaches rely on the whole text description of an event where relevant concepts are speci\ufb01ed; see example event descriptions used in these approaches in the Supplementary Materials (SM)1 ( explicitly de\ufb01ne the event explication, scene, objects, activities, and audio). In practice, however, we think that typical use of event queries under this setting should be similar to text-search, which is based on few words instead that we model their connection to the multimodal content in videos. The main question addressed in this paper is how to use an event text query (i.e. just the event title like \u201cbirthday party\u201d or \u201cfeeding an animal\u201d) to retrieve a ranked list of videos based on their content. In contrast to (Lampert, Nickisch, and Harmeling 2009; Liu et al. 2013), we do not manually assign relevant concepts for a given event query. Instead, we leverage information from a distributional semantic space (Mikolov et al. 2013b) trained on a large text corpus to embed event queries and videos to the same space, where similarity between both could be directly estimated. Furthermore, we only assume that query comes in the form of an \u201cunstructured\u201d few-keyword query (in contrast to (Wu et al. 2014; Jiang et al. 2014b; 2014a)). We abbreviate our method as EDiSE (Event-detection by Distributional Semantic Embedding of videos). Contributions. The contributions of this paper can be listed as follows: (1) Studying how to use few-keyword unstructured-text query to detect/retrieve videos based on their multimedia content, which is novel in this setting. We show how relevant concepts to that event query could be automatically retrieved through a distributional semantic space and got assigned a weight associated with the relevance; see Fig. 1 \u201cBirthday\u201d and \u201cGrooming an Animal\u201d example events. (2) To the best of our knowledge, our work is the \ufb01rst attempt to model the connection between few keywords and multimodal information in videos by distributional semantics . We study and propose different similarity metrics in the distributional semantic space to enable event retrieval based on (a) concepts, (b) ASR, and (c) OCR in videos. Our uni\ufb01ed framework is capable of embedding all of them into the same space; see Fig. 2. (3) Our method is also very fast, which makes it applicable to both large number of videos and concepts (i.e. 26.67 times faster than the state of the art (Jiang et al. 2014a)). Related Work Attribute methods for zero-shot learning are based on manually specifying the attributes for each category (e.g., (Lampert, Nickisch, and Harmeling 2009; Parikh and Grauman 2011)). Other methods focused on attribute discovery (Rohrbach, Stark, and Schiele 2011; Rohrbach, Ebert, and Schiele 2013) and then apply the same mechanism. Recently, several methods were proposed to perform zero shot recognition by representing unstructured text in document terms (e.g. (Elhoseiny, Saleh, and Elgammal 2013; Mensink, Gavves, and Snoek 2014)) One drawback of the TFIDF (Salton and Buckley 1988) in (Elho1Supplementary Materials (SM) could be found here https:// sites.google.com/site/mhelhoseiny/EDiSE_supp.zip seiny, Saleh, and Elgammal 2013) and hardly matching tag terms in (Mensink, Gavves, and Snoek 2014; Rohrbach et al. 2010) is that they do not capture semantically related terms that our model can relate in noisy videos instead of still images. Also, WordNet (Miller 1995), adopted in (Rohrbach et al. 2010), does not connect objects with actions (e.g., person blowing candle), making it hard to apply in our setting and heavily depending on prede\ufb01ned information like WordNet. There has been a recent interest especially in the computational linguistics\u2019 community in word-vector representation ( e.g., (Bengio et al. 2006)), which captures word semantics based on context. While word-vector representation is not new, recent algorithms (e.g. (Mikolov et al. 2013b; 2013a)) enabled learning these vectors from billions of words, which makes them much more semantically accurate. As a result, these models got recently adopted in several tasks including translation (Mikolov, Le, and Sutskever 2013) and web search (Shen et al. 2014). Several computer vision researchers explored using these wordvector representation to perform Zero-Shot learning in the object recognition (e.g. (Frome et al. 2013; Socher et al. 2013; Norouzi et al. 2014)). They embed the object class name into the word-vector semantic space learnt by models like (Mikolov et al. 2013b). It is worth mentioning that these zero-shot learning approaches (Frome et al. 2013; Socher et al. 2013) and also the aforementioned work (Elhoseiny, Saleh, and Elgammal 2013) assume that during training, there is a set of training classes and test classes. Hence, they learn a transformation to correlate the information between both domains (textual and visual). In contrast, zeroshot setting of event retrieval rely mainly on the event information without seeing any training events, as assumed in recent zero-shot event retrieval methods (e.g., (Dalton, Allan, and Mirajkar 2013; Jiang et al. 2014b; Wu et al. 2014; Liu et al. 2013)). Hence, there does not exist seen events to learn such transformation from. Differently, we also model multimodal connection from free text query to video information. In the context of videos, (Wu et al. 2014) proposed a method for zero-shot event detection by using the salient words in the whole structured event description, where relevant concept are already de\ufb01ned in the event structured text description; also see Eq. 1 in (Wu et al. 2014). Similarly, (Dalton, Allan, and Mirajkar 2013) adopted a MarkovRandom-Field language model proposed by (Metzler and Croft 2005). One drawback of this model is that it performs an intensive processing for each new concept. This is since it determines the relevance of the concept to a query event by creating a text document to represent each concept. This document is created by web-querying the concept name and some of its keywords and merging the top retrieved pages. In contrast, our model does not require this step to determine relevance of an event to a query. Once the language model is trained, any concept can be instantly added and captured in our multimodal semantic embedding of videos. In contrast to both (Wu et al. 2014) and (Dalton, Allan, and Mirajkar 2013), we focus on retrieving videos only with the event title (i.e., few-words query) and without semantic \fediting. The key difference is in modeling and embedding concepts to allow zero-shot event retrieval. In (Wu et al. 2014) and (Dalton, Allan, and Mirajkar 2013), the semantic space is a vector whose dimensionality is the number of the concepts. Our idea is to embed concepts, video information, and the event query into a distributional semantic space whose dimensionality is independent of the number of concepts. This property, together with the semantic properties captured by distributional semantics, feature our approach with two advantages (a) scalability to any concept size. Having new concepts does not affect the representation dimensionality (i.e., in all our experiments concepts, videos, event queries are embedded to M dimensional space; M is few hundreds in our experiments). (b) facilitating automatic determination of relevant concepts given an unstructured short event query: For example, being able to automatically determine that \u201cblowing a candle\u201d concept is a relevant concept to \u201cbirthday party\u201d event. (Wu et al. 2014) and (Dalton, Allan, and Mirajkar 2013) used the complete text description of an event for retrieval that explicitly speci\ufb01es relevant concepts. There is a class of models that improve zero-shot Event Detection performance by reranking. Jiang et al. proposed multimodal pseudo relevance feedback (Jiang et al. 2014b) and self-paced reranking (Jiang et al. 2014a) algorithms. The main assumption behind these models is that all unlabeled test examples are available and the top few examples by a given initial ranking have high top K precision (K \u223c 10). This means that reranking algorithms can not update con\ufb01dence of a video for an event without knowing the con\ufb01dences of the remaining videos to perform reranking. In contrast, our goal is different which is to directly model the probability of a few-keyword event-query given an arbitrary video. Hence, our work does not require an initial ranking and can compute the conditional probability of a video without any information about other videos. Our method is also 26.67 times faster, as detailed in our experiments. Method Problem De\ufb01nition Given an arbitrary event query e and a video v (e.g. just \u201dbirthday party\u201d), our objective is to model p(e|v). We start by de\ufb01ning the representation of event query e, the concept set c, the video v in our setting. Event-Query representation e: We use the unstructured event title to represent an event query for concept based retrieval. Our framework also allows additional terms specifically for ASR or OCR based retrieval. While we show retrieval on different modalities, concept based retrieval is our main focus in this work. The few-keyword event query for concept based retrieval is denoted by ec, while query keywords for OCR and ASR are denoted by eo and ea, respectively. Hence, under our setting e = {ec, eo, ea}. Concept Set c: We denote the whole concept set in our setting as c, which include visual concepts cv and audio concepts cd, i.e., c = {cv, cd}. The visual concepts include object, scene and action concepts. The audio concepts include acoustic related concepts like water sound. We performed an experiment on a set of audio concepts trained on MFCC audio features (Davis and Mermelstein 1980; Logan and others 2000). However, we found their performance \u22481% MAP, and hence we excluded them from our \ufb01nal experiments. Accordingly, our \ufb01nal performance mainly relies on the visual concepts for concept based retrieval; i.e., cd = \u2205. We denote each member ci \u2208c as the de\ufb01nition of the ith concept in c. ci is de\ufb01ned by the ith concept\u2019s name and optionally some related keywords; see examples in SM. Hence, c = {c1, \u00b7 \u00b7 \u00b7 , cN} is the the set of concept de\ufb01nitions, where N is the number concepts. Video Representation: For our zero-shot purpose, a video v is de\ufb01ned by three pieces of information, which are video OCR denoted by vo, video ASR denoted by va, and video concept representation denoted by vc. vo and va are the detected text in OCR and ASR, respectively. We used (Myers et al. 2005) to extract vo and (van Hout et al. 2013) to extract va. In this paper, we mainly focus on the visual video content, which is the most challenging. The video concept based representation vc is de\ufb01ned as vc = [p(c1|v), p(c2|v), \u00b7 \u00b7 \u00b7 , p(cN|v)] (1) where p(ci|v) is a conditional probability of concept ci given video v, detailed later. We denote p(ci|v) by vi c. In zero-shot event detection setting, we aim at recognizing events in videos without training examples based on its multimedia content including still-image concepts like objects and scenes, action concepts, OCR, and ASR2. Given a video v = {vc, vo, va}, our goal is to compute p(e|v) by embedding both the event query e and information of video v of different modalities (vc, vo, and vo) into a distributional semantic space, where relevance of v to e could be directly computed; see Fig. 2. Speci\ufb01cally, our approach is to model p(e|v) as a function F of \u03b8(e), \u03c8(vc), \u03b8(vo), and \u03b8(va), which are the distributional semantic embedding of e, vc, vo, and va, respectively p(e|v) \u221dF \u0000\u03b8(e), \u03c8(vc), \u03b8(vo), \u03b8(va) \u0001 (2) We remove the stop words from e, vo, va before applying the embedding \u03b8(\u00b7). The rest of this section is organized as follows. First, we present the distributional semantic manifold and the embedding function \u03b8(\u00b7) which is applied to e, va, vo, and the concept de\ufb01nitions c in our framework. Then, we show how to determine automatically relevant concepts to an event title query and assign a relevance weight to them, as illustrated in Fig. 1. We present this concept relevance weighting in a separate section since it might be generally useful for other applications. Finally, we present the details of p(e|v) where we derive vc embedding (i.e. \u03c8(vc)), which is based on the proposed concept relevance weighting. Distributional Semantic Model & \u03b8(\u00b7) Embedding We start by the distributional semantic model by (Mikolov et al. 2013b; 2013a) to train our semantic manifold. We denote the trained semantic manifold by Ms, and the vectorization function that maps a word to Ms space as vec(\u00b7). We denote the dimensionality of the real vector returned from 2Note that OCR and ASR are not concepts. They are rather detected text in video frames and speech \fFigure 2: EDiSE Approach Figure 3: PCA visualization in 3D of the \u201dGrooming an Animal\u201d event (in green) and its most 20 relevant concepts in Ms space using sp(\u00b7, \u00b7). The exact sp(\u03b8(\u201cGrooming An Animal\u201d), \u03b8(ci)) is shown between parenthesis vec(\u00b7) by M. These models learn a vector for each word wn, such that p(wn|(wi\u2212L, wi\u2212L+1, \u00b7 \u00b7 \u00b7 , wi+L\u22121, wi+L) is maximized over the training corpus; 2\u00d7L is the context window size. Hence similarity between vec(wi) and vec(wj) is high if they co-occurred a lot in context of size 2 \u00d7 L in the training text-corpus (i.e., semantically similar words share similar context). Based on the trained Ms space, we de\ufb01ne how to embed the event query e, and c. Each of ec, ea, and eo is set of one or more words. Each of these words can be directly embedded into Ms manifold by vec(\u00b7) function. Accordingly, we represent these sets of word vectors for each of ec, ea, and eo as \u03b8(ec), \u03b8(ea), and \u03b8(eo). We denote {\u03b8(ec), \u03b8(ea), \u03b8(eo)} by \u03b8(e). Regarding embedding of c, each concept c\u2217\u2208c is de\ufb01ned by its name and optionally some related keywords. Hence, the corresponding word vectors are then used to de\ufb01ne \u03b8(c\u2217) in Ms space. Relevance of Concepts to Event Query Let us de\ufb01ne a similarity function between \u03b8(c\u2217) and \u03b8(ec) as s(\u03b8(ec), \u03b8(c\u2217)). We propose two functions to measure the similarity between \u03b8(ec) and \u03b8(c\u2217). The \ufb01rst one is inspired by an example by (Mikolov et al. 2013b) to show the quality of their language model, where they indicated that vec(\u201cking\u201d) + vec(\u201cwoman\u201d) \u2212vec(\u201cman\u201d) is closest to vec(\u201cqueen\u201d). Accordingly, we de\ufb01ne a version of s(X, Y ), where the sets X and Y are \ufb01rstly pooled by the sum operation; we denote the sum pooling operation on a set by an overline. For instance, X = P i xi and Y = P i yj, where xi and yj are the word vectors of the ith element in X and the jth element in Y , respectively. Then, cosine similarity between X and Y is computed. We denote this version as sp(\u00b7, \u00b7); see Eq. 3. Fig. 3 shows how sp(\u00b7, \u00b7) could be used to retrieve the top 20 concepts relevant to \u03b8(\u201cGrooming An Animal\u201d) in Ms space. The \ufb01gure also shows embedding of the query and the relevant concept sets in 3D PCA visualization. \u03b8(ec =\u201cGrooming An Animal\u201d) and each of \u03b8(ci) for the most relevant 20 concepts are represented by their corresponding pooled vectors (\u03b8(ec) and \u03b8(ci))\u2200i), normalized to unit length under L2 norm. Another idea is to de\ufb01ne s(X, Y ) as a similarity function between the X and Y sets. For robustness (Torki and Elgammal 2010), we used percentile-based Hausdorff point set metric, where similarity between each pair of points is computed by the cosine similarity. We denote this version by st(X, Y ); see Eq. 3. We used l = 50% (i.e., median). sp(X, Y ) = (P i xi)T(P j yj) \u2225P i xi\u2225\u2225P j yj\u2225= X TY \u2225X\u2225\u2225Y \u2225 (3) st(X, Y ) = min{ l% min j m i ax xT i yj \u2225xi\u2225\u2225\u2225yj\u2225, l% min i m j ax xT i yj \u2225xi\u2225\u2225yj\u2225} Event Detection p(e|v) In practice, we decomposed p(e|v) into p(ec|v), p(eo|v), p(ea|v), which makes the problem reduces to deriving p(ec|v) (concept based retrieval), p(eo|v) (OCR based retrieval), and p(ea|v) (ASR based retrieval) under Ms. We start by p(ec|v) then we will how later in this section how p(eo|v), and p(ea|v) could be estimated. Estimating p(ec|v) : In our work, concepts are linguistic meanings that have corresponding detection functions given the video v. From Fig. 3, Ms space could be viewed as a space of meanings captured by a training text-corpus, where only sparse points in that space has a corresponding visual detection functions given v, which are the concepts c (e.g., \u201cblowing a candle\u201d). For zero shot event detection, we aim at exploiting these sparse points by the information captured by s(\u03b8(ec), \u03b8(ci \u2208c)) in Ms space. We derive p(ec|v) from probabilistic perspective starting from marginalizing p(ec|v) over the concept set c p(ec|v) \u221d X ci p(ec|ci)p(ci|v) \u221d X ci s(\u03b8(ec), \u03b8(ci))vi c (4) where p(e|ci)\u2200i are assumed to be proportional to s(\u03b8(ec), \u03b8(ci)) in our framework. From semantic embedding perspective, each video v is embedded into Ms by the \fset \u03c8(vc) = {\u03b8v(ci) = vi c\u03b8(ci), \u2200ci \u2208c}, where vi c\u03b8(ci) is a set of the same points in \u03b8(ci) scaled with vi c; \u03c8(vc) could be then directly compared with \u03b8(ec); see Eq. 5 p(ec|v) \u221d X ci s(\u03b8(ec), \u03b8(ci))vi c \u221ds\u2032(\u03b8(ec), \u03c8(vc) = {\u03b8v(ci), \u2200ci \u2208c}) (5) where s\u2032(\u03b8(ec), \u03c8(p(c|v)) = P i s(\u03b8(ec), \u03b8v(ci)) and s(\u00b7, \u00b7) could be replaced by sp(\u00b7, \u00b7), st(\u00b7, \u00b7), or any other measure in Ms space. An interesting observation is that when sp(\u00b7, \u00b7) is chosen, p(ec|v) \u221d\u03b8(ec) T \u2225\u03b8ec\u2225 \u0010 P i \u03b8(ci) \u2225\u03b8ci\u2225vi c \u0011 which is a direct similarity between \u03b8(ec) representing the query and the embedding of \u03c8(vc) as P i \u03b8(ci) \u2225\u03b8ci\u2225vi c; see proof in Appendix A. sp(\u00b7, \u00b7) performs consistently better than st(\u00b7, \u00b7) in our experiments. In practice, we only include \u03b8v(ci) in \u03c8(vc) such that ci is among the top R concepts with highest p(ec|ci). This is assuming that the remaining concepts are assigned p(ec|ci) = 0 which makes those items vanish; we used R=5. Hence, only a few concept detectors needs to be computed for on v which is a computational advantage. Estimating p(eo|v) and p(ea|v) : Both vo and va can be directly embedded into Ms since they are sets of words. Hence, we can model p(eo|v) and p(ea|v) as follows p(eo|v) \u221dsd(\u03b8(eo), \u03b8(vo)), p(ea|v) \u221dsd(\u03b8(ea) , \u03b8(va)) (6) where sd(X, Y ) = P ij xT i yj. We found this similarity function more appropriate for ASR/OCR text since they normally contains more text compared to concept de\ufb01nition. We also exploited an interesting property in Ms that nearest words to an arbitrary point can be retrieved. Hence, we automatically augment ea and eo with the nearest words to the event title in Ms using cosine similarity before retrieval. We found this trick effective in practice since it automatically retrieve relevant words that might appear in vo or va. Fusion: We fuse p(ec|v), p(eo|v), and p(ea|v) by weighted geometric mean with focus on visual concepts, i.e. p(e|v) = w+1 q p(ec|v)wp p(eo|v)p(ea|v)); w = 6. p(ec|v), p(ec|v), and p(ec|v) involves the similarity between \u03b8(e) and each of \u03c8(vc), \u03b8(vo), and \u03b8(va), leading to Eq. 2 view. Visual Concept Detection functions (p(c|v)) We leverage the information from three types of visual concepts in cv: object concepts co, action concepts ca, and scene concepts cs. Hence, c = cv = {co \u222aca \u222acs}; the list of concepts are attached in SM. We de\ufb01ne object and scene concept probabilities per video frame, and action concepts per video chunks. The rest of this section summarizes the concept detection for objects and scenes per frame f, and action concepts per video chunk u. Then, we show how they can be reduced to video level probabilities. Fig. 4 shows example high con\ufb01dence concepts in a \u201cBirthday Party\u201d video. Object Concepts p(oi|f), oi \u2208co: We involve 1000 Overfeat (Sermanet et al. 2014) object concept detectors which maps to 1000-ImageNet categories. We also adopt the concept detectors of face, car and person from a publicity available detector (i.e., (Felzenszwalb, McAllester, and Ramanan 2008)) Scene Concepts p(si|f), si \u2208cs: We represented scene concepts (p(si|f)) as bag of word representation on static features (i.e., SIFT (Lowe 2004) and HOG (Dalal and Triggs 2005)) with 10000 codebooks. We used TRECVID 500 SIN concepts concepts, including scene categories like \u201ccity\u201d and \u201chall\u201d way; these concepts are provided by provided by TRECVID2011 SIN track. Action Concepts p(ai|u), ai \u2208ca: We use both manually annotated (i.e. strongly supervised) and automatically annotated (i.e. weekly supervised) concepts; detailed in SM. We have \u223c500 action concepts; please refer to (Liu et al. 2013) for the action concept detection method that we adopt. Video level concept probabilities p(c|v) We represent probabilities of the cv set given a video v by a pooling operation over the the chunks or the frames of the videos similar to (Liu et al. 2013). In our experiments, we evaluated both max and average pooling. Speci\ufb01cally, p(oi|v) = \u03c1({p(oi|fk), fk \u2208v}), p(sl|v) = \u03c1({p(si|fk), fk \u2208v}), p(ak|v) = \u03c1({p(ai|uk), uk \u2208v}, where p(oi|v) and p(sl|v) are the video level probabilities of for the ith object and the lth scene concepts respectively, pooled over frames fk \u2208v. {fk \u2208v} are selected every M frames in v (M= 250). p(ak|v) is the video level probability of the kth action concept, pooled over a set of video chunks {uk \u2208v}. The chunk size is set to the mean chunk length of all concept training chunks. Finally, \u03c1 is the pooling function. We denote average and max pooling as \u03c1a(\u00b7) and \u03c1m(\u00b7), respectively. EDiSE Computational Performance Bene\ufb01ts Here we discuss the computational complexity of concept based EDISE, and ASR/OCR based EDiSE. The fusion part is negligible since it is constant time. Concept based EDiSE The computational complexity for computing p(ec|v) is mainly linear in the number of videos, denoted by |V |. We here detail why computational complexity of p(ec|v) is almost constant and hence video retrieval is almost O(|V |). From Eq. 5, p(ec|v) has a computational complexity of O(N \u00b7Q) for on e video, where Q is the computational complexity of computing s(\u00b7, \u00b7) and N is the number of concepts. We detail next the computational complexity of sp(\u00b7, \u00b7) and st(\u00b7, \u00b7) for the whole set of videos |V |. Complexity of p(ec|v) for sp(\u00b7, \u00b7) Let\u2019s assume that there \u03b8(ec) set has |ec| terms and \u03b8(ci) has |ci| terms. Then, the computational complexity of sp(\u03b8(ec), \u03b8(ci)) is Figure 4: Concept probabilities from videos (p(c|v)) \fO(M(|ec| + |ci|). |ci| and and |ec| are usually few terms in our case (< 10). Hence the computational complexity of sp(\u03b8(ec), \u03b8(ci)) is O(M), where M is the dimensionality of the word vectors. In our experiments M = 300. Given the complexity of sp(\u03b8(ec), \u03b8(ci)), the computational complexity of p(ec|v) will be O(N \u00b7 M), where N is the number of concepts. Hence, the computational complexity for computing p(ec|v) for |V | videos is O(|V | \u00b7 N \u00b7 M). However, for a given event, only few concepts are relevant, which are computed based on sp(\u03b8(ec), \u03b8(ci)) and only few concepts 5 in our case are suf\ufb01cient for event zero shot retrieval, retrieved by Nearest Neighbor search of ci \u2208c that is close the ec. Hence the computational complexity reduced to O(|V |\u00b7M), M = 300 for the GoogleNews word2vec model that we used. Hence, the complexity for |V | videos is basically linear O(|V |), given M is a constant and M << |V |. Complexity of p(ec|v) for st(\u00b7, \u00b7) The previous argument applies here in all elements except the complexity of the similarity function st(\u03b8(ec), \u03b8(ci)), which is O(M(|ec| \u00b7 |ci|). Assuming that |ec| \u00b7 |ci| is bounded by a constant, then the complexity of |V | videos is also O(|V | \u00b7 M), but with a bigger constant compared to sp(\u00b7, \u00b7) (linear in |V | for constant M << |V |). ASR/OCR based EDiSE The computational complexity of sd(\u03b8(eo), \u03b8(vo)) and sd(\u03b8(ea), \u03b8(va)) are O(|eo|\u00b7|vo|\u00b7M) and O(|ea|\u00b7|va|\u00b7M), respectively. There is no concepts for ASR/OCR based retrieval. Hence, the computational complexity of p(eo, v) and p(ea|v) are O(|V |\u00b7|eo|\u00b7|vo|\u00b7M) and O(|V |\u00b7|ea|\u00b7|va|\u00b7M), respectively. Since |eo| \u226a|V |, |vo| \u226a|V |, |ea| \u226a|V |, |va| \u226a|V |, and M \u226a|V |, the dominating factor in the complexity for both p(eo, v) and p(ea|v) will be |V |. Experiments We evaluated our method on the large TRECVID MED (Felzenszwalb, McAllester, and Ramanan 2013). We show the MAP (Mean Average Precision) and ROC AUC performance of the designated MEDTest set (Felzenszwalb, McAllester, and Ramanan 2013), containing more than 25,000 videos. Unless otherwise mentioned, our results are on TRECVID MED2013. There are two distributional semantic models in our experiments, trained on Wikipedia and GoogleNews using (Mikolov et al. 2013b). The Wikipedia model got trained on 1 billion words resulting in a vocabulary of size of\u2248120,000 words and word vectors of 250 dimensions. The GoogleNews model got trained on 100 billion words resulting in a vocabulary of size 3 million words and word vectors of 300 dimensions. The objective of having two models is to compare how well our EDiSE method performs depending on the size of the training corpus, used to train the language model. In the rest of this section, we present Concepts, OCR, ASR, and fusion results. Concept based Retrieval All the results in this section were generated by automatically retrieved concepts using only the event title. We start by comparing different settings of our method against (Dalton, Allan, and Mirajkar 2013). We used the language model in (Dalton, Allan, and Mirajkar 2013) for concept based retrieval to rank the concepts. This indicates that p(e|ci) in Eq. 4 is computed by the language model in (Metzler and Croft 2005) as adopted in (Dalton, Allan, and Mirajkar 2013), that we compare with under exactly the same setting. For our model, we evaluated the two pooling operations \u03c1m(\u00b7) and \u03c1p(\u00b7) and also the two different similarity measures on Ms space sp(\u00b7, \u00b7) and st(\u00b7, \u00b7). Furthermore, we evaluated the methods on both Wikipedia and GNews language models. In order to have conclusive experiments on these eight settings of our model compared to (Dalton, Allan, and Mirajkar 2013), we performed all of them on the four different sets of concepts (i.e. each has the same concept detectors; completely consistent comparison); see Table 1. Details about these concept sets are attached in SM. There are a number of observations. (1) using GNews (the bigger text corpus) language model is consistently better than using the Wikipedia language model. This indicates when the word embedding model is trained with a bigger text corpus, it captures more semantics and hence more accurate in our setting. (2) max pooling \u03c1m(\u00b7) behaves consistently better than average pooling \u03c1a(\u00b7). (3) sp(\u00b7, \u00b7) similarity measure is consistently better than st(\u00b7, \u00b7), which we see very interesting since this indicates that our hypothesis of using the vector operations on Ms manifold better represent p(e|ci). Hence, we recommend \ufb01nally to use the model trained on the larger corpus, \u03c1m(\u00b7) for concept pooling, and use sp(\u00b7, \u00b7) to measure the performance on Ms manifold. (4) our model\u2019s \ufb01nal setting is consistently better than (Dalton, Allan, and Mirajkar 2013). The \ufb01nal MED13 ROC AUC performance is 0.834. MAP for MED13 Events 31 to 40 (E31:40) is 5.97%. Detailed \ufb01gures are attached in SM. Our next experiment shows the \ufb01nal MAP performance using the recommended setting for our framework on the whole set of concepts, detailed earlier and in SM. Table 2 shows our \ufb01nal performance compared with (Dalton, Allan, and Mirajkar 2013) on the same concept detectors. It is not hard to see that our method performs more than double the Table 1: MED2013 MAP performance on four concept sets (event title query) Ours-Gnews Ours-Wiki (Dalton etal, 2013) TRECVID MED 2013 \u03c1m(\u00b7) \u03c1a(\u00b7) \u03c1m(\u00b7) \u03c1a(\u00b7) sp(\u00b7, \u00b7) st(\u00b7, \u00b7) sp(\u00b7, \u00b7) st(\u00b7, \u00b7) sp(\u00b7, \u00b7) st(\u00b7, \u00b7) sp(\u00b7, \u00b7) st(\u00b7, \u00b7) Concepts G1 (152 concepts) 4.29 3.94% 2.39% 2.38% 3.14% 2.13% 1.85% 1.70% 2.57% Concepts G2 (101 concepts) 1.74 1.20 1.56% 1.20% 1.09% 0.96% 0.66% 0.60% 1.17% Concepts G3 (60 concepts) 1.72 1.33% 1.28% 1.16% 1.21% 0.88% 0.88% 0.74% 1.54% Concepts G4 (56 concepts) 1.22 0.95 0.84% 0.69% 0.87% 0.76% 0.67% 0.56% 0.83% \fTable 2: MED2013 full concept set MAP Performance (auto-weighted versus manually-weighted concepts) Ours (auto) Dalton etal,13(auto) Dalton etal,13(manual) Overfeat 8.36% 3.40% 7.4% 2.43% SUN Object Rank Classeme CDDT WSCD\u2212SIF T Y ouT ube 0.48% 0.77% 0.84% 2.28% 3.48% MAP performance of (Dalton, Allan, and Mirajkar 2013) under the same concept set. Even when manual semantic editing is applied to (Dalton, Allan, and Mirajkar 2013), our performance is still better without semantic editing. We also show the performance on the same events of different concepts (i.e. SUN (Patterson and Hays 2012), Object Rank (Li et al. 2010), Classeme (Torresani, Szummer, and Fitzgibbon 2010)), and the best performing concepts in (Wu et al. 2014) (i.e., CDDT , WSCD\u2212SIF T Y ouT ube ). These numbers are as reported in (Wu et al. 2014). The results indicate the value of our concepts and approach compared to (Wu et al. 2014) and their concepts. We also report our performance using Overfeat concepts only to retrieve videos for the same events. This shows the value of involving action and scene concepts compared to only still image concepts like Overfeat for zeroshot event detection. We highlight that the results in (Wu et al. 2014) uses the whole event description which explicitly includes names of relevant concepts. ASR and OCR based Retrieval First, we compared our OCR and ASR retrieval trained on both Wikipedia and GoogleNews language model. Table 3 shows that the GoogleNews model MED13 MAP is better than the Wikipedia Model MAP in both ASR and OCR, which is consistent with our concept retrieval results. Fig. 5 shows the GoogleNews MED13 AP per event for both OCR and ASR. We further show our AP performance on MED14 events 31 to 40 in Fig. 5. In order to show the value our semantic modeling, we computed the performance of string matching method as a baseline, which basically increment the score for every exact match in the the detected text to words in the query. While, both our model and the matching model use the same query words and ASR/OCR detection, semantic properties captured by Ms boosts the performance compared to string matching; see table 3. This is since semantically relevant terms to the query have a high cosine similarity in Ms (i.e., high vec(wi)Tvec(wj) if wi is semantically related to Table 3: ASR & OCR Retrieval MAP on Ms using GNews, Wikipedia, and using word matching GNews MED2013 Wiki MED2013 matching MED2013 OCR 4.81% 3.85% 1.8% ASR 4.23% 1.50% 3.77% Table 4: ASR & OCR MAP performance using GNews corpus compared to (Wu et al. 2014)(pre\ufb01x E indicates Event) MED13 MED13 (Wu et al. 2014) MED14(E31:40) OCR 4.81% 4.30% 2.5% MED13 MED13 (Wu et al. 2014) MED14 (E31:40) ASR 4.23% 3.66% 5.97% (a) MED2013 (b) MED2014 (E31:40) Figure 5: ASR & OCR AP Performance (Google News) Figure 6: ASR & OCR AUCs on MED2013: Ours (GoogleNews) vs keyword Matching (the same query) \fwj). On the other hand, hard matching basically assumes that vec(wi)Tvec(wj) = 1 ifwi = wj, 0 otherwise. We also computed the ROC AUC metric for our method and the hard matching method on ASR and OCR; see Fig. 6. For ASR, average AUC is 0.623 for ours and 0.567 for Matching (9.9% gain). For OCR, average AUC is 0.621 for ours and 0.53 for Matching (17.1% gain). We report our GNews model results compared with (Wu et al. 2014) to indicate that, we achieve state-of-the-art MED13 MAP performance or even better for ASR/OCR; see table 4. The table also shows our ASR&OCR MED14 (E31:40) MAP. Fusion Experiments and Related Systems In table 5, we start by presenting a summary of our earlier ASR/OCR results on MED13 Test. Comparing OCR and ASR performances to Concepts performance, it is not hard to see that OCR/ASR have much lower average AUC zeroshot performance compared to concepts which are visual in our work. This indicates that OCR/ASR produces much higher false negatives compared to visual concepts. When we fused our all OCR and ASR con\ufb01dences, we achieved 10.7% MAP performance, however, the average AUC performance is as low as 0.67. We achieved lower MAP for our concepts 8.36% MAP but the average AUC performance is as high as 0.834. This indicates that measuring retrieval performance on MAP performance only is not informative, so one approach might achieve a high MAP but lower average AUC and vice versa. We further achieved the best performance of our system by fusing all Concepts, OCR, and ASR to achieve 13.1% MAP and 0.830 average AUC. We found our system achieves better than the state of the art system (Wu et al. 2014) 4.0% gain in MAP, but signi\ufb01cantly in average AUC; see 13.6% gain to (Wu et al. 2014) in table 5. We also discuss CPRF (Yang and Hanjalic 2010), MMPRF (Jiang et al. 2014b), and SPaR (Jiang et al. 2014a) reranking systems in contrast to our system that does not involve reranking. The initial retrieval performance is 3.9% MAP without reranking. Interestingly, we achieved a performance of 13.1% MAP also without reranking. The reranking methods assumes high top 5-10 precision of the initial ranking and that all test videos are available. Without any of these assumptions, our system without reranking performs 6.7%, 3.0%, and 0.2% better than CPRF (Yang and Hanjalic 2010), MMPRF (Jiang et al. 2014b), and SPaR (Jiang et al. 2014a) re-ranking systems; see table 5. Unfortunately, ROC AUC performances are not available for these method to compare with. Regarding ef\ufb01ciency, given vc representation of videos, our concept retrieval experiment on our whole concept set it takes \u2248270 seconds on a 16 cores Intel Xeon processor (64GB RAM) to the retrieval task on 20 events altogether. This is more than the time that SPaR (Jiang et al. 2014a) takes to rerank one event on an Intel Xeon processor(16GB RAM); see (Jiang et al. 2014a). Since, we detect the MED13 events in \u2248270 given vc representation of videos and as reported in (Jiang et al. 2014a), their average detection time per event for MED13 is \u22485 minutes assuming feature representation of videos (i.e., 360 seconds per event = 7200 seconds per 20 events). This indicates that our system Table 5: Fusion Experiments and Comparison to State of the Art Systems Method MAP AUC Our Concept retrieval (event title query) 8.36% 0.834 Concept retrieval (Dalton etal, 2013) (event title query) 3.4 % Concept retrieval (Dalton etal, 2013) (manual concepts) 7.4% Our ASR GNews 4.81% 0.623 Our OCR GNews 4.23% 0.621 Our ASR Matching 2.77% 0.567 Our OCR Matching 1.8% 0.536 Our ASR and OCR all fused 10.6 0.670 Our Full (Concepts+ASR+OCR) (No reranking) 13.1% 0.830 Our Full + SPaR reranking (Jiang et al. 2014a) 13.5% 0.790 Full system (Wu et al. 2014) 12.6 0.730 Reranking Systems Without Reranking (Jiang et al. 2014b) 3.9% CPRF (Yang and Hanjalic 2010) 6.4% Full Reranking system MMPRF(Jiang et al. 2014b) 10.1% Full Reranking system SPaR(Jiang et al. 2014a) 12.9% is 26.67X faster than (Jiang et al. 2014a) in MED13 detection. Finally, when we applied SPaR on our output as an initial ranking, we found that it improves MAP (from 13.1% to 13.5%) but hurts ROC AUC (from 0.83 to 0.79). This indicates that reranking has a limited/harmful effect on the performance of our method. We think is since our method already achieve a high performance without re-ranking; see SM for details about the features in this experiment."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1511.05175v6",
+                    "title": "Convolutional Models for Joint Object Categorization and Pose Estimation",
+                    "abstract": "In the task of Object Recognition, there exists a dichotomy between the\ncategorization of objects and estimating object pose, where the former\nnecessitates a view-invariant representation, while the latter requires a\nrepresentation capable of capturing pose information over different categories\nof objects. With the rise of deep architectures, the prime focus has been on\nobject category recognition. Deep learning methods have achieved wide success\nin this task. In contrast, object pose regression using these approaches has\nreceived relatively much less attention. In this paper we show how deep\narchitectures, specifically Convolutional Neural Networks (CNN), can be adapted\nto the task of simultaneous categorization and pose estimation of objects. We\ninvestigate and analyze the layers of various CNN models and extensively\ncompare between them with the goal of discovering how the layers of distributed\nrepresentations of CNNs represent object pose information and how this\ncontradicts with object category representations. We extensively experiment on\ntwo recent large and challenging multi-view datasets. Our models achieve better\nthan state-of-the-art performance on both datasets.",
+                    "authors": "Mohamed Elhoseiny, Tarek El-Gaaly, Amr Bakry, Ahmed Elgammal",
+                    "published": "2015-11-16",
+                    "updated": "2016-04-19",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Impressive progress has been made over the last decade towards solving the problems of object categorization, localization and detection. It is desirable for a vision system to address two tasks under general object recognition: object categorization and object pose estimation (estimating the relative pose of an object with respect to a camera). Pose estimation is crucial in many applications. These two broad tasks are contradicting in nature. An optimal object categorization system should be able to recognize the category of an object, independent of its viewpoint. This means that the system should be able to learn viewpoint-invariant representations of object categories. In contrast, a pose estimation system requires a representation that preserves the geometric and visual features of the objects in order to distinguish its pose. This gives rise to a fundamental question: should categorization and pose estimation be solved simultaneously, and if so, can one aid the other? Contrasting paradigms approach this question differently. Traditional instance-based 3D pose estimation approaches solve the instance-recognition and pose estimation problems simultaneously, given model bases of instances in 2D or 3D (Grimson & Lozano-Perez, 1985; Lamdan & Wolfson, 1988; Lowe, 1987; Shimshoni & Ponce, 1997). Most recent object pose estimation approaches solve the problem within the detection process, where category-speci\ufb01c object detectors that encode part geometry are trained (Savarese & Fei-Fei, 2007; Savarese & Li, 2008; Mei et al., 2011; Payet & Todorovic, 2011; Schels et al., 2012; Pepik et al., 2012). Since part-geometry is a function of the pose, these approaches are able to provide coarse estimate of the object pose with the detection. However the underlying assumption here is that the categorization is done a-priori, and the representation is view-variant. Other recent approaches try to solve the pose estimation simultaneously with categorization through learning dual representations: view-invariant category representation and view-variant category-invariant representation (Zhang et al., 2013a; Bakry & Elgammal, 2014). \u2217Equal contribution 1 arXiv:1511.05175v6  [cs.CV]  19 Apr 2016 \fWorkshop track ICLR 2016 With the rise of deep architectures, the main focus has been on category recognition. A wide success has been achieved on this task. In contrast, pose estimation has not received much attention. The impressive results of Convolutional Neural Networks (CNNs) in tasks of categorizations (Krizhevsky et al., 2012) and detection (Sermanet et al., 2013; Girshick et al., 2013) motivated many researchers to explore their applicability in different vision tasks. Several approaches recently showed successful results where they used networks that are pre-trained for a speci\ufb01c task (e.g.categorization) and then use the representation of higher layers as features for another task (Frome et al., 2013; Donahue et al., 2013; Zeiler & Fergus, 2013). This process is known as transfer learning. As pointed out by (Yosinski et al., 2014), this process is useful when the target task has signi\ufb01cantly smaller training data than what is needed to train the model. Typically the \ufb01rst n-layers are copied from the pre-trained network to initialize the corresponding layers for the target task. Within the CNN literature, typically the layers up until FC7 (which is the last layer before the output layer) are used for that purpose (Frome et al., 2013). Pose estimation is an example of a task that inherently suffers from lack of data. In fact the largest available dataset for multiview recognition and pose estimation has 51 object categories with a total of about 300 instances (Lai et al., 2011a). It is hard to imagine the availability of a dataset of thousands of objects where different views are sampled around each object in order to be able to train a learning machine such as a CNN with millions of parameters. Therefore, transfer learning is critical for this task. However the challenge lies in the contradicting objective that has been described in the \ufb01rst paragraph. Current CNN models are optimized for categorization, and therefore they are expected to achieve view invariant representation. Therefore it is not expected that feature representation at deeper layers are useful for pose estimation. However, feature representation in shallower layers tend to be more general and less category-speci\ufb01c and thus may hold enough information to discriminate between different poses. This is a key hypothesis that is explored in this paper and this work is the \ufb01rst exploration of the capability of CNNs on the task of object pose estimation. The contributions of this paper are: (1) we show how CNNs can be adapted to the task of simultaneous categorization and pose estimation of objects, (2) we investigate how each of these tasks affect the other, i.e.how category-speci\ufb01c information can help estimate the pose of an object and how a balance between these contrasting tasks can be achieved, (3) we analyze different CNN models and extensively compare between them to \ufb01nd an ef\ufb01cient balance between accurate categorization and robust pose estimation, (4) we validate our work by extensive experiments on two recent large and challenging multi-view datasets. We achieve better than state-of-the-art performance on both datasets. 2 RELATED WORK Due to the surge of work in deep architectures over the last few years, there has amassed a large number of research studies. Despite this, using CNNs for regression and capturing pose information is still a relatively unexplored area. This motivates the goals of this paper. We focus on the most relevant work, in particular, studies that focus on understanding the functions of CNN layers and CNNs that solve for pose information. We also brie\ufb02y touch upon previous approaches in object categorization and pose estimation. Although fundamentally different to object pose estimation, some research has explored using CNNs to recognize human pose (Toshev & Szegedy, 2013; LI et al., 2014; P\ufb01ster et al., 2014). Recently (Gkioxari et al., 2014) proposed joint optimization on human pose and activity. In human pose estimation, there is no problem getting millions of image of people at different postures. Human activities are correlated with human poses, while in the object-pose domain the category is independent of pose. This makes joint learning of category and pose more challenging than joint learning of human activity and pose. Some very recent work has explored joint detection and pose estimation using CNNs (Tulsiani & Malik, 2014). Recent in-depth studies explore the intricacies of CNNs; including the effects of transfer-learning and \ufb01ne-tuning, properties and dimensions of CNN layers and the study of invariances captured in CNN layers (e.g.(Yosinski et al., 2014; Zeiler & Fergus, 2013; Chat\ufb01eld et al., 2014)). A data-centric analysis of existing CNN models for object detection has appeared in (Pepik et al., 2015a). A comprehensive review of recent work in object recognition and pose estimation is detailed in (Savarese & Fei-fei, 2010). We highlight the most relevant research. Successful work have been done in estimating the object pose of a single object (Cyr & Kimia, 2004; Mei et al., 2011; Schels 2 \fWorkshop track ICLR 2016 et al., 2012). This model, referred to as single-instance 3D model, has the limitation of being category-speci\ufb01c and does not scalable to a large number of categories and deal with high intraclass variation. Recently, detection and pose have been solved simultaneously (e.g.(Tulsiani & Malik, 2014; Pepik et al., 2015b; Xiang et al., 2014; Savarese & Fei-Fei, 2007; Savarese & Li, 2008; Mei et al., 2011; Payet & Todorovic, 2011; Schels et al., 2012; Pepik et al., 2012; Lai et al., 2011b)). Most of these methods belongs to the category of limited-pose (discrete-pose) approaches since it uses classi\ufb01cation for pose estimation. Very few works formulate the pose estimation problem as regression over a continuous space. In (Lai et al., 2011b), an object pose tree is built for doing multilevel inference. This involves a classi\ufb01cation strategy for pose which results in coarse estimates and does not utilize information present in the continuous distribution of descriptor spaces. Work presented in (Zhang et al., 2013b) and (Torki & Elgammal, 2011) explicitly model the continuous pose variations of objects but the scalability of these models is limited. A more recent work (Bakry & Elgammal, 2014) proposes a feedforward approach to solve the two problems jointly by balancing between continuous and discrete modeling of pose in order to increase performance and scalability. In these models, the nonlinearity in the category representations is not modeled, which is mandatory for many applications. 3 MOTIVATION The \ufb01rst question we pose in this paper is how good are pre-trained representations of different CNN layers, without \ufb01ne-tuning, for the task of pose estimation? To answer this we analyzed a state-ofthe-art CNN trained on ImageNet (Krizhevsky et al., 2012) by testing it on dense multi-view images from the RGBD dataset (Lai et al., 2011a) to see how well it represented object view-manifolds and hence able to estimate object poses. This CNN is composed of 8 layers: Conv1, Pool1, Conv2, Pool2, Conv3, Conv4, Conv5, Pool5, FC6, FC7, FC8. Pool indicates Max-Pooling layers, Conv indicates convolutional layers and FC indicates fully connected layers. In order to quantitatively evaluate the representations of pose within the CNN, we trained both a pose regressor (using Kernel Ridge-Regression) and an SVM classi\ufb01er for categorization (linear one-vs-all) on the features extracted from each of the layers. Fig. 1-Left is the result of the regressor and classi\ufb01er. It clearly shows the con\ufb02ict in representation of the pre-trained network. For pose estimation, the performance increases until around Pool5 and then decreases. This con\ufb01rms that shallow layers that have suf\ufb01cient abstractive representation offer better feature encoding for pose estimation. It appears that Pool5 provides a representation that captures the best compromise in performance, between categorization and pose discrimination. In Fig 1-Left we report cross-evaluation of categorization using pose features and vice versa. FC8 (output) which is task speci\ufb01c, does not perform good pose estimation, while FC6/FC7 perform much better. It is interesting to observe the opposite is not true; when optimizing on pose, much of the category-speci\ufb01c information is still represented by the features of the CNN, as seen by the increase in performance of category recognition using the pose-optimized features. We further explored using other regressors on multiple views of a single object instance (GPR (Rasmussen & Williams, 2005), WKNN (H et al., 1996), SVR (H et al., 1996), KTA (H et al., 1996)). We use a coffee mug instance that has enough visual and shape features to discriminate its poses. Fig 1-Right shows the MAE of the pose regression. The results con\ufb01rm that the pose representation improves as we approach Pool5. This indicates that Pool5 has the best representation of the object\u2019s view-manifold. We also found that the performance of features based on Pool5 are the closest to correlate with the performance when using HOG features on the objects\u2019 multi-view images (Fig 1Right). This further proves that Pool5 has the abstraction capability to represent pose ef\ufb01ciently. It is important to point out here that, in addition to our analysis, in-depth manifold analysis was previously conducted to analyze the object-view manifolds and their representations within CNN layers. This can be found in (Bakry et al., 2015). This in-depth study corroborates the conclusions we make here. 4 ANALYZED MODELS We used a state-of-the-art CNN built by Krizhevsky et al. (2012) as our baseline network in our experiments (winner of the LSVRC-2012 Imagenet Challenge Russakovsky et al. (2014)). We refer to this model as Model0: base network. This model is pre-trained on Imagenet. The last fully connected layer (FC8) is fed to a 1000-way softmax which produces a distribution over the category labels. Dropout was employed during training and Recti\ufb01ed Linear Units (ReLU) were used for 3 \fWorkshop track ICLR 2016 faster training. Stochastic gradient descent is used for back propagation. Model0 is not \ufb01ne-tuned, and thus an analysis of it shows how the layers of a CNN trained on categorization of ImageNet lacks the ability to represent pose ef\ufb01ciently. Throughout this study we vary the architecture of the base network and the loss functions. All other models described are pre-trained on ImageNet and \ufb01ne-tuned on each of the two large dataset we experimented on. We propose and investigate four different CNN models: Parallel Model (PM), Cross-Product Model (CPM), Late Branching Model (LBM) and Early Branching Model (EBM). PM is a parallel version of the base network; two parallel and independent versions of the base network, one for categorization and one for pose. CPM has an output layer with units for each category and pose combination to jointly train (depicted in Figure 2-a). LBM and EBM models are also depicted in Figure 2. LBM branches into two last layers, one for categorization and one for pose. EBM performs early branching into two subnetworks, each specialized in categorization and pose estimation, respectively. The output layer FC8 is not merged but instead the LBM and EBM networks are optimized over two loss functions, one concerned with building view-invariance representations for categorization and the other with category-invariant representations for pose estimation. Because of the branching, this causes two units to be active, one in each branch, at the same time. The only work that has done something similar to this is the work byGkioxari et al. (2014). All losses are optimized by the multinomial logistic regression objective, similar to Krizhevsky et al. (2012) (Softmax loss). We denote softmax loss of label l \u2208{lc, lp} and image x as lossi(x, l), where i indicates if this loss is over category or pose modes, lp and lc are the labels for pose and category, respectively. In the following subsections we describe each model in detail. Parallel Model (PM): This model consists of two base networks running in parallel, each solving categorization and pose estimation independently. There is no sharing of information between the two networks. The goal of this model is to see how well the traditional CNN is capable of representing object-view manifolds and hence estimating object pose, independent of category-speci\ufb01c information. The category and pose losses minimized in this model are: lossc(x, lc) and lossp(x, lp), one for each of the tasks of categorization and pose estimation, respectively. Cross-Product Model (CPM): CPM explores a way to combine categorization and pose estimation by building a last layer capable of capturing both (see Fig2-a). We build a layer with the number of units equivalent to the number of combinations of category and pose, i.e.the cross-product of category and pose labels. The number of categories varies according to the dataset as we will see. The pose angles (in this case yaw or azimuth angle of an object) is discretized into angle bins across Figure 1: Left: Linear SVM categorization and pose regression performance based on feature encoding of different layers of a pre-trained CNN over all objects. The dotted lines are for cross-evaluation: for PM-Cat, LBM-Cat and EBM(800)-Cat represent the models\u2019 category representations evaluated on the task of pose estimation (to observe the effect of how category representations encode pose information). PM-Pose, LBMPose and EBM(800)-Pose are evaluated on the task of categorization to see how well pose representations in the CNN encodes categories. This is to show the complete pose-invariant representations of the layers when learning to categorize. Right: Pose regression on a single object showing the Mean Absolute Error (in degrees) of various regression algorithms from FC8 to Pool5. The horizontal lines represent the regression performance on HOG feature descriptors computed on the images. 4 \fWorkshop track ICLR 2016 Figure 2: The proposed models showing the joint loss layer in CPM, late branching in LBM and early branching in EBM. Blue layers correspond to layers with convolution, pooling and normalization. Violet colored layers correspond to layers with just convolution. Green layers correspond to fully-connected layers. the viewing circle. This is the case with all our pose estimating models. The loss function for CPM is the softmax loss over the cross-product of category and pose labels: loss(x, lp \u00d7 lc). Late Branching Model (LBM): We introduce a change in the architecture by splitting/branching the network into two last layers, each designed to be speci\ufb01c to the two tasks: categorization and pose estimation. Thus, this network has a shared representation for both category and pose information up until layer FC7 (see Fig2-b). The goal is to learn category and pose information simultaneously from the representations encoded in the previous layers of the CNN. The question behind this model is whether or not one last layer would be enough to recover the pose information from the previous layers, in other words untangle the object view-manifold and give accurate pose estimates. In other words, one can think of this as testing the ability of the deep distributed representations of a CNN in holding both categoryspeci\ufb01c pose-invariant information as well as pose-variant information. LBM is trained using a linear combination of losses over category and pose: \u03bb1 \u00b7 lossc(x, lc) + \u03bb2 \u00b7 lossp(x, lp) where \u03bb1, \u03bb2 are weights found empirically (see appendix G). Early Branching Model (EBM): The question of moving the branching to an earlier layer in the network poses itself here: Can the branching be moved earlier in the network to where the pose knowledge is still well preserved and in fact maximal across the layers? From our experiments (described later on) we observe that the objects\u2019 view-manifolds are maximally represented at Pool5. Thus, this network has a shared representation for both category and pose information up until layer Pool5. At Pool5 it branches out into two subnetworks, that are jointly optimized using a combined loss function (same as for LBM): \u03bb1 \u00b7 lossc(x, lc) + \u03bb2 \u00b7 lossp(x, lp). Similar to LBM, it is important to note that this network optimizes over two losses. This model achieves the most ef\ufb01cient balance between categorization and pose estimation and achieve stateof-the-art results on two large challenging datasets, as we shall see in Section 7. 5 DATASETS 5.1 RGBD DATASET One of the largest and challenging multi-view datasets available is the RGB-D dataset Lai et al. (2011a). It consists of 300 tabletop object instances over 51 different categories. Images are captured of objects rotating on a turn-table, resulting in dense views of each object. The camera is positioned at three different heights with elevation angles: 30\u25e6, 45\u25e6and 60\u25e6. In previous approaches the middle height (45\u25e6) is left out for testing Lai et al. (2011b); Zhang et al. (2013a); Bakry & Elgammal (2014); El-Gaaly et al. (2012). This means that object instances at test time have been seen before from other heights during training. For this dataset it was important to 5 \fWorkshop track ICLR 2016 experiment with an additional training-testing split of the data to give more meaningful results. We wanted to ensure that objects at test time had never seen before. We also wanted to make sure that the instances we are dealing with have non-degenerate view manifolds. We observed that many of the objects in the dataset are ill-posed, in the sense that the poses of the object are not distinct. This happens when the objects have no discriminating texture or shape to be able to identify its poses (e.g. a texture-less ball or orange). This causes object-view manifold degeneracy. For this reason, we select 34 out of the 51 categories as objects that possess variation across the viewpoints, and thus are not ill-posed with respect to pose estimation. We split the data into training, validation and testing. In this datasets, most categories have few instances; therefore we left out two random instances per category, one for validation and one for testing. In the case where a category has less than 5 instances, we form the validation set for that category by randomly sampling one object instance from the training set. We also left out all the middle height for testing. Thus, the testing set is composed of unseen instances and unseen heights and this allows us to more accurately evaluate the capability of CNNs in discriminating categories and poses of tabletop objects. We call this split, Split 1. In order to compare with state-of-the-art we also used the split used by previous approaches (we call this Split 2). 5.2 PASCAL3D+ DATASET We experiment on the recently released challenging dataset of multi-view images, called Pascal3D+ Xiang et al. (2014). Pascal3D+ is very challenging because it consists of images in the wild, in other words, images of object categories exhibiting high variability, captured under uncontrolled settings, in cluttered scenes and under many different poses. Pascal3D+ contains 12 categories of rigid objects selected from the PASCAL VOC 2012 dataset Everingham et al. (2010). These objects are annotated with pose information (azimuth, elevation and distance to camera). Pascal3D+ also adds pose annotated images of these 12 categories from the ImageNet dataset Russakovsky et al. (2014). The bottle category is omitted in state-of-the-art results. To be consistent, we do the same. This leaves 11 categories to experiment with. There are about 11,500 and 7,000 training images in ImageNet and Pascal3D+ subsets, respectively. We take a small portion of these images for validation and use the rest for training. For testing, there are about 11,200 and 6,900 testing images for ImageNet and Pascal3D+, respectively. On average there are about 3,000 object instances per category in Pascal3D+ captured in the wild, making it a challenging dataset for estimating object pose. 6 CNN LAYER ANALYSIS Similar to the analysis performed in Section 3, we do the same on all our described models. This gives insight into the ability of these models to represent pose and the intrinsic differences between them. We perform kernel Ridge-regression and SVM classi\ufb01cation on each layer of the CNN models. The results of this analysis on the two muti-view datasets are shown in Fig. 3 and 4. From Fig. 3 and 4, we can see that the base network monotonically decreases in pose accuracy after layer Pool5. Pool5 seems to again hold substantial pose information, before it is lost in the following layers. This is the premise behind the design of our EBM model. EBM is able to ef\ufb01ciently untangle the object-view manifold and achieve good pose estimation on the branch speci\ufb01c to pose estimation. From Fig. 3 and 4, it can be observed that the LBM is able to achieve a good boost in pose performance at its last layer. From 3-right, it can be observed that at the layers Conv4 and Pool5 EBM has slightly worse accuracy than LBM and PM on the RGBD dataset. This indicates that the optimization is putting emphasis on the category information just before branching to achieve better pose estimation at deeper layers. CPM does quite worse than the other models on both datasets, in both categorization and pose estimation. This can be seen in Fig. 3-Left and 3-Right and to some extent in Fig. 4. The reason for this lies in the fact that CPM shares information to jointly optimize over category and pose. The drop is more evident in the task of categorization, indicating again that category information aids in estimating the pose, but not the other way round. The drop is more on the RGBD dataset because there are a lot more categories than Pascal3D+ and thus a lot more inter-class confusion. This is analogous to using category labels to separate between objects of different categories which may help bring similar posed objects of the same category together in the latent space encoded in the layers. On the other hand there is no clear untangling of the object-view manifold, where the pose 6 \fWorkshop track ICLR 2016 Figure 3: Analysis of layers trained on the RGBD dataset. Left: the performance of linear SVM category classi\ufb01cation over the layers of different model. Right: the performance of pose regression over the layers of different models (including the category parts of some of the models this shows the lack of pose information encoded within the object category representations) Figure 4: Analysis of layers trained on the Pascal3D+ dataset. Left: the performance of linear SVM category classi\ufb01cation over the layers of different model. Right: the performance of pose regression over the layers of different models information is stored, and thus this lack of pose information negatively impacts the categorization of objects. k-NN Layer Analysis We conduct k \u2212NearestNeighbor pose estimation over the Pascal3D dataset on all the layers of the 4 models with varying neighborhood sizes (shown in Fig. 5). Comparing the two models (LBM and EBM), we gain slight improvement in categorization and a large improvement in pose estimation performance when using EBM. From Figure 5 and 9 in the Appendix, we conclude that as we go deeper into the network up to layer Conv5 we gain more category separation and object-view manifold preservation. This shows how the early branching better resolves the contradiction between the pose estimation and categorization tasks while sharing the low level \ufb01lter representations that are helpful for both tasks. After Conv5, there are two common layers in EBM. In these two layers, linear separability between categories increases (seen in Fig. 9), but the object-view manifolds collapse (as seen in Fig. 5). This hurts the pose estimation. At the same time, this supports the aforementioned claim that enforcing better categorization (\ufb01ne-tuning) hurts pose estimation. In our best performing model (EBM), in Figure 5, remarkable improvement to the pose object-view manifold is attained. For pose, the drop in KNN-classi\ufb01er as the K increases vanishes when going deeper in network; see FC6 and FC7 layers EBM in Fig. 5. KNN \ufb01gure for categorization on Pascal3D dataset could be seen in appendix A. In a similar behavior EBM behaves better than CPM and PM. An interesting behavior that CPM works clearly on Pascal3D dataset compared to RGBD; see Appendix A for KNN analysis on RGBD dataset. This is due that RGBD dataset has both dense poses and also larger number of categories ( 5 times Pas7 \fWorkshop track ICLR 2016 Figure 5: Comparison between EBM and LBM for categorization and pose estimation at each layer of the CNN using k-NN with varying k = {1, 3, 5, 7, 9}. This experiment was conducted on the Pascal3D dataset. cal3D). This increases the information/uncertainty to model that are beyond the capacity of CPM for RGBD dataset and generally as the number of categories and poses increase. Local Pose Measurement Analysis: In Appendix B, we further analyzed four local measurement pose analysis proposed in (Bakry et al., 2015) to analyze layers of the \ufb01ve models we study against the pose manifold. The purpose of this analysis is to show how the learning representations for each model is untangle to the circle manifold where the pose inhabits 7 EXPERIMENTS Here we describe the experimental setup and present the quantitative results of our experiments as well as comparisons with state-of-the-art. 7.1 TRAINING AND TESTING All models are trained by back propagation with Stochastic gradient descent. Refer to appendix E for parameter settings, e.g. learning rate, decay, etc. At training time, we randomly sample 227x227 patches from the down-scaled 256x256 images. At test time the center 227x227 patches are taken. All classi\ufb01cation losses are optimized by the multinomial logistic regression objective. Similar to (Krizhevsky et al., 2012), we optimized it by maximizing the average of the log-probability of the correct label under the prediction distribution across training cases. The pose softmax output (FC8) layer produces the pose probability distribution given the image. For each of the category and pose losses, the gradient with respect to the CNN parameters is computed which is then fed into CNN training for back propagation. All the results that are presented in the paper were based on the prediction of argmaxposep(pose|x), where pose is one of the 16 pose bins and x is the given image. In addition, we conduct an experiment where we predict the pose by computing the expected pose in the distribution of p(pose|x); see Eq. 1. E(p(pose|x)) = X i p(posei|x) \u00d7 \u03c6(posei) (1) where \u03c6(posei) is the center angle of the corresponding bin posei (the pose of the ith bin). The detailed de\ufb01nitions of the performance metrics used in our experiments are described in appendix D. 8 \fWorkshop track ICLR 2016 7.2 RESULTS Table 1 shows the category recognition and pose estimation performance for the different models on the two training-testing splits of the RGBD dataset. Table 2 shows our best performing model EBM compared to state-of-the-art approaches. Using the pose prediction rule of eq. 1, the pose accuracy of EBM(800) increased from 78.83% to 79.30% for the argmax prediction on split 2 of RGBD. Looking at the closest previous approach in Table 2, (Bakry & Elgammal, 2014) achieves 96.01% classi\ufb01cation accuracy. This is achieved using both visual and depth channels. We only used RGB (without depth) in our approach. (Bakry & Elgammal, 2014) achieves a lower 94.84% with RGB only, which shows the advantage of our CNN models for classi\ufb01cation. We achieve 2.3% increase in category recognition and about 2% increase in pose estimation (79.30%) using EBM(800), when compared with state-of-the-art. These measurements are likely to increase further when using EBM(4096), as we see slight improvement of EBM(4096) over EBM(800) in Table 2. It is also possible that running k-NN on top of the layer features could improve performance further. We achieved 97.14% categorization using EBM. We also achieved 99.0% classi\ufb01cation accuracy using Nearest Neighbor classi\ufb01cation on the Pool5 layer of EBM, which indicates that we learned better convolutional \ufb01lters. Table 3 shows the performance of our models on Pascal3D+. We compare the accuracy achieved by our models with state-of-the-art results by (Zhang et al., 2015; Xiang et al., 2014; Pepik et al., 2015b; Tulsiani & Malik, 2014). It must be noted here that we are solving slightly different problems to some of these approaches. In Xiang et al. (2014), the authors solve detection and pose estimation, assuming correct detection. On the other hand (Zhang et al., 2015) solve just pose estimation, assuming that the object categories are known. (Pepik et al., 2015b; Tulsiani & Malik, 2014) solve joint detection and pose estimation. In our case we are jointly solving both category recognition and pose estimation, which can be considered a harder problem than that of (Zhang et al., 2015) and (Xiang et al., 2014). Our pose estimation performance is better than all these previous approaches. For the sake of this comparison, we computed the pose performance using the metrics applied in (Zhang et al., 2015). These metrics are pose accuracy for images with pose errors < 22.5\u25e6and < 45\u25e6. Table 3 shows both our categorization and pose estimation results on Pascal3D compared against previous approaches. The table indicates 13.69% improvement of our method over (Zhang et al., 2015) (the best performing previous approach) in pose < 22.5\u25e6metric and 4% improvement in pose < 45\u25e6, which are signi\ufb01cant results. It is important to note that comparing to (Xiang et al., 2014; Tulsiani & Malik, 2014; Pepik et al., 2015b) is slightly unfair because these works solve for detection and pose simultaneously, while we do not solve detection. We also show our performance when including ImageNet images in the training set and also the test set see Table 3 (rows 7-8). The results show the bene\ufb01t of ImageNet training images which boosts pose performance to 76.9% (from 57.89%) and 88.26% (from 63.0%) for pose < 22.5\u25e6and pose < 45\u25e6, respectively. In Table 3, on the in the wild images of Pascal3D+, our EBM model achieves an impressive increase of \u223c8% and \u223c5% over the state-of-the-art models using the two pose accuracy metrics, respectively. 7.3 COMPUTATIONAL ANALYSIS AND CONVERGENCE We performed computational analysis on the convergence of the models and show that EBM converges substantially faster than all the other models. In Fig. 6 we show the convergence rates of the proposed models. EBM here is the larger EBM (4096) network. Despite having many more parameters than most of the other models (about \u223c112 million parameters compared to 60 million in the base network), EBM converges substantially faster than all the other models. This shows the ability of this particular network to specialize faster in the two tasks. The shared \ufb01rst \ufb01ve layers are able to build up the object-view manifolds, preserve them and enhance them in the pose subnetwork of the model, while the other subnetwork specializes in pose-invariant category recognition. 9 \fWorkshop track ICLR 2016 Table 1: A summary of all the results of the CNN models. Split 2 is the traditional RGBD dataset split. Split 1 is the one we describe that better evaluates our experiments. Split 2 is the state-of-the-art training-testing split. C indicates category performance and P indicates pose accuracy (where it is measured using 3 different metrics consistent with state-of-the-art. P (< 22.5\u25e6) indicates that the pose accuracy is measured for objects where the pose error was less than 22.5\u25e6 Model Split C% P% (< 22.5\u25e6) P% (< 45\u25e6) P% (AAAI) PM 1 89.63 69.58 81.09 81.21 CPM 1 80.68 63.46 75.45 77.35 LBM 1 91.48 68.25 79.31 79.94 EBM (4096) 1 89.94 71.49 82.19 82.00 EBM (800) 1 89.84 71.29 82.29 81.91 EBM (400) 1 89.77 70.80 81.73 81.65 EBM (200) 1 90.11 67.70 79.43 79.77 EBM (100) 1 90.34 69.15 80.09 80.36 EBM (800) 2 97.14 66.13 77.02 78.83 EBM (4096) 2 97.07 65.82 76.51 78.66 SVM/Kernel Reg Model 0 (best category FC6) 1 86.71 64.39 SVM/Kernel Reg Model 0 (best pose Conv4) 1 58.64 67.39 SVM/Kernel Reg HOG 1 80.26 27.95 Table 2: RGBD Dataset: Comparison with state-of-the-art approaches on category recognition and pose estimation (Ours use only RGB channel). Approach Category % Pose (AAAI) % (Lai et al., 2011b) 94.30 (RGB + Depth) 53.50 (Bakry & Elgammal, 2014) 94.84 (RGB only)/ 96.01 (RGB+ Depth) 76.01 (Zhang et al., 2013a) 92.00 (RGB only)/ 93.10 (RGB + Depth) 61.57 (Bakry et al., 2016) 85.00 77.31 Ours (EBM(800)) 97.14 79.30 Table 3: Pascal3D dataset (Xiang et al., 2014): Comparison with state-of-the-art approaches on category recognition and pose estimation. The AAAI pose metric is the performance metric used in(Zhang et al., 2015; Lai et al., 2011b; Zhang et al., 2013a). Train: Pascal12, Test: Pascal12 Approach Category Pose % (error < 22.5) Pose % (error < 45) Pose AAAI metric (Xiang et al., 2014) 15.60 18.70 (Pepik et al., 2012) 17.30 21.50 (Pepik et al., 2015b) 18.60 27.60 (Tulsiani & Malik, 2014) 36.00 44.50 (Zhang et al., 2015) 44.20 59.00 EBM (4096) 83.0 51.80 64.27 73.53 EBM (800) 83.10 51.37 64.20 73.26 LBM 82.69 48.38 60.11 70.88 CPM 76.35 49.39 61.90 71.80 PM 84.0 47.34 61.30 71.60 Train: Pascal12 + ImageNet, Test: Pascal12 EBM (800) 83.79 51.89 60.74 75.39 Train: Pascal12 + ImageNet, Test: Pascal12 + ImageNet EBM (800) 92.83 67.26 75.11 83.27 10 \fWorkshop track ICLR 2016 Figure 6: Comparison of convergence between the models. On the left is the category error and on the right is the pose error, on the validation set, respectively for each model (a) to (d). The error is computed per batch during each iteration. CPM shows the error for the joint category and pose. It can be seen that EBM (4096) converges much faster than the others which is another bene\ufb01t of early branching. This is despite have a lot more parameters than the other models. This indicates that each of the subnetworks of EBM are able to specialize in both categorization and pose estimation faster. Each iteration is computed on one batch of 100 training samples. 8 DISCUSSION Analysis of the layers of all the CNN models is shown in Fig. 3 to 9. We provide quantitative results over two challenging datasets and summarize them in Tables 1, 2 and 3. We compare our models with multiple baselines in Table 1: Linear SVM and Kernel RidgeRegression on HOG descriptors Dalal & Triggs (2005) as well as on features extracted from the best performing layers of the base network on each task. These baseline results were expected to be quite lower than our models\u2019 performance due to the lack of \ufb01ne-tuning in the base model and due to the sharing of the network layers between the two tasks of category recognition and pose estimation. Without \ufb01ne-tuning the base network does not represent the object-view manifold well enough to estimate the pose ef\ufb01ciently. After \ufb01ne-tuning on each of the respective datasets, we were able to achieve good category performance using the PM model. The downside of this model is its inability to perform robust pose estimation on the more challenging natural sparser-views of Pascal3D+. This is evident in the results shown in Table 3, where PM achieves less pose accuracy. In Tables 1 and 3, we see again that CPM does worse than the other models in both datasets. This is more evident in the task of categorization, e.g., a drop of \u223c7% and 2%-3% in category and pose accuracy on Pascal3D+, respectively, and similarly \u223c10% and \u223c6% on the RGBD dataset. This motivates the need for branching in the networks and branching at the particular layer that better represents both category and pose. Interestingly, we found that CPM performs relatively better on Pascal3D+. We argue that the reason is that object poses in natural images are dominated by a smaller range of viewpoints and hence most of the pose-bins have vanishing probability (easier to learn). In addition, Pascal3D+ has a smaller number of categories. LBM performs relatively well on RGBD, but not on Pascal3D+. This can be attributed to the fact that RGBD has many more categories and is composed of images of objects under controlled settings and not in-the-wild like in Pascal3D+. The images in the RGBD dataset are captured at dense views as the object rotates on a turn-table. This is why the pose information is more prevalent in the last layers. This is evident from the steep monotonically increasing curve of LBM in Fig. 3-Left. This is not the case in Pascal3D+ where the increase is more steady and in fact there is a decrease after layer FC6 (see Fig. 4-Right). The reason why EBM performs better than PM even though its weights are randomly initialized is that PM\u2019s FC6 and FC7 layers in the pose-speci\ufb01c branch are initialized with category-speci\ufb01c weights from pre-training. This adversely affects pose estimation since it is a contradictory task that requires view-variant representations and not view-invariant representations like that required in categorization. Therefore initializing FC6/FC7 by another network trained for a different task is not likely to help. We show that learning the convolutional \ufb01lters jointly with categories help make them discriminative for both tasks and thus achieves a good accuracy on both tasks (see Fig. 3 to 9 and Tables 1, 2 and 3). 11 \fWorkshop track ICLR 2016 Comparing between EBM and LBM, we see that early branching is able to achieve a good balance between categorization and pose estimation by sharing the representations up to where we found the layer representation still capture pose information. We see this in Tables 1, 3 and Figures 1 to 5, where better pose accuracy and slightly better categorization accuracy is achieved by EBM. We also see that in Figure 5 that the object view-manifold collapses in the last two layers (one layer before LBM) and thus achieves better pose discrimination than LBM. The slight effect of decreasing the size of the layers in the pose subnetwork of EBM can be observed from the results in Table 1 and 3. 9"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1506.08529v1",
+                    "title": "Tell and Predict: Kernel Classifier Prediction for Unseen Visual Classes from Unstructured Text Descriptions",
+                    "abstract": "In this paper we propose a framework for predicting kernelized classifiers in\nthe visual domain for categories with no training images where the knowledge\ncomes from textual description about these categories. Through our optimization\nframework, the proposed approach is capable of embedding the class-level\nknowledge from the text domain as kernel classifiers in the visual domain. We\nalso proposed a distributional semantic kernel between text descriptions which\nis shown to be effective in our setting. The proposed framework is not\nrestricted to textual descriptions, and can also be applied to other forms\nknowledge representations. Our approach was applied for the challenging task of\nzero-shot learning of fine-grained categories from text descriptions of these\ncategories.",
+                    "authors": "Mohamed Elhoseiny, Ahmed Elgammal, Babak Saleh",
+                    "published": "2015-06-29",
+                    "updated": "2015-06-29",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction We propose a framework to model kernelized classi\ufb01er prediction in the visual domain for categories with no training images, where the knowledge about these categories comes from a secondary domain. The side information can be in the form of textual, parse trees, grammar, visual representations, concepts in the ontologies, or any form; see Fig 1. Our work focuses on the unstructured text setting. We denote the side information as \u201cprivileged\u201d information, borrowing the notion from (Vapnik and Vashist, 2009). Our framework is an instance of the concept of Zero Shot Learning (ZSL)(Larochelle et al., 2008), aiming at transferring knowledge from seen classes to novel (unseen) classes. Most zero-shot learning applications in practice use symbolic or numeric visual attribute vectors (Lampert et al., 2014; Lampert et al., 2009). In contrast, recent works investigated other forms of descripFigure 1: Our setting where machine can predict unseen class from pure unstructured text tions, e.g. user provided feedback (Wah and Belongie, 2013), textual descriptions (Elhoseiny et al., 2013). It is common in zero-shot learning to introduce an intermediate layer that facilitates knowledge sharing between seen classes, hence the transfer of knowledge to unseen classes. Typically, visual attributes are being used for that purpose, since they provide a human-understandable representation, which enables specifying new categories (Lampert et al., 2014; Lampert et al., 2009; Farhadi et al., 2009; Palatucci et al., 2009; Akata et al., 2013; Li et al., 2014). A fundamental question in attribute-based ZSL models is how to de\ufb01ne attributes that are visually discriminative and human understandable. Researchers has explored learning attributes from text sources, e.g. (Rohrbach, 2014; Rohrbach et al., 2013; Rohrbach et al., 2010; Berg et al., 2010). Other works have explored interactive methodologies to learning visual attribute that are human understandable, e.g. (Parikh and Grauman, 2011). There are several differences between our proposed framework and the state-of-the-art zero-shot learning approaches. We are not restricted to use attributes as the interface to specify new classes. We can use any \u201cprivileged\u201d information available for each category. In particular in this paper we focus on the case of textual description of categories as the secondary domain. This difference is re\ufb02ected in our zero-shot classi\ufb01cation architecarXiv:1506.08529v1  [cs.CV]  29 Jun 2015 \fture. We learn a domain transfer model between the visual domain and the privileged information domain. This facilitates predicting explicit visual classi\ufb01ers for novel unseen categories given their privileged information. The difference in architecture becomes clear if we consider, for the sake of argument, attributes as the secondary domain in our framework, although this is not the focus of the paper. In that case we do not need explicit attribute classi\ufb01ers to be learned as an intermediate layer as typically done in attribute-based ZSL e.g. (Lampert et al., 2009; Farhadi et al., 2009; Palatucci et al., 2009), instead the visual classi\ufb01er are directly learned from the attribute labels. The need to learn an intermediate attribute classi\ufb01er layer in most attribute-based zero-shot learning approaches dictates using strongly annotated data, where each image comes with attribute annotation, e.g. CUBirds dataset (Welinder et al., 2010). In contrast, we do not need image-annotation pairs, and privileged information is only assumed at the category level; hence we denote our approach weakly supervised. This also directly facilitates using continuous attributes in our case, and does not assume independent between attributes. Another fundamental difference in our case is that we predict explicit kernel classi\ufb01er in the form de\ufb01ned in the representer theorem (Sch\u00a8 olkopf et al., 2001), from privileged information. Explicit classi\ufb01er prediction means that the output of our framework is classi\ufb01er parameters for any new category given text description, which can applied to any test image to predict its class. Predicting classi\ufb01er in kernelized form opens the door for using any kind of side information about classes, as long as kernels can be de\ufb01ned on them. The image features also do not need to be in a vectorized format. Kernelized classi\ufb01ers also facilitates combining different types of features through a multikernel learning (MKL) paradigm, where the fusion of different features can be effectively achieved. We can summarize the features of our proposed framework, hence the contribution as follows: 1) Our framework explicitly predicts classi\ufb01ers; 2) The predicted classi\ufb01ers are kernelized; 3) The framework facilitates any type of \u201cside\u201d information to be used; 4) The approach requires the side information at the class level, not at the image level, hence, it needs only weak annotation. 5) We propose a distributional semantic kernel between text description of visual classes that we show its value in the experiments. The structure of the paper is as follows. Sec 2 describes the relation to existing literature. Sec 3 and 4 explains the learning setting and our formulation. Sec 5 presents the proposed distributional semantic kernel for text descriptions. Sec 6 shows our experimental results. 2 Related Work We already discussed the relation to the zero-shot learning literature in the Introduction section. In this section, we focus on the relations to other volumes of literature. There has been increasing interest recently in the intersection between Language and Computer Vision. Most of the work on this area is focused on generating textual description from images (Farhadi et al., 2010; Kulkarni et al., 2011; Ordonez et al., 2011; Yang et al., 2011; Mitchell et al., 2012). In contrast, we focus on generating visual classi\ufb01ers from textual description or other side information at the category level. There are few recent works that involved unannotated text to improve visual classi\ufb01cation or achieve zero-shot learning. In (Frome et al., 2013; Norouzi et al., 2014) and (Socher et al., 2013), word embedding language models (e.g. (Mikolov et al., 2013b)) was adopted to represent class names as vectors. Their framework is based on mapping images into the learned language mode then perform classi\ufb01cation in that space. In contrast, our framework maps the text information to a classi\ufb01er in the visual domain, i.e. the apposite direction of their approach. There are several advantages in mapping textual knowledge into the visual domain. To perform ZSL, approaches such as (Norouzi et al., 2014; Frome et al., 2013; Socher et al., 2013) only embed new classes by their category names. This has clear limitations when dealing with \ufb01ne-grained categories (such as different bird species). Most of \ufb01ne-grained category names does not exist in current semantic models. Even if they exist, they will end up close to each other in the learned language models since they typically share similar contexts. This limits the discriminative power of such language models. In fact our baseline experiment using these models performed as low as random when applied to \ufb01negrained category; described in Sec 6.4. Moreover, our framework directly can use large text description of novel categories. In contrast to (Norouzi et al., 2014; Frome et al., 2013; Socher et al., 2013) \fwhich required a vectorized representation of images, our framework facilitates non-linear classi\ufb01cation using kernels. In (Elhoseiny et al., 2013), an approach was proposed to predict linear classi\ufb01ers from textual description, based on a domain transfer optimization method proposed in (Kulis et al., 2011). Although both of these works are kernelized, a close look reveals that kernelization was mainly used to reduce the size of the domain transfer matrix and the computational cost. The resulting predicted classi\ufb01er in (Elhoseiny et al., 2013) is still a linear classi\ufb01er. In contrast, our proposed formulation predicts kernelized visual classi\ufb01ers directly from the domain transfer optimization, which is a more general case. This directly facilitates using classi\ufb01ers that fused multiple visual cues such as Multiple Kernel Learning (MKL). 3 Problem De\ufb01nition We consider a zero-shot multi-class classi\ufb01cation setting on domain X as follows. At training, besides the data points from X and the class labels, each class is associated with privileged information in a secondary domain E in particular, however not limited to, a textual description. We assume that each class yi \u2208Ysc(training/seen labels), is associated with privileged information ei \u2208E. While, our formulation allows multiple pieces of privileged information per class (e.g. multiple class-level textual descriptions), we will use one per class for simplicity. Hence, we denote the training as Dtrain = {Sx = {(xi, yi)}N, Se = {yj, ej}Nsc}, where xi \u2208X, yi \u2208Ysc, yj \u2208Ysc, and Nsc and N are the number of the seen classes and the training examples/images respectively. We assume that each of the domains is equipped with a kernel function corresponding to a reproducing kernel Hilbert space (RKHS). Let us denote the kernel for X by k(\u00b7, \u00b7) and the kernel for E by g(\u00b7, \u00b7). At the zero-shot time, only the privileged information ez\u2217 is available for each novel unseen class z\u2217; see Fig 1. The common approach for multi-class classi\ufb01cation is to learn a classi\ufb01er for each class against the remaining classes (i.e., one-vs-all). According to the generalized representer theorem (Sch\u00a8 olkopf et al., 2001), a minimizer of a regularized empirical risk function over an RKHS could be represented as a linear combination of kernels, evaluated on the training set. Adopting the representer theorem on classi\ufb01cation risk function, we de\ufb01ne a kernel-classi\ufb01er of class y as follows fy(x\u2217) = N X i=1 \u03b2i yk(x\u2217, xi) + b = \u03b2y Tk(x\u2217), (1) where x\u2217 \u2208 X is the test point, xi \u2208 Sx, k(x\u2217) = [k(x\u2217, x1), \u00b7 \u00b7 \u00b7 , k(x\u2217, xN), 1]T, \u03b2y = [\u03b21 y \u00b7 \u00b7 \u00b7 \u03b2N y , b]T. Having learned fy(x\u2217) for each class y (for example using SVM classi\ufb01er), the class label of the test point x\u2217can be predicted as y\u2217= arg max y fy(x\u2217) (2) It is clear that fy(x\u2217) could be learned for all classes with training data y \u2208Ysc = y1 \u00b7 \u00b7 \u00b7 yNsc, since there are examples Sx for the seen classes; we denote the kernel-classi\ufb01er parameters of the seen classes as Bsc = {\u03b2y}Nsc, \u2200y \u2208Ysc. However, it is not obvious how to predict fz\u2217(x\u2217) for a new unseen class z\u2217\u2208Yus = z1 \u00b7 \u00b7 \u00b7 zNus. Our main notion is to use the privileged information ez\u2217\u2208E, associated with unseen class z\u2217, and the training data Dtrain to directly predict the unseen kernel-classi\ufb01er parameters. Hence, the classi\ufb01er of z\u2217is a function of ek\u2217and Dtrain; i.e. fz\u2217(x\u2217) = \u03b2(ez\u2217, Dtrain)T \u00b7 k(x\u2217), (3) fz\u2217(x\u2217) could be used to classify new points that belong to an unseen class as follows: 1) one-vsall setting fz\u2217(x\u2217) \u22770 ; or 2) in a Multi-class prediction as in Eq 2. 4 Approach Prediction of \u03b2(ez\u2217, Dtrain), which we denote as \u03b2(ez\u2217) for simplicity, is decomposed into training (domain transfer) and prediction phases. 4.1 Domain Transfer During training, we \ufb01rstly learn Bsc as SVMkernel classi\ufb01ers based on Sx. Then, we learn a domain transfer function to transfer the privileged information e \u2208E to kernel-classi\ufb01er parameters \u03b2 \u2208RN+1 in X domain. We call this function \u03b2DA(e), which has the form of TTg(e), where g(e) = [g(e, e1) \u00b7 \u00b7 \u00b7 g(e, eNsc)]T; T is an Nsc \u00d7 N + 1 matrix, which transforms e to kernel classi\ufb01er parameters for the class e represents. We aim to learn T, such that g(e)TTk(x) > l if e and x correspond to the same class, g(e)TTk(x) < u otherwise. Here l controls similarity lower-bound if e and x correspond to same \fclass, and u controls similarity upper-bound if e and x belong to different classes. In our setting, the term TTg(ei) should act as a classi\ufb01er parameter for class i of the training data. Therefore, we introduce penalization constraints to our minimization function if TTg(ei) is distant from \u03b2i \u2208Bsc, where ei corresponds to the class that \u03b2i classi\ufb01es. Inspired by domain adaptation optimization methods (e.g. (Kulis et al., 2011)), we model our domain transfer function as follows T\u2217= arg min T L(T) = [1 2r(T) + \u03bb1 X k ck(GTK)+ \u03bb2 Nsc X i=1 \u2225\u03b2i \u2212TTg(ei)\u22252 (4) where, G is an Nsc \u00d7 Nsc symmetric matrix, such that both the ith row and the ith column are equal to g(ei), ei \u2208Se; K is an N + 1 \u00d7 N matrix, such that the ith column is equal to k(xi), xi \u2208Sx. ck\u2019s are loss functions over the constraints de\ufb01ned as ck(GTK)) = (max(0, (l \u22121T i GTK1j)))2 for same class pairs of index i and j, or = r \u00b7 (max(0, (1T i GTK1j \u2212u)))2 otherwise, where 1i is an Nsc \u00d7 1 vector with all zeros except at index i, 1j is an N \u00d7 1 vector with all zeros except at index j. This leads to ck(GTK) = max(0, (l \u2212 g(ei)TTk(xj)))2 for same class pairs of index i and j, or = r\u00b7(max(0, (g(ei)TTk(xj)\u2212u)))2 otherwise, where u > l, r = nd ns such that nd and ns are the number of pairs (i, j) of different classes and similar pairs respectively. Finally, we used a Frobenius norm regularizer for r(T). The objective function in Eq 4 controls the involvement of the constraints ck by the term multiplied by \u03bb1, which controls its importance; we call it Cl,u(T). While, the trained classi\ufb01ers penalty is captured by the term multiplied by \u03bb2; we call it C\u03b2(T). One important observation on C\u03b2(T) is that it reaches zero when T = G\u22121BT, where B = [\u03b21 \u00b7 \u00b7 \u00b7 \u03b2Nsc], since it could be rewritten as C\u03b2(T) = \u2225BT \u2212GT\u22252 F . One approach to minimize L(T) is gradientbased optimization using a quasi-Newton optimizer. Our gradient derivation of L(T) leads to the following form \u03b4L(T) \u03b4T = T+\u03bb1\u00b7 X i,j g(ei)k(xj)Tvij +r\u00b7\u03bb2\u00b7(G2T\u2212GB) (5) where vij = \u22122 \u00b7 max(0, (l \u2212g(ei)TTk(xj))) if i and j correspond to the same class, 2 \u00b7 max(0, (g(ei)TTk(xj) \u2212u) otherwise. Another approach to minimize L(T) is through alternating projection using Bregman algorithm (Censor and Zenios, 1997), in which T is updated with respect to a single constraint every iteration. 4.2 Classi\ufb01er Prediction We propose two ways to predict the kernelclassi\ufb01er. (1) Domain Transfer (DT) Prediction, (2) One-class-SVM adjusted DT Prediction. Domain Transfer (DT) Prediction: Construction of an unseen category is directly computed from our domain transfer model as follows \u02dc \u03b2DT (ez\u2217) = T\u2217Tg(ez\u2217) (6) One-class-SVM adjusted DT (SVM-DT) Prediction: In order to increase separability against seen classes, we adopted the inverse of the idea of the one class kernel-svm, whose main idea is to build a con\ufb01dence function that takes only positive examples of the class. Our setting is the opposite scenario; seen examples are negative examples of the unseen class. In order introduce our proposed adjustment method, we start by presenting the one-class SVM objective function. The Lagrangian dual of the one-class SVM (Evangelista et al., 2007) can be written as \u03b2\u2217 + =argmin \u03b2 \u0002 \u03b2TK \u2032\u03b2 \u2212\u03b2T a \u0003 s.t. : \u03b2T 1 = 1, 0 \u2264\u03b2i \u2264C; i = 1 \u00b7 \u00b7 \u00b7 N (7) where K \u2032 is an N \u00d7 N matrix, K \u2032(i, j) = k(xi, xj), \u2200xi, xj \u2208Sx (i.e. in the training data), a is an N \u00d7 1 vector, ai = k(xi, xi), C is a hyperparameter . It is straightforward to see that, if \u03b2 is aimed to be a negative decision function instead, the objective function becomes in the form \u03b2\u2217 \u2212=argmin \u03b2 \u0002 \u03b2TK \u2032\u03b2 + \u03b2T a \u0003 s.t. : \u03b2T 1 = \u22121, \u2212C \u2264\u03b2i \u22640; i = 1 \u00b7 \u00b7 \u00b7 N (8) While \u03b2\u2217 \u2212 = \u2212\u03b2\u2217 +, the objective function in Eq 8 of the one-negative class SVM inspires us with the idea to adjust the kernel-classi\ufb01er parameters to increase separability of the unseen kernelclassi\ufb01er against the points of the seen classes, which leads to the following objective function \u02c6 \u03b2(ez\u2217) =argmin \u03b2 \u0002 \u03b2TK \u2032\u03b2 \u2212\u03b6 \u02c6 \u03b2DT (ez\u2217)TK \u2032\u03b2 + \u03b2T a \u0003 s.t. : \u03b2T 1 = \u22121, \u02c6 \u03b2 T DT K \u2032\u03b2 > l, \u2212C \u2264\u03b2i \u22640; \u2200i C, \u03b6, l: hyper-parameters, (9) \fwhere \u02c6 \u03b2DT is the \ufb01rst N elements in \u02dc \u03b2DT \u2208 RN+1, 1 is an N \u00d7 1 vector of ones. The objective function, in Eq 9, pushes the classi\ufb01er of the unseen class to be highly correlated with the domain transfer prediction of the kernel classi\ufb01er, while putting the points of the seen classes as negative examples. It is not hard to see that Eq 9 is a quadratic program in \u03b2, which could be solved using any quadratic solver; we used IBM CPLEX. It is worth to mention that, the approach in (Elhoseiny et al., 2013) predicts linear classi\ufb01ers by solving an optimization problem of size N + dX + 1 variables (dX + 1 linear-classi\ufb01er parameters and N slack variables); a similar limitation can be found in (Frome et al., 2013; Socher et al., 2013). In contrast, our objective function in Eq 9 solves a quadratic program of only N variables, and predicts a kernel-classi\ufb01er instead, with fewer parameters. Hence, if very highdimensional features are used, they will not affect our optimization complexity. 5 Distributional Semantic (DS) Kernel for text descriptions When E domain is the space of text descriptions, we propose a distributional semantic kernel g(\u00b7, \u00b7) = gDS(\u00b7, \u00b7) to de\ufb01ne the similarity between two text descriptions . We start by distributional semantic models by (Mikolov et al., 2013c; Mikolov et al., 2013a) to represent the semantic manifold Ms, and a function vec(\u00b7) that maps a word to a K \u00d7 1 vector in Ms. The main assumption behind this class of distributional semantic model is that similar words share similar context. Mathematically speaking, these models learn a vector for each word wn, such that p(wn|(wn\u2212L, wn\u2212L+1, \u00b7 \u00b7 \u00b7 , wn+L\u22121, wn+L) is maximized over the training corpus, where 2\u00d7L is the context window size. Hence similarity between vec(wi) and vec(wj) is high if they cooccurred a lot in context of size 2 \u00d7 L in the training text-corpus. We normalize all the word vectors to length 1 under L2 norm, i.e., \u2225vec(\u00b7)\u22252 = 1. Let us assume a text description D that we represent by a set of triplets D = {(wl, fl, vec(wl)), l = 1 \u00b7 \u00b7 \u00b7 M}, where wl is a word that occurs in D with frequency fl and its corresponding word vector is vec(wl) in Ms. We drop the stop words from D. We de\ufb01ne F = [f1, \u00b7 \u00b7 \u00b7 , fM]T and V = [vec(w1), \u00b7 \u00b7 \u00b7 , vec(wM)]T, where F is an M \u00d7 1 vector of term frequencies and V is an M \u00d7 K matrix of the corresponding term vectors. Given two text descriptions Di and Dj which contains M1 and M2 terms respectively. We compute Fi (Mi \u00d7 1) and Vi (Mi \u00d7 K) for Di and Fj (Mj \u00d7 1) and Vj (Mj \u00d7 K) for Dj. Finally gDS(Di, Dj) is de\ufb01ned as gDS(Di, Dj) = FT i ViVT j Fj (10) One advantage of this similarity measure is that it captures semantically related terms. It is not hard to see that the standard Term Frequency (TF) similarity could be thought as a special case of this kernel where vec(wl)Tvec(wm) = 1 if wl = wm, 0 otherwise, i.e., different terms are orthogonal. However, in our case the word vectors are learnt through a distributional semantic model which makes semantically related terms have higher dot product (vec(wl)Tvec(wm)). 6 Experiments 6.1 Datasets and Evaluation Methodology We validated our approach in a \ufb01ne-grained setting using two datasets: 1) The UCSD-Birds dataset (Welinder et al., 2010), which consists of 6033 images of 200 classes. 2) The Oxford-Flower dataset (Nilsback and Zisserman, 2008), which consists of 8189 images of 102 \ufb02ower categories. Both datasets were amended with class-level text descriptions extracted from different encyclopedias which is the same descriptions used in (Elhoseiny et al., 2013); see samples in the supplementary materials. We split the datasets to 80% of the classes for training and 20% of the classes for testing, with cross validations. We report multiple metrics while evaluating and comparing our approach to the baselines, detailed as follows Multiclass Accuracy of Unseen classes (MAU): Under this metric, we aim to evaluate the performance of the unseen classi\ufb01ers against each others. Firstly, the classi\ufb01ers of all unseen categories are predicted. Then, an instance x\u2217is classi\ufb01ed to the class z\u2217\u2208Yus of maximum con\ufb01dence for x\u2217 of the predicted classi\ufb01ers; see Eq 2. AUC: In order to measure the discriminative ability of our predicted one-vs-all classi\ufb01er for each unseen class, against the seen classes, we report the area under the ROC curve. Since unseen class positive examples are few compared to negative examples, a large accuracy could be achieved even if all unseen points are incorrectly classi\ufb01ed. \fHence, AUC is a more consistent measure. In this metric, we use the predicted classi\ufb01er of an unseen class as a binary separator against the seen classes. This measure is computed for each predicted unseen classi\ufb01er and the average AUC is reported. This is the only measure addressed in (Elhoseiny et al., 2013) to evaluate the unseen classi\ufb01ers, which is limiting in our opinion. |Nsc|to |Nsc + 1|Recall: Under this metric, we aim to check how the learned classi\ufb01ers of the seen classes confuse the predicted classi\ufb01ers, when they are involved in a multi-class classi\ufb01cation problem of Nsc + 1classes. We use Eq 2 to predict label of an instance x\u2217, such that the unknown label y\u2217\u2208Ysc \u222alus, such that lus is the label of the unseen class. We compute the recall under this setting. This metric is computed for each predicted unseen classi\ufb01er and the average is reported. 6.2 Comparisons to Linear Classi\ufb01er Prediction We compared our proposed approach to (Elhoseiny et al., 2013), which predicts a linear classi\ufb01er for zero-shot learning from textual descriptions ( E space in our framework). The aspects of the comparison includes 1) whether the predicted kernelized classi\ufb01er outperforms the predicted linear classi\ufb01er 2) whether this behavior is consistent on multiple datasets. We performed the comparison on both Birds and Flower dataset. For these experiments, in our setting, domain X is the visual domain and domain E is the textual domain, i.e. , the goal is to predict classi\ufb01ers from pure textual description. We used the same features on the visual domain and the textual domains as (Elhoseiny et al., 2013). That is, for the visual domain, we used classeme features (Torresani et al., 2010), extracted from images of the Bird and the Flower datasets. Classeme is a 2569dimensional features, which correspond to con\ufb01dences of a set of one-vs-all classi\ufb01ers, pre-trained on images from the web, as explained in (Torresani et al., 2010), not related to either the Bird nor the Flower datasets. The rationale behind using these features in (Elhoseiny et al., 2013) was that they offer a semantic representation. For the textual domain, we used the same textual feature extracted by (Elhoseiny et al., 2013). In that work, tf-idf (Term-Frequency Inverted Document Frequency)(Salton and Buckley, 1988) features were extracted from the textual articles were used, folTable 1: Recall, MAU, and average AUC on three seen/unseen splits on Flower Dataset and a seen/unseen split on Birds dataset Recall-Flower improvement Recall-Birds improvement SVM-DT kernel-rbf 40.34% (+/1.2) % 44.05 % Linear Classi\ufb01er 31.33 (+/2.22)% 27.8 % 36.56 % 20.4 % MAU-Flower improvement MAU-Birds improvement SVM-DT kernel-rbf 9.1 (+/2.77) % 3.4 % DT kernel-rbf 6.64 (+/4.1) % 37.93 % 2.95 % 15.25 % Linear Classi\ufb01er 5.93 (+/1.48)% 54.36 % 2.62 % 29.77 % Domain Transfer 5.79 (+/2.59)% 58.46 % 2.47 % 37.65 % AUC-Flower improvement AUC-Birds improvement SVM-DT kernel-rbf 0.653 (+/0.009) 0.61 DT kernel-rbf 0.623 (+/0.01) % 4.7 % 0.57 7.02 % Linear Classi\ufb01er 0.658 (+/0.034) 0.7 % 0.62 -1.61% Domain Transfer 0.644 (+/0.008) 1.28 % 0.56 8.93% lowed by a CLSI (Zeimpekis and Gallopoulos, 2005) dimensionality reduction phase. We denote our DT prediction and one class SVM adjust DT prediction approaches as DTkernel and SVM-DT-kernel respectively. We compared against the linear classi\ufb01er prediction by (Elhoseiny et al., 2013). We also compared against the direct domain transfer (Kulis et al., 2011), which was applied as a baseline in (Elhoseiny et al., 2013) to predict linear classi\ufb01ers. In our kernel approaches, we used Gaussian rbfkernel as a similarity measure in E and X spaces (i.e. k(d, d\u2032) = exp(\u2212\u03bb||d \u2212d\u2032||)). Recall metric : The recall of our approach is 44.05% for Birds and 40.34% for Flower, while it is 36.56% for Birds and 31.33% for Flower using (Elhoseiny et al., 2013). This indicates that the predicted classi\ufb01er is less confused by the classi\ufb01ers of the seen compared with (Elhoseiny et al., 2013); see table 1 (top part) MAU metric: It is worth to mention that the multiclass accuracies for the trained seen classi\ufb01ers are 51.3% and 15.4% using the classeme features on Flower dataset and Birds dataset1, respectively. Table 1 (middle part) shows the average MAU metric over three seen/unseen splits for Flower dataset and one split on Birds dataset, respectively. Furthermore, the relative improvements of our SVM-DT-kernel approach is reported against the baselines. On Flower dataset, it is interesting to see that our approach achieved 9.1% MAU, 182% improvement over the random guess performance, by predicting the unseen classi\ufb01ers using just textual features as privileged information (i.e. E domain). We also achieved also 13.4%, 268% the random guess performance, in one of the splits (the 9.1% is the average over 3 seen/unseen 1Birds dataset is known to be a challenging dataset for \ufb01ne-grained, even when applied in a regular multiclass setting as it is clear from the 15.4% performance on seen classes \f10 20 30 40 50 60 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 AUC for different class in Flower dataset AUC Object Class Index Figure 2: AUC of the 62 unseen classi\ufb01ers the \ufb02ower data-sets over three different splits (bottom part) and their Top 10 ROC-curves (top part) splits). Similarity on Birds dataset, we achieved 3.4% MAU from text features, 132% the random guess performance (further improved to 224% in next experiments). AUC metric: Fig 2 (top part) shows the ROC curves for our approach on the best predicted unseen classes from the Flower dataset. Fig 2 (bottom part) shows the AUC for all the classes on Flower dataset (over three different splits). More results and \ufb01gures are attached in the supplementary materials. Table 1 (bottom part) shows the average AUC on the two datasets, compared to the baselines. Looking at table 1, we can notice that the proposed approach performs marginally similar to the baselines from AUC perspective. However, there is a clear improvement in MAU and Recall metrics. These results show the advantage of predicting classi\ufb01ers in kernel space. Furthermore, the table shows that our SVM-DT-kernel approach outperforms our DT-kernel model. This indicates the advantage of the class separation, which is adjusted by the SVM-DT-kernel model. More details on the hyper-parameter selection are attached in the supplementary materials. 6.3 Multiple Kernel Learning (MKL) Experiment This experiment shows the added value of proposing a kernelized zero-shot learning approach. We conducted an experiment where the \ufb01nal kernel on the visual domain is produced by Multiple Kernel Learning (Gonen and Alpaydin, 2011). For the visual domain, we extracted kernel descriptors for Table 2: MAU on a seen-unseen split-Birds Dataset (MKL) MAU improvement SVM-DT kernel-rbf (text) 4.10 % Linear Classi\ufb01er 2.74 % 49.6 % Birds dataset. Kernel descriptors provide a principled way to turn any pixel attribute to patchlevel features, and are able to generate rich features from various recognition cues. We speci\ufb01cally used four types of kernels introduced by (Bo et al., 2010) as follows: Gradient Match Kernels that captures image variation based on prede\ufb01ned kernels on image gradients. Color Match Kernel that describes patch appearance using two kernels on top of RGB and normalized RGB for regular images and intensity for grey images. These kernels capture image variation and visual apperances. For modeling the local shape, Local Binary Pattern kernels have been applied. We computed these kernel descriptors on local image patches with \ufb01xed size 16 x 16 sampled densely over a grid with step size 8 in a spatial pyramid setting with four layers. The dense features are vectorized using codebooks of size 1000. This process ended up with a 120,000 dimensional feature for each image (30,000 for each type). Having extracted the four types of descriptors, we compute an rbf kernel matrix for each type separately. We learn the bandwidth parameters for each rbf kernel by cross validation on the seen classes. Then, we generate a new kernel kmkl(d, d\u2032) = P4 i=1 wiki(d, d\u2032), such that wi is a weight assigned to each kernel. We learn these weights by applying Bucak\u2019s Multiple Kernel Learning algorithm (Bucak et al., 2010). Then, we applied our approach where the MKL-kernel is used in the visual domain and rbf kernel on the text TFIDF features. To compare our approach to (Elhoseiny et al., 2013) under this setting, we concatenated all kernel descriptors to end up with 120,000 dimensional feature vector in the visual domain. As highlighted in the approach Sec 4, the approach in (Elhoseiny et al., 2013) solves a quadratic program of N +dX +1variables for each unseen class. Due to the large dimensionality of data (dX = 120, 000), this is not tractable. To make this setting applicable, we reduced the dimensionality of the feature vector into 4000 using PCA. This highlights the bene\ufb01t of our approach since it does not depend on the dimensionality of the data. Table 2 shows MAU for our approach under this setting \fTable 3: MAU on a seen-unseen split-Birds Dataset (CNN features, text description) MAU improvement SVM-DT kernel (X-rbf, E-DS kernel) 5.35 % SVM-DT kernel (X-rbf, E-rbf on TFIDF) 4.20 % 27.3% Linear Classi\ufb01er (TFIDF text) 2.65 % 102.0% (Norouzi et al., 2014) 2.3% 132.6% against (Elhoseiny et al., 2013). The results show the bene\ufb01ts of having a kernel approach for zero shot learning where kernel methods are applied to improve the performance. 6.4 Multiple Representation Experiment and Distributional Semantic(DS) Kernel The aim of this experiment is to show that our approach also work on different representations of text and visual domain. In this experiment, we extracted Convolutional Neureal Network(CNN) image features for the Visual domain. We used caffe (Jia et al., 2014) implementation of (Krizhevsky et al., 2012). Then, we extracted the sixth activation feature of the CNN since we found it works the best on the standard classi\ufb01cation setting. We found this consistent with the results of (Donahue et al., 2014) over different CNN layers. While using TFIDF feature of text description and CNN features for images, we achieved 2.65% for the linear version and 4.2% for the rbf kernel on both text and images. We further improved the performance to 5.35% by using our proposed Distributional Semantic (DS) kernel in the text domain and rbf kernel for images. In this DS experiment, we used the distributional semantic model by (Mikolov et al., 2013c) trained on GoogleNews corpus (100 billion words) resulting in a vocabulary of size 3 million words, and word vectors of K = 300 dimensions. This experiment shows both the value of having a kernel version and also the value of the proposed kernel in our setting. We also applied the zero shot learning approach in (Norouzi et al., 2014) which performs worse in our settings; see Table 3. 6.5 Attributes Experiment We emphasis that our main goal is not attribute prediction. However, it was interesting for us to see the behavior of our method where side information comes from attributes instead of text. In contrast to attribute-based models, which fully utilize attribute information to build attribute classi\ufb01ers, our approach do not learn attribute classi\ufb01ers. In this experiment, our method uses only the \ufb01rst moment of information of the attributes Table 4: MAU on a seen-unseen split-Birds Dataset (Attributes) MAU improvement SVM-DT kernel-rbf 5.6 % DT kernel-rbf 4.03 % 32.7 % Lampert DAP 4.8 % 16.6 % (i.e. the average attribute vector). We decided to compare to an attribute-based approach from this perspective. In particular, we applied the (DAP) attribute-based model (Lampert et al., 2014; Lampert et al., 2009), widely adopted in many applications (e.g., (Liu et al., 2013; Rohrbach et al., 2011)), to the Birds dataset. Details weak attribute representation in E space are attached in the supplementary materials due to space. For visual domain X, we used classeme features in this experiment (like table 1 experiment) An interesting result is that our approach achieved 5.6% MAU (224% the random guess performance); see Table 4. In contrast, we get 4.8% multiclass accuracy using DAP approach (Lampert et al., 2014). In this setting, we also measured the Nsc to Nsc + 1 average recall. We found the recall measure is 76.7% for our SVM-DT-kernel, while it is 68.1% on the DAP approach, which re\ufb02ects better true positive rate (positive class is the unseen one). We \ufb01nd these results interesting, since we achieved it without learning any attribute classi\ufb01ers, as in (Lampert et al., 2014). When comparing the results of our approach using attributes (Table 4) vs. textual description (Table 1)2 as the privileged information used for prediction, it is clear that the attribute features gives better prediction. This support our hypothesis that the more meaningful the E domain, the better the performance on X domain. 7"
+                }
+            ],
+            "Fabio Cuzzolin": [
+                {
+                    "url": "http://arxiv.org/abs/2401.09435v1",
+                    "title": "Reasoning with random sets: An agenda for the future",
+                    "abstract": "In this paper, we discuss a potential agenda for future work in the theory of\nrandom sets and belief functions, touching upon a number of focal issues: the\ndevelopment of a fully-fledged theory of statistical reasoning with random\nsets, including the generalisation of logistic regression and of the classical\nlaws of probability; the further development of the geometric approach to\nuncertainty, to include general random sets, a wider range of uncertainty\nmeasures and alternative geometric representations; the application of this new\ntheory to high-impact areas such as climate change, machine learning and\nstatistical learning theory.",
+                    "authors": "Fabio Cuzzolin",
+                    "published": "2023-12-19",
+                    "updated": "2023-12-19",
+                    "primary_cat": "math.ST",
+                    "cats": [
+                        "math.ST",
+                        "cs.AI",
+                        "stat.ML",
+                        "stat.TH",
+                        "62A01, 62A86, 60A05, 60C05, 60D05, 60E05"
+                    ],
+                    "main_content": "Introduction The theory of belief functions [162, 67] is a modelling language for representing and combining elementary items of evidence, which do not necessarily come in the form of sharp statements, with the goal of maintaining a mathematical representation of an agent\u2019s beliefs about those aspects of the world which the agent is unable to predict with reasonable certainty. While arguably a more appropriate mathematical description of uncertainty than classical probability theory, for the reasons we have thoroughly explored in [50], the theory of evidence is relatively simple to understand and implement, and does not require one to abandon the notion of an event, as is the case, for instance, for Walley\u2019s imprecise probability theory [193]. It is grounded in the beautiful mathematics of random sets, and exhibits strong relationships with many other theories of uncertainty. As mathematical objects, belief functions have fascinating properties in terms of their geometry, algebra [207] and combinatorics. Despite initial concerns about the computational complexity of a naive implementation of the theory of evidence, evidential reasoning can actually be implemented on large sample spaces [156] and in situations involving the combination of numerous pieces of evidence [74]. Elementary items of evidence often induce simple belief functions, which can be combined very efficiently with complexity O(n + 1). We do not 1 arXiv:2401.09435v1  [math.ST]  19 Dec 2023 \fneed to assign mass to all subsets, but we do need to be allowed to do so when necessary (e.g., in the case of missing data). Most relevantly, the most plausible hypotheses can be found without actually computing the overall belief function [151]. At any rate, Monte Carlo approximations [197] can easily be implemented when an explicit result for the combination is required. Last but not least, local propagation schemes [170] allow for the parallelisation of belief function reasoning, similarly to what happens with Bayesian networks. Statistical evidence can be represented in belief function theory in several ways, for example: \u2022 using likelihood-based belief functions [162, 3], in a way that generalises both likelihood-based and Bayesian inference; \u2022 via Dempster\u2019s inference approach, which makes use of auxiliary variables in a fiducial setting, and its natural evolution, weak belief [139, 206]; \u2022 in the framework of the generalised Bayesian theorem, proposed by Smets [175, 76]. Decision-making strategies based on intervals of expected utilities can be formulated which allow more cautious decisions than those based on traditional approaches, and are able to explain the empirical aversion to second-order uncertainty highlighted in Ellsberg\u2019s paradox [83]. A straightforward extension of the theory, originally formulated for finite sample spaces, to continuous domains [195] can be achieved via the Borel interval representation initially put forward by Strat [181] and Smets, when the analysis is restricted to intervals of real values [186]. In the more general case of arbitrary subsets of the real domain, the theory of random sets [148, 145] is, in our view, the natural mathematical framework to adopt. Finally, an extensive array of estimation, classification and regression tools based on the theory of belief functions has already been developed [157, 84], and further contributions can be envisaged. 1.1 Open issues According to Dempster and Shafer themselves [202], although flourishing by some measures, belief function theory had still not addressed questions such as deciding whether bodies of evidence were independent, or what to do if they were dependent. They lamented the ongoing confusion and disagreement about how to interpret the theory, and its limited acceptance in the field of mathematical statistics, where it first began. In response, they proposed an agenda to move the theory forward, centred on the following three elements: \u2022 a richer understanding of the uses of probability, for they believed the theory was best regarded as a way of using probability [165, 68, 166]; \u2022 a deeper understanding of statistical modelling itself [75], which would go beyond a traditional statistical analysis which begins by specifying probabilities that are supposed to be known except for certain parameters; 2 \f\u2022 in-depth examples of sensible Dempster\u2013Shafer analyses of a variety of problems of real scientific and technological importance. As for the last point, interesting examples in which the use of belief functions provides sound and elegant solutions to real-life problems, essentially characterised by \u2018missing\u2019 information, had been given by Smets in [177]. These include classification problems in which the training set is such that the classes are only partially known, an information retrieval system handling interdocument relationships, the combination of data from sensors providing data about partially overlapping frames, and the determination of the number of sources in a multisensor environment. The correct epistemic interpretation of belief function theory certainly needs to be clarified once and for all. In [50] we have argued that belief measures should be seen as random variables for set-valued observations. In an interesting, although rather overlooked, recent publication [81], instead, Dubucs and Shafer highlighted two ways of interpreting numerical degrees of belief in terms of betting, in a manner that echoes Shafer and Vovk\u2019s game-theoretical interpretation of probability [164, 167]: (i) you can offer to bet at the odds defined by the degrees of belief, or (ii) you can make a judgement that a strategy for taking advantage of such betting offers will not multiply the capital it risks by a large factor. Both interpretations, the authors argue, can be applied to ordinary probabilities and used to justify updating by conditioning, whereas only the second can be applied to belief functions and used to justify Dempster\u2019s rule of combination. The most appropriate mechanism for evidence combination is also still being debated [196]. The reason is that the choice seems to depend on meta-information about the reliability and independence of the sources involved, which is hardly accessible. As argued in [50], working with intervals of belief functions [71] may be the way forward, as this acknowledges the meta-uncertainty about the nature of the sources generating the evidence. The same holds for conditioning [199, 205, 121], as opposed to combining, belief functions. 1.2 A research programme Thus, in this paper we outline a potentially interesting research agenda for the future development of random set and belief function theory. For obvious reasons, we will only be able to touch on a few of the most interesting avenues of investigation, and even those will not be discussed beyond a certain level of detail. We hope, however, that this brief survey will stimulate the reader to pursue some of these research directions and to contribute to the further maturation of the theory in the near future. Although random set theory as a mathematical formalism is quite well developed, thanks in particular to the work of Ilya Molchanov [145, 146], a theory of statistical inference with random sets is not yet in sight. Therefore, in Section 3 We briefly consider the following questions: \u2022 the introduction of the notions of lower and upper likelihoods (Section 3.1), in order to move beyond belief function inference approaches which take the classical likelihood function at face value; 3 \f\u2022 the formulation of a framework for (generalised) logistic regression [198] with belief functions, making use of such generalised lower and upper likelihoods (Section 3.2) and their decomposition properties; \u2022 the generalisation of the law of total probability to random sets (Section 3.3), starting with belief functions [59, 76]; \u2022 the generalisation of classical limit theorems (central limit theorem, law of large numbers) to the case of random sets (Section 3.4): this would allow, for instance, a rigorous definition of Gaussian random sets and belief functions alternative to that proposed by Molchanov [146]; \u2022 the introduction (somewhat despite Dempster\u2019s aversion to them) of parametric models based on random sets (Section 3.5.1), which would potentially allow us to perform robust hypothesis testing (Section 3.5.2), thus laying the foundations for a theory of frequentist inference with random sets [138] (Section 3.5); \u2022 the development of a theory of random variables and processes in which the underlying probability space is replaced by a random-set space (Section 3.6): in particular, this requires the generalisation of the notion of the Radon\u2013Nikodym derivative to belief measures. The geometric approach to uncertainty, proposed by the author in [50], is also open to a number of further developments (Section 4), including: \u2022 the geometry of combination rules other than Dempster\u2019s (Section 4.3), and the associated conditioning operators (4.4); \u2022 an exploration of the geometry of continuous extensions of belief functions, starting from the geometry of belief functions on Borel intervals, to later tackle the general random-set representation; \u2022 a geometric analysis of uncertainty measures beyond those presented in [50], including e.g. capacities [100] and gambles [13] (Section 4.5); \u2022 newer geometric representations (Section 4.6), based on (among others) isoperimeters of convex bodies or exterior algebras [204]. More speculatively, the possibility exists of conducting inference in a purely geometric fashion, by finding a common representation for both belief measures and the data that drive the inference. More theoretical advances are necessary, in our view, including for instance: \u2022 a set of prescriptions for reasoning with intervals of belief functions, as [50] this seems to be a natural way to avoid the tangled issue of choosing a combination rule and handling metadata; \u2022 the further development of machine learning tools based on belief and randomset theory (see [50], Chapter 5), demonstrating the potential of these methods to model the epistemic uncertainty induced by forcibly limited training sets. 4 \fLast but not least, future work will need to be directed towards tackling high-impact applications by means of random set/belief function theory (Section 5). Here we will briefly consider, in particular: \u2022 the possible creation of a framework for climate change predictions based on random sets (Section 5.1), able to overcome the limitations of existing (albeit neglected) Bayesian approaches; \u2022 the generalisation of max-entropy classifiers [149] and log-linear models to the case of belief measures (Section 5.2.3), as an example of the fusion of machine learning with uncertainty theory; \u2022 new robust foundations for machine learning itself (Section 5.3), obtained by generalising Vapnik\u2019s probably approximately correct (PAC) analysis [187, 189] to the case in which the training and test distributions, rather than coinciding, come from the same random set. 2 Belief functions We first recall the basic definitions of belief function theory. 2.1 Belief and plausibility measures Definition 1. A basic probability assignment (BPA) [6] over a finite domain \u0398 is a set function [70] m : 2\u0398 \u2192[0, 1] defined on the collection 2\u0398 of all subsets of \u0398 s.t.: m(\u2205) = 0, X A\u2282\u0398 m(A) = 1. The quantity m(A) is called the basic probability number or \u2018mass\u2019 [126, 125] assigned to A. The elements of the power set 2\u0398 associated with non-zero values of m are called the focal elements (FEs) of m. Definition 2. The belief function (BF) associated with a basic probability assignment m : 2\u0398 \u2192[0, 1] is the set function Bel : 2\u0398 \u2192[0, 1] defined as: Bel(A) = X B\u2286A m(B). (1) The domain \u0398 on which a belief function is defined is usually interpreted as the set of possible answers to a given problem, exactly one of which is the correct one. For each subset (\u2018event\u2019) A \u2282\u0398 the quantity Bel(A) takes on the meaning of degree of belief that the truth lies in A, and represents the total belief committed to a set of possible outcomes A by the available evidence m. Another mathematical expression of the evidence generating a belief function Bel is the upper probability or plausibility of an event A: Pl(A) . = 1\u2212Bel( \u00af A), as opposed 5 \fto its lower probability Bel(A) [43]. The corresponding plausibility function Pl : 2\u0398 \u2192[0, 1] can be expressed as: Pl(A) = X B\u2229A\u0338=\u2205 m(B) \u2265Bel(A). 2.2 Evidence combination The issue of combining the belief function representing our current knowledge state with a new one encoding new evidence is central in belief theory. After an initial proposal by Dempster, other aggregation operators have been proposed, based on different assumptions on the nature of the sources of evidence to combine. Definition 3. The orthogonal sum or Dempster\u2019s combination Bel1 \u2295Bel2 : 2\u0398 \u2192 [0, 1] of two belief functions Bel1 : 2\u0398 \u2192[0, 1], Bel2 : 2\u0398 \u2192[0, 1] defined on the same domain \u0398 is the unique BF on \u0398 with as focal elements all the non-empty intersections of FEs of Bel1 and Bel2, and basic probability assignment: m\u2295(A) = m\u2229(A) 1 \u2212m\u2229(\u2205), (2) where mi denotes the BPA of the input BF Beli, and: m\u2229(A) = X B\u2229C=A m1(B)m2(C). Rather than normalising (as in (2)), Smets\u2019 conjunctive rule leaves the conflicting mass m(\u2205) with the empty set: m \u2229 \u20dd(A) = \u001a m\u2229(A) \u2205\u2286A \u2286\u0398, m\u2229(\u2205) A = \u2205, (3) and is thus applicable to \u2018unnormalised\u2019 beliefs [174]. In Dempster\u2019s rule, consensus between two sources is expressed by the intersection of the supported events. When the union is taken to express consensus, we obtain the disjunctive rule of combination [122, 203]: m \u222a \u20dd(A) = X B\u222aC=A m1(B)m2(C). (4) This yields more cautious inferences than conjunctive rules, by producing belief functions that are less \u2018committed\u2019, i.e., have larger focal sets. Under disjunctive combination, Bel1 \u222a \u20ddBel2(A) = Bel1(A) \u00b7 Bel2(A), i.e., input belief values are simply multiplied. Such a \u2018dual\u2019 combination rule was mentioned by Kramosil as well in [122]. The disjunctive rule is called \u2018combination by union\u2019 by Yamada (2008) [203]. 6 \f2.2.1 Conditioning Belief functions can also be conditioned, rather than combined, whenever we are presented hard evidence of the form \u2018A is true\u2019 [20, 88, 112, 97, 69, 205, 111]. In particular, Dempster\u2019s combination naturally induces a conditioning operator. Given a conditioning event A \u2282\u0398, the \u2018logical\u2019 (or \u2018categorical\u2019, in Smets\u2019 terminology) belief function BelA such that m(A) = 1 is combined via Dempster\u2019s rule with the a-priori belief function Bel. The resulting BF Bel \u2295BelA is the conditional belief function given A a la Dempster, denoted by Bel\u2295(A|B). 2.3 Multivariate analysis In many applications, we need to express uncertain information about a number of distinct variables (e.g., X and Y ) taking values in different domains (\u0398X and \u0398Y , respectively). The reasoning process needs then to take place in the Cartesian product of the domains associated with each individual variable. Let then \u0398X and \u0398Y be two sample spaces associated with two distinct variables, and let mXY be a mass function on \u0398XY = \u0398X \u00d7 \u0398Y . The latter can be expressed in the coarser domain \u0398X by transferring each mass mXY (A) to the projection A \u2193\u0398X of A on \u0398X. We obtain a marginal mass function on \u0398X, denoted by: mXY \u2193X (B) . = X {A\u2286\u0398XY ,A\u2193\u0398X=B} mXY (A), \u2200B \u2286\u0398X. (5) Conversely, a mass function mX on \u0398X can be expressed in \u0398X \u00d7 \u0398Y by transferring each mass mX(B) to the cylindrical extension B\u2191XY . = B \u00d7 \u2126Y of B. The vacuous extension of mX onto \u0398X \u00d7 \u0398Y will then be: m\u2191XY X (A) . = ( mX(B) if A = B \u00d7 \u2126Y , 0 otherwise. (6) The associated BF is denoted by Bel\u2191XY X . 2.4 Refinings An even more general treatment of belief functions over different but related sample spaces is provided by the notion of refining. Definition 4. Given two frames of discernment \u0398 and \u2126, a map \u03c1 : 2\u0398 \u21922\u2126is said to be a refining if it satisfies the following conditions: 1. \u03c1({\u03b8}) \u0338= \u2205\u2200\u03b8 \u2208\u0398. 2. \u03c1({\u03b8}) \u2229\u03c1({\u03b8\u2032}) = \u2205if \u03b8 \u0338= \u03b8\u2032. 3. \u222a\u03b8\u2208\u0398\u03c1({\u03b8}) = \u2126. The outer reduction associated with a refining \u03c1 is the map \u00af \u03c1 : 2\u2126\u21922\u0398 given by \u00af \u03c1(A) = n \u03b8 \u2208\u0398 \f \f \f\u03c1({\u03b8}) \u2229A \u0338= \u2205 o . (7) 7 \f3 A statistical random set theory 3.1 Lower and upper likelihoods The traditional likelihood function is a conditional probability of the data given a parameter \u03b8 \u2208\u0398, i.e., a family of PDFs over the measurement space X parameterised by \u03b8. Most of the work on belief function inference just takes the notion of a likelihood as a given, and constructs belief functions from an input likelihood function (see [50], Chapter 4). However, there is no reason why we should not formally define a \u2018belief likelihood function\u2019 mapping a sample observation x \u2208X to a real number, rather than use the conventional likelihood to construct belief measures. It is natural to define such a belief likelihood function as a family of belief functions on X, BelX(.|\u03b8), parameterised by \u03b8 \u2208\u0398. An expert in belief theory will note that such a parameterised family is the input to Smets\u2019s generalised Bayesian theorem [175], a collection of \u2018conditional\u2019 belief functions. Such a belief likelihood takes values on sets of outcomes, A \u2282\u0398, of which individual outcomes are just a special case. This seems to provide a natural setting for computing likelihoods of set-valued observations, in accordance with the random-set philosophy. 3.1.1 Belief likelihood function of repeated trials Let BelXi(A|\u03b8), for i = 1, 2, . . . , n be a parameterised family of belief functions on Xi, the space of quantities that can be observed at time i, depending on a parameter \u03b8 \u2208\u0398. A series of repeated trials then assumes values in X1\u00d7\u00b7 \u00b7 \u00b7\u00d7Xn, whose elements are tuples of the form \u20d7 x = (x1, . . . , xn) \u2208X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn. We call such tuples \u2018sharp\u2019 samples, as opposed to arbitrary subsets A \u2282X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn of the space of trials. Note that we are not assuming the trials to be equally distributed at this stage, nor do we assume that they come from the same sample space. Definition 5. The belief likelihood function BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn : 2X1\u00d7\u00b7\u00b7\u00b7\u00d7Xn \u2192[0, 1] of a series of repeated trials is defined as BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A|\u03b8) . = Bel\u2191\u00d7iXi X1 \u2299\u00b7 \u00b7 \u00b7 \u2299Bel\u2191\u00d7iXi Xn (A|\u03b8), (8) where Bel\u2191\u00d7iXi Xj is the vacuous extension (6) of BelXj to the Cartesian product X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn where the observed tuples live, A \u2282X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn is an arbitrary subset of series of trials and \u2299is an arbitrary combination rule. In particular, when the subset A reduces to a sharp sample, A = {\u20d7 x}, we can define the following generalisations of the notion of a likelihood. Definition 6. We call the quantities L(\u20d7 x) . = BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}|\u03b8), L(\u20d7 x) . = PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}|\u03b8) (9) the lower likelihood and upper likelihood, respectively, of the sharp sample A = {\u20d7 x} = {(x1, . . . , xn)}. 8 \f3.1.2 Binary trials: The conjunctive case Belief likelihoods factorise into simple products whenever conjunctive combination is employed (as a generalisation of classical stochastic independence) in Definition 5, and the case of trials with binary outcomes is considered. Focal elements of the belief likelihood Let us first analyse the case n = 2. We seek the Dempster sum BelX1 \u2295BelX2, where X1 = X2 = {T, F}. Figure 1(a) is a diagram of all the intersections of focal elements of the two input belief functions on X1 \u00d7 X2. There are 9 = 32 distinct non-empty intersections, which (a) (b) Figure 1: (a) Graphical representation of the Dempster combination BelX1 \u2295BelX2 on X1 \u00d7X2, in the binary case in which X1 = X2 = {T, F}. (b) Graphical representation of the disjunctive combination BelX1 \u222a \u20ddBelX2 there. correspond to the focal elements of BelX1 \u2295BelX2. According to (2), the mass of the focal element A1 \u00d7 A2, A1 \u2286X1, A2 \u2286X2, is then mBelX1\u2295BelX2 (A1 \u00d7 A2) = mX1(A1) \u00b7 mX2(A2). (10) Note that this result holds when the conjunctive rule (3) \u2229 \u20ddis used as well, for none of the intersections are empty, and hence no normalisation is required. Nothing is assumed about the mass assignment of the two belief functions BelX1 and BelX2. We can now prove the following lemma. Lemma 1. For any n \u2208Z, the belief function BelX1 \u2295\u00b7 \u00b7 \u00b7 \u2295BelXn, where Xi = X = {T, F}, has 3n focal elements, namely all possible Cartesian products A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An of n non-empty subsets Ai of X, with BPA mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An) = n Y i=1 mXi(Ai). 9 \fProof. The proof is by induction. The thesis was shown to be true for n = 2 in (10). In the induction step, we assume that the thesis is true for n, and prove it for n + 1. If BelX1 \u2295\u00b7 \u00b7 \u00b7 \u2295BelXn, defined on X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, has as focal elements the nproducts A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An with Ai \u2208 \b {T}, {F}, X \t for all i, its vacuous extension to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 Xn+1 will have as focal elements the n + 1-products of the form A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An \u00d7 Xn+1, with Ai \u2208 \b {T}, {F}, X \t for all i. The belief function BelXn+1 is defined on Xn+1 = X, with three focal elements {T}, {F} and X = {T, F}. Its vacuous extension to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 Xn+1 thus has the following three focal elements: X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 {T}, X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 {F} and X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 Xn+1. When computing (BelX1 \u2295\u00b7 \u00b7 \u00b7 \u2295BelXn) \u2295BelXn+1 on the common refinement X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 Xn+1, we need to compute the intersection of their focal elements, namely \u0000A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An \u00d7 Xn+1 \u0001 \u2229 \u0000X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 An+1 \u0001 = A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An \u00d7 An+1 for all Ai \u2286Xi, i = 1, . . . , n + 1. All such intersections are distinct for distinct focal elements of the two belief functions to be combined, and there are no empty intersections. By Dempster\u2019s rule (2), their mass is equal to the product of the original masses: mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn+1 (A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An \u00d7 An+1) = mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An) \u00b7 mBelXn+1 (An+1). Since we have assumed that the factorisation holds for n, the thesis easily follows. As no normalisation is involved in the combination BelX1 \u2295\u00b7 \u00b7 \u00b7 \u2295BelXn, Dempster\u2019s rule coincides with the conjunctive rule, and Lemma 1 holds for \u2229 \u20ddas well. Factorisation for \u2018sharp\u2019 tuples The following then becomes a simple corollary. Theorem 1. When either \u2229 \u20ddor \u2295is used as a combination rule in the definition of the belief likelihood function, the following decomposition holds for tuples (x1, . . . , xn), xi \u2208Xi, which are the singleton elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, with X1 = \u00b7 \u00b7 \u00b7 = Xn = {T, F}: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}|\u03b8) = n Y i=1 BelXi({xi}|\u03b8). (11) Proof. For the singleton elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, since {(x1, . . . , xn)} = {x1} \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 {xn}, (11) becomes BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}) = mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn ({(x1, . . . , xn)}) = n Y i=1 mXi({xi}) = n Y i=1 BelXi({xi}), where the mass factorisation follows from Lemma 1, as on singletons mass and belief values coincide. 10 \fThere is evidence to support the following conjecture as well. Conjecture 1. When either \u2229 \u20ddor \u2295is used as a combination rule in the definition of the belief likelihood function, the following decomposition holds for the associated plausibility values on tuples (x1, . . . , xn), xi \u2208Xi, which are the singleton elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, with X1 = . . . = Xn = {T, F}: PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}|\u03b8) = n Y i=1 PlXi({xi}|\u03b8). (12) In fact, we can write PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}) = 1 \u2212BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}c) = 1 \u2212 X B\u2286{(x1,...,xn)}c mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (B). (13) By Lemma 1, all the subsets B with non-zero mass are Cartesian products of the form A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An, \u2205\u0338= Ai \u2286Xi. We then need to understand the nature of the focal elements of BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn which are subsets of an arbitrary singleton complement {(x1, . . . , xn)}c. For binary spaces Xi = X = {T, F}, by the definition of the Cartesian product, each such A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An \u2286{(x1, . . . , xn)}c is obtained by replacing a number 1 \u2264 k \u2264n of components of the tuple (x1, . . . , xn) = {x1} \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 {xn} with a different subset of Xi (either {xi}c = Xi \\{xi} or Xi). There are \u0000n k \u0001 such sets of k components in a list of n. Of these k components, in general 1 \u2264m \u2264k will be replaced by {xi}c, while the other 1 \u2264k \u2212m < k will be replaced by Xi. Note that not all k components can be replaced by Xi, since the resulting focal element would contain the tuple {(x1, . . . , xn)} \u2208X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn. The following argument can be proved for (x1, ..., xn) = (T, ..., T), under the additional assumption that BelX1 \u00b7 \u00b7 \u00b7 BelXn are equally distributed with p . = BelXi({T}), q . = BelXi({F}) and r . = BelXi(Xi). If this is the case, for fixed values of m and k all the resulting focal elements have the same mass value, namely pn\u2212kqmrk\u2212m. As there are exactly \u0000 k m \u0001 such focal elements, (13) can be written as 1 \u2212 n X k=1 \u0012n k \u0013 k X m=1 \u0012 k m \u0013 pn\u2212kqmrk\u2212m, which can be rewritten as 1 \u2212 n X m=1 qm n X k=m \u0012n k \u0013\u0012 k m \u0013 pn\u2212krk\u2212m. A change of variable l = n\u2212k, where l = 0 when k = n, and l = n\u2212m when k = m, allows us to write this as 1 \u2212 n X m=1 qm n\u2212m X l=0 \u0012 n n \u2212l \u0013\u0012n \u2212l m \u0013 plr(n\u2212m)\u2212l, 11 \fsince k \u2212m = n \u2212l \u2212m, k = n \u2212l. Now, as \u0012 n n \u2212l \u0013\u0012n \u2212l m \u0013 = \u0012n m \u0013\u0012n \u2212m l \u0013 , we obtain 1 \u2212 n X m=1 qm \u0012n m \u0013 n\u2212m X l=0 \u0012n \u2212m l \u0013 plr(n\u2212m)\u2212l. By Newton\u2019s binomial theorem, the latter is equal to 1 \u2212 n X m=1 qm \u0012n m \u0013 (p + r)n\u2212m = 1 \u2212 n X m=1 qm \u0012n m \u0013 (1 \u2212q)n\u2212m, since p + r = 1 \u2212q. As Pn m=0 \u0000 n m \u0001 qm(1 \u2212q)n\u2212m = 1, again by Newton\u2019s binomial theorem, we get PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(T, . . . , T)}) = 1 \u2212[1 \u2212(1 \u2212q)n] = (1 \u2212q)n = n Y i=1 PlXi({T}). Question 1. Does the decomposition (13) hold for any sharp sample and arbitrary belief functions BelX1, . . . , BelXn defined on arbitrary frames of discernment X1, . . . , Xn? Factorisation for Cartesian products The decomposition (11) is equivalent to what Smets calls conditional conjunctive independence in his general Bayes theorem. In fact, for binary spaces the factorisation (11) generalises to all subsets A \u2286X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn which are Cartesian products of subsets of X1, . . . , Xn, respectively: A = A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An, Ai \u2286Xi for all i. Corollary 1. Whenever Ai \u2286Xi, i = 1, . . . , n, and Xi = X = {T, F} for all i, under conjunctive combination we have that BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An|\u03b8) = n Y i=1 BelXi(Ai|\u03b8). (14) Proof. As, by Lemma 1, all the focal elements of BelX1 \u2229 \u20dd\u00b7 \u00b7 \u00b7 \u2229 \u20ddBelXn are Cartesian products of the form B = B1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Bn, Bi \u2286Xi, BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An|\u03b8) = X B\u2286A1\u00d7\u00b7\u00b7\u00b7\u00d7An B=B1\u00d7\u00b7\u00b7\u00b7\u00d7Bn mX1(B1) \u00b7 . . . \u00b7 mXn(Bn). But {B \u2286A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An, B = B1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Bn} = {B = B1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Bn, Bi \u2286Ai\u2200i}, since, if Bj \u0338\u2282Aj for some j, the resulting Cartesian product would not be contained within A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An. Therefore, BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An|\u03b8) = X B=B1\u00d7\u00b7\u00b7\u00b7\u00d7Bn Bi\u2286Ai\u2200i mX1(B1) \u00b7 . . . \u00b7 mXn(Bn). 12 \fFor all those Ai\u2019s, i = i1, . . . , im that are singletons of Xi, necessarily Bi = Ai and we can write the above expression as m(Ai1) \u00b7 . . . \u00b7 m(Aim) X Bj\u2286Aj,j\u0338=i1,...,im Y j\u0338=i1,...,im mXj(Bj). If the frames involved are binary, Xi = {T, F}, those Ai\u2019s that are not singletons coincide with Xi, so that we have m(Ai1) \u00b7 . . . \u00b7 m(Aim) X Bj\u2286Xj,j\u0338=i1,...,im Y j mXj(Bj). The quantity X Bj\u2286Xj,j\u0338=i1,...,im Y j mXj(Bj) is, according to the definition of the conjunctive combination, the sum of the masses of all the possible intersections of (cylindrical extensions of) focal elements of BelXj, j \u0338= i1, . . . , im. Thus, it adds up to 1. In conclusion (forgetting the conditioning on \u03b8 in the derivation for the sake of readability), BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An) = m(Ai1) \u00b7 . . . \u00b7 m(Aim) \u00b7 1 \u00b7 . . . \u00b7 1 = BelXi1 (Ai1) \u00b7 . . . \u00b7 BelXim (Aim) \u00b7 BelXj1 (Xj1) \u00b7 . . . \u00b7 BelXjk (Xjk) = BelXi1 (Ai1) \u00b7 . . . \u00b7 BelXim (Aim) \u00b7 BelXj1 (Aj1) \u00b7 . . . \u00b7 BelXjk (Ajk), and we have (14). Corollary 1 states that conditional conjunctive independence always holds, for events that are Cartesian products, whenever the frames of discernment concerned are binary. 3.1.3 Binary trials: The disjunctive case Similar factorisation results hold when the (more cautious) disjunctive combination \u222a \u20dd is used. As in the conjunctive case, we first analyse the case n = 2. Structure of the focal elements We seek the disjunctive combination BelX1 \u222a \u20ddBelX2, where each BelXi has as focal elements {T}, {F} and Xi. Figure 1(b) is a diagram of all the unions of focal elements of the two input belief functions on their common refinement X1 \u00d7 X2. There are 5 = 22 + 1 distinct such unions, which correspond to the focal elements of BelX1 \u222a \u20ddBelX2, with mass values mBelX1\u2295BelX2 ({(xi, xj)}c) = mX1({xi}c) \u00b7 mX2({xj}c), mBelX1\u2295BelX2 (X1 \u00d7 X2) = 1 \u2212 X i,j mX1({xi}c) \u00b7 mX2({xj}c). We can now prove the following lemma. 13 \fLemma 2. The belief function BelX1 \u222a \u20dd\u00b7 \u00b7 \u00b7 \u222a \u20ddBelXn, where Xi = X = {T, F}, has 2n + 1 focal elements, namely all the complements of the n-tuples \u20d7 x = (x1, . . . , xn) of singleton elements xi \u2208Xi, with BPA mBelX1 \u222a \u20dd\u00b7\u00b7\u00b7 \u222a \u20ddBelXn ({(x1, . . . , xn)}c) = mX1({x1}c) \u00b7 . . . \u00b7 mXn({xn}c), (15) plus the Cartesian product X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn itself, with its mass value given by normalisation. Proof. The proof is by induction. The case n = 2 was proven above. In the induction step, we assume that the thesis is true for n, namely that the focal elements of BelX1 \u222a \u20dd\u00b7 \u00b7 \u00b7 \u222a \u20ddBelXn have the form A = n (x1, . . . , xn)}c = {(x\u2032 1, . . . , x\u2032 n) \f \f \f\u2203i : {x\u2032 i} = {xi}co , (16) where xi \u2208Xi = X. We need to prove it is true for n + 1. The vacuous extension of (16) has, trivially, the form A\u2032 = n (x\u2032 1, . . . , x\u2032 n, xn+1) \f \f \f\u2203i \u2208{1, . . . , n} : {x\u2032 i} = {xi}c, xn+1 \u2208X o . Note that only 2 = |X| singletons of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 are not in A\u2032, for any given tuple (x1, . . . , xn). The vacuous extension to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 of a focal element B = {xn+1} of BelXn+1 is instead B\u2032 = {(y1, . . . , yn, xn+1), yi \u2208X \u2200i = 1, . . . , n}. Now, all the elements of B\u2032, except for (x1, . . . , xn, xn+1), are also elements of A\u2032. Hence, the union A\u2032 \u222aB\u2032 reduces to the union of A\u2032 and (x1, . . . , xn, xn+1). The only singleton element of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 not in A\u2032 \u222aB\u2032 is therefore (x1, . . . , xn, x\u2032 n+1), {x\u2032 n+1} = {xn+1}c, for it is neither in A\u2032 nor in B\u2032. All such unions are distinct, and therefore, by the definition of disjunctive combination, have mass m({(x1, . . . , xn)}c)\u00b7 m({xn+1}c), which by the inductive hypothesis is equal to (15). Unions involving either Xn+1 or X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn are equal to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 by the property of the union operator. Factorisation Theorem 2. In the hypotheses of Lemma 2, when the disjunctive combination \u222a \u20ddis used in the definition of the belief likelihood function, the following decomposition holds: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}c|\u03b8) = n Y i=1 BelXi({xi}c|\u03b8). (17) Proof. Since {(x1, . . . , xn)}c contains only itself as a focal element, BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}c|\u03b8) = m({(x1, . . . , xn)}c|\u03b8). 14 \fBy Lemma 2, the latter becomes BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}c|\u03b8) = n Y i=1 mXi({xi}c|\u03b8) = n Y i=1 BelXi({xi}c|\u03b8), as {xi}c is a singleton element of Xi, and we have (17). Note that PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}c|\u03b8) = 1 for all tuples (x1, . . . , xn), as the set {(x1, . . . , xn)}c has non-empty intersection with all the focal elements of the belief function BelX1 \u222a \u20dd\u00b7 \u00b7 \u00b7 \u222a \u20ddBelXn. 3.1.4 General factorisation results A look at the proof of Lemma 1 shows that the argument is in fact valid for the conjunctive combination of belief functions defined on an arbitrary collection X1, . . . , Xn of finite spaces. Namely, we have the following theorem. Theorem 3. For any n \u2208Z, the belief function BelX1\u2295\u00b7 \u00b7 \u00b7\u2295BelXn, where X1, . . . , Xn are finite spaces, has as focal elements all the Cartesian products A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An of n focal elements A1 \u2286X1, . . . , An \u2286Xn, with BPA mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An) = n Y i=1 mXi(Ai). The proof is left to the reader. The corollary below follows. Corollary 2. When either \u2229 \u20ddor \u2295is used as a combination rule in the definition of the belief likelihood function, the following decomposition holds for tuples (x1, . . . , xn), xi \u2208Xi, which are the singleton elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, with X1, . . . , Xn being arbitrary discrete frames of discernment: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}|\u03b8) = n Y i=1 BelXi({xi}|\u03b8). (18) Proof. For the singleton elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, since {(x1, . . . , xn)} = {x1} \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 {xn}, (18) becomes BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, . . . , xn)}) = mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn ({(x1, ..., xn)}) = n Y i=1 mXi({xi}) = n Y i=1 BelXi({xi}), where the mass factorisation follows from Theorem 3, as mass and belief values coincide on singletons. This new approach to inference with belief functions, based on the notion of the belief likelihood, opens up a number of interesting avenues: from a more systematic comparison with other inference approaches, to the possible generalisation of the factorisation results obtained above, and to the computation of belief and plausibility likelihoods for other major combination rules. 15 \f3.1.5 Lower and upper likelihoods of Bernoulli trials In the case of Bernoulli trials, where not only is there a single binary sample space Xi = X = {T, F} and conditional independence holds, but also the random variables are assumed to be equally distributed, the conventional likelihood reads as pk(1 \u2212p)n\u2212k, where p = P(T), k is the number of successes (T) and n is as usual the total number of trials. Let us then compute the lower likelihood function for a series of Bernoulli trials, under the similar assumption that all the belief functions BelXi = BelX, i = 1, . . . , n, coincide (the analogous of equidistribution), with BelX parameterised by p = m({T}), q = m({F}) (where, this time, p + q \u22641). Corollary 3. Under the above assumptions, the lower and upper likelihoods of the sample \u20d7 x = (x1, . . . , xn) are, respectively, L(\u20d7 x) = n Y i=1 BelX({xi}) = pkqn\u2212k, L(\u20d7 x) = n Y i=1 PlX({xi}) = (1 \u2212q)k(1 \u2212p)n\u2212k. (19) The above decomposition for L(\u20d7 x), in particular, is valid under the assumption that Conjecture 1 holds, at least for Bernoulli trials, as the evidence seems to suggest. After normalisation, these lower and upper likelihoods can be seen as PDFs over the (belief) space B of all belief functions definable on X [33, 59]. Having observed a series of trials \u20d7 x = (x1, . . . , xn), xi \u2208{T, F}, one may then seek the belief function on X = {T, F} which best describes the observed sample, i.e., the optimal values of the two parameters p and q. In the next subsection we will address a similar problem in a generalised logistic regression setting. Example 1.Lower and upper likelihoods. Figure 2 plots the lower and upper likelihoods (19) for the case of k = 6 successes in n = 10 trials. Both subsume the traditional likelihood pk(1\u2212p)n\u2212k as their section for p+q = 1, although this is particularly visible for the lower likelihood (a). In particular, the maximum of the lower likelihood is the traditional maximum likelihood estimate p = k/n, q = 1 \u2212p. This makes sense, for the lower likelihood is highest for the most committed belief functions (i.e., for probability measures). The upper likelihood (b) has a unique maximum in p = q = 0: this is the vacuous belief function on {T, F}, with m({T, F}) = 1. The interval of belief functions joining max L with max L is the set of belief functions such that p q = k n\u2212k, i.e., those which preserve the ratio between the observed empirical counts. 3.2 Generalised logistic regression 3.2.1 Logistic regression Logistic regression allows us, given a sample Y = {Y1, ..., Yn}, X = {x1, ..., xn} where Yi \u2208{0, 1} is a binary outcome at time i and xi is the corresponding observed 16 \fp  q 6 4 (1\u0001\u0002p)  (1\u0001\u0002p) 6 4 (a) (b) Figure 2: Lower (a) and upper (b) likelihood functions plotted in the space of belief functions defined on the frame X = {T, F}, parameterised by p = m(T) (X axis) and q = m(F) (Y axis), for the case of k = 6 successes in n = 10 trials. measurement, to learn the parameters of a conditional probability relation between the two, of the form: P(Y = 1|x) = 1 1 + e\u2212(\u03b20+\u03b21x) , (20) where \u03b20 and \u03b21 are two scalar parameters. Given a new observation x, (20) delivers the probability of a positive outcome Y = 1. Logistic regression generalises deterministic linear regression, as it is a function of the linear combination \u03b20 + \u03b21x. The n trials are assumed independent but not equally distributed, for \u03c0i = P(Yi = 1|xi) varies with the time instant i of collection. The two scalar parameters \u03b20, \u03b21 in (20) are estimated by maximum likelihood of the sample. After denoting by \u03c0i = P(Yi = 1|xi) = 1 1 + e\u2212(\u03b20+\u03b21xi) , 1 \u2212\u03c0i = P(Yi = 0|xi) = e\u2212(\u03b20+\u03b21xi) 1 + e\u2212(\u03b20+\u03b21xi) (21) the conditional probabilities of the two outcomes, the likelihood of the sample can be expressed as: L(\u03b2|Y ) = Qn i=1 \u03c0Yi i (1 \u2212\u03c0i)Yi, where Yi \u2208{0, 1} and \u03c0i is a function of \u03b2 = [\u03b20, \u03b21]. Maximising L(\u03b2|Y ) yields a conditional PDF P(Y = 1|x). Unfortunately, logistic regression shows clear limitations when the number of samples is insufficient or when there are too few positive outcomes (1s) [119]. Moreover, inference by logistic regression tends to underestimate the probability of a positive outcome [119]. 3.2.2 Generalised formulation Based on Theorem 1 and Conjecture 1, we can also generalise logistic regression to a belief function setting by replacing the conditional probability (\u03c0i, 1 \u2212\u03c0i) on X = {T, F} with a belief function (pi = m({T}), qi = m({F}), pi + qi \u22641) on 2X. 17 \fNote that, just as in a traditional logistic regression setting (Section 3.2.1), the belief functions Beli associated with different input values xi are not equally distributed. Writing T = 1, F = 0, the lower and upper likelihoods can be computed as L(\u03b2|Y ) = n Y i=1 pYi i q1\u2212Yi i , L(\u03b2|Y ) = n Y i=1 (1 \u2212qi)Yi(1 \u2212pi)1\u2212Yi. The question becomes how to generalise the logit link between observations x and outputs y, in order to seek an analytical mapping between observations and belief functions over a binary frame. Just assuming pi = m(Yi = 1|xi) = 1 1 + e\u2212(\u03b20+\u03b21xi) , (22) as in the classical constraint (20), does not yield any analytical dependency for qi. To address this issue, we can just (for instance) add a parameter \u03b22 such that the following relationship holds: qi = m(Yi = 0|xi) = \u03b22 e\u2212(\u03b20+\u03b21xi) 1 + e\u2212(\u03b20+\u03b21xi) . (23) We can then seek lower and upper optimal estimates for the parameter vector \u03b2 = [\u03b20, \u03b21, \u03b22]\u2032: arg max \u03b2 L 7\u2192\u03b20, \u03b21, \u03b22, arg max \u03b2 L 7\u2192\u03b20, \u03b21, \u03b22. (24) Plugging these optimal parameters into (22), (23) will then yield an upper and a lower family of conditional belief functions given x (i.e., an interval of belief functions), BelX(.|\u03b2, x), BelX(.|\u03b2, x). Given any new test observation x\u2032, our generalised logistic regression method will then output a pair of lower and upper belief functions on X = {T, F} = {1, 0}, as opposed to a sharp probability value as in the classical framework. As each belief function itself provides a lower and an upper probability value for each event, both the lower and the upper regressed BFs will provide an interval for the probability P(T) of success, whose width reflects the uncertainty encoded by the training set of sample series. 3.2.3 Optimisation problem Both of the problems (24) are constrained optimisation problems (unlike the classical case, where qi = 1 \u2212pi and the optimisation problem is unconstrained). In fact, the parameter vector \u03b2 must be such that 0 \u2264pi + qi \u22641 \u2200i = 1, . . . , n. Fortunately, the number of constraints can be reduced by noticing that, under the analytical relations (22) and (23), pi + qi \u22641 for all i whenever \u03b22 \u22641. 18 \fThe objective function can be simplified by taking the logarithm. In the lowerlikelihood case, we get1 log L(\u03b2|Y ) = n X i=1 n \u2212log(1 + e\u2212(\u03b20+\u03b21xi)) + (1 \u2212Yi) \u0002 log \u03b22 \u2212(\u03b20 + \u03b21xi) \u0003o . We then need to analyse the Karush\u2013Kuhn\u2013Tucker (KKT) necessary conditions for the optimality of the solution of a nonlinear optimisation problem subject to differentiable constraints. Definition 7. Suppose that the objective function f : Rn \u2192R and the constraint functions gi : Rn \u2192R and hj : Rn \u2192R of a nonlinear optimisation problem arg max x f(x), subject to gi(x) \u22640 i = 1, . . . , m, hj(x) = 0 j = 1, . . . , l, are continuously differentiable at a point x\u2217. If x\u2217is a local optimum, under some regularity conditions there then exist constants \u00b5i (i = 1, . . . , m) and \u03bbj (j = 1, . . . , l), called KKT multipliers, such that the following conditions hold: 1. Stationarity: \u2207f(x\u2217) = Pm i=1 \u00b5i\u2207gi(x\u2217) + Pl j=1 \u03bbj\u2207hj(x\u2217). 2. Primal feasibility: gi(x\u2217) \u22640 for all i = 1, . . . , m, and hj(x\u2217) = 0, for all j = 1, . . . , l. 3. Dual feasibility: \u00b5i \u22650 for all i = 1, . . . , m. 4. Complementary slackness: \u00b5igi(x\u2217) = 0 for all i = 1, . . . , m. For the optimisation problem considered here, we have only inequality constraints, namely \u03b22 \u22641 \u2261 g0 = \u03b22 \u22121 \u22640, pi + qi \u22650 \u2261 gi = \u2212\u03b22 \u2212e\u03b20+\u03b21xi \u22640, i = 1, . . . , n. The Lagrangian becomes \u039b(\u03b2) = log L(\u03b2) + \u00b50(\u03b22 \u22121) \u2212 n X i=1 \u00b5i(\u03b22 + e\u03b20+\u03b21xi). The stationarity conditions thus read as \u2207\u039b(\u03b2) = 0, namely \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 n X i=1 h (1 \u2212pi) \u2212(1 \u2212Yi) \u2212\u00b5ie\u03b20+\u03b21xii = 0, n X i=1 h (1 \u2212pi) \u2212(1 \u2212Yi) \u2212\u00b5ie\u03b20+\u03b21xii xi = 0, n X i=1 \u00121 \u2212Yi \u03b22 \u2212\u00b5i \u0013 + \u00b50 = 0, (25) 1Derivations are omitted. 19 \fwhere, as usual, pi is as in (22). The complementary slackness condition reads as follows: \u001a \u00b50(\u03b22 \u22121) = 0, \u00b5i(\u03b22 + e\u03b20+\u03b21xi) = 0, i = 1, . . . , n. (26) Gradient descent methods can then be applied to the above systems of equations to find the optimal parameters of our generalised logistic regressor. Question 2. Is there an analytical solution to the generalised logistic regression inference problem, based on the above KKT necessity conditions? Similar calculations hold for the upper-likelihood problem. A multi-objective optimisation setting in which the lower likelihood is minimised as the upper likelihood is maximised can also be envisaged, in order to generate the most cautious interval of estimates. 3.2.4 Research questions As far as generalised logistic regression is concerned, an analysis of the validity of such a direct extension of logit mapping and the exploration of alternative ways of generalising it are research questions that are potentially very interesting to pursue. Question 3. What other parameterisations can be envisaged for the generalised logistic regression problem, in terms of the logistic links between the data and the belief functions Beli on {T, F}? Comprehensive testing of the robustness of the generalised framework on standard estimation problems, including rare-event analysis [89], needs to be conducted to further validate this new procedure. Note that the method, unlike traditional ones, can naturally cope with missing data (represented by vacuous observations xi = X), therefore providing a robust framework for logistic regression which can deal with incomplete series of observations. 3.3 The total probability theorem for random sets Spies and others have posed the problem of generalising the law of total probability, P(A) = N X i=1 P(A|Bi)P(Bi), where {B1, . . . , BN} is a disjoint partition of the sample space, to the case of belief functions. They mostly did so with the aim of generalising Jeffrey\u2019s combination rule [163] \u2013 nevertheless, the question goes rather beyond their original intentions, as it involves understanding the space of solutions to the generalised total probability problem. 20 \f3.3.1 The law of total belief The problem of generalising the total probability theorem to belief functions can be posed as follows (Fig. 3) [208]. Theorem 4. (Total belief theorem). Suppose \u0398 and \u2126are two frames of discernment, and \u03c1 : 2\u2126\u21922\u0398 the unique refining between them. Let Bel0 be a belief function defined over \u2126= {\u03c91, . . . , \u03c9|\u2126|}. Suppose there exists a collection of belief functions Beli : 2\u03a0i \u2192[0, 1], where \u03a0 = {\u03a01, . . . , \u03a0|\u2126|}, \u03a0i = \u03c1({\u03c9i}), is the partition of \u0398 induced by its coarsening \u2126. Then, there exists a belief function Bel : 2\u0398 \u2192[0, 1] such that: \u2022 (P1) Bel0 is the marginal of Bel to \u2126, Bel0 = Bel \u21be\u2126; \u2022 (P2) Bel \u2295Bel\u03a0i = Beli for all i = 1, . . . , |\u2126|, where Bel\u03a0i is the categorical belief function with BPA m\u03a0i(\u03a0i) = 1, m\u03a0i(B) = 0 for all B \u0338= \u03a0i. Note that a disjoint partition \u03a01, . . . , \u03a0|\u2126| of \u0398 defines a subalgebra A\u03c1 of 2\u0398 as a Boolean algebra with set operations, which is isomorphic to the set algebra A = 2\u2126. We use the notation Bel \u21be\u2126to denote the marginal of Bel to \u2126. \u03a9 \u03c9 \u03c1 Bel   = 2     [0,1] \u03a9 0 Bel  = 2       [0,1] \u03a0 i i i \u03a0\u0001\u0001\u0002\u0001\u03c1\u0003\u0004\u03c9\u0001\u0005\u0006 i i \u0398 Figure 3: Pictorial representation of the hypotheses of the total belief theorem (Theorem 4). 3.3.2 Proof of existence Theorem 4\u2019s proof makes use of the two lemmas that follow. Here, basically, we adapt Smets\u2019s original proof of the generalised Bayesian theorem [175] to the refinement framework presented here, which is more general than his multivariate setting, in which only Cartesian products of frames are considered. As usual, we denote by m0 and mi 21 \fthe mass functions of Bel0 and Beli, respectively. Their sets of focal elements are denoted by E\u2126and Ei, respectively. Suppose that \u0398\u2032 \u2287\u0398 and m is a mass function over \u0398. The mass function m can be identified with a mass function \u2212 \u2192 m\u0398\u2032 over the larger frame \u0398\u2032: for any E\u2032 \u2286\u0398\u2032: \u2212 \u2192 m\u0398\u2032(E\u2032) = \u001a m(E) E\u2032 = E \u222a(\u0398\u2032 \\ \u0398) 0 otherwise. Such an \u2212 \u2192 m\u0398\u2032 is called the conditional embedding of m into \u0398\u2032. When the context is clear, we can drop the subscript \u0398\u2032. It is easy to see that conditional embedding is the inverse of Dempster conditioning. For the collection of belief functions Beli : 2\u03a0i \u2192[0, 1], let \u2212 \u2212 \u2192 Beli be the conditional embedding of Beli into \u0398 and let \u2212 \u2212 \u2192 Bel denote the Dempster combination of all \u2212 \u2212 \u2192 Beli, i.e., \u2212 \u2212 \u2192 Bel = \u2212 \u2212 \u2192 Bel1 \u2295\u00b7 \u00b7 \u00b7 \u2295\u2212 \u2212 \u2212 \u2212 \u2192 Bel|\u2126|, with mass function \u2212 \u2192 m. Lemma 3. The belief function \u2212 \u2212 \u2192 Bel over \u0398 satisfies the following two properties: (1) each focal element \u2212 \u2192 e of \u2212 \u2212 \u2192 Bel is the union of exactly one focal element ei of each conditional belief function Beli; (2) the marginal \u2212 \u2212 \u2192 Bel \u21be\u2126on \u2126is the vacuous belief function over \u2126. Proof. Each focal element \u2212 \u2192 ei of \u2212 \u2212 \u2192 Beli is of the form (S j\u0338=i \u03a0j) \u222aei, where ei is some focal element of Beli. In other words, \u2212 \u2192 ei = (\u0398 \\ \u03a0i) \u222aei. Since \u2212 \u2212 \u2192 Bel is the Dempster combination of the \u2212 \u2212 \u2192 Belis, it is easy to see that each focal element \u2212 \u2192 e of \u2212 \u2212 \u2192 Bel is the union of exactly one focal element ei from each conditional belief function Beli. In other words, \u2212 \u2192 e = S|\u2126| i=1 ei, where ei \u2208Ei, and condition (1) is proven. Let \u2212 \u2192 E denote the set of all the focal elements of \u2212 \u2212 \u2192 Bel, namely \u2212 \u2192 E = n\u2212 \u2192 e \u2286\u0398 : \u2212 \u2192 e = S|\u2126| i=1 ei where ei is a focal element of Beli o . Note that ei\u2019s coming from different conditional belief functions Beli\u2019s are disjoint. For each \u2212 \u2192 e \u2208\u2212 \u2192 E , \u00af \u03c1(\u2212 \u2192 e ) = \u2126(where \u00af \u03c1 denotes the outer reduction (7)). It follows from (5) that \u2212 \u2192 m \u21be\u2126(\u2126) = 1 \u2013 hence the marginal of \u2212 \u2212 \u2192 Bel on \u2126is the vacuous belief function there. Let Bel\u2191\u0398 0 be the vacuous extension of Bel0 from \u2126to \u0398. We define the desired total function Bel to be the Dempster combination of Bel\u2191\u0398 0 and \u2212 \u2212 \u2192 Bel, namely Bel . = Bel\u2191\u0398 0 \u2295\u2212 \u2212 \u2192 Bel. (27) Lemma 4. The belief function Bel defined in (27) on the frame \u0398 satisfies the following two properties: 1. Bel\u2295Bel\u03a0i = Beli for all i = 1, . . . , |\u2126|, where Bel\u03a0i is again the categorical belief function focused on \u03a0i. 2. Bel0 is the marginal of Bel on \u2126, i.e., Bel0 = Bel \u21be\u2126. 22 \fThat is, Bel is a valid total belief function. Proof. Let \u2212 \u2192 m and mi be the mass functions corresponding to \u2212 \u2212 \u2192 Bel and Beli, respectively. For each \u2212 \u2192 e = S|\u2126| i=1 ei \u2208\u2212 \u2192 E , where ei \u2208Ei, we have that \u2212 \u2192 m(\u2212 \u2192 e ) = |\u2126| Y i=1 mi(ei). Let E\u2191\u0398 \u2126 denote the set of focal elements of Bel\u2191\u0398 0 . Since Bel\u2191\u0398 0 is the vacuous extension of Bel0, E\u2191\u0398 \u2126 = {\u03c1(e\u2126) : e\u2126\u2208E\u2126}. Each element of E\u2191\u0398 \u2126 is actually the union of some equivalence classes \u03a0i of the partition \u03a0. Since each focal element of Bel\u2191\u0398 0 has non-empty intersection with all the focal elements \u2212 \u2192 e \u2208\u2212 \u2192 E , it follows that X e\u2126\u2208E\u2126,\u2212 \u2192 e \u2208\u2212 \u2192 E ,\u03c1(e\u2126)\u2229\u2212 \u2192 e \u0338=\u2205 m\u2191\u0398 0 (\u03c1(e\u2126))\u2212 \u2192 m(\u2212 \u2192 e ) = 1. (28) Thus, the normalisation factor in the combination Bel\u2191\u0398 0 \u2295\u2212 \u2212 \u2192 Bel is equal to 1. Now, let E denote the set of focal elements of the belief function Bel = Bel\u2191\u0398 0 \u2295 \u2212 \u2212 \u2192 Bel. By the Dempster sum (2), each element e of E is the union of focal elements of some conditional belief functions Beli, i.e., e = ej1 \u222aej2 \u222a\u00b7 \u00b7 \u00b7 \u222aejK for some K such that {j1, . . . , jK} \u2286{1, . . . , |\u2126|} and ejl is a focal element of Beljl, 1 \u2264l \u2264K. Let m denote the mass function for Bel. For each such e \u2208E, e = \u03c1(e\u2126) \u2229\u2212 \u2192 e for some e\u2126\u2208E\u2126and \u2212 \u2192 e \u2208\u2212 \u2192 E , so that e\u2126= \u00af \u03c1(e). Thus we have (m\u2191\u0398 0 \u2295\u2212 \u2192 m)(e) = X e\u2126\u2208E\u2126,\u2212 \u2192 e \u2208\u2212 \u2192 E ,\u03c1(e\u2126)\u2229\u2212 \u2192 e =e m\u2191\u0398 0 (\u03c1(e\u2126))\u2212 \u2192 m(\u2212 \u2192 e ) = X e\u2126\u2208E\u2126,\u2212 \u2192 e \u2208\u2212 \u2192 E \u03c1(e\u2126)\u2229\u2212 \u2192 e =e m0(e\u2126)\u2212 \u2192 m(\u2212 \u2192 e ) = m0(\u00af \u03c1(e)) X \u2212 \u2192 e \u2208\u2212 \u2192 E ,\u03c1(\u00af \u03c1(e))\u2229\u2212 \u2192 e =e \u2212 \u2192 m(\u2212 \u2192 e ) = m0(\u00af \u03c1(e)) \u00b7 mj1(ej1) \u00b7 \u00b7 \u00b7 mjK(ejK) Y j\u0338\u2208{j1,...,jK} X e\u2208Ej mj(e) = m0(\u00af \u03c1(e)) K Y k=1 mjk(ejk), (29) as \u2212 \u2192 m(\u2212 \u2192 e ) = Qn i=1 mi(ei) whenever \u2212 \u2192 e = \u222an i=1ei. Without loss of generality, we will consider the conditional mass function m(e1|\u03a01), where e1 is a focal element of Bel1 and \u03a01 is the first partition class associated with the partition \u03a0, and show that m(e1|\u03a01) = m1(e1). In order to obtain m(e1|\u03a01), which is equal to X e\u2208E,e\u2229\u03a01=e1 m(e)/Pl(\u03a01), 23 \fin the following we separately compute P e\u2208E,e\u2229\u03a01=e1 m(e) and Pl(\u03a01). For any e \u2208E, if e \u2229\u03a01 \u0338= \u2205, \u00af \u03c1(e) is a subset of \u2126including \u03c91. Therefore, Pl(\u03a01) = X e\u2208E,e\u2229\u03a01\u0338=\u2205 m(e) = X C\u2286{\u03a02,...,\u03a0|\u2126|} \uf8eb \uf8ed X \u03c1(\u00af \u03c1(e))=\u03a01\u222a(S E\u2208C E) m(e) \uf8f6 \uf8f8 = X C\u2286{\u03a02,...,\u03a0|\u2126|} m\u2191\u0398 0   \u03a01 \u222a [ E\u2208C E !  X e1\u2208E1 m1(e1) Y \u03a0l\u2208C X el\u2208El ml(el) ! = X C\u2286{\u03a02,...,\u03a0|\u2126|} m\u2191\u0398 0   \u03a01 \u222a [ E\u2208C E ! = X e\u2126\u2208E\u2126,\u03c91\u2208e\u2126 m0(e\u2126) = Pl0({\u03c91}). (30) Similarly, X e\u2208E e\u2229\u03a01=e1 m(e) = X C\u2286{\u03a02,...,\u03a0|\u2126|} X \u03c1(\u00af \u03c1(e))=S E\u2208C E m(e1 \u222ae) = m1(e1) X C\u2286{\u03a02,...,\u03a0|\u2126|} m\u2191\u0398 0   \u03a01 \u222a [ E\u2208C E ! Y \u03a0l\u2208C X el\u2208El ml(el) = m1(e1) X C\u2286{\u03a02,...,\u03a0|\u2126|} m\u2191\u0398 0   \u03a01 \u222a [ E\u2208C E ! = m1(e1) X e\u2126\u2208E\u2126,\u03c91\u2208e\u2126 m0(e\u2126) = m1(e1)Pl0({\u03c91}). (31) From (30) and (31), it follows that m(e1|\u03a01) = P e\u2208E,e\u2229\u03a01=e1 m(e) Pl(\u03a01) = m1(e1). This proves property 1. Proving 2 is much easier. For any e\u2126. = {\u03c9j1, . . . , \u03c9jK} \u2208E\u2126, m \u21be\u2126(e\u2126) = X \u03c1(e)=e\u2126 m(e) = m\u2191\u0398 0 (\u03c1(e\u2126)) K Y l=1 X e\u2208Ejl mjl(e) = m\u2191\u0398 0 (\u03c1(e\u2126)) = m0(e\u2126). (32) It follows that Bel \u21be\u2126= Bel0, and hence the thesis. The proof of Theorem 4 immediately follows from Lemmas 3 and 4. Example 2. Suppose that the coarsening \u2126:= {\u03c91, \u03c92, \u03c93} considered induces a partition \u03a0 of \u0398: {\u03a01, \u03a02, \u03a03}. Suppose also that the conditional belief function 24 \fBel1 considered, defined on \u03a01, has two focal elements e1 1 and e2 1, that the conditional belief function Bel2 defined on \u03a02 has a single focal element e1 2 and that Bel3, defined on \u03a03, has two focal elements e1 3 and e2 3 (see Fig. 4). According to Lemma 3, the \u0398 \u03a01 e1 2 e1 1 e2 1 e3 1 e3 2 \u03a03 \u03a02 Figure 4: The conditional belief functions considered in our case study. The set-theoretical relations between their focal elements are immaterial to the solution. Dempster combination \u2212 \u2212 \u2192 Bel of the conditional embeddings of the Beli\u2019s has four focal elements, listed below: e1 = e1 1 \u222ae1 2 \u222ae1 3, e2 = e1 1 \u222ae1 2 \u222ae2 3, e3 = e2 1 \u222ae1 2 \u222ae1 3, e4 = e2 1 \u222ae1 2 \u222ae2 3. (33) The four focal total elements can be represented as \u2018elastic bands\u2019 as in Fig. 5. e1 2 e1 1 e2 1 e3 1 e3 2 e1 e2 e3 e4 Figure 5: Graphical representation of the four possible focal elements (33) of the total belief function (27) in our case study. 25 \fWithout loss of generality, we assume that a prior Bel0 on \u2126has each subset of \u2126as a focal element, i.e., E\u2126= 2\u2126. It follows that each focal element e of the total belief function Bel . = \u2212 \u2212 \u2192 Bel \u2295Bel\u2191\u0398 0 is the union of some focal elements from different conditional belief functions Beli. So, the set E of the focal elements of Bel is E = {e = [ 1\u2264i\u2264I ei : 1 \u2264I \u22643, ei \u2208Ei} and is the union of the following three sets: EI=1 : = E1 \u222aE2 \u222aE3, EI=2 : = {e \u222ae\u2032 : (e, e\u2032) \u2208Ei \u00d7 Ej, 1 \u2264i, j \u22643, i \u0338= j}, EI=3 : = {e1 \u222ae2 \u222ae3 : ei \u2208Ei, 1 \u2264i \u22643}. Hence |E| = 5 + 8 + 4 = 17. According to (29), it is very easy to compute the corresponding total mass function m. For example, for the two focal elements e1 1 \u222ae2 3 and e1 1 \u222ae1 2 \u222ae2 3, we have m(e1 1 \u222ae2 3) = m0({w1, w3})m1(e1 1)m3(e2 3), m(e1 1 \u222ae1 2 \u222ae2 3) = m0(\u2126)m1(e1 1)m2(e1 2)m3(e2 3). 3.3.3 Number of solutions The total belief function Bel obtained in Theorem 4 is not the only solution to the total belief problem (note that the Dempster combination (27) is itself unique, as the Dempster sum is unique). Assume that Bel\u2217is a total belief function posssessing the two properties in Theorem 4. Let m\u2217and E\u2217denote its mass function and the set of its focal elements, respectively. Without loss of generality, we may still assume that the prior Bel0 has every subset of \u2126as its focal element, i.e., E\u2126= 2\u2126. From property (P2), Bel\u2217\u2295Bel\u03a0i = Beli, 1 \u2264i \u2264|\u2126|, we derive that each focal element of Bel\u2217must be a union of focal elements of a number of conditional belief functions Beli. For, if e\u2217is a focal element of Bel\u2217and e\u2217= el \u222ae\u2032, where \u2205\u0338= el \u2286\u03a0l and e\u2032 \u2286\u0398 \\ \u03a0l for some 1 \u2264l \u2264|\u2126|, then ml(el) = (m\u2217\u2295m\u03a0l)(el) > 0 and hence el \u2208El. So we must have that E\u2217\u2286E, where E is the set of focal elements of the total belief function Bel (27) obtained in Theorem 4, E = \uf8f1 \uf8f2 \uf8f3 [ j\u2208J ej : J \u2286{1, . . . , |\u2126|}, ej \u2208Ej \uf8fc \uf8fd \uf8fe. In order to find Bel\u2217(or m\u2217), we need to solve a group of linear equations which correspond to the constraints dictated by the two properties, in which the mass m\u2217(e) of each focal element e \u2208E of the total solution (27) is treated as an unknown variable. There are |E| variables in the group. 26 \fFrom properties (P1) and (P2), we know that Pl0(\u03c9i) = Pl\u2217(\u03a0i), 1 \u2264i \u2264|\u2126|, where Pl0 and Pl\u2217are the plausibility functions associated with Bel0 and Bel\u2217, respectively. In addition, property (P1) implies the system of linear constraints \uf8f1 \uf8f2 \uf8f3 X e\u2229\u03a0i=ei,e\u2208E m\u2217(e) = mi(ei)Pl0(\u03c9i), \u2200i = 1, . . . , n, \u2200ei \u2208Ei. (34) The total number of such equations is P|\u2126| j=1 |Ej|. Since, for each 1 \u2264i \u2264|\u2126|, P e\u2208Ei mi(e) = 1, the system (34) includes a group of P|\u2126| j=1 |Ej| \u2212|\u2126| independent linear equations, which we denote as G1. From property (P2) (the marginal of Bel\u2217on \u2126is Bel0), we have the following constraints: \uf8f1 \uf8f2 \uf8f3 X e\u2208E,\u03c1(e)=C m\u2217(e) = m0(C), \u2200\u2205\u0338= C \u2286\u2126. (35) The total number of linear equations in (35) is the number of non-empty subsets of \u2126. Since P C\u2286\u2126m0(C) = 1, there is a subset of |2\u2126| \u22122 independent linear equations in (35), denoted by G2. As the groups of constraints G1 and G2 are independent, their union G := G1 \u222aG2 completely specifies properties (P1) and (P2) in Theorem 4. Since |G1| = P|\u2126| j=1 |Ej|\u2212 |\u2126| and |G2| = |2\u2126| \u22122, the cardinality of the union group G is |G| = P|\u2126| j=1 |Ej| \u2212 |\u2126| + |2\u2126| \u22122. From Theorem 4, we know that the system of equations G is solvable and has at least one positive solution (in which each variable has a positive value). This implies that |E| \u2265|G|, i.e., the number of variables must be no less than that of the independent linear equations in G. If |E| > |G|, in particular, we can apply the Fourier\u2013Motzkin elimination method [117] to show that G has another distinct positive solution Bel\u2217 (i.e., such that m\u2217(e) \u22650 \u2200e \u2208E). Example 3. We employ Example 2 to illustrate the whole process of finding such an alternative total belief function Bel\u2217. We assume further that m0 and mi, 1 \u2264i \u22643, take values m1(e1 1) = 1 2 = m1(e2 1), m2(e1 2) = 1, m3(e1 3) = 1 3, m3(e2 3) = 2 3, m0({\u03c91}) = m0({\u03c92}) = m0({\u03c93}) = 1 16, m0({\u03c91, \u03c92}) = 2 16, m0({\u03c92, \u03c93}) = 4 16, m0({\u03c91, \u03c93}) = 3 16, m0(\u2126) = 1 4. If we follow the process prescribed above to translate the two properties into a group G of linear equations, we obtain 17 unknown variables m\u2217(e), e \u2208E (|E| = 17) and eight independent linear equations (|G| = 8). From Theorem 4, we can construct a 27 \fpositive solution m defined according to (29). For this example, m({e1 1, e1 2, e1 3}) = m0(\u2126) m1(e1 1) m2(e1 2) m3(e1 3) = 1 24, m({e2 1, e1 2, e2 3}) = m0(\u2126) m1(e2 1) m2(e1 2) m3(e2 3) = 1 12. When solving the equation group G via the Fourier\u2013Motzkin elimination method, we choose m\u2217({e1 1, e1 2, e1 3}) and m\u2217({e2 1, e1 2, e2 3}) to be the last two variables to be eliminated. Moreover, there is a sufficiently small positive number \u03f5 such that m\u2217({e1 1, e1 2, e1 3}) = 1 24 \u2212\u03f5 > 0, m\u2217({e2 1, e1 2, e2 3}) = 1 12 + \u03f5, and all other variables also take positive values. It is easy to see that the m\u2217so obtained is different from the m obtained via Theorem 4. Clearly, this new solution has the same focal elements as the original one: other solutions may exist with a different set of focal elements (as can indeed be empirically shown). We will discuss this issue further in a couple of pages. 3.3.4 Case of Bayesian prior Note that when the prior belief function Bel0 is Bayesian, however, the total belief function obtained according to (29) is the unique one satisfying the two properties in Theorem 4. Corollary 4. For the belief function Bel0 over \u2126and the conditional belief functions Beli over \u03a0i in Theorem 4, if Bel0 is Bayesian (i.e., a probability measure), then there exists a unique total belief function Bel : 2\u0398 \u2192[0, 1] such that (P1) and (P2) are satisfied. Moreover, the total mass function m of Bel is m(e) = \u001a mi(e) m0(\u03c9i) if e \u2208Ei for some i, 0 otherwise. Proof. It is easy to check that the total mass function m defined above satisfies the two properties. Now we need to show that it is unique. Since Bel0 is Bayesian, E\u2126\u2286 {{\u03c91}, . . . , {\u03c9|\u2126|}}. In other words, all focal elements of Bel0 are singletons. It follows that E\u2191\u0398 \u2126 \u2286{\u03a01, . . . , \u03a0|\u2126|}. If e is a focal element of Bel, we obtain from (29) that e \u2208Ei for some i \u2208{1, . . . , |\u2126|} and m(e) = m0(\u03c9i)mi(e). This implies that, if e \u0338\u2208S|\u2126| i=1 Ei, i.e., e is not a focal element of any conditional belief function Beli, m(e) = 0. So we have shown that the total mass function m is the unique one satisfying (P1) and (P2). 3.3.5 Generalisation If the first requirement (P1) is modified to include conditional constraints with respect to unions of equivalence classes, the approach used to prove Theorem 4 is no longer valid. 28 \fFor each non-empty subset {i1, . . . , iJ} \u2286{1, . . . , |\u2126|}, let BelSJ j=1 \u03a0ij , Beli1\u00b7\u00b7\u00b7iJ denote the categorical belief function with SJ j=1 \u03a0ij as its only focal element, and a conditional belief function on SJ j=1 \u03a0ij, respectively. We can then introduce a new requirement by generalising property (P1) as follows: \u2022 (P1\u2032) : Bel \u2295BelSJ j=1 \u03a0ij = Beli1\u00b7\u00b7\u00b7iJ \u2200\u2205\u0338= {i1, . . . , iJ} \u2286{1, . . . , |\u2126|}. Let \u2212 \u2212 \u2192 Beli1\u00b7\u00b7\u00b7iJ denote the conditional embeddings of Beli1\u00b7\u00b7\u00b7iJ, and \u2212 \u2212 \u2192 Belnew the Dempster combination of all these conditional embeddings. Let Belnew = \u2212 \u2212 \u2192 Belnew \u2295Bel\u2191\u0398 0 . It is easy to show that Belnew satisfies neither (P1\u2032) nor (P2). 3.3.6 Relation to generalised Jeffrey\u2019s rules In spirit, our approach in this chapter is similar to Spies\u2019s Jeffrey\u2019s rule for belief functions in [180]. His total belief function is also the Dempster combination of the prior on the subalgebra generated by the partition \u03a0 of \u0398 with the conditional belief functions on all the equivalence classes \u03a0i. Moreover, he showed that this total belief function satisfies the two properties in Theorem 4. However, his definition of the conditional belief function is different from the one used here, which is derived from Dempster\u2019s rule of combination. His definition falls within the framework of random sets, in which a conditional belief function is a second-order belief function whose focal elements are conditional events, defined as sets of subsets of the underlying frame of discernment. The biggest difference between Spies\u2019s approach and ours is thus that his framework depends on probabilities, whereas ours does not. It would indeed be interesting to explore the connection of our total belief theorem with his Jeffrey\u2019s rule for belief functions. Smets [176] also generalised Jeffrey\u2019s rule within the framework of models based on belief functions, without relying on probabilities. Recall that \u03c1 is a refining mapping from 2\u2126to 2\u0398, and A\u03c1 is the Boolean algebra generated by the set of equivalence classes \u03a0i associated with \u03c1. Contrary to our total belief theorem, which assumes conditional constraints only with respect to the equivalence classes \u03a0i (the atoms of A\u03c1), Smets\u2019s generalised Jeffrey\u2019s rule considers constraints with respect to unions of equivalence classes, i.e., arbitrary elements of A\u03c1 (see the paragraph \u2018Generalisation\u2019 above). Given two belief functions Bel1 and Bel2 over \u0398, his general idea is to find a BF Bel3 there such that: \u2022 (Q1) its marginal on \u2126is the same as that of Bel1, i.e., Bel3 \u21be\u2126= Bel1 \u21be\u2126; \u2022 (Q2) its conditional constraints with respect to elements of A\u03c1 are the same as those of Bel2. 29 \fIn more detail, let m be a mass function over \u0398. Smets defines two kinds of conditioning for conditional constraints: for any E \u2208A\u03c1 and e \u2286E such that \u03c1(\u00af \u03c1(e)) = E, min(e|E) . = m(e) P \u03c1(\u00af \u03c1(e\u2032))=E m(e\u2032), mout(e|E) . = m(e|E) P \u03c1(\u00af \u03c1(e\u2032))=E m(e\u2032|E). The first one is the well-known geometric conditioning [178], whereas the second is called outer conditioning. Both are distinct from Dempster\u2019s rule of conditioning. From these two conditioning rules, he obtains two different forms of generalised Jeffrey\u2019s rule, namely, for any e \u2286\u0398, \u2022 min 3 (e) = min 1 (e|E)m2(E), where E = \u03c1(\u00af \u03c1(e)); \u2022 mout 3 (e) = mout 1 (e|E)m2(E). Both min 3 and mout 3 satisfy (Q1). As for (Q2), min 3 applies, whereas mout 3 does so only partially, since (mout 3 )in(e|E) = mout 1 (e|E) [209]. In [133], Ma et al. defined a new Jeffrey\u2019s rule where the conditional constraints are indeed defined according to Dempster\u2019s rule of combination, and with respect to the whole power set of the frame instead of a subalgebra as in Smets\u2019s framework. In their rule, however, the conditional constraints are not preserved by their total belief functions. 3.3.7 Exploring the space of solutions Lemma 3 ensures that the solution (27) of the total belief theorem has focal elements which are unions of one focal element ei of a subset of conditional belief functions Beli. The following result prescribes that any solution to the total belief problem must have focal elements which obey the above structure (see [59] for a proof). Proposition 1. Each focal element e of a total belief function Bel meeting the requirements of Theorem 4 is the union of exactly one focal element for each of the conditional belief functions whose domain \u03a0i is a subset of \u03c1(E), where E is the smallest focal element of the a priori belief function Bel0 such that e \u2282\u03c1(E). Namely, e = [ i:\u03a0i\u2282\u03c1(E) eji i , (36) where eji i \u2208Ei \u2200i, and Ei denotes the list of focal elements of Beli. The special total belief theorem If we require the a priori function Bel0 to have only disjoint focal elements (i.e., we force Bel0 to be the vacuous extension of a Bayesian function defined on some coarsening of \u2126), we have what we call the restricted or special total belief theorem [59]. The situation generalises Corollary 4. In this special case, it suffices to solve separately the |E\u2126| subproblems obtained by considering each focal element E of Bel0, and then combine the resulting partial solutions by simply weighting the resulting basic probability assignments using the a priori mass m0(E), to obtain a fully normalised total belief function. 30 \fCandidate solutions as linear systems For each individual focal element of Bel0, the task of finding a suitable solution translates into a linear algebra problem. Let N = |E| be the cardinality of E. A candidate solution to the subproblem of the restricted total belief problem associated with E \u2208E\u2126is the solution to a linear system with nmin = P i=1,...,N(ni \u22121) + 1 equations and nmax = Q i ni unknowns, A\u20d7 x = \u20d7 b, (37) where ni is the number of focal elements of the conditional belief function Beli. Each column of A is associated with an admissible (i.e., satisfying the structure of Proposition 1) focal element ej of the candidate total belief function with mass assignment m, \u20d7 x = [m(e1), . . . , m(en)]\u2032, and n = nmin is the number of equalities generated by the N conditional constraints. Minimal solutions Since the rows of the solution system (37) are linearly independent, any system of equations obtained by selecting nmin columns from A has a unique solution. A minimal (i.e., with the minimum number of focal elements) solution to the special total belief problem is then uniquely determined by the solution of a system of equations obtained by selecting nmin columns from the nmax columns of A. A class of linear transformations Let us represent each focal element of the candidate total function e = S i:\u03a0i\u2282\u03c1(E) eji i , where eji i \u2208Ei \u2200i (see (36)), as the vector of indices of its constituent focal elements e = [j1, . . . , jN]\u2032. Definition 8. We define a class T of transformations acting on columns e of a candidate minimal solution system via the following formal sum: e 7\u2192e\u2032 = \u2212e + X i\u2208C ei \u2212 X j\u2208S ej, (38) where C, |C| < N, is a covering set of \u2018companions\u2019 of e (i.e., such that every component of e is present in at least one of them), and a number of \u2018selection\u2019 columns S, |S| = |C| \u22122, are employed to compensate for the side effects of C to yield an admissible column (i.e., a candidate focal element satisfying the structure of Proposition 1). We call the elements of T column substitutions. A sequence of column substitutions induces a discrete path in the solution space: the values m(ei) of the solution components associated with the columns ei vary, and in a predictable way. If we denote by s < 0 the (negative) solution component associated with the old column e: 1. The new column e\u2032 has as its solution component \u2212s > 0. 2. The solution component associated with each companion column is decreased by |s|. 3. The solution component associated with each selection is increased by |s|. 31 \f4. All other columns retain the old values of their solution components. The proof is a direct consequence of the linear nature of the transformation (38). Clearly, if we choose to substitute the column with the most negative solution component, the overall effect is that the most negative component is changed into a positive one, components associated with selection columns become more positive (or less negative), and, as for companion columns, while some of them may end up being assigned negative solution components, these will be smaller in absolute value than |s| (since their initial value was positive). Hence we have the following proposition. Proposition 2. Column substitutions of the class T reduce the absolute value of the most negative solution component. An alternative existence proof We can then use Theorem 2 to prove that there always exists a selection of columns of A (the possible focal elements of the total belief function Bel) such that the resulting square linear system has a positive vector as a solution. This can be done in a constructive way, by applying a transformation of the type (38) recursively to the column associated with the most negative component, to obtain a path in the solution space leading to the desired solution. Such an existence proof for the special total belief theorem, as an alternative to that provided earlier in this section, exploits the effects on solution components of column substitutions of type T : 1. By Theorem 2, at each column substitution, the most negative solution component decreases. 2. If we keep replacing the most negative variable, we keep obtaining distinct linear systems, for at each step the transformed column is assigned a positive solution component and, therefore, if we follow the proposed procedure, cannot be changed back to a negative one by applying transformations of class T . 3. This implies that there can be no cycles in the associated path in the solution space. 4. The number \u0000nmax nmin \u0001 of solution systems is obviously finite, and hence the procedure must terminate. Unfortunately, counter-examples show that there are \u2018transformable\u2019 columns (associated with negative solution components) which do not admit a transformation of the type (38). Although they do have companions on every partition \u03a0i, such counterexamples do not admit a complete collection of \u2018selection\u2019 columns. Solution graphs The analysis of significant particular cases confirms that, unlike the classical law of total probability, the total belief theorem possesses more than one admissible minimal solution, even in the special case. In [59], we noticed that all the candidate minimal solution systems related to a total belief problem of a given size {ni, i = 1, . . . , N} can be arranged into a solution 32 \fgraph, whose structure and symmetries can be studied to compute the number of minimal total belief functions for any given instance of the problem. We focused on the group of permutations of focal elements of the conditional belief functions Beli, G = Sn1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 SnN , (39) namely the product of the permutation groups Sni acting on the collections of focal elements of each individual conditional belief function Beli. The group G acts on the solution graph by generating orbits, i.e., the set of all candidate solution systems (nodes of the graph) obtained by some permutation of the focal elements within at least some of the partition elements \u03a0i. We hypothesised that such orbits are in 1\u20131 correspondence with the number of admissible solutions. Future agenda A number of research directions remain open, based on the results obtained so far. The analysis of the special law of total belief needs to be completed, by finalising the alternative proof of existence and, through the latter, completing the analysis of solution graphs and the number of admissible solutions. Further down the road, the full description of all minimal solutions needs to be extended to the general case of an arbitrary prior Bel0, which our analysis earlier in this section sheds some light on but fails to provide a comprehensive understanding. Distinct versions of the law of total belief may arise from replacing Dempster conditioning with other accepted forms of conditioning for belief functions, such as credal [88], geometric [182], conjunctive and disjunctive [176] conditioning. As belief functions are a special type of coherent lower probabilities, which in turn can be seen as a special class of lower previsions (see [192], Section 5.13), marginal extension [143] can be applied to them to obtain a total lower prevision. The relationship between marginal extension and the law of total belief therefore needs to be understood. Finally, fascinating relationships exist between the total belief problem and transversal matroids [150], on the one hand, and the theory of positive linear systems [90], on the other, which we also plan to investigate in the near future. 3.4 Limit theorems for random sets The law of total probability is only one important result of classical probability theory that needs to be generalised to the wider setting of random sets. In order to properly define a Gaussian belief function, for instance, we would need to generalise the classical central limit theorem to random sets. The old proposal of Dempster and Liu merely transfers normal distributions on the real line by the Cartesian product with Rm [131]. In fact, both the central limit theorem and the law(s) of large numbers have already been generalised to imprecise probabilities [66, 25].2 Here we review the most recent and relevant attempts at formulating similar laws for belief functions. 2See http://onlinelibrary.wiley.com/book/10.1002/9781118763117. 33 \f3.4.1 Central limit theorems Chareka [19] and later Ter\u00b4 an [183] have conducted interesting work on the central limit theorem for capacities. More recently, specific attention has been directed at central limit theorem results for belief measures [86, 171]. Choquet\u2019s definition of belief measures Let \u0398 be a Polish space3 and B(\u0398) be the Borel \u03c3-algebra on \u0398. Let us denote by K(\u0398) the collection of compact subsets of \u0398, and by P(\u0398) the space of all probability measures on \u0398. The set K(\u0398) will be endowed with the Hausdorff topology4 generated by the topology of \u0398. Definition 9. A belief measure on (\u0398, B(\u0398)) is defined as a set function Bel : B(\u0398) \u2192 [0, 1] satisfying: \u2022 Bel(\u2205) = 0 and Bel(\u0398) = 1; \u2022 Bel(A) \u2264Bel(B) for all Borel sets A \u2282B; \u2022 Bel(Bn) \u2193Bel(B) for all sequences of Borel sets Bn \u2193B; \u2022 Bel(G) = sup{Bel(K) : K \u2282G, K \u2208K(\u0398)}, for all open sets G; \u2022 Bel is totally monotone (or \u221e-monotone ): for all Borel sets B1, . . . , Bn, Bel  n [ i=1 Bi ! \u2265 X \u2205\u0338=I\u2282{1,...,n} (\u22121)|I|+1Bel  \\ i\u2208I Bi ! . By [152], for all A \u2208B(\u0398) the collection {K \u2208K(\u0398) : K \u2282A} is universally measurable.5 Let us denote by Bu(K(\u0398)) the \u03c3-algebra of all subsets of K(\u0398) which are universally measurable. The following result is due to Choquet [23]. Proposition 3. The set function Bel : B(\u0398) \u2192[0, 1] is a belief measure if and only if there exists a probability measure PBel on (K(\u0398), B(K(\u0398))) such that Bel(A) = PBel({K \u2208K(\u0398) : K \u2282A}), \u2200A \u2208B(\u0398). Moreover, there exists a unique extension of PBel to (K(\u0398), Bu(K(\u0398))), which we can still denote by PBel. By comparison with the usual random-set formulation, we can appreciate that Proposition 3 confirms that a belief measure is the total probability induced by a probability measure on a source space \u2013 in this case, the source space \u2126= K(\u0398) is the collection of all compact subspaces of \u0398. 3A separable completely metrisable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. 4A Hausdorff space, or \u2018separated\u2019 space, is a topological space in which distinct points have disjoint neighbourhoods. It implies the uniqueness of limits of sequences, nets and filters. 5A subset A of a Polish space X is called universally measurable if it is measurable with respect to every complete probability measure on X that measures all Borel subsets of X. 34 \fFor any A \u2208B(\u0398\u221e), we define Bel\u221e(A) . = P \u221e Bel({K = K1 \u00d7 K2 \u00d7 \u00b7 \u00b7 \u00b7 \u2208(K(\u0398))\u221e: K \u2282A}), (40) where P \u221e Bel is the i.i.d. product probability measure. Bel\u221eis the unique belief measure on (\u0398\u221e, B(\u0398\u221e) induced by P \u221e Bel. Bernoulli variables A central limit theorem for belief functions, in their Choquet formulation, was recently proposed by Epstein and Seo for the case of Bernoulli random variables [86]. Let6 \u0398 = {T, F}, and consider the set \u0398\u221eof all infinite series of samples \u03b8\u221e= (\u03b81, \u03b82, . . .) extracted from \u0398. Let \u03a6n(\u03b8\u221e) be the empirical frequency of the outcome T in the first n experiments in sample \u03b8\u221e. Let Bel be a belief function on \u0398 induced by a measure PBel, and Bel\u221ethe belief measure (40) on \u0398\u221e. The law of large numbers asserts certainty that asymptotic empirical frequencies will lie in the interval [Bel(T), 1 \u2212Bel(F)], that is, Bel\u221en \u03b8\u221e: [lim inf \u03a6n(\u03b8\u221e), lim sup \u03a6n(\u03b8\u221e)] \u2282[Bel(T), 1 \u2212Bel(F)] o = 1. The authors of [86] proved the following proposition. Proposition 4. The following hold: limn\u2192\u221eBel\u221e  ( \u03b8\u221e: \u221an \u03a6n(\u03b8\u221e) \u2212Pl(T) p Bel(F)(1 \u2212Bel(F)) \u2264\u03b1 )! = N(\u03b1), limn\u2192\u221eBel\u221e  ( \u03b8\u221e: \u221an \u03a6n(\u03b8\u221e) \u2212Bel(T) p Bel(T)(1 \u2212Bel(T)) \u2264\u03b1 )! = 1 \u2212N(\u03b1), where N denotes the usual normal distribution on real numbers. The proof follows from showing that, for the events indicated, the minimising measures are i.i.d. As a result, classical limit theorems applied to these measures deliver corresponding limit theorems for the i.i.d. product Bel\u221e. The central result of [86], however, is the following. Proposition 5. Suppose that G : R \u2192R is bounded, quasi-concave and uppersemicontinuous. Then Z G(\u03a6n(\u03b8\u221e)) dBel\u221e(\u03b8\u221e) = E[G(X\u2032 1n), G(X\u2032 2n)] + O \u0012 1 \u221an \u0013 , where (X\u2032 1n, X\u2032 2n) is normally distributed with mean (Bel(T), Pl(T)) and covariance 1 n \u0014 Bel(T)(1 \u2212Bel(T)) Bel(T)Bel(F) Bel(T)Bel(F) (1 \u2212Bel(F))Bel(F) \u0015 . 6We adopt our own notation here, in place of Epstein and Seo\u2019s original one. 35 \fThat is, lim sup n\u2192\u221e \u221an \f \f \f \f Z G(\u03a6n(\u03b8\u221e)) dBel\u221e(\u03b8\u221e) \u2212E[G(X\u2032 1n), G(X\u2032 2n)] \f \f \f \f \u2264K for some constant K. Generalisation Xiaomin Shi [171] has recently (2015) generalised the result of Epstein and Seo [86] from Bernoulli random variables to general bounded random variables,7 thanks to a theorem in a seminar paper [23] by Choquet. Proposition 6. Let Yi, i = 1, . . ., be equally distributed random variables on (\u0398, B(\u0398)), bounded by a constant M, and let us define Xi(\u03b81, \u03b82, ..) . = Yi(\u03b8i), i \u22651. Then we have, for all \u03b1 \u2208R, lim n\u2192\u221eBel\u221e  Pn i=1 Xi \u2212n\u00b5 \u221an\u03c3 \u2265\u03b1 ! = 1 \u2212N(\u03b1), and lim n\u2192\u221eBel\u221e \u0012Pn i=1 Xi \u2212n\u00b5 \u221an\u03c3 < \u03b1 \u0013 = 1 \u2212N(\u03b1), where \u00b5 = EP \u221e Bel[Zi], \u00b5 = EP \u221e Bel[Zi] and Zi(K1 \u00d7 K2 \u00d7 \u00b7 \u00b7 \u00b7 ) = inf\u03b8i\u2208Ki Xi(\u03b81, \u03b82, . . .), Zi(K1 \u00d7 K2 \u00d7 \u00b7 \u00b7 \u00b7 ) = sup\u03b8i\u2208Ki Xi(\u03b81, \u03b82, . . .), for Ki \u2208K(\u0398), i \u22651. The expressions for \u03c3 and \u03c3 are given in [171], Theorem 3.1. A central limit theorem for two-sided intervals was also provided ([171], Theorem 4.1). Gaussian random sets One can note, however, that the above results do not really address the question of whether there exists, among all (compact) random sets, a special class (playing the role of Gaussian probability measures) characterised by all sums of independent and equally distributed random sets converging to an object of that class (which it would make sense to call a proper \u2018Gaussian random set\u2019). The issue remains open for future investigation. Question 4. Does there exist a (parameterisable) class of random sets, playing a role similar to that which Gaussian distributions play in standard probability, such that sample averages of (the equivalent of) i.i.d. random sets converge in some sense to an object of such a class? 7https://arxiv.org/pdf/1501.00771.pdf. 36 \f3.4.2 Laws of large numbers Several papers have been published on laws of large numbers for capacities [134], monotone measures [2] and non-additive measures [154, 184], although not strictly for belief measures. Molchanov [144], among others, studied the Glivenko\u2013Cantelli theorem for capacities induced by random sets. A strong law of large numbers for setvalued random variables in a G\u03b1 space was recently produced by Guan Li [103, 130], based on Taylor\u2019s result for single-valued random variables. The issue was also recently addressed by Kerkvliet in his PhD dissertation [116]. Definition 10. The weak law of large numbers states that the sample average converges in probability towards the expected value. Consider an infinite sequence of i.i.d. Lebesgue-integrable random variables X1, X2, . . . with expected value \u00b5 and sample average Xn = (X1 + \u00b7 \u00b7 \u00b7 + Xn)/n. Then, for any positive number \u03b5, lim n\u2192\u221eP \u0000|Xn \u2212\u00b5| > \u03b5 \u0001 = 0. (41) The strong law of large numbers states that the sample average converges almost surely to the expected value. Namely, P \u0010 lim n\u2192\u221eXn = \u00b5 \u0011 = 1. (42) When the probability measure is replaced by a set function \u03bd assumed to be completely monotone, the empirical frequencies fn(A) of an event A over a sequence of Bernoulli trials satisfy \u03bd \u0012 \u03bd(A) \u2264lim inf n fn(A) \u2264lim sup n fn(A) \u22641 \u2212\u03bd(Ac) \u0013 = 1. In order to replace the event by a random variable X on a measurable space (\u0398, B), one needs to calculate expectations with respect to \u03bd and \u03bd = 1 \u2212\u03bd(Ac) in such a way that E\u03bd[IA] = \u03bd(A), E\u03bd[IA] = 1 \u2212\u03bd(Ac), and both expectations equal the usual definition when \u03bd is a probability measure. That is achieved by using the Choquet integral [23]. The resulting type of limit theorem has already been considered by Marinacci [137] and Maccheroni and Marinacci [134], and in more recent papers [21, 66, 25, 154, 155]. In those papers, the weakening of the axiomatic properties of probability are balanced by the incorporation of extra technical assumptions about the properties of \u0398 and/or the random variables, for instance that \u0398 is a compact or Polish [134] topological space or that the random variables are continuous functions, are bounded, have sufficiently high moments, are independent in an unreasonably strong sense or satisfy ad hoc regularity requirements. In [184], it was shown that additivity can be replaced by complete monotonicity in the law of large numbers under the reasonable first-moment condition that both of the Choquet expectations E\u03bd[X] and E\u03bd[X] are finite, with no additional assumptions. Under the assumption that the sample space is endowed with a topology, the continuity condition for monotone sequences can also be relaxed. The main result reads as follows ([184], Theorem 1.1). 37 \fProposition 7. Let \u03bd be a completely monotone set function on a measurable space (\u0398, B). Let X be an integrable8 random variable, and let {Xn, n} be pairwise preindependent and identically distributed to X. If \u03bd is a capacity, then for every \u03f5 > 0, \u03bd \u0000E\u03bd[X] \u2212\u03f5 < Xn < E\u03bd[X] + \u03f5 \u0001 \u21921. If (\u0398, B) is a topological space with a Borel \u03c3-algebra, \u03bd is a topological capacity and the Xn are continuous functions on \u0398, then \u03bd \u0012 E\u03bd[X] \u2264lim inf n Xn \u2264lim sup n Xn \u2264E\u03bd[X] \u0013 = 1. 3.5 Frequentist inference with random sets Random sets are mathematical objects detached from any specific interpretation. Just as probability measures are used by both Bayesians and frequentists for their analyses, random sets can also be employed in different ways according to the interpretation they are provided with. In particular, it is natural to imagine a generalised frequentist framework in which random experiments are designed by assuming a specific random-set distribution, rather than a conventional one, in order to better cope with the ever-occurring set-valued observations (see the Introduction). 3.5.1 Parameterised families of random sets The first necessary step is to introduce parametric models based on random sets. \u03a9 \u0398 \u0393 \u03c9 \u0393\u0001\u03c9\u0002 Normal random set? Binomial random set? Normal Binomial Figure 6: Describing the family of random sets (right) induced by families of probability distributions in the source probability space (left) is a likely first step towards a generalisation of frequentist inference to random sets. Recall Dempster\u2019s random-set interpretation (Fig. 6). Should the multivalued mapping \u0393 which defines a random set be \u2018designed\u2019, or derived from the problem? For instance, in the cloaked die example discussed in the Introduction to [50], it is the 8That is, its lower and upper expectations E\u03bd[X] and E\u03bd[X] are finite. 38 \focclusion which generates the multivalued mapping and we have no control over it. In other situations, however, it may make sense to impose a parameterised family of mappings \u0393(.|\u03c0) : \u2126\u21922\u0398, \u03c0 \u2208\u03a0, which, given a (fixed) probability on the source space \u2126, would yield as a result a parameterised family of random sets. The alternative is to fix the multivalued mapping (e.g., when it is given by the problem), and model the source probability by a classical parametric model. A Gaussian or binomial family of source probabilities would then induce a family of \u2018Gaussian\u20199 or \u2018binomial\u2019 random sets (see Fig. 6 again). Question 5. What is the analytical form of random sets induced by say, Gaussian, binomial or exponential families of (source) probability distributions? 3.5.2 Hypothesis testing with random sets As we know, in hypothesis testing [9] designing an experiment amounts to choosing a family of probability distributions which is assumed to generate the observed data. If parameterised families of random sets can be constructed, they can then be plugged into the frequentist inference machinery, after an obvious generalisation of some of the steps involved. Hypothesis testing with random sets would then read as follows: \u2022 State the relevant null hypothesis H0 and the alternative hypothesis. \u2022 State the assumptions about the form of the random set (or mass assignment, as opposed to a classical distribution) describing the observations. \u2022 State the relevant test statistic T (a quantity derived from the sample) \u2013 only this time the sample may contain set-valued observations! \u2022 Derive the mass assignment (rather than probability distribution) of the test statistic under the null hypothesis (from the assumptions). \u2022 Set a significance level (\u03b1). \u2022 Compute from the observations the observed value tobs of the test statistic T \u2013 this will also now be set-valued. \u2022 Calculate the conditional belief value (playing the role of the classical p-value) under H0 of sampling a test statistic at least as extreme as the observed value. \u2022 Reject the null hypothesis, in favour of the alternative hypothesis, if and only if the above conditional belief value is less than the significance level. Question 6. How would results from random-set hypothesis testing relate to classical hypothesis testing applied to parameterised distributions within the assumed random set? 9Note that the Gaussian random sets so defined would be entirely distinct from Gaussian random sets as advocated in the context of the generalised central limit theorem setting. 39 \f3.6 Random-set random variables We know that random sets are set-valued random variables: nevertheless, the question remains as to whether one can build random variables on top of random-set (belief) spaces, rather than the usual probability space. It would be natural to term such objects random-set random variables. Just as in the classical case, we would need a mapping from \u0398 to a measurable space (e.g. the positive real half-line) f : \u0398 \u2192R+ = [0, +\u221e], where this time \u0398 is itself the codomain of a multivalued mapping \u0393 : \u2126\u21922\u0398, with source probability space \u2126. 3.6.1 Generalising the Radon\u2013Nikodym derivative For a classical continuous random variable X on a measurable space (\u0398, F(\u0398)), we can compute its PDF as its Radon\u2013Nikodym derivative (RND), namely the measurable function p : \u0398 \u2192[0, \u221e) such that for any measurable subset A \u2282\u0398, P[X \u2208A] = Z A p d\u00b5, where \u00b5 is a \u03c3-finite measure on (\u0398, F(\u0398)). We can pose ourselves the following research question. Question 7. Can a (generalised) PDF be defined for a random-set random variable as defined above? Answering Question 7 requires extending the notion of a Radon\u2013Nikodym derivative to random sets. Graf [101] pioneered the study of Radon\u2013Nikodym derivatives for the more general case of capacities. The extension of the RND to the even more general case of set functions was studied by Harding et al. in 1997 [106]. The problem was investigated more recently by Rebille in 2009 [153]. The following summary of the problem is abstracted from Molchanov\u2019s Theory of Random Sets [146]. 3.6.2 Absolute continuity Let us assume that the two capacities \u00b5, \u03bd are monotone, subadditive and continuous from below. Definition 11. A capacity \u03bd is absolutely continuous with respect to another capacity \u00b5 if, for every A \u2208F, \u03bd(A) = 0 whenever \u00b5(A) = 0. This definition is the same as for standard measures. However, whereas for standard measures absolute continuity is equivalent to the integral relation \u00b5 = R \u03bd (the existence of the corresponding RND), this is no longer true for general capacities. To understand this, consider the case of a finite \u0398, |\u0398| = n. Then any measurable function f : \u0398 \u2192R+ is determined by just n numbers, which do not suffice to uniquely define a capacity on 2\u0398 (which has 2n degrees of freedom) [146]. 40 \f3.6.3 Strong decomposition Nevertheless, a Radon\u2013Nikodym theorem for capacities can be established after introducing the notion of strong decomposition. Consider a pair of capacities (\u00b5, \u03bd) that are monotone, subadditive and continuous from below. Definition 12. The pair (\u00b5, \u03bd) is said to have the strong decomposition property if, \u2200\u03b1 \u22650, there exists a measurable set A\u03b1 \u2208F such that \u03b1(\u03bd(A) \u2212\u03bd(B)) \u2264\u00b5(A) \u2212\u00b5(B) if B \u2282A \u2282A\u03b1, \u03b1(\u03bd(A) \u2212\u03bd(A \u2229A\u03b1)) \u2265\u00b5(A) \u2212\u00b5(A \u2229A\u03b1) \u2200A. Roughly speaking, the strong decomposition condition states that, for each bound \u03b1, the \u2018incremental ratio\u2019 of the two capacities is bounded by \u03b1 in the sub-power set capped by some event A\u03b1. Note that all standard measures possess the strong decomposition property. 3.6.4 A Radon\u2013Nikodym theorem for capacities The following result was proved by Graf [101]. Theorem 5. For every two capacities \u00b5 and \u03bd, \u03bd is an indefinite integral of \u00b5 if and only if the pair (\u00b5, \u03bd) has the strong decomposition property and \u03bd is absolutely continuous with respect to \u00b5. Nevertheless, a number of issues remain open. Most relevantly to our ambition to develop a statistical theory based on random sets, the conditions of Theorem 5 (which holds for general capacities) need to be elaborated for the case of completely alternating capacities (distributions of random closed sets). As a first step, Molchanov notes ([146], page 75) that the strong decomposition property for two capacity functionals \u03bd = TX and \u00b5 = TY associated with the random sets X, Y implies that \u03b1TX(CB A) \u2264TY (CB A) if B \u2282A \u2282A\u03b1, and \u03b1TX(CA\u2229A\u03b1 A ) \u2265TY (CA\u2229A\u03b1 A ) \u2200A, where CB A = {C \u2208C, B \u2282C \u2282A}. A specific result on the Radon\u2013Nikodym derivative for random (closed) sets is still missing, and with it the possibility of generalising the notion of a PDF to these more complex objects. 4 Extending the geometric approach 4.1 The geometry of uncertainty As we argued in [50], geometry may be a unifying language, not only for random set and evidential theory, but for uncertainty measures more generally [49, 60, 61, 62, 41 \f65, 95, 99, 10, 120, 104, 194], possibly in conjunction with an algebraic perspective [58, 26, 35, 64, 30, 32, 46]. In our geometric approach to uncertainty [50], in particular, uncertainty measures are assimilated to points of a geometric space, where that can be processed via combination, conditioning and so on [27, 33]. Most efforts have focused on the geometry of belief functions, as elements of a belief space representable as either a simplex (the generalisation of a triangle in higher dimensions) or a recursive bundle structure [63, 53, 48, 59]. The approach can be applied to combination rules, such as Dempster\u2019s sum, resulting in rather elegant geometric procedures for them [56, 28]. The combinatorial features of plausibility and commonality functions, set functions which carry the same information as the associated BF, have also been touched upon [34, 43]. The geometric approach has been subsequently extended to other types uncertainty measures, such as possibility measures [42], and special classes of belief functions such as consistent BFs [45, 37, 55, 36], via the notion of simplicial complex [29]. The geometry of BFs as credal sets has also been investigated [38, 4, 15]. The geometry of the mappings between different forms of uncertainty measures has been considered [57, 40, 31, 41], specifically the task of mapping a belief function to a probability measure [24, 191, 173]. Belief functions can be approximated using possibility measures too [5], a problem which can be approached in geometric terms as well [39, 54, 47]. Namely, one can analytically compute and analyse approximations obtained by minimising standard Lp norms and compare with outer consonant approximations [79]. Consistent approximations of belief functions induced by classical Minkowski norms may also be derived [45] and compared with classical outer consonant approximations [79]. Geometric reasoning can be applied to most if not all aspects of making inferences from data in the framework of epistemic uncertainty, including conditioning [129], by associating a conditioning event A with a \u2018conditioning simplex\u2019 [52, 44]. Transformations mapping belief functions to probabilities can be interpreted in terms of credal sets [41]. 4.2 Open questions Nevertheless, the geometric approach to uncertainty still has much room for further development. For instance: Question 8. Can the inference problem can be posed in a geometric setting too, by representing both data and belief measures in a common geometric space? An answer to such an intriguing question would require a geometric representation general enough to encode both the data driving the inference and the (belief) measures possibly resulting from the inference, in such a way that the inferred measure minimises some sort of distance from the empirical data. The question of what norm is the most appropriate to minimise for inference purposes would also arise. On the other hand, while [50] has mainly concerned itself with the geometric representation of finite belief measures, as we move on from belief fuctions on finite frames to random sets on arbitrary domains, the nature of the geometry of these continuous formulations poses new questions. The formalism needs to tackle (besides probability, 42 \fpossibility and belief measures) other important mathematical descriptions of uncertainty, first of all general monotone capacities and gambles (or variations thereof). Finally, new, more sophisticated geometric representations of belief measures can be sought, in terms of either exterior algebras or areas of projections of convex bodies. Some of these aspects are briefly considered in this section. 4.3 Geometry of general combination The study of the geometry of the notion of evidence combination and belief updating, which started with the analysis of Dempster\u2019s rule provided in [28], will need to be extended to the geometric behaviour of the other main combination operators. A comparative geometric analysis of combination rules would allow us to describe the \u2018cone\u2019 of possible future belief states under stronger or weaker assumptions about the reliability and independence of sources. We can start by giving a general definition of a conditional subspace (see [50], Chapter 8). Definition 13. Given a belief function Bel \u2208B, we define the conditional subspace \u27e8Bel\u27e9\u2299as the set of all \u2299combinations of Bel with any other belief function on the same frame, where \u2299is an arbitrary combination rule, assuming such a combination exists. Namely, \u27e8Bel\u27e9\u2299. = n Bel \u2299Bel\u2032, Bel\u2032 \u2208B s.t. \u2203Bel \u2299Bel\u2032o . (43) 4.3.1 Geometry of Yager\u2019s and Dubois\u2019s rules Alternatives to the Dempster rule of combination were proposed by both Yager and Dubois. The latter\u2019s proposal is based on the view that conflict is generated by nonreliable information sources. In response, the conflicting mass (denoted here by m\u2229(\u2205)) is reassigned to the whole frame of discernment \u0398: mY (A) = \u001a m\u2229(A) \u2205\u0338= A \u228a\u0398, m\u2229(\u0398) + m\u2229(\u2205) A = \u0398, (44) where m\u2229(A) . = P B\u2229C=A m1(B)m2(C) and m1, m2 are the masses to be combined. The combination operator proposed by Dubois and Prade10 [78] comes from applying the minimum specificity principle [77] to cases in which the focal elements B, C of two input belief functions do not intersect. This results in assigning their product mass to B \u222aC, namely mD(A) = m\u2229(A) + X B\u222aC=A,B\u2229C=\u2205 m1(B)m2(C). (45) 10We follow here the same notation as in [128]. 43 \fAnalysis on binary frames On binary frames \u0398 = {x, y}, Yager\u2019s rule (44) and Dubois\u2019s rule (45) coincide, as the only conflicting focal elements there are {x} and {y}, whose union is \u0398 itself. Namely, m Y \u20dd(x) = m1(x)(1 \u2212m2(y)) + m1(\u0398)m2(x), m Y \u20dd(y) = m1(y)(1 \u2212m2(x)) + m1(\u0398)m2(y), m Y \u20dd(\u0398) = m1(x)m2(y) + m1(y)m2(x) + m1(\u0398)m2(\u0398). (46) Using (46), we can easily show that Bel Y \u20ddBelx = [m(x) + m(\u0398), 0, m(y)]\u2032, Bel Y \u20ddBely = [0, m(y) + m(\u0398), m(x)]\u2032, Bel Y \u20ddBel\u0398 = Bel = [m(x), m(y), m(\u0398)], (47) adopting the usual vector notation Bel = [Bel(x), Bel(y), Bel(\u0398)]\u2032. The global behaviour of Yager\u2019s (and Dubois\u2019s) rule in the binary case is then pretty clear: the conditional subspace \u27e8Bel\u27e9Y \u20ddis the convex closure \u27e8Bel\u27e9Y \u20dd= Cl(Bel, Bel Y \u20ddBelx, Bel Y \u20ddBely) of the points (47) in the belief space: see Fig. 7. Bel  = [0,0]' Bel  = [1,0]' Bel  = [0,1]' x y \u0398 Bel Bely Y Bel Belx Y Bel Bel Y Bel Figure 7: Conditional subspace \u27e8Bel\u27e9Y \u20ddfor Yager\u2019s (and Dubois\u2019s) combination rule on a binary frame \u0398 = {x, y}. Dempster\u2019s conditional subspace \u27e8Bel\u27e9is also shown for the sake of comparison. Comparing (46) with (47), it is easy to see that Bel1 Y \u20ddBel2 = m2(x)Bel1 Y \u20ddBelx + m2(y)Bel1 Y \u20ddBely + m2(\u0398)Bel1 Y \u20ddBel\u0398, i.e., the simplicial coordinates of Bel2 in the binary belief space B2 and of the Yager combination Bel1 Y \u20ddBel2 in the conditional subspace \u27e8Bel1\u27e9Y \u20ddcoincide. We can then conjecture the following. 44 \fConjecture 2. Yager combination and affine combination commute. Namely, Bel Y \u20dd  X i \u03b1iBeli ! = X i \u03b1iBel Y \u20ddBeli. As commutativity is the basis of our geometric analysis of Dempster\u2019s rule (see [50] Chapter 8), this opens the way for a similar geometric construction for Yager\u2019s rule. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 Bel Bel Bely Y Bely Belx Bel Belx Y Figure 8: In the case of the Yager combination, the images of constant-mass loci (dashed blue segments) do not converge to a focus, but are parallel lines (dashed purple lines; compare [50], Fig 8.5). Geometric construction However, as shown in Fig. 8, the images of constant-mass loci under Yager\u2019s rule are parallel, so that the related conditional subspaces have no foci. Indeed, from (46), lim m2(y)\u2192\u2212\u221e m Y \u20dd(y) m Y \u20dd(x) = m1(y)(1 \u2212m2(x)) + m1(\u0398)m2(y) m1(x)(1 \u2212m2(y)) + m1(\u0398)m2(x) = \u2212m1(\u0398) m1(x) , and similarly for the loci with m2(y) = const. Nevertheless, the general principle of geometric construction of intersecting the linear spaces which are images of constant-mass loci still holds. 45 \f4.3.2 Geometry of disjunctive combination Disjunctive combination is the natural, cautious dual of conjunctive/Dempster combination. The operator follows from the assumption that the consensus between two sources of evidence is best represented by the union of the supported hypotheses, rather than by their intersection. Again, here we will briefly analyse its geometry in the case of a binary frame, and make conjectures about its general behaviour. We will first consider normalised belief functions, to later extend our analysis to the unnormalised case. Global behaviour Let us first understand the shape of the disjunctive conditional subspace. By definition, m \u222a \u20dd(x) = m1(x)m2(x), m \u222a \u20dd(y) = m1(y)m2(y), m \u222a \u20dd(\u0398) = 1 \u2212m1(x)m2(x) \u2212m1(y)m2(y), (48) so that, adopting the usual vector notation [Bel(x), Bel(y), Bel(\u0398)]\u2032, Bel \u222a \u20ddBelx = [m(x), 0, 1 \u2212m(x)]\u2032, Bel \u222a \u20ddBely = [0, m(y), 1 \u2212m(y)]\u2032, Bel \u222a \u20ddBel\u0398 = Bel\u0398. (49) The conditional subspace \u27e8Bel\u27e9\u222a \u20ddis thus the convex closure \u27e8Bel\u27e9\u222a \u20dd= Cl(Bel, Bel \u222a \u20ddBelx, Bel \u222a \u20ddBely) of the points (49): see Fig. 9. As in the Yager case, Bel \u222a \u20dd[\u03b1Bel\u2032 + (1 \u2212\u03b1)Bel\u2032\u2032] = h m(x)(\u03b1m\u2032(x) + (1 \u2212\u03b1)m\u2032\u2032(x)), m(y)(\u03b1m\u2032(y) + (1 \u2212\u03b1)m\u2032\u2032(y)) i\u2032 = \u03b1Bel \u222a \u20ddBel\u2032 + (1 \u2212\u03b1)Bel \u222a \u20ddBel\u2032\u2032, i.e., \u222a \u20ddcommutes with affine combination, at least in the binary case. Question 9. Does disjunctive combination commute with affine combination in general belief spaces? Pointwise behaviour As in the Yager case, for disjunctive combination the images of constant-mass loci are parallel to each other. In fact, they are parallel to the corresponding constant-mass loci and the coordinate axes (observe, in Fig. 9, the locus m(x) = 1 3 and its image in the conditional subspace \u27e8Bel\u27e9\u222a \u20dd). We can prove the following. Theorem 6. In the binary case \u0398 = {x, y}, the lines joining Bel\u2032 and Bel \u222a \u20ddBel\u2032 for any Bel\u2032 \u2208B intersect in the point m(x) = m\u2032(x)m(x) \u2212m(y) 1 \u2212m(y) , m(y) = 0. (50) 46 \fBel  = [1,0]' Bel  = [0,1]' x y Bel Bely Bel Belx Bel Bel U U m(x) = 1/3 1/3 1 Bel Bel'' U Bel Bel' U Bel' Bel'' U Figure 9: Conditional subspace \u27e8Bel\u27e9\u222a \u20ddfor disjunctive combination on a binary frame. The pointwise behaviour of \u222a \u20ddis also illustrated (see text). Proof. Recalling the equation of the line joining two points (\u03c71, \u03c51) and (\u03c72, \u03c52) of R2, with coordinates (\u03c7, \u03c5), \u03c5 \u2212\u03c51 = \u03c52 \u2212\u03c51 \u03c72 \u2212\u03c71 (\u03c7 \u2212\u03c71), we can identify the line joining Bel\u2032 and Bel \u222a \u20ddBel\u2032 as \u03c5 \u2212m\u2032(y) = m(y)m\u2032(y) \u2212m\u2032(y) m(x)m\u2032(x) \u2212m\u2032(x)(\u03c7 \u2212m\u2032(x)). Its intersection with \u03c5 = 0 is the point (50), which does not depend on m\u2032(y) (i.e., on the vertical location of Bel\u2032 on the constant-mass loci). A valid geometric construction for the disjunctive combination Bel \u222a \u20ddBel\u2032 of two belief functions in B2 is provided by simple trigonometric arguments. Namely (see Fig. 10): 1. Starting from Bel\u2032, find its orthogonal projection onto the horizontal axis, with coordinate m\u2032(x) (point 1). 2. Draw the line with slope 45\u25e6passing through this projection, and intersect it with the vertical axis, at coordinate \u03c5 = m\u2032(x) (point 2). 3. Finally, take the line l passing through Bely and the orthogonal projection of Bel onto the horizontal axis, and draw a parallel line l\u2032 through point 2 \u2013 its 47 \fintersection with the horizontal axis (point 3) is the x coordinate m(x)m\u2032(x) of the desired combination. A similar construction (shown in magenta) allows us to locate the y coordinate of the combination (as also shown in Fig. 10). Bel  = [1,0]' Bel  = [0,1]' x y Bel 1 Bel Bel' U Bel' 2 3 l l' Figure 10: Geometric construction for the disjunctive combination of two belief functions Bel, Bel\u2032 on a binary frame. 4.3.3 Geometry of combination of unnormalised belief functions In the case of unnormalised belief functions, Dempster\u2019s rule is replaced by the conjunctive combination. The disjunctive combination itself needs to be reassessed for UBFs as well. In the unnormalised case, a distinction exists between the belief measure Bel(A) = X \u2205\u0338=B\u2286A m(B) and the believability (in Smets\u2019s terminology) measure of an event A, denoted by b(A), b(A) = X \u2205\u2286B\u2286A m(B). (51) 48 \fHere we analyse the geometric behaviour of the latter, in which case \u2205is not treated as an exception: the case of belief measures is left for future work. As Bel(\u0398) = b(\u0398) = 1, as usual, we neglect the related coordinate and represent believability functions as points of a Cartesian space of dimension |2\u0398| \u22121 (as \u2205cannot be ignored any more). Conjunctive combination on a binary frame In the case of a binary frame, the conjunctive combination of two belief functions Bel1 and Bel2, with BPAs m1, m2, yields m \u2229 \u20dd(\u2205) = m1(\u2205) + m2(\u2205) \u2212m1(\u2205)m2(\u2205) + m1(x)m2(y) + m1(y)m2(x), m \u2229 \u20dd(x) = m1(x)(m2(x) + m2(\u0398)) + m1(\u0398)m2(x), m \u2229 \u20dd(y) = m1(y)(m2(y) + m2(\u0398)) + m1(\u0398)m2(y), m \u2229 \u20dd(\u0398) = m1(\u0398)m2(\u0398). (52) Conditional subspace for conjunctive combination The global behaviour of \u2229 \u20ddin the binary (unnormalised) case can then be understood in terms of its conditional subspace. We have b \u2229 \u20ddb\u2205 = b\u2205= [1, 1, 1]\u2032, b \u2229 \u20ddbx = (m1(\u2205) + m1(y))b\u2205+ (m1(x) + m1(\u0398))bx = [m1(\u2205) + m1(y), 1, m1(\u2205) + m1(y)]\u2032 = b1(y)b\u2205+ (1 \u2212b1(y))bx, b \u2229 \u20ddby = (m1(\u2205) + m1(x))b\u2205+ (m1(y) + m1(\u0398))by = [m1(\u2205) + m1(x), m1(\u2205) + m1(x), 1]\u2032 = b1(x)b\u2205+ (1 \u2212b1(x))by, b \u2229 \u20ddb\u0398 = b, (53) as bx = [0, 1, 0]\u2032, by = [0, 0, 1]\u2032, b\u2205= [1, 1, 1]\u2032 and b\u0398 = [0, 0, 0]\u2032. From (53), we can note that the vertex b \u2229 \u20ddbx belongs to the line joining b\u2205and bx, with its coordinate given by the believability assigned by b to the other outcome y. Similarly, the vertex b \u2229 \u20ddby belongs to the line joining b\u2205and by, with its coordinate given by the believability assigned by b to the complementary outcome x (see Fig. 11). Conditional subspace for disjunctive combination As for the disjunctive combination, it is easy to see that in the unnormalised case we get b \u222a \u20ddb\u0398 = b\u0398, b \u222a \u20ddbx = b(x)bx + (1 \u2212b(x))b\u0398, b \u222a \u20ddb\u2205 = b, b \u222a \u20ddby = b(y)by + (1 \u2212b(y))b\u0398, so that the conditional subspace is as in Fig. 11. Note that, in the unnormalised case, there is a unit element with respect to \u222a \u20dd, namely b\u2205. In the unnormalised case, we can observe a clear symmetry between the conditional subspaces induced by disjunctive and conjunctive combination. 49 \fb b(y) b(x) bx b 1\u0001\u0002b(y) b(x) 1 \u0001 b(x) b b U b(y) b U \u0398 b bx U b by U by U b bx U b by Figure 11: Conditional subspaces induced by \u2229 \u20ddand \u222a \u20ddin a binary frame, for the case of unnormalised belief functions. 4.3.4 Research questions A number of questions remain open after this preliminary geometric analysis of other combination rules on binary spaces, and its extension to the case of unnormalised belief functions. Question 10. What is the general pointwise geometric behaviour of disjunctive combination, in both the normalised and the unnormalised case? A dual question concerns the conjunctive rule, as the alter ego of Dempster\u2019s rule in the unnormalised case. Question 11. What is the general pointwise geometric behaviour of conjunctive combination in the unnormalised case? Bold and cautious rules [72], which are also inherently defined for unnormalised belief functions, are still to be geometrically understood. Question 12. What is the geometric behaviour of bold and cautious rules? 50 \f4.4 Geometry of general conditioning The author\u2019s analysis of geometric conditioning (see [50], Chapter 15) is also still in its infancy. There, we analysed the case of classical Minkowski norms. It is natural to wonder what happens when we plug other norms into the optimisation problem that defines geometric conditioning: Beld(.|A) . = arg min Bel\u2032\u2208BA d(Bel, Bel\u2032), (54) where BA is the conditioning event A, i.e., the set of belief functions whose BPA assigns mass to subsets of A only, and d is a suitable distance function in the belief space. Even more relevantly, the question of whether geometric conditioning is a framework flexible enough to encompass general conditioning in belief calculus arises. Question 13. Can any major conditioning operator be interpreted as geometric (in the sense defined above) conditioning, i.e., as producing the belief function within the conditioning simplex at a minimum distance from the original belief function, with respect to an appropriate norm in the belief space? 4.5 A true geometry of uncertainty A true geometry of uncertainty will require the ability to manipulate in our geometric language any (or most) forms of uncertainty measures [51]. Probability and possibility measures are, as we know, special cases of belief functions: therefore, their geometric interpretation does not require any extension of the notion of a belief space. Most other uncertainty measures, however, are not special cases of belief functions \u2013 in fact, a number of them are more general than belief functions, such as probability intervals (2-monotone capacities), general monotone capacities and lower previsions. Tackling these more general measures requires, therefore, extending the concept of a geometric belief space in order to encapsulate the most general such representation. In the medium term, the aim is to develop a general geometric theory of imprecise probabilities, encompassing capacities, random sets induced by capacity functionals, and sets of desirable gambles. 4.5.1 Geometry of capacities and random sets Developing a geometric theory of general monotone capacities appears to be rather challenging. We can gain some insight into this issue, however, from an analysis of 2-monotone capacities (systems of probability intervals) and, more specifically, when they are defined on binary frames. Geometry of 2-monotone capacities Recall that a capacity \u00b5 is called 2-monotone iff, for any A, B in the relevant \u03c3-algebra \u00b5(A \u222aB) + \u00b5(A \u2229B) \u2265\u00b5(A) + \u00b5(B), 51 \fwhereas it is termed 2-alternating whenever \u00b5(A \u222aB) + \u00b5(A \u2229B) \u2264\u00b5(A) + \u00b5(B). Belief measures are 2-monotone capacities, while their dual plausibility measures are 2-alternating ones. More precisely, belief functions are infinitely monotone capacities, as they satisfy \u00b5 \uf8eb \uf8ed k [ j=1 Aj \uf8f6 \uf8f8\u2265 X \u2205\u0338=K\u2286[1,...,k] (\u22121)|K|+1\u00b5 \uf8eb \uf8ed\\ j\u2208K Aj \uf8f6 \uf8f8 (55) for every value of k \u2208N. A well-known lemma by Chateauneuf and Jaffray states the following. Proposition 8. A capacity \u00b5 on \u0398 is 2-monotone if and only if X {x,y}\u2286E\u2286A m(E) \u22650 \u2200x, y \u2208\u0398, \u2200A \u220bx, y, (56) where m is the M\u00a8 obius transform (57) of \u00b5: m(A) = X B\u2282A (\u22121)|A\u2212B|\u00b5(B). (57) We can use Proposition 8 to show that the following theorem is true. Theorem 7. The space of all 2-monotone capacities defined on a frame \u0398, which we denote by M2 \u0398, is convex. Proof. Suppose that \u00b51 and \u00b52, with M\u00a8 obius inverses m1 and m2, respectively, are 2monotone capacities (therefore satisfying the constraint (56)). Suppose that a capacity \u00b5 is such that \u00b5 = \u03b1\u00b51 + (1 \u2212\u03b1)\u00b52, 0 \u2264\u03b1 \u22641. Since M\u00a8 obius inversion is a linear operator, we have that X {x,y}\u2286E\u2286A m(E) = X {x,y}\u2286E\u2286A \u03b1m1 + (1 \u2212\u03b1)m2 = \u03b1 X {x,y}\u2286E\u2286A m1 + (1 \u2212\u03b1) X {x,y}\u2286E\u2286A m2 \u22650, since X {x,y}\u2286E\u2286A m1 \u22650, X {x,y}\u2286E\u2286A m2 \u22650 for all x, y \u2208\u0398, \u2200A \u220bx, y. Note that Theorem 7 states the convexity of the space of 2-monotone capacities: it is, instead, well known that every 2-monotone capacity corresponds to a convex set of probability measures in their simplex. 52 \fIt is useful to get an intuition about the problem by considering the case of 2monotone capacities defined on the ternary frame \u0398 = {x, y, z}. There, the constraints (56) read as \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 m({x, y}) \u22650, m({x, y}) + m(\u0398) \u22650, m({x, z}) \u22650, m({x, z}) + m(\u0398) \u22650, m({y, z}) \u22650, m({y, z}) + m(\u0398) \u22650, (58) whereas for \u0398 = {x, y, z, w} they become \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 m({xi, xj}) \u22650, m({xi, xj}) + m({xi, xj, xk}) \u22650, m({xi, xj}) + m({xi, xj, xl}) \u22650, m({xi, xj}) + m({xi, xj, xk}) + m({xi, xj, xl}) + m(\u0398) \u22650, for each possible choice of xi, xj, xk, xl \u2208\u0398. The constraints (58) are clearly equivalent to \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 m({x, y}) \u22650, m({x, z}) \u22650, m({y, z}) \u22650, m(\u0398) \u2265\u2212min n m({x, y}), m({x, z}), m({y, z}) o , (59) so that the space M2 of 2-monotone capacities on {x, y, z} is the set of capacities M2 {x,y,z} = \u001a \u00b5 : 2\u0398 \u2192[0, 1] \f \f \f \f X A\u2286\u0398 m(A) = 1, m(A) \u22650 \u2200A : |A| = 2, m(\u0398) \u2265\u2212min n m({x, y}), m({x, z}), m({y, z}) o\u001b . We can begin to understand the structure of this convex set and its vertices by studying the shape of the convex subset of R3 z \u2265\u2212min{x, y}, x, y \u22650, x + y + z = 1. (60) The latter is the intersection of the following loci: V = {x + y + z = 1}, H1 = {x \u22650}, H2 = {y \u22650}, H\u2032 3 = {z \u2265\u2212x, x \u2264y}, H\u2032\u2032 3 = {z \u2265\u2212y, y \u2264x}, the first of which is a linear space, whereas the others are half-spaces determined by a plane in R3. The resulting convex body is the convex closure Cl([1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, \u22121]) 53 \fdepicted in Fig. 12. The four vertices are associated with all possible binary configurations of the variables x and y, with the z component determined by the other constraints. One can notice that vertices are allowed to have negative mass values on some component (focal elements of \u0398). As a result, we can conjecture that the space of 2-monotone capacities on a frame \u0398, while being embedded in a Cartesian space of the same dimension as the belief space, will contain the belief space by virtue of its extra vertices (e.g. [1, 1, \u22121] in Fig. 12). x y z [1,0,0] [0,1,0] [0,0,1] [1,1,\u00011] Figure 12: The geometry of the convex body (60) in R3. As the geometry of 2-monotone capacities involves a hierarchical set of constraints involving events of increasing cardinality, however, the analysis of the toy problem (60) cannot be directly generalised to it. Geometry of capacities and random sets The geometry of capacities is related to that of random sets, via the capacity functional TX. Recall that Definition 14. A functional TX : K \u2192[0, 1] given by TX(K) = P({X \u2229K \u0338= \u2205}), K \u2208K, (61) is termed the capacity functional of a random closed set X. We will analyse this point further in the near future. 54 \f4.5.2 Geometry of functionals Any study of the geometry of gambles (widely employed in imprecise probability [192]) requires an analysis of the geometry of functionals, i.e., mappings from a space of functions to the real line. The same holds for other uncertainty representations which make use of functions on the real line, such as MV algebras [80] and belief measures on fuzzy events [172]. For a very interesting recent publication (2016) on the geometry of MV algebras in terms of rational polyhedra, see [147]. Geometric functional analysis [107, 108]11 is a promising tool from this perspective. A central question of geometric functional analysis is: what do typical n-dimensional structures look like when n grows to infinity? This is exactly what happens when we try to generalise the notion of a belief space to continuous domains. One of the main tools of geometric functional analysis is the theory of concentration of measure, which offers a geometric view of the limit theorems of probability theory. Geometric functional analysis thus bridges three areas: functional analysis, convex geometry and probability theory [190]. In more detail, a norm on a vector space X is a function \u2225\u2225: X \u2192R that satisfies (i) non-negativity, (ii) homogeneity and (iii) the triangle inequality. A Banach space [87] is a complete normed space. A convex set K in Rn is called symmetric if K = \u2212K (i.e., x \u2208K implies \u2212x \u2208K). Proposition 9. (Banach spaces and symmetric convex bodies) Let X = (Rn, \u2225\u00b7 \u2225) be a Banach space. Then its unit ball BX is a symmetric convex body in Rn. Further, let K be a symmetric convex body in Rn. Then K is the unit ball of some normed space with norm (its Minkowski functional, Fig. 13(a)) \u2225x\u2225K = inf n t > 0, x t \u2208K o . The correspondence between unit balls in Banach spaces and convex bodies established in Proposition 9 allows arguments from convex geometry to be used in functional analysis and vice versa. Using Proposition 9, one can show that subspaces of Banach spaces correspond to sections of convex bodies, and quotient spaces correspond to projections of convex bodies. The phenomenon of concentration of measure was the driving force for the early development of geometric functional analysis. It tells us that anti-intuitive phenomena occur in high dimensions. For example, the \u2018mass\u2019 of a high-dimensional ball is concentrated only in a thin band around any equator. This reflects the philosophy that metric and measure should be treated very differently: a set can have a large diameter but carry little mass [190]. The shape of random projections of symmetric convex bodies onto k-dimensional subspaces (Fig. 13(b)) can also be studied within geometric functional analysis. It is easy to see that a projection captures only the extremal points of K. Hence, one quantity affected by projections is the diameter of the body. 11https://link.springer.com/book/10.1007/978-1-4684-9369-6. 55 \fx x t K / Convex body K Subspace E1 Subspace E2 Projection P 2 Projection P 1 (a) (b) Figure 13: (a) The Minkowski functional of a symmetric convex body in R2 [190]. (b) Projections of a convex body. Proposition 10. (Diameters under random projections, [96]) Let K be a symmetric convex body in Rn, and let P be a projection onto a random subspace of dimension k. Then, with probability at least 1 \u2212e\u2212k, diam(PK) \u2264C   M \u2217(K) + r k ndiam(K) ! , where M \u2217(K) is the mean width of K. 4.5.3 Research questions As this is one of the most challenging topics for future research, most issues are still wide open, in particular the following. Question 14. What is the most appropriate framework for describing the geometry of monotone capacities? Question 15. Can we provide a representation in the theory of convex bodies of the set of 2-monotone capacities similar to that of the belief space, i.e., by providing an analytical expression for the vertices of this set? Question 16. How can the geometry of monotone capacities be exploited to provide a geometric theory of general random sets? Finally, we have the following question. Question 17. How can we frame the geometry of (sets of desirable) gambles and MV algebras in terms of functional spaces? 56 \f4.6 Fancier geometries Representing belief functions as mere vectors of mass or belief values is not entirely satisfactory. Basically, when this is done all vector components are indistinguishable, whereas they in fact correspond to values assigned to subsets of \u0398 of different cardinality. 4.6.1 Exterior algebras Other geometrical representations of belief functions on finite spaces can nevertheless be imagined, which take into account the qualitative difference between events of different cardinalities. For instance, the exterior algebra [102] is a promising tool, as it allows us to encode focal elements of cardinality k with exterior powers (k-vectors) of the form x1 \u2227x2 \u2227 \u00b7 \u00b7 \u00b7\u2227xk and belief functions as linear combinations of decomposable k-vectors, namely \u20d7 bel = X A={x1,...,xk}\u2282\u0398 m(A) x1 \u2227x2 \u2227\u00b7 \u00b7 \u00b7 \u2227xk. Under an exterior algebra representation, therefore, focal elements of different cardinalities are associated with qualitatively different objects (the k-vectors) which encode the geometry of volume elements of different dimensions (e.g. the area of parallelograms in R2). The question remains of how Dempster\u2019s rule may fit into such a representation, and what its relationship with the exterior product operator \u2227is. 4.6.2 Capacities as isoperimeters of convex bodies Convex bodies, as we have seen above, are also the subject of a fascinating field of study [14]. In particular, any convex body in Rn possesses 2n distinct orthogonal projections onto the 2n subspaces generated by all possible subsets of coordinate axes. It is easy to see that, given a convex body K in the Cartesian space Rn, endowed with coordinates x1, . . . , xn, the function \u03bd that assigns to each subset of coordinates S = {xi1, . . . , xik} the (hyper)volume \u03bd(S) of the orthogonal projection K|S of K onto the linear subspace generated by S = {xi1, . . . , xik} is potentially a capacity. In order for that to happen, volumes in linear spaces of different dimensions need to be compared, and conditions need to be enforced in order for volumes of projections onto higher-dimensional subspaces to be greater than those of projections on any of their subspaces. As we have just seen, symmetric convex bodies are in a 1\u20131 relationship with (balls in) Banach spaces, which can then be used as a tool to represent belief measures as projections of symmetric convex bodies. The following research questions arise. Question 18. Under what conditions is the above capacity monotone? Question 19. Under what conditions is this capacity a belief function (i.e. an infinitely monotone capacity)? 57 \f4.7 Relation to integral and stochastic geometry 4.7.1 Integral geometry, or geometric probability This is the theory of invariant measures [141] (with respect to continuous groups of transformations of a space onto itself) on sets consisting of submanifolds of the space (for example lines, planes, geodesics and convex surfaces; in other words, manifolds preserving their type under the transformations in question). Integral geometry has been constructed for various spaces, primarily Euclidean, projective and homogeneous spaces [160], and arose in connection with refinements of statements of problems in geometric probabilities. In order to introduce an invariant measure, one needs to identify a function depending on the coordinates of the space under consideration whose integral over some region of the space is not changed under any continuous coordinate transformation belonging to a specified Lie group. This requires finding an integral invariant of the Lie group. A classic problem of integral geometry is Buffon\u2019s needle problem (dating from 1733). Proposition 11. Some parallel lines on a wooden floor are a distance d apart from each other. A needle of length l < d is randomly dropped onto the floor (Fig. 14, left). Then the probability that the needle will touch one of the lines is P = 2l \u03c0d. d d d l Figure 14: Buffon\u2019s needle problem. Another classical problem is Firey\u2019s colliding dice problem (from 1974). Proposition 12. Suppose \u21261 and \u21262 are disjoint unit cubes in R3. In a random collision, the probability that the cubes collide edge-to-edge slightly exceeds the probability that the cubes collide corner-to-face. In fact, 0.54 \u224aP(edge-to-edge collision) > P(corner-to-face collision) \u224a0.46. 58 \fRoughly speaking, integral geometry is the problem of computing probabilities of events which involve geometric loci, as opposed to the geometric approach to uncertainty supported in [50], which is about understanding the geometric properties of (uncertainty) measures. The main technical tool is Poincar\u00b4 e\u2019s integral-geometric formula. After representing lines in the plane using coordinates (p, \u03b8), cos(\u03b8)x + sin(\u03b8)y = p, the kinematic measure for such lines is defined as dK = dp \u2227d\u03b8, as the measure on a set of lines which is invariant under rigid motion, i.e., the Jacobian of the coordinate transformation is equal to 1. Now, let C be a piecewise C1 curve in the plane. Given a line L in the plane, let n(L \u2229C) be the number of intersection points. If C contains a linear segment and L agrees with that segment, n(C \u2229L) = \u221e. Proposition 13. (Poincar\u00b4 e formula for lines, 1896) Let C be a piecewise C1 curve in the plane. Then the measure of unoriented lines meeting C, counted with multiplicity, is given by 2L(C) = Z L:L\u2229C\u0338=\u2205 n(C \u2229L) dK(L). Integral-geometric formulae hold for convex sets \u2126as well. Since n(L \u2229\u2202\u2126), where \u2202\u2126is the boundary of \u2126, is either zero or two for dK-almost all lines L, the measure of unoriented lines that meet the convex set \u2126is given by L(\u2202\u2126) = Z L:L\u2229\u2126\u0338=\u2205 dK. One can then prove the following. Proposition 14. (Sylvester\u2019s problem, 1889) Let \u0398 \u2282\u2126be two bounded convex sets in the plane. Then the probability that a random line meets \u0398 given that it meets \u2126is P = L(\u2202\u0398)/L(\u2202\u2126). Corollary 5. Let C be a piecewise C1 curve contained in a compact convex set \u2126. The expected number of intersections with C of all random lines that meet \u2126is E(n) = 2L(C) L(\u2202\u2126). Integral geometry potentially provides a powerful tool for designing new geometric representations of belief measures, on the one hand. On the other hand, we can foresee generalisations of integral geometry in terms of non-additive measures and capacities. Question 20. How does geometric probability generalise when we replace standard probabilities with other uncertainty measures? 4.7.2 Stochastic geometry Stochastic geometry is a branch of probability theory which deals with set-valued random elements. It describes the behaviour of random configurations such as random graphs, random networks, random cluster processes, random unions of convex sets, random mosaics and many other random geometric structures [109]. The topic is very 59 \fmuch intertwined with the theory of random sets, and Matheron\u2019s work in particular. The name appears to have been coined by David Kendall and Klaus Krickeberg [22]. Whereas geometric probability considers a fixed number of random objects of a fixed shape and studies their interaction when some of the objects move randomly, since the 1950s the focus has switched to models involving a random number of randomly chosen geometric objects. As a consequence, the notion of a point process started to play a prominent role in this field. Definition 15. A point process \u03b7 is a measurable map from some probability space (\u2126, F, P) to the locally finite subsets of a Polish space X (endowed with a suitable \u03c3algebra), called the state space. The intensity measure of \u03b7, evaluated at a measurable set A \u2282X, is defined by \u00b5(A) = E[\u03b7(A)], and equals the mean number of elements of \u03b7 lying in A. Typically, X is either Rn, the space of compact subsets of Rn or the set of all affine subspaces there (the Grassmannian). A point process can be written as \u03b7 = P\u03c4 i=1 \u03b4\u03b6i, where \u03c4 is a random variable taking values in N0 \u222a{\u221e}, and \u03b61, \u03b62, . . . is a sequence of random points in X (Fig. 15). \u2018Random point field\u2019 is sometimes considered a more appropriate terminology, as it avoids confusion with stochastic processes. A point process is, in fact, a special case of a random element, a concept introduced by Maurice Fr\u00b4 echet [91] as a generalisation of the idea of a random variable to situations in which the outcome is a complex structure (e.g., a vector, a function, a group [18], a series or a subset). (a) (b) (c) Figure 15: Examples of point processes (from http://stats.stackexchange.com/questions/16282/ how-to-generate-nd-point-process : (a) grid; (b) Poisson; (c) uniform. Definition 16. Let (\u2126, F, P) be a probability space, and (E, E) a measurable space. A random element with values in E is a function X : \u2126\u2192E which is (F, E)measurable, that is, a function X such that for any B \u2208E, the pre-image of B lies in F. Random elements with values in E are sometimes called E-valued random variables (hence the connection with random sets). Geometric processes [127] are defined as point processes on manifolds that represent spaces of events. Thus, processes of straight lines in the plane are defined as 60 \fpoint processes on a M\u00a8 obius strip (as the latter represents the space of straight lines in R2). Processes on manifolds form a more general concept; here, once again, stochastic geometry is linked with the theory of random sets [141]. The \u2018intrinsic volume\u2019 of convex bodies and Minkowski combinations of them (polyconvex bodies) is a topic of stochastic geometry which relates to the questions we posed in Section 4.6.2. A peculiarity which distinguishes stochastic geometry from the theory of random sets is the interest stochastic geometry has for geometric processes with distributions that are invariant relative to groups acting on the fundamental space X. For instance, one may examine the class of those processes of straight lines on R2 which are invariant relative to the group of Euclidean motions of the plane. 5 High-impact developments The theory of random sets is very well suited to tackling a number of high-profile applications, which range from the formulation of cautious climate change prediction models (Section 5.1) to the design of novel machine learning tool generalising classical mathematical frameworks (e.g. max-entropy classifiers and random forests, Section 5.2) and to the analysis of the very foundations of statistical machine learning (Section 5.3). Exciting work has recently been conducted on Bayesian formulations of the foundations of quantum mechanics [17, 8]. Further analyses of the problem in the extended framework of imprecise probabilities are likely to follow. 5.1 Climate change Climate change12 is a paramount example of a problem which requires predictions to be made under heavy uncertainty, due to the imprecision associated with any climate model, the long-term nature of the predictions involved and the second-order uncertainty affecting the statistical parameters involved. A typical question a policymaker may ask a climate scientist is, for example [159], \u201cWhat is the probability that a doubling of atmospheric CO2 from pre-industrial levels will raise the global mean temperature by at least 2\u25e6C?\u201d Rougier [159] nicely outlined a Bayesian approach to climate modelling and prediction, in which a predictive distribution for a future climate is found by conditioning the future climate on observed values for historical and current climates. More recent papers on the Bayesian modelling of climate change include, for instance, [142, 179, 115]. A number of challenges arise when modelling climate change in a Bayesian framework: \u2022 in climate prediction, the collection of uncertain quantities for which the climate scientist must specify prior probabilities can be large; 12https://link.springer.com/journal/10584. 61 \f\u2022 specifying a prior distribution over climate vectors is very challenging. According to Rougier, considering that people are spending thousands of hours on collecting climate data and constructing climate models, it is surprising that little attention is being devoted to quantifying our judgements about how the two are related. In this section, climate is represented as a vector of measurements y, collected at a given time. Its components include, for instance, the level of CO2 concentration on the various points of a grid. More precisely, the climate vector y = (yh, yf) collects together both historical and present (yh) and future (yf) climate values. A measurement error e is introduced to take into account errors due to, for instance, a seasick technician or atmospheric turbulence. The actual measurement vector is therefore z . = yh + e. The Bayesian treatment of the problem makes use of a number of assumptions. For starters, we have the following axioms. Axiom 1. Climate and measurement error are independent: e\u22a5y. Axiom 2. The measurement error is Gaussian distributed, with mean\u20d7 0 and covariance \u03a3e: e \u223cN(\u20d7 0, \u03a3e). Thanks to these assumptions, the predictive distribution for the climate given the measured values z = \u02dc z is p(y|z = \u02dc z) \u223cN(\u02dc z \u2212yh|\u20d7 0, \u03a3e)p(y), (62) which requires us to specify a prior distribution for the climate vector y itself. 5.1.1 Climate models The choice of such a prior p(y) is extremely challenging, because y is such a large collection of quantities, and these component quantities are linked by complex interdependencies, such as those arising from the laws of nature. The role of the climate model is then to induce a distribution for the climate itself, and plays the role of a parametric model in statistical inference. Namely, a climate model is a deterministic mapping from a collection of parameters x (equation coefficients, initial conditions, forcing functions) to a vector of measurements (the \u2018climate\u2019), namely x \u2192y = g(x), (63) where g belongs to a predefined \u2018model space\u2019 G. Climate scientists call an actual value of g(x), computed for some specific set of parameter values x, a model evaluation. The reason is that the analytical mapping (63) is generally not known, and only images of specific input parameters can be computed or sampled (at a cost). A climate scientist considers, on a priori grounds based on their past experience, that some choices of x are better than others, i.e., that there exists a set of parameter values x\u2217such that y = g(x\u2217) + \u03f5\u2217, where \u03f5\u2217is termed the \u2018model discrepancy\u2019. 62 \f5.1.2 Prediction via a parametric model The difference between the climate vector and any model evaluation can be decomposed into two parts: y \u2212g(x) = g(x\u2217) \u2212g(x) + \u03f5\u2217. The first part is a contribution that may be reduced by a better choice of the model g; the second part is, in contrast, an irreducible contribution that arises from the model\u2019s own imperfections. Note that x\u2217is not just a statistical parameter, though, for it relates to physical quantities, so that climate scientists have a clear intuition about its effects. Consequently, scientists may be able to exploit their expertise to provide a prior p(x\u2217) on the input parameters. In this Bayesian framework for climate prediction, two more assumptions are needed. Axiom 3. The \u2018best\u2019 input, discrepancy and measurement error are mutually (statistically) independent: x\u2217\u22a5\u03f5\u2217\u22a5e. Axiom 4. The model discrepancy \u03f5\u2217is Gaussian distributed, with mean \u20d7 0 and covariance \u03a3\u03f5. Axioms 3 and 4 then allow us to compute the desired climate prior as p(y) = Z N(y \u2212g(x\u2217)|\u20d7 0, \u03a3\u03f5)p(x\u2217) dx\u2217, (64) which can be plugged into (62) to yield a Bayesian prediction of future climate values. In practice, as we have said, the climate model function g(.) is not known \u2013 we only possess a sample of model evaluations {g(x1), . . . , g(xn)}. We call the process of tuning the covariances \u03a3\u03f5, \u03a3e and checking the validity of the Gaussianity assumptions (Axioms 2 and 4) model validation. This can be done by using (62) to predict past/present climates p(z), and applying some hypothesis testing to the result. If the observed value \u02dc z is in the tail of the distribution, the model parameters (if not the entire set of model assumptions) need to be corrected. As Rougier admits [159], responding to bad validation results is not straightforward. 5.1.3 Model calibration Assuming that the model has been validated, it needs to be \u2018calibrated\u2019, i.e., we need to find the desired \u2018best\u2019 value x\u2217of the model\u2019s parameters. Under Axioms 3\u20136, we can compute p(x\u2217|z = \u02dc z) = p(z = \u02dc z|x\u2217) = N(\u02dc z = g(x\u2217)|\u20d7 0, \u03a3\u03f5 + \u03a3e)p(x\u2217). As we know, maximum a posteriori estimation could be applied to the above posterior distribution: however, the presence of multiple modes might make it ineffective. As an alternative, we can apply full Bayesian inference to compute p(yf|z = \u02dc z) = Z p(yf|x\u2217, z = \u02dc z)p(x\u2217|z = \u02dc z) dx\u2217, (65) 63 \fwhere p(yf|x\u2217, z = \u02dc z) is Gaussian with a mean which depends on \u02dc z \u2212g(x). The posterior prediction (65) highlights two routes for climate data to impact on future climate predictions: \u2022 by concentrating the distribution p(x\u2217|z = \u02dc z) relative to the prior p(x\u2217), depending on both the quantity and the quality of the climate data; \u2022 by shifting the mean of p(yf|x\u2217, z = \u02dc z) away from g(x), depending on the size of the difference \u02dc z \u2212g(x). 5.1.4 Role of model evaluations Let us go back to the initial question: what is the probability that a doubling of atmospheric CO2 will raise the global mean temperature by at least 2\u25e6C by 2100? Let Q \u2282Y be the set of climates y for which the global mean temperature is at least 2\u25e6C higher in 2100. The probability of the event of interest can then be computed by integration, as follows: Pr(yf \u2208Q|z = \u02dc z) = Z f(x\u2217)p(x\u2217|z = \u02dc z) dx\u2217. The quantity f(x) . = R Q N(yf|\u00b5(x), \u03a3) dyf can be computed directly, while the overall integral in dx\u2217requires numerical integration, for example via a naive Monte-Carlo approach, Z \u223c = Pn i=1 f(xi) n , xi \u223cp(x\u2217|z = \u02dc z), or by weighted sampling Z \u223c = Pn i=1 wif(xi) n , xi \u223cp(x\u2217|z = \u02dc z), weighted by the likelihood wi \u221dp(z = \u02dc z|x\u2217= xi). A trade-off appears, in the sense that sophisticated models which take a long time to evaluate may not provide enough samples for the prediction to be statistically significant \u2013 however, they may make the prior p(x\u2217) and covariance \u03a3\u03f5 easier to specify. 5.1.5 Modelling climate with belief functions As we have seen, a number of issues arise in making climate inferences in the Bayesian framework. Numerous assumptions are necessary to make calculations practical, which are only weakly related to the actual problem to be solved. Although the prior on climates is reduced to a prior p(x\u2217) on the parameters of a climate model, there is no obvious way of picking the latter (confirming our criticism of Bayesian inference in the Introduction). In general, it is far easier to say what choices are definitively wrong (e.g. uniform priors) than to specify \u2018good\u2019 ones. Finally, significant parameter tuning is required (e.g. to set the values of \u03a3\u03f5, \u03a3e and so on). Formulating a climate prediction method based on the theory of belief functions will require quite a lot of work, but a few landmark features can already be identified: 64 \f\u2022 such a framework will have to avoid committing to priors p(x\u2217) on the \u2018correct\u2019 climate model parameters; \u2022 it will likely use a climate model as a parametric model to infer either a belief function on the space of climates Y \u2022 or a belief function on the space of parameters (e.g. covariances, etc.) of the distribution on Y. Question 21. Can a climate model allowing predictions about future climate be formulated in a belief function framework, in order to better describe the huge Knightian uncertainties involved? In recent times, efforts to model climate change using belief functions have been conducted [1, 123], for instance by Kriegler and co-authors [124], but more could and should be done. 5.2 Machine learning Machine learning provides a clear case for the mathematical modelling of epistemic uncertainty, in particular via random set and belief function theory, as the issue of learning models capable of making predictions which are robust to variations in the process generating the data is foundational in machine learning. 5.2.1 Modelling uncertainty in machine learning A significant amount of work has been done in this sense over the years. Denoeux and co-authors, together with others, [92], have much contributed to unsupervised learning and clustering, in particular [140]. The combination of the outcomes of multiple classifiers has been investigated within the theory of evidence [200] [158], and evidential approaches to decision trees [85] and K-nearest neighbour classifiers [73] have been proposed. However, most of the sparse attempts made so far to incorporate uncertainty in neural predictions [110, 93] have been made from a Bayesian stance, which only captures aleatory uncertainty, and do not fully model epistemic uncertainty. A recent wave of papers have brought forward various distinct approaches to evidential deep classification [185], in particular via a second-order Dirichlet parameter representation [161], imprecise neural networks [82], interval neural networks [168] or convolutional neural networks outputting random sets [136, 135] rather than conventional probability vectors. Here, we focus on a specific type of classifier as a representative of the issues involved. 5.2.2 Maximum entropy classification Maximum entropy classifiers13 are a widespread tool in machine learning. There, the Shannon entropy of a sought joint (or conditional) probability distribution p(Ck|x) of 13http://www.kamalnigam.com/papers/maxent-ijcaiws99.pdf. 65 \fobserved data x and an associated class Ck \u2208C = {C1, . . . , CK} is maximised, following the maximum entropy principle that the least informative distribution matching the available evidence should be chosen. Once one has picked a set of feature functions, chosen to efficiently encode the training information, the joint distribution is subject to the constraint that the empirical expectation of the feature functions equals the expectation associated with it. Clearly, the two assumptions that (i) the training and test data come from the same probability distribution, and that (ii) the empirical expectation of the training data is correct, and the model expectation should match it, are rather strong, and work against generalisation power. A more robust learning approach can instead be sought by generalising max-entropy classification in a belief function setting. We take the view here that a training set does not provide, in general, sufficient information to precisely estimate the joint probability distribution of the class and the data. We assume instead that a belief measure can be estimated, providing lower and upper bounds on the joint probability of the data and class. As in the classical case, an appropriate measure of entropy for belief measures is maximised. In opposition to the classical case, however, the empirical expectation of the chosen feature functions is assumed to be merely \u2018compatible\u2019 with the lower and upper bounds associated with the sought belief measure. This leads to a constrained optimisation problem with inequality constraints, rather than equality ones, which needs to be solved by looking at the Karush-Kuhn-Tucker conditions. A shorter version of what follows was published in [49]. 5.2.3 Generalised max-entropy classifiers Given a training set in which each observation is attached to a class, namely D = n (xi, yi), i = 1, . . . , N \f \f \fxi \u2208X, yi \u2208C o , (66) a set M of feature maps is designed, \u03d5(x, Ck) = [\u03d51(x, Ck), . . . , \u03d5M(x, Ck)]\u2032, (67) whose values depend on both the object observed and its class. Each feature map \u03d5m : X \u00d7 C \u2192R is then a random variable, whose expectation is obviously E[\u03d5m] = X x,k p(x, Ck)\u03d5m(x, Ck). (68) In opposition, the empirical expectation of \u03d5m is \u02c6 E[\u03d5m] = X x,k \u02c6 p(x, Ck)\u03d5m(x, Ck), (69) where \u02c6 p is a histogram constructed by counting occurrences of the pair (x, Ck) in the training set: \u02c6 p(x, Ck) = 1 N X (xi,yi)\u2208D \u03b4(xi = x \u2227yi = Ck). 66 \fThe theoretical expectation (68) can be approximated by decomposing via Bayes\u2019 rule p(x, Ck) = p(x)p(Ck|x), and approximating the (unknown) prior of the observations p(x) by the empirical prior \u02c6 p, i.e., the histogram of the observed values in the training set: \u02dc E[\u03d5m] = X x,k \u02c6 p(x)p(Ck|x)\u03d5m(x, Ck). For values of x that are not observed in the training set, \u02c6 p(x) is obviously equal to zero. Definition 17. Given a training set (66) related to the problem of classifying x \u2208X as belonging to one of the classes C = {C1, . . . , CK}, the max-entropy classifier is the conditional probability p\u2217(Ck|x) which solves the following constrained optimisation problem: p\u2217(Ck|x) . = arg max p(Ck|x) Hs(P), (70) where Hs is the traditional Shannon entropy, subject to \u02dc Ep[\u03d5m] = \u02c6 E[\u03d5m] \u2200m = 1, . . . , M. (71) The constraint (71) requires the classifier to be consistent with the empirical frequencies of the features in the training set, while seeking the least informative probability distribution that does so (70). The solution of the maximum entropy classification problem (Definition 17) is the so-called log-linear model, p\u2217(Ck|x) = 1 Z\u03bb(x)e P m \u03bbm\u03d5m(x,Ck), (72) where \u03bb = [\u03bb1, . . . , \u03bbM]\u2032 are the Lagrange multipliers associated with the linear constraints (71) in the above constrained optimisation problem, and Z\u03bb(x) is a normalisation factor. The classification function associated with (72) is y(x) = arg max k X m \u03bbm\u03d5m(x, Ck), i.e., x is assigned to the class which maximises the linear combination of the feature functions with coefficients \u03bb. Generalised maximum entropy optimisation problem When generalising the maximum entropy optimisation problem (Definition 17) to the case of belief functions, we need to (i) choose an appropriate measure of entropy for belief functions as the objective function, and (ii) revisit the constraints that the (theoretical) expectations of the feature maps are equal to the empirical ones computed over the training set. As for (i), manifold generalisations of the Shannon entropy to the case of belief measures exist ([50], Chapter 4). As for (ii), it is sensible to require that the empirical 67 \fexpectation of the feature functions is bracketed by the lower and upper expectations associated with the sought belief function Bel : 2X\u00d7C \u2192[0, 1], namely X (x,Ck)\u2208X\u00d7C Bel(x, Ck)\u03d5m(x, Ck) \u2264\u02c6 E[\u03d5m] \u2264 X (x,Ck)\u2208X\u00d7C Pl(x, Ck)\u03d5m(x, Ck). (73) Note that this makes use only of the 2-monotonicity of belief functions, as only probability intervals on singleton elements of X \u00d7 C are considered. Question 22. Should constraints of this form be enforced on all possible subsets A \u2282 X \u00d7 C, rather than just singleton pairs (x, Ck)? This is linked to the question of what information a training set actually carries (see Section 5.3). More general constraints would require extending the domain of feature functions to set values \u2013 we will investigate this idea in the near future. We can thus pose the generalised maximum-entropy optimisation problem as follows [49]. Definition 18. Given a training set (66) related to the problem of classifying x \u2208X as belonging to one of the classes C = {C1, . . . , CK}, the maximum random set entropy classifier is the joint belief measure Bel\u2217(x, Ck) : 2X\u00d7C \u2192[0, 1] which solves the following constrained optimisation problem: Bel\u2217(x, Ck) . = arg max Bel(x,Ck) H(Bel), (74) subject to the constraints X x,k Bel(x, Ck)\u03d5m(x, Ck) \u2264\u02c6 E[\u03d5m] \u2264 X x,k Pl(x, Ck)\u03d5m(x, Ck) \u2200m = 1, . . . , M, (75) where H is an appropriate measure of entropy for belief measures. Question 23. An alternative course of action is to pursue the least committed, rather than the maximum-entropy, belief function satisfying the constraints (75). This is left for future work. KKT conditions and sufficiency As the optimisation problem in Definition 18 involves inequality constraints (75), as opposed to the equality constraints of traditional max-entropy classifiers (71), we need to analyse the KKT (recall Definition 7) [114] necessary conditions for a belief function Bel to be an optimal solution to the problem. Interestingly, the KKT conditions are also sufficient whenever the objective function f is concave, the inequality constraints gi are continuously differentiable convex functions and the equality constraints hj are affine.14 This provides us with a way to tackle the generalised maximum entropy problem. 14More general sufficient conditions can be given in terms of invexity [7] requirements. A function f : Rn \u2192R is \u2018invex\u2019 if there exists a vector-valued function g such that f(x) \u2212f(u) \u2265g(x, u)\u2207f(u). Hanson [105] proved that if the objective function and constraints are invex with respect to the same function g, then the KKT conditions are sufficient for optimality. 68 \fConcavity of the entropy objective function As for the objective function, we recalled that several generalisations of the standard Shannon entropy to random sets are possible [113]. It is a well-known fact that the Shannon entropy is a concave function of probability distributions, represented as vectors of probability values.15 A number of other properties of concave functions are well known: any linear combination of concave functions is concave, a monotonic and concave function of a concave function is still concave, and the logarithm is a concave function. Relevantly, the transformations which map mass vectors to vectors of belief (and commonality) values are linear, as they can be expressed in the form of matrices. In particular, \u20d7 bel = BfrM \u20d7 m, where BfrM is a matrix whose (A, B) entry is BfrM(A, B) = \u001a 1 B \u2286A, 0 otherwise. The same can be said of the mapping \u20d7 q = QfrM \u20d7 m between a mass vector and the associated commonality vector. As a consequence, belief, plausibility and commonality are all linear (and therefore concave) functions of a mass vector. Using this matrix representation, it is easy to conclude that several of the entropies proposed by various authors for belief measures are indeed concave. In particular, Smets\u2019s specificity measure Ht = X A log(1/Q(A)) is concave, as a linear combination of concave functions. Nguyen\u2019s entropy Hn = \u2212 X A m(A) log(m(A)) = Hs(m) is also concave as a function of m, as the Shannon entropy of a mass assignment. Dubois and Prade\u2019s measure Hd = P A m(A) log(|A|) is also concave with respect to m, as a linear combination of mass values. Straighforward applications of the Shannon entropy function to Bel and Pl, HBel[m] = Hs[Bel] = X A\u2286\u0398 Bel(A) log \u0012 1 Bel(A) \u0013 (76) and HP l[m] = Hs[Pl] = X A\u2286\u0398 Pl(A) log \u0012 1 Pl(A) \u0013 , are also trivially concave, owing to the concavity of the entropy function and to the linearity of the mapping from m to Bel, Pl. Drawing conclusions about other entropy measures is less immediate, as they involve products of concave functions (which are not, in general, guaranteed to be concave). 15http://projecteuclid.org/euclid.lnms/1215465631 69 \fConvexity of the interval expectation constraints As for the constraints (75) of the generalised maximum entropy problem, we first note that (75) can be decomposed into the following pair of constraints: g1 m(m) . = X x,k Bel(x, Ck)\u03d5m(x, Ck) \u2212\u02c6 E[\u03d5m] \u22640, g2 m(m) = X x,k \u03d5m(x, Ck)[\u02c6 p(x, Ck) \u2212Pl(x, Ck)] \u22640, for all m = 1, . . . , M. The first inequality constraint is a linear combination of linear functions of the sought mass assignment m\u2217: 2X\u00d7C \u2192[0, 1] (since Bel\u2217results from applying a matrix transformation to m\u2217). As \u20d7 pl = 1 \u2212J \u20d7 bel = 1 \u2212JBfrM \u20d7 m, the constraint on g2 m is also a linear combination of mass values. Hence, as linear combinations, the constraints on g1 m and g2 m are both concave and convex. KKT conditions for the generalised maximum entropy problem We can conclude the following. Theorem 8. If either Ht, Hn, Hd, HBel or HP l is adopted as the measure of entropy, the generalised maximum entropy optimisation problem (Definition 18) has a concave objective function and convex constraints. Therefore, the KKT conditions are sufficient for the optimality of its solution(s). Let us then compute the KKT conditions (see Definition 7) for the Shannon-like entropy (76). Condition 1 (stationarity), applied to the sought optimal belief function Bel\u2217: 2X\u00d7C \u2192[0, 1], reads as \u2207HBel(Bel\u2217) = M X m=1 \u00b51 m\u2207g1 m(Bel\u2217) + \u00b52 m\u2207g2 m(Bel\u2217). The components of \u2207HBel(Bel) are the partial derivatives of the entropy with respect to the mass values m(B), for all B \u2286\u0398 = X \u00d7 C. They read as \u2202HBel \u2202m(B) = \u2202 \u2202m(B) X A\u2287B \uf8ee \uf8f0\u2212 \uf8eb \uf8edX B\u2286A m(B) \uf8f6 \uf8f8log \uf8eb \uf8edX B\u2286A m(B) \uf8f6 \uf8f8 \uf8f9 \uf8fb = \u2212 X A\u2287B \uf8ee \uf8f01 \u00b7 log \uf8eb \uf8edX B\u2286A m(B) \uf8f6 \uf8f8+ \uf8eb \uf8edX B\u2286A m(B) \uf8f6 \uf8f8 1 \u0010P B\u2286A m(B) \u0011 \uf8f9 \uf8fb = \u2212 X A\u2287B \uf8ee \uf8f01 + log \uf8eb \uf8edX B\u2286A m(B) \uf8f6 \uf8f8 \uf8f9 \uf8fb= \u2212 X A\u2287B [1 + log Bel(A)]. 70 \fFor \u2207g1 m(Bel), we easily have \u2202g1 m \u2202m(B) = \u2202 \u2202m(B) X (x,Ck)\u2208\u0398 Bel(x, Ck)\u03d5m(x, Ck) \u2212\u02c6 E[\u03d5m] = \u2202 \u2202m(B) X (x,Ck)\u2208\u0398 m(x, Ck)\u03d5m(x, Ck) \u2212\u02c6 E[\u03d5m] = \u001a \u03d5m(x, Ck) B = {(x, Ck)}, 0 otherwise. (77) Note that, if we could define feature functions over non-singleton subsets A \u2286\u0398, (77) would simply generalise to \u2202g1 m \u2202m(B) = \u03d5(B) \u2200B \u2286\u0398. As for the second set of constraints, \u2202g2 m \u2202m(B) = \u2202 \u2202m(B) X (x,Ck)\u2208\u0398 \u03d5m(x, Ck)[\u02c6 p(x, Ck) \u2212Pl(x, Ck)] = \u2202 \u2202m(B) X (x,Ck)\u2208\u0398 \u03d5m(x, Ck) \uf8ee \uf8f0\u02c6 p(x, Ck) \u2212 X B\u2229{(x,Ck)}\u0338=\u2205 m(B) \uf8f9 \uf8fb = \u2202 \u2202m(B) \uf8eb \uf8ed\u2212 X (x,Ck)\u2208\u0398 \u03d5m(x, Ck) X B\u2287{(x,Ck)} m(B) \uf8f6 \uf8f8 = \u2212 X (x,Ck)\u2208B \u03d5m(x, Ck). Putting everything together, the KKT stationarity conditions for the generalised maximum entropy problem amount to the following system of equations: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \u2212 X A\u2287B [1 + log Bel(A)] = M X m=1 \u03d5m(x, Ck)[\u00b51 m \u2212\u00b52 m], B = {(x, Ck)}, \u2212 X A\u2287B [1 + log Bel(A)] = M X m=1 \u00b52 m X (x,Ck)\u2208B \u03d5m(x, Ck), B \u2282X \u00d7 C, |B| > 1. (78) The other conditions are (75) (primal feasibility), \u00b51 m, \u00b52 m \u22650 (dual feasibility) and 71 \fcomplementary slackness, \u00b51 m \uf8ee \uf8f0 X (x,Ck)\u2208\u0398 Bel({(x, Ck)})\u03d5m(x, Ck) \u2212\u02c6 E[\u03d5m] \uf8f9 \uf8fb= 0, \u00b52 m \uf8ee \uf8f0 X (x,Ck)\u2208\u0398 \u03d5m(x, Ck) h \u02c6 p(x, Ck) \u2212Pl({(x, Ck)}) i \uf8f9 \uf8fb= 0 (79) In the near future we will address the analytical form of the solution of the above system of constraints, and its alternative versions associated with different entropy measures. Question 24. Derive the analytical form of the solution to the above generalised maximum entropy classification problem. Question 25. How does this analytical solution compare with the log-linear model solution to the traditional maximum entropy problem? Question 26. Derive and compare the constraints and analytical solutions for the alternative generalised maximum entropy frameworks obtained by plugging in other concave generalised entropy functions. 5.3 Generalising statistical learning theory Current machine learning paradigms focus on explaining the observable outputs in the training data (\u2018overfitting\u2019), which may, for instance, lead an autonomous driving system to perform well in validation training sessions but fail catastrophically when tested in the real world. Methods are often designed to handle a specific set of testing conditions, and thus are unable to predict how a system will behave in a radically new setting (e.g., how would a smart car [94] cope with driving in extreme weather conditions?). With the pervasive deployment of machine learning algorithms in \u2018mission-critical\u2019 artificial intelligence systems for which failure is not an option (e.g. smart cars navigating a complex, dynamic environment, robot surgical assistants capable of predicting the surgeon\u2019s needs even before they are expressed, or smart warehouses monitoring the quality of the work conducted: see Fig. 16), it is imperative to ensure that these algorithms behave predictably \u2018in the wild\u2019. Vapnik\u2019s classical statistical learning theory [187, 189, 188] is effectively useless for model selection, as the bounds on generalisation errors that it predicts are too wide to be useful, and rely on the assumption that the training and testing data come from the same (unknown) distribution. As a result, practitioners regularly resort to crossvalidation16 to approximate the generalisation error and identify the parameters of the optimal model. 16https://en.wikipedia.org/wiki/Cross-validation (statistics). 72 \fFigure 16: New applications of machine learning to unconstrained environments (e.g. smart vehicles, left, or robotic surgical assistants, right) require new, robust foundations for statistical learning theory. 5.3.1 PAC learning Statistical learning theory [187] considers the problem of predicting an output y \u2208Y given an input x \u2208X, by means of a mapping h : X \u2192Y, h \u2208H called a model or hypothesis, which lives in a specific model space H. The error committed by a model is measured by a loss function l : (X \u00d7 Y) \u00d7 H \u2192R, for instance the zero\u2013one loss l((x, y), h) = I[y \u0338= h(x)]. The input\u2013output pairs are assumed to be generated a probability distribution p\u2217. The expected risk of a model h, L(h) . = E(x,y)\u223cp\u2217[l((x, y), h)], (80) is measured as its expected loss l assuming that the pairs (x1, y1), (x2, y2), . . . are sampled i.i.d. from the probability distribution p\u2217. The expected risk minimiser, h\u2217. = arg min h\u2208H L(h), (81) is any hypothesis in the given model space that minimises the expected risk (not a random quantity at all). Given n training examples, also drawn i.i.d. from p\u2217, the empirical risk of a hypothesis h is the average loss over the available training set D = {(x1, y1), . . . , (xn, yn)}: \u02c6 L(h) . = 1 n n X i=1 l((xi, yi), h). (82) We can then define the empirical risk minimiser (ERM), namely \u02c6 h . = arg min h\u2208H \u02c6 L(h), (83) which is instead a random variable which depends on the collection of training data. Statistical learning theory is interested in the expected risk of the ERM (which is the only thing we can compute from the available training set). The core notion is that of probably approximately correct algorithms. 73 \fDefinition 19. A learning algorithm is probably approximately correct (PAC) if it finds with probability at least 1 \u2212\u03b4 a model h \u2208H which is \u2018approximately correct\u2019, i.e., it makes a training error of no more than \u03f5. According to this definition, PAC learning aims at providing bounds of the kind P[L(\u02c6 h) \u2212L(h\u2217) > \u03f5] \u2264\u03b4, on the difference between the loss of the ERM and the minimum theoretical loss for that class of models. This is to account for the generalisation problem, due to the fact that the error we commit when training a model is different from the error one can expect in general cases in which data we have not yet observed are presented to the system. As we will see in more detail later on, statistical learning theory makes predictions about the reliability of a training set, based on simple quantities such as the number of samples n. In particular, the main result of PAC learning, in the case of finite model spaces H, is that we can relate the required size n of a training sample to the size of the model space H, namely log |H| \u2264n\u03f5 \u2212log 1 \u03b4 , so that the minimum number of training examples given \u03f5, \u03b4 and |H| is n \u22651 \u03f5 \u0012 log |H| + log 1 \u03b4 \u0013 . 5.3.2 Vapnik\u2013Chervonenkis dimension Obviously, for infinite-dimensional hypothesis spaces H, the previous relation does not make sense. However, a similar constraint on the number of samples can be obtained after we introduce the following notion. Definition 20. The Vapnik\u2013Chervonenkis (VC) dimension17 of a model space H is the maximum number of points that can be successfully \u2018shattered\u2019 by a model h \u2208H (i.e., they can be correctly classified by some h \u2208H for all possible binary labellings of these points). Unfortunately, the VC dimension dramatically overestimates the number of training instances required. Nevertheless, arguments based on the VC dimension provide the only justification for max-margin linear SVMs, the dominant force in classification for the last two decades, before the advent of deep learning. As a matter of fact, for the space Hm of linear classifiers with margin m, the following expression holds: V CSVM = min \u001a d, 4R2 m2 \u001b + 1, where d is the dimensionality of the data and R is the radius of the smallest hypersphere enclosing all the training data. As the VC dimension of Hm decreases as m grows, it is then desirable to select the linear boundaries which have maximum margin, as in support vector machines. 17http://www.svms.org/vc-dimension/. 74 \f5.3.3 Towards imprecise-probabilistic foundations for machine learning These issues with Vapnik\u2019s traditional statistical learning theory have recently been recognised by many scientists. There is exciting research exploring a variety of options, for instance risk-sensitive reinforcement learning [169]; approaches based on situational awareness, i.e., the ability to perceive the current status of an environment in terms of time and space, comprehend its meaning and behave with a corresponding risk attitude; robust optimisation using minimax learning [210]; and budgeted adversary models [118]. The latter, rather than learning models that predict accurately on a target distribution, use minimax optimisation to learn models that are suitable for any target distribution within a \u2018safe\u2019 family. A portfolio of models have been proposed [16], including Satzilla [201] and IBM Watson (http://www.ibm.com/watson/), which combines more than 100 different techniques for analysing natural language, identifying sources, finding and generating hypotheses, finding and scoring evidence, and merging and ranking hypotheses. By an ensemble of models, however, these methods mean sets of models of different nature (e.g. SVMs, random forests) or models of the same kind learned from different slices of data (e.g. boosting). Exciting recent progress in domain adaptation includes multiview learning [11], multifaceted understanding [12] and learning of transferable components [98, 132]. Some consensus exists that robust approaches should provide worst-case guarantees, as it is not possible to rule out completely unexpected behaviours or catastrophic failures. The minimax approach, in particular, evokes the concept of imprecise probability, at least in its credal, robust-statistical incarnation. As we know, imprecise probabilities naturally arise whenever the data are insufficient to allow the estimation of a probability distribution. Indeed, training sets in virtually all applications of machine learning constitute a glaring example of data which is both: \u2022 insufficient in quantity (think of a Google routine for object detection from images, trained on a few million images, compared with the thousands of billions of images out there), and \u2022 insufficient in quality (as they are selected based on criteria such as cost, availability or mental attitudes, therefore biasing the whole learning process). Uncertainty theory therefore has the potential to address model adaptation in new, original ways, so providing a new paradigm for robust statistical learning. A sensible programme for future research could then be based on the following developments (see Fig. 17): 1. Addressing the domain adaptation problem by allowing the test data to be sampled from a different probability distribution than the training data, under the weaker assumption that both belong to the same convex set of distributions (credal set) (Fig. 17(a)). 2. Extending further the statistical-learning-theory framework by moving away from the selection of a single model from a class of hypotheses to the identification of a convex set of models induced by the available training data (Fig. 17(b)). 75 \f3. Applying the resulting convex-theoretical learning theory (with respect to both the data-generating probabilities and the model space) to the functional spaces associated with convolutional neural networks (e.g., series of convolutions and max-pooling operations, Fig. 17(c)) in order to lay solid theoretical foundations for deep learning. Credal set of distributions Training distribution Test distribution Extrema of the associated  convex set of linear boundaries Training set (a) (b) Deep neural nets... ...induce complex decision boundaries (c) Figure 17: (a) Training and test distributions from a common random set. (b) A training set of separable points determines only a convex set of admissible linear boundaries. (c) Going against conventional wisdom, deep neural models have very good generalisation power while generating, at the same time, complex decision boundaries. 5.3.4 Bounds for realisable finite hypothesis classes To conclude this essay, we consider the domain adaptation issue in the case of realisable finite hypothesis classes, and outline a credal solution for it. Under the assumption that (1) the model space H is finite, and (2) there exists a hypothesis h\u2217\u2208H that obtains zero expected risk, that is, L(h\u2217) = E(x,y)\u223cp\u2217[l((x, y), h\u2217)] = 0, (84) 76 \fa property called realisability, the following result holds18. Theorem 9. Let H be a hypothesis class, where each hypothesis h \u2208H maps some X to Y; let l be the zero\u2013one loss, l((x, y), h) = I[y \u0338= h(x)]; let p\u2217be any distribution over X \u00d7 Y; and let \u02c6 h be the empirical risk minimiser (83). Assume that (1) and (2) hold. Then the following two equivalent statements hold, with probability at least 1\u2212\u03b4: L(\u02c6 h) \u2264log |H| + log(1/\u03b4) n , (85) n \u2265log |H| + log(1/\u03b4) \u03f5 \u21d2L(\u02c6 h) \u2264\u03f5. (86) Proof. Let B = {h \u2208H : L(h) > \u03f5} be the set of bad hypotheses. We wish to upper-bound the probability19 of selecting a bad hypothesis: P[L(\u02c6 h) > \u03f5] = P[\u02c6 h \u2208B]. (87) Recall that the empirical risk of the ERM is always zero, i.e., \u02c6 L(\u02c6 h) = 0, since at least \u02c6 L(h\u2217) = L(h\u2217) = 0. So if we have selected a bad hypothesis (\u02c6 h \u2208B), then some bad hypothesis must have zero empirical risk: \u02c6 h \u2208B implies \u2203h \u2208B : \u02c6 L(h) = 0, which is equivalent to saying that P[\u02c6 h \u2208B] \u2264P[\u2203h \u2208B : \u02c6 L(h) = 0]. (88) Firstly, we need to bound P[\u02c6 L(h) = 0] for any fixed h \u2208B. For each example, hypothesis h does not err, with probability 1 \u2212L(h). Since the training examples are i.i.d. and since L(h) > \u03f5 for h \u2208B, P[\u02c6 L(h) = 0] = (1 \u2212L(h))n \u2264(1 \u2212\u03f5)n \u2264e\u2212\u03f5n \u2200h \u2208B, where the first equality comes from the examples being i.i.d., and the last step follows since 1 \u2212a \u2264e\u2212a. Note that the probability P[\u02c6 L(h) = 0] decreases exponentially with n. Secondly, we need to show that the above bound holds simultaneously for all h \u2208 B. Recall that P(A1 \u222a\u00b7 \u00b7 \u00b7 \u222aAK) \u2264P(\u222ak k=1Ai). When applied to the (non-disjoint) events Ah = {\u02c6 L(h) = 0}, the union bound yields P[\u2203h \u2208B : \u02c6 L(h) = 0] \u2264 X h\u2208B P[\u02c6 L(h) = 0]. Finally, P[L(\u02c6 h) \u2264\u03f5] = P[\u02c6 h \u2208B] \u2264 P[\u2203h \u2208B : \u02c6 L(h) = 0] \u2264 X h\u2208B P[\u02c6 L(h) = 0] \u2264 |B|e\u2212en \u2264|H|e\u2212en . = \u03b4, and by rearranging we have \u03f5 = (log |H| + log(1/\u03b4))/n, i.e., (85). 18https://web.stanford.edu/class/cs229t/notes.pdf. 19Note that this is a probability measure on the space of models H, completely distinct from the datagenerating probability p\u2217. 77 \fThe inequality (85) should be interpreted in the following way: with probability at least 1 \u2212\u03b4, the expected loss of the empirical risk minimiser \u02c6 h (the particular model selected after training on n examples) is bounded by the ratio on the right-hand side, so that L(\u02c6 h) = O(log |H|/n). The inequality (85) amounts to saying that if I need to obtain an expected risk of at most \u03f5 with confidence at least 1 \u2212\u03b4, I need to train the model on at least (log |H| + log(1/\u03b4))/\u03f5 examples. The result is distribution-free, as it is independent of the choice of p\u2217(x, y). However, it does rely on the assumption that the training and test distributions are the same. 5.3.5 Bounds under credal generalisation A credal generalisation of Theorem 9 would thus read as follows. Theorem 10. Let H be a finite hypothesis class, let P be a credal set over X \u00d7 Y and let a training set of samples D = {(xi, yi), i = 1, . . . , n} be drawn from one of the distributions \u02c6 p \u2208P in this credal set. Let \u02c6 h . = arg min h\u2208H \u02c6 L(h) = arg min h\u2208H 1 n n X i=1 l((xi, yi), h) be the empirical risk minimiser, where l is the 0\u20131 loss. Assume that \u2203h\u2217\u2208H, p\u2217\u2208P : Ep\u2217[l] = Lp\u2217(h\u2217) = 0 (89) (credal realisability) holds. Then, with probability at least 1 \u2212\u03b4, the following bound holds: P \u0014 max p\u2208P Lp(\u02c6 h) > \u03f5 \u0015 \u2264\u03f5(H, P, \u03b4), (90) where \u03f5 is a function of the size of the model space H, the size of the credal set P and \u03b4. How does the proof of Theorem 9 generalise to the credal case? We can note that, if we define B . = \u001a h : max p\u2208P Lp(h) > \u03f5 \u001b as the set of models which do not guarantee the bound in the worst case, we get, just as in the classical result (87), P \u0014 max p\u2208P Lp(\u02c6 h) > \u03f5 \u0015 = P[\u02c6 h \u2208B]. However, whereas in the classical case \u02c6 L(\u02c6 h) = 0, in the credal case we can only say that \u02c6 L(h\u2217) = L\u02c6 p(h\u2217) \u2265min p\u2208P Lp(h\u2217) = 0, by the credal realisability assumption (89). Thus, (88) does not hold, and the above argument which tries to bound P[\u02c6 L(h) = 0] does not apply here. 78 \fOne option is to assume that \u02c6 p = p\u2217= arg minp Lp(h\u2217). This would allow (88) to remain valid. A different generalisation of realisability can also be proposed, in place of (89): \u2200p \u2208P, \u2203h\u2217 p \u2208H : Ep[l(h\u2217 p)] = Lp(h\u2217 p) = 0, (91) which we can call uniform credal realisability. Under the latter assumption, we would get \u02c6 L(\u02c6 h) = min h\u2208H \u02c6 L(h) = min h\u2208H L\u02c6 p(h) = L\u02c6 p(h\u2217 \u02c6 p) = 0. We will complete this analysis and extend it to the case of infinite, non-realisable model spaces in the near future. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2201.01729v1",
+                    "title": "The intersection probability: betting with probability intervals",
+                    "abstract": "Probability intervals are an attractive tool for reasoning under uncertainty.\nUnlike belief functions, though, they lack a natural probability transformation\nto be used for decision making in a utility theory framework. In this paper we\npropose the use of the intersection probability, a transform derived originally\nfor belief functions in the framework of the geometric approach to uncertainty,\nas the most natural such transformation. We recall its rationale and\ndefinition, compare it with other candidate representives of systems of\nprobability intervals, discuss its credal rationale as focus of a pair of\nsimplices in the probability simplex, and outline a possible decision making\nframework for probability intervals, analogous to the Transferable Belief Model\nfor belief functions.",
+                    "authors": "Fabio Cuzzolin",
+                    "published": "2022-01-05",
+                    "updated": "2022-01-05",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "math.PR",
+                        "math.ST",
+                        "stat.TH",
+                        "62A01, 62A86, 60A05, 60A86, 60D05, 60E05"
+                    ],
+                    "main_content": "Introduction An estimation or a decision problem Q usually involvesknowing in which state we are, where the possible states of the world are often assumed to belong to a \ufb01nite set \u0398 = {x1, ..., xn}. Our uncertainty about the outcome of Q can be described in many ways: the classical option is to assume a probability distribution on \u0398. This, however, can only model the aleatory uncertainty about the problem, in which the outcome is random, but the probability distribution that governs the process is fully known. For instance, if a person plays a fair roulette wheel they will not, by any means, know the outcome in advance, but they will nevertheless be able to predict the long-term frequency with which each outcome manifests itself (1/36). In opposition, in practical situations we are required to incorporate imprecise measurements and people\u2019s opinions in our knowledge state, or need to cope with missing or scarce information. A more cautious approach is therefore to assume that we have no access to the \u2018correct\u2019 probability distribution, but that the available evidence provides us with some constraint on this unknown distribution. This more fundamental type of uncertainty is referred to as epistemic [83], and is caused by lack of knowledge about 1 arXiv:2201.01729v1  [cs.AI]  5 Jan 2022 \fthe very process that generates the data. Suppose that the player is presented with ten different doors, which lead to rooms each containing a roulette wheel modelled by a different probability distribution. They will then be uncertain about the very game they are supposed to play. How will this affect their betting behaviour, for instance? Numerous mathematical theories of epistemic uncertainty have been proposed, starting from de Finetti\u2019s pioneering work on subjective probability [61]. To list just the most impactful efforts, we could mention possibility theory [120, 72], credal sets [90, 88], monotone capacities [117], random sets [94] and imprecise probability theory [114]. New original foundations of subjective probability in behavioural terms [115] or by means of game theory [97] have been put forward. One of the simplest such approaches is given by probability intervals [109, 60, 82]: the probability values p(x) of the elements of the decision space \u0398 are assumed to belong to an interval l(x) \u2264p(x) \u2264u(x) delimited by a lower bound l(x) and an upper bound u(x). This encodes a possible (convex) set of probabilities, usually called credal set [90], from which decision can then be taken. When considering an associated decision problem Q, many extensions of the classical expected utility rule proposes to extract not one, but multiple potentially optimal decisions [111]. There are many situations, however, in which one must converge to a unique decision. Decision rules producing one optimal decision in the imprecise-probabilistic literature either focus on speci\ufb01c bounds (e.g., maxi-min [113, 111] or maxi-max rules), not accounting for the whole representation, or require solving complex optimisation problems (e.g., selecting the maximal entropy distribution). An alternative approach which has been often studied within the belief function theory [11] consists in approximating a complex uncertainty measure (such as a belief function) by a single probability distribution, from which a unique optimal decision can be deduced. This is, for instance, the case of the Transferable Belief Model [98], in which the so called pignistic transformation [103] is employed for this purpose. Similarly to the case of belief functions, it could be useful to apply such a transformation to reduce a set of probability intervals to a single probability distribution prior to actually making a decision. However, this problem has been quite neglected so far. One could of course pick a representative from the corresponding credal set, but it makes sense to wonder whether a transformation inherently designed for probability intervals as such could be found. In this paper we argue that the natural candidate for the role of a probability transform of probability intervals is the intersection probability, originally identi\ufb01ed in the context of the geometric approach to uncertainty [20]. Even though originally introduced as a probability transform for belief functions [20], the intersection probability is (as we show here) inherently associated with probability intervals, in which context its rationale clearly emerges. It can be shown that the intersection probability is the only probability distribution that behave homogeneously in each element x of the frame \u0398, i.e., it assigns the same fraction of the available probability interval to each element of the decision space. 2 \f1.1 Contributions and paper outline We \ufb01rst recall the basic notions of the theories of lower and upper probabilities, interval probabilities and belief functions (Section 2), to then quickly review the prior art on probability transform and its geometric interpretation (Section 3). We then formally de\ufb01ne the intersection probability and its rationale (Section 4), showing that it can be de\ufb01ned for any interval probability system as the unique probability distribution obtained by assigning the same fraction of the uncertainty interval to all the elements of the domain. We compare it with other possible representatives of interval probability systems, and recall its geometric interpretation in the space of belief functions and the justi\ufb01cation for its name that derives from it (Section 5). In Section 6 we extensively illustrate the credal rationale for the intersection probability as focus of the pair of lower and upper simplices associated with the interval probability system. As a belief function determines itself an interval probability system, the intersection probability exists for belief functions too and can therefore be compared with classical approximations of belief functions like the pignistic function [98] and relative plausibility and belief [24, 27] of singletons, or more recent approximations proposed by Sudano [105]. In Section 7 we thus analyse the relations of intersection probability with other probability transforms of belief functions, while in Section 8 we discuss its properties with respect to af\ufb01ne combination and convex closure. The potential use of the intersection probability to bet on interval probability systems, in a framework analogous to the Transferable Belief Model, is outlined in Section 9, which concludes the paper. 2 Uncertainty theories 2.1 Lower and upper probabilities A lower probability P is a function from 2\u0398, the power set of \u0398, to the unit interval [0, 1]. A lower probability is associated with a dual upper probability P, de\ufb01ned for any A \u2286\u0398 as P(A) = 1 \u2212P(Ac), where Ac is the complement of A (this means that we can simply focus on one of the two measures, provided it is de\ufb01ned for all subsets). A lower probability P can also be associated with a (closed convex) set P(P) = n p : P(A) \u2265P(A), \u2200A \u2286\u0398 o (1) of probability distributions p whose measure P dominates P. Such a polytope or convex set of probability distributions is usually called a credal set. A lower probability P will be called consistent if P(P) and tight if infp\u2208P(P ) P(A) = P(A) (respectively it \u2018avoids sure loss\u2019 and is \u2018coherent\u2019 in Peter Walley\u2019s [114] terms). Consistency means that lower bound constraints P(A) can be satis\ufb01ed, while tightness means that P is the lower envelope on subsets of P(P). Note that not all convex sets of probabilities can be described by only focusing on events (see Walley [114]), however they will be suf\ufb01cient here. The notation P will simply denote the set of all possible probabilities on \u0398. 3 \f2.2 Probability intervals Dealing with general lower probabilities de\ufb01ned on 2\u0398 can be dif\ufb01cult when \u0398 is big, and it may be interesting in applications to focus on simpler models. One popular and practical model used to model such kind of uncertainty are probability intervals. A set of probability intervals or interval probability system is a system of constraints on the probability values of a probability distribution p : \u0398 \u2192[0, 1] on a \ufb01nite domain \u0398 of the form P(l, u) . = n p : l(x) \u2264p(x) \u2264u(x), \u2200x \u2208\u0398 o . (2) Probability intervals have been introduced as a tool for uncertain reasoning in [60], where combination and marginalization of intervals were studied in detail. In [60] the authors also studied the speci\ufb01c constraints for such intervals to be consistent and tight. As pointed out for instance in [84], a typical way in which probability intervals arise is through measurement errors. As a matter of fact, measurements can be inherently of interval nature (due to the \ufb01nite resolution of the instruments). In that case the probability interval of interest is the class of probability measures consistent with the measured interval. A set of constraints of the form (2) determines a credal set, which is just a sub-class of sets described by lower and upper probabilities. The lower and upper probabilities induced by P(l, u) on any subset A \u2286\u0398 from bounds (l, u) can be obtained using the simple formulas: P(A) = max \uf8f1 \uf8f2 \uf8f3 X x\u2208A l(x), 1 \u2212 X x\u0338\u2208A u(x) \uf8fc \uf8fd \uf8fe, P(A) = min \uf8f1 \uf8f2 \uf8f3 X x\u2208A u(x), 1 \u2212 X x\u0338\u2208A l(x) \uf8fc \uf8fd \uf8fe. (3) 2.3 Belief functions A special class of lower and upper probabilities is provided by belief and plausibility measures. Namely, a basic probability assignment (BPA) [96] is a set function [65, 69] m : 2\u0398 \u2192[0, 1] such that m(\u2205) = 0, X A\u2282\u0398 m(A) = 1. Subsets of \u0398 whose mass values are non-zero are called focal elements of m. The belief function (BF) associated with a BPA m : 2\u0398 \u2192[0, 1] is the set function Bel : 2\u0398 \u2192[0, 1] de\ufb01ned as Bel(A) = X B\u2286A m(B). (4) The corresponding plausibility function is Pl(A) . = X B\u2229A\u0338=\u2205 m(B) \u2265Bel(A). Note that belief functions can also be equivalently de\ufb01ned in axiomatic terms [96]. 4 \fClassical probability measures on \u0398 are a special case of belief functions (those assigning mass to singletons only), termed Bayesian belief functions. A BF is said to be consonant if its focal elements A1, ..., Am are nested: A1 \u2282\u00b7 \u00b7 \u00b7 \u2282Am, and corresponds to a possibility measure [70, 73, 96, 43]. 2.3.1 Combination In belief theory conditioning is replaced by the notion of (associative) combination of any number of belief functions. The Dempster combination Bel1 \u2295Bel2 of two belief functions on \u0398 is the unique BF there with as focal elements all the non-empty intersections of focal elements of Bel1 and Bel2, and basic probability assignment m\u2295(A) = m\u2229(A) 1 \u2212m\u2229(\u2205), (5) where m\u2229(A) = X B\u2229C=A m1(B)m2(C) (6) and mi is the BPA of the input belief function Beli. Nevertheless, Dempster\u2019s combination naturally induces a conditioning operator. Given a conditioning event A \u2282\u0398, the \u2018logical\u2019 or categorical belief function BelA such that m(A) = 1 is combined via Dempster\u2019s rule with the a-priori belief function Bel. The resulting BF Bel \u2295BelA is the conditional belief function given A a la Dempster, denoted by Bel\u2295(A|B). Many alternative combination rules have since been de\ufb01ned [86, 119, 71, 67], often associated with a distinct approach to conditioning [64, 74, 108]. An exhaustive review of these proposals can be found in [43], Section 4.3. Rather than normalising (as in Dempster\u2019s rule) or reassigning the con\ufb02icting mass m\u2229(\u2205) to other non-empty subsets, Philippe Smets\u2019s conjunctive rule leaves the con\ufb02icting mass with the empty set, m \u2229 \u20dd(A) = \u001a m\u2229(A) \u2205\u0338= A \u2286\u0398, m\u2229(\u2205) A = \u2205, (7) and thus is applicable to unnormalised belief functions [100]. In Dempster\u2019s original random-set idea, consensus between two sources is expressed by the intersection of the supported events (6). When the union of the supported propositions is taken to represent such a consensus instead, we obtain what Smets called the disjunctive rule of combination, m \u222a \u20dd(A) = X B\u222aC=A m1(B)m2(C). (8) It is interesting to note that under disjunctive combination, Bel1 \u222a \u20ddBel2(A) = Bel1(A) \u2217Bel2(A), i.e., the belief values of the input belief functions are simply multiplied. 5 \f2.3.2 Belief functions and other measures Each belief function Bel uniquely identi\ufb01es a credal set [88] P[Bel] = {P \u2208P : P(A) \u2265Bel(A)} (9) (where P is the set of all probabilities one can de\ufb01ne on \u0398), of which it is its lower envelope: Bel(A) = P(A). Belief functions are thus a special case of lower probabilities (Section 2.1). The corresponding plausibility measure is the upper probability of an event A: Pl(A) = P(A). The probability intervals resulting from Dempster\u2019s updating of the credal set associated with a BF, however, are included in those resulting from Bayesian updating [88]. Belief functions are also in\ufb01nitely monotone capacities [8, 107], and a special case of coherent lower previsions [114, 115]. Finally, every belief function speci\ufb01es a unique probability box [75], i.e., a class of cumulative distribution functions delimited by two lower and upper bounds. 2.4 The geometry of uncertainty measures Geometry has been proposed by this author and others as a unifying language for the \ufb01eld [42, 53, 54, 55, 58, 76, 77, 5, 85, 78, 116], possibly in conjunction with an algebraic view [51, 12, 23, 57, 16, 18, 38]. Indeed, uncertainty measures can be seen as points of a suitably complex geometric space, and there manipulated (e.g. combined, conditioned and so on) [13, 21, 43]. Much work has been focusing on the geometry of belief functions, which live in a convex space termed the belief space, which can be described both in terms of a simplex (a higher-dimensional triangle) and in terms of a recursive bundle structure [56, 45, 41, 52]. The analysis can be extended to Dempster\u2019s rule of combination by introducing the notion of a conditional subspace and outlining a geometric construction for Dempster\u2019s sum [49, 14]. The combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function, have also been studied [22, 34]. The corresponding spaces are simplices which are congruent to the belief space. Subsequent work extended the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) [32] and consistent belief functions [36, 48, 25], in terms of simplicial complexes [15]. Analyses of belief functions in terms credal sets have also been conducted [26, 1, 6]. The geometry of the relationship between measures of different kinds has also been extensively studied [50, 30, 17, 31], with particular attention to the problem of transforming a belief function into a classical probability measure [9, 112, 99] (see Section 3). One can distinguish between an af\ufb01ne family of probability transformations [20] (those which commute with af\ufb01ne combination in the belief space), and an epistemic family of transforms [19], formed by the relative belief and relative plausibility of singletons [28, 27, 37, 46, 33], which possess dual properties with respect to Dempster\u2019s sum [24]. The problem of \ufb01nding the possibility measure which best approximates a given belief function [2] can also be approached in geometric terms [29, 47, 39, 40]. In particular, approximations induced by classical Minkowski norms can be derived and 6 \fcompared with classical outer consonant approximations [72]. Minkowski consistent approximations of belief functions in both the mass and the belief space representations can also be derived [36]. The geometric approach to uncertainty can also be applied to the conditioning problem [89]. Conditional belief functions can be de\ufb01ned as those which minimise an appropriate distance between the original belief function and the \u2018conditioning simplex\u2019 associated with the conditioning event [44, 35]. Recent papers on this topic include [93, 95, 91]. 3 Probability transform 3.1 Probability transforms of belief functions The relation between belief and probability in the theory of evidence has been and continues to be an important subject of study[118, 87, 3, 4, 66, 68, 81]. A probability transform mapping belief functions to probability measures can be instrumental in addressing a number of issues: mitigating the inherently exponential complexity of belief calculus [3], making decisions via the probability distributions obtained in a utility theory framework [99] and obtaining pointwise estimates of quantities of interest from belief functions (e.g., the pose of an articulated object in computer vision: see [52], Chapter 8, or [54]). As both belief and probability measures can be assimilated into points of a Cartesian space [43], the problem can (as mentioned) be posed in a geometric setting. Without loss of generality, we can de\ufb01ne a probability transform as a mapping from the space of belief functions B on the domain of interest to the probability simplex P there, PT : B \u2192 P, Bel \u2208B 7\u2192 PT [Bel] \u2208P, such that an appropriate distance function or similarity measure d from Bel is minimised [59]: PT [Bel] = arg min P \u2208P d(Bel, P). (10) A minimal, sensible requirement is for the probability which results from the transform to be compatible with the upper and lower bounds that the original belief function Bel enforces on the singletons only, rather than on all the focal sets. Thus, this does not require probability transforms to adhere to the upper\u2013lower probability semantics of belief functions. As a matter of fact, some important transforms of this kind are not compatible with such semantics. Many such transformations have been proposed, according to different criteria [92, 110, 4, 66, 68, 3]. In Smets\u2019s transferable belief model [98, 104], in particular, decisions are made by resorting to the pignistic probability: BetP[Bel](x) = X A\u2287{x} m(A) |A| , (11) which is the output of the pignistic transform. 7 \fAn interesting approach to the problem seeks approximations which enjoy commutativity properties with respect to a speci\ufb01c combination rule, in particular Dempster\u2019s sum [63, 62]. This is the case of the relative plausibility of singletons [112], the unique probability that, given a belief function Bel with plausibility Pl(A) = 1 \u2212Bel(Ac), assigns to each singleton its normalized plausibility:1 \u02dc Pl[Bel](x) = Pl(x) P y\u2208\u0398 Pl(y). (12) Its properties have been later analyzed by Cobb and Shenoy [9, 10]. Voorbraak proved that his (in our terminology) relative plausibility of singletons \u02dc Pl[Bel] is a perfect representative of Bel when combined with other probabilities P \u2208P through Dempster\u2019s rule \u2295: \u02dc Pl[Bel] \u2295P = Bel \u2295P \u2200P \u2208P. (13) Dually, a relative belief transform \u02dc Bel : B \u2192P, Bel 7\u2192 \u02dc Bel[Bel] mapping each belief function Bel to the corresponding relative belief of singletons [24, 28, 80, 59], \u02dc Bel[Bel](x) = Bel(x) P y\u2208\u0398 Bel(y), (14) can be de\ufb01ned. The notion of a relative belief transform (under the name of \u2018normalised belief of singletons\u2019) was \ufb01rst proposed by Daniel in [59]. Some analyses of the relative belief transform and its close relationship with the (relative) plausibility transform were presented in [24, 28]. 3.2 Geometric approaches Only a few authors have in the past posed the study of the connections between belief functions and probabilities in a geometric setting. In particular, Ha and Haddawy [79] proposed an \u2018af\ufb01ne operator\u2019, which can be considered a generalisation of both belief functions and interval probabilities, and can be used as a tool for constructing convex sets of probability distributions. In their work, uncertainty is modelled as sets of probabilities represented as \u2018af\ufb01ne trees\u2019, while actions (modi\ufb01cations of the uncertain state) are de\ufb01ned as tree manipulators. In a later publication [78], the same authors presented an interval generalisation of the probability cross-product operator, called the \u2018convex-closure\u2019 (cc) operator, analysed the properties of the cc operator relative to manipulations of sets of probabilities and presented interval versions of Bayesian propagation algorithms based on it. Probability intervals were represented there in a computationally ef\ufb01cient fashion by means of a data structure called a \u2018pcc-tree\u2019, in which branches are annotated with intervals, and nodes with convex sets of probabilities. The intersection probability introduced in this paper is somewhat related to Ha\u2019s cc operator, as it commutes (at least under certain conditions) with af\ufb01ne combination, and is therefore part of the af\ufb01ne family of Bayesian transforms of which Smets\u2019s pignistic transform [102] is the foremost representative. 1With a harmless abuse of notation, we will often denote the values of belief functions and plausibility functions on a singleton x by m(x), Pl(x) rather than by m({x}), Pl({x}). 8 \f4 The intersection probability When our uncertainty is described by imprecise probabilities, it may be desirable for some reasons to transform this knowledge into a classical unique probability. Such reasons include the need to take a unique optimal decision, the need to use classical probabilistic calculus (e.g., for ef\ufb01ciency), or more simply the will to obtain a unique probability from partial probabilistic information. Existing proposals are general in scope but rather complex, e.g., they imply solving a convex optimization problem. 4.1 De\ufb01nition There are clearly many ways of selecting a single measure to represent a collection of probability intervals (2). Note, however, that each of the intervals [l(x), u(x)], x \u2208\u0398, carries the same weight within the system of constraints (2), as there is no reason for the different elements x of the domain to be treated differently. It is then sensible to require that the desired representative probability should behave homogeneously in each element x of the frame \u0398. Mathematically, this translates into seeking a probability distribution p : \u0398 \u2192[0, 1] such that p(x) = l(x) + \u03b1(u(x) \u2212l(x)) for all the elements x of \u0398, and some constant value \u03b1 \u2208[0, 1] (see Fig. 1). This value needs to be between 0 and 1 in order for the sought probability distribution p to belong to the interval. It is easy to see that there is indeed a unique solution to this problem. It \u03b1 \u03b1 \u03b1 l(x) u(x) l(y) u(y) l(z) u(z) Figure 1: An illustration of the notion of the intersection probability for an interval probability system (l, u) on \u0398 = {x, y, z} (2). suf\ufb01ces to enforce the normalisation constraint X x p(x) = X x h l(x) + \u03b1(u(x) \u2212l(x)) i = 1 to understand that the unique value of \u03b1 is given by \u03b1 = \u03b2[(l, u)] . = 1 \u2212P x\u2208\u0398 l(x) P x\u2208\u0398 \u0000u(x) \u2212l(x) \u0001. (15) De\ufb01nition 1. The intersection probability p[(l, u)] : \u0398 \u2192[0, 1] associated with the interval probability system (2) is the probability distribution p[(l, u)](x) = \u03b2[(l, u)]u(x) + (1 \u2212\u03b2[(l, u)])l(x), (16) with \u03b2[(l, u)] given by (15). 9 \fThe ratio \u03b2[(l, u)] (15) measures the fraction of each interval [l(x), u(x)] which we need to add to the lower bound l(x) to obtain a valid probability function (adding up to one). It is easy to see that when (l, u) are a pair of belief/plausibility measures (Bel, Pl), we can de\ufb01ne the intersection probability for belief functions as well. Although originally de\ufb01ned by geometric means [20], the intersection probability is thus in fact \u2018the\u2019 rational probability transform for general interval probability systems. Note that p[(l, u)] can also be written as p[(l, u)](x) = l(x) +   1 \u2212 X x l(x) ! R[(l, u)](x), (17) where R[(l, u)](x) . = u(x) \u2212l(x) P y\u2208\u0398(u(y) \u2212l(y)) = \u2206(x) P y\u2208\u0398 \u2206(y). (18) Here \u2206(x) measures the width of the probability interval for x, whereas R[(l, u)] : \u0398 \u2192[0, 1] measures how much the uncertainty in the probability value of each singleton \u2018weighs\u2019 on the total width of the interval system (2). We thus term it the relative uncertainty of singletons. Therefore, we can say that p[(l, u)] distributes the mass (1 \u2212P x l(x)) to each singleton x \u2208\u0398 according to the relative uncertainty R[(l, u)](x) it carries for the given interval. Example 1. Consider as an example an interval probability system on a domain \u0398 = {x, y, z} of size 3: 0.2 \u2264p(x) \u22640.8, 0.4 \u2264p(y) \u22641, 0.3 \u2264p(z) \u22640.3. (19) Notice that there is no uncertainty at all on the value of p(z) = 0.3. The widths of the corresponding intervals are \u2206(x) = 0.6, \u2206(y) = 0.6, \u2206(z) = 0 respectively. The relative uncertainty on each singleton (18) is therefore: R[(l, u)](x) = \u2206(x) P w\u2208\u0398 \u2206(w) = 0.6 1.2 = 1 2, R[(l, u)](y) = 1 2, R[(l, u)](z) = \u2206(z) P w\u2208\u0398 \u2206(w) = 0 1.2 = 0. (20) Computing the intersection probability is then really easy. By Equation (15) the fraction of the uncertainty u(x) \u2212l(x) on p(x) we need to add to the lower bound l(x) to get an admissible, normalized probability is \u03b2 = 1 \u22120.2 \u22120.4 \u22120.3 0.6 + 0.6 = 0.1 1.2 = 1 12. The intersection probability (17) has therefore values: p[(l, u)](x) = 0.2 + 1 120.6 = 0.25, p[(l, u)](y) = 0.4 + 1 120.6 = 0.45, p[(l, u)](z) = 0.3 + 1 120 = 0.3. 10 \fNotice that the fact of having a zero-width interval for one of the singletons does not pose a problem for the intersection probability, which falls as expected inside the probability interval for all the elements of the domain. According to its interpretation of Equation (17), p[(l, u)] is also the result of distributing the necessary mass (1 \u2212P x l(x)) = 1 \u22120.2 \u22120.4 \u22120.3 = 0.1 to each singleton in proportion to the relative uncertainty R[(l, u)] (Equation (20)) of their intervals: p[(l, u)](x) = 0.2 + 0.1 1 2 = 0.25, p[(l, u)](y) = 0.4 + 0.1 1 2 = 0.45, p[(l, u)](z) = 0.3 + 0.1 \u00b7 0 = 0.3. 4.2 Comparison with other interval representatives It can be useful to brie\ufb02y compare the proposed intersection probability with other possible representatives of an interval probability system (2). 4.2.1 Comparison with the center of mass of P[(l, u)] The naive choice of picking the barycenter of each interval [l(x), u(x)] to represent an interval probability system (l, u), for instance, does not yield in general a valid probability function, for X x\u2208\u0398 \u0014 l(x) + 1 2(u(x) \u2212l(x)) \u0015 \u0338= 1. This marks the difference with the case of belief functions, for which the pignistic function has a strong interpretation as barycenter of the associated credal set. 4.2.2 Comparison normalised lower and upper bounds For the probability interval system (21) determined by a belief function, (Bel, Pl) . = \b p \u2208P : Bel(x) \u2264p(x) \u2264Pl(x), \u2200x \u2208\u0398 \t , (21) the probabilities we obtain by normalizing lower \u02dc l(x) = l(x)/ P y l(y) or upper bound \u02dc u(x) = u(x)/ P y u(y) are not guaranteed to be consistent with the interval itself. For instance, if there exists an element x \u2208\u0398 such that Bel(x) = Pl(x) (the interval has width zero for that element) we have that \u02dc Bel(x) = m(x) P y m(y) > Pl(x), \u02dc Pl(x) = Pl(x) P y Pl(y) < Bel(x). Therefore, both relative belief and plausibility of singletons fall outside the interval system (21). This holds for a general collection of probability intervals (2), again marking the contrast with the behavior of the intersection probability. 11 \f4.2.3 Comparison with Sudano\u2019s proposal In the belief functions framework, Sudano proposed in [106] the following four probability transforms: PrPl[Bel](x) . = X A\u2287{x} m(A) Pl(x) P y\u2208A Pl(y), (22) PrBel[Bel](x) . = X A\u2287{x} m(A) Bel(x) P y\u2208A Bel(y) = X A\u2287{x} m(A) m(x) P y\u2208A m(y), (23) PrNPl[Bel](x) . = 1 \u2206 X A\u2229{x}\u0338=\u2205 m(A) = \u02dc Pl[Bel](x), (24) PraPl[Bel](x) . = Bel(x) + \u03f5 \u00b7 Pl(x), \u03f5 = 1 \u2212P y\u2208\u0398 Bel(y) P y\u2208\u0398 Pl(y) = 1 \u2212kBel kP l , (25) where kBel . = X x\u2208\u0398 Bel(x), kP l . = X x\u2208\u0398 Pl(x). (26) The \ufb01rst two transformations are clearly inspired by the pignistic function (11). While in the latter case the mass m(A) of each focal element is redistributed homogeneously to all its elements x \u2208A, PrPl[Bel] (22) redistributes m(A) proportionally to the relative plausibility of a singleton x inside A. Similarly, PrBel[Bel] (23) redistributes m(A) proportionally to the relative belief of a singleton x within A. The fourth transformation (25), PraPl[Bel], is more related to the case of probability intervals and to the intersection probability. By Equation (16), p[Bel](x) = (1 \u2212\u03b2[Bel]) \u02dc Bel(x)kBel + \u03b2[Bel] \u02dc Pl(x)kP l, (27) where (1 \u2212\u03b2[Bel])kBel + \u03b2[Bel]kP l = kP l \u22121 kP l \u2212kBel kBel + 1 \u2212kBel kP l \u2212kBel kP l = 1, i.e., p[Bel] lies on the line joining the relative plausibility \u02dc Pl and the relative belief \u02dc Bel of singletons. Here \u03b2[Bel] is the value of (15) for a system of probability intervals associated with a belief function Bel, namely: \u03b2[Bel] . = 1 \u2212P x\u2208\u0398 Bel(x) P x\u2208\u0398 \u0000Pl(x) \u2212Bel(x) \u0001. (28) Just like the intersection probability (27) and the relative uncertainty of singletons [20], PraPl[b] can also be expressed as an af\ufb01ne combination of relative belief and plausibility of singletons: PraPl[Bel](x) = m(x) + 1 \u2212kBel kP l Pl(x) = kBel \u02dc Bel(x) + (1 \u2212kBel) \u02dc Pl(x). (29) 12 \fMore to the point, as its de\ufb01nition only involves belief and plausibility values of singletons, it is more correct to think of PraPl[b] as of a probability transformation of a probability interval system (rather than an approximation of a belief function) PraPl[(l, u)] . = l(x) + 1 \u2212P y l(y) P y u(y) u(x) just like the intersection probability. However, it is easier to point out its weakness as a representative of probability intervals when put in the above form. Just like in the case of relative belief and plausibility of singletons, PraPl[(l, u)] is not in general consistent with the original probability interval system (l, u). If there exists an element x \u2208\u0398 such that l(x) = u(x) (the interval has width \u2206(x) equal to zero for that element) we have that PraPl[(l, u)](x) = l(x) + 1 \u2212P y l(y) P y u(y) u(x) = u(x) + 1 \u2212P y l(y) P y u(y) u(x) = u(x) \u00b7 P y u(y) + 1 \u2212P y l(y) P y u(y) > u(x) as P y u(y)+1\u2212P y l(y) P y u(y) > 1, and PraPl[(l, u)] falls outside the interval. Another fundamental objection against PraPl[(l, u)] arises when we compare it to p[(l, u)]. While the latter adds to the lower bound l(x) an equal fraction of the uncertainty u(x) \u2212l(x) for all singletons (17), PraPl[(l, u)] adds to the lower bound l(x) an equal fraction of the upper bound u(x), effectively counting twice the evidence represented by the lower bound l(x) (29). In the case of belief functions, this amounts to adding to the mass value m(x) of x yet another fraction of m(x) itself, instead of distributing only the remaining mass Pl(x) \u2212m(x) allowed to be assigned to x. 5 Geometric interpretation Despite having being de\ufb01ned as a representative for systems of probability intervals, the intersection probability was \ufb01rst identi\ufb01ed in the context of the geometric analysis of belief measures [21, 13, 56, 52]. As we brie\ufb02y recall here, its very name derives from its geometry in the space of belief functions, or belief space [21, 49]. 5.1 Geometry in the belief space 5.1.1 Belief space Given a frame of discernment \u0398, a belief function Bel : 2\u0398 \u2192[0, 1] is completely speci\ufb01ed by its N \u22122 belief values {Bel(A), \u2205\u228aA \u228a\u0398}, N . = 2|\u0398| (as Bel(\u2205) = 0, Bel(\u0398) = 1 for all BFs), and can then be seen as a point of RN\u22122. 13 \fThe belief space associated with \u0398 is the set of points B \u2282RN\u22122 which correspond to admissible belief functions [21]. This turns out to be the simplex determined by the convex closure of all the categorical belief functions BelA, namely B = Cl(BelA, \u2205\u228aA \u2286\u0398), (Bel\u0398 included). The faces of a simplex are all the simplices generated by a subset of its vertices. The set of all the Bayesian belief functions on \u0398, P = Cl(Belx, x \u2208\u0398), is then a face of B. Plausibility functions, also determined by their N \u22122 values {Pl(A), \u2205\u228aA \u228a\u0398}, can too be seen as points of RN\u22122. We call plausibility space [24, 22] the corresponding region PL of RN\u22122, again, a simplex [45]. P' Bel P Pl p [Bel] a(P) \u03c0 [Bel] [Bel] \u03c2 Figure 2: The geometry of the line a(Bel, Pl) and the relative locations of p[Bel], \u03c2[Bel] and of the orthogonal projection \u03c0[Bel] [20] for a frame of discernment of arbitrary size. Each belief function Bel and the related plausibility function Pl lie on opposite sides of the hyperplane P\u2032 of all Bayesian pseudo belief functions [20], which divides the space RN\u22122 of all such functions into two halves. The line a(Bel, Pl) connecting them always intersects P\u2032, but not necessarily a(P) (vertical line). This intersection \u03c2[Bel] is naturally associated with a probability p[Bel] (in general distinct from the orthogonal projection \u03c0[Bel] of Bel onto P), having the same components in the base {Belx, x \u2208\u0398} of a(P). P is a simplex (a segment in the \ufb01gure) in a(P): \u03c0[Bel] and p[Bel] are both \u2018true\u2019 probabilities. 5.1.2 Dual line The intersection probability for a belief function Bel can be shown to be the unique probability distribution determined by the intersection of the dual line joining a belief function Bel and the related plausibility function Pl with the region of Bayesian (pseudo) belief functions [20]. It can be shown that this dual line a(Bel, Pl) is always orthogonal to P, but it does not intersect the probabilistic subspace in general. It does always intersect, however, 14 \fthe region of Bayesian pseudo belief functions (or \u2018normalised sum functions\u2019) in a point \u03c2[Bel] . = Bel + \u03b2[Bel](Pl \u2212Bel) = a(Bel, Pl) \u2229P\u2032 (30) (where P\u2032 denotes the set of all Bayesian normalised sum functions in RN\u22122). \u03c2[Bel] is a Bayesian pseudo BF but is not guaranteed to be a \u2018proper\u2019 Bayesian belief function. But, of course, since P x m\u03c2[Bel](x) = 1, \u03c2[Bel] is naturally associated with a Bayesian belief function assigning an equal amount of mass to each singleton and 0 to each A : |A| > 1. Namely, we can de\ufb01ne the probability measure p[Bel] . = X x\u2208\u0398 m\u03c2[Bel](x)Belx, (31) where m\u03c2[Bel](x) is given by m\u03c2[Bel](x) = m(x) + \u03b2[Bel] X A\u228bx m(A). (32) This Bayesian BF p[Bel] is nothing but the intersection probability associated with the probability interval system induced by Bel. The relative geometry of \u03c2[Bel] and p[Bel] with respect to the regions of Bayesian belief and normalised sum functions, respectively, is outlined in Fig. 2. 5.2 Justi\ufb01cation for the name The pseudo probability \u03c2[Bel] (30) provides the justi\ufb01cation for the name \u2018intersection probability\u2019. It turns out that p[Bel] and \u03c2[Bel] are equivalent when combined with a Bayesian belief function. We \ufb01rst need to recall the following result [14]. Proposition 1. The orthogonal sum Bel \u2295(\u03b11Bel1 + \u03b12Bel2) of a belief function Bel and any af\ufb01ne combination \u03b11Bel1 + \u03b12Bel2, \u03b11 + \u03b12 = 1 of other two belief functions Bel1, Bel2 on the same frame reads as Bel \u2295(\u03b11Bel1 + \u03b12Bel2) = \u03b31(Bel \u2295Bel1) + \u03b32(Bel \u2295Bel2), (33) where \u03b3i = \u03b1ik(Bel, Beli) \u03b11k(Bel, Bel1) + \u03b12k(Bel, Bel2) and k(Bel, Beli) is the normalisation factor of the orthogonal sum Bel \u2295Beli. Similar results can be proven for both conjunctive and disjunctive rules. Lemma 1. Af\ufb01ne combination commutes with both conjunctive and disjunctive rules: Bel \u2229 \u20dd(\u03b11Bel1 + \u03b12Bel2) = \u03b11(Bel \u2229 \u20ddBel1) + \u03b12(Bel \u2229 \u20ddBel2), Bel \u222a \u20dd(\u03b11Bel1 + \u03b12Bel2) = \u03b11(Bel \u222a \u20ddBel1) + \u03b12(Bel \u222a \u20ddBel2) whenever \u03b11 + \u03b12 = 1. 15 \fProof. By de\ufb01nition (8), we have that Bel \u2229 \u20dd(\u03b11Bel1 + \u03b12Bel2) has basic probability assignment: mBel \u2229 \u20dd(\u03b11Bel1+\u03b12Bel2)(A) = X B\u2229C=A m(B)m\u03b11Bel1+\u03b12Bel2(C) = X B\u2229C=A m(B) \u0000\u03b11m1(C) + \u03b12m2(C) \u0001 = \u03b11 X B\u2229C=A m(B)m1(C) + \u03b12 X B\u2229C=A m(B)m2(C) = \u03b11mBel \u2229 \u20ddBel1(A) + \u03b12mBel \u2229 \u20ddBel2(A). An analogous proof holds for (7). Theorem 1. The combinations of p[Bel] and \u03c2[Bel] with any probability function p \u2208 P coincide under both the Dempster (5) and conjunctive (8) rules: p[Bel] \u2295p = \u03c2[Bel] \u2295p, p[Bel] \u2229p = \u03c2[Bel] \u2229p, \u2200p \u2208P. (34) Proof. Let us de\ufb01ne by \u00b5(A) = X B\u2286A (\u22121)|A\u2212B|Pl(B) (35) the Moebius inverse of a plausibility function Pl (see [22]). It can be proven that [34]: X A\u2287{x} \u00b5(A) = m(x). (36) Now, applying Equation (33) to \u03c2 \u2295p yields \u03c2 \u2295p = \u0002 \u03b2[Bel]Pl + (1 \u2212\u03b2[Bel])Bel \u0003 \u2295p = \u03b2[Bel]k(p, Pl)Pl \u2295p + (1 \u2212\u03b2[Bel])k(p, Bel)Bel \u2295p \u03b2[Bel]k(p, Pl) + (1 \u2212\u03b2[Bel])k(p, Bel) (37) where k(p, Pl) = X x\u2208\u0398 p(x) \uf8eb \uf8edX A\u2287{x} \u00b5(A) \uf8f6 \uf8f8 = X x\u2208\u0398 p(x)m(x), k(p, Bel) = X x\u2208\u0398 p(x) \uf8eb \uf8edX A\u2287{x} m(A) \uf8f6 \uf8f8 = X x\u2208\u0398 p(x)Pl(x). by Equation (36), and by the de\ufb01nition of the plausibility of singletons (P A\u2287{x} m(A) = Pl(x)). On the other hand, recalling Equation (27), we can write p[Bel] = \u03b2[Bel] \u00af Pl + (1 \u2212\u03b2[Bel]) \u00af Bel (38) 16 \fwhere we call the quantities \u00af Pl . = X x\u2208\u0398 Pl(x)Belx \u00af Bel = X x\u2208\u0398 m(x)Belx (39) plausibility of singletons and belief of singletons, respectively [27, 24, 33, 37], for sake of consistency of nomenclature. However, \u00af Pl is traditionally referred to as the contour function. Therefore when we apply (33) to p[Bel] \u2295p, instead, we get (by Equation (38)): p[Bel] \u2295p = \u0002 \u03b2[Bel] \u00af Pl + (1 \u2212\u03b2[Bel]) \u00af Bel \u0003 \u2295p = \u03b2[Bel]k(p, \u00af Pl) \u00af Pl \u2295p + (1 \u2212\u03b2[Bel])k(p, \u00af Bel) \u00af Bel \u2295p \u03b2[Bel]k(p, \u00af Pl) + (1 \u2212\u03b2[Bel])k(p, \u00af Bel) . (40) By de\ufb01nition of Dempster\u2019s combination (5): \u00af Pl \u2295p = X x\u2208\u0398 Belxp(x)(Pl(x) + 1 \u2212kP l) k(p, \u00af Pl) , \u00af Bel \u2295p = X x\u2208\u0398 Belxp(x)(m(x) + 1 \u2212kBel) k(p, \u00af Bel) . Hence k(p, \u00af Pl) \u00af Pl \u2295p = X x\u2208\u0398 Belxp(x)Pl(x) + (1 \u2212kP l) X x\u2208\u0398 Belxp(x) = k(Bel, p)Bel \u2295p + (1 \u2212kP l)p, k(p, \u00af Bel) \u00af Bel \u2295p = X x\u2208\u0398 Belxp(x)m(x) + (1 \u2212kBel) X x\u2208\u0398 Belxp(x) = k(Pl, p)Pl \u2295p + (1 \u2212kBel)p, as: 1. P x\u2208\u0398 Belxp(x) = p; 2. in the calculation of Bel \u2295p, each singleton x is assigned mass p(x) X A\u2287{x} m(A) = p(x)Pl(x); 3. in the calculation of Pl \u2295p, each singleton x is assigned mass p(x) X A\u2287{x} \u00b5(A) = p(x)m(x) (again by Equation (36)). 17 \fAfter replacing these expressions in the numerator of (40) we can notice that, as \u03b2[Bel] = 1 \u2212kBel kP l \u2212kBel , 1 \u2212\u03b2[Bel] = kP l \u22121 kP l \u2212kBel , the contributions of p vanish, leaving expression (37) for \u03c2 \u2295p. As conjunctive rule and af\ufb01ne combination commute, and k(Bel1, Bel2) = 1 for each pair of pseudo belief functions Bel1, Bel2 under conjunctive combination, the proof holds for \u2229 \u20ddtoo. Even though p[Bel] is not the actual intersection \u03c2[Bel] of the line a(Bel, Pl) with the region of pseudo probabilities in the belief space, it behaves exactly like it when aggregated to a probability distribution. Notice that Theorem 1 is not a simple consequence of Voorbraak\u2019s representation theorem [24]: Bel \u2295p = \u02dc Pl \u2295p. It is indeed easy to prove that the relative plausibility of singletons of \u03c2[Bel] is not p[Bel]. As \u03c2[Bel](A) = Bel(A) + \u03b2[Bel][Pl \u2212Bel](A) we have: Pl\u03c2[Bel](x) = 1 \u2212\u03c2[Bel]({x}c) = 1 \u2212Bel({x}c) \u2212\u03b2[Bel] [Pl({x}c) \u2212Bel({x}c)] = Pl(x) \u2212\u03b2[Bel] [1 \u2212Bel(x) \u22121 + Pl(x)] = Pl(x) \u2212\u03b2[Bel] [Pl(x) \u2212Bel(x)] = \u03b2[Bel]Bel(x) + (1 \u2212\u03b2[Bel])Pl(x) = \u03b2[Bel]m(x) + (1 \u2212\u03b2[Bel])Pl(x). Its normalization factor is P x Pl\u03c2[Bel](x) = X x Pl(x) \u2212\u03b2[Bel] \"X x Pl(x) \u2212 X x m(x) # = X x Pl(x) \u22121 + X x m(x). Clearly then \u02dc Pl\u03c2[Bel](x) = Pl\u03c2[Bel](x) P y Pl\u03c2[Bel](y) = \u03b2[Bel]m(x) + (1 \u2212\u03b2[Bel])Pl(x) P y Pl(y) \u22121 + P y m(y) is different from p[Bel](x) = \u03b2[Bel]Pl(x) + (1 \u2212\u03b2[Bel])m(x). 6 Credal rationale Probability interval systems admit a credal representation, which for intervals associated with belief functions is also strictly related to the credal set P[Bel] of all consistent probabilities [31, 43]. By the de\ufb01nition (9) of P[Bel], it follows that the polytope of consistent probabilities can be decomposed into a number of component polytopes, namely P[Bel] = n\u22121 \\ i=1 Pi[Bel], (41) 18 \fwhere Pi[Bel] is the set of probabilities that satisfy the lower probability constraint for size-i events, Pi[Bel] . = n P \u2208P : P(A) \u2265Bel(A), \u2200A : |A| = i o . (42) Note that for i = n the constraint is trivially satis\ufb01ed by all probability measures P: Pn[Bel] = P. 6.1 Lower and upper simplices A simple and elegant geometric description of interval probability systems can be provided if, instead of considering the polytopes (42), we focus on the credal sets T i[Bel] . = n P \u2032 \u2208P\u2032 : P \u2032(A) \u2265Bel(A), \u2200A : |A| = i o . Here P\u2032 denotes the set of all pseudo-probability measures P \u2032 on \u0398, whose distribution p\u2032 : \u0398 \u2192R satisfy the normalisation constraint P x\u2208\u0398 p\u2032(x) = 1 but not necessarily the non-negativity one \u2013 there may exist an element x such that p\u2032(x) < 0. In particular, we focus here on the set of pseudo-probability measures which satisfy the lower constraint on singletons, T 1[Bel] . = n p\u2032 \u2208P\u2032 : p\u2032(x) \u2265Bel(x) \u2200x \u2208\u0398 o , (43) and the set T n\u22121[Bel] of pseudo-probability measures which satisfy the analogous constraint on events of size n \u22121, T n\u22121[Bel] . = n P \u2032 \u2208P\u2032 : P \u2032(A) \u2265Bel(A) \u2200A : |A| = n \u22121 o = n P \u2032 \u2208P\u2032 : P \u2032({x}c) \u2265Bel({x}c) \u2200x \u2208\u0398 o = n p\u2032 \u2208P\u2032 : p\u2032(x) \u2264Pl(x) \u2200x \u2208\u0398 o , (44) i.e., the set of pseudo-probabilities which satisfy the upper constraint on singletons. 6.2 Simplicial form The extension to pseudo-probabilities allows us to prove that the credal sets (43) and (44) have the form of simplices (see [31] and [43], Chapter 16). Theorem 2. The credal set T 1[Bel], or lower simplex, can be written as T 1[Bel] = Cl(t1 x[Bel], x \u2208\u0398), (45) namely as the convex closure of the vertices t1 x[Bel] = X y\u0338=x m(y)Bely + \uf8eb \uf8ed1 \u2212 X y\u0338=x m(y) \uf8f6 \uf8f8Belx. (46) 19 \fDually, the upper simplex T n\u22121[Bel] reads as the convex closure T n\u22121[Bel] = Cl(tn\u22121 x [Bel], x \u2208\u0398) (47) of the vertices tn\u22121 x [Bel] = X y\u0338=x Pl(y)Bely + \uf8eb \uf8ed1 \u2212 X y\u0338=x Pl(y) \uf8f6 \uf8f8Belx. (48) By (46), each vertex t1 x[Bel] of the lower simplex is a pseudo-probability that adds the total mass 1\u2212kBel of non-singletons to that of the element x, leaving all the others unchanged: mt1 x[Bel](x) = m(x) + 1 \u2212kBel, mt1 x[Bel](y) = m(y) \u2200y \u0338= x. In fact, as mt1 x[Bel](z) \u22650 for all z \u2208\u0398 and for all x \u2208\u0398 (all t1 x[Bel] are actual probabilities), we have that T 1[Bel] = P1[Bel], (49) and T 1[Bel] is completely included in the probability simplex. On the other hand, the vertices (48) of the upper simplex are not guaranteed to be valid probabilities. Each vertex tn\u22121 x [Bel] assigns to each element of \u0398 different from x its plausibility Pl(y) = pl(y), while it subtracts from Pl(x) the \u2018excess\u2019 plausibility kP l \u22121: mtn\u22121 x [Bel](x) = Pl(x) + (1 \u2212kP l), mtn\u22121 x [Bel](y) = Pl(y) \u2200y \u0338= x. Now, as 1 \u2212kP l can be a negative quantity, mtn\u22121 x [Bel](x) can also be negative and tn\u22121 x [Bel] is not guaranteed to be a \u2018true\u2019 probability. We will have con\ufb01rmation of this fact in the example in Section 6.4. 6.3 Lower and upper simplices and probability intervals By comparing (2), (43) and (44), it is clear that the credal set P[(l, u)] associated with a set of probability intervals (l, u) is nothing but the intersection P[(l, u)] = T[l] \u2229T[u] of the lower and upper simplices (50) associated with its lowerand upper-bound constraints, respectively: T[l] . = n p : p(x) \u2265l(x) \u2200x \u2208\u0398 o , T[u] . = n p : p(x) \u2264u(x) \u2200x \u2208\u0398 o . (50) In particular, when these lower and upper bounds are those enforced by a pair of belief and plausibility measures on the singleton elements of a frame of discernment, l(x) = Bel(x) and u(x) = Pl(x), we get P[(Bel, Pl)] = T 1[Bel] \u2229T n\u22121[Bel]. 20 \f6.4 Ternary case Let us consider the case of a frame of cardinality 3, \u0398 = {x, y, z}, and a belief function Bel with mass assignment m(x) = 0.2, m(y) = 0.1, m(z) = 0.3, m({x, y}) = 0.1, m({y, z}) = 0.2, m(\u0398) = 0.1. (51) Figure 3 illustrates the geometry of the related credal set P[Bel] in the simplex, denoted by Cl(Px, Py, Pz), of all the probability measures on \u0398. It is well known [7, 26] that the credal set associated with a belief function is a polytope whose vertices are associated with all possible permutations of singletons. Proposition 2. Given a belief function Bel : 2\u0398 \u2192[0, 1], the simplex P[Bel] of the probability measures consistent with Bel is the polytope P[Bel] = Cl(P \u03c1[Bel] \u2200\u03c1), where \u03c1 is any permutation {x\u03c1(1), . . . , x\u03c1(n)} of the singletons of \u0398, and the vertex P \u03c1[Bel] is the Bayesian belief function such that P \u03c1[Bel](x\u03c1(i)) = X A\u220bx\u03c1(i),A\u0338\u220bx\u03c1(j) \u2200j<i m(A). (52) By Proposition 2, for the example belief function (51), P[Bel] has as vertices the probabilities P \u03c11, P \u03c12, P \u03c13, P \u03c14, P \u03c15[Bel] identi\ufb01ed by purple squares in Fig. 3, namely \u03c11 = (x, y, z) : P \u03c11[Bel](x) = .4, P \u03c11[Bel](y) = .3, P \u03c11[Bel](z) = .3, \u03c12 = (x, z, y) : P \u03c12[Bel](x) = .4, P \u03c12[Bel](y) = .1, P \u03c12[Bel](z) = .5, \u03c13 = (y, x, z) : P \u03c13[Bel](x) = .2, P \u03c13[Bel](y) = .5, P \u03c13[Bel](z) = .3, \u03c14 = (z, x, y) : P \u03c14[Bel](x) = .3, P \u03c14[Bel](y) = .1, P \u03c14[Bel](z) = .6, \u03c15 = (z, y, x) : P \u03c15[Bel](x) = .2, P \u03c15[Bel](y) = .2, P \u03c15[Bel](z) = .6 (53) (as the permutations (y, x, z) and (y, z, x) yield the same probability distribution). We can notice a number of relevant facts: 1. As pointed out above, P[Bel] (the polygon delimited by the purple squares) is the intersection of the two triangles (two-dimensional simplices) T 1[Bel] and T 2[Bel]. 2. The relative belief of singletons, \u02dc Bel(x) = .2 .6 = 1 3, \u02dc Bel(y) = .1 .6 = 1 6, \u02dc Bel(z) = .3 .6 = 1 2, is the intersection of the lines joining the corresponding vertices of the probability simplex P and the lower simplex T 1[Bel]. 21 \fPx 1 0 0.5 0.5 0 0 0.5 1 1 T 1 T 2 Bel ~ P [Bel] Pl ~ p[Bel] Py Pz Figure 3: The polytope P[Bel] of probabilities consistent with the belief function Bel (51) de\ufb01ned on {x, y, z} is shown in a darker shade of purple, in the probability simplex P = Cl(Px, Py, Pz) (shown in light blue). Its vertices (purple squares) are given in (53). The intersection probability, relative belief and relative plausibility of singletons (green squares) are the foci of the pairs of simplices {T 1[Bel], T 2[Bel]}, {T 1[Bel], P} and {P, T 2[Bel]}, respectively. In the ternary case, T 1[Bel] and T 2[Bel] (light purple) are regular triangles. Geometrically, their focus is the intersection of the lines joining their corresponding vertices (dashed lines for {T 1[Bel], P},{P, T 2[Bel]}; solid lines for {T 1[Bel], T 2[Bel]}). 3. The relative plausibility of singletons, \u02dc Pl(x) = m(x) + m({x, y}) + m(\u0398) kP l \u2212kBel = .4 .4 + .5 + .6 = 4 15, \u02dc Pl(y) = .5 .4 + .5 + .6 = 1 3, \u02dc Pl(z) = 2 5, is the intersection of the lines joining the corresponding vertices of the probability simplex P and the upper simplex T 2[Bel]. 4. Finally, the intersection probability, p[Bel](x) = m(x) + \u03b2[Bel] \u0000m({x, y}) + m(\u0398) \u0001 = .2 + .4 \u22170.2 1.5 \u22120.4 = .27, p[Bel](y) = .1 + .4 1.10.4 = .245, p[Bel](z) = .485, is the unique intersection of the lines joining the corresponding vertices of the upper and lower simplices T 2[Bel] and T 1[Bel]. 22 \fAlthough Fig. 3 suggests that \u02dc Bel, \u02dc Pl and p[Bel] might be consistent with Bel, this is a mere artefact of this ternary example, for it can be proved ([43], Chapter 12) that neither the relative belief of singletons nor the relative plausibility of singletons necessarily belongs to the credal set P[Bel]. Indeed, the point here is that the epistemic transforms \u02dc Bel, \u02dc Pl, p[Bel] are instead consistent with the interval probability system P[(Bel, Pl)] associated with the original belief function Bel: \u02dc Bel, \u02dc Pl, p[Bel] \u2208P[(Bel, Pl)] = T 1[Bel] \u2229T n\u22121[Bel]. Their geometric behaviour, as described by facts 2, 3 and 4, holds in the general case as well. 6.5 Focus of a pair of simplices De\ufb01nition 2. Consider two simplices in Rn\u22121, denoted by S = Cl(s1, . . . , sn) and T = Cl(t1, . . . , tn), with the same number of vertices. If there exists a permutation \u03c1 of {1, . . . , n} such that the intersection n \\ i=1 a(si, t\u03c1(i)) (54) of the lines joining corresponding vertices of the two simplices exists and is unique, then p = f(S, T) . = Tn i=1 a(si, t\u03c1(i)) is termed the focus of the two simplices S and T. Not all pairs of simplices admit a focus. For instance, the pair of simplices (triangles) S = Cl(s1 = [2, 2]\u2032, s2 = [5, 2]\u2032, s3 = [3, 5]\u2032) and T = Cl(t1 = [3, 1]\u2032, t2 = [5, 6]\u2032, t3 = [2, 6]\u2032) in R2 does not admit a focus, as no matter what permutation of the order of the vertices we consider, the lines joining corresponding vertices do not intersect. Geometrically, all pairs of simplices admitting a focus can be constructed by considering all possible stars of lines, and all possible pairs of points on each line of each star as pairs of corresponding vertices of the two simplices. De\ufb01nition 3. We call a focus special if the af\ufb01ne coordinates of f(S, T) on the lines a(si, t\u03c1(i)) all coincide, namely \u2203\u03b1 \u2208R such that f(S, T) = \u03b1si + (1 \u2212\u03b1)t\u03c1(i) \u2200i = 1, . . . , n. (55) Not all foci are special. As an example, the simplices S = Cl(s1 = [\u22122, \u22122]\u2032, s2 = [0, 3]\u2032, s3 = [1, 0]\u2032) and T = Cl(t1 = [\u22121, 0]\u2032, t2 = [0, \u22121]\u2032, t3 = [2, 2]\u2032) in R2 admit a focus, in particular for the permutation \u03c1(1) = 3, \u03c1(2) = 2, \u03c1(3) = 1 (i.e., the lines a(s1, t3), a(s2, t2) and a(s3, t1) intersect in 0 = [0, 0]\u2032). However, the focus f(S, T) = [0, 0]\u2032 is not special, as its simplicial coordinates in the three lines are \u03b1 = 1 2, \u03b1 = 1 4 and \u03b1 = 1 2, respectively. On the other hand, given any pair of simplices in Rn\u22121 S = Cl(s1, . . . , sn) and T = Cl(t1, . . . , tn), for any permutation \u03c1 of the indices there always exists a linear 23 \fvariety of points which have the same af\ufb01ne coordinates in both simplices, \uf8f1 \uf8f2 \uf8f3p \u2208Rn\u22121 \f \f \f \f \fp = n X i=1 \u03b1isi = n X j=1 \u03b1jt\u03c1(j), n X i=1 \u03b1i = 1. \uf8fc \uf8fd \uf8fe (56) Indeed, the conditions on the right-hand side of (56) amount to a linear system of n equations in n unknowns (\u03b1i, i = 1, . . . , n). Thus, there exists a linear variety of solutions to such a system, whose dimension depends on the rank of the matrix of constraints in (56). It is rather easy to prove the following theorem. Theorem 3. Any special focus f(S, T) of a pair of simplices S, T has the same af\ufb01ne coordinates in both simplices, i.e., f(S, T) = n X i=1 \u03b1isi = n X j=1 \u03b1jt\u03c1(j), n X i=1 \u03b1i = 1, where \u03c1 is the permutation of indices for which the intersection of the lines a(si, t\u03c1(i)), i = 1, . . . , n exists. Note that the af\ufb01ne coordinates {\u03b1i, i} associated with a focus can be negative, i.e., the focus may be located outside one or both simplices. Notice also that the barycentre itself of a simplex is a special case of a focus. In fact, the centre of mass b of a d-dimensional simplex S is the intersection of the medians of S, i.e., the lines joining each vertex with the barycentre of the opposite ((d \u22121)dimensional) face (see Fig. 4). But those barycentres, for all (d \u22121)-dimensional faces, themselves constitute the vertices of a simplex T. t S 1 T b t2 t3 s2 s1 s3 Figure 4: The barycentre of a simplex is itself a special case of a focus, as shown here for a two-dimensional example. 24 \f6.6 Intersection probability as a focus Theorem 4. For each belief function Bel, the intersection probability p[Bel] has the same af\ufb01ne coordinates in the lower and upper simplices T 1[Bel] and T n\u22121[Bel], respectively. Theorem 5. The intersection probability is the special focus of the pair of lower and upper simplices {T 1[Bel], T n\u22121[Bel]}, with its af\ufb01ne coordinate on the corresponding intersecting lines equal to \u03b2[Bel] (28). The fraction \u03b1 = \u03b2[Bel] of the width of the probability interval that generates the intersection probability can be read in the probability simplex as its coordinate on any of the lines determining the special focus of {T 1[Bel], T n\u22121[Bel]}. Similar results hold for the relative belief and relative plausibility of singletons, which are the (special) foci associated with the lower and upper simplices T 1[Bel] and T n\u22121[Bel], the geometric incarnations of the lower and upper constraints on singletons. Those two simplices can also be interpreted as the sets of probabilities consistent with the plausibility and belief of singletons, respectively (see [43], Chapter 12). Just as the pignistic function adheres to sensible rationality principles and, as a consequence, it has a clear geometrical interpretation as the centre of mass of the credal set associated with a belief function Bel, the intersection probability has an elegant geometric behaviour with respect to the credal set associated with an interval probability system, being the (special) focus of the related upper and lower simplices. Now, selecting the special focus of two simplices representing two different constraints (i.e., the point with the same convex coordinates in the two simplices) means adopting the single probability distribution which satis\ufb01es both constraints in exactly the same way. If we assume homogeneous behaviour in the two sets of constraints {P(x) \u2265Bel(x) \u2200x}, {P(x) \u2264Pl(x) \u2200x} as a rationality principle for the probability transformation of an interval probability system, then the intersection probability necessarily follows as the unique solution to the problem. Another interesting results stems from the fact that the pignistic function and af\ufb01ne combination commute: BetP[\u03b11Bel1 + \u03b12Bel2] = \u03b11BetP[Bel1] + \u03b12BetP[Bel2] whenever \u03b11 + \u03b12 = 1. Proposition 3. [43] The intersection probability is the convex combination of the barycentres t1[Bel], tn\u22121[Bel] of the lower and upper simplices T 1[Bel], T n\u22121[Bel] associated with a belief function Bel, with coef\ufb01cient (28) p[Bel] = \u03b2[Bel]tn\u22121[Bel] + (1 \u2212\u03b2[Bel])t1[Bel]. 7 Relations with other probability transforms In Section 4.2 we discussed how the intersection probability relates to alternative probability transforms for probability intervals, including Sudano\u2019s proposals. 25 \fSince the intersection probability can also be interpreted (as we have seen) as a probability transform which applies to belief functions, it is quite natural to wonder how it relates to the two main classes of such transforms: the epistemic family and the af\ufb01ne family (see [43], Chapters 11 and 12). 7.1 Epistemic transforms From (17) it follows that, for a belief function Bel, p[Bel](x) = m(x) + (1 \u2212kBel)Pl(x) \u2212m(x) kP l \u2212kBel , (57) which can be rewritten as: p[Bel] = kBel \u02dc Bel + (1 \u2212kBel)R[Bel]. (58) Since kBel = P x\u2208\u0398 m(x) \u22641, (58) implies that the intersection probability p[Bel] belongs to the segment linking the relative uncertainty of singletons R[Bel] to the relative belief of singletons \u02dc Bel. Its convex coordinate on this segment is the total mass of singletons kBel. The relative plausibility function \u02dc Pl can also be written in terms of \u02dc Bel and R[Bel] as, by de\ufb01nition (12), R[Bel](x) = Pl(x) \u2212m(x) kP l \u2212kBel = Pl(x) kP l \u2212kBel \u2212 m(x) kP l \u2212kBel = \u02dc Pl(x) kP l kP l \u2212kBel \u2212\u02dc Bel(x) kBel kP l \u2212kBel , since \u02dc Pl(x) = Pl(x)/kP l and \u02dc Bel(x) = m(x)/kBel. Therefore, \u02dc Pl = \u0012kBel kP l \u0013 \u02dc Bel + \u0012 1 \u2212kBel kP l \u0013 R[Bel]. (59) In summary, both the relative plausibility of singletons \u02dc Pl and the intersection probability p[Bel] belong to the segment Cl(R[Bel], \u02dc Bel) joining the relative belief \u02dc Bel and the relative uncertainty of singletons R[Bel] (see Fig. 5). The convex coordinate of \u02dc Pl in Cl(R[Bel], \u02dc Bel) (59) measures the ratio between the total mass and the total plausibility of the singletons, while that of \u02dc Bel measures the total mass of the singletons kBel. Since kP l = P A\u2282\u0398 m(A)|A| \u22651, we have that kBel/kP l \u2264kBel: hence the relative plausibility function of singletons \u02dc Pl is closer to R[Bel] than p[Bel] is (Figure 5 again). Obviously, when kBel = 0 (the relative belief of singletons \u02dc Bel does not exist, for Bel assigns no mass to singletons), the remaining probability approximations coincide: p[Bel] = \u02dc Pl = R[Bel] by (57). 26 \fP p [Bel] R [Bel] Bel ~ Pl ~ Figure 5: Location in the probability simplex P of the intersection probability p[Bel] and the relative plausibility of singletons \u02dc Pl with respect to the relative uncertainty of singletons R[Bel]. They both lie on the segment joining R[Bel] and the relative belief of singletons \u02dc Bel, but \u02dc Pl is closer to R[Bel] than p[Bel] is. 7.2 Pignistic function To shed even more light on p[Bel], and to get an alternative interpretation of the intersection probability, it is useful to compare p[Bel] as expressed in (57) with the pignistic function, BetP[Bel](x) . = X A\u2287x m(A) |A| = m(x) + X A\u2283x,A\u0338=x m(A) |A| . In BetP[Bel] the mass of each event A, |A| > 1, is considered separately, and its mass m(A) is shared equally among the elements of A. In p[Bel], instead, it is the total mass P |A|>1 m(A) = 1 \u2212kBel of non-singleton focal sets which is considered, and this total mass is distributed proportionally to their non-Bayesian contribution to each element of \u0398. Now, if |A| > 1, \u03b2[BelA] = P |B|>1 m(B) P |B|>1 m(B)|B| = 1 |A|, so that both p[Bel](x) and BetP[Bel](x) assume the form m(x) + X A\u2283x,A\u0338=x m(A)\u03b2A, where \u03b2A = const = \u03b2[Bel] for p[Bel], while \u03b2A = \u03b2[BelA] in the case of the pignistic function. Under what conditions do the intersection probability and pignistic function coincide? A suf\ufb01cient condition can be easily given for a special class of belief functions. Theorem 6. The intersection probability and pignistic function coincide for a given belief function Bel whenever the focal elements of Bel have size 1 or k only. Proof. The desired equality p[Bel] = BetP[Bel] is equivalent to m(x) + X A\u228bx m(A)\u03b2[Bel] = m(x) + X A\u228bx m(A) |A| , 27 \fwhich in turn reduces to X A\u228bx m(A)\u03b2[Bel] = X A\u228bx m(A) |A| . If \u2203k : m(A) = 0 for |A| \u0338= k, |A| > 1, then \u03b2[Bel] = 1/k and the equality is satis\ufb01ed. In particular, this is true when Bel is 2-additive (its focal elements have cardinality \u22642). Example 1. Let us brie\ufb02y discuss these two interpretations of p[Bel] in a simple example. Consider a ternary frame \u0398 = {x, y, z}, and a belief function Bel with BPA m(x) = 0.1, m(y) = 0, m(z) = 0.2, m({x, y}) = 0.3, m({x, z}) = 0.1, m({y, z}) = 0, m(\u0398) = 0.3. (60) The related basic plausibility assignment is, according to (35), \u00b5(x) = (\u22121)|x|+1 X B\u2287{x} m(B) = m(x) + m({x, y}) + m({x, z}) + m(\u0398) = 0.8, \u00b5(y) = 0.6, \u00b5(z) = 0.6, \u00b5({x, y}) = \u22120.6, \u00b5({x, z}) = \u22120.4, \u00b5({y, z}) = \u22120.3, \u00b5(\u0398) = 0.3. Figure 6 depicts the subsets of \u0398 with non-zero BPA (left) and BPlA (middle) induced by the belief function (60): dashed ellipses indicate a negative mass. The total mass that (60) accords to singletons is kBel = 0.1+0+0.2 = 0.3. Thus, the line coordinate \u03b2[Bel] of the intersection \u03c2[Bel] of the line a(Bel, Pl) with P\u2032 is \u03b2[Bel] = 1 \u2212kBel m({x, y})|{x, y}| + m({x, z})|{x, z}| + m(\u0398)|\u0398| = 0.7 1.7. By (32), the mass assignment of \u03c2[Bel] is therefore m\u03c2[Bel](x) = m(x) + \u03b2[Bel](\u00b5(x) \u2212m(x)) = 0.1 + 0.7 \u00b7 0.7 1.7 = 0.388, m\u03c2[Bel](y) = 0 + 0.6 \u00b7 0.7 1.7 = 0.247, m\u03c2[Bel](z) = 0.2 + 0.4 \u00b7 0.7 1.7 = 0.365, m\u03c2[Bel]({x, y}) = 0.3 \u22120.9 \u00b7 0.7 1.7 = \u22120.071, m\u03c2[Bel]({x, z}) = 0.1 \u22120.5 \u00b7 0.7 1.7 = \u22120.106, m\u03c2[Bel]({y, z}) = 0 \u22120.3 \u00b7 0.7 1.7 = \u22120.123, m\u03c2[Bel](\u0398) = 0.3 + 0 \u00b7 0.7 1.7 = 0.3. 28 \fWe can verify that all singleton masses are indeed non-negative and add up to one, while the masses of the non-singleton events add up to zero, \u22120.071 \u22120.106 \u22120.123 + 0.3 = 0, con\ufb01rming that \u03c2[Bel] is a Bayesian normalised sum function (pseudo belief function). Its mass assignment has signs which are still described by Fig. 6 (middle) although, as m\u03c2[Bel] is a weighted average of m and \u00b5, its mass values are closer to zero. x y z \u0398 x y z x y z Figure 6: Signs of non-zero masses assigned to events by the functions discussed in Example 1. Left: BPA of the belief function (60), with \ufb01ve focal elements. Middle: the associated BPlA assigns positive masses (solid ellipses) to all events of size 1 and 3, and negative ones (dashed ellipses) to all events of size 2. This is the case for the mass assignment associated with \u03c2 (32) too. Right: the intersection probability p[Bel] (31) retains, among the latter, only the masses assigned to singletons. In order to compare \u03c2[Bel] with the intersection probability, we need to recall (57): the non-Bayesian contributions of x, y, z are, respectively, Pl(x) \u2212m(x) = m(\u0398) + m({x, y}) + m({x, z}) = 0.7, Pl(y) \u2212m(y) = m({x, y}) + m(\u0398) = 0.6, Pl(z) \u2212m(z) = m({x, z}) + m(\u0398) = 0.4, so that the relative uncertainty of singletons is R(x) = 0.7/1.7, R(y) = 0.6/1.7, R(z) = 0.4/1.7. For each singleton \u03b8, the value of the intersection probability results from adding to the original BPA m(\u03b8) a share of the mass of the non-singleton events 1 \u2212kBel = 0.7 proportional to the value of R(\u03b8) (see Fig. 6, right): p[Bel](x) = m(x) + (1 \u2212kBel)R(x) = 0.1 + 0.7 \u22170.7/1.7 = 0.388, p[Bel](y) = m(y) + (1 \u2212kBel)R(y) = 0 + 0.7 \u22170.6/1.7 = 0.247, p[Bel](z) = m(z) + (1 \u2212kBel)R(z) = 0.2 + 0.7 \u22170.4/1.7 = 0.365. We can see that p[Bel] coincides with the restriction of \u03c2[Bel] to singletons. Equivalently, \u03b2[Bel] measures the share of Pl(x)\u2212m(x) assigned to each element of the frame of discernment: p[Bel](x) = m(x) + \u03b2[Bel](Pl(x) \u2212m(x)) = 0.1 + 0.7/1.7 \u22170.7, p[Bel](y) = m(y) + \u03b2[Bel](Pl(y) \u2212m(y)) = 0 + 0.7/1.7 \u22170.6, p[Bel](z) = m(z) + \u03b2[Bel](Pl(z) \u2212m(z)) = 0.2 + 0.7/1.7 \u22170.4. 29 \f8 Operators To complete our analysis, it can be useful to understand the way the intersection probability relates to two major (geometric) operators in the space of belief functions: af\ufb01ne combination and convex closure. 8.1 Af\ufb01ne combination We have seen in Section 7 that p[Bel] and BetP[Bel] are closely related probability transforms, linked by the role of the quantity \u03b2[Bel]. It is natural to wonder whether p[Bel] exhibits a similar behaviour with respect to the convex closure operator (see [43], Chapter 4). Indeed, although the situation is a little more complex in this second case, p[Bel] turns also out to be related to Cl(.) in a rather elegant way. Let us introduce the notation \u03b2[Beli] = Ni/Di. Theorem 7. Given two arbitrary belief functions Bel1, Bel2 de\ufb01ned on the same frame of discernment, the intersection probability of their af\ufb01ne combination \u03b11 Bel1 + \u03b12 Bel2 is, for any \u03b11 \u2208[0, 1], \u03b12 = 1 \u2212\u03b11, p[\u03b11 Bel1 + \u03b12 Bel2] = \\ \u03b11D1 \u0000\u03b11p[Bel1] + \u03b12T[Bel1, Bel2] \u0001 +\\ \u03b12D2 \u0000\u03b11T[Bel1, Bel2]) + \u03b12p[Bel2] \u0001 , (61) where [ \u03b1iDi = \u03b1iDi \u03b11D1+\u03b12D2 , T[Bel1, Bel2] is the probability with values T[Bel1, Bel2](x) . = \u02c6 D1p[Bel2, Bel1] + \u02c6 D2p[Bel1, Bel2], (62) with \u02c6 Di . = Di D1+D2 , and p[Bel2, Bel1](x) . = m2(x) + \u03b2[Bel1] \u0000Pl2(x) \u2212m2(x) \u0001 , p[Bel1, Bel2](x) . = m1(x) + \u03b2[Bel2] \u0000Pl1(x) \u2212m1(x) \u0001 . (63) Geometrically, p[\u03b11 Bel1 + \u03b12 Bel2] can be constructed as in Fig. 7 as a point of the simplex Cl(T[Bel1, Bel2], p[Bel1], p[Bel2]). The point \u03b11T[Bel1, Bel2] + \u03b12p[Bel2] is the intersection of the segment Cl(T, p[Bel2]) with the line l2 passing through \u03b11p[Bel1] + \u03b12p[Bel2] and parallel to Cl(T, p[Bel1]). Dually, the point \u03b12T[Bel1, Bel2] + \u03b11p[Bel1] is the intersection of the segment Cl(T, p[Bel1]) with the line l1 passing through \u03b11p[Bel1] + \u03b12p[Bel2] and parallel to Cl(T, p[Bel2]). Finally, p[\u03b11Bel1 + \u03b12Bel2] is the point of the segment Cl(\u03b11T + \u03b12p[Bel2], \u03b12T + \u03b11p[Bel1]) with convex coordinate \\ \u03b11D1 (or, equivalently, \\ \u03b12D2). 30 \fp[Bel ] T [Bel ,Bel ] 1 2 p[                        ] D T +     p [Bel ]  \u03b12 2 p[Bel ] 1 Bel   1 Bel2 l2 l1 \u03b11 \u03b12 \u03b12 \u03b12 \u03b12 \u03b11 + \u03b11 \u03b11 D \u03b11 1 2 \u03b11 \u03b12 T +     p [Bel ]  \u03b11 \u03b12 1 2 Bel   1 Bel2 \u03b12 \u03b11 + Figure 7: Behaviour of the intersection probability p[Bel] under af\ufb01ne combination. \u03b12T + \u03b11p[Bel1] and \u03b11T +\u03b12p[Bel2] lie in inverted locations on the segments joining T[Bel1, Bel2] and p[Bel1], p[Bel2], respectively: \u03b1ip[Beli] + \u03b1jT is the intersection of the line Cl(T, p[Beli]) with the line parallel to Cl(T, p[Belj]) passing through \u03b11p[Bel1]+\u03b12p[Bel2]. The quantity p[\u03b11Bel1 + \u03b12Bel2] is, \ufb01nally, the point of the segment joining them with convex coordinate [ \u03b1iDi. 8.1.1 Location of T[Bel1, Bel2] in the binary case As an example, let us consider the location of T[Bel1, Bel2] in the binary belief space B2 (Fig. 8), where \u03b2[Bel1] = \u03b2[Bel2] = mi(\u0398) 2mi(\u0398) = 1 2 \u2200Bel1, Bel2 \u2208B2, and p[Bel] always commutes with the convex closure operator. Accordingly, T[Bel1, Bel2](x) = m1(\u0398) m1(\u0398) + m2(\u0398) \u0014 m2(x) + m2(\u0398) 2 \u0015 + m2(\u0398) m1(\u0398) + m2(\u0398) \u0014 m1(x) + m1(\u0398) 2 \u0015 = m1(\u0398) m1(\u0398) + m2(\u0398)p[Bel2] + m2(\u0398) m1(\u0398) + m2(\u0398)p[Bel1]. Looking at Figure 8, simple trigonometric considerations show that the segment de\ufb01ned by Cl(p[Beli], T[Bel1, Bel2]) has length mi(\u0398)/( \u221a 2 tan \u03c6), where \u03c6 is the 31 \fBel Bel  = [0,1]' y Bel   = [0,0]' \u0398 Bel  = [1,0]' x Pl Pl2 p[Bel ] T [Bel ,Bel ] m  (\u0398) m (\u0398) Bel2 p[Bel ] 2 1 2 2 1 1 1 1 Figure 8: Location of the probability function T[Bel1, Bel2] in the binary belief space B2. angle between the segments Cl(Beli, T) and Cl(p[Beli], T). T[Bel1, Bel2] is then the unique point of P such that the angles \\ Bel1Tp[Bel1] and \\ Bel2Tp[Bel2] coincide, i.e., T is the intersection of P with the line passing through Beli and the re\ufb02ection of Belj through P. As this re\ufb02ection (in B2) is nothing but Plj, T[Bel1, Bel2] = Cl(Bel1, Pl2) \u2229P = Cl(Bel2, Pl1) \u2229P. 8.2 Convex closure Although the intersection probability does not commute with af\ufb01ne combination (Theorem 7), p[Bel] can still be assimilated into the orthogonal projection and the pignistic function. Theorem 8 states the conditions under which p[Bel] and convex closure (Cl) commute. Theorem 8. The intersection probability and convex closure commute iff T[Bel1, Bel2] = \u02c6 D1p[Bel2] + \u02c6 D2p[Bel1] or, equivalently, either \u03b2[Bel1] = \u03b2[Bel2] or R[Bel1] = R[Bel2]. Geometrically, only when the two lines l1, l2 in Fig. 7 are parallel to the af\ufb01ne space a(p[Bel1], p[Bel2]) (i.e., T[Bel1, Bel2] \u2208Cl(p[Bel1], p[Bel2]); compare the above) does the desired quantity p[\u03b11Bel1 + \u03b12Bel2] belong to the line segment Cl(p[Bel1], p[Bel2]) (i.e., it is also a convex combination of p[Bel1] and p[Bel2]). 32 \fTheorem 8 re\ufb02ects the two complementary interpretations of p[Bel] we gave in terms of \u03b2[Bel] and R[Bel]: p[Bel] = m(x) + (1 \u2212kBel)R[Bel](x), p[Bel] = m(x) + \u03b2[Bel](Pl(x) \u2212m(x)). If \u03b2[Bel1] = \u03b2[Bel2], both belief functions assign to each singleton the same share of their non-Bayesian contribution. If R[Bel1] = R[Bel2], the non-Bayesian mass 1 \u2212kBel is distributed in the same way to the elements of \u0398. A suf\ufb01cient condition for the commutativity of p[.] and Cl(.) can be obtained via the following decomposition of \u03b2[Bel]: \u03b2[Bel] = P |B|>1 m(B) P |B|>1 m(B)|B| = Pn k=2 P |B|=k m(B) Pn k=2 k \u00b7 P |B|=k m(B) = \u03c32 + \u00b7 \u00b7 \u00b7 + \u03c3n 2\u03c32 + \u00b7 \u00b7 \u00b7 + n\u03c3n , (64) where \u03c3k . = P |B|=k m(B). Theorem 9. If the ratio between the total masses of focal elements of different cardinality is the same for all the belief functions involved, namely \u03c3l 1 \u03c3m 1 = \u03c3l 2 \u03c3m 2 \u2200l, m \u22652 s.t. \u03c3m 1 , \u03c3m 2 \u0338= 0, (65) then the intersection probability (considered as an operator mapping belief functions to probabilities) and convex combination commute. 9"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2104.10651v1",
+                    "title": "A geometric approach to conditioning belief functions",
+                    "abstract": "Conditioning is crucial in applied science when inference involving time\nseries is involved. Belief calculus is an effective way of handling such\ninference in the presence of epistemic uncertainty -- unfortunately, different\napproaches to conditioning in the belief function framework have been proposed\nin the past, leaving the matter somewhat unsettled. Inspired by the geometric\napproach to uncertainty, in this paper we propose an approach to the\nconditioning of belief functions based on geometrically projecting them onto\nthe simplex associated with the conditioning event in the space of all belief\nfunctions. We show here that such a geometric approach to conditioning often\nproduces simple results with straightforward interpretations in terms of\ndegrees of belief. This raises the question of whether classical approaches,\nsuch as for instance Dempster's conditioning, can also be reduced to some form\nof distance minimisation in a suitable space. The study of families of\ncombination rules generated by (geometric) conditioning rules appears to be the\nnatural prosecution of the presented research.",
+                    "authors": "Fabio Cuzzolin",
+                    "published": "2021-04-21",
+                    "updated": "2021-04-21",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "math.PR",
+                        "math.ST",
+                        "stat.TH",
+                        "62A01, 62A86, 60A05, 60A86, 60D05, 60E05"
+                    ],
+                    "main_content": "Introduction Decision making and estimation are common problems in applied science, as people or machines need to make inferences about the state of the external \u2217F. Cuzzolin is with the School of Engineering, Computing and Mathematics, Oxford Brookes University, Oxford, UK. Email: Fabio.Cuzzolin@brookes.ac.uk. 1 arXiv:2104.10651v1  [cs.AI]  21 Apr 2021 \fworld and take appropriate actions. Such state is typically assumed to be described by a probability distribution over a set of alternative hypotheses, which in turn needs to be inferred from the available data. Sometimes, however, as in the case of extremely rare events (e.g., a volcanic eruption), few samples are available to drive such inference. Part of the data can be missing. In addition, under the law of large numbers, probability distributions are the outcome of an in\ufb01nite process of evidence accumulation whereas, in all practical cases, the available evidence can only provide some sort of constraint on the unknown, \u2018true\u2019 probability governing the process. Di\ufb00erent kinds of constraints are associated with di\ufb00erent generalisations of probabilities, formulated to model \u2018epistemic\u2019 uncertainty at the level of probability distributions [67]. The simplest such generalisations are, arguably, interval probabilities [61] and convex sets of probabilities or \u2018credal sets\u2019 [102]. A whole battery of di\ufb00erent uncertainty theories [143] has been developed in the last century or so, starting from De Finetti\u2019s pioneering work [62]. In particular, Shafer\u2019s theory of belief functions (b.f.s) [124], based on A. Dempster\u2019s [63] seminal work, allows us to express partial belief by providing lower and upper bounds to probability values [145]. The widespread presence and in\ufb02uence of uncertainty at di\ufb00erent levels explains why belief functions have been increasingly applied to \ufb01elds as diverse as robotics [121], economics, computer vision [7, 104], communications [1, 123] or cybersecurity [138]. Powerful tools for decision making [137, 133, 141] and classi\ufb01cation [105] with belief functions have also been proposed \u2013 see [66] for a recent review. When observations come from time series, however, or when conditional independence assumptions are necessary to simplify the structure of the joint (belief) distribution to estimate, the need for generalising the classical results on total probability to belief calculus arises. This is the case, for instance, in computer vision problems such as image segmentation [112], where conditional independence is crucial to make large-scale optimisation problems tractable. In target tracking, conditional constraints on the targets\u2019 future positions given their past locations are available. If such constraints are described as belief functions [56, 80, 55, 59, 83], predicting current target locations requires combining conditional belief functions into a single \u2018total\u2019 belief function [14, 113, 148, 68, 69], in a generalisation of the total probability theorem. Unfortunately, in opposition to what happens in classical probability theory, di\ufb00erent de\ufb01nitions of conditional belief functions [99] can be imagined 2 \fand have indeed been proposed in the past [76, 131, 139, 95, 136, 97, 140, 101]. Quite recently, the idea of formulating the problem geometrically has emerged. Lehrer [101], in particular, has proposed such a geometric approach to determine the conditional expectation of non-additive probabilities (such as belief functions). The notion of generating conditional belief functions by minimising a suitable distance function between the original b.f. b and the \u2018conditioning region\u2019 BA associated with the conditioning event A, i.e., the set of belief functions whose b.b.a. assigns mass to subsets of A only bd(.|A) = arg min b\u2032\u2208BA d(b, b\u2032), (1) naturally arises in the context of the geometric approach to uncertainty [44]. This original angle has a clear potential, as it expands our arsenal of approaches to the problem and, we argue here, is a promising candidate to the role of general framework for conditioning. 1.1 Contributions and paper outline In particular, in this paper we explore the geometric conditioning problem as de\ufb01ned above in the mass space representation in which belief functions are represented by the vectors of their mass values, and adopt distance measures of the classical Minkowski (Lp) family. We \ufb01rst recall in Section 2 the state of the art concerning evidence theory (Section 2.1), the conditioning of belief functions (Section 2.2), the geometric approach to uncertainty (Section 2.3) and the use of distances in evidence theory (Section 2.4). The core of the paper is Section 3, in which our main results on conditioning belief functions in the mass space are derived. We \ufb01rst recall in Section 3.1 the basic notions of the geometric approach to belief functions. In particular, we show how each b.f. can be represented by either the vector of its belief values in the belief space or the vector of its mass values in the mass space. The notion of geometric conditional belief function and the associated minimisation problem are formalised in Section 3.2. In Sections 3.3, 3.4 and 3.5, instead, we prove the analytical forms of the L1, L2 and L\u221e conditional belief functions in the mass space, respectively. The results obtained are discussed in Section 5, by summarising the main results (Section 5.1), highlighting the properties of geometric conditional 3 \fbelief functions (Section 5.2), hinting at an interesting connection with Lewis\u2019 imaging [103] generalized to b.f.s (Section 5.3) and comparing the results of geometrically conditioning belief functions in the mass as opposed to the belief space (Section 5.4). Finally, in Section 6 we prospect a number of future developments for the geometric approach to conditioning. 2 State of the art 2.1 Belief functions A basic probability assignment (b.p.a.) over a \ufb01nite set (frame of discernment [124]) \u0398 is a function mb : 2\u0398 \u2192[0, 1] on its power set 2\u0398 = {A \u2286\u0398} such that mb(\u2205) = 0, X A\u2286\u0398 mb(A) = 1. Subsets of \u0398 associated with non-zero values of mb are called focal elements. The belief function b : 2\u0398 \u2192[0, 1] associated with a basic probability assignment mb on \u0398 is de\ufb01ned as b(A) = X B\u2286A mb(B). (2) The core Cb of a b.f. b is the union of its focal elements. A dual mathematical representation of the evidence encoded by a belief function b is the plausibility function plb : 2\u0398 \u2192[0, 1], A 7\u2192plb(A), where the plausibility value plb(A) of an event A is given by plb(A) . = 1 \u2212b(Ac) = X B\u2229A\u0338=\u2205 mb(B) and expresses the amount of evidence not against A. The orthogonal sum b1 \u2295b2 : 2\u0398 \u2192[0, 1] of two belief functions b1 : 2\u0398 \u2192[0, 1], b2 : 2\u0398 \u2192[0, 1] de\ufb01ned on the same frame \u0398 is the unique belief function on \u0398 whose focal elements are all the possible intersections A = B \u2229C of focal elements B and C of b1 and b2, respectively, and whose 4 \fbasic probability assignment is given by mb1\u2295b2(A) = X B\u2229C=A mb1(B)mb2(C) 1 \u2212 X B\u2229C=\u2205 mb1(B)mb2(C) . 2.2 Conditioning in belief calculus 2.2.1 Dempster conditioning The original proposal is due to Dempster himself [63]. He formulated it in his original model, in which belief functions are induced by multi-valued mappings \u0393 : \u2126\u21922\u0398 of probability distributions de\ufb01ned on a set \u2126onto the power set of another set (\u2018frame\u2019) \u0398. Given a conditioning event B \u2282\u0398, the \u2018logical\u2019 (or \u2018categorical\u2019, in Smets\u2019s terminology) belief function bB such that mb(B) = 1 is combined via Dempster\u2019s rule with the a priori BF b. The resulting measure b \u2295bB is the conditional belief function given B ` a la Dempster, which we denote by b\u2295(A|B). Dempster conditioning was almost immediately and strongly criticised from a Bayesian standpoint. In response to these objections a number of approaches to conditioning in belief calculus have been proposed [100, 10, 76, 88, 81, 64, 147, 87, 140] along the years, in di\ufb00erent mathematical setups. 2.2.2 Credal conditioning Fagin and Halpern, in particular, argued that Dempster conditioning behaves unreasonably in the context of the classical \u2018three prisoners\u2019 example [76], originally discussed by Diaconis [70]. They thus proposed a notion of conditional belief [76] as the lower envelope of a family of conditional probability functions, and provided a closed-form expression for it. The notion, quite related to the concept of \u2018inner measure\u2019, was considered by various other authors [100]. Namely, a b.f. b can be interpreted as the lower envelope of a family P[b] of probability distributions, i.e. b(A) = inf P\u2208P[b] P(A). 5 \fFagin and Halpern thus de\ufb01ned the conditional belief function associated with b as the lower envelope (that is, the in\ufb01mum) of the family of conditional probability functions P(A|B), where P is consistent with b: bCr(A|B) . = inf P\u2208P[b] P(A|B), plCr(A|B) . = sup P\u2208P[b] P(A|B). (3) This \u2018credal\u2019 conditioning is more conservative than Dempster conditioning, as the associated probability interval is included in that resulting from Dempster conditioning. 2.2.3 Spies\u2019s de\ufb01nition In the (original) framework of multi-valued mappings, instead, Spies [136] de\ufb01ned conditional events as sets of equivalent events under conditioning. By applying multi-valued mapping to such events, conditional belief functions can be introduced. An updating rule, which is equivalent to the law of total probability if all beliefs are probabilities, was also introduced. Importantly, in this de\ufb01nition a conditional belief function is not a b.f. on the subalgebra of the subsets of the conditioning event A. It can be proven that Spies\u2019s conditional belief functions are closed under Dempster\u2019s rule of combination and therefore coherent with the random-set interpretation of the theory. In related work by Slobodova [127], a multi-valued extension of conditional belief functions was introduced [128] and its properties examined. 2.2.4 Geometric conditioning Under the focusing principle of belief revision [73, 75, 74], no new information is introduced as we merely focus on a speci\ufb01c subset of the original hypothesis space. When applied to belief functions, this yields Suppes and Zanotti\u2019s geometric conditioning [139]: bG(A|B) = b(A \u2229B) b(B) , plG(A|B) = b(B) \u2212b(B \\ A) b(B) . (4) This was proved by Smets using Pearl\u2019s \u2018probability of provability\u2019 [116] interpretation of belief functions. It is interesting to note that geometric conditioning is somewhat dual to Dempster conditioning, as it amounts to replacing probability values with 6 \fbelief values in Bayes\u2019 rule: pl\u2295(A|B) = plb(A \u2229B) plb(B) \u2194 bG(A|B) = b(A \u2229B) b(B) . Unlike Dempster and conjunctive conditioning (see the next paragraph), geometric conditioning does not seem to be linked to an underlying combination operator, although the issue remains open. 2.2.5 Smets\u2019s conjunctive rule of conditioning Another way of dealing with the classical Bayesian criticism of Dempster\u2019s rule is to abandon all notions of multivalued mapping and de\ufb01ne belief directly on the power set of the frame as in Smets\u2019 Transferable Belief Model [130] as in (2). To signal the fact that in his TBM no multivalued mapping is required Smets called mass functions basic belief assignments (b.b.a.s). In particular, the conditional b.f. bU(B|A) with b.b.a. mU(B|A) = X C\u2286Ac m(B \u222aC), B \u2286A turns out to be the minimal commitment specialisation of b such that the plausibility of the complementary event Ac is nil [95]. In [131], Smets pointed out the distinction between revision and focussing in the conditional process, and the way they lead to unnormalized and geometric [139] conditioning, respectively. In these two scenarios he proposed some generalizations of Je\ufb00rey\u2019s rule of conditioning [89, 125] to belief calculus. 2.2.6 Other work The topic of conditioning in evidence theory has been studied by a number of other authors [82]. Klopotek and Wierzchon [97], for instance, have provided a frequencybased interpretation for conditional belief functions. Tang and Zheng [140] have discussed the issue of conditioning in a multidimensional space. Lehrer [101] has proposed a geometric approach to determining the conditional expectation of non-additive probabilities. Such a conditional expectation can then be applied for updating, and to introduce a notion of independence. 7 \fAmong the most recent work in the area, Meester and Kerkvliet have recently proposed to look at the issue from a di\ufb00erent perspective [111, 110], by redeveloping and rederiving various notions of conditional belief functions, using a relative frequencies stance similar to that of [97]. The authors call the two main forms of conditioning contingent and necessary conditioning, respectively. Matuszewski and Klopotek [109], on their hand, have investigated the empirical nature of Dempster\u2019s rule of combination, providing an original interpretation of conditional belief functions as belief functions given the manipulation of the original empirical data. Novel Je\ufb00rey-like conditioning rules have been proposed in [85], whereas the e\ufb03cient computation of belief-theoretic conditionals, in particular Dempster and Fagin-Halpern conditionals, has been investigated by Polpitiya et al. in [120, 119]. Both Dezert et al. [68] and Zhou and Cuzzolin [148] have recently approached the problem of generalising the total probability theorem for belief functions. The former have proposed their own fomulation of the total belief theorem, and used it to derive formal expressions for conditional belief functions, whereas the latter have shown a proof of existence for total belief functions under the assumption that Dempster conditioning is used. From the point of view of manipulating conditionals, a message passing algorithm [114] that approximates belief updating on evidential networks with conditional belief functions has been proposed by Nguyen. The various approaches to conditioning have been recently reviewed by Coletti et al. [12], who considered the problem of Bayesian inference under imprecise prior information in the form of a conditional belief function. 2.2.7 Conditioning: A summary Table 1 summarises the behaviour of the main conditioning operators (including Smets\u2019s conjunctive b \u2229 \u25cb(\u00b7|B) and disjunctive b \u222a \u25cb(\u00b7|B) conditioning) in terms of the degrees of belief and plausibility of the resulting conditional belief function. Conditioning operators form a nested family, from the most committal to the least: b \u222a \u25cb(\u00b7|B) \u2264bCr(\u00b7|B) \u2264b\u2295(\u00b7|B) \u2264b \u2229 \u25cb(\u00b7|B) \u2264pl \u2229 \u25cb(\u00b7|B) \u2264pl\u2295(\u00b7|B) \u2264plCr(\u00b7|B) \u2264pl \u222a \u25cb(\u00b7|B). 8 \fTable 1: Belief and plausibility values of the outcomes of various conditioning operators Operator Belief value Plausibility value Dempster\u2019s \u2295 plb(B) \u2212plb(B \\ A) plb(B) plb(A \u2229B) plb(B) Credal Cr b(A \u2229B) b(A \u2229B) + plb( \u00af A \u2229B) plb(A \u2229B) plb(A \u2229B) + b( \u00af A \u2229B) Geometric G b(A \u2229B) b(B) b(B) \u2212b(B \\ A) b(B) Conjunctive \u2229 \u25cb b(A \u222a\u00af B), A \u2229B \u0338= \u2205 plb(A \u2229B), A \u0338\u2283B Disjunctive \u222a \u25cb b(A), A \u2283B plb(A), A \u2229B = \u2205 2.3 Geometry approach to uncertainty The geometry of set functions and other uncertainty measures has been studied by several authors [5, 60, 108]. Indeed, geometry may potentially be a unifying language for the \ufb01eld of uncertainty theory [43, 54, 5, 94, 84, 144], possibly in conjunction with an algebraic view [52, 13, 24, 58, 17, 19, 39]. Recent papers on this topic include [107, 115, 106]. In the geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and there manipulated (e.g. combined, conditioned and so on) [14, 22, 44]. Much work has been focusing on the geometry of belief functions, which live in a convex space termed the belief space, which can be described both in terms of a simplex (a higher-dimensional triangle) and in terms of a recursive bundle structure [57, 46, 42, 53]. The analysis can be extended to Dempster\u2019s rule of combination by introducing the notion of a conditional subspace and outlining a geometric construction for Dempster\u2019s sum [50, 15]. The combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function, have also been studied [23, 35]. The corresponding spaces are simplices which are congruent to the belief space. Subsequent work has extended the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) [33] and consistent belief functions [37, 49, 26], in terms of 9 \fsimplicial complexes [16]. Analyses of belief functions in terms credal sets have also been conducted [27, 2, 9]. The geometry of the relationship between measures of di\ufb00erent kinds has also been extensively studied [51, 31, 18, 32], with particular attention to the problem of transforming a belief function into a classical probability measure [11, 142, 129]. One can distinguish between an \u2018a\ufb03ne\u2019 family of probability transformations [21] (those which commute with a\ufb03ne combination in the belief space), and an \u2018epistemic\u2019 family of transforms [20], formed by the relative belief and relative plausibility of singletons [28, 29, 38, 47, 34], which possess dual properties with respect to Dempster\u2019s sum [25]. Semantics for the main probability transforms can be provided in terms of credal sets, i.e., convex sets of probabilities [32]. The problem of \ufb01nding the possibility measure which best approximates a given belief function [3] can also be approached in geometric terms [30, 48, 40, 41]. In particular, analogously to what done in this paper, approximations induced by classical Minkowski norms can be derived and compared with classical outer consonant approximations [72]. Minkowski consistent approximations of belief functions in both the mass and the belief space representations can also be derived [37]. Preliminary studies on the application of the geometric approach to conditioning have been conducted by the author in [45, 36]. 2.4 Distances in evidence theory A number of norms for belief functions have been introduced [71, 90, 93, 126], as a tool for assessing the level of con\ufb02ict between di\ufb00erent bodies of evidence, for approximating a b.f. using a di\ufb00erent uncertainty measure and so on. Most relevantly to the proposed, Jousselme et al [92] have conducted a very interesting survey of all the distances and similarity measures so far introduced in belief calculus, and proposed a number of generalisations. Generalisations to belief functions of the classical Kullback\u2013Leibler divergence DKL(P|Q) = Z \u221e \u2212\u221e p(x) log(p(x)/q(x))dx of two probability distributions P, Q, for instance, have been proposed, together with measures based on information theory, such as \ufb01delity, or entropybased norms [78]. 10 \fThe most popular and most cited measure of dissimilarity was proposed by Jousselme et al. [91]. Jousselme\u2019s measure assumes that mass functions m are represented as vectors \u20d7 m, and reads as dJ(m1, m2) . = r 1 2(\u20d7 m1 \u2212\u20d7 m2)TD(\u20d7 m1 \u2212\u20d7 m2), where D(A, B) = |A\u2229B| |A\u222aB| for all A, B \u22082\u0398. Jousselme\u2019s distance so de\ufb01ned (1) is positive de\ufb01nite (as proved by Bouchard et al. in [8]), and thus de\ufb01nes a metric distance; (2) takes into account the similarity among subsets (focal elements); and (3) is such that D(A, B) < D(A, C) if C is \u2018closer\u2019 to A than B. Other similarity measures between belief functions have been proposed by Shi et al [126], Jiang et al [90], and others [93, 71, 90]. Among others, it is worth mentioning the following proposals. \u2022 The Dempster con\ufb02ict \u03ba and Ristic\u2019s closely related additive global dissimilarity measure [122]: \u2212log(1 \u2212\u03ba). \u2022 The \u2018\ufb01delity\u2019 or Bhattacharia coe\ufb03cient [4] extended to belief functions, namely \u221a\u20d7 m1 TW\u221a\u20d7 m2, where W is positive de\ufb01nite and \u221a \u20d7 m is the vector obtained by taking the square roots of each component of \u20d7 m. \u2022 Perry and Stephanou\u2019s distance [118], dPS(m1, m2) = |E1 \u222aE2| \u0012 1 \u2212E1 \u2229E2 E1 \u222aE2 \u0013 + (\u20d7 m12 \u2212\u20d7 m1)T(\u20d7 m12 \u2212\u20d7 m2), where E is, as usual, the collection of focal elements of the belief function with mass assignment m and \u20d7 m12 is the mass vector of the Dempster combination of m1 and m2. \u2022 Blackman and Popoli\u2019s attribute distance [6], dBP(m1, m2) = \u22122 log \u0014 1 \u2212\u03ba(m1, m2) 1 \u2212maxi{\u03ba(mi, mi)} \u0015 + (\u20d7 m1 + \u20d7 m2)T\u20d7 gA \u2212\u20d7 mT 1 G\u20d7 m2, where \u20d7 gA is the vector with elements \u20d7 gA(A) = |A|\u22121 |\u0398|\u22121, A \u2282\u0398, and G(A, B) = (|A| \u22121)(|B| \u22121) (|\u0398| \u22121)2 , A, B \u2282\u0398. 11 \f\u2022 Fixen and Mahler\u2019s Bayesian percent attribute miss [77]: \u20d7 m\u2032 1P \u20d7 m2, where P(A, B) = p(A\u2229B) p(A)p(B) and p is an a priori probability on \u0398. \u2022 Zouhal and Denoeux\u2019s inner product of pignistic functions [149]. \u2022 The family of information-based distances also proposed by Denoeux [65], in which the distance between m1 and m2 is quanti\ufb01ed by the di\ufb00erence between their information contents U(m1), U(m2), dU(m1, m2) = |U(m1) \u2212U(m2)|, where U is any uncertainty measure for belief functions. Many if not all of these measures could be in principle plugged in the minimisation problem (1) to de\ufb01ne conditional belief functions. Nevertheless, in this paper we will restrict ourselves to the class of Minkowski measures. Note that the L1 distance was used earlier by Klir [96] and Harmanec [86]). 3 Geometric conditional belief functions 3.1 The geometry of belief functions Given a frame \u0398, each belief function b : 2\u0398 \u2192[0, 1] is completely speci\ufb01ed by its N \u22122 belief values {b(A), \u2205\u228aA \u228a\u0398}, N . = 2n (n . = |\u0398|), (as b(\u2205) = 0, b(\u0398) = 1 for all b.f.s) and can therefore be represented as a vector of RN\u22122 \u20d7 b = [b(A), \u2205\u228aA \u228a\u0398]\u2032. If we denote by bA the categorical [135] belief function assigning all the mass to a single subset A \u2286\u0398, mbA(A) = 1, mbA(B) = 0 \u2200B \u2286\u0398, B \u0338= A, we can prove that [57, 22] the set of points of RN\u22122 which correspond to a b.f. or \u2018belief space\u2019 B coincides with the convex closure Cl of all the vectors representing categorical belief functions B = Cl(\u20d7 bA, \u2205\u228aA \u2286\u0398), 12 \fwhere Cl(\u20d7 b1, ...,\u20d7 bk) = ( \u20d7 b \u2208B : \u20d7 b = \u03b11\u20d7 b1 + \u00b7 \u00b7 \u00b7 + \u03b1k\u20d7 bk, X i \u03b1i = 1, \u03b1i \u22650 \u2200i ) . The belief space B is a simplex [14, 53, 44], and each vector \u20d7 b \u2208B representing a belief function b can be written as a convex sum as: \u20d7 b = X \u2205\u228aB\u2286\u0398 mb(B)\u20d7 bB. (5) In the same way, each belief function is uniquely associated with the related set of \u2018mass\u2019 values mb(A). It can therefore be seen also as a point of RN\u22121, the vector \u20d7 mb = [mb(A), \u2205\u228aA \u2286\u0398]\u2032 (\u0398 this time included) of its N \u22121 mass components, which can be decomposed as \u20d7 mb = X \u2205\u228aB\u2286\u0398 mb(B)\u20d7 mB, (6) where \u20d7 mB is the vector of mass values associated with the categorical belief function bB. Note that in RN\u22121 \u20d7 m\u0398 = [0, ..., 0, 1]\u2032 cannot be neglected. 3.2 Notion of geometric conditional belief functions Similarly, the vector \u20d7 ma associated with any belief function a whose mass supports only focal elements {\u2205\u228aB \u2286A} included in a given event A can be decomposed as: \u20d7 ma = X \u2205\u228aB\u2286A ma(B)\u20d7 mB. (7) The set of such vectors is a simplex MA . = Cl(\u20d7 mB, \u2205\u228aB \u2286A). We call MA the conditioning simplex in the mass space. Given a belief function b, we call geometric conditional belief function induced by a distance function d in M the b.f.(s) bd,M(.|A) which minimize(s) the distance d(\u20d7 mb, MA) between the mass vector representing b and the conditioning simplex associated with A in M. As recalled above, a large number of proper distance functions or mere dissimilarity measures between belief functions have been proposed in the 13 \fpast, and many others can be imagined or designed [92]. We consider here as distance functions the three major Lp norms d = L1, d = L2 and d = L\u221e. This is not to claim that these are the distance functions of choice for this problem. In recent times, however, Lp norms have been successfully employed in di\ufb00erent problems such as probability [21] and possibility [37, 48] transformation/approximation. For vectors \u20d7 mb, \u20d7 mb\u2032 \u2208M representing the b.p.a.s of two belief functions b, b\u2032, such norms read as \u2225\u20d7 mb \u2212\u20d7 mb\u2032\u2225L1 . = X \u2205\u228aB\u2286\u0398 \f \f \fmb(B) \u2212mb\u2032(B) \f \f \f, \u2225\u20d7 mb \u2212\u20d7 mb\u2032\u2225L\u221e. = max \u2205\u228aB\u2286\u0398 \f \f \fmb(B) \u2212mb\u2032(B) \f \f \f \u2225\u20d7 mb \u2212\u20d7 mb\u2032\u2225L2 . = s X \u2205\u228aB\u2286\u0398 (mb(B) \u2212mb\u2032(B))2. (8) 3.3 Conditioning by L1 norm Given a belief function b with basic probability assignment mb collected in a vector \u20d7 mb \u2208M, its L1 conditional version(s) bL1,M(.|A) has/have basic probability assignment mL1,M(.|A) s.t.: \u20d7 mL1,M(.|A) . = arg min \u20d7 ma\u2208MA \u2225\u20d7 mb \u2212\u20d7 ma\u2225L1. (9) Using the expression (8) of the L1 norm in the mass space M, (9) becomes: arg min \u20d7 ma\u2208MA \u2225\u20d7 mb \u2212\u20d7 ma\u2225L1 = arg min \u20d7 ma\u2208MA X \u2205\u228aB\u2286\u0398 |mb(B) \u2212ma(B)|. By exploiting the fact that the candidate solution \u20d7 ma is an element of MA (Equation (7)) we can greatly simplify this expression. Lemma 1. The di\ufb00erence vector \u20d7 mb \u2212\u20d7 ma in M has the form: \u20d7 mb \u2212\u20d7 ma = X \u2205\u228aB\u228aA \u03b2(B)\u20d7 mB + \u0010 b(A) \u22121 \u2212 X \u2205\u228aB\u228aA \u03b2(B) \u0011 \u20d7 mA + X B\u0338\u2282A mb(B)\u20d7 mB (10) where \u03b2(B) . = mb(B) \u2212ma(B). 14 \fIn the L1 case therefore: \u2225\u20d7 mb \u2212\u20d7 ma\u2225L1 = X \u2205\u228aB\u228aA |\u03b2(B)| + \f \f \fb(A) \u22121 \u2212 X \u2205\u228aB\u228aA \u03b2(B) \f \f \f, (11) plus the constant P B\u0338\u2282A |mb(B)|. This is a function of the form X i |xi| + \f \f \f \u2212 X i xi \u2212k \f \f \f, k \u22650 (12) which has an entire simplex of minima, namely: xi \u22640 \u2200i, P i xi \u2265\u2212k. See Figure 1 for the case of two variables, x1 and x2 (corresponding to the L1 conditioning problem on an event A of size |A| = 2). \u0001k 0 \u0001k x1 x2 Figure 1: The minima of a function of the form (12) with two variables x1, x2 form the triangle x1 \u22640, x2 \u22640, x1 + x2 \u2265\u2212k depicted here. A similar behaviour takes place in the general case too. Theorem 1. Given a belief function b : 2\u0398 \u2192[0, 1] and an arbitrary nonempty focal element \u2205\u228aA \u2286\u0398, the set of L1 conditional belief functions bL1,M(.|A) with respect to A in M is the set of b.f.s with core in A such that 15 \ftheir mass dominates that of b over all the proper subsets of A: bL1,M(.|A) = n a : 2\u0398 \u2192[0, 1] : Ca \u2286A, ma(B) \u2265mb(B) \u2200\u2205\u228aB \u2286A o . (13) As in the toy example of Figure 1, the set of L1 conditional belief function in M has geometrically the form of a simplex. Theorem 2. Given a b.f. b : 2\u0398 \u2192[0, 1] and an arbitrary non-empty focal element \u2205\u228aA \u2286\u0398, the set of L1 conditional belief functions bL1,M(.|A) with respect to A in M is the simplex ML1,A[b] = Cl(\u20d7 m[b]|B L1A, \u2205\u228aB \u2286A) whose vertex \u20d7 m[b]|B L1A, \u2205\u228aB \u2286A, has coordinates {ma(B)} such that \u001a ma(B) = mb(B) + 1 \u2212b(A) = mb(B) + plb(Ac), ma(X) = mb(X) \u2200\u2205\u228aX \u228aA, X \u0338= B. (14) It is important to notice that all the vertices of the L1 conditional simplex fall inside MA proper. In principle, some of them could have fallen in the linear space generated by MA but outside the simplex MA, i.e., some of the solutions ma(B) could have been negative. This is indeed the case for geometrical b.f.s induced by other norms, as we will see in the following. 3.4 Conditioning by L2 norm Let us now compute the analytical form of the L2 conditional belief function(s) in the mass space. We make use of the form (10) of the di\ufb00erence vector \u20d7 mb \u2212\u20d7 ma, where again \u20d7 ma is an arbitrary vector of the conditional simplex MA. In this case, though, it is convenient to recall that the minimal L2 distance between a point \u20d7 p and a vector space is attained by the point \u02c6 q of the vector space V s.t. the di\ufb00erence vector \u20d7 p \u2212\u02c6 q is orthogonal to all the generators \u20d7 gi of V : arg min \u20d7 q\u2208V \u2225\u20d7 p \u2212\u20d7 q\u2225L2 = \u02c6 q \u2208V : \u27e8\u20d7 p \u2212\u02c6 q,\u20d7 gi\u27e9= 0 \u2200i whenever \u20d7 p \u2208Rm, V = span(\u20d7 gi, i). This fact is used in the proof of Theorem 3. 16 \fTheorem 3. Given a belief function b : 2\u0398 \u2192[0, 1] and an arbitrary nonempty focal element \u2205\u228aA \u2286\u0398, the unique L2 conditional belief function bL2,M(.|A) with respect to A in M is the b.f. whose b.p.a. redistributes the mass 1 \u2212b(A) to each focal element B \u2286A in an equal way: \u2200\u2205\u228aB \u2286A mL2,M(B|A) = mb(B) + 1 2|A| \u22121 X B\u0338\u2282A mb(B) = mb(B) + plb(Ac) 2|A| \u22121. (15) According to Equation (15) the L2 conditional belief function is unique, and corresponds to the mass function which redistributes the mass the original belief function assigns to focal elements not included in A to each and all the subsets of A in an equal, even way. L2 and L1 conditional belief functions in M display a strong relationship. Theorem 4. Given a belief function b : 2\u0398 \u2192[0, 1] and an arbitrary nonempty focal element \u2205\u228aA \u2286\u0398, the L2 conditional belief function bL2,M(.|A) with respect to A in M is the center of mass of the simplex ML1,A[b] of L1 conditional belief functions with respect to A in M. Proof. By de\ufb01nition the center of mass of ML1,A[b], whose vertices are given by (14), is the vector 1 2|A| \u22121 X \u2205\u228aB\u2286A \u20d7 m[b]|B L1A whose entry B is given by 1 2|A| \u22121 h mb(B)(2|A|\u22121)+(1\u2212b(A)) i , i.e., (15). 3.5 Conditioning by L\u221enorm Similarly, we can use Equation (10) to minimize the L\u221edistance between the original mass vector \u20d7 mb and the conditioning subspace MA. Let us recall it here for sake of readability: \u20d7 mb \u2212\u20d7 ma = X \u2205\u228aB\u228aA \u03b2(B)\u20d7 mB + X B\u0338\u2282A mb(B)\u20d7 mB + \u0012 b(A) \u22121 \u2212 X \u2205\u228aB\u228aA \u03b2(B) \u0013 \u20d7 mA. Its L\u221enorm reads as \u2225\u20d7 mb \u2212\u20d7 ma\u2225L\u221e= = max ( |\u03b2(B)|, \u2205\u228aB \u228aA; |mb(B)|, B \u0338\u2282A; \f \f \f \fb(A) \u22121 \u2212 X \u2205\u228aB\u228aA \u03b2(B) \f \f \f \f ) . 17 \fAs \f \f \f \fb(A) \u22121 \u2212 X \u2205\u228aB\u228aA \u03b2(B) \f \f \f \f = \f \f \f \f X B\u0338\u2282A mb(B) + X \u2205\u228aB\u228aA \u03b2(B) \f \f \f \f the above norm simpli\ufb01es as max ( |\u03b2(B)|, \u2205\u228aB \u228aA; max B\u0338\u2282A{mb(B)}; \f \f \f \f X B\u0338\u2282A mb(B) + X \u2205\u228aB\u228aA \u03b2(B) \f \f \f \f ) . (16) This is a function of the form f(x1, ..., xm\u22121) = max ( |xi| \u2200i, \f \f \f \f X i xi + k1 \f \f \f \f, k2 ) , (17) with 0 \u2264k2 \u2264k1 \u22641. Consider the case m = 3. Such a function has two possible behaviors in terms of its minimal region in the plane x1, x2. If k1 \u22643k2 its contour function has a set of minimal points given by xi \u2265\u2212k2, x1 + x2 \u2264k2 \u2212k1. In the opposite case k1 > 3k2 the contour function admits a single minimal point, located in [\u22121/3k1, \u22121/3k1]. For an arbitrary number m \u22121 of variables x1, ..., xm\u22121, the \ufb01rst case is such that k2 \u2265k1/m, in which situation the set of minimal points of a function of the form (17) is such that xi \u2265\u2212k2, X i xi \u2264k2 \u2212k1, and forms a simplex with m vertices. Each vertex vi, i \u0338= m has components vi(j) = \u2212k2 \u2200j \u0338= i, vi(i) = \u2212k1 + (m \u22121)k2, while obviously vm = [\u2212k2, \u00b7 \u00b7 \u00b7 , \u2212k2]\u2032. In the opposite case the unique minimal point is located in [(\u22121/m)k1, \u00b7 \u00b7 \u00b7 , (\u22121/m)k1]\u2032. This analysis applies to the norm (16) as follows. 18 \fTheorem 5. Given a belief function b : 2\u0398 \u2192[0, 1] with b.p.a. mb, and an arbitrary non-empty focal element \u2205\u228aA \u2286\u0398, the set of L\u221econditional belief functions mL\u221e,M(.|A) with respect to A in M forms the simplex ML\u221e,A[b] = Cl(\u20d7 m[b]| \u00af B L\u221eA, \u00af B \u2286A) with vertices \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 \u20d7 m[b]| \u00af B L\u221e(B|A) = mb(B) + max C\u0338\u2282A mb(C) \u2200B \u2286A, B \u0338= \u00af B \u20d7 m[b]| \u00af B L\u221e( \u00af B|A) = mb( \u00af B) + X C\u0338\u2282A mb(C) \u2212(2|A| \u22122) max C\u0338\u2282A mb(C) (18) whenever max C\u0338\u2282A mb(C) \u2265 1 2|A| \u22121 X C\u0338\u2282A mb(C). It reduces to the single belief function mL\u221e,M(B|A) = mb(B) + 1 2|A| \u22121 X C\u0338\u2282A mb(C) \u2200B \u2286A whenever max C\u0338\u2282A mb(C) < 1 2|A| \u22121 X C\u0338\u2282A mb(C). The latter is the barycenter of the simplex of L\u221econditional b.f.s in the former case, and coincides with the L2 conditional belief function (15). Note that, as (18) is not guaranteed to be non-negative, the simplex of L\u221econditional belief functions in M does not necessarily fall entirely inside the conditioning simplex MA, i.e., it may include pseudo belief functions [134]. Looking at (18) we can observe that vertices are obtained by assigning the maximum mass not in the conditioning event to all its subsets indi\ufb00erently. Normalisation is then achieved, in opposition to what happens in Dempster\u2019s rule, by subtracting the total mass in excess of 1 in the speci\ufb01c component \u00af B. This behavior is exhibited by other geometric conditional b.f. as shown in the following. 19 \f4 A case study: the ternary frame If |A| = 2, A = {x, y}, the conditional simplex is 2-dimensional, with three vertices \u20d7 mx, \u20d7 my and \u20d7 mx,y. For a belief function b on \u0398 = {x, y, z} Theorem 1 states that the vertices of the simplex ML1,A of L1 conditional belief functions in M are: \u20d7 m[b]|{x} L1 {x, y} = \u0002 mb(x) + plb(z), mb(y), mb(x, y) \u0003\u2032, \u20d7 m[b]|{y} L1 {x, y} = \u0002 mb(x), mb(y) + plb(z), mb(x, y) \u0003\u2032, \u20d7 m[b]|{x,y} L1 {x, y} = \u0002 mb(x), mb(y), mb(x, y) + plb(z) \u0003\u2032. Figure 2 shows such simplex in the case of a belief function b on the ternary frame \u0398 = {x, y, z} and basic probability assignment \u20d7 m = [0.2, 0.3, 0, 0, 0.5, 0]\u2032, (19) i.e., mb(x) = 0.2, mb(y) = 0.3, mb(x, z) = 0.5. In the case of the belief function (19) of the above example, by (15) its L2 conditional belief function in M has b.p.a. m(x) = mb(x) + 1 \u2212b(x, y) 3 = mb(x) + plb(z) 3 , m(y) = mb(y) + plb(z) 3 , m(x, y) = mb(x, y) + plb(z) 3 . (20) Figure 2 visually con\ufb01rms that such L2 conditional belief function lies in the barycenter of the simplex of the related L1 conditional b.f.s. For what concerns L\u221econditional belief functions, the b.f. (19) is such that max C\u0338\u2282A mb(C) = max n mb(z), mb(x, z), mb(y, z), mb(\u0398) o = mb(x, z) = 0.5 \u2265 1 2|A| \u22121 X C\u0338\u2282A mb(C) = 1 3mb(x, z) = 0.5 3 . We hence fall within case 1, and there is a whole simplex of L\u221econditional belief function (in M). According to Equation (18) such simplex has 2|A| \u2212 20 \fmx m{x,y} my 1 1 1 0 0 0 Figure 2: The simplex (solid red triangle) of L1 conditional belief functions in M associated with the belief function with mass assignment (19) in \u0398 = {x, y, z}. The related unique L2 conditional belief function in M is plotted as a red square, and coincides with the center of mass of the L1 set. The set of L\u221econditional (pseudo) belief functions is a simplex (triangle) with three vertices, depicted in green. 1 = 3 vertices, namely (taking into account the nil masses in (19)) \u20d7 m[b]|{x} L\u221e,M{x, y} = \u0002 mb(x) \u2212mb(x, z), mb(y) + mb(x, z), mb(x, z) \u0003\u2032, \u20d7 m[b]|{y} L\u221e,M{x, y} = \u0002 mb(x) + mb(x, z), mb(y) \u2212mb(x, z), mb(x, z) \u0003\u2032, \u20d7 m[b]|{x,y} L\u221e,M{x, y} = \u0002 mb(x) + mb(x, z), mb(y) + mb(x, z), \u2212mb(x, z) \u0003\u2032. (21) We can notice that the set of L\u221econditional (pseudo) b.f.s is not entirely admissible, but its admissible part contains the set of L1 conditional b.f.s, which amounts therefore a more conservative approach to conditioning. Indeed, the latter is the triangle inscribed in the former, determined by its 21 \fmedian points. Note also that both the L1 and L\u221esimplices have the same barycenter in the L2 conditional belief function (20). 5 Discussion 5.1 Summary of results To summarise, given a belief function b : 2\u0398 \u2192[0, 1] and an arbitrary nonempty focal element \u2205\u228aA \u2286\u0398, we have the following. 1. The set of L1 conditional belief functions bL1,M(.|A) with respect to A in M is the set of b.f.s with core in A such that their mass dominates that of b over all the subsets of A: bL1,M(.|A) = n a : Ca \u2286A, ma(B) \u2265mb(B) \u2200\u2205\u228aB \u2286A o . Such a set is a simplex ML1,A[b] = Cl(\u20d7 m[b]|B L1A, \u2205\u228aB \u2286A) whose vertices \u20d7 ma = \u20d7 m[b]|B L1A have b.p.a.: \u001a ma(B) = mb(B) + 1 \u2212b(A) = mb(B) + plb(Ac), ma(X) = mb(X) \u2200\u2205\u228aX \u228aA, X \u0338= B. 2. The unique L2 conditional belief function bL2,M(.|A) with respect to A in M is the b.f. whose b.p.a. redistributes the mass 1\u2212b(A) = plb(Ac) to each focal element B \u2286A in an equal way: mL2,M(B|A) = mb(B) + plb(Ac) 2|A| \u22121, (22) \u2200\u2205\u228aB \u2286A, and corresponds to the center of mass of the simplex ML1,A[b] of L1 conditional b.f.s. 3. The L\u221econditional b.f. either coincides with the L2 one, or forms a simplex obtained by assigning the maximal mass outside A (rather than the sum of such masses plb(Ac)) to all subsets of A (but one) indi\ufb00erently. L1 and L2 conditioning are strictly related in the mass space, the latter being the barycenter of the former, and they have a compelling interpretation in terms of general imaging [117, 79], as we argue next. 22 \f5.2 Properties of geometric conditional belief functions From our analysis a number of facts arise. \u2022 Lp conditional belief functions, albeit obtained by minimising purely geometric distances, possess very simple and elegant interpretations in terms of degrees of belief. \u2022 While some of them correspond to pointwise conditioning, some others form entire polytopes of solutions whose vertices also have simple interpretations. \u2022 Conditional belief functions associated with the major L1, L2 and L\u221e norms are strictly related to each other. \u2022 In particular, while distinct, both the L1 and L\u221esimplices have barycenter in (or coincide with, in case 2) the L2 conditional belief function. \u2022 They are all characterized by the fact that, in the way they re-assign mass from focal elements B \u0338\u2282A not in A to focal elements in A, they do not distinguish between subsets which have non-empty intersection with A and those which have not. The last point is quite interesting: mass-based geometric conditional b.f.s do not seem to care about the contribution focal elements make to the plausibility of the conditioning event A, but only to whether they contribute or not to the degree of belief of A. The reason is, roughly speaking, that in mass vectors \u20d7 mb the mass of a given focal element appears only in the corresponding entry of \u20d7 mb. In opposition, belief vectors \u20d7 b are such that each entry \u20d7 b(B) = X X\u2286B mb(X) contains information about the mass of all the subsets of B. As a result, it could be expected that geometric conditioning in the belief space B will see the mass redistribution process function in a manner linked to the contribution of each focal element to the plausibility of the conditioning event A. This is discussed in Section 5.4.1. 23 \f5.3 Interpretation as general imaging The form of geometric conditional belief functions in the mass space can be naturally interpreted in the framework of an interesting approach to belief revision, known as imaging [117]. We will illustrate this notion and how it relates to our results using the example proposed in [117]. Suppose we brie\ufb02y glimpse at a transparent urn \ufb01lled with black or white balls, and are asked to assign a probability value to the possible \u2018con\ufb01gurations\u2019 of the urn. Suppose also that we are given three options: 30 black balls and 30 white balls (state a); 30 black balls and 20 white balls (state b); 20 black balls and 20 white balls (state c). Hence, \u0398 = {a, b, c}. Since the observation only gave us the vague impression of having seen approximately the same number of black and white balls, we would probably deem the states a and c equally likely, but at the same time we would tend to deem the event \u201da or c\u201d twice as likely as the state b. Hence, we assign probability 1/3 to each of the states. Now, we are told that state c is false. How do we revise the probabilities of the two remaining states a and b? Lewis [103] argued that, upon observing that a certain state x \u2208\u0398 is impossible, we should transfer the probability originally allocated to x to the remaining state deemed the \u2018most similar\u2019 to x. In this case, a is the state most similar to c, as they both consider an equal number of black and white balls. We obtain (2/3, 1/3) as probability values of a and b, respectively. Peter G\u00a8 ardenfors further extended Lewis\u2019 idea (general imaging) by allowing to transfer a part \u03bb of the probability 1/3, initially assigned to c, towards state a, and the remaining part 1 \u2212\u03bb to state b. These fractions should be independent of the initial probabilistic state of belief. Now, what happens when our state of belief is described by a belief function, and we are told that A is true? In the general imaging framework we need to re-assign the mass m(C) of each focal element not included in A to all the focal elements B \u2286A, according to some weights {\u03bb(B), B \u2286A}. Suppose there is no reason to attribute larger weights to any focal element in A, as, for instance, we have no meaningful similarity measure (in the given context for the given problem) between the states described by two di\ufb00erent focal elements. We can then proceed in two di\ufb00erent ways. One option is to represent our complete ignorance about the similarities between C and each B \u2286A as a vacuous belief function on the set of weights. If applied to all the focal elements C not included in A, this results in an entire polytope of revised belief functions, each associated with an arbitrary 24 \fnormalized weighting. It is not di\ufb03cult to see that this coincides with the set L1 conditional belief functions bL1,M(.|A) of Theorem 1. On the other hand, we can represent the same ignorance as a uniform probability distribution on the set of weights {\u03bb(B), B \u2286A}, for all C \u0338\u2282A. Again, it is easy to see that general imaging produces in this case a single revised b.f., the L2 conditional belief function bL2,M(.|A) of Theorem 3. As a \ufb01nal remark, the \u2018information order independence\u2019 axiom of belief revision [117] states that the revised belief should not depend on the order in which the information is made available. In our case, the revised (conditional) b.f.s obtained by observing \ufb01rst an event A and later another event A\u2032 should be the same as the ones obtained by revising \ufb01rst with respect to A\u2032 and then A. Both the L1 and L2 geometric conditioning operators presented here meet such axiom, supporting the case for their rationality. 5.4 Comparison with conditioning in the belief space 5.4.1 Results of conditioning in the belief space Conditional belief functions can be derived by minimising Minkowski distances in the belief space representation as well [36]. Unfortunately the results are of more di\ufb03cult interpretation \u2013 nevertheless a comparison with the more intuitive results obtained in the mass space is in place. Namely [36, 44], the L2 conditional belief function and the barycentre L\u221eof the L\u221econditional belief functions computed in the belief space are, respectively, mL2,B(B|A) = mb(B) + X C\u2286Ac mb(B + C)2\u2212|C| + (\u22121)|B|+1 X C\u2286Ac mb(C)2\u2212|C|, mL\u221e,B(B|A) = mb(B) + 1 2 X \u2205\u228aC\u2286Ac mb(B + C) + 1 2(\u22121)|B|+1b(Ac). The L2 result appears to be related to the process of mass redistribution (called by some authors specialisation, [95, 98]) among all subsets, as happens with the (L2-induced) orthogonal projection of a belief function onto the probability simplex [18, 21]. In both expressions above, we can note that normalisation is achieved by alternately subtracting and summing a quantity, rather than via a ratio or, as in (22), by reassigning the mass of all B \u0338\u2282A to each B \u228aA on an equal basis. 25 \fmx m{x,y} my 1 1 1 0 0 0 Figure 3: The set of L\u221e, B conditional belief functions is drawn here as a blue quadrangle for the belief function with mass assignment (19) in \u0398 = {x, y, z}, with conditioning event A = {x, y}. The region falls partly outside the conditioning simplex (gray triangle). The set of L1 conditional belief functions in B is a line segment (in pink) with its barycentre in the L2 conditional BF (pink square). In the ternary case, L2, B is the barycentre of the L\u221e, B rectangle. The Lp conditional belief functions in the mass space, already considered in Fig. 2, are still visible in the background. We can interpret the barycentre of the set of L\u221econditional belief functions, instead, as follows: the mass of all the subsets whose intersection with A is C \u228aA is re-assigned by the conditioning process half to C, and half to A itself. In the case of C = A itself, by normalisation, all the subsets D \u2287A, including A, have their whole mass reassigned to A, consistently with the above interpretation. The mass b(Ac) of the subsets which have no relation to the conditioning event A is used to guarantee the normalisation of the resulting mass distribution. As a result, the mass function obtained 26 \fis not necessarily non-negative: once again such a version of geometrical conditioning may generate pseudo-belief functions. In the L1 case obtaining a general analytic expression for the resulting conditional b.f.s appears impossible [44]. In the special cases in which this is possible, however, the result has potentially interesting interpretations. 5.4.2 Ternary example Figure 3 illustrates the di\ufb00erent geometric conditional belief functions as computed in the belief space, for a conditioning event A = {x, y} and a belief function with masses as in (19), i.e. mb(x) = 0.2, mb(y) = 0.3, mb(x, z) = 0.5. We already know that mL2,M(.|A) lies in the barycentre of the simplex of the L1, M conditional belief functions. The same is true (at least in the ternary case) for mL2,B(.|A) (the pink square), which is the barycentre of the (blue) polytope of mL\u221e,B(.|A) conditional belief functions. We can note that, as pointed out above, the latter does not fall entirely in the admissible conditional simplex C(\u20d7 bx,\u20d7 by,\u20d7 bx,y) (although a signi\ufb01cant portion does). Finding the admissible part of mL\u221e,B(.|A) remains an open problem. The set of L1 conditional belief functions in B is, instead, a line segment (drawn in pink) whose barycentre is mL2,B(.|A). Such a set is: \u2022 entirely included in the set of L\u221eapproximations in both B and M, thus representing a more conservative approach to conditioning; \u2022 entirely admissible. It seems that, hard as it is to compute (see [44], Chapter 15), L1 conditioning in the belief space delivers interesting results. A number of interesting cross-relations between conditional belief functions in the two representation domains appear to exist: 1. mL\u221e,B(.|A) seems to contain mL1,M(.|A); 2. the two L2 conditional belief functions mL2,M(.|A) and mL2,B(.|A) appear to both lie on a line joining opposite vertices of mL\u221e,B(.|A); 3. mL\u221e,B(.|A) (the blue polytope in Fig. 3) and mL\u221e,M(.|A) (the green triangle) have several vertices in common. 27 \fThere is probably more to these conditioning approaches than has been shown by the simple comparison done here. We will investigate these aspects further in the near future. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2104.06839v1",
+                    "title": "Uncertainty measures: The big picture",
+                    "abstract": "Probability theory is far from being the most general mathematical theory of\nuncertainty. A number of arguments point at its inability to describe\nsecond-order ('Knightian') uncertainty. In response, a wide array of theories\nof uncertainty have been proposed, many of them generalisations of classical\nprobability. As we show here, such frameworks can be organised into clusters\nsharing a common rationale, exhibit complex links, and are characterised by\ndifferent levels of generality. Our goal is a critical appraisal of the current\nlandscape in uncertainty theory.",
+                    "authors": "Fabio Cuzzolin",
+                    "published": "2021-04-14",
+                    "updated": "2021-04-14",
+                    "primary_cat": "math.ST",
+                    "cats": [
+                        "math.ST",
+                        "cs.AI",
+                        "math.PR",
+                        "stat.TH",
+                        "62A01, 62A86, 60A05, 60A86, 60D05, 68T37"
+                    ],
+                    "main_content": "Introduction Uncertainty is of paramount importance in arti\ufb01cial intelligence, applied science, and many other areas of human endevour [48]. Uncertainty can be understood as lack of information about an issue of interest for a certain agent. Sometimes less perceived by scientists themselves, a iatus exists between two fundamentally distinct forms of uncertainty. Somewhat \u2018predictable\u2019 variations (e.g., the outcomes of a carnival wheel) are typically encoded by probability distributions. Unpredictable variations, on the other hand, re\ufb02ect a more fundamental (sometimes called \u2018Knightian\u2019) uncertainty about the laws which themselves govern the outcome (e.g., what happens if part of the wheel is cloaked?). The mainstream mathematical theory of (\ufb01rst order) uncertainty is measure-theoretical probability, mainly due to Russian mathematician Andrey Kolmogorov [119]. A probability measure over a \u03c3-algebra F = F(\u0398) \u22822\u0398 . = {A \u2286\u0398}, associated with a sample space \u0398, is a function P : F \u2192[0, 1] such that: 1. P(\u2205) = 0; 2. P(\u0398) = 1; 3. if A\u2229B = \u2205, A, B \u2208F then P(A\u222aB) = P(A)+P(B) (additivity). A sample space \u0398 together with a \u03c3-algebra F of its subsets and a probability measure P on F forms a probability space: (\u0398, F, P). A (measurable) function X from a sample space \u0398 (endowed with a probability space) to a measurable space E (usually the real line R) is termed random variable (RV). Bayesian reasoning or frequentist inference is then applied to infer additive probabilities from the available data. A long series of students, however, have argued that Kolmogorov\u2019s measure-theoretical probability theory is incapable of handling higher-order uncertainty, associated with widespread phenomena such as set-valued observations (which arise from missing or incomplete data), the representation of ignorance, inherently propositional data, and so on. In response, various mathematical theories of uncertainty have been proposed [102; 137]. A theory of uncertainty is typically composed of two elements: (i) a mathematical object encoding an uncertain state of the world, and (ii) an operator which allows us to reason with uncertain states (e.g., the Choquet integral for monotone capacities, or Dempster\u2019s rule for belief functions). The \ufb01eld has been (partially) surveyed by various authors [117; 160], who focused, for instance, on the difference between imprecision and uncertainty, or the applicability of various models of uncertainty. A number of uni\ufb01cation attempts have also been made [74], most notably Walley\u2019s theory of imprecise probability [169; 170], Klir\u2019s generalised information theory [117], and Zadeh\u2019s generalized theory of uncertainty [186]. A fascinating effort to identify a unifying framework in the language of modal logic can be found in [149]. 1.1 Structure and contributions of the paper In this paper we will \ufb01rst recall the attempts to de\ufb01ne the notion of uncertainty, and summarise the rationale behind the need to move beyond classical probability theory (Section 2). Subsequently, in the core of the paper we will review the basic concepts of each approach to uncertainty theory. Although characterised by various levels of generality, uncertainty theories cannot be simply arranged into a \u2018linear\u2019 hierarchy but, we argue here, should rather be clustered according to their common rationale. A \ufb01rst group of approaches (Section 3) attempt to \u2018robustify\u2019 probability theory. Most of them are encompassed by \u2018imprecise probability\u2019, a behavioural approach (Section 4) inspired by de Finetti\u2019s pioneering work. More recently, a game-theoretical approach by Vovk and Shafer has received considerable attention. A number of frameworks, including possibility theory or rough sets and more recently Zadeh\u2019s generalised theory of uncertainty (Section 5) follow from different generalisations of set arXiv:2104.06839v1  [math.ST]  14 Apr 2021 \fTable 1: Imprecise-probabilistic theories: a timeline Approach Proposer(s) Seminal paper Year Interval probabilities John Maynard Keynes A treatise on probability 1921 Subjective probability Bruno de Finetti Sul signi\ufb01cato soggettivo della probabilit` a 1931 Theory of previsions Bruno de Finetti La pr\u00b4 evision: ses lois logiques, ses sources subjectives 1937 Theory of capacities Gustave Choquet Theory of capacities 1953 Fuzzy theory Lot\ufb01Zadeh, Dieter Klaua Fuzzy sets 1965 Theory of evidence Arthur Dempster, Glenn Shafer Upper and lower probabilities induced by a multivalued mapping; A mathematical theory of evidence 1967, 1976 Fuzzy measures Michio Sugeno Theory of fuzzy integrals and its applications 1974 Credal sets Isaac Levi The enterprise of knowledge 1980 Possibility theory Didier Dubois, Henri Prade Th\u00b4 eorie des possibilit\u00b4 es 1985 Imprecise probability Peter Walley Statistical reasoning with imprecise probabilities 1991 Game-theoretical probability Glenn Shafer, Vladimir Vovk Probability and \ufb01nance: It\u2019s only a game! 2001 theory, upon which probability is built. On a more general plane lie theories which generalise measure theory, the very foundation of mathematical probability (Section 6), including the theory of monotone capacities and Liu\u2019s uncertainty theory. Some proposals (Section 7), such as Harper\u2019s Popperian approach and Groen\u2019s extension aim at generalising Bayesian reasoning in terms of either measures or inference mechanisms. At the intersection of all these like the theory of random sets (or belief functions, in the \ufb01nite case), which generalise the notion of set, are monotone capacities and generalise Bayesian reasoning (Section 8). Methods replacing events with functions in a functional space are discussed in Section 9. Due to lack of space, we neglect relatively minor proposals such as Cohen\u2019s endorsements [15; 14], Laskey\u2019s assumptions [125], Shastri\u2019s evidential reasoning in semantic networks [157] or evidential con\ufb01rmation theory [99]. We also entirely neglect (imprecise-)probabilistic logic [151; 152; 108; 101; 84; 174; 103], which is also connected to uncertainty theory, especially in its modal logic form, but would deserve a separate survey paper. Another relevant paradigm not covered here is info-gap theory [6]. A critical appraisal of the uncertainty theory landscape concludes the document. 2 Uncertainty 2.1 Notion of uncertainty Uncertainty is of paramount importance in arti\ufb01cial intelligence, applied science, and many other areas of human endeavour. Whilst each and every one of us possesses some intuitive grasp of what uncertainty is, providing a formal definition can prove elusive. Uncertainty can be understood as a lack of information about an issue of interest for a certain agent (e.g., a human decision maker or a machine), a condition of limited knowledge in which it is impossible to exactly describe the state of the world or its future evolution. According to Dennis Lindley [132]: \u201c There are some things that you know to be true, and others that you know to be false; yet, despite this extensive knowledge that you have, there remain many things whose truth or falsity is not known to you. We say that you are uncertain about them. You are uncertain, to varying degrees, about everything in the future; much of the past is hidden from you; and there is a lot of the present about which you do not have full information. Uncertainty is everywhere and you cannot escape from it \u201d. What is sometimes less clear to scientists themselves is the existence of a hiatus between two fundamentally distinct forms of uncertainty. The \ufb01rst level consists of somewhat \u2018predictable\u2019 variations, which are typically encoded as probability distributions. For instance, if a person plays a fair roulette wheel they will not, by any means, know the outcome in advance, but they will nevertheless be able to predict the frequency with which each outcome manifests itself (1/36), at least in the long run. The second level is about \u2018unpredictable\u2019 variations, which re\ufb02ect a more fundamental uncertainty about the laws themselves which govern the outcome. Continuing with our example, suppose that the player is presented with ten different doors, which lead to rooms each containing a roulette wheel modelled by a different probability distribution. They will then be uncertain about the very game they are supposed to play. How will this affect their betting behaviour, for instance? Lack of knowledge of the second kind is often called Knightian uncertainty [118; 106], from the Chicago economist Frank Knight. He would famously distinguish \u2018risk\u2019 from \u2018uncertainty\u2019: \f\u201cUncertainty must be taken in a sense radically distinct from the familiar notion of risk, from which it has never been properly separated ...The essential fact is that \u2018risk\u2019 means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating ...It will appear that a measurable uncertainty, or \u2018risk\u2019 proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all.\u201d In Knight\u2019s terms, \u2018risk\u2019 is what people normally call probability or chance, while the term \u2018uncertainty\u2019 is reserved for second-order uncertainty. The latter has a measurable consequence on human behaviour: people are demonstrably averse to unpredictable variations (as highlighted by Ellsberg\u2019s paradox [81]). This difference between predictable and unpredictable variation is one of the fundamental issues in the philosophy of probability, and is sometimes referred to as the distinction between common cause and special cause [161]. Different interpretations of probability treat these two aspects of uncertainty in different ways, as debated by economists such as John Maynard Keynes [113] and G. L. S. Shackle. 2.2 Probability Measure-theoretical probability, due to the Russian mathematician Andrey Kolmogorov [119], is the mainstream mathematical theory of (\ufb01rst-order) uncertainty. In Kolmogorov\u2019s mathematical approach probability is simply an application of measure theory, and uncertainty is modelled using additive measures. A number of authors, however, have argued that measuretheoretical probability theory is not quite up to the task when it comes to encoding second-order uncertainty. In particular, as we discuss in the Introduction, additive probability measures cannot properly model missing data or data that comes in the form of sets. Probability theory\u2019s frequentist interpretation is utterly incapable of modelling \u2018pure\u2019 data (without \u2018designing\u2019 the experiment which generates it). In a way, it cannot even properly model continuous data (owing to the fact that, under measure-theoretical probability, every point of a continuous domain has zero probability), and has to resort to the \u2018tail event\u2019 contraption to assess its own hypotheses. Scarce data can only be effectively modelled asymptotically. Bayesian reasoning is also plagued by many serious limitations: (i) it just cannot model ignorance (absence of data); (ii) it cannot model pure data (without arti\ufb01cially introducing a prior, even when there is no justi\ufb01cation for doing so); (iii) it cannot model \u2018uncertain\u2019 data, i.e., information not in the form of propositions of the kind \u2018A is true\u2019; and (iv) again, it is able to model scarce data only asymptotically, thanks to the Bernstein\u2013von Mises theorem [166]. 2.3 Beyond probability Similar considerations have led a number of scientists to recognise the need for a coherent mathematical theory of uncertainty able to properly tackle all these issues. Both alternatives to and extensions of classical probability theory have been proposed, starting from de Finetti\u2019s pioneering work on subjective probability [67]. Formalisms include possibilityfuzzy set theory [185; 79], probability intervals [102], credal sets [127; 122], monotone capacities [173], random sets [140] and imprecise probability theory [169]. New original foundations of subjective probability in behavioural terms [170] or by means of game theory [156] have been put forward. Table 1 presents a sketchy timeline of the various existing approaches to the mathematics of uncertainty. Sometimes collectively referred to as imprecise probabilities (as most of them comprise classical probabilities as a special case), these theories in fact form an entire hierarchy of encapsulated formalisms. In what follows we will attempt to classify those formalisms into clusters characterised by a common theme. 3 Probability robusti\ufb01ed 3.1 Lower and upper probabilities A lower probability (LP) [88] P is a function from a sigmaalgebra to the unit interval [0, 1] such that: P(A \u222aB) \u2265P(A) + P(B) \u2200A \u2229B = \u2205 (super-additivity). P is an upper probability (UP) whenever P(A \u222aB) \u2264P(A) + P(B), A \u2229B = \u2205. With a lower probability P is associated a dual UP P(A) = 1 \u2212P(Ac), Ac being the complement of A \u2286\u0398. 3.2 Probability envelopes or credal sets Each P is also associated with a closed convex set of probability measures P(P) = {P : P(A) \u2265P(A), \u2200A \u2286\u0398}, termed a credal set [127]. Not all credal sets, however, can be described by merely focusing on events [169]. A lower probability P on \u0398 is called \u2018consistent\u2019 (\u2018avoids sure loss\u2019 in Walley\u2019s terminology) if P(P) \u0338= \u2205, i.e.: sup \u03b8\u2208\u0398 n X i=1 1Ai(\u03b8) \u2265 n X i=1 P(Ai), whenever n \u2208N+, A1, ..., An \u2208F are events, and 1Ai is the indicator function of Ai in \u0398. A lower probability P is \u2018tight\u2019 (\u2018coherent\u2019 for Walley) if infP \u2208P(P ) P(A) = P(A), i.e.: sup \u03b8\u2208\u0398 [ n X i=1 1Ai(\u03b8) \u2212m1A0(\u03b8)] \u2265 n X i=1 P(Ai) \u2212m \u00b7 P(A0) whenever n, m \u2208N+ and A0, A1, ..., An \u2208F. Consistency means that the lower bound constraints P(A) can indeed be satis\ufb01ed by some probability measure, while tightness indicates that P is the lower envelope on subsets of P(P). Any coherent lower probability is monotone, i.e., P(A) \u2264P(B) for A \u2282B. As shown by [122], one can operate with credal sets in a robust statistical fashion, by simply applying Bayes\u2019 rule to their (\ufb01nite) set of vertices. \f3.3 Probability intervals One can instead provide lower and upper constraints on the probabilities of elements x \u2208\u0398. A set of probability intervals [111; 165; 65] is a system of constraints P(l, u) . = {p : l(x) \u2264p(x) \u2264u(x), \u2200x \u2208\u0398} on the values of a probability distribution p : \u0398 \u2192[0, 1] on a \ufb01nite domain \u0398. Such a set of constraints determines a credal set, of a sub-class of those generated by lower and upper probabilities. De Campos et al. studied the speci\ufb01c constraints such intervals need satisfy in order to be consistent and tight. Their vertices can be computed as in [65], p. 174. In particular, a set of probability intervals is feasible if and only if for each x \u2208\u0398 and every value v(x) \u2208[l(u), u(x)] there exists a probability distribution function p : \u0398 \u2192[0, 1] for which p(x) = v(x). One can then obtain the lower and upper probabilities on any subset A \u2286\u0398 via: P(A) = max \u001a X x\u2208A l(x), 1 \u2212 X x\u0338\u2208A u(x) \u001b , P(A) = min \u001a X x\u2208A u(x), 1 \u2212 X x\u0338\u2208A l(x) \u001b . Lower and upper probabilities associated with a feasible set of probability intervals are 2-monotone capacities ([65], Prop 5; see Section 6.1). Belief functions (Section 8.1) also form a special class of interval probabilites. Combination, marginalisation and conditioning operators for probability intervals can be de\ufb01ned, typically using operators acting on lower and upper probabilities. In [65], credal conditioning [83] was suggested: P(A|B) = P(A \u2229B) P(A \u2229B) + P(Ac \u2229B) \u2200A, B \u2286\u0398. A generalised Bayesian inference framework based on interval probabilities was also proposed in [144]. 3.4 Probability boxes Probability boxes [86; 1; 110] arise in engineering and reliability whenever the available information is insuf\ufb01cient to identify a sought joint probability density function (PDF). Let P be a probability measure on the real line R. Its cumulative distribution function (CDF) is a non-decreasing mapping FP from R to [0, 1], such that for any \u03b1 \u2208R, FP (\u03b1) = P((\u2212\u221e, \u03b1]). A probability box or p-box [86] \u27e8F, F\u27e9is a class of cumulative distribution functions delimited by two lower and upper bounds F and F: \u27e8F, F\u27e9= {F CDF|F \u2264F \u2264F}. P-boxes can be convolved by \ufb01rst discretising them into belief functions (cfr. Section 8.1) and applying a parametric copula [180] to the latter. A copula C : [0, 1]d \u2192[0, 1] is a multivariate probability distribution for which the marginal probability distribution of each variable is uniform. Copulas are used to describe the dependence between RVs. The resulting BF is \ufb01nally transformed back into a p-box. Generalised p-boxes [74] have also been proposed. 4 Behavioural probability In behavioural probability, pioneered by [67; 68], probability has a behavioural rationale which derives from equalling \u2018belief\u2019 to \u2018inclination to act\u2019. An agent believes in a certain outcome to the extent they are willing to accept a bet on that outcome. 4.1 De Finetti\u2019s previsions A person who wants to summarise his degree of belief in a random event A \u2282\u0398 by a number p is supposed to accept any bet on A with gain c(p \u22121A), where 1A denotes the indicator function of A and c is any real number chosen by an opponent. Since c may be positive or negative, there is no advantage for the person in question in choosing a value p such that cp and c(p\u22121) are both strictly positive for some c. Hence, p is an admissible evaluation of the probability of A if it meets a principle of coherence: p has to ensure that \u2204c \u2208R so that the realisations of c(p \u22121A) are all strictly positive (or strictly negative). Coherence means that the agent is fully aware of the consequences of its betting rates. Probabilities The concept of coherence is extended to a class F of events as follows. The real-valued function P on F \u22822\u0398 is said to be a probability if, for any \ufb01nite subclass {A1, ..., An} of F and any choice of (c1, ..., cn) in Rn, n = 1, 2, ... the gain G = n X k=1 (P(Ak) \u22121Ak) is such that inf G \u22640 \u2264sup G, where inf and sup are taken over all constituents of {A1, ..., An}. Gambles and previsions Now, let a gamble X \u2208L(\u0398) [66] be a bounded real-valued function on \u0398: X : \u0398 \u2192R, \u03b8 7\u2192X(\u03b8) yielding different utilities for different outcomes \u03b8 \u2208\u0398. P is termed a prevision if, for every \ufb01nite subclass {X1, ..., Xn} of L(\u0398) and for every n-tuple (c1, ..., cn) of real numbers, one has inf n X k=1 ck(P(Xk) \u2212Xk) \u22640 \u2264sup n X k=1 ck(P(Xk) \u2212Xk), i.e., there is no (\ufb01nite) betting system which makes uniformly strictly negative the gain of the agent adopting P. The probability of an event A coincides with the prevision of 1A and the theory of probability is included in that of previsions. A prevision P is coherent iff 1. P(\u03bbX + \u00b5Y ) = \u03bbP(X) + \u00b5P(Y ); 2. if X > 0 then P(X) \u22650; 3. P(\u0398) = 1. 4.2 Imprecise probability Inspired by de Finetti\u2019s work, imprecise probability [170; 169] (IP, see [138] for a survey) aims at unifying all approaches to mathematical uncertainty in a single coherent setting, with a generality comparable to the theory of credal sets [16]. \fSets of desirable gambles There, an agent\u2019s set of desirable gambles D \u2286L(\u0398) is used as a model of their uncertainty about the problem. D is coherent iff it is a convex cone, i.e., 1. 0 (the constant gamble X(\u03b8) = 0 \u2200\u03b8) \u0338\u2208D; 2. if X > 0 (i.e., X(\u03b8) > 0 for all \u03b8) then X \u2208D; 3. if X, Y \u2208D, then X + Y \u2208D; 4. if X \u2208D and \u03bb > 0 then \u03bbX \u2208D. As a consequence, if X \u2208D and Y > X then Y \u2208D. Lower and upper previsions Suppose the agent buys a gamble X for a price \u03b1. This yields a new gamble X \u2212\u03b1. The lower prevision P(X) of a gamble X, P(X) . = sup{\u03b1 : X \u2212\u03b1 \u2208D}, is the supremum acceptable price for buying X. Selling a gamble X for a price \u03b1 yields a new gamble \u03b1 \u2212X. The upper prevision P(X) . = inf{\u03b1 : \u03b1 \u2212X \u2208D} is the supremum acceptable price for selling X. By de\ufb01nition, P(X) = \u2212P(\u2212X). When lower and upper previsions coincide, P(X) = P(X) = P(X) is called the (precise) prevision of X in de Finetti\u2019s sense [67]. Rational rules of behaviour such as \u2018avoiding sure loss\u2019 and coherence can be applied to lower previsions too, by replacing indicator functions with general gambles. One consequence of avoiding sure loss is that P(A) \u2264P(A). From coherence it follows that lower previsions are subadditive. Natural and marginal extension Natural extension is used to extend a coherent lower prevision de\ufb01ned on a collection of gambles to a lower prevision on all gambles, assumed coherent and conservative. The natural extension of a set of gambles D is the smallest coherent set of desirable gambles that includes D. Let P be a lower probability on \u0398 that avoids sure loss, and let L be the set of all bounded functions on \u0398. The functional E de\ufb01ned on L as E(f) = sup \u001a n X i=1 \u03bbiP(Ai) + c | f \u2265 n X i=1 \u03bbi1Ai + c, n \u22650, Ai \u2286\u0398, \u03bbi \u22650, c \u2208{\u2212\u221e, +\u221e} \u001b for all f \u2208L is called the natural extension of P. A similar de\ufb01nition can be given for lower previsions. When P is a classical (\u2018precise\u2019) probability, the natural extension agrees with the expectation. Also, E(1A) = P(A) for all A iff P is coherent. A marginal extension operator [138] can be introduced for the aggregation of conditional lower previsions. 4.3 Game-theoretical probability In game-theoretic probability [156] probabilistic predictions are proven by constructing a betting strategy which allows to multiply capital inde\ufb01nitely if the prediction fails, in a gametheoretical setting. Probabilistic theories are tested by betting against its predictions. Mathematically, the approach builds on the predating theory of prequential probability [64]. Dawid\u2019s prequential principle states that the forecasting success of a probability distribution for a sequence of events should be evaluated using only the actual outcomes and the sequence of forecasts (conditional probabilities) to which these outcomes give rise, without reference to other aspects. In fact, the notion goes back to the work of Blaise Pascal, (\u2018Probability is about betting\u2019), Antoine Cournot (\u2018Events of small probability do not happen\u2019), and Jean Ville [155]. The latter, in particular, showed in 1939 that the laws of probability can be derived from this principle: You will not multiply the capital you risk by a large factor (the so-called Cournot\u2019s principle). The Ville/Vovk perfect-information protocol for probability can be stated in terms of a game involving three players: Forecaster, Skeptic and Reality, as in Algorithm 1. Using Shafer and Vovk\u2019s weather forecasting example, Forecaster may be a very complex computer program that escapes precise mathematical de\ufb01nition as it is constantly under development, which uses information external to the model to announce every evening a probability for rain on the following day. Skeptic decides whether to bet for or against rain and how much, and Reality decides whether it rains. As shown by Ville, the fundamental hypothesis that Skeptic cannot get rich can be tested by any strategy for betting at Forecaster\u2019s odds. The bottom line of Shafer and Vovk\u2019s theory if that Forecaster can beat Skeptic [168]. Algorithm 1 Game theoretical probability protocol 1: procedure GAMETHEORETICALPROTOCOL 2: K0 = 1 3: for n = 1, \u00b7 \u00b7 \u00b7 , N do 4: Forecaster announces prices for various payoffs. 5: Skeptic decides which payoffs to buy. 6: Reality determines the payoffs. 7: Skeptic\u2019s capital changes as Kn = Kn\u22121+ net gain or loss. 8: end for 9: end procedure 5 Generalising set theory 5.1 Fuzzy and possibility theory The concept of fuzzy set was introduced in [184] as an extension of the notion of set. Possibility theory Possibility theory [77] equips fuzzy set theory to provide a graded semantics to natural language statements. A possibility measure on \u0398 is a function \u03a0 : 2\u0398 \u2192[0, 1] such that \f1. \u03a0(\u2205) = 0; 2. \u03a0(\u0398) = 1; 3. the condition \u03a0 \u0012 [ i Ai \u0013 = sup i \u03a0(Ai) is satis\ufb01ed for every family of subsets {Ai \u22082\u0398}. Each possibility measure is uniquely characterised by a membership function \u03c0 : \u0398 \u2192[0, 1] s.t. \u03c0(x) . = \u03a0({x}) via the formula \u03a0(A) = sup x\u2208A \u03c0(x). The dual quantity N(A) = 1 \u2212\u03a0(Ac) is called necessity measure. A further extension called vague set [93] imposes a pair of lower and upper bounds on the membership function of a fuzzy set. A conditional possibility measure \u03c0(.|A) can be de\ufb01ned such that \u03a0(A\u2229B) = min{\u03a0(B|A), \u03a0(A)}. Its least speci\ufb01c solution is \u03a0(B|A) = 1 if \u03a0(A\u2229B) = \u03a0(A), \u03a0(A\u2229B) else. Other solutions [80] are possible. Other fuzzy-inspired frameworks Other uncertainty frameworks based on fuzzy theory have been proposed, including fril-fuzzy [4], granular computing [181], interval fuzzy reasoning [182] and neighborhood systems [131]. 5.2 Rough sets In opposition, rough sets [145] are strongly linked to the idea of partition of the universe of hypotheses. Let \u0398 be a \ufb01nite universe, and R \u2286\u0398 \u00d7 \u0398 be an equivalence relation which partitions it into a family of disjoint subsets \u0398/R, called elementary sets. We can then call measurable sets \u03c3(\u0398/R) the unions of one or more elementary sets, plus the empty set \u2205. The lower approximation apr(A) of A \u2282\u0398 is formed by the measurable elements whose equivalence classes are contained in A. Its upper approximation apr(A) is composed by the elements whose equivalence classes have nonempty intersection with A. Rough sets are useful to describe the probabilistic properties of decision rules [146], and are quite related to belief functions. Given a \u03c3-algebra F of subsets of \u0398, one can construct a rough set algebra such that F = \u03c3(\u0398/R). Any probability P on F can be extended to 2\u0398 using inner and outer measures as follows: P\u2217(A) = sup{P(X)|X \u2208\u03c3(\u0398/R), X \u2286A}, P \u2217(A) = sup{P(X)|X \u2208\u03c3(\u0398/R), X \u2287A}. These are a pair of belief and plausibility functions (Section 8.1). Generalized theory of uncertainty (GTU) Generalised theory of uncertainty (GTU) [186] describes information via generalised constraints of the form GC(X) : XisrR, where r \u2208{ blank, probabilistic, veristic, random set, fuzzy graph, etc } is a label which determines the type of constraint, and R a constraining relation of that type (e.g. a probability distribution, a random set, etc). This allows us to operate on information such as: \u2018Usually Robert returns from work at about 6 p.m.\u2019, to which generalised constraint propagation is employed as a reasoning mechanism. In the GTU everything is or is allowed to be a matter of degree (fuzzy). A generalised constraint language is de\ufb01ned as the set of all generalised constraints together with the rules governing syntax, semantics and generation. Examples are: (X is small) is likely; ((X, Y ) isp A) \u2227(X is B), where \u2018isp\u2019 denotes a probabilistic constraint, \u2018is\u2019 a possibilistic constraint, and \u2227conjunction. In GTU, inference is treated as an instance of question-answering. Given a system of natural language propositions and a query q, GTU performs generalised constraint propagation governed by deduction rules. 6 Generalising measure theory 6.1 Capacities (fuzzy measures) The theory of capacities or fuzzy measure theory [172; 163; 97] generalises measure theory by replacing additivity by monotonicity [12; 163]. Many uncertainty measures are special cases of fuzzy measures, including belief functions, possibilities and probabilities [124]. Capacities Given a domain \u0398 and a non-empty family F of subsets of \u0398, a monotone capacity or fuzzy measure \u00b5 on (\u0398, F) is a function \u00b5 : F \u2192[0, 1] such that 1. \u00b5(\u2205) = 0; 2. if A \u2286B then \u00b5(A) \u2264\u00b5(B) (monotonicity); Sometimes the following additional conditions are imposed: 3. for any increasing sequence A1 \u2286A2 \u2286\u00b7 \u00b7 \u00b7 in F, if S\u221e i=1 Ai \u2208F then lim i\u2192\u221e\u00b5(Ai) = \u00b5 \u0012 \u221e [ i=1 Ai \u0013 (\u2018continuity from below\u2019); 4. for any decreasing sequence A1 \u2287A2 \u2287\u00b7 \u00b7 \u00b7 , if T\u221e i=1 Ai \u2208F and \u00b5(A1) < \u221e, then lim i\u2192\u221e\u00b5(Ai) = \u00b5 \u0012 \u221e \\ i=1 Ai \u0013 (\u2018continuity from above\u2019). When \u0398 is \ufb01nite the last two requirements are trivially satis\ufb01ed and can be disregarded. Monotone decreasing measures can also be de\ufb01ned. Choquet integral For any nonnegative measurable function f on (\u0398, F), the Choquet integral of f on any A \u2208F is de\ufb01ned as inf A fd\u00b5 . = Z \u221e 0 \u00b5(F\u03b1 \u2229A)d\u03b1, where F\u03b1 = {x \u2208\u0398|f(x) \u2265\u03b1}, \u03b1 \u2208[0, \u221e). Both the Choquet integral of monotone capacities and the natural extension of lower probabilities are generalisations of the Lebesgue integral with respect to \u03c3-additive measures. The Choquet integral is monotone in both arguments f and \u00b5, but is not a linear functional. \fOrder of capacities A capacity \u00b5 is termed of order k if \u00b5 \u0012 k [ j=1 Aj \u0013 \u2265 X \u2205\u0338=K\u2286[1,...,k] (\u22121)|K|+1\u00b5 \u0012 \\ j\u2208K Aj \u0013 for all collections of k subsets Aj, j \u2208K of \u0398. If k\u2032 > k the resulting theory is less general than a theory of capacities of order k (i.e., it contemplates fewer measures). A capacity is k-alternating when its dual is k-monotone. When a capacity is k-alternating (monotone) for every k, it is called \u221e-alternating (monotone). Monotone capacities of order 2 are those such that \u00b5(A \u222aB) + \u00b5(A \u2229B) \u2265\u00b5(A) + \u00b5(B) \u2200A, B \u2286\u0398. Their conjugate measures \u00b5c . = 1 \u2212\u00b5(Ac) are such that \u00b5c(A \u222aB) + \u00b5c(A \u2229B) \u2264\u00b5c(A) + \u00b5c(B). A 2-monotone capacity is a lower probability (but not viceversa). Feasible probability intervals are 2-monotone capacities. Sugeno \u03bb-measures Sugeno \u03bb-measures [163], are special monotone measures g\u03bb such that g\u03bb(A \u222aB) = g\u03bb(A) + g\u03bb(B) + \u03bbg\u03bb(A)g\u03bb(B) for any given pair of disjoint sets A, B \u22082\u0398, where \u03bb \u2208 (\u22121, \u221e) is a parameter. Each \u03bb-measure is uniquely determined by values g\u03bb(\u03b8), \u03b8 \u2208\u0398 [172]: \u2022 if P \u03b8 g\u03bb(\u03b8) < 1, then g\u03bb is a lower probability; \u2022 if P \u03b8 g\u03bb(\u03b8) = 1 then g\u03bb is a probability measure and \u03bb = 0; \u2022 if P \u03b8 g\u03bb(\u03b8) > 1, g\u03bb is an upper probability. Lower and upper probabilities based on \u03bb-measures are special belief and plausibility measures [7]. 6.2 Comparative probability Probability measures can also be generalised by replacing numerical values with order relations. Fine\u2019s comparative probability (CP) [87] asserts that, given any two propositions A and B, all that is necessary is to say is whether P(A) is greater than, lower than or equal to P(B). A CP relation is required: (i) to be a linear, complete and simple order; (ii) to be non-trivial (\u0398 \u227b\u2205); (iii) to satisfy the \u2018improbability of impossibility\u2019 condition, A \u227b\u2205for all A; (iv) to be such that A \u2229(B \u222aC) = \u2205implies B \u227bC iff A \u222aB \u227bA \u222aC. Wong et al. later generalised this to belief structures [175], and showed that within this system there are belief functions which \u2018almost agree\u2019 with comparative relations, i.e., A \u227bB \u21d2Bel(A) > Bel(B) \u2200A, B \u22082\u0398. 6.3 Baoding Liu\u2019s Uncertainty Theory Liu\u2019s uncertainty theory (UT) [133; 134] is based on the notion of uncertain measure, a function M on F(\u0398) such that: 1. M(\u0398) = 1 (normality); 2. M(A) \u2264M(B) if A \u2282B (monotonicity); 3. M(A) + M(Ac) = 1 (\u2018self-duality\u2019); 4. for every countable sequence of events {Ai} we have that M  \u221e [ i=1 Ai ! \u2264 \u221e X i=1 M(Ai) (countable subadditivity). Clearly, uncertain measures are monotone capacities. Just as clearly, probability measures do satisfy these axioms. However, Liu claims, probability theory is not a special case of UT since probabilities do not satisfy the product axiom: M  \u221e Y k=1 Ak ! = \u221e ^ k=1 Mk(Ak) for all Cartesian products of events from individual uncertain spaces (\u0398k, Fk, Mk). However, the product axiom was only introduced in [134] (much after Liu\u2019s introduction of uncertain theory). The extension of uncertain measures to any subset of a product algebra is rather cumbersome and unjusti\ufb01ed (Equation (1.10) or Figure 1.1 in [133]). Axioms are not well justi\ufb01ed. Nevertheless, based on such measures a generalisation of random variables can be de\ufb01ned (\u2018uncertain variables\u2019), as measurable (in the usual sense) functions from an uncertainty space (\u0398, F, M) to the set of real numbers. 7 Generalising Bayesian reasoning 7.1 Harper\u2019s Popperian approach [104] proposed to use Popper\u2019s propensity treatment of probability to extend Bayesian reasoning so that revision of previously accepted evidence is allowed for. A function P : F \u00d7 F \u2192[0, 1] assigning values to pairs of events, endowed with a binary operation AB and a unary one \u00af A, is a Popper probability function if: 1. 0 \u2264P(B|A) \u2264P(A|A) = 1; 2. if P(A|B) = 1 = P(B|A) then P(C|A) = P(C|B); 3. if P(C|A) \u0338= 1 then P( \u00af B|A) = 1 \u2212P(B|A); 4. P(AB|C) = P(A|C) \u00b7 P(B|AC); 5. P(AB|C) \u2264P(B|C). In Popperian probability, conditional probability is primitive, whereas absolute one is derived as P(A) . = P(A|T), T = A \u00af A. Note that on traditional \u03c3-algebras T = (A \u2229Ac)c = \u0398. When P(A), as de\ufb01ned above, is greater than 0 the Popper conditional probability is the ratio of the absolute probabilities: P(B|A) = P(AB)/P(B), i.e, it is a classical conditional probability. In Popper\u2019s theory, however, P(B|A) exists even when P(A) = 0. As a result, previously accepted \fevidence can be revised to condition on events which currently have 0 probability. Harper\u2019s results included an epistemic semantics for the theory of counterfactual conditionals [128; 129]. 7.2 Groen\u2019s extension of Bayesian theory [98] proposed an extension of Bayesian theory in which observations are used to rule out possible valuations of the variables. The extension is different from probabilistic approaches such as Jeffrey\u2019s rule, in which certainty in a single proposition A is replaced by a probability on a disjoint partition of the universe, and Cheeseman\u2019s rule of distributed meaning [11], while non-probabilistic analogues are found in evidence and possibility theory. Interpretations An interpretation I of an observation O is de\ufb01ned as the union of values of a variable of interest X (e.g., an object\u2019s luminosity) that are not contradicted by O (e.g., the object is \u2018very dark\u2019). Uncertainty regarding the interpretation of an observation can be expressed as either a PDF on the space I of possible interpretations, \u03c0(I|H), I \u2208I, or by introducing an interpretation function \u03c1(x) . = Pr(x \u2208I|H), 0 \u2264\u03c1(x) \u22641, where x \u2208I denotes the event that value x is not contradicted by the observation, \u03c1(x) is the probability that this is true, and H our prior knowledge. The two de\ufb01nitions are related via: \u03c1(x) = X I\u2208I Pr(x \u2208I|I, H)\u03c0(I|H) = X I\u2208I:x\u2208I \u03c0(I|H). Prior to making an observation, our initial state of uncertainty about the pair \u2018observable variable\u2019 x \u2208X, \u2018variable of interest\u2019 \u03b8 \u2208\u0398, can be represented by the probability distribution: \u03c0(x, \u03b8|H) = \u03c0(x|\u03b8, H)\u03c0(\u03b8|H) de\ufb01ned over X \u00d7 \u0398. Full Bayesian inference can then be modi\ufb01ed by introducing the above interpretation function into Bayes\u2019 rule, yielding the posterior joint: \u03c0(x, \u03b8|H, O) = \u03c1(x)\u03c0(x|\u03b8, H)\u03c0(\u03b8|H) R x,\u03b8 \u03c1(x)\u03c0(x|\u03b8, H)\u03c0(\u03b8|H)dxd\u03b8. The rationale is that observations provide no basis for a preference among representations not contradicted by them. Thus, the relative likelihoods of the remaining representations should not be affected by it. 8 Set-valued probability 8.1 Belief functions Let us denote by \u2126and \u0398 the sets of outcomes of two different but related problems Q1 and Q2, respectively. Given a probability measure P on \u2126, we want to derive a \u2018degree of belief\u2019 Bel(A) that A \u2282\u0398 contains the correct response to Q2. If we call \u0393(\u03c9) the subset of outcomes of Q2 compatible with \u03c9 \u2208\u2126, \u03c9 tells us that the answer to Q2 is in A whenever \u0393(\u03c9) \u2282A. The degree of belief Bel(A) of an event A \u2282\u0398 is then the total probability (in \u2126) of all the outcomes \u03c9 of Q1 that satisfy the above condition, namely [69]: Bel(A) = P({\u03c9|\u0393(\u03c9) \u2282A}) = X \u03c9\u2208\u2126:\u0393(\u03c9)\u2282A P({\u03c9}). The map \u0393 : \u2126\u21922\u0398 = {A \u2286\u0398} is called a multivalued mapping from \u2126to \u0398. Such a mapping, together with a probability measure P on \u2126, induces a belief function on 2\u0398. Belief and plausibility measures A basic probability assignment (BPA) [154] is a set function [71; 76] m : 2\u0398 \u2192[0, 1] s.t.: 1. m(\u2205) = 0; 2. P A\u2282\u0398 m(A) = 1. The \u2018mass\u2019 m(A) assigned to A is in fact the probability P({\u03c9 \u2208\u2126: \u0393(\u03c9) = A}). Subsets of \u0398 whose mass values are non-zero are called focal elements of m. The belief function (BF) associated with a BPA m : 2\u0398 \u2192 [0, 1] is the set function Bel(A) = X B\u2286A m(B). The corresponding plausibility function is Pl(A) . = X B\u2229A\u0338=\u2205 m(B) \u2265Bel(A). Belief functions can be de\ufb01ned axiomatically, however, without resorting to the above random set interpretation [154]. Classical probability measures on \u0398 are a special case of belief functions (those assigning mass to singletons only), termed Bayesian belief functions. A BF is said to be consonant if its focal elements A1, ..., Am are nested: A1 \u2282\u00b7 \u00b7 \u00b7 \u2282 Am, and corresponds to a possibility measure. Combination Dempster\u2019s combination Bel1 \u2295Bel2 of two belief functions on \u0398 is the unique BF there with as focal elements all the non-empty intersections of focal elements of Bel1 and Bel2, and basic probability assignment m\u2295(A) = m\u2229(A) 1 \u2212m\u2229(\u2205), where m\u2229(A) = X B\u2229C=A m1(B)m2(C) and mi is the BPA of the input BF Beli. Dempster\u2019s combination naturally induces a conditioning operator. Given a conditioning event A \u2282\u0398, the \u2018logical\u2019 or \u2018categorical\u2019 belief function BelA such that m(A) = 1 is combined via Dempster\u2019s rule with the a-priori belief function Bel. The resulting BF Bel \u2295BelA is the conditional belief function given A a la Dempster, denoted by Bel\u2295(A|B). Many alternative combination rules have since been de\ufb01ned [115; 178; 78; 73], often associated with a distinct approach to conditioning [70; 83; 164]. An exhaustive review of these proposals can be found in [48], Section 4.3. \fBelief functions and other measures Each belief function Bel uniquely identi\ufb01es a credal set [122] P[Bel] = {P \u2208P : P(A) \u2265Bel(A)} (where P is the set of all probabilities one can de\ufb01ne on \u0398), of which it is its lower envelope: Bel(A) = P(A). The probability intervals resulting from Dempster\u2019s updating of the credal set associated with a BF, however, are included in those resulting from Bayesian updating [122]. Belief functions are in\ufb01nitely monotone capacities, and a special case of coherent lower previsions. On the other hand, possibility measures are equivalent to consonant BFs [48]. Belief functions can also be generalised to assume values on fuzzy sets, rather than traditional \u2018crisp\u2019 ones [8]. Indeed, when we recall that belief measures are induced by mass assignments on subsets of a frame we realise that generalisations of belief functions which are de\ufb01ned on fuzzy sets can (and indeed have) been proposed . Following Zadeh\u2019s work, Ishizuka et al [107], Ogawa and Fu [141], Yager [176], Yen [183] and recently Biacino [8] extended belief theory to fuzzy sets by de\ufb01ning an appropriate measure of inclusion for them. Every BF speci\ufb01es a unique p-box, whereas each p-box speci\ufb01es an entire equivalence class of belief functions [148]. Authors such as Heilpern [105], Yager [179; 177], Palacharla [142], Romer [150], Kreinovich [120], Denoeux [72] and many others [147; 96; 85; 130; 90; 153] have also studied the connection between fuzzy and Dempster\u2013Shafer theory. A framework similar to belief functions\u2019 is Spohn\u2019s, in which some propositions are believed to be true, others are believed to be false, and the remainder are neither believed nor disbelieved [162]. Behavioural interpretation An intriguing behavioural interpretation of belief functions has been recently advanced in [112]. A betting function R : L \u2192[0, 1] is a binary function on the set of gambles such that \u2200X \u2208L \u2203\u03b1X \u2208R such that R(X + \u03b1) = 0 for \u03b1 < \u03b1X, while R(X + \u03b1) = 1 for \u03b1 \u2265\u03b1X. Given a betting function, BuyR(X) . = max{\u03b1 \u2208R : R(X \u2212\u03b1) = 1} is the maximum price an agent is willing to pay for the gamble X. In Walley\u2019s terminology, the betting function determines the set of desirable gambles D = {X : R(X) = 1}, whereas BuyR(X) coincides with (the lower prevision P(X) of X). A belief valuation B is a belief function BelB, such that BelB(A) = 1 if A \u2287B, 0 otherwise. For any belief valuation B, the guaranteed revenue GB : L \u2192R is de\ufb01ned as GB(X) . = max A\u2287B min \u03b8\u2208A X(\u03b8). A betting function R is B-consistent if for all X1, ..., XN \u2208 L, Y1, ..., YM \u2208L such that N X i=1 GB(Xi) \u2264 M X j=1 GB(Xj) for every belief valuation B, we have N X i=1 BuyR(Xi) \u2264 M X j=1 BuyR(Xj). Bel is a belief function if and only if there exists a coherent (in Walley\u2019s sense) and B-consistent R such that Bel(A) = BuyR(1A) [112]. Also, if R is a coherent betting function, then R is B-consistent if and only if there is a BPA m such that: BuyR(X) = P A\u2286\u0398 m(A) min\u03b8\u2208A X(\u03b8) for all X \u2208L. In other words, adding B-consistency to imprecise probabilities\u2019 rationality axioms, lower previsions reduce to belief functions. 8.2 Random sets When the sample space is continuous, belief functions generalise to random sets. Let (\u2126, F, P) be a probability space [139]. A map X : \u2126\u2192C, where C is the space of closed subsets of a certain topological space E, is called a random closed set if, for every compact set K \u2208K in E {\u03c9 : X(\u03c9) \u2229K \u0338= \u2205} \u2208F. Alternatively, we can de\ufb01ne the Borel \u03c3-algebra B(C) generated by {C \u2208C : C \u2229X \u0338= \u2205} for all compact sets K, and call X a random closed set if it is measurable with respect to the Borel \u03c3-algebra on C with respect to the Fell topology, namely: X\u22121(X) = {\u03c9 : X(\u03c9) \u2208X} \u2208F for each X \u2208B(C). The distribution of a random closed set X is determined by P(X) = P({\u03c9 : X(\u03c9) \u2208X}) for all X \u2208B(C). In particular we can consider the case in which X = CK . = {C \u2208C : C \u2229K \u0338= \u2205} and P({X(\u03c9) \u2208CK}) = P({X \u2229K \u0338= \u2205}), since the families CK, K \u2208K, generate the Borel \u03c3-algebra B(C). A functional TX : K \u2192[0, 1] given by TX(K) = P({X \u2229K \u0338= \u2205}), K \u2208K is termed the capacity functional of X. In particular, if X = {\u03be} is a random singleton, i.e., a classical random variable, then TX(K) = P({\u03be \u2208K}), so that the capacity functional is the probability distribution of the RV \u03be. The functional TX is indeed a capacity in the sense of Section 6.1. The links betwen random closed sets and belief functions, upper and lower probabilities, and contaminated models in statistics is brie\ufb02y hinted at in [139], Chapter 1. 9 Measures on functional spaces Possibly the most general group of approaches clusters those advocating the need to go beyond the notion of event entirely. These methods replace the usual \u03c3-algebra of events with a functional space, the space of all functions of a certain class that can be de\ufb01ned over the universe of discourse \u0398, and attach numerical quanti\ufb01ers to these functions. Lower and upper previsions (see Section 4.2) are an example of such quanti\ufb01ers, as they are de\ufb01ned on gambles rather than events. \f9.1 Higher-order probabilities Second-order probabilities An intuitive approach is to model uncertainty about probabilities using probability itself [75]. As pointed out by [5], a second-order probability Q(P) may be understood as the probability that the \u2018true\u2019 probability of something has the value P. Baron proceeded to derive a rule for combining evidence from two independent sources generating each a second-order probability Qi(P), and showed that Dempster\u2019s rule is a special case representing a restriction of a full Bayesian analysis. In Fung and Chong\u2019s metaprobability theory, belief update is still based on Bayes\u2019 rule [91], namely p2(p|D, Pr) \u221dp2(D|p, Pr) \u00b7 p2(p|Pr), where D is the data (evidence) and Pr a prior on space of (\ufb01rst order) probability distributions p. Josang\u2019s Dirichlet distributions Josang showed that a bijective mapping exists between Dirichlet distributions [109], a special class of second-order probability distributions, and belief functions whose focal elements are of size 1 or |\u0398|. Such a link can be exploited to apply belief based reasoning to statistical data, or to apply statistical and probabilistic analysis to belief functions. Gaifman\u2019s higher order probability spaces [92] axiomatically de\ufb01ned a higher order probability space (HOP) as consisting of a probability space and an operation PR, such that, for every event A and every real closed interval \u2206, PR(A, \u2206) is the event that A\u2019s \u2018true\u2019 probability (in the sense described above) lies in \u2206. In a general HOP the operation PR includes an additional argument ranging over an ordered set of time-points, so that PR(A, t, \u2206) is the event that A\u2019s probability at time t lies in \u2206. Various connections with modal logic were pointed out. Kyburg\u2019s analysis Whereas [158] argued that higher order probabilities are essential for a correct representation of belief, [123] claimed that higher order probabilities can always be replaced by the marginal distributions of a joint probability distribution de\ufb01ned on I \u00d7 \u0398, where \u0398 is the sample space of lower order probabilities, and I is a \ufb01nite set indexing all possible probability distributions Pi (i.e., the sample space for the secondorder probability Q). This follows from the principle that [158] P(\u03b8) = X i\u2208I Q(Pi)Pi(\u03b8) = E[Pi(\u03b8)], combined with the use of expected utilities. As a result, Kyburg claimed, higher order probabilities do not appear to offer any conceptual or computational advantage. 9.2 MV-algebras Various extensions of probability measures of belief functions to fuzzy events, rather than traditional, \u2018crisp\u2019 events, have also been proposed. As fuzzy events amount to membership functions on the universe \u2126, they can also be assimilated to measures on functional spaces. Another remarkable example is provided by MV-algebras [121; 89]. An MV-algebra is an algebra \u27e8M, \u2295, \u00ac, 0\u27e9of many-valued events, upon which upper and lower probabilities can be de\ufb01ned. M is endowed with a binary operation \u2295, a unary operation \u00ac and a constant 0 such that \u27e8M, \u2295, 0\u27e9is an Abelian monoid and the following equations hold true for every f, g \u2208 M: \u00ac\u00acf = f, f \u2295\u00ac0 = \u00ac0, \u00ac(\u00acf \u2295g)\u2295g = \u00ac(\u00acg\u2295f)\u2295f. Boolean algebras are, in fact, a special case of MV algebras. A state is a mapping s : M \u2192[0, 1] such that s(1) = 1 and s(f + g) = s(f) + s(g) whenever f \u2299g = 0, where f \u2299g . = \u00ac(\u00acf \u2295\u00acg). Clearly, states are generalisations of \ufb01nitely additive probability measures, once we replace events A, B with continuous functions f, g onto [0, 1] and \u2229with \u2299. Considering the MV algebra [0, 1]2\u0398 of all functions 2\u0398 \u2192 [0, 1], where X is \ufb01nite, one de\ufb01ne a belief function Bel : [0, 1]2\u0398 \u2192[0, 1] there if there is a state on the MV algebra [0, 1]P(X) such that s(1\u2205) = 0 and Bel(f) = s(\u03c1(f)), for every f \u2208[0, 1]X. 10 Discussion 10.1 A tentative classi\ufb01cation Figure 1 provides a visual illustration of our tentative classi\ufb01cation of uncertainty theories into clusters of approaches, depending on what objects they attach values to and, related to this, their rationale. The diagram also shows the dependency between the various theories, based on the results available so far. An arrow between formalism 1 and formalism 2 means that the former is less general than the latter. The \u2018quality\u2019 of the various proposals differs radically. For instance, whereas an uncertain entropy and an uncertain calculus are built within Liu\u2019s UT, the general lack of rigour and convincing justi\ufb01cation leaves us quite unimpressed with this work. As far as Zadeh\u2019s GTU is concerned, which was not reviewed here in detail, generality is achieved there in a rather nomenclative way, which explains the complexity and lack of clear rationale for the formalism. Imprecise probability is remarkable for its generality, as special cases of coherent lower previsions include probabilities, de Finetti previsions, Choquet capacities, possibility and belief measures, random sets but also probability boxes, (lower and upper envelopes of) credal sets, and robust Bayesian models. Nevertheless, IP has so far failed to achieved the expected impact in both mathematical statistics and application \ufb01elds. This may lead to the re\ufb02ection that generality, after all, is not the real thing. As discussed, whether very general settings such as meta-probability offer any conceptual or practical advantage is unclear. It is also somewhat puzzling that coherent lower previsions end up being as expressive as credal sets, which can be handled without resorting to complex linear programming optimisation in a robust Bayesian fashion. Moreover, the linear utility assumptions in IP are very strong, which weakens their case as a \u2018natural\u2019 formalisation of subjective reasoning. Other recent and philosophically interesting frameworks such as Vovk and Shafer\u2019s game theoretical probability (also \fGENERALISED SETS GENERALISED MEASURES ON EVENTS ROBUST PROBABILITY PROBABILITY BEHAVIOURAL MEASURES ON FUNCTIONS CAPACITIES Belief functions  on fuzzy sets Possibility Rough sets P-boxes Feasible probability intervals Credal sets Probability HIGHER-ORDER PROBABILITIES Dirichlet Vovk-Shafer MV-algebras Meta-probability Probability  on fuzzy sets Probability intervals probabilities Zadeh's GTU 2-monotone  capacities Random Lower/upper Belief functions sets Liu's UT Granular previsions De Finetti's previsions Lower/upper Figure 1: Illustration of how the uncertainty theories reviewed in this paper cluster into a number of groups, associated with the rationale and class of objects they attempt to mathematically model. Arrows represent implications, from the less general model to the more general. related to imprecise probability, [138]) have encountered similar issues with rationale and motivation. In addition, to this author\u2019s understanding, a further extension of this reasoning to non-repetitive situations is still sought. 10.2 Geometry as unifying language Geometry may be a unifying language for the \ufb01eld [47; 58; 59; 60; 63; 94; 95; 9; 114; 100; 171], possibly in conjunction with an algebraic view [56; 17; 28; 62; 21; 23; 43]. In the geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and there manipulated (e.g. combined, conditioned and so on) [18; 26; 48]. Much work has been focusing on the geometry of belief functions, which live in a convex space termed the belief space, which can be described both in terms of a simplex (a higher-dimensional triangle) and in terms of a recursive bundle structure [61; 50; 46; 57]. The analysis can be extended to Dempster\u2019s rule of combination by introducing the notion of a conditional subspace and outlining a geometric construction for Dempster\u2019s sum [54; 19]. The combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function, have also been studied [27; 39]. The corresponding spaces are simplices which are congruent to the belief space. Subsequent work extended the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) [37] and consistent belief functions [41; 53; 30], in terms of simplicial complexes [20]. Analyses of belief functions in terms credal sets have also been conducted [31; 2; 10]. The geometry of the relationship between measures of different kinds has also been extensively studied [55; 35; 22; 36], with particular attention to the problem of transforming a belief function into a classical probability measure [13; 167; 159]. One can distinguish between an \u2018af\ufb01ne\u2019 family of probability transformations [25] (those which commute with af\ufb01ne combination in the belief space), and an \u2018epistemic\u2019 family of transforms [24], formed by the relative belief and relative plausibility of singletons [32; 33; 42; 51; 38], which possess dual properties with respect to Dempster\u2019s sum [29]. The problem of \ufb01nding the possibility measure which best approximates a given belief function [3] can also be approached in geometric terms [34; 52; 44; 45]. In particular, approximations induced by classical Minkowski norms can be derived and compared with classical outer consonant approximations [79]. Minkowski consistent approximations of belief functions in both the mass and the belief space representations can also be derived [41]. In fact, the geometric approach to uncertainty can be applied to various other elements of the inference chain. For instance, the conditioning problem can be posed geometrically [126]. Conditional belief functions can be de\ufb01ned as those which minimise an appropriate distance between the original belief function and the \u2018conditioning simplex\u2019 associated with the conditioning event [49; 40]. A semantics for the main probability transforms can be provided in terms of credal sets, i.e., convex sets of probabilities [36]. \fRecent papers on this topic include [136; 143; 135]. 10.3 What is the \u2018right\u2019 formalism to use? A current of thought supports the notion that there is no such a thing as the best mathematical model [82; 116]. The choice of the most suitable methodology, instead, should depend on the actual problem at hand. We are not quite in favour of such a position. In our view, evidence from the applications should inform the choice of the most suitable formalism, general enough to provide a comprehensive theory of uncertainty, but no so general as to become useless or overcomplicated. Uncertainty theory should avoid resorting to axioms whenever possible, as this inevitably opens the way to arguments on their validity, especially when claims are made regarding human rationality. In this sense, strong evidence that observations are inherently set-valued provides support for the theory of random sets. The latter is very general, belongs to most of the above conceptual clusters (amounts to a robust probability theory, generalises set and measure theory as well as Bayesian reasoning, has signi\ufb01cant behavioural interpretations), and naturally encodes ignorance, missing, vague and propositional data, and is compatible with both Bayesian and frequentist interpretations of probability [48]. Theoretical advances have been made in recent times, but more are needed. Despite its \ufb02aws additive probability is simple to understand and easy to use. To overcome the conservatorism of experts and practitioners alike, uncertainty theory needs to provide a similar compelling, simple message."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1808.02560v2",
+                    "title": "Belief likelihood function for generalised logistic regression",
+                    "abstract": "The notion of belief likelihood function of repeated trials is introduced,\nwhenever the uncertainty for individual trials is encoded by a belief measure\n(a finite random set). This generalises the traditional likelihood function,\nand provides a natural setting for belief inference from statistical data.\nFactorisation results are proven for the case in which conjunctive or\ndisjunctive combination are employed, leading to analytical expressions for the\nlower and upper likelihoods of `sharp' samples in the case of Bernoulli trials,\nand to the formulation of a generalised logistic regression framework.",
+                    "authors": "Fabio Cuzzolin",
+                    "published": "2018-08-07",
+                    "updated": "2018-08-20",
+                    "primary_cat": "math.ST",
+                    "cats": [
+                        "math.ST",
+                        "cs.AI",
+                        "stat.ME",
+                        "stat.TH"
+                    ],
+                    "main_content": "Introduction Logistic regression [4] is a popular statistical method for modelling data in which one or more independent observed variables determine an outcome, represented by a binary variable. The framework can also be extended to the multinomial case, and is widely used in various \ufb01elds, including machine learning, medical diagnosis, and social sciences, to cite a few. Despite its successes, the method has serious limitations. In particular, it has been shown to consistently and sharply underestimate the probability of \u2018rare\u2019 events [19]. The term [25] denotes cases in which the training data are of insuf\ufb01cient quality, in the sense that they do not represent well enough the underlying distribution. As a result, scientists are forced to infer probability distributions using information captured in \u2018normal\u2019 times (e.g. while a nuclear power plant is working nominally), whereas these distributions are later used to extrapolate results at the \u2018tail\u2019 of the curve. Although corrections to logistic regression have been proposed [19], the root cause of the problem, in our view, our very models of uncertainty are themselves affected by uncertainty: a phenomenon often called \u2018Knightian\u2019 uncertainty. The latter can be explicitly modelled by considering convex sets of probability distributions, or \u2018credal sets [24, 23]. Random sets [26, 27, 29, 15], in particular, are a sub-class of credal sets induced by probability distributions on the collection of all subsets of the sample space. In the \ufb01nite case random sets are often called belief functions, a term introduced by Glenn Shafer [28] from a subjective probability perspective. As we show here, the logistic regression framework can indeed be generalised to the case of belief functions, which themselves generalise classical discrete probability measures. Given a sample space X, the traditional likelihood function is equal to the conditional probability of the data given a parameter \u03b8 \u2208\u0398, i.e., a family of probability distribution functions (PDFs) over X parameterised by \u03b8: L(\u03b8|X) . = p(X|\u03b8), \u03b8 \u2208\u0398. As originally proposed by Shafer and Wasserman [28, 32, 33], belief functions can indeed be built from traditional likelihood functions. However, as we argue here, one can directly de\ufb01ne a belief likelihood function, mapping a sample observation x \u2208X to a real number, as a natural set-valued generalisation of the conventional likelihood. It is natural to de\ufb01ne such a belief likelihood function as family of belief functions on X, BelX(.|\u03b8), parameterised by \u03b8 \u2208\u0398. As the latter take values on sets of outcomes, A \u2282X, of which singleton outcomes are mere special cases, they provide a natural setting for computing likelihoods of set-valued observations, in accordance with the random set philosophy. When applied to samples generated by series of independent trials, under a generalisation of stochastic independence, belief likelihoods factorise into simple products. The resulting lower and upper likelihoods can be easily computed for series of Bernoulli trials, and allows us to formulate a generalised logistic regression framework, in which the mass values of individual trials are constrained to follow a logistic dependence on scalar parameters. The values of the parameters which optimise the lower arXiv:1808.02560v2  [math.ST]  20 Aug 2018 \fand upper likelihoods induce a pair of \u2018lower\u2019 and \u2018upper\u2019 belief functions on the parameter space, whose interval effectively encodes the uncertainty associated with the amount of data at our disposal. Every new observation, possibly in areas of the sample space not previously explored, is mapped to a pair of lower and upper logistic belief functions, which together provide lower and upper estimates for the belief values of each event. 1.1 Contributions The contributions of the paper are thus as follows: (1) a belief likelihood function for repeated trials is de\ufb01ned, whenever the uncertainty on individual trials is assumed to be encoded by a belief measure; (2) elegant factorisation properties are proven for events that are Cartesian products, whenever belief measures are combined by conjunctive rule, leading to the notions of lower and upper likelihoods; (3) factorisation results are also provided in the case in which the dual, disjunctive combination is used to compute belief and plausibility likelihoods; (4) analitical expressions of lower and upper likelihoods are provided for the case of Bernoulli trials; (4) \ufb01nally, a generalised logistic regression based on lower and upper likelihoods is formulated and analysed, as an alternative inference mechanism to generate belief functions from statistical data. 1.2 Paper outline After reviewing in Section 2 the logistic regression framework, we recall in Section 3 the necessary notions of the theory of belief functions. In Section 4 the belief likelihood function of repeated trials is de\ufb01ned. In Section 5, the belief likelihood of a series of binary trials is analysed in the conjunctive case. Factorisation results are shown which reduce upper and lower likelihoods of \u2018sharp\u2019 samples to products of belief values of individual binary observations, and can be generalised to arbitrary Cartesian products of focal elements. In Section 6 an analysis of the belief likelihood function in the disjunctive case is conducted. General factorisation results holding for series of observations from arbitrary sample spaces are illustrated in Section 7, while analytical expressions for the Bernoulli case are given in Section 8. Finally, a generalised logistic regression framework is outlined (Section 9) in which the masses of the two outcomes are constrained to have a logistic dependency, and dual optimisation problems lead to a pair of lower and upper estimates for the belief measure of the outcomes. Section 10 concludes the paper and points at future work. 2 Logistic regression Logistic regression allows us, given a sample Y = {Y1, ..., Yn}, X = {x1, ..., xn} where Yi \u2208{0, 1} is a binary outcome at time i and xi is the corresponding observed measurement, to learn the parameters of a conditional probability relation between the two, of the form: P(Y = 1|x) = 1 1 + e\u2212(\u03b20+\u03b21x) , (1) where \u03b20 and \u03b21 are two scalar parameters. Given a new observation x, (1) delivers the probability of a positive outcome Y = 1. Logistic regression generalises deterministic linear regression, as it is a function of the linear combination \u03b20+\u03b21x. The n trials are assumed independent but not equally distributed, for \u03c0i = P(Yi = 1|xi) varies with the time instant i of collection. The two scalar parameters \u03b20, \u03b21 in (1) are estimated by maximum likelihood of the sample. After denoting by \u03c0i = P(Yi = 1|xi) = 1 1 + e\u2212(\u03b20+\u03b21xi) , 1 \u2212\u03c0i = P(Yi = 0|xi) = e\u2212(\u03b20+\u03b21xi) 1 + e\u2212(\u03b20+\u03b21xi) (2) the conditional probabilities of the two outcomes, the likelihood of the sample can be expressed as: L(\u03b2|Y ) = Qn i=1 \u03c0Yi i (1 \u2212\u03c0i)Yi, where Yi \u2208{0, 1} and \u03c0i is a function of \u03b2 = [\u03b20, \u03b21]. Maximising L(\u03b2|Y ) yields a conditional PDF P(Y = 1|x). Unfortunately, logistic regression shows clear limitations when the number of samples is insuf\ufb01cient or when there are too few positive outcomes (1s) [19]. Moreover, inference by logistic regression tends to underestimate the probability of a positive outcome [19]. 3 Belief functions 3.1 Belief and plausibility measures De\ufb01nition 1. A basic probability assignment (BPA) [1] over a \ufb01nite domain \u0398 is a set function [10, 11] m : 2\u0398 \u2192[0, 1] de\ufb01ned on the collection 2\u0398 of all subsets of \u0398 s.t.: m(\u2205) = 0, X A\u2282\u0398 m(A) = 1. The quantity m(A) is called the basic probability number or \u2018mass\u2019 [22, 21] assigned to A. The elements of the power set 2\u0398 associated with non-zero values of m are called the focal elements of m. De\ufb01nition 2. The belief function (BF) associated with a basic probability assignment m : 2\u0398 \u2192[0, 1] is the set function Bel : 2\u0398 \u2192[0, 1] de\ufb01ned as: Bel(A) = X B\u2286A m(B). (3) \fThe domain \u0398 on which a belief function is de\ufb01ned is usually interpreted as the set of possible answers to a given problem, exactly one of which is the correct one. For each subset (\u2018event\u2019) A \u2282\u0398 the quantity Bel(A) takes on the meaning of degree of belief that the truth lies in A, and represents the total belief committed to a set of possible outcomes A by the available evidence m. Another mathematical expression of the evidence generating a belief function Bel is the upper probability or plausibility of an event A: Pl(A) . = 1 \u2212Bel( \u00af A), as opposed to its lower probability Bel(A) [6]. The corresponding plausibility function Pl : 2\u0398 \u2192[0, 1] conveys the same information as Bel, and can be expressed as: Pl(A) = X B\u2229A\u0338=\u2205 m(B) \u2265Bel(A). 3.2 Evidence combination The issue of combining the belief function representing our current knowledge state with a new one encoding the new evidence is central in belief theory. After an initial proposal by Dempster, several other aggregation operators have been proposed, based on different assumptions on the nature of the sources of evidence to combine. De\ufb01nition 3. The orthogonal sum or Dempster\u2019s combination Bel1 \u2295Bel2 : 2\u0398 \u2192[0, 1] of two belief functions Bel1 : 2\u0398 \u2192[0, 1], Bel2 : 2\u0398 \u2192[0, 1] de\ufb01ned on the same domain \u0398 is the unique BF on \u0398 with as focal elements all the non-empty intersections of focal elements of Bel1 and Bel2, and basic probability assignment: m\u2295(A) = m\u2229(A) 1 \u2212m\u2229(\u2205), (4) where mi denotes the BPA of the input BF Beli, and: m\u2229(A) = P B\u2229C=A m1(B)m2(C). Rather than normalising (as in (4)), Smets\u2019 conjunctive rule leaves the con\ufb02icting mass m(\u2205) with the empty set: m \u2229 \u20dd(A) = m\u2229(A) \u2205\u2286A \u2286\u0398, (5) and is thus applicable to \u2018unnormalised\u2019 beliefs [31]. In Dempster\u2019s rule, consensus between two sources is expressed by the intersection of the supported events. When the union is taken to express consensus we obtain the disjunctive rule of combination [20, 34]: m \u222a \u20dd(A) = X B\u222aC=A m1(B)m2(C), (6) which yields more cautious inferences than conjunctive rules, by producing belief functions that are less \u2018committed\u2019, i.e., have larger focal sets. Under disjunctive combination: Bel1 \u222a \u20ddBel2(A) = Bel1(A) \u00b7 Bel2(A), input belief values are simply multiplied. 3.3 Conditioning Belief functions can also be conditioned, rather than combined, whenever we are presented hard evidence of the form \u2018A is true\u2019 [3, 12, 18, 14, 9, 35, 17]. In particular, Dempster\u2019s combination naturally induces a conditioning operator. Given a conditioning event A \u2282 \u0398, the \u2018logical\u2019 (or \u2018categorical\u2019, in Smets\u2019 terminology) belief function BelA such that m(A) = 1 is combined via Dempster\u2019s rule with the a-priori belief function Bel. The resulting BF Bel \u2295BelA is the conditional belief function given A a la Dempster, denoted by Bel\u2295(A|B). 3.4 Multivariate analysis In many applications, we need to express uncertain information about a number of distinct variables (e.g., X and Y ) taking values in different domains (\u0398X and \u0398Y , respectively). The reasoning process needs then to take place in the Cartesian product of the domains associated with each individual variable. Let then \u0398X and \u0398Y be two sample spaces associated with two distinct variables, and let mXY be a mass function on \u0398XY = \u0398X \u00d7 \u0398Y . The latter can be expressed in the coarser domain \u0398X by transferring each mass mXY (A) to the projection A \u2193\u0398X of A on \u0398X. We obtain a marginal mass function on \u0398X, denoted by: mXY \u2193X (B) . = X {A\u2286\u0398XY ,A\u2193\u0398X=B} mXY (A), \u2200B \u2286\u0398X. Conversely, a mass function mX on \u0398X can be expressed in \u0398X \u00d7 \u0398Y by transferring each mass mX(B) to the cylindrical extension B\u2191XY . = B \u00d7 \u2126Y of B. The vacuous extension of mX onto \u0398X \u00d7 \u0398Y will then be: m\u2191XY X (A) . = ( mX(B) if A = B \u00d7 \u2126Y , 0 otherwise. (7) The associated BF is denoted by Bel\u2191XY X . 4 Belief likelihood of repeated trials Let BelXi(A|\u03b8), for i = 1, 2, ..., n be a parameterised family of belief functions on Xi, the space of quantities that can be observed at time i, depending on a parameter \u03b8 \u2208\u0398. A series of repeated trials then assumes values in X1\u00d7\u00b7 \u00b7 \u00b7\u00d7Xn, whose elements are tuples of the form \u20d7 x = (x1, ..., xn) \u2208X1\u00d7\u00b7 \u00b7 \u00b7\u00d7Xn. We call such tuples \u2018sharp\u2019 samples, as opposed to arbitrary subsets A \u2282X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn of the space of trials. Note that we are not assuming the trials to be equally distributed at this stage, nor we assume that they come from the same sample space. \fDe\ufb01nition 4. The belief likelihood function BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn : 2X1\u00d7\u00b7\u00b7\u00b7\u00d7Xn \u2192[0, 1] of a series of repeated trials is de\ufb01ned as: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A|\u03b8) . = Bel\u2191\u00d7iXi X1 \u2299\u00b7 \u00b7 \u00b7\u2299Bel\u2191\u00d7iXi Xn (A|\u03b8), (8) where Bel\u2191\u00d7iXi Xj is the vacuous extension (7) of BelXj to the Cartesian product X1 \u00d7\u00b7 \u00b7 \u00b7\u00d7Xn where the observed tuples live, and \u2299is an arbitrary combination rule. In particular, when the subset A reduces to a sharp sample, A = {\u20d7 x}, we can de\ufb01ne the following generalisations of the notion of likelihood. De\ufb01nition 5. We call the quantities L(\u20d7 x) . = BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}|\u03b8), L(\u20d7 x) . = PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}|\u03b8) (9) lower likelihood and upper likelihood, respectively, of A = {\u20d7 x} = {(x1, ..., xn)}. 5 Binary trials: the conjunctive case Belief likelihoods factorise into simple products, whenever conjuctive combination is employed (as a generalisation of classical stochastic independence) in De\ufb01nition 4, and trials with binary outcomes are considered. 5.1 Focal elements of the belief likelihood Let us \ufb01rst analyse the case n = 2. We seek the Dempster\u2019s sum BelX1 \u2295BelX2, where X1 = X2 = {T, F}. Figure 1 is a diagram of all the intersections of focal elements of the two input BF on X1 \u00d7X2. There are 9 = 32 Figure 1: Graphical representation of Dempster\u2019s combination BelX1 \u2295BelX2 on X1 \u00d7 X2. distinct, non-empty intersections, which correspond to the focal elements of BelX1\u2295BelX2. According to Equation (4), the mass of focal element A1 \u00d7 A2, A1 \u2286X1, A2 \u2286X2, is then: mBelX1\u2295BelX2 (A1 \u00d7 A2) = mX1(A1) \u00b7 mX2(A2). (10) Note that the result holds when using the conjunctive rule \u2229 \u20ddas well (5), for none of the intersections is empty, hence no normalisation is required. Nothing is assumed about the mass assignment of BelX1 and BelX2. We can now prove the following Lemma. Lemma 1. For any n \u2208Z the belief function BelX1 \u2295 \u00b7 \u00b7 \u00b7 \u2295BelXn, where Xi = X = {T, F}, has 3n focal elements, namely all possible Cartesian products A1 \u00d7 ... \u00d7 An of n non-empty subsets Ai of X, with BPA: mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (A1 \u00d7 ... \u00d7 An) = n Y i=1 mXi(Ai). Proof. The proof is by induction. The thesis was shown to be true for n = 2 in Equation (10). In the induction step, we assume that the thesis is true for n, and prove it for n+1. If BelX1 \u2295\u00b7 \u00b7 \u00b7\u2295BelXn, de\ufb01ned on X1 \u00d7\u00b7 \u00b7 \u00b7\u00d7 Xn, has as focal elements the n-products A1 \u00d7 ... \u00d7 An with Ai \u2208 \b {T}, {F}, X \t for all i, its vacuous extension to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 Xn+1 will have as focal elements the n + 1-products of the form: A1 \u00d7 ... \u00d7 An \u00d7 Xn+1, with Ai \u2208 \b {T}, {F}, X \t for all i. The belief function BelXn+1 is de\ufb01ned on Xn+1 = X, with three focal elements: {T}, {F} and X = {T, F}. Its vacuous extension to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 Xn+1 thus has the following three focal elements: X1\u00d7\u00b7 \u00b7 \u00b7\u00d7Xn\u00d7{T}, X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 {F} and X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 Xn+1. When computing (BelX1 \u2295\u00b7 \u00b7 \u00b7 \u2295BelXn) \u2295BelXn+1 on the common re\ufb01nement X1\u00d7\u00b7 \u00b7 \u00b7\u00d7Xn\u00d7Xn+1 we need to compute the intersection of their focal elements, namely: \u0000A1 \u00d7 ... \u00d7 An \u00d7 Xn+1 \u0001 \u2229 \u0000X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn \u00d7 An+1 \u0001 = A1 \u00d7 ... \u00d7 An \u00d7 An+1 for all Ai \u2286Xi, i = 1, ..., n + 1. All such intersections are distinct for distinct focal elements of the two belief functions to combine, and there are no empty intersection. By Dempster\u2019s rule (4) their mass is equal to the product of the original masses, i.e.: mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn+1 (A1 \u00d7 ... \u00d7 An \u00d7 An+1) = mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (A1 \u00d7 ... \u00d7 An) \u00b7 mBelXn+1 (An+1). Since we assumed that the factorisation holds for n, the thesis easily follows. As no normalisation is involved in the combination BelX1 \u2295\u00b7 \u00b7 \u00b7 \u2295BelXn, Dempster\u2019s rule coincides with the conjunctive rule and Lemma 1 holds for \u2229 \u20ddas well. \f5.2 Factorisation for \u2018sharp\u2019 tuples The following becomes then a simple corollary. Theorem 1. When using either \u2229 \u20ddor \u2295as a combination rule in the de\ufb01nition of belief likelihood function, the following decomposition holds for tuples (x1, ..., xn), xi \u2208 Xi, which are the singletons elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, with X1 = ... = Xn = {T, F}: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}|\u03b8) = n Y i=1 BelXi({xi}|\u03b8), (11) Proof. For the singleton elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, since {(x1, ..., xn)} = {x1} \u00d7 ... \u00d7 {xn}, Equation (11) becomes: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}) = mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn ({(x1, ..., xn)}) = Qn i=1 mXi({xi}) = Qn i=1 BelXi({xi}), where the mass factorisation follows from Lemma 1, as on singletons mass and belief values coincide. There is evidence to support the following as well. Conjecture 1. When using either \u2229 \u20ddor \u2295as a combination rule in the de\ufb01nition of belief likelihood function, the following decomposition holds for the associated plausibility values on tuples (x1, ..., xn), xi \u2208Xi, which are the singletons elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, with X1 = ... = Xn = {T, F}: PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}|\u03b8) = n Y i=1 PlXi({xi}|\u03b8). (12) Indeed we can write: PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}) = = 1 \u2212BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}c) = = 1 \u2212 X B\u2286{(x1,...,xn)}c mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (B). (13) By Lemma 1 all the subsets B with non-zero mass are Cartesian products of the form A1 \u00d7 ... \u00d7 An, \u2205\u0338= Ai \u2286 Xi. We then need understand the nature of the focal elements of BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn which are subsets of an arbitrary singleton complement {(x1, ..., xn)}c. For binary spaces Xi = X = {T, F}, by de\ufb01nition of Cartesian product, each such B = A1 \u00d7 ... \u00d7 An \u2286 {(x1, ..., xn)}c is obtained by replacing a number 1 \u2264 k \u2264n of components of the tuple (x1, ..., xn) = {x1} \u00d7 ... \u00d7 {xn} with a different subset of Xi (either {xi}c = Xi\\{xi} or Xi). There are \u0000n k \u0001 such sets of k components in a list of n. Of these k components, in general 1 \u2264m \u2264 k will be replaced by {xi}c, while the other 1 \u2264k\u2212m < k will be replaced by Xi. Note that not all k components can be replaced by Xi, since the resulting focal element would contain the tuple {(x1, ..., xn)} \u2208X1 \u00d7 ... \u00d7 Xn. The following argument can be proved for (x1, ..., xn) = (T, ..., T), under the additional assumption that BelX1 \u00b7 \u00b7 \u00b7 BelXn are equally distributed with p . = BelXi({T}), q . = BelXi({F}) and r . = BelXi(Xi). If this is the case, for \ufb01xed values of m and k all the resulting focal elements have the same mass value, namely: pn\u2212kqmrk\u2212m, where p . = BelXi({T}), q . = BelXi({F}) and r . = BelXi(Xi). As there are exactly \u0000 k m \u0001 such focal elements, (13) can be written as: 1 \u2212 n X k=1 \u0012n k \u0013 k X m=1 \u0012 k m \u0013 pn\u2212kqmrk\u2212m. which can be rewritten as: 1 \u2212 n X m=1 qm n X k=m \u0012n k \u0013\u0012 k m \u0013 pn\u2212krk\u2212m. A change of variable l = n\u2212k, where l = 0 when k = n, l = n \u2212m when k = m, allows us to write it as: 1 \u2212 n X m=1 qm n\u2212m X l=0 \u0012 n n \u2212l \u0013\u0012n \u2212l m \u0013 plr(n\u2212m)\u2212l, since k \u2212m = n \u2212l \u2212m, k = n \u2212l. Now, as \u0012 n n \u2212l \u0013\u0012n \u2212l m \u0013 = \u0012n m \u0013\u0012n \u2212m l \u0013 we obtain: 1\u2212 n X m=1 qm \u0012n m \u0013 n\u2212m X l=0 \u0012n \u2212m l \u0013 plr(n\u2212m)\u2212l. By Newton\u2019s binomial, the latter is equal to 1 \u2212 n X m=1 qm \u0012n m \u0013 (1 \u2212q)n\u2212m, since 1 \u2212q = p + r. Again, by Newton\u2019s binomial, we get: PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(T, ..., T)}) = 1 \u2212[1 \u2212(1 \u2212q)n] = (1 \u2212q)n = n Y i=1 PlXi({T}). 5.3 Factorisation for Cartesian products Decomposition (11) is equivalent to what Smets calls conditional conjunctive independence [30]. In fact, for binary spaces factorisation (11) generalises to all subsets A \u2286X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn of samples which are Cartesian products of subsets of X1, ..., Xn, respectively: A = A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An, Ai \u2286Xi for all i. Corollary 1. Whenever Ai \u2286Xi = {T, F}, i = 1, ..., n, under conjunctive combination we have that: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An|\u03b8) = n Y i=1 BelXi(Ai|\u03b8). (14) \fProof. As by Lemma 1 all the focal elements of BelX1 \u2229 \u20dd\u00b7 \u00b7 \u00b7 \u2229 \u20ddBelXn are Cartesian products of the form B = B1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Bn, Bi \u2286 Xi, it follows that BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 An|\u03b8) is equal to: X B\u2286A1\u00d7\u00b7\u00b7\u00b7\u00d7An,B=B1\u00d7\u00b7\u00b7\u00b7\u00d7Bn mX1(B1) \u00b7 ... \u00b7 mXn(Bn). But {B \u2286A1 \u00d7\u00b7 \u00b7 \u00b7\u00d7An, B = B1 \u00d7\u00b7 \u00b7 \u00b7\u00d7Bn} = {B = B1\u00d7\u00b7 \u00b7 \u00b7\u00d7Bn, Bi \u2286Ai\u2200i}, since if Bj \u0338\u2282Aj for some j the resulting Cartesian product would not be a subset of A1 \u00d7\u00b7 \u00b7 \u00b7\u00d7An. Thus, BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1 \u00d7\u00b7 \u00b7 \u00b7\u00d7An|\u03b8) = = X B=B1\u00d7\u00b7\u00b7\u00b7\u00d7Bn,Bi\u2286Ai\u2200i mX1(B1)\u00b7...\u00b7mXn(Bn). (15) For all Ai\u2019s, i = i1, ..., im that are singletons of Xi, necessarily Bi = Ai and we can write (15) as: m(Ai1)\u00b7...\u00b7m(Aim) X Bj\u2286Aj,j\u0338=i1,...,im Y j\u0338=i1,...,im mXj(Bj). If the frames are binary, Xi = {T, F}, those Ai\u2019s that are not singletons coincide with Xi, so that we have: m(Ai1) \u00b7 ... \u00b7 m(Aim) X Bj\u2286Xj,j\u0338=i1,...,im Y j mXj(Bj). The quantity X Bj\u2286Xj,j\u0338=i1,...,im Y j mXj(Bj) is, according to the de\ufb01nition of conjunctive combination, the sum of the masses of all the possible intersections of (cylindrical extensions of) focal elements of BelXj, j \u0338= i1, ..., im, thus they add up to 1. In conclusion (forgetting the conditioning on \u03b8 in the derivation for sake of readability): BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn(A1\u00d7\u00b7 \u00b7 \u00b7\u00d7An) = m(Ai1)\u00b7...\u00b7m(Aim)\u00b71\u00b7 ... \u00b7 1 = BelXi1 (Ai1) \u00b7 ... \u00b7 BelXim(Aim) \u00b7 BelXj1 (Xj1) \u00b7 ... \u00b7 BelXjk (Xjk) = BelXi1 (Ai1) \u00b7 ... \u00b7 BelXim(Aim) \u00b7 BelXj1 (Aj1) \u00b7 ... \u00b7 BelXjk (Ajk) and we have (14). Corollary 1 states that conditional conjunctive independence always holds for events that are Cartesian products, whenever the involved frames are binary. 6 Binary trials: the disjunctive case Similar factorisation results hold when using the (more cautious) disjunctive combination \u222a \u20dd. 6.1 Structure of the focal elements As in the conjunctive case, we \ufb01rst analyse the case n = 2. We seek the disjunctive combination BelX1 \u222a \u20ddBelX2, where each BelXi has as focal elements {T}, {F} and Figure 2: Graphical representation of the disjunctive combination BelX1 \u222a \u20ddBelX2 on X1 \u00d7 X2. Xi. Figure 2 is a diagram of all the unions of focal elements of the two input BFs on their common re\ufb01nement X1 \u00d7 X2. There are 5 = 22 + 1 distinct such unions, the focal elements of BelX1 \u222a \u20ddBelX2, with masses: m({(xi, xj)}c) = mX1({xi}c) \u00b7 mX2({xj}c), m(X1 \u00d7 X2) = 1 \u2212 X i,j mX1({xi}c) \u00b7 mX2({xj}c). We can now prove the following Lemma. Lemma 2. The belief function BelX1 \u222a \u20dd\u00b7 \u00b7 \u00b7 \u222a \u20ddBelXn, where Xi = X = {T, F}, has 2n + 1 focal elements, namely all the complements of the n-tuples \u20d7 x = (x1, ..., xn) of singleton elements xi \u2208Xi, with BPA: mBelX1 \u222a \u20dd\u00b7\u00b7\u00b7 \u222a \u20ddBelXn ({(x1, ..., xn)}c) = = mX1({x1}c) \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 mXn({xn}c), (16) plus the Cartesian product X1\u00d7\u00b7 \u00b7 \u00b7\u00d7Xn itself, with mass value given by normalisation. Proof. The proof is by induction. The case n = 2 was proven above. In the induction step, we assume that the thesis is true for n, namely that the focal elements of BelX1 \u222a \u20dd\u00b7 \u00b7 \u00b7 \u222a \u20ddBelXn have the form: A = {(x1, ..., xn)}c = {(x\u2032 1, ..., x\u2032 n)|\u2203i : {x\u2032 i} = {xi}c}, (17) where xi \u2208Xi = X. We need to prove it true for n + 1. The vacuous extension of (17) has trivially the form: A\u2032 = {(x\u2032 1, ..., x\u2032 n, xn+1)|\u2203i : {x\u2032 i} = {xi}c, xn+1 \u2208X}. Note that only 2 = |X| singletons of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 are not in A\u2032, for any given tuple (x1, ..., xn). The vacuous extension to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 of a focal element B = {xn+1} of BelXn+1 is instead: B\u2032 = {(y1, \u00b7 \u00b7 \u00b7 , yn, xn+1)|yi \u2208X \u2200i = 1, ..., n}. \fNow, all the elements of B\u2032, except for (x1, ..., xn, xn+1), are also elements of A\u2032. Hence, the union A\u2032 \u222aB\u2032 reduces to the union of A\u2032 and (x1, ..., xn, xn+1). The only singleton element of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 not in A\u2032 \u222aB\u2032 is therefore (x1, ..., xn, x\u2032 n+1), {x\u2032 n+1} = {xn+1}c, for it is neither in A\u2032 nor in B\u2032. All such unions are distinct. Thus, by de\ufb01nition of \u222a \u20dd, their mass is m({(x1, ..., xn)}c) \u00b7 m({xn+1}c) which by inductive hypothesis is equal to (16). Unions involving either Xn+1 or X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn are equal to X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn+1 by the property of the union operator. 6.2 Factorisation Theorem 2. In the hypotheses of Lemma 2, when using disjunctive combination \u222a \u20ddin the de\ufb01nition of belief likelihood function, the following decomposition holds: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}c|\u03b8) = n Y i=1 BelXi({xi}c|\u03b8). (18) Proof. As {(x1, ..., xn)}c contains only itself as a focal element: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}c|\u03b8) = m({(x1, ..., xn)}c|\u03b8). By Lemma 2 the latter becomes BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}c|\u03b8) = n Y i=1 mXi({xi}c|\u03b8) = n Y i=1 BelXi({xi}c|\u03b8), as {xi}c is a singleton element of Xi, and we have (18). Note that PlX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}c|\u03b8) = 1 for all tuples (x1, ..., xn), as the set {(x1, ..., xn)}c has non-empty intersection with all the focal elements of BelX1 \u222a \u20dd\u00b7 \u00b7 \u00b7 \u222a \u20ddBelXn. 7 General factorisation results The argument of Lemma 1 is in fact valid for the conjunctive combination of belief functions de\ufb01ned on an arbitrary collection X1, ..., Xn of \ufb01nite spaces. Theorem 3. For any n \u2208Z the belief function BelX1 \u2295 \u00b7 \u00b7 \u00b7 \u2295BelXn, where X1, ..., Xn are \ufb01nite spaces, has as focal elements all the Cartesian products A1 \u00d7 ... \u00d7 An of n focal elements A1 \u2286X1, ..., An \u2286Xn, with BPA: mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn (A1 \u00d7 ... \u00d7 An) = n Y i=1 mXi(Ai). The proof is similar to that of Lemma 1, and is omitted for lack of space. It follows that: Corollary 2. When using either \u2229 \u20ddor \u2295as a combination rule in the de\ufb01nition of belief likelihood function, the following decomposition holds for tuples (x1, ..., xn), xi \u2208Xi, which are the singletons elements of X1 \u00d7\u00b7 \u00b7 \u00b7\u00d7 Xn, with X1, ..., Xn any \ufb01nite frames of discernment: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}|\u03b8) = n Y i=1 BelXi({xi}|\u03b8). (19) Proof. For the singleton elements of X1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 Xn, since {(x1, ..., xn)} = {x1} \u00d7 ... \u00d7 {xn}, Equation (19) becomes: BelX1\u00d7\u00b7\u00b7\u00b7\u00d7Xn({(x1, ..., xn)}) = mBelX1\u2295\u00b7\u00b7\u00b7\u2295BelXn ({(x1, ..., xn)}) = Qn i=1 mXi({xi}) = Qn i=1 BelXi({xi}), where the mass factorisation follows from Theorem 3, as on singletons mass and belief values coincide. 8 Lower and upper likelihoods of Bernoulli trials In the case of Bernoulli trials, where not only there is a single binary sample space, Xi = X = {T, F} and conditional independence holds, but the random variables are assumed equally distributed, the conventional likelihood reads as pk(1 \u2212p)n\u2212k, where p = P(T), k is the number of successes (T) and n the total number of trials. Let us then compute the lower likelihood function for a series of Bernoulli trials, under the assumption that all the BFs BelXi = BelX, i = 1, ..., n, coincide (the analogous of equidistribution), with BelX parameterised by p = m({T}), q = m({F}) (where, this time, p+q \u22641). Corollary 3. Under the above assumptions, the lower and upper likelihoods of the sample \u20d7 x = (x1, ..., xn) are: L(\u20d7 x) = n Y i=1 BelX({xi}) = pkqn\u2212k; L(\u20d7 x) = n Y i=1 PlX({xi}) = (1 \u2212q)k(1 \u2212p)n\u2212k. (20) The above decomposition for L(\u20d7 x), in particular, is valid under the assumption that Conjecture 1 holds, at least for Bernoulli trials, as the evidence seems to suggest. After normalisation, these can be seen as probability distribution functions (PDFs) over the (belief) space B of all belief functions de\ufb01nable on X [5, 7, 8]. Having observes a series of trials, \u20d7 x = (x1, ..., xn), xi \u2208 {T, F}, one may then seek the belief function on X = {T, F} which best describes the observed sample, i.e., the optimal values of the two parameters p and q. \fFigure 3: Lower (top) and upper (bottom) likelihood functions plotted over the space of belief functions on X = {T, F}, parameterised by p = m(T) (X axis) and q = m(F) (Y axis), for the case of k = 6 successes over n = 10 trials. Figure 3 plots the both lower and upper likelihoods (20) for the case of k = 6 successes over n = 10 trials. Both subsume the traditional likelihood pk(1 \u2212p)n\u2212k, as their section for p + q = 1, although this is particularly visible for the lower likelihood (top). In particular, the maximum of the lower likelihood is the traditional ML estimate p = k/n, q = 1 \u2212p. This makes sense, for the lower likelihood is highest for the most committed belief functions (i.e., for probability measures). The upper likelihood (bottom) has a unique maximum in p = q = 0: this is the vacuous belief function on {T, F}, with m({T, F}) = 1. The interval of belief functions joining max L with max L is the set of belief functions such that p q = k n\u2212k, i.e., those which preserve the ratio between the observed empirical counts. 9 Generalising logistic regression Based on Theorem 1 and Conjecture 1, we can also generalise logistic regression (Section 2) to a belief function setting by replacing the conditional probability (\u03c0i, 1 \u2212 \u03c0i) on X = {T, F} with a belief function (pi = m({T}), qi = m({F}), pi + qi \u22641) on 2X. Note that, just as in a traditional logistic regression setting (Section 2), the belief functions Beli associated with different input values xi are not equally distributed. After writing T = 1, F = 0, the lower and upper likelihoods can be expressed as: L(\u03b2|Y ) = n Y i=1 pYi i q1\u2212Yi i , L(\u03b2|Y ) = n Y i=1 (1 \u2212qi)Yi(1 \u2212pi)1\u2212Yi. The question becomes how to generalise the logit link between observations x and outputs y, in order to seek an analytical mapping between observations and belief functions over a binary frame. Just assuming: pi = P(Yi = 1|xi) = 1 1 + e\u2212(\u03b20+\u03b21xi) , (21) as in the classical contraint (2), does not yield any analytical dependency for qi. To address this issue we can, for instance, add a parameter \u03b22 such that the following relationship holds: qi = m(Yi = 0|xi) = \u03b22 e\u2212(\u03b20+\u03b21xi) 1 + e\u2212(\u03b20+\u03b21xi) . (22) We can then seek lower and upper optimal estimates for the parameter vector \u03b2 = [\u03b20, \u03b21, \u03b22]\u2032: arg max \u03b2 L 7\u2192\u03b20, \u03b21, \u03b22, arg max \u03b2 L 7\u2192\u03b20, \u03b21, \u03b22. (23) Plugging these optimal parameters into (21), (22) will then yield a lower and an upper family of conditional belief functions given x (i.e., an interval of belief functions): BelX(.|\u03b2, x), BelX(.|\u03b2, x). Given any new test observation x\u2032, this generalised logistic regression method will then output a pair of lower and upper belief functions on X = {T, F}, as opposed to a sharp probability value as in the classical framework. As each belief function itself provides a lower and an upper probability for each event, both the lower and the upper regressed BFs will provide an interval for the probability P(T) of success, whose width will re\ufb02ect the uncertainty encoded by the training set of sample series. \f9.1 Optimisation The problems (23) are both constrained optimisation ones (contrarily to the classical case, where qi = 1 \u2212pi and the optimisation problem is unconstrained). Indeed, the parameter vector \u03b2 must be such that: 0 \u2264pi + qi \u22641 \u2200i = 1, ..., n. Fortunately the number of constraints can be reduced by noticing that, if pi and qi have the analytical forms (21) and (22), respectively, then pi+qi \u22641 for all i whenever \u03b22 \u22641. The objective function can also be simpli\ufb01ed by taking the logarithm. In the lower likelihood case we get1: log L(\u03b2|Y ) = n X i=1 n \u2212log(1 + e\u2212(\u03b20+\u03b21xi)) +(1 \u2212Yi) \u0002 log \u03b22 \u2212(\u03b20 + \u03b21xi) \u0003o . We then need to analyse the Karush-Kuhn-Tucker (KKT) necessary conditions for the optimality of the solution of a nonlinear optimisation problem arg maxx f(x) subject to differentiable constraints: gi(x) \u22640 i = 0, ..., n. If x\u2217is a local optimum, under some regularity conditions then there exist constants \u00b5i, (i = 0, . . . , n), called KKT multipliers, such that the following conditions hold: 1. \u2207f(x\u2217) \u2212Pm i=1 \u00b5i\u2207gi(x\u2217) = 0 (Stationarity); 2. Primal feasibility: gi(x\u2217) \u22640 for all i = 0, . . . , n; 3. Dual feasibility: \u00b5i \u22650 for all i = 0, . . . , n; 4. Complementary slackness: \u00b5igi(x\u2217) = 0 for all i. In our case the constraints are: \u03b21 \u22641 \u2261 g0 = \u03b22 \u22121 \u22640; pi + qi \u22650 \u2261 gi = \u2212\u03b22 \u2212e\u03b20+\u03b21xi \u22640 i = 1, ..., n, and the Lagrangian becomes: \u039b(\u03b2) = log L(\u03b2)+\u00b50(\u03b22 \u22121)\u2212 n X i=1 \u00b5i(\u03b22 +e\u03b20+\u03b21xi). The stationarity conditions thus read as \u2207\u039b(\u03b2) = 0, i.e.: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 n X i=1 h (1 \u2212pi) \u2212(1 \u2212Yi) \u2212\u00b5ie\u03b20+\u03b21xii = 0, n X i=1 h (1 \u2212pi) \u2212(1 \u2212Yi) \u2212\u00b5ie\u03b20+\u03b21xii xi = 0, n X i=1 \u00121 \u2212Yi \u03b22 \u2212\u00b5i \u0013 + \u00b50 = 0, (24) 1Derivations are omitted due to lack of space. where, as usual, pi is as in (21). Complementary slackness reads instead as follows: \u001a \u00b50(\u03b22 \u22121) = 0, \u00b5i(\u03b22 + e\u03b20+\u03b21xi) = 0, i = 1, ..., n. (25) Standard gradient descent methods can be applied to the above systems of equations to get the optimal parameters of our generalised logistic regressor. Similar calculations hold for the upper likelihood problem. A multi-objective optimisation setting in which the lower likelihood is minimised as the upper likelihood is maximised can be envisaged, to generate the most cautious interval of estimates. 10"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1407.8151v1",
+                    "title": "Consistent transformations of belief functions",
+                    "abstract": "Consistent belief functions represent collections of coherent or\nnon-contradictory pieces of evidence, but most of all they are the counterparts\nof consistent knowledge bases in belief calculus. The use of consistent\ntransformations cs[.] in a reasoning process to guarantee coherence can\ntherefore be desirable, and generalizes similar techniques in classical logic.\nTransformations can be obtained by minimizing an appropriate distance measure\nbetween the original belief function and the collection of consistent ones. We\nfocus here on the case in which distances are measured using classical Lp\nnorms, in both the \"mass space\" and the \"belief space\" representation of belief\nfunctions. While mass consistent approximations reassign the mass not focussed\non a chosen element of the frame either to the whole frame or to all supersets\nof the element on an equal basis, approximations in the belief space do\ndistinguish these focal elements according to the \"focussed consistent\ntransformation\" principle. The different approximations are interpreted and\ncompared, with the help of examples.",
+                    "authors": "Fabio Cuzzolin",
+                    "published": "2014-07-30",
+                    "updated": "2014-07-30",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "stat.ME"
+                    ],
+                    "main_content": "Introduction Belief functions (b.f.s) [1, 2] are complex objects, in which di\ufb00erent and sometimes contradictory bodies of evidence may coexist, as they mathematically describe the fusion of possibly con\ufb02icting expert opinions or measurements. This is the case, for instance, of the pose estimation problem in computer vision [? ]. There, image features do not necessary come from the object of interest, but may be extracted from the background: in such a case, Preprint submitted to Information Sciences September 1, 2021 arXiv:1407.8151v1  [cs.AI]  30 Jul 2014 \fthe belief functions inferred from those feature will con\ufb02ict [? ]. Indeed, con\ufb02ict and combinability play a central role in the theory of evidence [3, 4, 5], and have been recently subject to novel analyses [6, 7, 8]. Consistent knowledge bases and belief functions. Making decisions based on objects collecting possibly inconsistent evidence, such as belief functions, is therefore less than trivial. This is a well known problem in classical logics, where the application of inference rules to inconsistent sets of assumptions or \u201cknowledge bases\u201d may lead to incompatible conclusions, depending on the set of assumptions we start our reasoning from [9]. A variety of approaches have been proposed in this context to solve the problem of inconsistent knowledge bases, such as: fragmenting the latter into maximally consistent subsets; limiting the power of the formalism; or, adopting non-classical semantics [10, 11]. Even when a knowledge base is formally inconsistent, though, it may contain potentially useful information. Paris [9] tackles the problem by not assuming each proposition in the knowledge base as a fact, but by attributing to it a certain degree of belief in a probability logic approach. This leads to something similar to a belief function. Indeed, extensions of classical logics in which propositions are assigned belief rather than truth values exist [33]. In this \u201cbelief logic\u201d framework, a belief function can be interpreted as the analogous of a knowledge base [12]. But then, what are the counterparts of consistent knowledge bases in the theory of evidence? To answer that we need to specify the notion of a belief function \u201cimplying\u201d a certain proposition. As we show here, under a rather sensible de\ufb01nition of such an implication, the class of belief functions which generalize consistent knowledge bases is uniquely determined as the set of b.f.s whose non-zero mass \u201cfocal elements\u201d have non-empty intersection. We are therefore allowed to call them consistent belief functions (cs.b.f.s). Consistent transformation. Analogously to consistent knowledge bases, consistent b.f.s are characterized by null internal con\ufb02ict. It may be therefore be desirable to transform a generic belief function to a consistent one prior to making a decision, or picking a course of action. A similar \u201ctransformation\u201d problem has been widely studied in both the probabilistic [13, 14, 15, 16] and possibilistic [17, 18, 19] cases. A sensible approach, in particular, consists of studying the geometry [20, 21] of the class of b.f.s of interest and projecting the original belief function onto the corresponding geometric locus. Indeed, consistent transformations can be built by solving a minimization 2 \fproblem of the form: cs[b] = arg min cs\u2208CS dist(b, cs) (1) where b is the original belief function, dist an appropriate distance measure between belief functions, and CS denotes the collection of all consistent b.f.s. By plugging in di\ufb00erent distance functions in (1) we get di\ufb00erent consistent transformations. Indeed, Jousselme et al [22] have recently conducted a very nice survey of the distance or similarity measures so far introduced in belief calculus, come out with an interesting classi\ufb01cation, and proposed a number of generalizations of known measures. Many of these measures could be in principle employed to de\ufb01ne indi\ufb00erently consistent transformations or conditional belief functions [23], or approximate belief functions by necessity or probability measures. Other similarity measures between belief functions have been proposed by Shi et al [24], Jiang et al [25], and others [26, 27]. In this paper we focus on what happens when employing classical Lp norms in the approximation problem. The L\u221enorm, in particular, is closely related to consistent and consonant belief functions. The region of consistent b.f.s can be expressed as: CS = n b : max x\u2208\u0398 plb(x) = 1 o , i.e., the set of b.f.s for which the L\u221enorm of the \u201cplausibility distribution\u201d or \u201dcontour function\u201d plb(x) is equal to 1. In addition, cs.b.f.s relate to possibility distributions, and possibility measures Pos are inherently associated with L\u221eas Pos(A) = maxx\u2208A Pos(x). In recent times, Lp norms have been successfully employed in di\ufb00erent problems such as probability [16] and possibility [28] transformation/approximation, or conditioning [23, 29]. As the author has proven [30], geometrically, consistent (as well as consonant [31]) belief functions live in a collection of simplices or \u201csimplicial complex\u201d CS. Each maximal simplex CSx of the consistent complex is associated with associated with an \u201cultra\ufb01lter\u201d {A \u2287{x}}, x \u2208\u0398 of focal elements. A partial solution has therefore to be found separately for each maximal simplex of the consistent complex: these partial solutions are later compared to determine the global approximation(s). In addition, geometric approximation can be performed in di\ufb00erent Cartesian spaces. A belief function can be represented either by the vector \u20d7 b = [b(A), \u2205\u228aA \u228a\u0398]\u2032 of its belief values, or the vector of its mass values \u20d7 mb = [mb(A), \u2205\u228aA \u228a\u0398]\u2032. 3 \fWe call the set of vectors of the \ufb01rst kind belief space B [21, 32], and the collection of vectors of the second kind mass space M [23]. Contribution and outline. In this paper, we solve the Lp consistent transformation problem in full generality in both the mass and the belief space, and discuss the semantics of the results. After brie\ufb02y recalling a few basis notions of the theory of evidence, we prove that consistent belief functions are the counterparts of consistent knowledge bases in belief logic (Section 2.2). As we investigate the transformation problem in a geometric framework, we brie\ufb02y recall in Section 2.7 the geometry of the simplicial complex of consistent belief functions, and explain how we need to solve the approximation/transformation problem separately for each maximal simplex of this complex. We then proceed to solve the L1-, L2and L\u221e-consistent approximation problems in full generality, in both the mass (Section 3) and the belief (Section 4) space representations. In Section 5 we compare and interpret the outcomes of Lp approximations in the two frameworks, with the help of the ternary example. We conclude and discuss the natural prosecution of this line of research in Section 6. 2. Theory: consistent approximations of belief functions 2.1. Belief functions and belief logic A basic probability assignment (b.p.a.) on a \ufb01nite set or frame of discernment \u0398 is a set function mb : 2\u0398 \u2192[0, 1] on 2\u0398 . = {A \u2286\u0398} s.t. mb(\u2205) = 0, P A\u2286\u0398 mb(A) = 1. Subsets of \u0398 associated with non-zero values of mb are called focal elements (f.e.), and their intersection is called the core: Cb . = \\ A\u2286\u0398:mb(A)\u0338=0 A. The belief function (b.f.) b : 2\u0398 \u2192[0, 1] associated with a basic probability assignment mb on \u0398 is de\ufb01ned as: b(A) = X B\u2286A mb(B). A dual mathematical representation of the evidence encoded by a belief function b is the plausibility function (pl.f.) plb : 2\u0398 \u2192[0, 1], A 7\u2192plb(A) where the plausibility value plb(A) of an event A is given by plb(A) . = 1 \u2212b(Ac) = P B\u2229A\u0338=\u2205mb(B) and expresses the amount of evidence not against A. Generalizations of classical logic in which propositions are assigned probability values have been proposed in the past. As belief functions naturally 4 \fgeneralize probability measures, it is quite natural to de\ufb01ne non-classical logic frameworks in which propositions are assigned belief values, rather than probability values. This approach has been brought forward in particular by Sa\ufb03otti [33], Haenni [12], and others. In propositional logic, propositions or formulas are either true or false, i.e., their truth value T is either 0 or 1 [34]. Formally, an interpretation or model of a formula \u03c6 is a valuation function mapping \u03c6 to the truth value \u201ctrue\u201d (1). Each formula can therefore be associated with the set of interpretations or models under which its truth value is 1. If we de\ufb01ne the frame of discernment of all the possible interpretations, each formula \u03c6 is associated with the subset A(\u03c6) of this frame which collects all its interpretations. If the available evidence allows to de\ufb01ne a belief function on this frame of possible interpretations, to each formula A(\u03c6) \u2286\u0398 is then naturally assigned a degree of belief b(A(\u03c6)) between 0 and 1 [33, 12], measuring the total amount of evidence supporting the proposition \u201c\u03c6 is true\u201d. 2.2. Semantics of consistent belief functions In classical logic, a set \u03a6 of formulas or \u201cknowledge base\u201d is said to be consistent if and only if there does not exist another formula \u03c6 such that the knowledge base implies both the formula and its negation: \u03a6 \u22a2\u03c6, \u03a6 \u22a2\u00ac\u03c6. Hence, it is impossible to derive incompatible conclusions from the set of propositions which form a consistent knowledge base. This is crucial if we want to derive univocal, non-contradictory conclusions from a given body of evidence. A knowledge base in propositional logic \u03a6 = {\u03c6 : T(\u03c6) = 1} (where T(\u03c6) denotes the truth value of proposition \u03c6) corresponds in belief logic [33] to a belief function, i.e., a set of propositions together with their non-zero belief values: b = {A \u2286\u0398 : b(A) \u0338= 0}. To determine what consistency amounts to in such a framework, we need to formalize the notion of proposition implied by a belief function. One option is to decide that b \u22a2B \u2286\u0398 if B is implied by all the propositions supported by b: b \u22a2B \u21d4A \u2286B \u2200A : b(A) \u0338= 0. (2) An alternative de\ufb01nition requires the proposition B itself to receive non-zero support by the belief function b: b \u22a2B \u21d4b(B) \u0338= 0. (3) 5 \fWhatever de\ufb01nition we choose for such implication relation, we can de\ufb01ne the class of consistent belief functions as the set of b.f.s which cannot imply contradictory propositions. De\ufb01nition 1. A belief function b is consistent if there exists no proposition A such that both A and its negation Ac are implied by b. When adopting the implication relation (2), it is easy to see that A \u2286B \u2200A : b(A) \u0338= 0 is equivalent to T b(A)\u0338=0 A \u2286B. Furthermore, as each proposition with non-zero belief value must by de\ufb01nition contain a focal element C s.t. mb(C) \u0338= 0, the intersection of all non-zero belief propositions reduces to that of all focal elements of b, i.e., the core of b: \\ b(A)\u0338=0 A = \\ \u2203C\u2286A:mb(C)\u0338=0 A = \\ mb(C)\u0338=0 C = Cb. Indeed, no matter our de\ufb01nition of implication, the class of consistent belief functions corresponds to the set of b.f.s whose core is not empty. De\ufb01nition 2. A belief function is said consistent if its core is non-empty. We can prove that, under either de\ufb01nition (2) or de\ufb01nition (3) of the implication b \u22a2B, De\ufb01nitions 1 and 2 are equivalent. Theorem 1. A belief function b : 2\u0398 \u2192[0, 1] has non-empty core if and only if there do not exist two complementary propositions A, Ac \u2286\u0398 which are both implied by b in the sense (2). Proof. A proposition A is implied (2) by b i\ufb00Cb \u2286A. Accordingly, in order for both A and Ac to be implied by b we would need Cb = \u2205. \u25a1 Theorem 2. A belief function b : 2\u0398 \u2192[0, 1] has non-empty core if and only if there do not exist two complementary propositions A, Ac \u2286\u0398 which both enjoy non-zero support from b, b(A) \u0338= 0, b(Ac) \u0338= 0 (i.e., they are implied by b in the sense (3)). Proof. By De\ufb01nition 1, in order for a subset (or proposition, in a propositional logic interpretation) A \u2286\u0398 to have non-zero belief value it has to contain the core of b: A \u2287Cb. In order to have both b(A) \u0338= 0, b(Ac) \u0338= 0 we need both to contain the core, but in that case A \u2229Ac \u2287Cb \u0338= \u2205which is absurd as A \u2229Ac = \u2205. \u25a1 6 \f2.3. Achieving consistency in belief logic: consistent approximation Belief functions are complex objects, in which di\ufb00erent and sometimes contradictory bodies of evidence coexist, as they may result from the fusion of possibly con\ufb02icting expert opinions and/or measurements. It is reasonable to conjecture that taking belief functions at face value may lead in some situations to incorrect/inappropriate decisions. A variety of approaches have been proposed in the context of classical logics to solve the analogous problem with inconsistent knowledge bases: we mentioned some of them in the introduction [10, 11]. Paris\u2019 approach [9] is particularly interesting as it tackles the problem by attributing to each proposition in the knowledge base a certain degree of belief, leading to something similar to a belief function. As consistent belief functions represent consistent knowledge bases in belief logic, such a mechanism can be be described there as an operator cs : B \u2192CS, b 7\u2192cs[b] (4) where B and CS denote respectively the set of all belief functions, and that of all cs.b.f.s. Consistent transformations (4) can be built by posing a minimization problem of the form cs[b] = arg min cs\u2208CS dist(b, cs) where b is the belief function to approximate and dist is some distance measure between b.f.s. It is natural to pose this problem in a geometric setup. 2.4. Geometry of belief functions: belief and mass vectors As belief functions b : 2\u0398 \u2192[0, 1], b(A) = P B\u2286A mb(B) are set functions de\ufb01ned on a the power set 2\u0398 of a \ufb01nite space \u0398, they are obviously completely de\ufb01ned by the associate set of 2|\u0398|\u22122 belief values, that we can collect in a vector \u20d7 b = [b(A), \u2205\u228aA \u228a\u0398]\u2032 (since b(\u2205) = 0, b(\u0398) = 1 for all b.f.s). They can therefore be represented as points of RN\u22122, N = 2|\u0398| [21]. The set B of points of RN\u22122 which correspond to belief functions is an N-dimensional triangle or simplex called belief space [21], namely: B = Cl(\u20d7 bA, \u2205\u228aA \u2286\u0398), where Cl denotes the convex closure operator Cl(\u20d7 b1, ...,\u20d7 bk) = n \u20d7 b \u2208B : \u20d7 b = \u03b11\u20d7 b1 + \u00b7 \u00b7 \u00b7 + \u03b1k\u20d7 bk, X i \u03b1i = 1, \u03b1i \u22650 \u2200i o 7 \fand \u20d7 bA is the vector associated with the categorical [35] belief function bA assigning all the mass to a single subset A \u2286\u0398: mbA(A) = 1, mbA(B) = 0 for all B \u0338= A. The vector \u20d7 b \u2208B that corresponds to a belief function b has coordinates mb(A) in the simplex B : \u20d7 b = X \u2205\u228aA\u2286\u0398 mb(A)\u20d7 bA. In the same way, each b.f. is uniquely associated with the related set of mass values {mb(A), \u2205\u228aA \u2286\u0398} (\u0398 this time included). It can therefore be seen also as a point of RN\u22121, the vector \u20d7 mb of its N \u22121 mass components: \u20d7 mb = X \u2205\u228aA\u2286\u0398 mb(A)\u20d7 mA, (5) where \u20d7 mA is the vector of mass values associated with the categorical b.f. bA. The collection M of points which are valid basic probability assignments is also a simplex, which we can call mass space: M = Cl(\u20d7 mA, \u2205\u228aA \u2282\u0398). 2.4.1. Binary example As an example let us consider a frame of discernment formed by just two elements, \u03982 = {x, y}. Each b.f. b : 2\u03982 \u2192[0, 1] is completely determined by its mass/belief values mb(x) = b(x), mb(y) = b(y), as mb(\u0398) = 1 \u2212mb(x) \u2212 mb(y) and mb(\u2205) = 0. We can therefore collect them in a vector of RN\u22122 = R2 (since N = 22 = 4): \u20d7 mb = \u20d7 b = [mb(x), mb(y)]\u2032 = \u20d7 b \u2208R2. In this example mass space and belief space coincide. Since mb(x) \u22650, mb(y) \u22650, and mb(x) + mb(y) \u22641 we can easily infer that the set B2 = M2 of all the possible basic probability assignments (belief functions) on \u03982 can be depicted as the triangle in the Cartesian plane of Figure 1, whose vertices are the points \u20d7 b\u0398 = \u20d7 m\u0398 = [0, 0]\u2032, \u20d7 bx = \u20d7 mx = [1, 0]\u2032, \u20d7 by = \u20d7 my = [0, 1]\u2032, which correspond respectively to the vacuous belief function b\u0398 (mb\u0398(\u0398) = 1), the Bayesian b.f. bx with mbx(x) = 1, and the Bayesian b.f. by with mby(y) = 1. The region P2 of all Bayesian b.f.s on \u03982 is the diagonal line segment Cl(\u20d7 mx, \u20d7 my) = Cl(\u20d7 bx,\u20d7 by). In the binary case consistent belief functions can have as list of focal elements either {{x}, \u03982} or {{y}, \u03982}. Therefore the space of cs.b.f.s CS2 is the union of two line segments: CS2 = CSx \u222aCSy = Cl(\u20d7 m\u0398, \u20d7 mx) \u222aCl(\u20d7 m\u0398, \u20d7 my) = Cl(\u20d7 b\u0398,\u20d7 bx) \u222aCl(\u20d7 b\u0398,\u20d7 by). It is easy to recognize that CS2 can be synthetically 8 \fFigure 1: Both mass M2 and belief B2 spaces for a binary frame \u0398 = {x, y} coincide with the triangle of R2 whose vertices are the mass (belief) vectors associated with the categorical b.f.s focused on {x}, {y} and \u0398, respectively. Consistent b.f.s live in the union of the two segments CSx = Cl(\u20d7 m\u0398, \u20d7 mx) = Cl(\u20d7 b\u0398,\u20d7 bx) and CSy = Cl(\u20d7 m\u0398, \u20d7 my) = Cl(\u20d7 b\u0398,\u20d7 by). written in terms of the L\u221e(max) norm as CS2 = {b : min{b(x), b(y)} = 0} = {b : max{plb(x), plb(y)} = 1}. (6) 2.5. The consistent complex In the general case the geometry of consistent belief functions can be described by resorting to the notion of simplicial complex [36]. A simplicial complex is a collection \u03a3 of simplices of arbitrary dimensions possessing the following properties: 1. if a simplex belongs to \u03a3, then all its faces of any dimension belong to \u03a3; 2. the intersection of any two simplices is a face of both the intersecting simplices. It has been proven that [30, 37] the region CS of consistent belief functions in the belief space is a simplicial complex, the union CSB = [ x\u2208\u0398 Cl(\u20d7 bA, A \u220bx). 9 \fof a number of (maximal) simplices, each associated with a \u201cmaximal ultra\ufb01lter\u201d {A \u2287{x}}, x \u2208\u0398 of subsets of \u0398 (those containing a given element x). It is not di\ufb03cult to see that the same holds in the mass space, where the consistent complex is the union CSM = [ x\u2208\u0398 Cl(\u20d7 mA, A \u220bx) of maximal simplices Cl(\u20d7 mA, A \u220bx) formed by the mass vectors associated with belief functions whose core contains a particular element x of \u0398. 2.6. Using Lp norms The geometry of the binary case hints to a close relationship between consistent belief functions and Lp norms, in particular the L\u221eone (Equation (6)). It is easy to realize that this holds in general as, since the plausibility of all elements of their core is 1, plb(x) = P A\u2287{x} mb(A) = 1 \u2200x \u2208Cb, the region of consistent b.f.s can be expressed as CS = n b : max x\u2208\u0398 plb(x) = 1 o , i.e., the set of b.f.s for which the L\u221enorm of the \u201dcontour function\u201d plb(x) is equal to 1. This argument is strengthened by the observation that cs.b.f.s relate to possibility distributions, and possibility measures Pos are inherently related to L\u221eas Pos(A) = maxx\u2208A Pos(x). It makes therefore sense to conjecture that a consistent transformation obtained by picking as distance function in the approximation problem (1) one of the classical Lp norms maybe be meaningful. For vectors \u20d7 mb, \u20d7 mb\u2032 \u2208M representing the b.p.a.s of two belief functions b, b\u2032, such norms read as: \u2225\u20d7 mb \u2212\u20d7 mb\u2032\u2225L1 . = X \u2205\u228aB\u2286\u0398 |mb(B) \u2212mb\u2032(B)|; \u2225\u20d7 mb \u2212\u20d7 mb\u2032\u2225L2 . = s X \u2205\u228aB\u2286\u0398 (mb(B) \u2212mb\u2032(B))2; \u2225\u20d7 mb \u2212\u20d7 mb\u2032\u2225L\u221e . = max \u2205\u228aB\u2286\u0398 |mb(B) \u2212mb\u2032(B)|, (7) while the same norms in the belief space read as: \u2225\u20d7 b \u2212\u20d7 b\u2032\u2225L1 . = X \u2205\u228aB\u2286\u0398 |b(B) \u2212b\u2032(B)|; \u2225\u20d7 b \u2212\u20d7 b\u2032\u2225L2 . = s X \u2205\u228aB\u2286\u0398 (b(B) \u2212b\u2032(B))2; \u2225\u20d7 b \u2212\u20d7 b\u2032\u2225L\u221e. = max \u2205\u228aB\u2286\u0398 |b(B) \u2212b\u2032(B)|. (8) 10 \fIn recent times, Lp norms have been employed in di\ufb00erent problems such as probability [16] and possibility [28] transformation/approximation, or conditioning [23, 29]. In the probability transformation problem [13, 14, 38], p[b] = arg minp\u2208P dist(b, p), the use of Lp norms leads indeed to quite interesting results. The L2 approximation produces the so-called \u201corthogonal projection\u201d of b onto P [16]. In addition, the set of L1/L\u221eprobabilistic approximations of b coincide with the set of probabilities consistent with b: {p : p(A) \u2265b(A)} (at least in the binary case). Consonant approximations of belief functions obtained by minimizing Lp distances in the mass space have simple interpretations in terms of redistribution of the mass outside a desired chain A1 \u2282\u00b7 \u00b7 \u00b7 \u2282An, |Ai| = i of focal elements to a single element of the chain, or all in equal terms [28]. Conditional b.f.s can also be de\ufb01ned by minimizing Lp distances from a \u201cconditioning simplex\u201d BA = Cl(\u20d7 bB, \u2205\u228aB \u2286A) \u000e MA = Cl(\u20d7 mB, \u2205\u228aB \u2286A) in the belief/mass space determined by the conditioning event A [23]. In the mass space, the obtained conditional b.f.s have natural interpretation in terms of Lewis\u2019 \u201cgeneral imaging\u201d [39, 40] applied to belief functions. Figure 2: Left: To minimize the distance of a point from a simplicial complex, we need to \ufb01nd all the partial solutions (9) on all the maximal simplices in the complex (empty circles), to later compare these partial solutions and select a global optimum (black circle). 2.7. Consistent approximation on the complex As the consistent complex CS is a collection of linear spaces (better, simplices which generate a linear space), solving the problem (1) involves 11 \f\ufb01nding a number of partial solutions in the belief/mass space csx B,Lp[b] = arg min \u20d7 cs\u2208CSx B \u2225\u20d7 b \u2212\u20d7 cs\u2225Lp, csx M,Lp[mb] = arg min \u20d7 mcs\u2208CSx M \u2225\u20d7 m \u2212\u20d7 mcs\u2225Lp, (9) respectively (see Figure 2-left). Then, the distance of b from all such partial solutions has to be assessed in order to select a global approximation. 3. Calculation: consistent approximation in M Let us compute the analytical form of all Lp consistent approximations in the mass space. We start by describing the di\ufb00erence vector \u20d7 mb \u2212\u20d7 mcs between the original mass vector and its approximation. Using the notation \u20d7 mcs = X B\u2287{x},B\u0338=\u0398 mcs(B)\u20d7 mB, \u20d7 mb = X B\u228a\u0398 mb(B)\u20d7 mB (as in RN\u22122 mb(\u0398) is not included by normalization) the di\ufb00erence vector is \u20d7 mb \u2212\u20d7 mcs = X B\u2287{x},B\u0338=\u0398 (mb(B) \u2212mcs(B))\u20d7 mB + X B\u0338\u2283{x} mb(B)\u20d7 mB (10) so that its classical Lp norms read as \u2225\u20d7 mb \u2212\u20d7 mcs\u2225M L1 = X B\u2287{x},B\u0338=\u0398 |mb(B) \u2212mcs(B)| + X B\u0338\u2283{x} |mb(B)|, \u2225\u20d7 mb \u2212\u20d7 mcs\u2225M L2 = s X B\u2287{x},B\u0338=\u0398 |mb(B) \u2212mcs(B)|2 + X B\u0338\u2283{x} |mb(B)|2, \u2225\u20d7 mb \u2212\u20d7 mcs\u2225M L\u221e = max n max B\u2287{x},B\u0338=\u0398 |mb(B) \u2212mcs(B)|, max B\u0338\u2283{x} |mb(B)| o . (11) 3.1. L1 approximation Let us tackle \ufb01rst the L1 case. After introducing the variables \u03b2(B) . = mb(B) \u2212mcs(B) we can write the L1 norm of the di\ufb00erence vector as \u2225\u20d7 mb \u2212\u20d7 mcs\u2225M L1 = X B\u2287{x},B\u0338=\u0398 |\u03b2(B)| + X B\u0338\u2283{x} |mb(B)|, (12) which is obviously minimized by \u03b2(B) = 0, for all B \u2287{x}, B \u0338= \u0398. Hence: 12 \fTheorem 3. Given an arbitrary belief function b : 2\u0398 \u2192[0, 1] and an element x \u2208\u0398 of the frame, its unique partial L1 consonant approximation csx M,L1[mb] in M with core containing x is the consonant b.f. whose mass distribution coincides with that of b on all the subsets containing x: mcsx M,L1[mb](B) = \u001a mb(B) \u2200B \u2287{x}, B \u0338= \u0398 mb(\u0398) + b({x}c) B = \u0398. (13) The mass value for B = \u0398 comes from normalization, as follows: mcsx M,L1[mb](\u0398) = 1 \u2212 X B\u2287{x},B\u0338=\u0398 mcsx M,L1[mb](B) = mb(\u0398) + b({x}c). The mass of all the subsets not in the desired \u201cprincipal ultra\ufb01lter\u201d {B \u2287 {x}} is simply re-assigned to \u0398. A similarity emerges with the case of L1 conditional belief functions [23], when we recall that the set of L1 conditional belief functions bL1,M(.|A) with respect to A in M is the simplex whose vertices are each associated with a subset \u2205\u228aB \u2286A of the conditional event A, and have b.p.a. (compare Equation (13)): \u001a m\u2032(B) = mb(B) + 1 \u2212b(A), m\u2032(X) = mb(X) \u2200\u2205\u228aX \u228aA, X \u0338= B. In the L1 conditional case, each vertex of the set of solutions is obtained by re-assigning the mass not in the conditional event A to a single subset of A, just as in L1 consistent approximation all the mass not in the principal ultra\ufb01lter {B \u2287{x}} is re-assigned to the top of the ultra\ufb01lter, \u0398. 3.1.1. Global approximation The global L1 consistent approximation in M coincides with the partial approximation (13) at minimal distance from the original mass vector \u20d7 mb. By (12) the partial approximation focussed on x has distance b({x}c) = P B\u0338\u2283{x} mb(B) from \u20d7 mb. The global L1 approximation(s) form therefore the union of the partial approximation(s) associated with the maximal plausibility singleton(s): csL1,M[mb] = [ arg minx b(xc)=arg maxx plb(x) csx M,L1[mb]. (14) This is in accordance with our intuition, as it makes sense to focus on the singletons which are supported by the strongest evidence. 13 \f3.1.2. A running example Consider as an example the belief function b with mass assignment mb(x) = 0.2, mb(y) = 0.1, mb(z) = 0, mb(x, y) = 0.4, mb(x, z) = 0, mb(y, z) = 0.3, mb(\u0398) = 0 (15) on the ternary frame \u0398 = {x, y, z}. Suppose that we seek the L1 approximation of b with core {x}. By Equation (13) we get the following consonant b.f.: m\u2032 b(x) = 0.2, m\u2032 b(x, y) = 0.4, m\u2032 b(x, z) = 0, m\u2032 b(\u0398) = 0.4. If, instead, we focus on {y} or {z} we get, respectively (see Figure 3): m\u2032 b(y) = 0.1, m\u2032 b(x, y) = 0.4, m\u2032 b(y, z) = 0.3, m\u2032 b(\u0398) = 0.2; m\u2032 b(z) = 0, m\u2032 b(x, z) = 0, m\u2032 b(y, z) = 0.3, m\u2032 b(\u0398) = 0.7. Note that the partial approximation on CSz has {y, z} as actual core, and Figure 3: (a) The belief function (15); (b) its partial L1 consistent approximation in M focussed on {x}; (c) partial approximation on {y}; (d) partial approximation on {z}. that (b) and (d) are actually consonant (a special case of consistent b.f.), stressing the close relation between consonant and consistent approximation. Since plb(x) = 0.6, plb(y) = 0.8, plb(z) = 0.3, by (14) the global L1 approximation of (15) is the partial approximation with core containing {y}. We can notice how the global approximation is visually much closer to the original b.f. than the partial ones, in terms of the structure of its focal elements. 3.2. L\u221eapproximation In the L\u221ecase \u2225\u20d7 mb \u2212\u20d7 mcs\u2225M L\u221e= max n max B\u2287{x},B\u0338=\u0398 |\u03b2(B)|, max C\u0338\u2283{x} mb(C) o . The L\u221enorm of the di\ufb00erence vector is obviously minimized by {\u03b2(B)} such that: |\u03b2(B)| \u2264maxC\u0338\u2283{x} mb(C) for all B \u2287{x}, B \u0338= \u0398, i.e., \u2212max C\u0338\u2283{x} mb(C) \u2264mb(B) \u2212mcs(B) \u2264max C\u0338\u2283{x} mb(C) \u2200B \u2287{x}, B \u0338= \u0398. 14 \fTheorem 4. Given an arbitrary belief function b : 2\u0398 \u2192[0, 1] and an element x \u2208\u0398 of the frame, its partial L\u221econsistent approximations csx M,L\u221e[mb] with core containing x in M are those whose mass values on all the subsets containing x di\ufb00er from the original ones by the maximum mass of the subsets not in the ultra\ufb01lter: for all B \u2283{x}, B \u0338= \u0398 mb(B) \u2212max C\u0338\u2283{x} mb(C) \u2264mcsx M,L\u221e[mb](B) \u2264mb(B) + max C\u0338\u2283{x} mb(C). (16) Clearly this set of solutions can also include pseudo belief functions. Also, a comparison of Equations (16) and (13) shows that the barycenter of the partial L\u221eapproximations coincides with the partial L1 approximation, reassigning all the mass outside the ultra\ufb01lter to the whole frame. 3.2.1. Global approximation Once again, the global L\u221econsistent approximation in M coincides with the partial approximation (16) at minimal distance from the original b.p.a. \u20d7 mb. The partial approximation focussed on x has distance maxC\u0338\u2283{x} mb(C) from \u20d7 mb. The global L\u221eapproximation is therefore the (union of the) partial approximation(s) associated with the singleton(s) which minimize the maximal mass outside the ultra\ufb01lter: csM,L\u221e[mb] = [ arg minx maxC\u0338\u2283{x} mb(C) csx M,L\u221e[mb]. (17) Global L\u221esolutions are not totally unrelated to their L1 peers, as maximizing the plausibility of the core as in (14) involves minimizing the total mass outside the ultra\ufb01lter. 3.2.2. Running example For the b.f. (15) of our running example, the maximal mass outside the ultra\ufb01lter in x is max C\u0338\u2283x mb(C) = max \b mb(y), mb(z), mb(y, z) \t = mb(y, z) = 0.3, so that the set of L\u221econsistent approximations with core containing {x} is such that \uf8f1 \uf8f2 \uf8f3 mb(x) \u2212mb(y, z) \u2264m\u2032 b(x) \u2264mb(x) + mb(y, z) mb(x, y) \u2212mb(y, z) \u2264m\u2032 b(x, y) \u2264mb(x, y) + mb(y, z) mb(x, z) \u2212mb(y, z) \u2264m\u2032 b(x, z) \u2264mb(x, z) + mb(y, z) \u2261 \uf8f1 \uf8f2 \uf8f3 \u22120.1 \u2264m\u2032 b(x) \u22640.5 0.1 \u2264m\u2032 b(x, y) \u22640.7 \u22120.3 \u2264m\u2032 b(x, z) \u22640.3. (18) 15 \fThis set is clearly not entirely admissible. The same can be veri\ufb01ed for partial approximations with cores containing {y} and {z}, for which max C\u0338\u2283y mb(C) = mb(x) = 0.2, max C\u0338\u2283z mb(C) = mb(x, y) = 0.4. Therefore, the global L\u221eapproximations of (15) are the partial ones whose core contains y (as it was the case for the global L1 approximation). 3.3. L2 approximation In order to \ufb01nd the L2 consistent approximation(s) in M it is convenient to recall that the minimal L2 distance between a point and a vector space is attained by the point of the vector space V such that the di\ufb00erence vector is orthogonal to all the generators \u20d7 gi of V : arg min \u20d7 q\u2208V \u2225\u20d7 p \u2212\u20d7 q\u22252 = \u02c6 q \u2208V : \u27e8\u20d7 p \u2212\u02c6 q,\u20d7 gi\u27e9= 0 \u2200i whenever \u20d7 p \u2208Rm, V = span(\u20d7 gi, i). Instead of minimizing the L2 norm of the di\ufb00erence vector \u2225\u20d7 mb \u2212\u20d7 mcs\u2225M L2 we can just impose the orthogonality of the di\ufb00erence vector itself \u20d7 mb \u2212\u20d7 mcs and the subspace CSx M associated with consistent mass functions focused on {x}. Clearly the generators of such linear space are the vectors in M: \u20d7 mB \u2212\u20d7 m{x}, for all B \u228b{x}. Theorem 5. Consider an arbitrary belief function b : 2\u0398 \u2192[0, 1] and an element x \u2208\u0398 of the frame. When using the (N \u22122)-dimensional representation of mass vectors (5), its unique L2 partial consistent approximation in M with core containing x coincides with its partial L1 approximation: csx M,L2[mb] = csx M,L1[mb]. When representing belief functions as mass vectors \u20d7 mb of RN\u22121 (B = \u0398 included) \u20d7 mb = X \u2205\u228aB\u2286\u0398 mb(B)\u20d7 mB (19) the partial L2 approximation of b is obtained by equally redistributing to each element of the ultra\ufb01lter {B \u2287{x}} an equal fraction of the mass of focal elements originally not in it: mcsx M,L2[mb](B) = mb(B) + b({x}c) 2|\u0398|\u22121 \u2200B \u2287{x}. (20) The partial L2 approximation in RN\u22121 redistributes the mass equally to all the elements of the ultra\ufb01lter. 16 \f3.3.1. Global approximation The global L2 consistent approximation in M is, as usual, given by the partial approximation (20) at minimal L2 distance from \u20d7 mb. In the N \u22122 representation, by de\ufb01nition of L2 norm in M (11), the partial approximation focussed on x has distance from \u20d7 mb (b(xc))2 + X B\u0338\u2283{x} (mb(B))2 = \u0010 X B\u0338\u2283{x} mb(B) \u00112 + X B\u0338\u2283{x} (mb(B))2, which is minimized by the singleton(s) arg min x X B\u0338\u2283{x} (mb(B))2. Therefore: csM,L2[mb] = [ arg minx P B\u0338\u2283{x}(mb(B))2 csx M,L2[mb]. In the N \u22121-dimensional representation, instead, X B\u2287{x},B\u0338=\u0398 h mb(B) \u2212 \u0010 mb(B) + b(xc) 2|\u0398|\u22121 \u0011i2 + X B\u0338\u2283{x} (mb(B))2 = X B\u2287{x},B\u0338=\u0398 \u0010 b(xc) 2|\u0398|\u22121 \u00112 + X B\u0338\u2283{x} (mb(B))2 = (P B\u0338\u2283{x} mb(B))2 2|\u0398|\u22121 + X B\u0338\u2283{x} (mb(B))2 (21) which is minimized by the same singleton(s). In any case, even though (in the N \u22122 representation) the partial L1 and L2 approximations coincide, the global approximations in general may fall on di\ufb00erent components of the consonant complex. 3.3.2. Running example For the usual example b.f. (15), by Equation (20) the L2 partial approximations with cores containing x, y and x respectively have mass assignments: m\u2032 b(x) = 0.3, m\u2032 b(x, y) = 0.5, m\u2032 b(x, z) = 0.1, m\u2032 b(\u0398) = 0.1 m\u2032 b(y) = 0.15, m\u2032 b(x, y) = 0.45, m\u2032 b(y, z) = 0.35, m\u2032 b(\u0398) = 0.05; m\u2032 b(z) = 0.175, m\u2032 b(x, z) = 0.175, m\u2032 b(y, z) = 0.475, m\u2032 b(\u0398) = 0.175, since b(xc) = 1\u2212plb(x) = 0.4, b(xc) = 1\u2212plb(x) = 0.4, b(xc) = 1\u2212plb(x) = 0.4 and 2|\u0398|\u22121 = 4 (see Figure 4). Note that, unlike the corresponding L1 approximation, the partial L2 approximation on {z} has indeed the singleton as core. Also, as L2 approximation redistributes mass evenly to all the focal elements in the desired 17 \fFigure 4: (a) The b.f. (15); (b) its partial L2 consistent approximation in M, focussed on {x}; (c) L2 partial approximation on {y}; (d) L2 partial approximation on {z}. ultra\ufb01lter, all partial approximations contain all possible f.e.s. As for the global solution (b(xc))2 = 0.16, X B\u0338\u2283x (mb(B))2 = 0.12 + 0.32 = 0.01 + 0.09 = 0.1, (b(yc))2 = 0.04, X B\u0338\u2283y (mb(B))2 = 0.22 = 0.04, (b(zc))2 = 0.49, X B\u0338\u2283z (mb(B))2 = 0.22 + 0.12 + 0.42 = 0.04 + 0.01 + 0.16 = 0.21, and the L2 norm (21) is minimized, once again, by the partial solution on {y} (as visually con\ufb01rmed by Figure 4). 4. Calculation: approximation in the belief space 4.1. L1/L2 approximations We have seen that in the mass space (at least in its N \u22122 representation, Theorem 5) the L1 and L2 approximations coincide. This is true in the belief space in the general case as well. We will gather some intuition on the general solution by considering \ufb01rst the slightly more complex case of a ternary frame: \u0398 = {x, y, z}. In this Section we will use the notation: \u20d7 cs = X B\u2287{x} mcs(B)\u20d7 bB, \u20d7 b = X B\u228a\u0398 mb(B)\u20d7 bB. In the case of an arbitrary frame a cs.b.f. cs \u2208CSx B is a solution of the L2 approximation problem if, again, \u20d7 b \u2212\u20d7 cs is orthogonal to all generators {\u20d7 bB \u2212\u20d7 b\u0398 = \u20d7 bB, {x} \u2286B \u228a\u0398} of CSx B: \u27e8\u20d7 b \u2212\u20d7 cs,\u20d7 bB\u27e9= 0 \u2200B : {x} \u2286B \u228a\u0398. 18 \fAs \u20d7 b \u2212\u20d7 cs = P A\u228a\u0398(mb(A) \u2212mcs(A))\u20d7 bA we get \uf8f1 \uf8f2 \uf8f3 X A\u2287{x} \u03b2(A)\u27e8\u20d7 bA,\u20d7 bB\u27e9+ X A\u0338\u2283{x} mb(A)\u27e8\u20d7 bA,\u20d7 bB\u27e9= 0 \u2200B : {x} \u2286B \u228a\u0398, (22) where once again \u03b2(A) = mb(A) \u2212mcs(A). In the L1 case, the minimization problem is (using again the notation \u20d7 cs = P B\u2287{x} mcs(B)\u20d7 bB) arg min mcs(.) \u001a X B\u2287{x} \f \f \f \f X A\u2286B mb(A) \u2212 X A\u2286B,A\u2287{x} mcs(A) \f \f \f \f \u001b = arg min \u03b2(.) \u001a X B\u2287{x} \f \f \f \f X A\u2286B,A\u2287{x} \u03b2(A) + X A\u2286B,A\u0338\u2283{x} mb(A) \f \f \f \f \u001b which is clearly solved by setting all addenda to zero, obtaining \uf8f1 \uf8f2 \uf8f3 X A\u2286B,A\u2287{x} \u03b2(A) + X A\u2286B,A\u0338\u2283{x} mb(A) = 0 \u2200B : {x} \u2286B \u228a\u0398. (23) An interesting fact emerges when comparing the linear systems for L1 and L2 in the ternary case \u0398 = {x, y, x}: \uf8f1 \uf8f2 \uf8f3 3\u03b2(x) + \u03b2(x, y) + \u03b2(x, z) + mb(y) + mb(z) = 0 \u03b2(x) + \u03b2(x, y) + mb(y) = 0 \u03b2(x) + \u03b2(x, z) + mb(z) = 0 \uf8f1 \uf8f2 \uf8f3 \u03b2(x) = 0 \u03b2(x) + \u03b2(x, y) + mb(y) = 0 \u03b2(x) + \u03b2(x, z) + mb(z) = 0. (24) The solution is the same for both, due to the fact that the second linear system is obtained from the \ufb01rst one by a linear transformation of rows. We just need to replace the \ufb01rst equation e1 in the \ufb01rst system with the di\ufb00erence: e1 7\u2192e1 \u2212e2 \u2212e3. This holds in the general case, too. Lemma 1. X C\u2287B \u27e8\u20d7 bC,\u20d7 bA\u27e9(\u22121)|C\\B| = 1 whenever A \u2286B, 0 otherwise. 19 \fCorollary 1. The linear system (22) can be reduced to the system (23) through the following linear transformation of rows: rowB 7\u2192 X C\u2287B rowC(\u22121)|C\\B|. (25) Proof. If we apply the linear transformation (25) to the system (22) we get X C\u2287B \u0014 X A\u2287{x} \u03b2(A)\u27e8\u20d7 bA,\u20d7 bC\u27e9+ X A\u0338\u2283{x} mb(A)\u27e8\u20d7 bA,\u20d7 bC\u27e9 \u0015 (\u22121)|C\\B| = X A\u2287{x} \u03b2(A) X C\u2287B \u27e8\u20d7 bA,\u20d7 bC\u27e9(\u22121)|C\\B| + X A\u0338\u2283{x} mb(A) X C\u2287B \u27e8\u20d7 bA,\u20d7 bC\u27e9(\u22121)|C\\B| \u2200B : {x} \u2286B \u228a\u0398. Therefore by Lemma 1 we get \u001a X A\u2287{x},A\u2286B \u03b2(A) + X A\u0338\u2283{x},A\u2286B mb(A) = 0 \u2200B : {x} \u2286B \u228a\u0398, i.e., the system of equations (23). \u25a1 4.1.1. Form of the solution To obtain both the L2 and the L1 consistent approximations of b it then su\ufb03ces to solve the system (23) associated with the L1 norm. Theorem 6. The unique solution of the linear system (23) is: \u03b2(A) = \u2212mb(A \\ {x}) for all A : {x} \u2286A \u228a\u0398. We can prove it by simple substitution, as system (23) becomes: \u2212 X B\u2286A,B\u2287{x} mb(B \\ {x}) + X B\u2286A,B\u0338\u2283{x} mb(B) = \u2212 X C\u2286A\\{x} mb(C) + X B\u2286A,B\u0338\u2283{x} mb(B) = 0. Therefore, according to what discussed in Section 2.7, the partial consistent approximations of b on the maximal component CSx B of the consistent complex have b.p.a. mcsx B,L1[b](A) = mcsx B,L2[b](A) = mb(A) \u2212\u03b2(A) = mb(A) + mb(A \\ {x}) 20 \ffor all events A such that {x} \u2286A \u228a\u0398. As for the mass of \u0398, by normalization: mcsx B,L1/L2[b](\u0398) = = 1 \u2212 X {x}\u2286A\u228a\u0398 mcsx B,L1/L2[b](A) = 1 \u2212 X {x}\u2286A\u228a\u0398 \u0010 mb(A) + mb(A \\ {x}) \u0011 = 1 \u2212 X {x}\u2286A\u228a\u0398 mb(A) \u2212 X {x}\u2286A\u228a\u0398 mb(A \\ {x}) = 1 \u2212 X A\u0338=\u0398,{x}c mb(A) = mb({x}c) + mb(\u0398) as all events B \u0338\u2283{x} can be written as B = A \\ {x} for A = B \u222a{x}. Corollary 2. mcsx B,L1[b](A) = mcsx B,L2[b](A) = mb(A) + mb(A \\ {x}) \u2200x \u2208\u0398, and for all A s.t. {x} \u2286A \u2286\u0398. 4.1.2. Global L1 and L2 approximations To \ufb01nd the consistent approximation of b we need to work out which of the partial approximations csx L1/2[b] has minimal distance from b. Theorem 7. Given an arbitrary belief function b : 2\u0398 \u2192[0, 1], its global L1 consistent approximation in the belief space is its partial approximation associated with the singleton: arg min x n X A\u2286{x}c b(A), x \u2208\u0398 o . (26) In the binary case (\u0398 = {x, y}) the optimal singleton cores (26) simplify as: arg min x X A\u2286{x}c b(A) = arg min x mb({x}c) = arg max x plb(x), and the global approximation falls on the component of the consistent complex associated with the element of maximal plausibility. Unfortunately, in the case of an arbitrary frame \u0398 the element (26) is not necessarily the maximal plausibility element: arg min n X A\u2286{x}c b(A), x \u2208\u0398 o \u0338= arg max{plb(x), x \u2208\u0398 \t , as a simple counterexample can prove. As for the L2 case: 21 \fTheorem 8. Given an arbitrary belief function b : 2\u0398 \u2192[0, 1], its global L2 consistent approximation in the belief space is its partial approximation associated with the singleton: arg min x n X A\u2286{x}c \u0000b(A) \u00012, x \u2208\u0398 o . (27) Once again, in the binary case the optimal singleton cores (27) specialize as: arg min x X A\u2286{x}c (b(A))2 = arg min x (mb({x}c))2 = arg max x plb(x) and the global approximation for L2 also falls on the component of the consistent complex associated with the element of maximal plausibility, while this is not generally true for an arbitrary frame. 4.1.3. Running example According to Equation (33), for the belief function (15) of our running example the partial L1/L2 consistent approximations in B with core containing {x}, {y} and {z}, respectively, have mass assignments (see Figure 5): m\u2032 b(x) = 0.2, m\u2032 b(x, y) = 0.5, m\u2032 b(x, z) = 0, m\u2032 b(\u0398) = 0.3; m\u2032 b(y) = 0.1, m\u2032 b(x, y) = 0.6, m\u2032 b(y, z) = 0.3, m\u2032 b(\u0398) = 0; m\u2032 b(z) = 0, m\u2032 b(x, z) = 0.2, m\u2032 b(y, z) = 0.4, m\u2032 b(\u0398) = 0.4, (28) We can note that kind of approximation tends to penalize the smallest, Figure 5: (a) The belief function (15); (b) its partial L1/L2 consistent approximation in B, focussed on {x}; (c) partial L1/L2 partial approximation on {y}; (d) partial L1/L2 partial approximation on {z}. singleton element of the ultra\ufb01lter {B \u2287x}: this is due to the fact that m\u2032 b(x) = mb(x), while the masses of all the other elements of the \ufb01lter are 22 \fincreased. As for global approximations we have, for L1: P A\u2286{x}c b(A) = b(y) + b(z) + b(y, z) = 0.1 + 0 + 0.4 = 0.5, P A\u2286{y}c b(A) = b(x) + b(z) + b(x, z) = 0.2 + 0 + 0.2 = 0.4, P A\u2286{z}c b(A) = b(x) + b(y) + b(x, y) = 0.2 + 0.1 + 0.7 = 1, which is minimized by the partial solution in y. For L2: P A\u2286{x}c \u0000b(A) \u00012 = 0.01 + 0.16 = 0.17, P A\u2286{y}c \u0000b(A) \u00012 = 0.04 + 0.04 = 0.08, P A\u2286{z}c \u0000b(A) \u00012 = 0.04 + 0.01 + 0.49 = 0.54, which is also minimized by the same partial solution. This is not necessarily the case, in general. 4.2. L\u221econsistent approximation While the partial L1and L2-consistent approximations in the belief space are pointwise and coincide, the set of partial L\u221e-approximations for each component CSx B of the consistent complex form a polytope whose center of mass is exactly equal to csx B,L1[b] = csx B,L2[b]. 4.2.1. General solution By applying the expression (8) of the L\u221enorm to the di\ufb00erence vector \u20d7 b\u2212\u20d7 cs in the belief space, where cs is a consistent b.f. with core containing x, we obtain maxA\u228a\u0398 \b\f \f P B\u2286A mb(B) \u2212P B\u2286A,B\u2287{x} mcs(B) \f \f\t . Therefore, the partial L\u221econsistent approximation with core containing x is: csx B,L\u221e[b] = arg min mcs(.) max A\u228a\u0398 \u001a\f \f \f \f X B\u2286A mb(B) \u2212 X B\u2286A,B\u2287{x} mcs(B) \f \f \f \f \u001b . Again, maxA\u228a\u0398 has as lower limit the value associated with the largest norm which does not depend on mcs(.), i.e., max A\u228a\u0398 \u001a\f \f \f \f X B\u2286A mb(B) \u2212 X B\u2286A,B\u2287{x} mcs(B) \f \f \f \f \u001b \u2265b({x}c) or equivalently max A\u228a\u0398 \u001a\f \f \f \f X B\u2286A,B\u2287{x} \u03b2(B) + X B\u2286A,B\u0338\u2283{x} mb(B) \f \f \f \f \u001b \u2265b({x}c). In the above constraint only the expressions associated with B \u2287{x} contain 23 \fvariable terms. Therefore the desired optimal values of the variables {\u03b2(B)} are such that: \uf8f1 \uf8f2 \uf8f3 \f \f \f \f X B\u2286A,B\u2287{x} \u03b2(B) + X B\u2286A,B\u0338\u2283{x} mb(B) \f \f \f \f \u2264b({x}c) \u2200A : {x} \u2286A \u228a\u0398. (29) After introducing the change of variables \u03b3(A) . = X B\u2286A,B\u2287{x} \u03b2(B) (30) system (29) trivially reduces to n\f \f \f\u03b3(A) + X B\u2286A,B\u0338\u2283{x} mb(B) \f \f \f \u2264b({x}c) for all A such that {x} \u2286A \u228a\u0398, whose solution \u2212b(xc) \u2212 X B\u2286A,B\u0338\u2283{x} mb(B) \u2264\u03b3(A) \u2264b(xc) \u2212 X B\u2286A,B\u0338\u2283{x} mb(B) (31) de\ufb01nes a high-dimensional \u201crectangle\u201d in the space of the solutions {\u03b3(A), {x} \u2286 A \u228a\u0398}. In the mass assignment mcs(.) of the desired approximations, as we can clearly see in the running ternary example, the solution set is polytope whose vertices do not appear to have straightforward interpretations. On the other hand, the barycenter of this polytope is easy to compute and interpret. 4.2.2. Barycenter of the L\u221esolution and global approximation The center of mass of the set of solutions (31) to the L\u221econsistent approximation problem is clearly \u03b3(A) = \u2212 X B\u2286A,B\u0338\u2283{x} mb(B), {x} \u2286A \u228a\u0398, which reads in the space of the variables {\u03b2(A), {x} \u2286A \u228a\u0398} as \u001a X B\u2286A,B\u2287{x} \u03b2(B) = \u2212 X B\u2286A,B\u0338\u2283{x} mb(B), {x} \u2286A \u228a\u0398. But this is exactly the linear system (23) which determines the L1/L2 consistent approximation csx B,L1/2[b] of b onto CSx B. Besides, the L\u221edistance between b and CSx B is minimal for the element x which minimizes \u2225\u20d7 b \u2212\u20d7 csx B,L\u221e\u2225\u221e= b({x}c). In conclusion, 24 \fTheorem 9. Given a belief function b : 2\u0398 \u2192[0, 1], and an element of its frame x \u2208\u0398, its partial L1/L2 approximation onto any given component CSx B of the consistent complex CSB in the belief space is also the geometric barycenter of the set its L\u221econsistent approximations on the same component. Its global L\u221econsistent approximations in B form the union of the partial L\u221e approximations associated with the maximal plausibility element(s) x \u2208\u0398. 4.3. Running example For the usual belief function (15) on the ternary frame, the set of partial L\u221esolutions (31) focussed on x becomes: b(x) \u2212b(xc) \u2264 mcs(x) \u2264b(x) + b(xc), b(x, y) \u2212b(xc) \u2264 mcs(x) + mcs(x, y) \u2264b(x, y) + b(xc), b(x, z) \u2212b(xc) \u2264 mcs(x) + mcs(x, z) \u2264b(x, z) + b(xc), whose 23 = 8 vertices (represented as vectors [mcs(x), mcs(x, y), mcs(x, z), mcs(\u0398)]\u2032) \u0002 b(x) \u2212b(xc), b(x, y) \u2212b(x), b(x, z) \u2212b(x), 1 + b(x) + b(xc) \u2212b(x, y) \u2212b(x, z) \u0003 , \u0002 b(x) \u2212b(xc), b(x, y) \u2212b(x), b(x, z) \u2212b(x) + 2b(xc), 1 + b(x) \u2212b(x, y) \u2212b(x, z) \u2212b(xc) \u0003 , \u0002 b(x) \u2212b(xc), b(x, y) \u2212b(x) + 2b(xc), b(x, z) \u2212b(x), 1 + b(x) \u2212b(x, y) \u2212b(x, z) \u2212b(xc) \u0003 , \u0002 b(x) \u2212b(xc), b(x, y) \u2212b(x) + 2b(xc), b(x, z) \u2212b(x) + 2b(xc), 1 + b(x) \u2212b(x, y) \u2212b(x, z) \u22123b(xc) \u0003 , \u0002 b(x) + b(xc), b(x, y) \u2212b(x) \u22122b(xc), b(x, z) \u2212b(x) \u22122b(xc), 1 + b(x) \u2212b(x, y) \u2212b(x, z) + 3b(xc) \u0003 , \u0002 b(x) + b(xc), b(x, y) \u2212b(x) \u22122b(xc), b(x, z) \u2212b(x), 1 + b(x) \u2212b(x, y) \u2212b(x, z) + b(xc) \u0003 , \u0002 b(x) + b(xc), b(x, y) \u2212b(x), b(x, z) \u2212b(x) \u22122b(xc), 1 + b(x) \u2212b(x, y) \u2212b(x, z) + b(xc) \u0003 , \u0002 b(x) + b(xc), b(x, y) \u2212b(x), b(x, z) \u2212b(x), 1 + b(x) \u2212b(x, y) \u2212b(x, z) \u2212b(xc) \u0003 , do not appear to be particularly meaningful. Their barycenter, instead, \u0002 b(x), b(x, y) \u2212b(x), b(x, z) \u2212b(x), 1 + b(x) \u2212b(x, y) \u2212b(x, z) \u0003 = \u0002 mb(x), mb(x, y) + mb(y), mb(x, z) + mb(z), mb(y, z) + mb(\u0398) \u0003 (32) clearly coincides with the focussed consistent approximation on x. The same holds for the barycenters of the L\u221eapproximations focussed on y and z (see Equation (28)). 25 \f5. Discussion Let us conclude by discussing the interpretation of the consistent approximations derives in the mass and belief space, and by comparing their features and geometry, with the help of the usual ternary example. 5.1. Approximation in the mass space and general imaging in belief revision As it is the case for geometric conditioning in the mass space [23], consistent approximations in the mass space can be interpreted as a generalization of Lewis\u2019 imaging approach to belief revision, originally formulated in the context of probabilities [39]. The idea behind imaging is that, upon observing that some state x \u2208\u0398 is impossible, you transfer the probability initially assigned to x completely towards the remaining state you deem the most similar to x [41]. Peter G\u00a8 ardenfors [40] extended Lewis\u2019 idea by allowing a fraction \u03bbi of the probability of such state x to be re-distributed to all remaining states xi (P i \u03bbi = 1). In the case of partial consistent approximation of belief functions, the mass mb(A) of each focal element not in the desired ultra\ufb01lter {A \u2287{x}} should be re-assigned to the \u201cclosest\u201d focal element in the latter. The partial L1 consistent approximation in M amounts to admitting ignorance about the closeness of focal elements, and re-assigning all the mass outside the \ufb01lter to the whole frame, therefore increasing the uncertainty of the state. If such ignorance is expressed by assigning instead equal weight \u03bb(A) to each A \u2208C, the resulting partial consistent approximation is the unique partial L2 approximation, the barycenter of the polytope of L1 partial approximations. 5.2. Partial L1/L2 approximations as focused consistent transformations The expression of the the basic probability assignment of the L1/L2 consistent approximations of b (Corollary 2) is simple and elegant. It also has a straightforward interpretation: to get a consistent belief function focused on a singleton x, the mass contribution of all events B such that B \u222a{x} = A coincide is assigned indeed to A. But there are just two such events: A itself, and A \\ {x}. The partial consistent approximation of a belief function on a frame \u0398 = {x, y, z, w} with core {x} is illustrated in Figure 6. The b.f. with focal elements {y}, {y, z}, and {x, z, w} is transformed by the map {y} 7\u2192 {x} \u222a{y} = {x, y}, {y, z} 7\u2192 {x} \u222a{y, z} = {x, y, z}, {x, z, w} 7\u2192 {x} \u222a{x, z, w} = {x, z, w} 26 \fFigure 6: A belief function (a) and its L1/2 consistent approximation in B with core {x} (b). into the consistent b.f. with focal elements {x, y}, {x, y, z}, and {x, z, w} and the same b.p.a. The partial solutions to the L1/L2 consistent approximation problem turn out to be related to the classical inner consonant approximations of a b.f. b, i.e., the set of consonant belief functions co such that co(A) \u2265b(A) \u2200A \u2286\u0398 (or equivalently plco(A) \u2264plb(A) \u2200A). Dubois and Prade [17] proved indeed that such an approximation exists i\ufb00b is consistent. However, when b is not consistent a \u201cfocused consistent transformation\u201d can be applied to get a new belief function b\u2032 such that mb\u2032(A \u222axi) = mb(A) \u2200A \u2286\u0398 (33) and xi is the element of \u0398 with highest plausibility. Clearly then, the results of Theorem and Corollary say that the L1/L2 consistent approximation onto each component CSx B of CSB generates the consistent transformation focused on x. 5.3. Improper solutions and unnormalized belief functions In some cases, improper partial solutions (in the sense that they potentially include negative mass assignments) can be generated by the Lp minimization process. This situation is not entirely new. For instance, outer consonant approximations [17] also include in\ufb01nitely many improper solutions: nevertheless, only the subset of acceptable solutions is retained. As in the consonant case [28], the set of all (admissible and not) solutions is typically much simpler to describe geometrically, in terms of simplices or polytopes. Computing the set of proper approximations in all cases requires signi\ufb01cant further e\ufb00ort, which for reasons of clarity and length we reserve for the near future. 27 \fAdditionally, in this work only \u201cnormalized\u201d belief functions, i.e., b.f.s whose mass of the empty set is nil, are considered. Unnormalized b.f.s, however, play an important role in the TBM [42] as the mass of the empty set is an indicator of con\ufb02icting evidence. The analysis of the unnormalized case is also left to future work for lack of su\ufb03cient space here. 5.4. Massversus beliefconsistent approximations Summarizing, in the mass space: \u2022 the partial L1 consistent approximation focussed on a certain element x of the frame is obtained by reassigning all the mass b(xc) outside the \ufb01lter to \u0398; \u2022 the global approximation is associated, as expected, with cores containing the maximal plausibility element(s) of \u0398; \u2022 the L\u221eapproximation generates a \u201crectangle\u201d of partial approximations, with barycenter in the L1 partial approximation; \u2022 the corresponding global approximations spans the component(s) focussed on the element(s) x such that maxB\u0338\u2283x mb(B) is minimal; \u2022 the L2 partial approximation coincides with the L1 one in the N \u22122 representation; \u2022 in the N \u22121 representation the L2 partial approximation reassigns the mass outside the desired \ufb01lter (b(xc)) to each element of the \ufb01lter focussed on x on equal basis; \u2022 global approximations in the L2 case are of more di\ufb03cult interpretation. In the belief space: \u2022 partial L1/L2 approximations coincide on each component of the consistent complex; \u2022 such partial L1/L2 approximation turns out to be the consistent transformation [17] focused on the considered element of the frame: for all events A such that {x} \u2286A \u2286\u0398 mcsx B,L1[b](A) = mcsx B,L2[b](A) = mb(A) + mb(A \\ {x}); 28 \f\u2022 the corresponding global solutions have not in general as core the maximal plausibility element, and may lie in general on di\ufb00erent components of CS; the L1 global consistent approximation is associated with the singleton(s) x \u2208\u0398 such that: \u02c6 x = arg minx P A\u2286{x}c b(A), while the L2 global approximation is associated with \u02c6 x = arg minx P A\u2286{x}c(b(A))2 which do not appear to have simple epistemic interpretations; \u2022 the set of partial L\u221esolutions form a polytope on each component of the consistent complex, whose center of mass lies on the partial L1/L2 approximation; \u2022 the global L\u221esolutions fall on the component(s) associated with the maximal plausibility element(s), and their center of mass, when such element is unique, is the consistent transformation focused on the maximal plausibility singleton [17]. By comparing the behavior of the two classes of approximations we can notice a general pattern. Approximations in both mass and belief space reassign the mass outside the \ufb01lter focussed on x, in di\ufb00erent ways. However, mass consistent approximations reassign this mass b(xc) either to no focal element in the \ufb01lter (i.e., to \u0398) or to all on an equal basis. They do not distinguish between focal elements w.r.t. their set-theoretic relationships with subsets B \u0338\u2283x outside the \ufb01lter. In opposition, approximations in the belief space do distinguish them according to the focussed consistent transformation principle. 5.5. Comparison on a ternary example To conclude, let us graphically illustrate the di\ufb00erent approximations for the belief function of the running ternary example. Figure 7 illustrates the di\ufb00erent partial consistent approximations in the simplex Cl(\u20d7 mx, \u20d7 mx,y, \u20d7 mx,z, \u20d7 m\u0398) of consistent belief functions focussed on x in a ternary frame, for the belief function (15). This is the solid black tetrahedron of Figure 7. The set of partial L\u221eapproximations in M (18) is represented by the green cube in the Figure. As expected, it does not entirely fall inside the tetrahedron of admissible consistent belief functions. Its barycenter (the green star) coincides with the L1 partial consistent approximation in M. The L2 approximation in M does also coincide, as expected, with the L1 approximation. There seems to be a strong case for the latter approximation, whose natural interpretation in terms of mass assignment is the following: 29 \fall the mass outside the desired ultra\ufb01lter {B \u2287x} is reassigned to \u0398, increasing the overall uncertainty of the belief state. The L2, M partial approximation in the N \u22121 representation is distinct from the previous ones, but still falls inside the polytope of L\u221epartial approximations (green cube) and is admissible, as it falls in the interior of the simplicial component Cl(\u20d7 mx, \u20d7 mx,y, \u20d7 mx,z, \u20d7 m\u0398). It itself possesses a strong interpretation, as the overall mass not in the desired ultra\ufb01lter focused on x is equally redistributed to all elements of the \ufb01lter. Figure 7: The simplex (solid black tetrahedron) Cl(\u20d7 mx, \u20d7 mx,y, \u20d7 mx,z, \u20d7 m\u0398) of consistent belief functions focussed on x, and the related Lp partial consistent approximations of a generic b.f. on \u0398 = {x, y, z}. Finally, the unique L1/L2 partial approximation in B of Equation (28) (which is also the barycenter (32) of the partial L\u221eapproximations in B) is shown as a red square. Just as the L1/L2 approximation in M (green star), it attributes zero mass to {x, z}, which fails to be supported by any focal element of the original belief function. As a result, they both fall on the border of the tetrahedron of admissible consistent b.f.s focussed on x (highlighted in yellow). 6."
+                }
+            ],
+            "Yongqin Xian": [
+                {
+                    "url": "http://arxiv.org/abs/1903.10132v1",
+                    "title": "f-VAEGAN-D2: A Feature Generating Framework for Any-Shot Learning",
+                    "abstract": "When labeled training data is scarce, a promising data augmentation approach\nis to generate visual features of unknown classes using their attributes. To\nlearn the class conditional distribution of CNN features, these models rely on\npairs of image features and class attributes. Hence, they can not make use of\nthe abundance of unlabeled data samples. In this paper, we tackle any-shot\nlearning problems i.e. zero-shot and few-shot, in a unified feature generating\nframework that operates in both inductive and transductive learning settings.\nWe develop a conditional generative model that combines the strength of VAE and\nGANs and in addition, via an unconditional discriminator, learns the marginal\nfeature distribution of unlabeled images. We empirically show that our model\nlearns highly discriminative CNN features for five datasets, i.e. CUB, SUN, AWA\nand ImageNet, and establish a new state-of-the-art in any-shot learning, i.e.\ninductive and transductive (generalized) zero- and few-shot learning settings.\nWe also demonstrate that our learned features are interpretable: we visualize\nthem by inverting them back to the pixel space and we explain them by\ngenerating textual arguments of why they are associated with a certain label.",
+                    "authors": "Yongqin Xian, Saurabh Sharma, Bernt Schiele, Zeynep Akata",
+                    "published": "2019-03-25",
+                    "updated": "2019-03-25",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Learning with limited labels has been an important topic of research as it is unrealistic to collect suf\ufb01cient amounts of labeled data for every object. Recently, generating visual features of previously unseen classes [58, 5, 28, 11] has shown its potential to perform well on extremely imbalanced image collections. However, current feature generation approaches have still shortcomings. First, they rely on simple generative models which are not able to capture complex data distributions. Second, in many cases, they do not truly generalize to the under represented classes. Third, although classi\ufb01ers trained on a combination of real and generated features obtain state-of-the-art results, generated features may not be easily interpretable.        \u2026 it     has      a    brown       center  and yellow petals .   Textual Explanation \u2026 because     it     has      a         brown center and  yellow      petals       . This  flower has  a large  brown  center and  its petals  are long. f-VAEGAN-D2 Sunflower Visualization Transductive  Learning (D2) Feature  Generator  (f-WGAN) Feature  Reconstruction  (f-VAE) real  image  \u2208 This is a sunflower because ... feature  space Figure 1: Our any-shot feature generating framework learns discriminative and interpretable CNN features from both labeled data of seen and unlabeled data of novel classes. Our main focus in this work is a new model that generates visual features of any class, utilizing labeled samples when they are available and generalizing to unknown concepts whose labeled samples are not available. Prior work used GANs for this task [58, 11] as they directly optimize the divergence between real and generated data, but they suffer from mode collapse issues [3]. On the other hand, feature generation with VAE [28] is more stable. However, VAE optimizes the lower bound of log likelihood rather than the likelihood itself [23]. Our model combines the strengths of VAE and GANs by assembling them to a conditional feature generating model, called f-VAEGAN-D2, that synthesizes CNN image features from class embeddings, i.e. class-level attributes or word2vec [35]. Thanks to its additional discriminator that distinguishes real and generated features, our f-VAEGAN-D2 is able to use unlabeled data from previously unseen classes without any condition. The features learned by our model, e.g. Figure 1, are disciminative in that they boost the performance of any-shot learning as well as being visually and textually interpretable. Our main contributions are as follows. (1) We propose 1 arXiv:1903.10132v1  [cs.CV]  25 Mar 2019 \fthe f-VAEGAN-D2 model that consists of a conditional encoder, a shared conditional decoder/generator, a conditional discriminator and a non-conditional discriminator. The \ufb01rst three networks aim to learn the conditional distribution of CNN image features given class embeddings optimizing VAE and WGAN losses on labeled data of seen classes. The last network learns the marginal distribution of CNN image features on the unlabeled features of novel classes. Once trained, our model synthesizes discriminative image features that can be used to augment softmax classi\ufb01er training. (2) Our empirical analysis on CUB, AWA2, SUN, FLO, and large-scale ImageNet shows that our generated features improve the state-of-the-art in low-shot regimes, i.e. (generalized) zeroand few shot learning in both the inductive and transductive settings. (3) We demonstrate that our generated features are interpretable by inverting them back to the raw pixel space and by generating visual explanations. 2. Related Work In this section, we discuss related works on zeroand few-shot learning as well as generative models. Zero-shot Learning. We are interested in both zero-shot learning (ZSL) that aims to predict unseen classes and generalized zero-shot learning (GZSL) that predicts both seen and unseen classes. The required knowledge transfer from seen classes to unseen classes relies on the semantic embedding, e.g. attributes annotated by humans, word embeddings learned on text corpus, hierarchy embeddings obtained from label hierarchy, sentence embeddings from a language model. Unlike the instance-level image features, the semantic embedding is usually class-level, i.e. we use class embedding and semantic embedding interchangeably. Early works [29, 22] associate seen and unseen classes by learning attribute classi\ufb01ers. Most of recent zero-shot learning works [1, 27, 49, 13, 60] learn a compatibility function between the image and semantic embedding spaces. [61, 40, 6] represents image and class embeddings as a mixture of seen class proportions. SYNC [6] and [10, 32] learn to predict linear classi\ufb01er weights of unseen classes. [54] proposes to combine the semantic embedding and knowledge graph with graph convolutional network [24]. An orthogonal direction is generative model [52, 38], where class-conditional distribution is learned based on the Gaussian assumption. In contrast to those inductive approaches that only use labeled data from seen classes, transductive zero-shot learning methods additionally leverage unlabeled data from unseen classes. PST [48] and DSZSL [59] project image embedding to the semantic embedding space followed by label propagation. TMV [14] combines multiple semantic embeddings and performs hypergraph label propagation. [26, 16] exploit semantic manifold learning. GFZSL [52] treats unknown labels of unseen class images as latent variables and applies Expectation-Maximization (EM). As the prediction is biased to seen classes in GZSL, UE [51] maximizes the probability of predicting unlabeled images as unseen classes. Our model operates in both inductive and transductive zero-shot settings. However, unlike most of other transductive approaches that rely on label propagation, we propose to learn a feature generator with labeled data of seen classes and unlabeled data of unseen classes. Few-shot Learning. The task of few-shot learning is to train a model with only a few training samples. Directly optimizing the standard model with few samples will have high risk of over-\ufb01tting. The general idea is to train a model on classes with enough training samples and generalize to classes with few samples without learning new parameters. Siamese neural networks [25] proposes a CNN architecture that computes similarity between an image pair. Matching network [53] and prototypical networks [50] predict an image label based on support sets and apply the episode training strategy that mimics the few-shot testing. MetaLSTM [45] learns the exact optimization algorithm used to train the few-shot classi\ufb01er. MAML [12] proposes to learn good weight initialization that can be adapted to small dataset ef\ufb01ciently. [20, 55] propose a large scale low-shot benchmark on ImageNet and generate features for novel classes. Imprinting[42] directly copies the normalized image embedding as classi\ufb01er weights, while [43] predicts classi\ufb01er weights from image features with a learned neural network. In contrast to those prior works that only rely on visual information, we also leverage class-level semantic information, i.e. attribute or word2vec [35]. Generative Models. Generative modeling aims to learn the probability distribution of data points such that we can randomly sample data from it that can be used as a data augmentation mechanism. Generative Adversarial Networks (GANs)[17, 36, 44] consist of a generator that synthesizes fake data and a discriminator that distinguishes fake and real data. The instable training issues of GANs have been studied by [19, 3, 37]. An interesting application of GANs is CycleGAN [62] that translates an image from one domain to another domain. [47] generates natural images from text descriptions, and SRGAN[31] solves single image super-resolution. Variational Autoencoder (VAE) [23] employs an encoder that represents the input as a latent variable with Gaussian distribution assumption and a decoder that reconstructs the input from the latent variable. GMMN [33] optimizes the maximum mean discrepancy (MMD) [18] between real and generated distribution. Recently, generative models [5, 63, 28, 58] have been applied to solve generalized zero-shot learning by synthesizing CNN features of unseen classes from semantic embeddings. Among those, [5] uses GMMN [33], [63, 58] use GANs[17] and [28] em2 \fEncoder (E) Decoder/Generator(G) Cape May  Warbler Discriminator1 (D1) Discriminator2 (D2) VAE GAN D2 D2 f-WGAN f-VAE Figure 2: Our any-shot feature generating network (f-VAEGAN-D2) consist of a feature generating VAE (f-VAE), a feature generating WGAN (f-WGAN) with a conditional discriminator (D1) and a transductive feature generator with a nonconditional discriminator (D2) that learns from both labeled data of seen classes and unlabeled data of novel classes. ploys VAE [23]. Our model combines the advantages of both VAE and GAN with an additional discriminator to use unlabeled data of unseen classes which lead to more discriminative features. 3. f-VAEGAN-D2 Model Existing models that operate on sparse data regimes are either trained with labeled data from a set of classes which is disjoint from the set of classes at test time, i.e. inductive zero-shot setting [29, 13], or the samples can come from all classes but then their labels are not known, i.e. transductive zero-shot setting [15, 48]. Recent works [58, 28, 11] address generalized zero-shot learning by generating synthetic CNN features of unseen classes followed by training softmax classi\ufb01ers, which alleviates the imbalance between seen and unseen classes. However, we argue that those feature generating approaches are not expressive enough to capture complicated feature distributions in real world. In addition, since they have no access to any real unseen class features, there is no guarantee on the quality of generated unseen class features. As shown in Figure 2, we proposes to enhance the feature generator by combining VAE and GANs with shared decoder and generator, and adding another discriminator (D2) to distinguish real or generated features without applying any condition. Intuitively, in transductive zero-shot setting, by feeding real unlabeled features of unseen classes, D2 will be able to learn the manifold of unseen class such that more realistic features can be generated. Hence, the key to our approach is the ability to generate semantically rich CNN feature distributions, which is generalizes to any-shot learning scenarios ranging from (generalized) zero-shot to (generalized) fewshot to (generalized) many-shot learning. Setup. We are given a set of images X = {x1, . . . , xl} \u222a {xl+1, . . . , xt} encoded in the image feature space X, a seen class label set Y s, a novel label set Y n, a.k.a unseen class label set Y u in the zero-shot learning literature. The set of class embeddings C = {c(y)|\u2200y \u2208Y s \u222aY n} are encoded in the semantic embedding space C that de\ufb01nes high level semantic relationships between classes. The \ufb01rst l points xs(s \u2264l) are labeled as one of the seen classes ys \u2208Y s and the remaining points xn(l + 1 \u2264n \u2264t) are unlabeled, i.e. may come from seen or novel classes. In the inductive setting, the training set contains only labeled samples of seen class images, i.e. {x1, . . . , xl}. On the other hand, in the transductive setting, the training set contains both labeled and unlabeled samples, i.e. {x1, . . . , xl, xl+1, . . . , xt}. For both inductive and transductive settings the inference is the same. In zero-shot learning, the task is to predict the label of those unlabeled points that belong to novel classes, i.e. fzsl : X \u2192Yn, while in the generalized zero-shot learning, the goal is to classify those unlabeled points that can be either from seen or novel classes, i.e. fgzsl : X \u2192Ys \u222aYn. Few-shot and generalized few-shot learning are de\ufb01ned similarly. Our framework can be thought of as a data augmentation scheme where arbitrarily many synthetic features of sparsely populated classes aid in improving the disciminative power of classi\ufb01ers. In the following, we only detail our feature generating network structure as the classi\ufb01er is unconstrained (we use linear softmax classi\ufb01ers). 3.1. Baseline Feature Generating Models In feature generating networks (f-WGAN) [58] the generator G(z, c) generates a CNN feature \u02c6 x in the input feature space X from random noise zp and a condition c, and the 3 \fdiscriminator D(x, c) takes as input a pair of input features x and a condition c and outputs a real value, optimizing: Ls W GAN =E[D(x, c)] \u2212E[D(\u02dc x, c)] (1) \u2212\u03bbE[(||\u2207\u02c6 xD(\u02c6 x, c)||2 \u22121)2], where \u02dc x = G(z, c) is the generated feature and \u02c6 x = \u03b1x + (1 \u2212\u03b1x) with \u03b1 \u223cU(0, 1) and \u03bb is the penalty coef\ufb01cient. The feature generating VAE [23] (f-VAE) consists of an encoder E(x, c), which encodes an input feature x and a condition c to a latent variable z, and a decoder Dec(z, c), which reconstructs the input x from the latent z and condition c optimizing: Ls V AE = KL(q(z|x, c)||p(z|c)) (2) \u2212Eq(z|x,c)[log p(x|z, c)], where the conditional distribution q(z|x, c) is modeled as E(x, c), p(z|c)) is assumed to be N(0, 1), KL is the Kullback-Leibler divergence, and p(x|z, c) is equal to Dec(z, c). 3.2. Our f-VAEGAN-D2 Model It has been shown that ensembling a VAE and a GAN leads to better image generation results [30]. We hypothesize that VAE and GAN learn complementary information for feature generation as well. This is likely when the target data follows a complicated multi-modal distribution where two losses are able to capture different modes of the data. To combine f-VAE and f-WGAN, we introduce an encoder E(x, c) : X \u00d7 C \u2192Z, which encodes a pair of feature and class embedding to a latent representation, and a discriminator D1 : X \u00d7 C \u2192R maps this embedding pair to a compatibility score, optimizing: Ls V AEGAN = Ls V AE + \u03b3Ls W GAN (3) where the generator G(z, c) of the GAN and decoder Dec(z, c) of the VAE share the same parameters. The superscript s indicates that the loss is applied to feature and class embedding pair of seen classes. \u03b3 is a hyperparameter to control the weighting of VAE and GAN losses. Furthermore, when unlabeled data of novel classes becomes available, we propose to add a non-conditional discriminator D2 (D2 in f-VAEGAN-D2) which distinguishes between real and generated features of novel classes. This way D2 learns the feature manifold of novel classes. Formally, our additional non-conditional discriminator D2 : X \u2192R distinguishes real and synthetic unlabeled samples using a WGAN loss: Ln W GAN =E[D2(xn)] \u2212E[D2(\u02dc xn)]\u2212 (4) \u03bbE[(||\u2207\u02c6 xnD2(\u02c6 xn)||2 \u22121)2], where \u02dc xn = G(z, yn) with yn \u2208Y n, \u02c6 xn = \u03b1xn+(1\u2212\u03b1xn) with \u03b1 \u223cU(0, 1). Since Ls W GAN is trained to learn CNN features using labeled data conditioned on class embeddings of seen classes and class embeddings encode shared properties across classes, we expect these CNN features to be transferable across seen and novel classes. However, this heavily relies on the quality of semantic embeddings and suffers from domain shift problems. Intuitively, Ln W GAN captures the marginal distribution of CNN features and provides useful signals of novel classes to generate transferable CNN features. Hence, our uni\ufb01ed f-VAEGAN-D2 model optimizes the following objective function: min G,E max D1,D2 Ls V AEGAN + Ln W GAN (5) Implementation Details. Our generator (G) and discriminators (D1 and D2) are implemented as multilayer perceptron (MLP). The random Gaussian noise z \u223cN(0, 1) and class embedding c(y) are concatenated and fed into the generator, which is composed of 2 fully connected layers with 4096 hidden units. We \ufb01nd dimension of noise dz = dc, i.e. dimension of class embeddings, works well. Similarly, the discriminators take input as the concatenation of image feature and class embedding and have 2 fully connected layers with 4096 hidden units. We use LeakyReLU as the nonlinear activation function except for the output layer of G, for which Sigmoid is used because we apply binary cross entropy loss as LREC and input features are rescaled to be in [0, 1]. We \ufb01nd \u03b2 = 1 and \u03b3 = 1000 works well across all the datasets. Gradient penalty coef\ufb01cient is set to \u03bb = 10 and generator is updated every 5 discriminator iterations as suggested in WGAN paper [4]. As for the optimization, we use Adam optimizer with constant learning rate 0.001 and early stopping on the validation set. 4. Experiments In this section, we validate our approach in both zeroshot and few-shot learning. The details of the settings are provided in their respective sections. 4.1. (Generalized) Zero-shot Learning We validate our model on \ufb01ve widely-used datasets for zero-shot learning, i.e. Caltech-UCSD-Birds (CUB) [56], Oxford Flowers (FLO) [39], SUN Attribute (SUN) [41] and Animals with Attributes2 (AWA2) [57]. Among those, CUB, FLO and SUN are medium scale, \ufb01ne-grained datasets. AWA2, on the other hand, is a coarse-grained dataset. Finally we evaluate our model also on ImageNet [7] with more than 14 million images and 21K classes as a large-scale and \ufb01ne-grained dataset. We follow the exact ZSL and GZSL splits as well as the evaluation protocol of [57] and for fair comparison we 4 \fModel ZSL GZSL INDUCTIVE GAN 59.1 52.3 VAE 58.4 52.5 VAE-GAN 61.0 53.7 TRANSDUCTIVE GAN 67.3 61.6 VAE 68.9 59.6 VAE-GAN 71.1 63.2 Table 1: Ablating different generative models on CUB (using attribute class embedding and image features with no \ufb01ne-tuning). ZSL: top-1 accuracy on unseen classes, GZSL: harmonic mean of seen and unseen class accuracies. 2H 3H All 0 2 4 6 8 10 12 14 16 Top-1 Acc. (in %) ZSL CLSWGAN Ours 2H 3H All 0 2 4 6 8 10 12 14 16 Top-1 Acc. (in %) GZSL CLSWGAN Ours Figure 3: Top-1 ZSL results on ImageNet. We follow the splits in [57] and compare our results with the state-of-theart feature generating model CLSWGAN [58]. use the same image and class embeddings for all models. Brie\ufb02y, image (with no image cropping or \ufb02ipping) features are extracted from the 2048-dim top pooling units of 101layer ResNet pretrained on ImageNet 1K. For comparative studies, we also \ufb01ne-tune ResNet-101 on the seen class images of each dataset. As for class embeddings, unless otherwise speci\ufb01ed, we use class-level attributes for CUB (312dim), AWA2 (85-dim) and SUN(102-dim). For CUB and FLO, we also extract 1024-dim sentence embeddings of character-based CNN-RNN model [46] from \ufb01ne-grained visual descriptions (10 sentences per image). Ablation study. We ablate our model with respect to the generative model, i.e. using GAN, VAE or VAE-GAN in both inductive and transductive settings. Our conclusions from Table 1, are as follows. In the inductive setting VAE-GAN has an edge over both VAE and GAN, i.e. 59.1% and 58.4% vs 61.0% in ZSL setting. Adding unlabeled samples to the training set, i.e. transductive learning setting, is bene\ufb01cial for all the generative models. As in the inductive setting VAE and GAN achieve similar results, i.e 67.3% and 68.9% for ZSL. Our VAE-GAN model leads to the state-of-the-art results, i.e. 71.1% in ZSL and 63.2% in GZSL con\ufb01rming that VAE and GAN learn complementary representations. As VAE-GAN gives the highest accuracy in all settings, it is employed in all remaining results of the paper. Comparing with the state-of-the-art. In Table 2 we compare our model with the best performing recent methods on four zero-shot learning datasets on ZSL and GZSL settings. In the inductive ZSL setting, our model both with and without \ufb01ne-tuning outperforms the state-of-the art for all datasets. Our model with \ufb01ne-tuned features establishes the new state-of-the-art, i.e. 72.9% on CUB, 70.4% on FLO, 65.6% on SUN and 70.3% on AWA. For the transductive ZSL setting, our model without \ufb01ne-tuning on CUB is surpassed by UE-\ufb01netune of [51], i.e. 71.1% vs 72.1%. However, when we also \ufb01ne-tune our features, we establish the new state-of-the-art on the transductive ZSL setting as well, i.e. 82.6% on CUB, 95.4% on FLO, 72.6% on SUN and 89.3% on AWA. In the GZSL setting, we observe that feature generating methods, i.e. our model, CLSWGAN [58], SE-GZSL [28], Cycle-CLSWGAN [11] achieve better results than others. This is due to the fact that data augmentation through feature generation leads to a more balanced data distribution such that the learned classi\ufb01er is not biased to seen classes. Note that although UE [51] is not a feature generating method, it leads to strong results as this model uses additional information, i.e. it assumes that unlabeled test samples always come from unseen classes. Nevertheless, our model with \ufb01ne-tuning leads to 77.3% harmonic mean (H) on CUB, 94.1% H on FLO, 47.2% H on SUN and 87.5% H on AWA achieving signi\ufb01cantly higher results than all the prior works. Large-scale experiments. Although most of the prior work presented in Table 2 has not been evaluated in ImageNet, this dataset serves a challenging and interesting test bed for (G)ZSL research. Hence, we compare our model with CLSWGAN [58] on ImageNet using the same evaluation protocol. As shown in Figure 3 our model signi\ufb01cantly improves over the state-of-the-art in both ZSL and GZSL settings in 2H, 3H and All splits determined by considering the classes 2 hops or 3 hops away from 1000 classes of Imagenet as well as all the remaining classes. These experiments are important for two reasons. First, they show that our feature generation model is scalable to the largest scale setting available. Second, our model is applicable to the situations even when human annotated attributes are not available, i.e. for ImageNet classes attributes are not available hence we use per-class word2vec representations. 4.2. (Generalized) Few-shot Learning In few-shot or low-shot learning scenarios, classes are divided into base classes that have a large number of labeled training samples and novel classes that contain only few labeled samples per category. In the plain FSL setting, the goal is to achieve good performance on novel classes whereas in GFSL setting good performance must generalize to all classes. 5 \fZero-Shot Learning Generalized Zero-Shot Learning CUB FLO SUN AWA CUB FLO SUN AWA Method T1 T1 T1 T1 u s H u s H u s H u s H IND ALE [2] 54.9 48.5 58.1 59.9 23.7 62.8 34.4 13.3 61.6 21.9 21.8 33.1 26.3 16.8 76.1 27.5 CLSWGAN [58] 57.3 67.2 60.8 68.2 43.7 57.7 49.7 59.0 73.8 65.6 42.6 36.6 39.4 57.9 61.4 59.6 SE-GZSL [28] 59.6 63.4 69.2 41.5 53.3 46.7 40.9 30.5 34.9 58.3 68.1 62.8 Cycle-CLSWGAN [11] 58.6 70.3 59.9 66.8 47.9 59.3 53.0 61.6 69.2 65.2 47.2 33.8 39.4 59.6 63.4 59.8 Ours 61.0 67.7 64.7 71.1 48.4 60.1 53.6 56.8 74.9 64.6 45.1 38.0 41.3 57.6 70.6 63.5 Ours-finetuned 72.9 70.4 65.6 70.3 63.2 75.6 68.9 63.3 92.4 75.1 50.1 37.8 43.1 57.1 76.1 65.2 TRAN ALE-tran [57] 54.5 48.3 55.7 70.7 23.5 45.1 30.9 13.6 61.4 22.2 19.9 22.6 21.2 12.6 73.0 21.5 GFZSL [52] 50.0 85.4 64.0 78.6 24.9 45.8 32.2 21.8 75.0 33.8 0.0 41.6 0.0 31.7 67.2 43.1 DSRL [59] 48.7 57.7 56.8 72.8 17.3 39.0 24.0 26.9 64.3 37.9 17.7 25.0 20.7 20.8 74.7 32.6 UE-finetune [51] 72.1 58.3 79.7 74.9 71.5 73.2 33.6 54.8 41.7 93.1 66.2 77.4 Ours 71.1 89.1 70.1 89.8 61.4 65.1 63.2 78.7 87.2 82.7 60.6 41.9 49.6 84.8 88.6 86.7 Ours-finetuned 82.6 95.4 72.6 89.3 73.8 81.4 77.3 91.0 97.4 94.1 54.2 41.8 47.2 86.3 88.7 87.5 Table 2: Comparing with the-state-of-the-art. Top: inductive methods (IND), Bottom: transductive methods (TRAN). Fine tuning is performed only on seen class images as this does not violate the zero-shot condition. We measure top-1 accuracy (T1) in ZSL setting, Top-1 accuracy on seen (s) and unseen (s) classes as well as their harmonic mean (H) in GZSL setting. # training samples per class 1 2 5 10 20 Top-1 Acc. (in %) 40 50 60 70 80 90 CUB Ours-tran Ours-ind Imprint[42] Softmax Analogy[20] ALE-tran[57] # training samples per class 1 2 5 10 20 Top-1 Acc. (in %) 70 75 80 85 90 95 100 FLO Ours-tran Ours-ind Imprint[42] Softmax Analogy[20] ALE-tran[57] (a) Few-Shot Learning (FSL) # training samples per class 1 2 5 10 20 Top-1 Acc. (in %) 45 50 55 60 65 70 75 80 85 CUB Ours-tran Ours-ind Imprint[42] Softmax Analogy[20] ALE-tran[57] # training samples per class 1 2 5 10 20 Top-1 Acc. (in %) 60 65 70 75 80 85 90 95 100 FLO Ours-tran Ours-ind Imprint[42] Softmax Analogy[20] ALE-tran[57] (b) Generalized Few-Shot Learning (GFSL) Figure 4: FSL and GFSL results on CUB and FLO with increasing number of training samples per novel class. Left: FSL plots show the top-1 accuracy on novel classes. Right: GZSL plots show the top-1 accuracy on all classes. Among the classic ZSL datasets, CUB has been used for few-shot learning in [42] by taking the \ufb01rst 100 classes as base classes and the rest as novel classes. However, as ImageNet 1K contains some of those novel classes and feature extractors are pretrained on it, we use the class splits from the standard ZSL setting, i.e. 150 base and 50 novel. For FLO we also follow the same class splits as in ZSL. As for features, we use the same \ufb01ne-tuned ResNet-101 features and attribute class embeddings used in zero-shot learning experiments. For fairness, we repeat all the experiments for [42] and [20] with the same image features. Comparing with the state-of-the-art. As shown in Figure 4 both for FSL and GFSL settings and for both datasets both our inductive and transductive models have a signi\ufb01cant edge over all the competing methods when the number of samples from novel classes is small, e.g. 1,2 and 5. This shows that our model generates highly discriminative features even with only few real samples are present. In fact, only with one real sample per class, our model achieves almost the full accuracy obtained with 20 samples per class. Going towards the full supervised learning, e.g. with 10 or 20 samples per class, all methods perform similarly. This is expected since in the setting where a large number of labeled samples per class is available, then a simple softmax classi\ufb01er that uses real ResNet-101 features achieves the state-of-the-art. In the inductive FSL setting, our model that uses one labeled sample per class reaches the accuracy as softmax that uses \ufb01ve samples per class. In the transductive FSL setting, our model that uses one labeled sample per class reaches the accuracy of softmax obtained with 10 samples per class. Furthermore, the inductive GFSL setting, our model with two samples per class achieves the same accuracy as softmax trained with ten samples per class on CUB. In the transductive GFSL setting, for FLO, for our model only one labeled sample is enough to reach the accuracy obtained with 20 labeled samples with softmax. Note that the same behavior is observed on SUN and AWA as well. Due to space restrictions we present them in the supplementary material. 6 \f# training samples per class 1 2 5 10 20 Top-5 Acc. (in %) 20 30 40 50 60 70 80 90 FSL Ours-tran Ours-ind PMN w/G*[55] Softmax Analogy[20] # training samples per class 1 2 5 10 20 Top-5 Acc. (in %) 20 30 40 50 60 70 80 90 GFSL Ours-tran Ours-ind PMN w/G*[55] Softmax Analogy[20] Figure 5: Few Shot Learning results on ImageNet with increasing number of training samples per novel class (Top-5 Accuracy). Left: FSL setting, Right: GFSL setting. Large-scale experiments. Regarding few-shot learning results on ImageNet, we follow the procedure in [20] where 1K ImageNet categories are randomly divided into 389 base and 611 novel classes. To facilitate cross validation, base classes are further split into C1 base (193 classes) and C2 base (196 classes), and novel classes into C1 novel (300 classes) and C2 novel (311 classes). The cross validation of hyperparameters is performed on C1 base and C1 novel and the \ufb01nal results are reported on C2 base and C2 novel. Here, we extract image features from the ResNet-50 pretrained on C1 base \u222aC2 base, which is provided by the benchmark [20]. Since there is no attribute annotation on ImageNet, we use 300-dim word2vec [35] embeddings as the class embedding. Following [55], we measure the averaged top-5 accuracy on test examples of novel classes with the model restricted to only output novel class labels, and the averaged top-5 accuracy on test examples of all classes with the model that predicts both base and novel classes. Our baselines are PMN w/G* [55] combining metalearning and feature generation, analogy generator [20] learning an analogy-based feature generator and softmax classi\ufb01er learned with uniform class sampling. For, fewshot learning results in Figure 5(left), we observe that our model in the transductive setting, i.e. Ours-tran improves the state-of-the-art PMN w/G* [55] signi\ufb01cantly when the number of training samples is small, i.e. 1,2 and 5. Notably, we achieve 60.6% vs 54.7% state-of-the art at 1 shot, 70.3 vs 66.8% at 2 shots. This indicates that our model generates highly discriminative features by leveraging unlabeled data and word embeddings. In the challenging generalized few-shot learning setting (Figure 5 right), although PMN /G* [55] is quite strong by applying meta-learning [50], our model still achieves comparable results with the state-of-the-art. It is also worth noting that PMN w/G* [55] cannot be directly applied to zero-shot learning. Hence, our approach is more versatile. 4.3. Interpreting Synthesized Features In this section, we show that our generated features on FLO are visually discriminative and textually explainable. Visualising generated features. A number of methods [8, 34, 9] have explored strategies to generate images by inverting feature embeddings. We follow a strategy similar to [8] and train a deep upconvolutional neural network to invert feature embeddings to the image pixel space. We impose a L1 loss between the ground truth image and the inverted image, as well as a perceptual loss, by passing both images through a pre-trained Resnet101, and taking an L2 loss on the feature vectors at conv5 4 and average pooling layers. We also utilize an adversarial loss, by feeding the image and feature embedding to a discriminator, to improve our image quality. Our generator consists of a fully connected layer followed by 5 upconvolutional blocks. Each upconvolutional block contains an Upsampling layer, a 3x3 convolution, BatchNorm and ReLu non-linearity. The \ufb01nal size of the reconstructed image is 64x64. The discriminator processes the image through 4 downsampling blocks, the feature embedding is sent to a linear layer and spatially replicated and concatenated with the image embedding, and this \ufb01nal embedding is passed through a convolutional and sigmoid layer to get the probability that the sample is real or fake. We train this model on all the real feature-image pairs of the 102 classes, and use the trained generator to invert images from synthetic features. In Figure 6, we show generated images from real and synthetic features for comparison. We observe that images generated from synthetic features contain the essential attributes required for classi\ufb01cation, such as the general color distribution and sometimes even features like the petal and stamen are visible. Also, the image quality is similar for the images generated from real and synthetic features. Interestingly, the synthetic features of unseen classes generated by our model without observing any real features from that class, i.e. \u201cUnseen classes\u201d and \u201cS\u201d row, also yield pleasing reconstructions. As shown in \u201cChallenging Classes\u201d of Figure 6, in some cases the generated images from synthetic features lack a certain level of detail, e.g. see images for \u201cBalloon Flower\u201d and in some cases the colors do not match with the real image, e.g. see images for \u201cSweat Pea\u201d. We noticed that these correspond to classes with high inter class variation. Explaining visual features. We also explore generating textual explanations of our synthetic features. For this, we choose a language model [21], that produces an explanation of why an image belongs to a particular class, given a feature embedding and a class label. The architecture of our model is similar to [21], we use a linear layer for the feature embedding, and feed it as the start token for a LSTM. At every step in the sequence, we also feed the class embedding, to produce class relevant captions. The class embedding is obtained by training a LSTM to generate captions from images, and taking the average hidden state for images of that class. A softmax cross entropy loss is imposed on the out7 \f\u2026 this flower has a wide  brown center and tapered  yellow petals. \u2026 this flower has a wide  center and layers of wide,  tapered yellow petals. This is a Sunflower  because ... \u2026 this flower has petals that  are white and has a bushy  yellow center \u2026 the flower is big with  white petals, and a bulb of  yellow colored anthers. This is a Tree Poppy  because ... \u2026 this flower has simple rows of  overlapping orange petals with a  notched tip of yellow stamen in  the center. Seen Classes Unseen Classes This is a Marigold  because ... \u2026 this flower has layers of  long tapered pale yellow  petals surrounding orange and  red stamen.  \u2026 this flower is pink in color,  and has petals that are drooping  downward. \u2026 this flower has pink petals  that are pointed down, and a  lot of red stamen in the center This is a Purple Coneflower  because ... R S R S \u2026 this flower has red  petals that have yellow  tips.  \u2026 this flower has petals  that are red with yellow  edges  This is a Blanket Flower  because ... \u2026 the petals of the  flower are light pink,  while the anthers are  white and yellow. \u2026 this flower is pink and  white in color, with petals  that are rounded. This is a Pink Primrose  because ... \u2026 the petals on this flower are  mostly lavender in color and  the inner stamen is the color  purple. \u2026 this flower is green, white,  and purple in color, and has  petals that are oval shaped. This is a Passion Flower  because ... \u2026 this flower has petals that  are red with pointy tips \u2026 this flower has a lot of very  thin red petals and a lot of  white stamen on it This is a King Protea  because ... Challenging Classes R S \u2026 this flower has wide  trumpet shaped purple flowers  with a star shape. This is a Canterburry Bells  because \u2026  \u2026 this flower has broad  alternating leaves, and its  pink colored petals are  lighter pink. This is a Sweat Pea because \u2026 \u2026 the flowers color of  the flower are visible. The  stamen and pistil <unk>  from it. This is a Balloon Flower  because \u2026  \u2026 this flower has petals that  are pink and white with  green pedicel.  \u2026 the petals on this flower are  mostly bulb shaped purple. This is a Cameilla  because \u2026  \u2026 the flower has five  purple petals with white  stamen and a white pistil. \u2026 this red flower has  rounded petals and yellow  stamen with yellow anthers.  \u2026 the petals of the flower are  layered in layers while the  anthers and are yellow in  color.  Figure 6: Interpretability: visualizations by generating images and textual explanations from real or synthetic features. For every block, the top is the target, the middle is reconstructed from the real feature (R) of the target, the bottom is reconstructed from a synthetic feature (S) from the same class. We also generate visual explanations conditioned with the predicted class and the reconstructed real or synthetic images. Top (Middle): Features come from seen (unseen) classes. Bottom: classes with a large inter-class variation lead to poorer visualizations and explanations. put using the ground truth caption. Also, a discriminative loss that encourages the generated sentence to belong to the relevant class is imposed by sampling a sentence from the LSTM and sending it to a pre-trained sentence classi\ufb01er. The model is trained on the dataset from [46]. As before, we train this model on all the real feature-caption pairs, and use it to obtain explanations for synthetic features. In Figure 6, we show explanations obtained from real and synthetic features. We observe that the model generates image relevant and class speci\ufb01c explanations for synthetic features of both seen and unseen classes. For instance, a \u201cKing Protea\u201d feature contains information about \u201cred petals and pointy tips\u201d while \u201cPurple Cone\ufb02ower\u201d feature has information on \u201cpink in color and petals that are drooping downward\u201d which are the most visually distinguishing properties of this \ufb02ower. On the other hand, as shown at the bottom of the \ufb01gure, for classes where image features lack a certain level of detail, the generated explanations have some issues such as repetitions, e.g. \u201ctrumpet shaped\u201d and \u201cstar shape\u201d in the same sentence and unknown words, e.g. see the explanation for \u201cBalloon Flower\u201d. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1712.00981v2",
+                    "title": "Feature Generating Networks for Zero-Shot Learning",
+                    "abstract": "Suffering from the extreme training data imbalance between seen and unseen\nclasses, most of existing state-of-the-art approaches fail to achieve\nsatisfactory results for the challenging generalized zero-shot learning task.\nTo circumvent the need for labeled examples of unseen classes, we propose a\nnovel generative adversarial network (GAN) that synthesizes CNN features\nconditioned on class-level semantic information, offering a shortcut directly\nfrom a semantic descriptor of a class to a class-conditional feature\ndistribution. Our proposed approach, pairing a Wasserstein GAN with a\nclassification loss, is able to generate sufficiently discriminative CNN\nfeatures to train softmax classifiers or any multimodal embedding method. Our\nexperimental results demonstrate a significant boost in accuracy over the state\nof the art on five challenging datasets -- CUB, FLO, SUN, AWA and ImageNet --\nin both the zero-shot learning and generalized zero-shot learning settings.",
+                    "authors": "Yongqin Xian, Tobias Lorenz, Bernt Schiele, Zeynep Akata",
+                    "published": "2017-12-04",
+                    "updated": "2018-04-12",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Deep learning has allowed to push performance considerably across a wide range of computer vision and machine learning tasks. However, almost always, deep learning requires large amounts of training data which we are lacking in many practical scenarios, e.g. it is impractical to annotate all the concepts that surround us, and have enough of those annotated samples to train a deep network. Therefore, training data generation has become a hot research topic [10, 18, 11, 37, 48, 41]. Generative Adversarial Networks [18] are particularly appealing as they allow generating realistic and sharp images conditioned, for instance, on object categories [37, 48]. However, they do not yet generate images of suf\ufb01cient quality to train deep learning architectures as demonstrated by our experimental results. In this work, we are focusing on arguably the most extreme case of lacking data, namely zero-shot learning [24, 46, 9], where the task is to learn to classify when no labeled examples of certain classes are available during training. Head color: re Back color: bl Crown color:  Wing shape: s z ~ N(0 G(z, a) seen unseen ResNet space ? x xg f-CLSWGAN CNN CNN CNN feature  space synthetic  image real  image Head color: brown        Belly color: yellow Bill shape: pointy Figure 1: CNN features can be extracted from: 1) real images, however in zero-shot learning we do not have access to any real images of unseen classes, 2) synthetic images, however they are not accurate enough to improve image classi\ufb01cation performance. We tackle both of these problems and propose a novel attribute conditional feature generating adversarial network formulation, i.e. f-CLSWGAN, to generate CNN features of unseen classes. We argue that this scenario is a great testbed for evaluating the robustness and generalization of generative models. In particular, if the generator learns discriminative visual data with enough variation, the generated data should be useful for supervised learning. Hence, one contribution of our paper is a comparison of various existing GAN-models and another competing generative model, i.e. GMMN, for visual feature generation. In particular, we look into both zero-shot learning (ZSL) where the test time search space is restricted to unseen class labels and generalized zero-shot learning (GZSL) for being a more realistic scenario as at test time the classi\ufb01er has to decide between both seen and unseen class labels. In this context, we propose a novel GANmethod \u2013 namely f-CLSWGAN that generates features instead of images and is trained with a novel loss improving over alternative GAN-models. We summarize our contributions as follows. (1) We propose a novel conditional generative model f-CLSWGAN that synthesizes CNN features of unseen classes by optimizing the Wasserstein distance regularized by a classi\ufb01cation loss. (2) Across \ufb01ve datasets with varying granularity and sizes, we consistently improve upon the state of the art in both the ZSL and GZSL settings. We demonstrate a prac1 arXiv:1712.00981v2  [cs.CV]  12 Apr 2018 \ftical application for adversarial training and propose GZSL as a proxy task to evaluate the performance of generative models. (3) Our model is generalizable to different deep CNN features, e.g. extracted from GoogleNet or ResNet, and may use different class-level auxiliary information, e.g. sentence, attribute, and word2vec embeddings. 2. Related work In this section we review some recent relevant literature on Generative Adversarial Networks, Zero-Shot Learning (ZSL) and Generalized Zero-Shot (GZSL) Learning. Generative Adversarial Network. GAN [18] was originally proposed as a means of learning a generative model which captures an arbitrary data distribution, such as images, from a particular domain. The input to a generator network is a \u201cnoise\u201d vector z drawn from a latent distribution, such as a multivariate Gaussian. DCGAN [34] extends GAN by leveraging deep convolution neural networks and providing best practices for GAN training. [43] improves DCGAN by factorizing the image generation process into style and structure networks, InfoGAN [12] extends GAN by additionally maximizing the mutual information between interpretable latent variables and the generator distribution. GAN has also been extended to a conditional GAN by feeding the class label [29], sentence descriptions [36, 37, 48], into both the generator and discriminator. The theory of GAN is recently investigated in [4, 5, 19], where they show that the Jenson-Shannon divergence optimized by the original GAN leads to instability issues. To cure the unstable training issues of GANs, [5] proposes Wasserstein-GAN (WGAN), which optimizes an ef\ufb01cient approximation of the Wasserstein distance. While WGAN attains better theoretical properties than the original GAN, it still suffers from vanishing and exploding gradient problems due to weight clipping to enforce the 1-Lipschitz constraint on the discriminator. Hence, [19] proposes an improved version of WGAN enforcing the Lipschitz constraint through gradient penalty. Although those papers have demonstrated realistic looking images, they have not applied this idea to image feature generation. In this paper, we empirically show that images generated by the state-of-the-art GAN [19] are not ready to be used as training data for learning a classi\ufb01er. Hence, we propose a novel GAN architecture to directly generate CNN features that can be used to train a discriminative classi\ufb01er for zeroshot learning. Combining the powerful WGAN [19] loss and a classi\ufb01cation loss which enforces the generated features to be discriminative, our proposed GAN architecture improves the original GAN [18] by a large margin and has an edge over WGAN [19] thanks to our regularizer. ZSL and GZSL. In the zero-shot learning setting, the set of classes seen during training and evaluated during test are disjoint [22, 24, 25, 39, 47]. As supervised learning methods can not be employed for this task, [24, 39] proposed to solve it by solving related sub-problems. [50, 31, 8] learn unseen classes as a mixture of seen class proportions, and [2, 3, 14, 42, 45, 40, 16, 33, 1, 6, 17, 23] learn a compatibility between images and classes. On the other hand, instead of using only labeled data, [15, 38, 26] leverage unlabeled data from unseen classes in the transductive setting. While zero-shot learning has attracted a lot of attention, there has been little work [42, 9] in the more realistic generalized zero-shot learning setting, where both seen and unseen classes appear at test time. In this paper, we propose to tackle generalized zero-shot learning by generating CNN features for unseen classes via a novel GAN model. Our work is different from [20] because they generate additional examples for data-starved classes from feature vectors alone, which is unimodal and do not generalize to unseen classes. Our work is closer to [7] in which they generate features via GMMN [27]. Hence, we directly compare with them on the latest zeroshot learning benchmark [46] and show that WGAN [5] coupled with our proposed classi\ufb01cation loss can further improve GMMN in feature generation on most datasets for both ZSL and GZSL tasks. 3. Feature Generation & Classi\ufb01cation in ZSL Existing ZSL models only see labeled data from seen classes during training biasing the predictions to seen classes. The main insight of our proposed model is that by feeding additional synthetic CNN features of unseen classes, the learned classi\ufb01er will also explore the embedding space of unseen classes. Hence, the key to our approach is the ability to generate semantically rich CNN feature distributions conditioned on a class speci\ufb01c semantic vector e.g. attributes, without access to any images of that class. This alleviates the imbalance between seen and unseen classes, as there is no limit to the number of synthetic CNN features that our model can generate. It also allows to directly train a discriminative classi\ufb01er, i.e. Softmax classi\ufb01er, even for unseen classes. We begin by de\ufb01ning the problem of our interest. Let S = {(x, y, c(y))|x \u2208X, y \u2208Ys, c(y) \u2208C} where S stands for the training data of seen classes, x \u2208Rdx is the CNN features, y denotes the class label in Ys = {y1, . . . , yK} consisting of K discrete seen classes, and c(y) \u2208Rdc is the class embedding, e.g. attributes, of class y that models the semantic relationship between classes. In addition, we have a disjoint class label set Yu = {u1, . . . , uL} of unseen classes, whose class embedding set U = {(u, c(u))|u \u2208Yu, c(u) \u2208C} is available but images and image features are missing. Given S and U, the task of ZSL is to learn a classi\ufb01er fzsl : X \u2192Yu and in GZSL we learn a classi\ufb01er fgzsl : X \u2192Ys \u222aYu. \f3.1. Feature Generation In this section, we begin our discussion with Generative Adversarial Networks (GAN) [18] for it being the basis of our model. GAN consists of a generative network G and a discriminative network D that compete in a two player minimax game. In the context of generating image pixels, D tries to accurately distinguish real images from generated images, while G tries to fool the discriminator by generating images that are mistakable for real. Following [29], we extend GAN to conditional GAN by including a conditional variable to both G and D. In the following we give the details of the conditional GAN variants that we develop. Our novelty lies in that we develop three conditional GAN variants, i.e. f-GAN, f-WGAN and f-CLSWGAN, to generate image features rather than image pixels. It is worth noting that our models are only trained with seen class data S but can also generate image features of unseen classes. f-GAN. Given the train data S of seen classes, we aim to learn a conditional generator G : Z \u00d7 C \u2192X, which takes random Gaussian noise z \u2208Z \u2282Rdz and class embedding c(y) \u2208C as its inputs, and outputs a CNN image feature \u02dc x \u2208 X of class y. Once the generator G learns to generate CNN features of real images, i.e. x, conditioned on the seen class embedding c(y) \u2208Ys, it can also generate \u02dc x of any unseen class u via its class embedding c(u). Our feature generator f-GAN is learned by optimizing the following objective, min G max D LGAN =E[log D(x, c(y))]+ (1) E[log (1 \u2212D(\u02dc x, c(y)))], with \u02dc x = G(z, c(y)). The discriminator D : X \u00d7 C \u2192 [0, 1] is a multi-layer perceptron with a sigmoid function as the last layer. While D tries to maximize the loss, G tries to minimizes it. Although GAN has been shown to capture complex data distributions, e.g. pixel images, they are notoriously dif\ufb01cult to train [4]. f-WGAN. We extend the improved WGAN [19] to a conditional WGAN by integrating the class embedding c(y) to both the generator and the discriminator. The loss is, LW GAN =E[D(x, c(y))] \u2212E[D(\u02dc x, c(y))]\u2212 (2) \u03bbE[(||\u2207\u02c6 xD(\u02c6 x, c(y))||2 \u22121)2], where \u02dc x = G(z, c(y)), \u02c6 x = \u03b1x + (1 \u2212\u03b1)\u02dc x with \u03b1 \u223c U(0, 1), and \u03bb is the penalty coef\ufb01cient. In contrast to the GAN, the discriminative network here is de\ufb01ned as D : X \u00d7 C \u2192R, which eliminates the sigmoid layer and outputs a real value. The log in Equation 1 is also removed since we are not optimizing the log likelihood. Instead, the \ufb01rst two terms in Equation 2 approximate the Wasserstein distance, and the third term is the gradient penalty which enforces the gradient of D to have unit norm along the straight line CNN f-CLSWGAN z ~ N(0, 1) Head color: brown        Belly color: yellow Bill shape: pointy discriminator generator Head color: brown        Belly color: yellow Bill shape: pointy Figure 2: Our f-CLSWGAN: we propose to minimize the classi\ufb01cation loss over the generated features and the Wasserstein distance with gradient penalty. between pairs of real and generated points. Again, we solve a minmax optimization problem, min G max D LW GAN (3) f-CLSWGAN. f-WGAN does not guarantee that the generated CNN features are well suited for training a discriminative classi\ufb01er, which is our goal. We conjecture that this issue could be alleviated by encouraging the generator to construct features that can be correctly classi\ufb01ed by a discriminative classi\ufb01er trained on the input data. To this end, we propose to minimize the classi\ufb01cation loss over the generated features in our novel f-CLSWGAN formulation. We use the negative log likelihood, LCLS = \u2212E\u02dc x\u223cp\u02dc x[log P(y|\u02dc x; \u03b8)], (4) where \u02dc x = G(z, c(y)), y is the class label of \u02dc x, P(y|\u02dc x; \u03b8) denotes the probability of \u02dc x being predicted with its true class label y. The conditional probability is computed by a linear softmax classi\ufb01er parameterized by \u03b8, which is pretrained on the real features of seen classes. The classi\ufb01cation loss can be thought of as a regularizer enforcing the generator to construct discriminative features. Our full objective then becomes, min G max D LW GAN + \u03b2LCLS (5) where \u03b2 is a hyperparameter weighting the classi\ufb01er. 3.2. Classi\ufb01cation Given c(u) of any unseen class u \u2208Yu, by resampling the noise z and then recomputing \u02dc x = G(z, c(u)), arbitrarily many visual CNN features \u02dc x can be synthesized. After repeating this feature generation process for every unseen class, we obtain a synthetic training set \u02dc U = {(\u02dc x, u, c(u))}. We then learn a classi\ufb01er by training either a multimodal embedding model or a softmax classi\ufb01er. Our generated features allow to train those methods on the combinations of real seen class data S and generated unseen class data \u02dc U. \fMultimodal Embedding. Many zero-shot learning approaches, e.g. ALE [2], DEVISE [14], SJE [3], ESZSL [40] and LATEM [45], learn a multimodal embedding between the image feature space X and the class embedding space C using seen classes data S. With our generated features, those methods can be trained with seen classes data S together with unseen classes data \u02dc U to learn a more robust classi\ufb01er. The embedding model F(x, c(y); W), parameterized by W, measures the compatibility score between any image feature x and class embedding c(y) pair. Given a query image feature x, the classi\ufb01er searches for the class embedding with the highest compatibility via: f(x) = argmax y F(x, c(y); W), (6) where in ZSL, y \u2208Yu and in GZSL, y \u2208Ys \u222aYu. Softmax. The standard softmax classi\ufb01er minimizes the negative log likelihood loss, min \u03b8 \u22121 |T | X (x,y)\u2208T log P(y|x; \u03b8), (7) where \u03b8 \u2208Rdx\u00d7N is the weight matrix of a fully connected layer which maps the image feature x to N unnormalized probabilities with N being the number of classes, and P(y|x; \u03b8) = exp(\u03b8T y x) PN i exp(\u03b8T i x). Depending on the task, T = \u02dc U if it is ZSL and T = S \u222a\u02dc U if it is GZSL. The prediction function is: f(x) = arg max y P(y|x; \u03b8), (8) where in ZSL, y \u2208Yu and in GZSL, y \u2208Ys \u222aYu. 4. Experiments First we detail our experimental protocol, then we present (1) our results comparing our framework with the state of the art for GZSL and ZSL tasks on four challenging datasets, (2) our analysis of f-xGAN 1 under different conditions, (3) our large-scale experiments on ImageNet and (4) our comparison of image and image feature generation. Datasets. Caltech-UCSD-Birds 200-2011 (CUB) [44], Oxford Flowers (FLO) [30] and SUN Attribute (SUN) [32] are all \ufb01ne-grained datasets. CUB contains 11,788 images from 200 different types of birds annotated with 312 attributes. FLO dataset 8189 images from 102 different types of \ufb02owers without attribute annotations. However, for both CUB and FLO we use the \ufb01ne-grained visual descriptions collected by [35]. SUN contains 14,340 images from 717 scenes annotated with 102 attributes. Finally, Animals with Attributes (AWA) [24] is a coarse-grained dataset with 1We denote our f-GAN, f-WGAN, f-CLSWGAN as f-xGAN Dataset att stc |Ys| + |Yu| |Ys| |Yu| CUB [44] 312 Y 200 100 + 50 50 FLO [30] \u2013 Y 102 62 + 20 20 SUN [32] 102 N 717 580 + 65 72 AWA [24] 85 N 50 27 + 13 10 Table 1: CUB, SUN, FLO, AWA datasets, in terms of number of attributes per class (att), sentences (stc), number of classes in training + validation (Ys) and test classes (Yu). 30,475 images, 50 classes and 85 attributes. Statistics of the datasets are presented in Table 1. We use the zero-shot splits proposed by [46] for AWA, CUB and SUN insuring that none of the training classes are present in ImageNet [13]2. For FLO, we use the standard split provided by [35]. Features. As real CNN features, we extract 2048-dim toplayer pooling units of the 101-layered ResNet [21] from the entire image. We do not do any image pre-processing such as cropping or use any other data augmentation techniques. ResNet is pre-trained on ImageNet 1K and not \ufb01ne-tuned. As synthetic CNN features, we generate 2048-dim CNN features using our f-xGAN model. As the class embedding, unless it is stated otherwise, we use per-class attributes for AWA (85-dim), CUB (312-dim) and SUN (102-dim). Furthermore, for CUB and Flowers, we extract 1024-dim character-based CNN-RNN [35] features from \ufb01ne-grained visual descriptions (10 sentences per image). None of the Yu sentences are seen during training the CNN-RNN. We build per-class sentences by averaging the CNN-RNN features that belong to the same class. Evaluation Protocol. At test time, in the ZSL setting, the aim is to assign an unseen class label, i.e. Yu to the test image and in GZSL setting, the search space includes both seen or unseen classes, i.e. Ys \u222aYu. We use the uni\ufb01ed evaluation protocol proposed in [46]. In the ZSL setting, the average accuracy is computed independently for each class before dividing their cumulative sum by the number of classes; i.e., we measure average per-class top-1 accuracy (T1). In the GZSL setting, we compute average per-class top-1 accuracy on seen classes (Ys) denoted as s, average per-class top-1 accuracy on unseen classes (Yu) denoted as u and their harmonic mean, i.e. H = 2 \u2217(s \u2217u)/(s + u). Implementation details. In all f-xGAN models, both the generator and the discriminator are MLP with LeakyReLU activation. The generator consists of a single hidden layer with 4096 hidden units. Its output layer is ReLU because we aim to learn the top max-pooling units of ResNet-101. While the discriminator of f-GAN has one hidden layer with 1024 hidden units in order to stabilize the GAN training, the discriminators of f-WGAN and f-CLSWGAN have 2as ImageNet is used for pre-training the ResNet [21] \fZero-Shot Learning Generalized Zero-Shot Learning CUB FLO SUN AWA CUB FLO SUN AWA Classi\ufb01er FG T1 T1 T1 T1 u s H u s H u s H u s H DEVISE [14] none 52.0 45.9 56.5 54.2 23.8 53.0 32.8 9.9 44.2 16.2 16.9 27.4 20.9 13.4 68.7 22.4 f-CLSWGAN 60.3 60.4 60.9 66.9 52.2 42.4 46.7 45.0 38.6 41.6 38.4 25.4 30.6 35.0 62.8 45.0 SJE [3] none 53.9 53.4 53.7 65.6 23.5 59.2 33.6 13.9 47.6 21.5 14.7 30.5 19.8 11.3 74.6 19.6 f-CLSWGAN 58.4 67.4 56.5 66.9 48.1 37.4 42.1 52.1 56.2 54.1 36.7 25.0 29.7 37.9 70.1 49.2 LATEM [45] none 49.3 40.4 55.3 55.1 15.2 57.3 24.0 6.6 47.6 11.5 14.7 28.8 19.5 7.3 71.7 13.3 f-CLSWGAN 60.8 60.8 61.3 69.9 53.6 39.2 45.3 47.2 37.7 41.9 42.4 23.1 29.9 33.0 61.5 43.0 ESZSL [40] none 53.9 51.0 54.5 58.2 12.6 63.8 21.0 11.4 56.8 19.0 11.0 27.9 15.8 6.6 75.6 12.1 f-CLSWGAN 54.7 54.3 54.0 63.9 36.8 50.9 43.2 25.3 69.2 37.1 27.8 20.4 23.5 31.1 72.8 43.6 ALE [2] none 54.9 48.5 58.1 59.9 23.7 62.8 34.4 13.3 61.6 21.9 21.8 33.1 26.3 16.8 76.1 27.5 f-CLSWGAN 61.5 71.2 62.1 68.2 40.2 59.3 47.9 54.3 60.3 57.1 41.3 31.1 35.5 47.6 57.2 52.0 Softmax none \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 \u2013 f-CLSWGAN 57.3 67.2 60.8 68.2 43.7 57.7 49.7 59.0 73.8 65.6 42.6 36.6 39.4 57.9 61.4 59.6 Table 2: ZSL measuring per-class average Top-1 accuracy (T1) on Yu and GZSL measuring u = T1 on Yu, s = T1 on Ys, H = harmonic mean (FG=feature generator, none: no access to generated CNN features, hence softmax is not applicable). f-CLSWGAN signi\ufb01cantly boosts both the ZSL and GZSL accuracy of all classi\ufb01cation models on all four datasets. one hidden layer with 4096 hidden units as WGAN [19] does not have instability issues thus a stronger discriminator can be applied here. We do not apply batch normalization our empirical evaluation showed a signi\ufb01cant degradation of the accuracy when batch normalization is used. The noise z is drawn from a unit Gaussian with the same dimensionality as the class embedding. We use \u03bb = 10 as suggested in [19] and \u03b2 = 0.01 across all the datasets. 4.1. Comparing with State-of-the-Art In a \ufb01rst set of experiments, we evaluate our f-xGAN features in both the ZSL and GZSL settings on four challenging datasets: CUB, FLO, SUN and AWA. Unless it is stated otherwise, we use att for CUB, SUN, AWA and stc for FLO (as att are not available). We compare the effect of our feature generating f-xGAN to 6 recent stateof-the-art methods [46]. ZSL with f-CLSWGAN. We \ufb01rst provide ZSL results with our f-CLSWGAN in Table 2 (left). Here, the test-time search space is restricted to unseen classes Yu. First, our f-CLSWGAN in all cases improves the state of the art that is obtained without feature generation. The overall accuracy improvement on CUB is from 54.9% to 61.5%, on FLO from 53.4% to 71.2%, on SUN from 58.1% to 62.1% and on AWA from 65.6% to 69.9%, i.e. all quite signi\ufb01cant. Another observation is that feature generation is applicable to all the multimodal embedding models and softmax. These results demonstrate that indeed our f-CLSWGAN generates generalizable and strong visual features of previously unseen classes. GZSL with f-CLSWGAN. Our main interest is GZSL where the test time search space contains both seen and unseen classes, Ys \u222aYu, and at test time the images come both from seen and unseen classes. Therefore, we evaluate both seen and unseen class accuracy, i.e. s and u, as well as their harmonic mean (H). The GZSL results with f-CLSWGAN in Table 2 (right) demonstrate that for all datasets our f-xGAN signi\ufb01cantly improves the H-measure over the state-of-the-art. On CUB, f-CLSWGAN obtains 49.7% in H measure, signi\ufb01cantly improving the state of the art (34.4%), on FLO it achieves 65.6% (vs. 21.9%), on SUN it reaches 39.4% (vs. 26.3%), and on AWA it achieves 59.6% (vs. 27.5%). The accuracy boost can be attributed to the strength of the f-CLSWGAN generator learning to imitate CNN features of unseen classes although not having seen any real CNN features of these classes before. We also observe that without feature generation on all models the seen class accuracy is signi\ufb01cantly higher than unseen class accuracy, which indicates that many samples are incorrectly assigned to one of the seen classes. Feature generation through f-CLSWGAN \ufb01nds a balance between seen and unseen class accuracies by improving the unseen class accuracy while maintaining the accuracy on seen classes. Furthermore, we would like to emphasize that the simple softmax classi\ufb01er beats all the models and is now applicable to GZSL thanks to our CNN feature generation. This shows the true potential and generalizability of feature generation to various tasks. ZSL and GZSL with f-xGAN. The generative model is an important component of our framework. Here, we evalu\fDEVISE SJE LATEM ESZSL ALE Softmax Classification Model 0 20 40 60 80 100 Top-1 Acc. (in %) FLO NONE f-GAN f-GMMN f-WGAN f-CLSWGAN DEVISE SJE LATEM ESZSL ALE Softmax Classification Model 0 20 40 60 80 100 Top-1 Acc. (in %) CUB NONE f-GAN f-GMMN f-WGAN f-CLSWGAN (a) Zero-Shot Learning DEVISE SJE LATEM ESZSL ALE Softmax Classification Model 0 20 40 60 80 100 Harmonic Mean (in %) FLO NONE f-GAN f-GMMN f-WGAN f-CLSWGAN DEVISE SJE LATEM ESZSL ALE Softmax Classification Model 0 20 40 60 80 100 Harmonic Mean (in %) CUB NONE f-GAN f-GMMN f-WGAN f-CLSWGAN (b) Generalized Zero-Shot Learning Figure 3: Comparing f-xGAN versions with f-GMMN as well as comparing multimodal embedding methods with softmax. 0 100 200 300 Epoch 0 20 40 60 80 100 Top-1 Acc. (in %) FLO f-GAN f-GMMN f-WGAN f-CLSWGAN Real Data 0 100 200 300 Epoch 0 20 40 60 80 100 Top-1 Acc. (in %) CUB f-GAN f-GMMN f-WGAN f-CLSWGAN Real Data Figure 4: Measuring the seen class accuracy of the classi\ufb01er trained on generated features of seen classes w.r.t. the training epochs (with softmax). ate all versions of our f-xGAN and f-GMMN for it being a strong alternative. We show ZSL and GZSL results of all classi\ufb01cation models in Figure 3. We selected CUB and FLO for them being \ufb01ne-grained datasets, however we provide full numerical results and plots in the supplementary which shows that our observations hold across datasets. Our \ufb01rst observation is that for both ZSL and GZSL settings all generative models improve in all cases over \u201cnone\u201d with no access to the synthetic CNN features. This applies to the GZSL setting and the difference between \u201cnone\u201d and f-xGAN is strikingly signi\ufb01cant. Our second observation is that our novel f-CLSWGAN model is the best performing generative model in almost all cases for both datasets. Our \ufb01nal observation is that although f-WGAN rarely performs lower than f-GMMN, e.g. ESZL on FLO, our f-CLSWGAN which uses a classi\ufb01cation loss in the generator recovers from it and achieves the best result among all these generative models. We conclude from these experiments that generating CNN features to support the classi\ufb01er when there is missing data is a technique that is \ufb02exible and strong. We notice that recently [49] has shown great performance on the old splits of AWA and CUB datasets. We compare our method with [49] using the same evaluation protocol as our paper, i.e same data splits and evaluation metrics. On AWA, in ZSL task, the comparison is 66.1% vs 69.9% (ours) and in GZSL task, it is 41.4% vs 59.6% (ours). On CUB, in ZSL task, the comparison is 50.1% vs 61.5% (ours) and in GZSL task it is 29.2% vs 49.7% (ours). 1 2 6 10 30 50 100 200 300 # of generated features per class 10 20 30 40 50 60 70 Top-1 Acc. (in %) FLO f-GAN f-GMMN f-WGAN f-CLSWGAN 1 2 6 10 30 50 100 200 300 # of generated features per class 10 20 30 40 50 60 70 Top-1 Acc. (in %) CUB f-GAN f-GMMN f-WGAN f-CLSWGAN Figure 5: Increasing the number of generated f-xGAN features wrt unseen class accuracy (with softmax) in ZSL. 4.2. Analyzing f-xGAN Under Different Conditions In this section, we analyze f-xGAN in terms of stability, generalization, CNN architecture used to extract real CNN features and the effect of class embeddings on two \ufb01ne-grained datasets, namely CUB and FLO. Stability and Generalization. We \ufb01rst analyze how well different generative models \ufb01t the seen class data used for training. Instead of using Parzen window-based loglikelihood [18] that is unstable, we train a softmax classi\ufb01er with generated features of seen classes and report the classi\ufb01cation accuracy on a held-out test set. Figure 4 shows the classi\ufb01cation accuracy w.r.t the number of training epochs. On both datasets, we observe a stable training trend. On FLO, compared to the supervised classi\ufb01cation accuracy obtained with real images, i.e. the upper bound marked with dashed line, f-GAN remains quite weak even after convergence, which indicates that f-GAN has under\ufb01tting issues. A strong alternative is f-GMMN leads to a signi\ufb01cant accuracy boost while our f-WGAN and f-CLSWGAN improve over f-GMMN and almost reach the supervised upper bound. After having established that our f-xGAN leads to a stable training performance and generating highly descriptive features, we evaluate the generalization ability of the f-xGAN generator to unseen classes. Using the pre-trained model, we generate CNN features of unseen classes. We then train a softmax classi\ufb01er using these synthetic CNN features of unseen classes with real CNN features of seen \fCNN FG u s H GoogLeNet none 20.2 35.7 25.8 f-CLSWGAN 35.3 38.7 36.9 ResNet-101 none 23.7 62.8 34.4 f-CLSWGAN 43.7 57.7 49.7 Table 3: GZSL results with GoogLeNet vs ResNet-101 features on CUB (CNN: Deep Feature Encoder Network, FG: Feature Generator, u = T1 on Yu, s = T1 on Ys, H = harmonic mean, \u201cnone\u201d= no generated features). C FG u s H Attribute (att) none 23.7 62.8 34.4 f-CLSWGAN 43.7 57.7 49.7 Sentence (stc) none 38.8 53.8 45.1 f-CLSWGAN 50.3 58.3 54.0 Table 4: GZSL results with conditioning f-xGAN with stc and att on CUB (C: Class embedding, FG: Feature Generator, u = T1 on Yu, s = T1 on Ys, H = harmonic mean, \u201cnone\u201d= no generated features). classes. On the GZSL task, Figure 5 shows that increasing the number of generated features of unseen classes from 1 to 100 leads to a signi\ufb01cant boost of accuracy, e.g. 28.2% to 56.5% on CUB and 37.9% to 66.5% on FLO. As in the case for generating seen class features, here the ordering is f-GAN < f-WGAN < f-GMMN < f-CLSWGAN on CUB and f-GAN < f-GMMN < f-WGAN < f-CLSWGAN on FLO. With these results, we argue that if the generative model can generalize well to previously unseen data distributions, e.g. perform well on GZSL task, they have practical use in a wide range of real-world applications. Hence, we propose to quantitatively evaluate the performance of generative models on the GZSL task. Effect of CNN Architectures. The aim of this study is to determine the effect of the deep CNN encoder that provides real features to our f-xGAN discriminator. In Table 3, we \ufb01rst observe that with GoogLeNet features, the results are lower compared to the ones obtained with ResNet features. This indicates that ResNet features are stronger than GoogLeNet, which is expected. Besides, most importantly, with both CNN architectures we observe that our f-xGAN outperforms the \u201cnone\u201d by a large margin. Speci\ufb01cally, the accuracy increases from 25.8% to 36.9% for GoogleNet features and 34.4% to 49.7% for ResNet features. Those results are encouraging as they demonstrate that our f-xGAN is not limited to learning the distribution of ResNet-101 features, but also able to learn other feature distributions. Effect of Class Embeddings. The conditioning variable, 2H 3H M500 M1K M5K L500 L1K L5K All Top-1 Acc. (in %) 0 2 4 6 8 10 12 14 16 ZSL ALE Ours 2H 3H M500 M1K M5K L500 L1K L5K All Top-1 Acc. (in %) 0 1 2 3 4 5 GZSL ALE Ours Figure 6: ZSL and GZSL results on ImageNet (ZSL: T1 on Yu, GZSL: T1 on Yu). The splits, ResNet features and Word2Vec are provided by [46]. \u201cOurs\u201d = feature generator: f-CLSWGAN, classi\ufb01er: softmax. i.e. class embedding, is an important component of our f-xGAN. Therefore, we evaluate two different class embeddings, per-class attributes (att) and per-class sentences (stc) on CUB as this is the only dataset that has both. In Table 4, we \ufb01rst observe that f-CLSWGAN features generated with att not only lead to a signi\ufb01cantly higher result (49.7% vs 34.4%), s and u are much more balanced (57.7% and 43.7% vs. 62.8% and 23.7%) compared to the state-ofthe-art, i.e. \u201cnone\u201d. This is because generated CNN features help us explore the space of unseen classes whereas the state of the art learns to project images closer to seen class embeddings. Finally, f-CLSWGAN features generated with per-class stc signi\ufb01cantly improve results over att, achieving 54.0% in H measure, and also leads to a notable u of 50.3% without hurting s (58.3%). This is due to the fact that stc leads to high quality features [35] re\ufb02ecting the highly descriptive semantic content language entails and it shows that our f-CLSWGAN is able to learn higher quality CNN features given a higher quality conditioning signal. 4.3. Large-Scale Experiments Our large-scale experiments follow the same zero-shot data splits of [46] and serve two purposes. First, we show the generalizability of our approach by conducting ZSL and GZSL experiments on ImageNet [13] for it being the largest-scale single-label image dataset, i.e. with 21K classes and 14M images. Second, as ImageNet does not contain att, we use as a (weak) conditioning signal Word2Vec [28] to generate f-CLSWGAN features. Figure 6 shows that softmax as a classi\ufb01er obtains the state-of-theart of ZSL and GZSL on ImageNet, signi\ufb01cantly improving over ALE [2]. These results show that our f-CLSWGAN is able to generate high quality CNN features also with Word2Vec as the class embedding. For ZSL, for instance, with the 2H split \u201cOurs\u201d almost doubles the performance of ALE (5.38% to 10.00%) and in one of the extreme cases, e.g. with L1K split, the accuracy improves from 2.85% to 3.62%. For GZSL the same \fCUB FLO Generated Data u s H u s H none 38.8 53.8 45.1 13.3 61.6 21.9 Image (with [48]) 23.8 48.5 31.9 39.4 64.9 49.0 CNN feature (Ours) 50.3 58.3 54.0 59.0 73.8 65.6 Table 5: Summary Table (u = T1 on Yu, s = T1 accuracy on Ys, H = harmonic mean, class embedding = stc). \u201cnone\u201d: ALE with no generated features. observations hold, i.e. the gap between ALE and \u201cOurs\u201d is 2.18 vs 4.38 with 2H split and 1.21 vs 2.50 with L1K split. Note that, [46] reports the highest results with SYNC [8] and \u201cOurs\u201d improves over SYNC as well, e.g. 9.26% vs 10.00% with 2H and 3.23% vs 3.56% with L1K. With these results we emphasize that with a supervision as weak as a Word2Vec signal, our model is able to generate CNN features of unseen classes and operate at the ImageNet scale. This does not only hold for the ZSL setting which discards all the seen classes from the test-time search space assuming that the evaluated images will belong to one of the unseen classes. It also holds for the GZSL setting where no such assumption has been made. Our model generalizes to previously unseen classes even when the seen classes are included in the search space which is the most realistic setting for image classi\ufb01cation. 4.4. Feature vs Image Generation As our main goal is solving the GZSL task which suffers from the lack of visual training examples, one naturally thinks that image generation serves the same purpose. Therefore, here we compare generating images and image features for the task of GZSL. We use the StackGAN [48] to generate 256 \u00d7 256 images conditioned on sentences. In Table 5, we compare GZSL results obtained with \u201cnone\u201d, i.e. with an ALE model trained on real images of seen classes, Image, i.e. image features extracted from 256 \u00d7 256 synthetic images generated by StackGAN [48] and CNN feature, i.e. generated by our f-CLSWGAN. Between \u201cnone\u201d and \u201cImage\u201d, we observe that generating images of unseen classes improves the performance i.e. harmonic mean on FLO (49.0% for \u201cImage\u201d vs 21.9% for \u201cnone\u201d), but hurts the performance on CUB (31.9% for \u201cImage\u201d vs 45.1% for \u201cnone\u201d). This is because generating birds is a much harder task than generating \ufb02owers. Upon visual inspection, we have observed that although many images have an accurate visual appearance as birds or \ufb02owers, they lack the necessary discriminative details to be classi\ufb01ed correctly and the generated images are not class-consistent. On the other hand, generating CNN features leads to a signi\ufb01cant boost of accuracy, e.g. 54.0% on CUB and 65.6% on FLO which is clearly higher than having no generation, i.e. \u201cnone\u201d, and image generation. We argue that image feature generation has the following advantages. First, the number of generated image features is limitless. Second, the image feature generation learns from compact invariant representations obtained by a deep network trained on a large-scale dataset such as ImageNet, therefore the feature generative network can be quite shallow and hence computationally ef\ufb01cient. Third, generated CNN features are highly discriminative, i.e. they lead to a signi\ufb01cant boost in performance of both ZSL and GZSL. Finally, image feature generation is a much easier task as the generated data is much lower dimensional than high quality images necessary for discrimination. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1703.04394v2",
+                    "title": "Zero-Shot Learning -- The Good, the Bad and the Ugly",
+                    "abstract": "Due to the importance of zero-shot learning, the number of proposed\napproaches has increased steadily recently. We argue that it is time to take a\nstep back and to analyze the status quo of the area. The purpose of this paper\nis three-fold. First, given the fact that there is no agreed upon zero-shot\nlearning benchmark, we first define a new benchmark by unifying both the\nevaluation protocols and data splits. This is an important contribution as\npublished results are often not comparable and sometimes even flawed due to,\ne.g. pre-training on zero-shot test classes. Second, we compare and analyze a\nsignificant number of the state-of-the-art methods in depth, both in the\nclassic zero-shot setting but also in the more realistic generalized zero-shot\nsetting. Finally, we discuss limitations of the current status of the area\nwhich can be taken as a basis for advancing it.",
+                    "authors": "Yongqin Xian, Bernt Schiele, Zeynep Akata",
+                    "published": "2017-03-13",
+                    "updated": "2020-09-23",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Zero-shot learning aims to recognize objects whose instances may not have been seen during training [17, 21, 22, 29, 40]. The number of new zero-shot learning methods proposed every year has been increasing rapidly, i.e. the good aspects as our title suggests. Although each new method has been shown to make progress over the previous one, it is dif\ufb01cult to quantify this progress without an established evaluation protocol, i.e. the bad aspects. In fact, the quest for improving numbers has lead to even \ufb02awed evaluation protocols, i.e. the ugly aspects. Therefore, in this work, we propose to extensively evaluate a signi\ufb01cant number of recent zero-shot learning methods in depth on several small to large-scale datasets using the same evaluation protocol both in zero-shot, i.e. training and test classes are disjoint, and the more realistic generalized zero-shot learning settings, i.e. training classes are present at test time. We benchmark and systematically evaluate zero-shot learning w.r.t. three aspects, i.e. methods, datasets and evaluation protocol. The crux of the matter for all zeroshot learning methods is to associate observed and non observed classes through some form of auxiliary information which encodes visually distinguishing properties of objects. Different \ufb02avors of zero-shot learning methods that we evaluate in this work are linear [11, 2, 4, 31] and nonlinear [39, 33] compatibility learning frameworks whereas an orthogonal direction is learning independent attribute [21] classi\ufb01ers and \ufb01nally others [42, 7, 25] propose a hybrid model between independent classi\ufb01er learning and compatibility learning frameworks. We thoroughly evaluate the second aspect of zero-shot learning, by using multiple splits of several small to largescale datasets [27, 37, 21, 10, 9]. We emphasize that it is hard to obtain labeled training data for \ufb01ne-grained classes of rare objects recognizing which requires expert opinion. Therefore, we argue that zero-shot learning methods should be evaluated mainly on least populated or rare classes. We propose a uni\ufb01ed evaluation protocol to address the third aspect of zero-shot learning which is arguably the most important one. We emphasize the necessity of tuning hyperparameters of the methods on a validation class split that is disjoint from training classes as improving zero-shot learning performance via tuning parameters on test classes violates the zero-shot assumption. We argue that per-class averaged top-1 accuracy is an important evaluation metric when the dataset is not well balanced with respect to the number of images per class. We point out that extracting image features via a pre-trained deep neural network (DNN) on a large dataset that contains zero-shot test classes also violates the zero-shot learning idea as image feature extraction is a part of the training procedure. Moreover, we argue that demonstrating zero-shot performance on smallscale and coarse grained datasets, i.e. aPY [10] is not conclusive. We recommend to abstract away from the restricted nature of zero-shot evaluation and make the task more practical by including training classes in the search space, i.e. generalized zero-shot learning setting. Therefore, we argue that our work plays an important role in advancing the zeroshot learning \ufb01eld by analyzing the good and bad aspects of the zero-shot learning task as well as proposing ways to eliminate the ugly ones. 1 arXiv:1703.04394v2  [cs.CV]  23 Sep 2020 \f2. Related Work We review related work on zero-shot and generalized zero-shot learning, we present previous evaluations on the same task and describe the unique aspects of our work. Zero-Shot Learning. In zero-shot learning setting test and training class sets are disjoint [17, 21, 22, 29, 40] which can be tackled by solving related sub-problems, e.g. learning intermediate attribute classi\ufb01ers [21, 29, 30] and learning a mixture of seen class proportions [42, 43, 25, 7], or by a direct approach, e.g. compatibility learning frameworks [3, 4, 11, 15, 26, 31, 33, 39, 31, 12, 28, 1, 6, 23, 13, 20]. Among these methods, in our evaluation we choose to use DAP [21] for being one of the most fundamental methods in zeroshot learning research; CONSE [25] for being one of the most widely used representatives of learning a mixture of class proportions; SSE [42] for being a recent method with a public implementation; SJE [4], ALE [3], DEVISE [11] for being recent compatibility learning methods with similar loss functions; ESZSL [31] for adding a regularization term to unregularized compatibility learning methods; [39] and CMT [33] proposing non-linear extensions to bilinear compatibility learning framework and \ufb01nally SYNC [7] for reporting the state-of-the-art on several benchmark datasets. Generalized Zero-shot Learning. This setting [32] generalizes the zero-shot learning task to the case with both seen and unseen classes at test time. [18] argues that although ImageNet classi\ufb01cation challenge performance has reached beyond human performance, we do not observe similar behavior of the methods that compete at the detection challenge which involves rejecting unknown objects while detecting the position and label of a known object. [11] uses label embeddings to operate on the generalized zero-shot learning setting whereas [41] proposes to learn latent representations for images and classes through coupled linear regression of factorized joint embeddings. On the other hand, [5] introduces a new model layer to the deep net which estimates the probability of an input being from an unknown class and [33] proposes a novelty detection mechanism. We evaluate [33] and [11] for being the most widely used. Previous Evaluations of Zero-Shot Learning. In the literature some zero-shot vs generalized zero-shot learning evaluation works exist [29, 8]. Among these, [29] proposes a model to learn the similarity between images and semantic embeddings on the ImageNet 1K by using 800 classes for training and 200 for test. [8] provides a comparison between \ufb01ve methods evaluated on three datasets including ImageNet with three standard splits and proposes a metric to evaluate generalized zero-shot learning performance. Our work. We evaluate ten zero-shot learning methods on \ufb01ve datasets with several splits both for zero-shot and generalized zero-shot learning settings, provide statistical signi\ufb01cancy and robustness tests, and present other valuable insights that emerge from our benchmark. In this sense, ours is a more extensive evaluation compared to prior work. 3. Evaluated Methods We start by formalizing the zero-shot learning task and then we describe the zero-shot learning methods that we evaluate in this work. Given a training set S = {(xn, yn), n = 1...N}, with yn \u2208Ytr belonging to training classes, the task is to learn f : X \u2192Y by minimizing the regularized empirical risk: 1 N N X n=1 L(yn, f(xn; W)) + \u2126(W) (1) with L(.) being the loss function and \u2126(.) being the regularization term. Here, the mapping f : X \u2192Y from input to output embeddings is de\ufb01ned as: f(x; W) = argmax y\u2208Y F(x, y; W) (2) At test time, in zero-shot learning setting, the aim is to assign a test image to an unseen class label, i.e. Yts \u2282Y and in generalized zero-shot learning setting, the test image can be assigned either to seen or unseen classes, i.e. Ytr+ts \u2282Y with the highest compatibility score. 3.1. Learning Linear Compatibility Attribute Label Embedding (ALE) [3], Deep Visual Semantic Embedding (DEVISE) [11] and Structured Joint Embedding (SJE) [4] use bi-linear compatibility function to associate visual and auxiliary information: F(x, y; W) = \u03b8(x)T W\u03c6(y) (3) where \u03b8(x) and \u03c6(y), i.e. image and class embeddings, both of which are given. F(.) is parameterized by the mapping W, to be learned. Embarassingly Simple Zero Shot Learning (ESZSL) [31] adds a regularization term to this objective. In the following, we provide a uni\ufb01ed formulation of these four zero-shot learning methods. DEVISE [11] uses pairwise ranking objective that is inspired from unregularized ranking SVM [19]: X y\u2208Ytr [\u2206(yn, y) + F(xn, y; W) \u2212F(xn, yn; W)]+ (4) ALE [3] uses weighted approximate ranking objective [36]: X y\u2208Ytr lr\u2206(xn,yn) r\u2206(xn,yn) [\u2206(yn, y)+F(xn, y; W)\u2212F(xn, yn; W)]+ (5) 2 \fwhere lk = Pk i=1 \u03b1i and r\u2206(xn,yn) is de\ufb01ned as: X y\u2208Ytr 1(F(xn, y; W) + \u2206(yn, y) \u2265F(xn, yn; W)) (6) Following the heuristic in [38], [3] selects \u03b1i = 1/i which puts high emphasis on the top of the rank list. SJE [4] gives full weight to the top of the ranked list and is inspired from the structured SVM [35]: [ max y\u2208Ytr(\u2206(yn, y) + F(xn, y; W)) \u2212F(xn, yn; W)]+ (7) ESZSL [31] adds the following regularization term to the unregularized risk minimization formulation: \u03b3\u2225W\u03c6(y)\u22252 F ro + \u03bb\u2225\u03b8(x)T W\u22252 F ro + \u03b2\u2225W\u22252 F ro (8) where \u03b3, \u03bb, \u03b2 are parameters of this regularizer. 3.2. Learning Nonlinear Compatibility Latent Embeddings (LATEM) [39] and Cross Modal Transfer (CMT) [33] encode an additional non-linearity in compatibility learning framework. LATEM [39] constructs a piece-wise linear compatibility: F(x, y; Wi) = max 1\u2264i\u2264K \u03b8(x)T Wi\u03c6(y) (9) where every Wi models a different visual characteristic of the data and the selection of which matrix to use to do the mapping is a latent variable. LATEM uses the ranking loss formulated in Equation 4. CMT [33] \ufb01rst maps images into a semantic space of words, i.e. class names, where a neural network with tanh nonlinearity learns the mapping: X y\u2208Ytr X x\u2208Xy \u2225\u03c6(y) \u2212W1 tanh(W2.\u03b8(x)\u2225 (10) where (W1, W2) are weights of the two layer neural network. This is followed by a novelty detection mechanism that assigns images to unseen or seen classes. The novelty is detected either via thresholds learned using the embedded images of the seen classes or the outlier probabilities are obtained in an unsupervised way. 3.3. Learning Intermediate Attribute Classi\ufb01ers Although Direct Attribute Prediction (DAP) [21] has been shown to perform poorly compared to compatibility learning frameworks [3], we include it to our evaluation for being historically one of the most widely used methods in the literature. DAP [21] learns probabilistic attribute classi\ufb01ers and makes a class prediction by combining scores of the learned attribute classi\ufb01ers. A novel image is assigned to one of the unknown classes using: f(x) = argmax c M Y m=1 p(ac m|x) p(ac m) . (11) with M being the total number of attributes. We train a one-vs-rest SVM with log loss that gives probability scores of attributes with respect to training classes. 3.4. Hybrid Models Semantic Similarity Embedding (SSE) [42], Convex Combination of Semantic Embeddings (CONSE) [25] and Synthesized Classi\ufb01ers (SYNC) [7] express images and semantic class embeddings as a mixture of seen class proportions, hence we group them as hybrid models. SSE [42] leverages similar class relationships both in image and semantic embedding space. An image is labeled with: argmax u\u2208U \u03c0(\u03b8(x))T \u03c8(\u03c6(yu)) (12) where \u03c0, \u03c8 are mappings of class and image embeddings into a common space. Speci\ufb01cally, \u03c8 is learned by sparse coding and \u03c0 is by class dependent transformation. CONSE [25] learns the probability of a training image belonging to a training class: f(x, t) = argmax y\u2208Ytr ptr(y|x) (13) where y denotes the most likely training label (t=1) for image x. Combination of semantic embeddings (s) is used to assign an unknown image to an unseen class: 1 Z T X i=1 ptr(f(x, t)|x).s(f(x, t)) (14) where Z = PT i=1 ptr(f(x, t)|x), f(x, t) denotes the tth most likely label for image x and T controls the maximum number of semantic embedding vectors. SYNC [7] learns a mapping between the semantic class embedding space and a model space. In the model space, training classes and a set of phantom classes form a weighted bipartite graph. The objective is to minimize distortion error: min wc,vr \u2225wc \u2212 R X r=1 scrvr\u22252 2. (15) Semantic and model spaces are aligned by embedding real (wc) and phantom classes (vr) in the weighted graph (scr). 3 \fNumber of Classes Number of Images At Training Time At Evaluation Time SS PS SS PS Dataset Size Detail Att Y Ytr Yts Total Ytr Yts Ytr Yts Ytr Yts Ytr Yts SUN [27] medium \ufb01ne 102 717 580 + 65 72 14K 12900 0 10320 0 0 1440 2580 1440 CUB [37] medium \ufb01ne 312 200 100 + 50 50 11K 8855 0 7057 0 0 2933 1764 2967 AWA [21] medium coarse 85 50 27 + 13 10 30K 24295 0 19832 0 0 6180 4958 5685 aPY [10] small coarse 64 32 15 + 5 12 15K 12695 0 5932 0 0 2644 1483 7924 Table 1: Statistics for attribute datasets: SUN [27], CUB [37], AWA [21], aPY [10] in terms of size of the datasets, \ufb01negrained or coarse-grained, number of attributes, number of classes in training + validation (Ytr) and test classes (Yts), number of images at training and test time for standard split (SS) and our proposed splits (PS). 4. Datasets and Evaluation Protocol In this section, we provide several components of previously used and our proposed zero-shot and generalized zero-shot learning evaluation protocols, e.g. datasets, image and class encodings and the evaluation protocol. 4.1. Dataset Statistics Among the most widely used datasets for zero-shot learning, we select two coarse-grained, one small and one medium-scale, and two \ufb01ne-grained, both medium-scale, datasets with attributes and one large-scale dataset without. Here, we consider between 10K and 1M images, and, between 100 and 1K classes as medium-scale. Attribute Datasets. Statistics of the attribute datasets are presented in Table 1. Attribute Pascal and Yahoo (aPY) [10] is a small-scale coarse-grained dataset with 64 attributes. Among the total number of 32 classes, 20 Pascal classes are used for training (we randomly select 5 for validation) and 12 Yahoo classes are used for testing. Animals with Attributes (AWA) [21] is a coarse-grained dataset that is medium-scale in terms of the number of images, i.e. 30, 475 and small-scale in terms of number of classes, i.e. 50. [21] introduces a standard zero-shot split with 40 classes for training (we randomly select 13 for validation) and 10 for testing. AWA has 85 attributes. Caltech-UCSD-Birds 200-2011 (CUB) [37] is a \ufb01ne-grained and medium scale dataset with respect to both number of images and number of classes, i.e. 11, 788 images from 200 different types of birds annotated with 312 attributes. [3] introduces the \ufb01rst zero-shot split of CUB with 150 training (50 validation classes) and 50 test classes. SUN [27] is a \ufb01ne-grained and medium-scale dataset with respect to both number of images and number of classes, i.e. SUN contains 14340 images coming from 717 types of scenes annotated with 102 attributes. Following [21] we use 645 classes of SUN for training (we randomly select 65 for val) and 72 for testing. Large-Scale ImageNet. We also evaluate the performance of methods on the large scale ImageNet [9]. Among the total of 21K classes, 1K classes are used for training (we use 200 classes for validation) and the test split is either all the remaining 21K classes or a subset of it, e.g. we determine these subsets based on the hierarchical distance between classes and the population of classes. 4.2. Proposed Evaluation Protocol We present our proposed uni\ufb01ed protocol for image and class embeddings, dataset splits and evaluation criteria. Image and Class Embedding. We extract image features from the entire image for SUN, CUB, AWA and ImageNet, with no image pre-processing. For aPY, as proposed in [10], we extract image features from bounding boxes. Our image embeddings are 2048-dim top-layer pooling units of the 101-layered ResNet [16] as we found that it performs better than 1, 024-dim top-layer pooling units of GoogleNet [34]. ResNet is pre-trained on ImageNet 1K and not \ufb01ne-tuned. In addition to ResNet features, we evaluate methods with their published image features. As class embeddings, for aPY, AWA, CUB and SUN, we use per-class attributes. For ImageNet we use Word2Vec [24] provided by [7] as it does not contain attribute annotation for all the classes. Dataset Splits. Zero-shot learning assumes disjoint training and test classes with the presence of all the images of training classes and the absence of any image from test classes during training. On the other hand, as deep neural network (DNN) training for image feature extraction is actually a part of model training, the dataset used to train DNNs, e.g. ImageNet, should not include any of the test classes. However, we notice from the standard splits (SS) of aPY and AWA datasets that 7 aPY test classes out of 12 (monkey, wolf, zebra, mug, building, bag, carriage), 6 AWA test classes out of 10 (chimpanzee, giant panda, leopard, persian cat, pig, hippopotamus), are among the 1K classes of ImageNet, i.e. are used to pre-train ResNet. On the other hand, the mostly widely used splits, i.e. we term them as standard splits (SS), for SUN from [21] and CUB from [2] shows us that 1 CUB test class out of 50 (Indigo Bunting), 4 \fand 6 SUN test classes out of 72 (restaurant, supermarket, planetarium, tent, market, bridge), are also among the 1K classes of ImageNet. We noticed that the accuracy for all methods on those overlapping test classes are higher than others. Therefore, we propose new dataset splits, i.e. proposed splits (PS), insuring that none of the test classes appear in ImageNet 1K, i.e. used to train the ResNet model. We present the differences between the standard splits (SS) and the proposed splits (PS) in Table 1. While in SS and PS no image from test classes is present at training time, at test time SS does not include any images from training classes however our PS does. We designed the PS this way as evaluating accuracy on both training and test classes is crucial to show the generalization of methods. Note that we introduce Proposed Split Version 2.01. ImageNet with thousands of classes provides possibilities of constructing several zero-shot evaluation splits. Following [7], our \ufb01rst two standard splits consider all the classes that are 2-hops and 3-hops away from the original 1K classes according to the ImageNet label hierarchy, corresponding to 1509 and 7678 classes. This split measures the generalization ability of the models with respect to the hierarchical and semantic similarity between classes. Our proposed split considers 500, 1K and 5K most populated classes among the remaining 21K classes of ImageNet with \u22481756, \u22481624 and \u22481335 images per class on average. Similarly, we consider 500, 1K and 5K least-populated classes in ImageNet which correspond to most \ufb01ne-grained subsets of ImageNet with \u22481, \u22483 and \u224851 images per class on average. Our \ufb01nal split considers all the remaining \u224820K classes of ImageNet with at least 1 image per-class, \u2248631 images per class on average. Evaluation Criteria. Single label image classi\ufb01cation accuracy has been measured with Top-1 accuracy, i.e. the prediction is accurate when the predicted class is the correct one. If the accuracy is averaged for all images, high performance on densely populated classes is encouraged. However, we are interested in having high performance also on sparsely populated classes. Therefore, we average the correct predictions independently for each class before dividing their cumulative sum w.r.t the number of classes, i.e. we measure average per-class top-1 accuracy. In generalized zero-shot learning setting, the search space at evaluation time is not restricted to only test classes, but includes also the training classes, hence this setting is more practical. As with our proposed split at test time we have access to some images from training classes, after having computed the average per-class top-1 accuracy on training and test classes, we compute the harmonic mean of training and test accuracies: H = 2 \u2217(accYtr \u2217accYts)/(accYtr + accYts) (16) 1http://www.mpi-inf.mpg.de/zsl-benchmark SUN AWA Model R O R O DAP [21] 22.1 22.2 41.4 41.4 SSE [42] 83.0 82.5 64.9 76.3 LATEM [39] \u2013 \u2013 71.2 71.9 SJE [4] \u2013 \u2013 67.2 66.7 ESZSL [31] 64.3 65.8 48.0 49.3 SYNC [7] 62.8 62.8 69.7 69.7 Table 2: Reproducing zero-shot results: O = Original results published in the paper, R = Reproduced using provided image features and code. We measure top-1 accuracy in %. where accYtr and accYts represent the accuracy of images from seen (Ytr), and images from unseen (Yts) classes respectively. We choose harmonic mean as our evaluation criteria and not arithmetic mean because in arithmetic mean if the seen class accuracy is much higher, it effects the overall results signi\ufb01cantly. Instead, our aim is high accuracy on both seen and unseen classes. 5. Experiments We \ufb01rst provide zero-shot learning results on attribute datasets SUN, CUB, AWA and aPY and then on the largescale ImageNet dataset. Finally, we present results for the generalized zero-shot learning setting. 5.1. Zero-Shot Learning Results On attribute datasets, i.e. SUN, CUB, AWA and aPY, we \ufb01rst reproduce the results of each method using their evaluation protocol, then provide a uni\ufb01ed evaluation protocol using the same train/val/test class splits, followed by our proposed train/val/test class splits. We also evaluate the robustness of the methods to parameter tuning and visualize the ranking of different methods. Finally, we evaluate the methods on the large-scale ImageNet dataset. Reproducing Results. For sanity-check, we re-evaluate methods [21, 42, 39, 4, 31, 7] 2 using provided features and code. We chose SUN and AWA as two representative of \ufb01ne-grained and non-\ufb01ne-grained datasets having been widely used in the literature. We observe from the results in Table 2 that our reproduced results and the reported results of DAP and SYNC are identical to the reported number in their original publications. For LATEM, we obtain slightly different results which can be explained by the nonconvexity and thus the sensibility to initialization. Similarly for SJE random sampling in SGD might lead to slightly different results. ESZSL has some variance because its algorithm randomly picks a validation set during each run, which leads to different hyperparameters. Notable obser2[33] has public code available, but is not evaluated on SUN or AWA. 5 \fSUN CUB AWA aPY Method SS PS SS PS SS PS SS PS DAP [21] 38.9 39.9 37.5 40.0 57.1 44.1 35.2 33.8 CONSE [25] 44.2 38.0 36.7 33.6 63.6 46.3 25.9 26.4 CMT [33] 41.9 40.1 37.3 34.6 58.9 39.5 26.9 28.0 SSE [42] 54.5 51.5 43.7 43.9 68.8 60.1 31.1 35.0 LATEM [39] 56.9 55.3 49.4 49.6 74.8 55.1 34.5 36.8 ALE [3] 59.1 58.1 53.2 54.9 78.6 59.9 30.9 39.7 DEVISE [11] 57.5 56.5 53.2 52.0 72.9 54.2 35.4 37.0 SJE [4] 57.1 52.7 55.3 53.9 76.7 65.6 32.0 31.7 ESZSL [31] 57.3 54.5 55.1 51.9 74.7 58.2 34.4 38.3 SYNC [7] 59.1 56.2 54.1 56.0 72.2 51.8 39.7 23.9 Table 3: Zero-shot on SS = Standard Split, PS = Proposed Split Version 2.0 using ResNet features (top-1 accuracy). 5 1 2 1 2 1 3 4 3 1 1 5 1 1 4 1 2 3 1 2 3 1 1 4 3 1 3 1 1 3 4 1 2 1 2 9 2 3 6 1 2 10 5 6 1 1 2 3 4 5 6 7 8 9 10 Rank SYNC [2.6] DEVISE [3.7] ALE [3.8] SJE [3.9] LATEM [4.2] ESZSL [4.2] SSE [6.3] DAP [8.2] CONSE [9.0] CMT [9.0] 4 2 3 1 2 5 4 1 1 1 1 1 3 1 4 2 2 2 4 3 1 1 4 4 1 2 2 4 3 2 1 2 5 3 1 1 1 1 4 3 3 1 3 5 3 2 1 3 6 1 2 3 4 5 6 7 8 9 10 Rank ALE [2.1] DEVISE [3.2] SJE [4.2] LATEM [4.7] ESZSL [4.7] SSE [5.2] SYNC [5.3] DAP [7.7] CMT [8.8] CONSE [9.2] Figure 1: Ranking 10 models by setting parameters on three validation splits on the standard (SS, left) and proposed (PS Version 2.0, right) setting. Element (i, j) indicates number of times model i ranks at jth over all 4 \u00d7 3 observations. Models are ordered by their mean rank ( in brackets). vations on SSE [42] results are as follows. The published code has hard-coded hyperparameters operational on aPY, i.e. number of iterations, number of data points to train SVM, and one regularizer parameter \u03b3 which lead to inferior results than the ones reported here, therefore we set these parameters on validation sets. On SUN, SSE uses 10 classes (instead of 72) and our results with validated parameters got an improvement of 0.5% that may be due to random sampling of training images. On AWA, our reproduced result being 64.9% is signi\ufb01cantly lower than the reported result (76.3%). However, we could not reach the reported result even by tuning parameters on the test set, i.e. we obtain 73.8% in this case. Reproduced Results vs Standard Split (SS). In addition to [21, 42, 39, 4, 31, 7, 33], we re-implement [25, 11, 3] based on the original publications. We use train, validation, test splits as provided in Table 1 and report results on Table 3 with deep ResNet features. DAP [21] uses hand-crafted image features and thus reproduced results with those features are signi\ufb01cantly lower than the results with deep features (22.1% vs 38.9%). When we investigate the results in detail, we noticed two irregularities with reported results on SUN. First, SSE [42] and ESZSL [31] report results on a test split with 10 classes whereas the standard split of SUN contains 72 test classes (74.5% vs 54.5% with SSE [42] and 64.3% vs 57.3% with ESZSL [31]). Second, after careful examination and correspondence with the authors of SYNC [7], we detected that SUN features were extracted with a MIT Places [44] pre-trained model. As MIT Places dataset intersects with both training and test classes of SUN dataset, it is expected to lead to signi\ufb01cantly better results than ImageNet pre-trained model (62.8% vs 59.1%). Results on Standard (SS) and Proposed Splits (PS). We propose new dataset splits (see details in section 4) insuring that test classes do not belong to the ImageNet1K that is used to pre-train ResNet. We compare these results (PS) with previously published standard split (SS) results in Table 3. Our \ufb01rst observation is that the results on PS is signi\ufb01cantly lower than SS for AWA. This is expected as most of the test classes in SS is included in ImageNet 1K. On the other hand, for \ufb01ne-grained datasets CUB and SUN, the results are not signi\ufb01cantly effected. Our second observation regarding the method ranking is as follows. On SS, SYNC [7] is the best performing method on SUN (59.1%) and aPY (39.7%) datasets whereas SJE [4] performs the best on CUB (55.3%) and ALE [3] performs the best on AWA (78.6%) dataset. On PS, ALE [3] performs the best on SUN (58.1%) and aPY (39.7%), SYNC [7] on CUB (56.0%) and SJE [4] on AWA (65.6%). Note that ALE, SJE and DEVISE all use max-margin bi-linear compatibility learning framework. Robustness. We evaluate robustness of 10 methods to parameters by setting them on 3 different validation splits while keeping the test split intact. We report results on SS (Figure 2, top) and PS (Figure 2, bottom). On SUN and CUB, the results are stable across methods and across splits. This is expected as these datasets have balanced number of images across classes and due to their \ufb01ne-grained nature, the validation splits are similar. On the other hand, AWA and aPY being small and coarse-grained datasets have several issues. First, many of the test classes on AWA and aPY are included in ImageNet1K. Second, they are not well balanced, i.e. different validation class splits contain signi\ufb01cantly different number of images. Third, the class embeddings are far from each other, i.e. objects are semantically different, therefore different validation splits learn a different mapping between images and classes. Visualizing Method Ranking. We rank the 10 methods based on their per-class top-1 accuracy using the nonparametric Friedman test [14], which does not assume a distribution on performance but rather uses algorithm ranking. Each entry of the rank matrix on Figure 1 indicates the number of times the method is ranked at the \ufb01rst to tenth rank. We then compute the mean rank of each method and order them based on that. Our general observation is that the 6 \f1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) SUN 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) CUB 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) AWA 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) APY 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) APY DAP CONSE CMT SSE LATEM ALE DEVISE SJE ESZSL SYNC 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) SUN 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) CUB 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) AWA 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) APY 1 2 3 4 5 6 7 8 9 10 Models 0 20 40 60 80 Top-1 Acc. (in %) APY DAP CONSE CMT SSE LATEM ALE DEVISE SJE ESZSL SYNC Figure 2: Robustness of 10 methods evaluated on SUN, CUB, AWA, aPY using 3 validation set splits (results are on the same test split). Top: original split, Bottom: proposed split (Image embeddings = ResNet). We measure top-1 accuracy in %. Hierarchy Most Populated Least Populated All Method 2 H 3 H 500 1K 5K 500 1K 5K 20K CONSE [25] 7.63 2.18 12.33 8.31 3.22 3.53 2.69 1.05 0.95 CMT [33] 2.88 0.67 5.10 3.04 1.04 1.87 1.08 0.33 0.29 LATEM [39] 5.45 1.32 10.81 6.63 1.90 4.53 2.74 0.76 0.50 ALE [3] 5.38 1.32 10.40 6.77 2.00 4.27 2.85 0.79 0.50 DEVISE [11] 5.25 1.29 10.36 6.68 1.94 4.23 2.86 0.78 0.49 SJE [4] 5.31 1.33 9.88 6.53 1.99 4.93 2.93 0.78 0.52 ESZSL [31] 6.35 1.51 11.91 7.69 2.34 4.50 3.23 0.94 0.62 SYNC [7] 9.26 2.29 15.83 10.75 3.42 5.83 3.52 1.26 0.96 Table 4: ImageNet with different splits: 2/3 H = classes with 2/3 hops away from 1K Ytr, 500/1K/5K most populated classes, 500/1K/5K least populated classes, All=20K categories of ImageNet. We measure top-1 accuracy in %. highest ranked method on the standard split (SS) is SYNC while on the proposed split (PS) it is ALE. These results indicate the importance of choosing zero-shot splits carefully. On the proposed split, the three highest ranked methods are compatibility learning methods, i.e. ALE, DEVISE and SJE whereas the three lowest ranked methods are attribute classi\ufb01er learning or hybrid methods, i.e. DAP, CMT and CONSE. Therefore, max-margin compatibility learning methods lead to consistently better results in the zero-shot learning task compared to learning independent classi\ufb01ers. Results on ImageNet. ImageNet scales the methods to a truly large-scale setting, thus these experiments provide further insights on how to tackle the zero-shot learning problem from the practical point of view. Here, we evaluate 8 methods. We exclude DAP as attributes are not available for all ImageNet classes and SSE due to scalability issues of the public implementation of the method. Table 4 shows that the best performing method is SYNC [7] which may indicate that it performs well in large-scale setting or it can learn under uncertainty due to usage of Word2Vec instead of attributes. Another possibility is Word2Vec may be tuned for SYNC as it is provided by the same authors however making a strong claim requires a full evaluation on class embeddings which is out of the scope of this paper. Our general observation from all the methods is that in the most populated classes, the results are higher than the least populated classes which indicates that \ufb01ne-grained subsets are more dif\ufb01cult. We consistently observe a large drop in accuracy between 1K and 5K most populated classes which is expected as 5K contains \u22486.6M images, making the problem much more dif\ufb01cult than 1K (\u22481624 images). On the other hand, All 20K results are poor for all methods which indicates the dif\ufb01culty of this problem where there is a large room for improvement. 5.2. Generalized Zero-Shot Learning Results In real world applications, image classi\ufb01cation systems do not have access to whether a novel image belongs to a seen or unseen class in advance. Hence, generalized zeroshot learning is interesting from a practical point of view. Here, we use same models trained on zero-shot learning setting on our proposed splits (PS). We evaluate performance on both Ytr and Yts, i.e. using held-out images from Yts. As shown in Table 5, generalized zero-shot results are signi\ufb01cantly lower than zero-shot results as training classes are included in the search space. Another interesting observation is that compatibility learning frameworks, e.g. ALE, DEVISE, SJE, perform well on test classes. However, methods that learn independent attribute or object classi\ufb01ers, e.g. DAP and CONSE, perform well on training classes. Due to this discrepancy, we evaluate the harmonic 7 \fSUN CUB AWA aPY Method ts tr H ts tr H ts tr H ts tr H DAP [21] 4.2 25.1 7.2 1.7 67.9 3.3 0.0 88.7 0.0 4.8 78.3 9.0 CONSE [25] 6.8 35.9 11.4 2.0 70.6 3.9 0.4 89.6 0.8 0.0 91.2 0.0 CMT [33] 8.1 21.8 11.8 7.2 49.8 12.6 0.9 87.6 1.8 1.4 85.2 2.8 CMT* [33] 8.7 28.0 13.3 4.7 60.1 8.7 8.4 86.9 15.3 10.9 74.2 19.0 SSE [42] 2.1 36.4 4.0 8.5 46.9 14.4 7.0 80.5 12.9 0.3 78.4 0.6 LATEM [39] 14.7 28.8 19.5 15.2 57.3 24.0 7.3 71.7 13.3 1.3 71.4 2.6 ALE [3] 21.8 33.1 26.3 23.7 62.8 34.4 16.8 76.1 27.5 4.6 73.7 8.7 DEVISE [11] 16.9 27.4 20.9 23.8 53.0 32.8 13.4 68.7 22.4 3.5 78.4 6.7 SJE [4] 14.4 29.7 19.4 23.5 59.2 33.6 11.3 74.6 19.6 1.3 71.4 2.6 ESZSL [31] 11.0 27.9 15.8 14.7 56.5 23.3 6.6 75.6 12.1 2.4 70.1 4.6 SYNC [7] 7.9 43.3 13.4 11.5 70.9 19.8 9.0 88.9 16.3 7.4 66.3 13.3 Table 5: Generalized Zero-Shot Learning on Proposed Split Version 2.0 (PS) measuring ts = Top-1 accuracy on Yts, tr=Top-1 accuracy on Ytr+ts), H = harmonic mean (CMT*: CMT with novelty detection). We measure top-1 accuracy in %. 5 3 1 3 4 5 1 2 1 3 4 3 1 4 1 4 1 2 2 1 6 1 1 1 1 1 1 1 3 2 1 1 1 1 2 2 2 2 3 1 2 6 3 1 3 1 3 1 3 1 1 2 4 4 1 1 5 5 1 2 3 4 5 6 7 8 9 10 11 Rank ALE [2.2] DEVISE [2.3] SJE [4.0] ESZSL [5.2] LATEM [5.5] CMT* [5.6] SYNC [6.0] CMT [7.9] SSE [8.0] DAP [9.1] CONSE [10.2] 6 4 1 1 6 3 3 3 2 1 2 4 1 2 4 1 2 2 3 1 3 2 1 1 1 4 1 2 4 1 3 1 2 4 2 1 1 2 4 3 1 1 2 2 5 2 3 3 3 1 1 1 1 3 3 1 4 1 2 3 4 5 6 7 8 9 10 11 Rank CONSE [1.5] SYNC [4.6] ALE [5.1] DAP [5.5] CMT* [5.8] CMT [6.3] ESZSL [6.9] SSE [7.0] LATEM [7.7] DEVISE [7.8] SJE [7.9] 7 1 1 3 2 4 3 2 1 1 6 3 1 1 3 1 1 5 2 2 1 5 2 1 1 2 2 4 2 1 1 1 2 2 2 5 1 2 1 8 3 1 2 4 2 1 2 4 5 1 1 5 5 1 2 3 4 5 6 7 8 9 10 11 Rank ALE [2.0] DEVISE [2.7] SJE [3.8] ESZSL [5.3] SYNC [5.4] CMT* [5.5] LATEM [5.5] SSE [8.1] CMT [8.3] DAP [9.1] CONSE [10.2] Figure 3: Ranking 11 models on the proposed split (PS) in generalized zero-shot learning setting. Top-Left: on unseen cla sses (ts) accuracy, Top-Right: on seen classes (tr) accuracy, Bottom: on Harmonic mean (H). mean which takes a weighted average of training and test class accuracy. H measure ranks ALE as the best performing method on SUN, CUB and AWA datasets whereas on aPY dataset CMT* performs the best. Note that CMT* has an integrated novelty detection phase for which the method receives another supervision signal determining if the image belongs to a train or a test class. As a summary, generalized zero-shot learning setting provides one more level of detail on the performance of zero-shot learning methods. Our take-home message is that the accuracy of training classes is as important as the accuracy of test classes in real world scenarios. Therefore, methods should be designed in a way that they are able to predict labels well in train and test classes. Visualizing Method Ranking. Similar to the analysis in the previous section, we rank the 11 methods based on perclass top-1 accuracy on train classes, test classes and based on Harmonic mean of the two. Looking at the rank matrix obtained by evaluating on test classes, i.e. Figure 3 top left, highest ranked 5 methods are the same as in Figure 1, i.e. ALE, DEVISE, SJE, LATEM, ESZSL while overall the absolute numbers are lower. Looking at the rank matrix obtained by evaluating the harmonic mean, i.e. Figure 3 bottom, the highest ranked 3 methods are the same as in Figure 1, i.e. ALE, DEVISE, SJE. Looking at the rank matrix obtained by evaluating on train classes, i.e. Figure 3 top right, our observations are different from Figure 1. ALE is ranked the 3rd but other highest ranked methods are at the bottom of this rank list. These results clearly suggest that we should not only optimize for test class accuracy but also for train class accuracy when evaluating zero-shot learning. Our \ufb01nal observation from Figure 3 is that CMT* is better than CMT in all cases which supports the argument that a simple novelty detection scheme helps to improve results. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1603.08895v2",
+                    "title": "Latent Embeddings for Zero-shot Classification",
+                    "abstract": "We present a novel latent embedding model for learning a compatibility\nfunction between image and class embeddings, in the context of zero-shot\nclassification. The proposed method augments the state-of-the-art bilinear\ncompatibility model by incorporating latent variables. Instead of learning a\nsingle bilinear map, it learns a collection of maps with the selection, of\nwhich map to use, being a latent variable for the current image-class pair. We\ntrain the model with a ranking based objective function which penalizes\nincorrect rankings of the true class for a given image. We empirically\ndemonstrate that our model improves the state-of-the-art for various class\nembeddings consistently on three challenging publicly available datasets for\nthe zero-shot setting. Moreover, our method leads to visually highly\ninterpretable results with clear clusters of different fine-grained object\nproperties that correspond to different latent variable maps.",
+                    "authors": "Yongqin Xian, Zeynep Akata, Gaurav Sharma, Quynh Nguyen, Matthias Hein, Bernt Schiele",
+                    "published": "2016-03-29",
+                    "updated": "2016-04-10",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Zero-shot classi\ufb01cation [14, 20, 21, 30, 42] is a challenging problem. The task is generally set as follows: training images are provided for certain visual classes and the classi\ufb01er is expected to predict the presence or absence of novel classes at test time. The training and test classes are connected via some auxiliary, non visual source of information e.g. attributes. Combining visual information with attributes [7, 8, 11, 17, 20, 28, 27] has also supported \ufb01ne grained classi\ufb01cation. In \ufb01ne grained image collections, images that belong to different classes are visually similar to each other, e.g. different bird species. Image labeling for such collections is a costly process, as it requires either expert opinion or a large number of attributes. To overcome this limitation, recent works have explored distributed text representations [23, 29, 24] which are learned from general (or domain speci\ufb01c) text corpora. \u2217Currently with CSE, Indian Institute of Technology Kanpur. Majority of this work was done at Max Planck Institute for Informatics. cerulean  warbler W3 W2 Label space pine  grosbeak cardinal W1 Image space Text space Figure 1: LatEm learns multiple Wi\u2019s that maximize the compatibility between the input embedding (image, text space) and the output embedding (label space) of all training examples. The different Wi\u2019s may capture different visual characteristics of objects, i.e. color, beak shape etc. and allow distribution of the complexity among them, enabling the model to do better classi\ufb01cation. Substantial progress has been made for image classi\ufb01cation problem in the zero-shot setting on \ufb01ne-grained image collections [2]. This progress can be attributed to (i) strong deep learning based image features [19, 37] and (ii) learning a discriminative compatibility function between the structured image and class embeddings [1, 2, 12, 32]. The focus of this work is on the latter, i.e. on improving the compatibility learning framework, in particular via unsupervised auxiliary information. The main idea of structured embedding frameworks [1, 2, 12, 32] is to \ufb01rst represent both the images and the classes in some multi-dimensional vector spaces. Image embeddings are obtained from state-of-the-art image representations e.g. those from convolutional neural networks [19, 37]. Class embeddings can either (i) be obtained using manually speci\ufb01ed side information e.g. attributes [20], or (ii) extracted automatically [23, 29] from an unlabeled large text corpora. A discriminative bilinear compatibility function is then learned that pulls images from the same class close to each other and pushes images from different classes away from each other. Once learned, such a compatibility 1 arXiv:1603.08895v2  [cs.CV]  10 Apr 2016 \ffunction can be used to predict the class (more precisely, the embedding) of any given image (embedding). In particular, this prediction can be done for images from both seen and unseen classes, hence enabling zero-shot classi\ufb01cation. We address the \ufb01ne-grained zero-shot classi\ufb01cation problem while being particularly interested in more \ufb02exible unsupervised text embeddings. The state-of-the-art methods [1, 2, 12, 32] use a unique, globally linear compatibility function for all types of images. However, learning a linear compatibility function is not particularly suitable for the challenging \ufb01ne-grained classi\ufb01cation problem. For \ufb01ne-grained classi\ufb01cation, a model that can automatically group objects with similar properties together and then learn for each group a separate compatibility model is required. For instance, two different linear functions that separate blue birds with brown wings and from other blue birds with blue wings can be learned separately. To that end, we propose a novel model for zero-shot setting which incorporates latent variables to learn a piecewise linear compatibility function between image and class embeddings. The approach is inspired by many recent advances in visual recognition that utilize latent variable models, e.g. in object detection [10, 15], human pose estimation [40] and face detection [44] etc. Our contributions are as follows. (1) We propose a novel method for zero-shot learning. By incorporating latent variables in the compatibility function our method achieves factorization over such (possibly complex combinations of) variations in pose, appearance and other factors. Instead of learning a single linear function, we propose to learn a collection of linear models while allowing each image-class pair to choose from them. This effectively makes our model non-linear, as in different local regions of the space the decision boundary, while being linear, is different. We use an ef\ufb01cient stochastic gradient descent (SGD) based learning method. (2) We propose a fast and effective method for model selection, i.e. through model pruning. (3) We evaluate our novel piecewise linear model for zero-shot classi\ufb01cation on three challenging datasets. We show that incorporating latent variables in the compatibility learning framework consistently improves the state-of-the-art. The rest of the paper is structured as follows. In Sec. 3 we detail the bilinear compatibility learning framework that we base our method on. In Sec. 4 we present our novel Latent Embedding (LatEm) method. In Sec. 5 we present our experimental evaluation and in Sec. 6 we conclude. 2. Related Work We are interested in the problem of zero-shot learning where the test classes are disjoint from the training classes [14, 20, 21, 30, 42, 43]. As visual information from such test classes is not available during training, zero-shot learning requires secondary information sources to make up for the missing visual information. While secondary information can come from different sources, usually they are derived from either large and unrestricted, but freely available, text corpora, e.g. word2vec [23], glove [29], or structured textual sources e.g. wordnet hierarchies [24], or costly human annotations e.g. manually speci\ufb01ed attributes [7, 8, 11, 17, 20, 28, 27]. Attributes, such as \u2018furry\u2019, \u2018has four legs\u2019 etc. for animals, capture several characteristics of objects (visual classes) that help associate some and differentiate others. They are typically collected through costly human annotation [7, 17, 28] and have shown promising results [1, 3, 6, 20, 22, 33, 34, 41] in various computer vision problems. The image classi\ufb01cation problem, with a secondary stream of information, could be either solved by solving related sub-problems, e.g. attribute prediction [20, 30, 31], or by a direct approach, e.g. compatibility learning between embeddings [1, 12, 39]. One such factorization could be by building intermediate attribute classi\ufb01ers and then making a class prediction using a probabilistic weight of each attribute for each sample [20]. However, these methods, based on attribute classi\ufb01ers, have been shown to be suboptimal [1]. This is due to their reliance on binary mappings (by thresholding attribute scores) between attributes and images which causes loss in information. On the other hand, solving the problem directly, by learning a direct mapping between images and their classes (represented as numerical vectors) has been shown to be better suited. Such label embedding methods [1, 2, 12, 13, 25, 26, 32, 35] aim to \ufb01nd a mapping between two embedding spaces, one each for the two streams of information e.g. visual and textual. Among these methods, CCA [13] maximizes the correlation between these two embedding spaces, [26] learns a linear compatibility between an fMRI-based image space and the semantic space, [35] learns a deep non-linear mapping between images and tags, ConSe [25] uses the probabilities of a softmax-output layer to weight the vectors of all the classes, SJE [2] and ALE [1] learn a bilinear compatibility function using a multiclass [4] and a weighted approximate ranking loss [16] respectively. DeViSE [12] does the same, however, with an ef\ufb01cient ranking formulation. Most recently, [32] proposes to learn this mapping by optimizing a simple objective function which has closed form solution. We build our work on multimodal embedding methods. However, instead of learning a linear compatibility function, we propose a nonlinear compatibility framework that learns a collection of such linear models making the overall function piecewise linear. 3. Background: Bilinear Joint Embeddings In this section, we describe the bilinear joint embedding framework [39, 1, 2], on which we build our Latent Embedding Model (LatEm) (Sec. 4). \fWe work in a supervised setting where we are given an annotated training set T = {(x, y)|x \u2208X \u2282Rdx, y \u2208 Y \u2282Rdy} where x is the image embedding de\ufb01ned in an image feature space X, e.g. CNN features [19], and y is the class embedding de\ufb01ned in a label space Y that models the conceptual relationships between classes, e.g. attributes [9, 20]. The goal is to learn a function f : X \u2192Y to predict the correct class for the query images. In previous work [39, 1, 2], this is done via learning a function F : X \u00d7 Y \u2192R that measures the compatibility between a given input embedding (x \u2208X) and an output embedding (y \u2208 Y). The prediction function then chooses the class with the maximum compatibility, i.e. f(x) = arg max y\u2208Y F(x, y). (1) In general, the class embeddings re\ufb02ect the common and distinguishing properties of different classes using sideinformation that is extracted independently of images. Using these embeddings, the compatibility can be computed even with those unknown classes which have no corresponding images in the training set. Therefore, this framework can be applied to zero-shot learning [1, 2, 26, 32, 35]. In previous work, the compatibility function takes a simple form, F(x, y) = x\u22a4Wy (2) with the matrix W \u2208Rdx\u00d7dy being the parameter to be learnt from training data. Due to the bilinearity of F in x and y, previous work [1, 2, 39] refer to this model as a bilinear model, however one can also view it as a linear one since F is linear in the parameter W. In the following, these two terminologies will be used interchangeably depending on the context. 4. Latent Embeddings Model (LatEm) In general, the linearity of the compatibility function (Eq. (2)) is a limitation as the problem of image classi\ufb01cation is usually a complex nonlinear decision problem. A very successful extension of linear decision functions to nonlinear ones, has been through the use of piecewise linear decision functions. This idea has been applied successfully to various computer vision tasks e.g. mixture of templates [15] and deformable parts-based model [10] for object detection, mixture of parts for pose estimation [40] and face detection [44]. The main idea in most of such models, along with modeling parts, is that of incorporating latent variables, thus making the decision function piecewise linear, e.g. the different templates in the mixture of templates [15] and the different \u2018components\u2019 in the deformable parts model [10]. The model then becomes a collection of linear models and the test images pick one from these linear models, with the selection being latent. Intuitively, this factorizes the decision function into components which focus on distinctive \u2018clusters\u2019 in the data e.g. one component may focus on the pro\ufb01le view while another on the frontal view of the object. Objective. We propose to construct a nonlinear, albeit piecewise linear, compatibility function. Parallel to the latent SVM formulation, we propose a non-linear compatibility function as follows, F(x, y) = max 1\u2264i\u2264K \u02dc w\u22a4 i (x \u2297y), (3) where i = 1, . . . , K, with K \u22652, indexes over the latent choices and \u02dc wi \u2208Rdxdy are the parameters of the individual linear components of the model. This can be rewritten as a mixture of bilinear compatibility functions from Eq. (2) as F(x, y) = max 1\u2264i\u2264K x\u22a4Wiy. (4) Our main goal is to learn a set of compatibility spaces that minimizes the following empirical risk, 1 N |T | X n=1 L(xn, yn), (5) where L : X \u00d7 Y \u2192R is the loss function de\ufb01ned for a particular example (xn, yn) as L(xn, yn) = X y\u2208Y max{0, \u2206(yn, y)+F(xn, y)\u2212F(xn, yn)} (6) where \u2206(y, yn) = 1 if y \u0338= yn and 0 otherwise. This ranking-based loss function has been previously used in [12, 39] such that the model is trained to produce a higher compatibility between the image embedding and the class embedding of the correct label than between the image embedding and class embedding of other labels. Optimization. To minimize the empirical risk in Eq. (5), one \ufb01rst observes that the ranking loss function L from Eq. (6) is not jointly convex in all the Wi\u2019s even though F is convex. Thus, \ufb01nding a globally optimal solution as in the previous linear models [1, 2] is out of reach. To solve this problem, we propose a simple SGD-based method that works in the same fashion as in the convex setting. It turns out that our algorithm works well in practice and achieves state-of-the-art results as we empirically show in Sec. 5. We explain the details of our Algorithm 1 as follows. We loop through all our samples for a certain number of epochs T. For each sample (xn, yn) in the training set, we randomly select a y that is different from yn (step 3 of Algorithm 1). If the randomly selected y violates the margin (step 4 in Algorithm 1), then we update the Wi matrices (steps 5 \u221213 in Algorithm 1). In particular, we \ufb01nd the Wi \fAlgorithm 1 SGD optimization for LatEm T = {(x, y)|x \u2208Rdx, y \u2208Rdy} 1: for all t = 1 to T do 2: for all n = 1 to |T | do 3: Draw (xn,yn) \u2208T and y \u2208Y \\ {yn} 4: if F(xn, y) + 1 > F(xn, yn) then 5: i\u2217\u2190arg max 1\u2264k\u2264K x\u22a4 n Wky 6: j\u2217\u2190arg max 1\u2264k\u2264K x\u22a4 n Wkyn 7: if i\u2217= j\u2217then 8: W t+1 i\u2217 \u2190W t i\u2217\u2212\u03b7txn(y \u2212yn)\u22a4 9: end if 10: if i\u2217\u0338= j\u2217then 11: W t+1 i\u2217 \u2190W t i\u2217\u2212\u03b7txny\u22a4 12: W t+1 j\u2217 \u2190W t j\u2217+ \u03b7txny\u22a4 n 13: end if 14: end if 15: end for 16: end for that leads to the maximum score for y and the Wj that gives the maximum score for y. If the same matrix gives the maximum score (step 7 in Algorithm 1), we update that matrix. If two different matrices lead to the maximum score (step 9 in Algorithm 1), we update them both using SGD. Model selection. The number of matrices K in the model is a free parameter. We use two strategies to select the number of matrices. As the \ufb01rst method, we use a standard cross-validation strategy \u2013 we split the dataset randomly into disjoint parts (in a zero-shot setup) and choose the K with the best cross-validation performance. While this is a well established strategy which we \ufb01nd to work well experimentally, we also propose a pruning based strategy which is competitive while being faster to train. As the second method, we start with a large number of matrices and prune them as follows. As the training proceeds, each sampled training examples chooses one of the matrices for scoring \u2013 we keep track of this information and build a histogram over the number of matrices counting how many times each matrix was chosen by any training example. In particular, this is done by increasing the counter for Wj\u2217by 1 after step 6 of Algorithm 1. With this information, after \ufb01ve passes over the training data, we prune out the matrices which were chosen by less than 5% of the training examples, so far. This is based on the intuition that if a matrix is being chosen only by a very small number of examples, it is probably not critical for performance. With this approach we have to train only one model which adapts itself, instead of training multiple models for cross-validating K and then training a \ufb01nal model with the chosen K. Discussion. LatEm builds on the idea of Structured Joint Embeddings (SJE) [2]. We discuss below the differences between LatEm and SJE and emphasize our technical contributions. LatEm learns a piecewise linear compatibility function through multiple Wi matrices whereas SJE [2] is linear. With multiple Wi\u2019s the compatibility function has the freedom to treat different types of images differently. Let us consider a \ufb01xed class \u02c6 y and two substantially visually different types of images x1, x2, e.g. the same bird \ufb02ying and swimming. In SJE [2] these images will be mapped to the class embedding space with a single mapping W \u22a4x1, W \u22a4x2. On the other hand, LatEm will have learned two different matrices for the mapping i.e. W \u22a4 1 x1, W \u22a4 2 x2. While in the former case a single W has to map two visually, and hence numerically, very different vectors (close) to the same point, in the latent case such two different mappings are factorized separately and hence are arguably easier to perform. Such factorization is also expected to be advantageous when two classes sharing partial visual similarity are to be discriminated e.g. while blue birds could be relative easily distinguished from red birds, to do so for different types of blue birds is harder. In such cases, one of the Wi\u2019s could focus on color while another one could focus on the beak shape (in Sec. 5.2 we show that this effect is visible). The task of discrimination against different bird species would then be handled only by the second one, which would also arguably be easier. LatEm uses the ranking based loss [39] in Eq. (6) whereas SJE [2] uses the multiclass loss of Crammer and Singer [4] which replaces the P in Eq. (6) with max. The SGD algorithm for multiclass loss of Crammer and Singer [4] requires at each iteration a full pass over all the classes to search for the maximum violating class. Therefore it can happen that some matrices will not be updated frequently. On the other hand, the ranking based loss in Eq. (6) used by our LatEm model ensures that different latent matrices are updated frequently. Thus, the ranking based loss in Eq. (6) is better suited for our piecewise linear model. 5. Experiments We evaluate the proposed model on three challenging publicly available datasets of Birds, Dogs and Animals. First, we describe the datasets, then give the implementation details and \ufb01nally report the experimental results. Datasets. Caltech-UCSD Birds (CUB), Stanford Dogs (Dogs) are standard benchmarks of \ufb01ne-grained recognition [7, 5, 38, 18] and Animals With Attributes (AWA) is another popular and challenging benchmark dataset [20]. All these three datasets have been used for zero-shot learning [2, 30, 17, 42]. Tab. 1 gives the statistics for them. In zero-shot setting, the dataset is divided into three dis\fTotal train+val test imgs cls imgs cls imgs cls CUB 11786 200 8855 150 2931 50 AWA 30473 50 24293 40 6180 10 Dogs 19499 113 14681 85 4818 28 Table 1: The statistics of the three datasets used. CUB and Dog are \ufb01ne-grained datasets whereas AWA is a more general concept dataset. joint sets of train, val and test. For comparing with previous works, we follow the same train/val/test set split used by [2]. In zero-shot learning, where training and test classes are disjoint sets, to get a more stable estimate in our own results, we make four more splits by random sampling, while keeping the number of classes the same as before. We average results over the total of \ufb01ve splits. The average performance over the \ufb01ve splits is the default setting reported in all experiments, except where mentioned otherwise, e.g. in comparison with previous methods. Image and class embeddings. In our latent embedding (LatEm) model, the image embeddings (image features) and class embeddings (side information) are two essential components. To facilitate direct comparison with the state-of-the-art, we use the embeddings provided by [2]. Brie\ufb02y, as image embeddings we use the 1, 024 dimensional outputs of the top-layer pooling units of the pre-trained GoogleNet [37] extracted from the whole image. We do not do any task speci\ufb01c pre-processing on images such as cropping foreground objects. As class embeddings we evaluate four different alternatives, i.e. attributes (att), word2vec (w2v), glove (glo) and hierarchies (hie). Attributes [20, 9] are distinguishing properties of objects that are obtained through human annotation. For \ufb01ne-grained datasets such as CUB and Dogs, as objects are visually very similar to each other, a large number of attributes are needed. Among the three datasets used, CUB contains 312 attributes, AWA contains 85 attributes while Dogs does not contain annotations for attributes. Our attribute class embedding is a vector per-class measuring the strength of each attribute based on human judgment. In addition to human annotation the class embeddings can be constructed automatically from either a large unlabeled text corpora or through hierarchical relationship between classes. This has certain advantages such as we do not need any costly human annotation, however as a drawback, they tend not to perform as well as supervised attributes. One of our motivations for this work is that the class embeddings captured from a large text corpora contains latent relationships between classes and we would like to automatically learn these. Therefore, we evaluCUB AWA Dogs SJE LatEm SJE LatEm SJE LatEm att 50.1 45.5 66.7 71.9 N/A N/A w2v 28.4 31.8 51.2 61.1 19.6 22.6 glo 24.2 32.5 58.8 62.9 17.8 20.9 hie 20.6 24.2 51.2 57.5 24.3 25.2 Table 2: Comparison of Latent Embeddings (LatEm) method with the state-of-the-art SJE [2] method. We report average per-class Top-1 accuracy on unseen classes. We use the same data partitioning, same image features and same class embeddings as SJE [2]. We cross-validate the K for LatEm. ate three common methods for building unsupervised text embeddings. Word2Vec [23] is a two-layer neural network which predicts words given the context within a skip window slided through a text document. It builds a vector for each word in a learned vocabulary. Glove [29] is another distributed text representation method which uses co-occurrence statistics of words within a document. We use the pre-extracted word2vec and glove vectors from wikipedia provided by [2]. Finally, another way of building a vectorial structure for our classes is to use a hierarchy such as WordNet [24]. Our hierarchy vectors are based on the hierarchical distance between child and ancestor nodes, in WordNet, corresponding to our class names. For a direct comparison, we again use the hierarchy vectors provided by [2]. In terms of size, w2v and glo are 400 dimensional whereas hie is \u2248200 dimensional. Implementation details. Our image features are z-score normalized such that each dimension has zero mean and unit variance. All the class embeddings are l2 normalized. The matrices Wi are initialized at random with zero mean and standard deviation 1 \u221adx [1]. The number of epochs is \ufb01xed to be 150. The learning rates for the CUB, AWA and Dog datasets are chosen as \u03b7t = 0.1, 0.001, 0.01, respectively, and kept constant over iterations. For each dataset, these parameters are tuned on the validation set of the default dataset split and kept constant for all other dataset folds and for all class embeddings. As discussed in Sec 4, we perform two strategies for selecting the number of latent matrices K: cross-validation and pruning. When using crossvalidation, K is varied in {2, 4, 6, 8, 10} and the optimal K is chosen based the accuracy on a validation set. For pruning, K is initially set to be 16, and then at every \ufb01fth epoch during training, we prune all those matrices that support less than 5% of the data points. 5.1. Comparison with State-of-the-Art We now provide a direct comparison between our LatEm and the state-of-the-art SJE [2] method. SJE (Sec. 3) learns \fCUB AWA Dogs SJE LatEm SJE LatEm SJE LatEm w 51.7 47.4 73.9 76.1 N/A N/A w/o 29.9 34.9 60.1 66.2 35.1 36.3 Table 3: Combining embeddings either including or not including supervision in the combination. w: the combination includes attributes, w/o: the combination does not include attributes. a bilinear function that maximizes the compatibility between image and class embeddings. Our LatEm on the other hand learns a nonlinear, i.e. piece-wise linear function, through multiple compatibility functions de\ufb01ned between image and class embeddings. The results are presented in Tab. 2. Using the text embeddings obtained through human annotation, i.e. attributes (att), LatEm improves over SJE on AWA (71.9% vs. 66.7%) signi\ufb01cantly. However, as our aim is to reduce the accuracy gap between supervised and unsupervised class embeddings, we focus on unsupervised embeddings, i.e. w2v, glo and hie. On all datasets, LatEm with w2v, glo and hie improves the state-of-the-art SJE [2] signi\ufb01cantly. With w2v, LatEm achieves 31.8% accuracy (vs 28.4%) on CUB, 61.1% accuracy (vs 51.2%) on AWA and \ufb01nally 22.6% (vs 19.6%) on Dogs. Similarly, using glo, LatEm achieves 32.5% accuracy (vs 24.2%) on CUB, 62.9% accuracy (vs. 58.8%) on AWA and 20.9% accuracy (vs. 17.8%) on Dogs. Finally, while LatEm with hie on Dogs improves the result to 25.2% from 24.3%, the improvement is more signi\ufb01cant on CUB (24.2% from 20.6%) and on AWA (57.5% from 51.2%). These results establish our novel Latent Embeddings (LatEm) as the new state-ofthe-art method for zero-shot learning on three datasets in ten out of eleven test settings. They are encouraging, as they quantitatively show that learning piecewise linear latent embeddings indeed capture latent semantics on the class embedding space. Following [2] we also include a comparison when combining supervised and unsupervised embeddings. The results are given in Tab 3. First, we combine all the embeddings, i.e. att,w2v,glo,hie for AWA and CUB. LatEm improves the results over SJE signi\ufb01cantly on AWA (76.1% vs 73.9%). Second, we combine the unsupervised class embeddings, i.e. w2v,glo,hie, for all datasets. LatEm consistently improves over the combined embeddings obtained with SJE in this setting. On CUB combining w2v,glo,hie achieves 34.9% (vs 29.9%), on AWA, it achieves 66.2% (vs 60.1%) and on Dogs, it obtains 36.3% (vs 35.1%). These experiments show that the embeddings contain non-redundant information, therefore the results tend to improve by combining them. CUB AWA Dogs SJE LatEm SJE LatEm SJE LatEm att 49.5 45.6 70.7 72.5 N/A N/A w2v 27.7 33.1 49.3 52.3 23.0 24.5 glo 24.8 30.7 50.1 50.7 14.8 20.2 hie 21.4 23.7 43.4 46.2 24.6 25.6 Table 4: Average per-class top-1 accuracy on unseen classes (the results are averaged on \ufb01ve folds). SJE: [2], LatEm: Latent embedding model (K is cross-validated). Stability evaluation of zero-shot learning. Zero-shot learning is a challenging problem due to the lack of labeled training data. In other words, during training time, neither images nor class relationships of test classes are seen. As a consequence, zero-shot learning suffers from the dif\ufb01culty in parameter selection on a zero-shot set-up, i.e. train, val and test classes belong to disjoint sets. In order to get stable estimates of our predictions, we experimented on additional (in our case four) independently and randomly chosen data splits in addition to the standard one. Both with our LatEm and the publicly available implementation of SJE [2] we repeated the experiments \ufb01ve times. The results are presented in Tab 4. For all datasets, all the result comparisons between SJE and LatEm hold and therefore the conclusions are the same. Although the SJE outperforms LatEm with supervised attributes on CUB, LatEm outperforms the SJE results with supervised attributes on AWA and consistently outperforms all the SJE results obtained with unsupervised class embeddings. The details of our results are as follows. Using supervised class embeddings, i.e. attributes, on AWA, LatEm obtains an impressive 72.5% (vs 70.5%) and using unsupervised embeddings the highest accuracy is observed with w2v with 52.3% (vs 49.3%). On CUB, LatEm with w2v obtains the highest accuracy among the unsupervised class embeddings with 33.1% (vs 27.7%) On Dogs, LatEm with hie obtains the highest accuracy among all the class embeddings, i.e. 25.6% (vs 24.6%). These results insure that our accuracy improvements reported in Tab 2 were not due to a dataset bias. By augmenting the datasets with four more splits, our LatEm obtains a consistent improvement on all the class embeddings on all datasets over the state-of-the-art. Note that, for completion, in this section we provided a full comparison with the state-of-the-art on all class embeddings, including supervised attributes. However, there are two disadvantages of using attributes. First, since \ufb01negrained object classes share many common properties, we need a large number of attributes which is costly to obtain. Second, attribute annotations need to be done on a dataset basis, i.e. the attributes collected for birds do not work with dogs. Consequently, attribute based methods are not gen\fsmall bird with mostly yellow plumage sea bird with red eyes blue plumage with brown wings Glove embedding Word2vec embedding long and pointy beak brown head, light breast, small bird completely black plumage Figure 2: Top images ranked by the matrices using word2vec and glove on CUB dataset, each row corresponds to different matrix in the model. Qualitative examples support our intuition \u2013 each latent variable captures certain visual aspects of the bird. Note that, while the images may not belong to the same \ufb01ne-grained class, they share common visual properties. eralizable across datasets. Therefore, we are interested in the unsupervised text embeddings settings, i.e. w2v, glo, hie. Moreover, with these unsupervised embeddings, our LatEm outperforms the SJE on nine out of nine cases in all our datasets. For the following sections, we will present results only with w2v, glo and hie. 5.2. Interpretability of latent embeddings In Sec. 5.1, we have demonstrated that our novel latent embedding method improves the state-of-the-art of zeroshot classi\ufb01cation on two \ufb01ne-grained datasets of birds and dogs, i.e. CUB and Dogs, and one dataset of Animals, i.e. AWA. In this section, we zoom into the challenging CUB dataset and aim to investigate if individual Wi\u2019s learn visually consistent and interpretable latent relationships between images and classes. We use word2vec and glove as text embeddings. Fig 2 shows the top scoring images retrieved by three different Wi for the two embeddings i.e. w2v and glo. For w2v, we observe that the images scored highly by the same Wi (each row) share some visual aspect. The images in the \ufb01rst row are consistently of birds which have long and pointy beaks. Note that they belong to different classes; having a long and pointy beak is one of the shared aspect of birds of these classes. Similarly, for the second row, the retrieved images are of small birds with brown heads and light colored breasts and the last row contains large birds with completely black plumage. These results are interesting because although w2v is trained on wikipedia in an unsupervised manner with no notion of attributes, our LatEm is able to (1) infer hidden common properties of classes and (2) support them with visual evidence, leading to a data clustering which is optimized for classi\ufb01cation, however also performs well in retrieval. For glo, similar to the results with w2v, the top-scoring images using the same Wi consistently show distinguishing visual properties of classes. The \ufb01rst row shows blue birds although belonging to different species, are clustered together which indicates that this matrix captures the \u201cblue\u201dness of the objects. The second row has exclusively aquatic birds, surrounded by water. Finally, the third row has yellow birds only. Similar to w2v, although glo is trained in an unsupervised manner, our LatEm is able to bring out the latent information that re\ufb02ect object attributes and support this with its visual counterpart. These results clearly demonstrate that our model factorizes the information with visually interpretable relations between classes. 5.3. Pruning vs. cross-validation for model selection In this section we evaluate the performances obtained with the number of matrices in the model is \ufb01xed with pruning vs. cross-validation. Tab. 5 presents the number of matrices selected by two methods along with their performances on three datasets. In terms of performance, both methods are competitive. Pruning outperforms cross validation on \ufb01ve cases and is outperformed on the remaining six cases. The performance gaps are usually within 1-2% absolute, with the exception of AWA dataset with att and w2v with 72.5% vs. 70.7% and 52.3% vs. 49.3%, respectively for cross validation and pruning. Hence neither of the methods has a clear advantage in terms of performance, however cross validation is slightly better. In terms of the model size, cross validation seems to have a slight advantage. It selects a smaller model, hence more space and time ef\ufb01cient one, seven cases out of eleven. The trend is consistent for all class embeddings for the AwA dataset but is mixed for CUB and Dogs. The advantage of pruning over cross-validation is that it is much faster to train \u2013 while cross validation requires training and testing with multiple models (once each per every possible choice \fCUB AWA Dogs PR CV PR CV PR CV att 3 4 7 2 N/A N/A w2v 8 10 8 4 6 8 glo 6 10 7 6 9 4 hie 8 2 7 2 11 10 CUB AWA Dogs PR CV PR CV PR CV att 43.8 45.6 63.2 72.5 N/A N/A w2v 33.9 33.1 48.9 52.3 25.0 24.5 glo 31.5 30.7 51.6 50.7 18.8 20.2 hie 23.8 23.7 45.5 46.2 25.2 25.6 Table 5: (Left) Number of matrices selected (on the original split) and (right) average per-class top-1 accuracy on unseen classes (averaged over \ufb01ve splits). PR: proposed model learnt with pruning, CV: with cross validation. K 1 2 4 6 8 10 Top-1 Acc. (in %) 20 25 30 35 w2v glo hie Figure 3: Effect of latent variable K, with unsupervised class embeddings (on CUB dataset with \ufb01ve splits). of K), pruning just requires training once. There is however another free parameter in pruning i.e. choice of the amount of training data supporting a matrix for it to survive pruning. Arguably, it is more intuitive than setting directly the number of matrices to use instead of cross validating. 5.4. Evaluating the number of latent embeddings In Sec. 5.1, when we use multiple splits of the data, although the relative performance difference between the state of the art and our method has not changed, for some cases we observe a certain increase or decrease in accuracy. In this section, we investigate the experiments performed with \ufb01ve-folds on the CUB dataset and provide further analysis for a varying number of K. For completeness of the analysis, we also evaluate the single matrix case, namely K \u2208{1, 2, 4, 6, 8, 10} using unsupervised embeddings, i.e. w2v, glo, hie. Fig. 3 shows the performance of the model with a different number of matrices. We observe that the performance generally increases with increasing K, initially, and then the patterns differ with different embeddings. With w2v the performance keeps increasing until K = 6 and then starts decreasing, probably due to model over\ufb01tting. With glo the performance increases until K = 10 where the \ufb01nal accuracy is \u22485% higher than with K = 1. With the hie embedding the standard errors do not increase signi\ufb01cantly in any of the cases, are similar for all values of K and there is no clear trend in the performance. In conclusion, the variation in performance with K seems to depend of the embeddings used, however, in the zero-shot setting, depending on the data distribution the results may vary up to 5%. 6."
+                }
+            ],
+            "Xiaohua Zhai": [
+                {
+                    "url": "http://arxiv.org/abs/2303.15343v4",
+                    "title": "Sigmoid Loss for Language Image Pre-Training",
+                    "abstract": "We propose a simple pairwise Sigmoid loss for Language-Image Pre-training\n(SigLIP). Unlike standard contrastive learning with softmax normalization, the\nsigmoid loss operates solely on image-text pairs and does not require a global\nview of the pairwise similarities for normalization. The sigmoid loss\nsimultaneously allows further scaling up the batch size, while also performing\nbetter at smaller batch sizes. Combined with Locked-image Tuning, with only\nfour TPUv4 chips, we train a SigLiT model that achieves 84.5% ImageNet\nzero-shot accuracy in two days. The disentanglement of the batch size from the\nloss further allows us to study the impact of examples vs pairs and negative to\npositive ratio. Finally, we push the batch size to the extreme, up to one\nmillion, and find that the benefits of growing batch size quickly diminish,\nwith a more reasonable batch size of 32k being sufficient. We release our\nmodels at https://github.com/google-research/big_vision and hope our research\nmotivates further explorations in improving the quality and efficiency of\nlanguage-image pre-training.",
+                    "authors": "Xiaohua Zhai, Basil Mustafa, Alexander Kolesnikov, Lucas Beyer",
+                    "published": "2023-03-27",
+                    "updated": "2023-09-27",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Contrastive pre-training using weak supervision from image-text pairs found on the web is becoming the go-to method for obtaining generic computer vision backbones, slowly replacing pre-training on large labelled multi-class datasets. The high-level idea is to simultaneously learn an aligned representation space for images and texts using paired data. Seminal works CLIP [36] and ALIGN [23] established the viability of this approach at a large scale, and following their success, many large image-text datasets became available privately [59, 13, 21, 49] and publicly [40, 6, 15, 7, 41]. The standard recipe to pre-train such models leverages the image-text contrastive objective. It aligns the image and \u22c6equal contribution Table 1: SigLiT and SigLIP results. Sigmoid loss is memory ef\ufb01cient, allows larger batch sizes (BS) that unlocks language image pre-training with a small number of chips. SigLiT model with a frozen public B/8 checkpoint [42], trained on the LiT image-text dataset [59] using four TPUv4 chips for one day, achieves 79.7% 0-shot accuracy on ImageNet. The same setup with a g/14 checkpoint [58] leads to 84.5% accuracy, trained for two days. With a public unlocked B/16 image checkpoint [42], trained on the WebLI dataset [13], SigLIP achieves 71.0% 0-shot accuracy using 16 TPU-v4 chips for three days. The last two rows show results with randomly initialized models. Image Text BS #TPUv4 Days INet-0 SigLiT B/8 L\u2217 32 k 4 1 79.8 SigLiT g/14 L 20 k 4 2 84.5 SigLIP B/16 B 16 k 16 3 71.0 SigLIP B/16 B 32 k 32 2 72.1 SigLIP B/16 B 32 k 32 5 73.4 \u2217We use a variant of the L model with 12 layers. text embeddings for matching (positive) image-text pairs while making sure that unrelated (negative) image-text pairs are dissimilar in the embedding space. This is achieved via a batch-level softmax-based contrastive loss, applied twice to normalize the pairwise similarity scores across all images, then all texts. A naive implementation of the softmax is numerically unstable; it is usually stabilized by subtracting the maximum input value before applying the softmax [18], which requires another pass over the full batch. In this paper, we propose a simpler alternative: the sigmoid loss. It does not require any operation across the full batch and hence greatly simpli\ufb01es the distributed loss implementation and boosts ef\ufb01ciency. Additionally, it conceptually decouples the batch size from the de\ufb01nition of the task. We compare the proposed sigmoid loss with the standard softmax loss across multiple setups. In particular, we investigate sigmoid-based loss with two promi1 \fnent approaches for image-text learning: CLIP [36] and LiT [59], which we call sigmoid language image pretraining (SigLIP) and sigmoid LiT (SigLiT), respectively. We \ufb01nd that the sigmoid loss performs signi\ufb01cantly better than the softmax loss when the batch size is smaller than 16 k. As the train batch size grows, the gap closes. Importantly, the sigmoid loss is symmetric, requires just a single pass, and a typical implementation requires less memory than the softmax loss. This enables successful training of a SigLiT model at a batch size of one million. However, we \ufb01nd that the performance saturates with growing batch size, both for softmax and sigmoid. The good news is that a reasonable batch size, i.e. 32 k, is suf\ufb01cient for image-text pretraining. This conclusion also holds for multilingual SigLIP training on over 100 languages. In Table 1, we present setups for image-text pre-training that require a moderate amount of TPUv4 chips for training. SigLiT is surprisingly ef\ufb01cient, reaching 79.7% zero-shot accuracy on ImageNet in just a single day on four chips. SigLIP\u2019s more demanding from-scratch training reaches 73.4% zero-shot accuracy in 5 days with 32 TPUv4 chips. This compares favorably to prior works such as FLIP [30] and CLIP [36], which require approximately 5 and 10 days respectively on 256 TPUv3 cores. When \ufb01ne-tuning a pretrained vision backbone in SigLIP, denoted as in Table 1, we found that disabling the weight decay on the pre-trained backbone leads to better results (see Figure 4 for details). We hope our work paves the way for making the nascent language-image pre-training \ufb01eld more accessible. 2. Related Work Contrastive learning with the sigmoid loss. One prior work proposes a similar sigmoid loss for the task of unsupervised dimensionality reduction [19]; in the scope of contrastive image-text learning, the vast majority of works rely on the softmax-based InfoNCE loss as popularized by [46]. In supervised classi\ufb01cation, the sigmoid loss has already been shown to be slightly more effective and robust than the softmax loss [3, 51]. Contrastive language-image pre-training has become popular since CLIP [36] and ALIGN [23] applied softmax contrastive learning [60, 46, 10, 24] to large-scale imagetext datasets. Both models perform very well on zero-shot transfer tasks, including classi\ufb01cation and retrieval. Followup works show that contrastively pre-trained models produce good representations for \ufb01ne-tuning [53, 16], linear regression [23], object detection [31], semantic segmentation [33] and video tasks [57]. Generative language-image pre-training Besides softmax contrastive pre-training, various alternatives have been proposed. GIT [49], SimVLM [50], and LEMON [21] successfully pre-train models using a generative text decoder Algorithm 1 Sigmoid loss pseudo-implementation. 1 # img_emb : image model embedding [n, dim] 2 # txt_emb : text model embedding [n, dim] 3 # t_prime, b : learnable temperature and bias 4 # n : mini-batch size 5 6 t = exp(t_prime) 7 zimg = l2_normalize(img_emb) 8 ztxt = l2_normalize(txt_emb) 9 logits = dot(zimg, ztxt.T) * t + b 10 labels = 2 * eye(n) ones(n) # -1 with diagonal 1 11 l = -sum(log_sigmoid(labels * logits)) / n instead, while CoCa [56] adds such a decoder to the discriminative CLIP/ALIGN setup, thus combining the pros and cons of both approaches into a single very capable model. BLIP [28] further proposes CapFilt which uses the generative decoder to create better captions and the discriminative part of the model to \ufb01lter pairs. Language-Image pre-training is a very active \ufb01eld and surveys [8] rapidly become outdated. Ef\ufb01cient language-image pre-training On the other hand, few works have tried making language image pre-training more ef\ufb01cient. LiT [59] and FLIP [30] are notable attempts, the former requires a pre-trained and locked backbone, and the latter sacri\ufb01ces quality by randomly dropping visual tokens. BASIC [35] and LAION [52] look at scaling batchsize but only go up to 16 k and 160 k respectively, by using many hundreds of chips, and for the former also mixing in a large private classi\ufb01cation dataset [35, 55]. The recent Lion optimizer [12] claims to be able to reduce the training cost to reach similar quality. 3. Method In this section, we \ufb01rst review the widely-used softmaxbased contrastive loss. We then introduce the pairwise sigmoid loss and discuss its ef\ufb01cient implementation. Given a mini-batch B = {(I1, T1), (I2, T2), . . . } of image-text pairs, the contrastive learning objective encourages embeddings of matching pairs (Ii, Ti) to align with each other, while pushing embeddings of unmatched pairs (Ii, Tj\u0338=i) apart. For practical purposes, it is assumed that for all images i, the text associated with a different image j is not related to i, and vice-versa. This assumption is usually noisy and imperfect. 3.1. Softmax loss for language image pre-training When using the softmax loss to formalize this objective, an image model f(\u00b7) and a text model g(\u00b7) are trained to 2 \fDevice 1 Device 2 Device 3 I\u2081 I\u2082 I\u2083 I\u2084 I\u2085 I\u2086 I\u2087 I\u2088 I\u2089 I\u2081\u2080 I\u2081\u2081 I\u2081\u2082 Device 1 T\u2081 T\u2082 T\u2083 T\u2084 Device 2 T\u2085 T\u2086 T\u2087 T\u2088 Device 3 T\u2089 T\u2081\u2080 T\u2081\u2081 T\u2081\u2082 (a) Initially each device holds 4 image and 4 text representations. Each device needs to see the representations from other devices to calculate the full loss. Device 1 Device 2 Device 3 I\u2081 I\u2082 I\u2083 I\u2084 I\u2085 I\u2086 I\u2087 I\u2088 I\u2089 I\u2081\u2080 I\u2081\u2081 I\u2081\u2082 Device 1 T\u2081 + \u2013 \u2013 \u2013 T\u2082 \u2013 + \u2013 \u2013 T\u2083 \u2013 \u2013 + \u2013 T\u2084 \u2013 \u2013 \u2013 + Device 2 T\u2085 + \u2013 \u2013 \u2013 T\u2086 \u2013 + \u2013 \u2013 T\u2087 \u2013 \u2013 + \u2013 T\u2088 \u2013 \u2013 \u2013 + Device 3 T\u2089 + \u2013 \u2013 \u2013 T\u2081\u2080 \u2013 + \u2013 \u2013 T\u2081\u2081 \u2013 \u2013 + \u2013 T\u2081\u2082 \u2013 \u2013 \u2013 + \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 loss 33% 33% 33% 33% 33% 33% 33% 33% 33% 33% 33% 33% Device 1 Device 2 Device 3 (b) They each compute the component of the loss (highlighted) for their representations, which includes the positives. Device 1 Device 2 Device 3 I\u2081 I\u2082 I\u2083 I\u2084 I\u2085 I\u2086 I\u2087 I\u2088 I\u2089 I\u2081\u2080 I\u2081\u2081 I\u2081\u2082 Device 3 T\u2081 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 T\u2082 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 T\u2083 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 T\u2084 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 Device 1 T\u2085 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2086 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2087 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2088 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 Device 2 T\u2089 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2081\u2080 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2081\u2081 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2081\u2082 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 loss 66% 66% 66% 66% 66% 66% 66% 66% 66% 66% 66% 66% Device 1 Device 2 Device 3 (c) Texts are swapped across the devices, so device 1 now has I1:4 and T5:8 etc. The new loss is computed and accumulated with the previous. Device 1 Device 2 Device 3 I\u2081 I\u2082 I\u2083 I\u2084 I\u2085 I\u2086 I\u2087 I\u2088 I\u2089 I\u2081\u2080 I\u2081\u2081 I\u2081\u2082 Device 2 T\u2081 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2082 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2083 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 T\u2084 \u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713 Device 3 T\u2085 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 T\u2086 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 T\u2087 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 T\u2088 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 \u2013 \u2013 \u2013 \u2013 Device 1 T\u2089 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 T\u2081\u2080 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 T\u2081\u2081 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 T\u2081\u2082 \u2013 \u2013 \u2013 \u2013 \u2713\u2713\u2713\u2713\u2713\u2713\u2713\u2713 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 loss \u2713 \u2713 \u2713 \u2713 \u2713 \u2713 \u2713 \u2713 \u2713 \u2713 \u2713 \u2713 Device 1 Device 2 Device 3 \u2198 \u2193 \u2199 Cross Device \u03a3 (d) This repeats till every image & text pair have interacted, e.g. device 1 has the loss of I1:4 and T1:12. A \ufb01nal cross-device sum brings everything together. Figure 1: Ef\ufb01cient loss implementation demonstrated via a mock setup with 3 devices and a global batch size of 12. There are no all-gathers, and at any point in time only the bright yellow square (size 4 \u00d7 4) is materialized in memory. minimize the following objective: \u22121 2|B| |B| X i=1 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed image\u2192text softmax z }| { log etxi\u00b7yi P|B| j=1 etxi\u00b7yj + text\u2192image softmax z }| { log etxi\u00b7yi P|B| j=1 etxj\u00b7yi \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 where xi = f(Ii) \u2225f(Ii)\u22252 and yi = g(Ti) \u2225g(Ti)\u22252 . In this paper, we adopt the vision transformer architecture [17] for images and the transformer architecture [47] for texts. Note that due to the asymmetry of the softmax loss, the normalization is independently performed two times: across images and across texts [36]. The scalar t is parametrized as exp(t\u2032), where t\u2032 is a global freely learnable parameter. 3.2. Sigmoid loss for language image pre-training Instead of the softmax-based contrastive loss, we propose a simpler alternative that does not require computing global normalization factors. The sigmoid-based loss processes every image-text pair independently, effectively turning the learning problem into the standard binary classi\ufb01cation on the dataset of all pair combinations, with a positive labels for the matching pairs (Ii, Ti) and negative labels for all other pairs (Ii, Tj\u0338=i). It is de\ufb01ned as follows: \u22121 |B| |B| X i=1 |B| X j=1 log 1 1 + ezij(\u2212txi\u00b7yj+b) | {z } Lij where zij is the label for a given image and text input, which equals 1 if they are paired and \u22121 otherwise. At initialization, the heavy imbalance coming from the many negatives dominates the loss, leading to large initial optimization steps attempting to correct this bias. To alleviate this, we introduce an additional learnable bias term b similar to the temperature t. We initialize t\u2032 and b to log 10 and \u221210 respectively. This makes sure the training starts roughly close to the prior and does not require massive over-correction. Algorithm 1 presents a pseudocode implementation of the proposed sigmoid loss for language image pre-training. 3.3. Ef\ufb01cient \u201cchunked\u201d implementation Contrastive training typically utilizes data parallelism. Computing the loss when data is split across D devices necessitates gathering all embeddings [59] with expensive all-gathers and, more importantly, the materialization of a memory-intensive |B| \u00d7 |B| matrix of pairwise similarities. The sigmoid loss, however, is particularly amenable to a memory ef\ufb01cient, fast, and numerically stable implementation that ameliorates both these issues. Denoting the perdevice batch size as b = |B| D , the loss is reformulated as: \u22121 |B| D X di=1 |{z} A: \u2200device di B: swap negs across devices z}|{ D X dj=1 C: per device loss z }| { b(di+1) X i=bdi | {z } all local positives b(dj+1) X j=bdj | {z } negs from next device Lij This is particularly simple for the sigmoid loss as each pair is an independent term in the loss. Figure 1 illustrates this 3 \f2 8 32 262 1024 Batch Size (k) 81 82 83 84 85 ImageNet 0-shot SigLiT 4 8 16 32 98 307 Batch Size (k) 66 68 70 72 74 SigLIP Sigmoid Softmax 16 32 65 131 245 Batch Size (k) 30 31 32 33 34 35 36 XM T\u2192I 36 lang. avg. mSigLIP Figure 2: The effect of pre-training batch size. Left: SigLiT results, trained for 18B seen examples. Sigmoid loss outperforms the softmax loss signi\ufb01cantly with small batch sizes, and performs similarly at larger batch sizes. We successfully trained an SigLiT model with up to one million batch size. However, performance for both sigmoid and softmax saturate at around 32 k batch size. Middle: SigLIP results, trained for 9B seen examples. Both sigmoid loss and softmax loss saturate at a reasonable batch size, while the peak of the sigmoid loss comes earlier and slightly outperforms the peak of the softmax loss. A very large batch size hurts both losses. Right: mSigLIP results, trained for 30B seen examples. With a multilingual setup using over 100 languages, 32 k batch size is surprisingly suf\ufb01cient and scaling beyond that hurts performance on a 36-language cross-modal retrieval task. method. In words, we \ufb01rst compute the component of the loss corresponding to the positive pairs, and b \u22121 negative pairs. We then permute representations across devices, so each device takes negatives from its neighbouring device (next iteration of sum B). The loss is then calculated with respect to this chunk (sum C). This is done independently in each device, such that each device computes the loss with respect to its local batch b. Losses can then simply be summed across all devices (sum A). Individual collective permutes (for sum B) are fast (and indeed D collective permutes is typically faster than two all-gathers between D devices), and the memory cost at any given moment is reduced from |B|2 to b2 (for sum C). Usually b is constant as scaling |B| is achieved by increasing the number of accelerators. Due to being quadratic with respect to the batch size, the vanilla loss computation rapidly bottlenecks scaling up. This chunked approach enabled training with batch sizes over 1 million on relatively few devices. 4. Results In this section, we evaluate the proposed SigLiT and SigLIP models across a wide range of batch sizes. We discuss what can be achieved with a small number of accelerator chips, using both SigLiT and SigLIP recipes. We also brie\ufb02y discuss the impact of batch size on multilingual language image pre-training. We ablate the importance of our large-batch stabilization modi\ufb01cation and the introduced learned bias term and present a study on the effect of positive and negative pairs ratio in the sigmoid loss. Lastly, we explore SigLIP\u2019s data noise robustness. To validate our models, we report zero-shot transfer results on the ImageNet dataset [14] and zero-shot retrieval results across 36 languages on the XM3600 dataset [44]. We use the ScalingViT-Adafactor optimizer [58] by default for all our experiments. 4.1. SigLiT: Scaling batch size to the limit Following [59], we use the same precomputed embeddings for the images using a ViT-g vision model, and train a base size text tower from scratch with the same hyperparameters using the LiT image-text dataset [59]. We perform a study over a wide range of batch sizes, from 512 to 1 M, demonstrating the impact of batch size for contrastive learning. Results are presented in Figure 2 (left). When the batch size is smaller than 16 k, sigmoid loss outperforms softmax loss by a large margin. With growing batch sizes, we observe that softmax loss quickly catches up and potentially slightly underperforms sigmoid loss with a large enough batch size. Overall, we recommend using the SigLIP recipe for large batch sizes as well, due to the simplicity, compute savings, and straightforward memory ef\ufb01cient implementation. There is a consensus that contrastive learning bene\ufb01ts from large batch sizes, while most of the existing studies stop at 64 k batch size [59, 35, 10]. We successfully trained an SigLiT model at one million batch size, to explore the limit of contrastive learning. To our surprise, the performance saturates at 32 k batch size, further scaling up the batch size only gives a minor boost, and the model peaks at 4 \f450 900 3000 18'000 Examples Seen [M] 78 79 80 81 82 83 84 85 ImageNet 0-shot     8k 262k Sigmoid Softmax Figure 3: SigLiT ImageNet 0-shot transfer results with different training durations. Large batch size results in a big performance boost, but needs a suf\ufb01ciently long schedule to ramp up, as for short schedules, very large batch size results in a small number of gradient update steps. 256 k batch size. Our best SigLiT with a B-sized text mode achieves 84.7% zero-shot transfer accuracy on ImageNet, while the original LiT paper reports a slightly better 85.2% score with a 10 times larger g-sized text model. Figure 3 presents the impact of training duration for different batch sizes. It demonstrates that large, 262 k batch size signi\ufb01cantly outperforms smaller 8 k batch size when trained for a suf\ufb01ciently long time. Note, that for short training durations, large batch size leads to the fewer absolute number of update steps and thus needs more time to ramp up. 4.2. SigLIP: Sigmoid loss is bene\ufb01cial for languageimage pre-training We pre-train SigLIP models on the WebLI dataset [13], using only English image and text pairs. We use CLIP (WebLI) to denote the CLIP baseline pre-trained on WebLI with the standard softmax loss. We use moderately-sized models: B/16 ViT for image embeddings and B-sized transformer for text embeddings. The input images are resized to 224\u00d7224 resolution. The text is tokenized by a 32 k vocabulary sentencepiece tokenizer [27] trained on the English C4 dataset [37], and a maximum of 16 text tokens are kept. Figure 2 middle plot shows SigLIP results, With less than 32 k batch size, SigLIP outperforms CLIP (WebLI) baselines. On the other end of the scale, the memory ef\ufb01ciency of the sigmoid loss enabled much larger batch sizes. For example, with four TPU-v4 chips, we could \ufb01t a batch size of 4096 with a Base SigLIP but only 2048 with a corresponding CLIP model. The two advantages together demonstrate signi\ufb01cant bene\ufb01ts of the sigmoid loss for language image pre-training with \ufb01xed resources, which will be discussed in Section 4.5. 16 k 32 k 64 k 128 k 240 k INet-0 71.6 73.2 73.2 73.2 73.1 XM avg 34.8 34.9 34.4 33.6 32.7 XM de 54.7 54.8 55.4 54.3 54.7 XM en 46.5 46.2 46.5 46.6 46.6 XM hi 9.1 8.5 7.9 8.1 7.3 XM ru 50.1 49.9 49.7 48.6 49.3 XM zh 30.7 32.5 32.0 30.6 23.7 Table 2: Multilingual SigLIP results with various batch sizes, pre-trained for 30 billion seen examples. We report zero-shot transfer results on ImageNet (INet-0) and averaged text to image retrieval results across 36 languages on the crossmodal 3600 dataset (XM). The full table on 36 languages can be found in Appendix. As batch size increases, the gap between the sigmoid and the softmax losses diminish. SigLIP performs best at batch size 32 k, whereas the softmax loss required 98 k for optimal performance and still didn\u2019t outperform the sigmoid based variant. Scaling further, a larger batch size like 307 k hurts both losses. 4.3. mSigLIP: Multi-lingual pre-training We further scale up the training data by keeping all the 100 languages from the WebLI dataset [13]. With multilingual data, one usually needs to use a larger international vocabulary. We \ufb01rst verify the impact of two tokenizers: a small multilingual vocabulary with 32 k tokens [37], and a large multilingual vocabulary with 250 k tokens [54]. We train B-sized ViT and text models for 900 M total examples seen, and observe slightly more than 1% improvement when using a larger vocabulary. However, the token embeddings become huge for very large vocabulary sizes. Following the standard setup, we would need to store a N \u00d7W token embedding lookup table to train the multilingual model, where N is the vocabulary size mentioned above and W is the embedding dimension of the text model. To save memory, we propose to use a \u201cbottlenecked\u201d token embedding. We use N \u00d7 K embedding matrix and additional K \u00d7 W projection, where the bottleneck K is much smaller than W. In our experiments, we observed that using a large multilingual vocabulary with a bottleneck can be scaled up as ef\ufb01ciently as using a small multilingual vocabulary. Specifically, by enabling the bottleneck of size K = 96 for Base architecture with W = 768, we only see about a half percent quality drop on ImageNet zero-shot transfer, compared to using the full 250k vocabulary. 5 \f1 2 4 8 16 24 0 10 20 30 40 50 60 70 INet 0-shot 1 2 4 8 16 24 Examples Seen [100M] 0 10 20 30 40 50 60 70 INet 10-shot from-scratch fine-tune fine-tune w/o enc.wd Figure 4: Top: SigLIP with pre-trained encoders ramps up quickly. However, only disabling weight decay on the pretrained encoder weights leads to stable behavior and good ImageNet 0-shot transfer results. Bottom: ImageNet 10shot transfer results, where decaying the pre-trained weights leads to deterioration of the pre-trained model visual representation quality. Disabling weight decay \ufb02attens the curve. With the memory improvements, we train mSigLIP models for various batch sizes, for a total of 30 billion examples seen. Table 2 and Figure 2 (right plot) show the results. We were expecting a large batch size to improve multilingual pre-training, where the model sees more examples from the same language as hard negatives in a single mini-batch. However, we didn\u2019t observe clear improvements with a batch size larger than 32 k. A batch size of 32 k is suf\ufb01cient for a multilingual setup as well. On the XM3600 cross-modal retrieval tasks, we found that going beyond 32 k batch size leads to worse results on average while on ImageNet zero-shot transfer it stays \ufb02at. mSigLIP sets the new state-of-the-art on XM3600 text to image retrieval task, with only a Base size model. Our best result is 34.9%, which is more than 6% higher than the previously reported result 28.5% [13] with a standard LiT model [59] using a much larger four billion ViT-e model. We further scale up mSigLIP training in Section 4.6. 4.4. SigLiT with four TPU-v4 chips For many practitioners, the important question usually is \u201cwhat can be trained with a limited amount of resources?\u201d We explore the usage of SigLiT models in this section with only four TPU-v4 chips, as the memory ef\ufb01cient sigmoid loss is suitable for this application scenario. 3 4 5 6 Loss L \u03b22 =0.999 \u03b22 =0.95 1 10 ||\u2207wL|| 1B 2B 3B 4B 5B Examples seen 2 4 ||\u0394w|| Figure 5: The effect of Adam and AdaFactor\u2019s \u03b22. As we increase batch-size, we observe more frequent training instability. This instability seen in the loss curves (top) is caused by spikes in gradient norm (middle) leading to large parameter updates (bottom). Decreasing the \u03b22 momentum stabilizes training. Occasional gradient spikes still happen (see step at 2B), but do not destabilize the training process. We follow the same setup as in section 4.1. We use the publicly available ViT-AugReg-B/8 [42] model as the frozen ( ) vision tower, and precompute embeddings to accelerate the training [59]. The text model is a Large Transformer, but with a depth of only 12 layers (instead of 24). It is trained using the LION [12] optimizer with decoupled weight decay 1 \u00d7 10\u22127, linearly warm-up of learning rate over 6.5k steps up to a peak of 1 \u00d7 10\u22124, followed by a cosine decay to 0. We train for a total of 65 000 steps with a batch size of 32k \u2013 this leads to just under one day of training. Table 1 shows the results when training a model on four chips for one day, achieving 79.7% 0-shot ImageNet classi\ufb01cation accuracy; very competitive in this limited resource regime. With a ViT-g/14 [58] model as the vision tower and a Large text tower, we can train at 20 k batch size on four chips for 107 k steps in under two days. This further pushes the 0-shot ImageNet classi\ufb01cation accuracy up to 84.5%. 4.5. SigLIP with a small amount of TPU-v4 chips It\u2019s resource demanding to train a CLIP model fromscratch in general, with SigLIP it\u2019s possible to \ufb01t a larger train batch size with fewer amount of chips. In this section, we explore ways to train SigLIP models ef\ufb01ciently with pretrained weights. We use pre-trained weights to initialize the image model to accelerate the pre-training, which was orig6 \f1 : 16k 1 : 1.6k 1 : 164 1 : 16 1 : 1.6 60 70 80 ImageNet 0-shot Random Hard Hard, matched pairs Easy 1 : 16k 1 : 1.6k 1 : 164 1 : 16 1 : 1.6 \u221215 \u221210 \u22125 0 Learned bias 1 : 16k 1 : 1.6k 1 : 164 1 : 16 1 : 1.6 \u221220 \u221215 \u221210 \u22125 0 5 Average logit of pos and neg Figure 6: The effect of batch composition. We simulate various batch compositions by masking out negatives, either randomly, keeping only the hardest, or the easiest. With no masking, we have 16 k negatives for each positive in the batch (1:16 k) and the strongest masking we apply (1:1.6) results in almost balanced minibatches. In one setting we match total pairs seen by training for signi\ufb01cantly longer. We observe ImageNet 0-shot score, the \ufb01nal value of the learned bias, and the average logits of positive and negative pairs. Overall, the imbalance does not seem to be detrimental, but \ufb01nding an ef\ufb01cient way of mining negatives might be bene\ufb01cial. inally discussed in [59]. We use the public and unlocked ViT-AugReg-B/16 [42] model to initialize our vision tower and \ufb01ne-tune on the same WebLI English data as used for SigLIP. In all the experiments, we apply a 0.1 learning rate multiplier to the pre-trained image tower to make it suitable for \ufb01ne-tuning. Figure 4 presents unlocked \ufb01ne-tuning results alongside from-scratch randomly initialized baselines. We used 16 TPU-v4 chips and train at 16 k batch size for 2.4 B examples seen. We found that the \ufb01ne-tuning setup doesn\u2019t perform well out-of-the-box; this is consistent with prior works [59] where \ufb01netuning image models degraded visual representation quality. This is evidenced by ImageNet 10shot linear classi\ufb01cation, where in Figure 4 the \ufb01ne-tuned setup is barely better than the from-scratch baseline. We hypothesize that the default weight decay applied to the pre-trained weights reduces their effectiveness. Motivated by the \ufb01ne-tuning recipe from [17, 58, 25], that uses no weight decay, we also propose disabling weight decay on the pre-trained weights for SigLIP training. Weight decay is therefore only applied to the randomly initialized weights in the text model. This simple modi\ufb01cation signi\ufb01cantly improved SigLIP results. Figure 4 shows that with our improved recipe, SigLIP reaches 71% 0-shot accuracy on ImageNet, using 16k batch size, trained on 16 chips for three days. We also present from-scratch results in the bottom rows of Table 1: with 32 TPUv4 chips for only two days, SigLIP achieves 72.1% 0-shot accuracy. This presents a signi\ufb01cant training cost reduction e.g. compared to CLIP (approx. 2500 TPUv3-days for 72.6%) reported in [30]. 4.6. Scaling up SigLIP and mSigLIP In this section, we scale up SigLIP by \u201covertraining\u201d the model [45, 1]. We present results in Table 3 using ViT-B, ViT-L or So-400m [1] as the vision encoder, with a text encoder of the same size (B, L and So-400m respectively). Following the recipe described in Section 4.2, we train both models for 40 billion examples seen at batch size 32 k, but use (256/16)2 = 256 image patches and 64 text tokens (instead of 16). To get SigLIP models for different resolutions, we train for 5 billion more examples at the target resolution, with a 100x smaller learning rate and no weight decay. In Table 3, we report zero-shot classi\ufb01cation results on ImageNet [14], ObjectNet [2], ImageNet-v2 [39], ImageNet ReaL [3], and zero-shot image-to-text (I\u2192T) retrieval, textto-image (I\u2192T) retrieval results on MSCOCO [11]. We also scale up the multilingual mSigLIP ViT-B model in the same way. We report image-text retrieval results across 36 languages on the XM3600 benchmark [44]. The scaled-up mSigLIP ViT-B model achieves the state-of-theart 42.6% image retrieval recall@1 and 54.1% text retrieval recall@1 for a Base model. This is slightly outperformed by the Large model in [48] getting 42.96% image retrieval recall@1. Detailed results are provided in Appendix Table 9 and Figure 8, denoted as *32 k. 4.7. Stabilizing large-batch training As we move to large batch sizes, the language image pretraining using transformers becomes increasingly more unstable, even when using a modestly-sized model (e.g. Base size). The reason for these instabilities is large spikes in the 7 \fMethod Image Encoder ImageNet-1k COCO R@1 ViT size # Patches Validation v2 ReaL ObjectNet I \u2192T T \u2192I CLIP B 196 68.3 61.9 55.3 52.4 33.1 OpenCLIP B 196 70.2 62.3 56.0 59.4 42.3 EVA-CLIP B 196 74.7 67.0 62.3 58.7 42.2 SigLIP B 196 76.2 69.6 82.8 70.7 64.4 47.2 SigLIP B 256 76.7 70.0 83.1 71.3 65.1 47.4 SigLIP B 576 78.6 72.1 84.5 73.8 67.5 49.7 SigLIP B 1024 79.2 73.0 84.9 74.7 67.6 50.4 CLIP L 256 75.5 69.0 69.9 56.3 36.5 OpenCLIP L 256 74.0 61.1 66.4 62.1 46.1 CLIPA-v2 L 256 79.7 72.8 71.1 64.1 46.3 EVA-CLIP L 256 79.8 72.9 75.3 63.7 47.5 SigLIP L 256 80.5 74.2 85.9 77.9 69.5 51.1 CLIP L 576 76.6 72.0 70.9 57.9 37.1 CLIPA-v2 L 576 80.3 73.5 73.1 65.5 47.2 EVA-CLIP L 576 80.4 73.8 78.4 64.1 47.9 SigLIP L 576 82.1 75.9 87.0 81.0 70.6 52.7 OpenCLIP G (2B) 256 80.1 73.6 73.0 67.3 51.4 CLIPA-v2 H (630M) 576 81.8 75.6 77.4 67.2 49.2 EVA-CLIP E (5B) 256 82.0 75.7 79.6 68.8 51.1 SigLIP SO (400M) 729 83.2 77.2 87.5 82.9 70.2 52.0 Table 3: Comparison with other publicly released models. Our SigLIP models outperform all prior models, e.g. OpenCLIP [22] and CLIP [36], by a signi\ufb01cant margin on both zero-shot classi\ufb01cation and retrieval tasks. Compared to the concurrent EVA-CLIP [43] and CLIPA-v2 [29], our SigLIP-L performs better across the board, in both the low and high resolution cases. Especially noteworthy is the Shape-Optimized 400M parameter ViT [1] architecture, which outperforms all signi\ufb01cantly larger models. We publicly release our models: https://github.com/google-research/big_vision. gradient norms, which translate to large-magnitude changes in the weights that may destabilize the training process, see Figure 5. We observe that reducing \u03b22 in Adam and AdaFactor from its default 0.999 to 0.95 (which was suggested in [20, 9]) is enough to stabilize the training. Intuitively, this allows recovering from gradient spikes quicker. We opt for setting \u03b22 = 0.95 for all our experiments. 4.8. Negative ratio in sigmoid loss One question which arises when shifting the perspective from the softmax\u2019s \u201cpick the right class\u201d view to the sigmoid\u2019s \u201crate this pair\u201d view, is the imbalance in positive versus negative pairs. For a batch size |B|, the batch contains |B| positive pairs, but |B|2 \u2212|B| negative examples. In the modest batch-size of 16 k, there are actually 268 M negative examples for only 16 k positive ones. At the same time, because the sigmoid loss decomposes into a sum of per-example losses, we can perform controlled experiments to study the effect of the mini-batch composition and distribution of examples visited. We run experiments in the SigLiT setup at batch-size 16 k for 900 M steps and vary the composition of the batch by masking out (i.e. ignoring) enough negative examples to reach a target \u201cpositive : negative\u201d ratio, masking in the following ways: \u2022 Random: Randomly choose negative pairs to mask. \u2022 Hard: Keep hardest negative pairs (highest loss). \u2022 Easy: Keep easiest negatives pairs (lowest loss). \u2022 Hard + matching total pairs seen: Masking examples while training for a \ufb01xed number of steps does decrease the total number of pairs seen during training. Hence in the matched pairs setting, we increase the number of training steps by the masking ratio in order to keep the number of pairs seen constant. Figure 6 shows the effect of the various masking strategies. Randomly removing negatives to rebalance does deteriorate performance. Keeping the easiest examples does not work at all, while keeping the hardest negatives does almost 8 \f0.0 0.2 0.4 0.50 0.52 0.54 0.56 ImageNet 0shot Image 0.0 0.2 0.4 Text Sigmoid Softmax 0.0 0.2 0.4 p(corruption) Batch 0.0 0.1 0.2 Image & Text 0.0 0.1 0.2 Image, Text & Batch Figure 7: Sigmoid-training increases robustness to data noise. Titles show the type of corruption applied, and x-axes show the probability with which they are applied. With increasing corruption severity, M-scale models trained with sigmoid loss for 3.6 billion examples retain superiority over corresponding softmax baseline. maintain the quality, indicating that, as could be expected, a lot of the learning on the negative side comes from the harder examples. This is further con\ufb01rmed by the slightly increased performance of training longer on the hardest examples in order to match the total pairs seen. We also look at the value of the learned bias at the end of training as well as the average logit value for positive and negative examples across these settings, and \ufb01nd the result mostly follows what one would expect: as fewer negatives are present, the bias and logits become more positive overall. Interestingly, when training with more hard negative pairs, the average logits of positive pairs stays mostly \ufb02at. This study con\ufb01rms that (1) the imbalance does not seem to be a major reason for concern, while at the same time (2) coming up with an ef\ufb01cient way of including more negative examples can be promising but is not trivial. 4.9. Bias term in sigmoid loss We ablate the bias term in the loss function, using the Base architecture with an 8 k batch size, trained for 900M examples with the SigLIP setup. Zero-shot transfer results are reported on ImageNet [14], Oxford-iiit pet [34] and Cifar100 [26]. Table 4 presents results with and without a bias term in the sigmoid loss. Table 4: Bias (b) and temperature (t\u2032) initialization. Results are reported using Base architecture, 8 k batch size, trained for 900M examples. Enabling the bias term b with \u221210 initialization improves results consistently. b t\u2032 INet-0 Pet-0 C100-0 n/a log 10 62.0 81.8 59.9 -10 log 10 63.0 82.4 61.0 -10 log 1 61.0 80.0 60.4 0 log 10 61.7 79.9 59.0 0 log 1 53.7 73.2 53.8 Enabling the bias term with a \u221210 initialization consistently improves performance across all tasks. This is because the bias term ensures that the training starts close to the prior, preventing dramatic over-correction in early optimization. In contrast, a randomly chosen bias term initialization, such as the 0 initialization in Table 4, fails to address the over-correction issue, leading to signi\ufb01cantly worse results. This effect is particularly noticeable when using a small temperature t\u2032 initialization. We set the bias and temperature initialization to b = \u221210 and t\u2032 = log 10 (hence t = 10) as the default for all experiments. 4.10. Label noise robustness Prior works demonstrated improved robustness against label noise when using the sigmoid loss for classi\ufb01cation models [3]. This property would be particularly useful here in the face of the famously noisy nature of popular largescale image-text datasets. In order to study this for SigLIP, we train M/16 image models alongside an M text model at batch size 16384 for 3.6 billion seen examples. We corrupt the training data using one of the following methods: \u2022 Image: With probability p, replace the image with uniform random noise. \u2022 Text: With probability p, replace tokenized text with a new sequence of randomly sampled tokens, up to some (sampled) sequence length. \u2022 Batch alignment: Randomly shuf\ufb02e the ordering of p% of the batch. \u2022 Image & text: Apply both with probability p each. \u2022 Image, text & batch: Alongside (4), also shuf\ufb02e fraction p of alignments. Results from varying the likelihood of the corruption are shown in Figure 7. Models trained with sigmoid loss are increasingly robust to all kinds of added noise. 9 \f5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2111.07991v3",
+                    "title": "LiT: Zero-Shot Transfer with Locked-image text Tuning",
+                    "abstract": "This paper presents contrastive-tuning, a simple method employing contrastive\ntraining to align image and text models while still taking advantage of their\npre-training. In our empirical study we find that locked pre-trained image\nmodels with unlocked text models work best. We call this instance of\ncontrastive-tuning \"Locked-image Tuning\" (LiT), which just teaches a text model\nto read out good representations from a pre-trained image model for new tasks.\nA LiT model gains the capability of zero-shot transfer to new vision tasks,\nsuch as image classification or retrieval. The proposed LiT is widely\napplicable; it works reliably with multiple pre-training methods (supervised\nand unsupervised) and across diverse architectures (ResNet, Vision Transformers\nand MLP-Mixer) using three different image-text datasets. With the\ntransformer-based pre-trained ViT-g/14 model, the LiT model achieves 85.2%\nzero-shot transfer accuracy on the ImageNet test set, and 82.5% on the\nchallenging out-of-distribution ObjectNet test set.",
+                    "authors": "Xiaohua Zhai, Xiao Wang, Basil Mustafa, Andreas Steiner, Daniel Keysers, Alexander Kolesnikov, Lucas Beyer",
+                    "published": "2021-11-15",
+                    "updated": "2022-06-22",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.CL",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Transfer learning [45] has been a successful paradigm in computer vision [33,34,43]. Zero-shot learning [36,37,66] is an alternative approach aiming to develop models that can handle a new task without task-speci\ufb01c data or adaptation protocols. Recently it was demonstrated that web-sourced paired image-text data can be used to pre-train strong models for zero-shot transfer [31, 46]. Zero-shot transfer differs from classical zero-shot learning in that the transfer setup may see relevant supervised information during pretraining; it is zero-shot insofar as no supervised examples are used during the transfer protocol. GPT-3 [4] explored a similar zero-shot transfer setup using model prompting via natural language. In [31,46] authors propose a contrastive learning framework where an image model (or image tower) is trained simultaneously with a text model (or text tower). Both towers are trained to minimize a contrastive loss, which encourages \u22c6equal technical contribution, \u2020equal advising 0 250 500 750 1000 Image-text pairs seen [M] 30 40 50 60 ImageNet 0-shot accuracy From-scratch Fine-tuned LiT CLIP Design choice comparison (Data: yfcc100m subset) 0 5 10 15 20 Image-text pairs seen [B] 75 80 85 90 CLIP ALIGN Supervised fine-tuning LiT SOTA 0-shot comparison (Data: private) Figure 1. Comparison to the previous SOTA methods. Left: results on public YFCC100m subset, with from-scratch, \ufb01ne-tuned from a pre-trained image model, and LiT with a pre-trained image model. The proposed LiT improves over 30% ImageNet zero-shot transfer accuracy on YFCC100m subset. Right: results on privately gathered data, LiT halves the gap between previous fromscratch methods CLIP [46], ALIGN [31] and supervised \ufb01netuning [13,69]. representations of paired images and texts to be similar, and representations of non-paired images and texts to be dissimilar. At test time, the resulting model can be used for zero-shot image classi\ufb01cation by comparing the image embedding with embeddings of textual class descriptions. In this paper, we adopt a contrastive learning framework and propose a more dataand compute-ef\ufb01cient strategy named contrastive-tuning. The key idea is to tune the text tower using image-text data, while using a pre-trained, strong image model as the image tower. During training, both towers\u2019 weights can be locked or unlocked, leading to different design choices that are illustrated in Figure 2. Speci\ufb01cally, we \ufb01nd that locking the image tower works best, as shown in Figure 1. We call this speci\ufb01c instance of contrastive-tuning \u201cLocked-image Tuning\u201d (LiT), which just teaches a text model to read out suitable representations from a pre-trained image model. LiT achieves better results compared with the from-scratch CLIP [46] or ALIGN [31] models. With the pre-trained model ViT-g/14 [69], LiT achieves 85.2% zero-shot transfer accuracy on ImageNet, halving the gap between previous best zero-shot transfer re1 arXiv:2111.07991v3  [cs.CV]  22 Jun 2022 \fsults [31,46] and supervised \ufb01ne-tuning results [13,69]. The best LiT model also sets new state-of-the-art on several outof-distribution (OOD) ImageNet test variants, compared to previous supervised and unsupervised methods. For example, it achieves 82.5% accuracy on the challenging ObjectNet test set [1], outperforming the previous state-of-the-art method [46] by 10.2%. We believe the reason that LiT works well lies in its decoupling of data sources and techniques for learning image descriptors and vision-language alignment. Image-text data can be great for learning correspondences between natural language and the visual world, but, at the same time, it may not be precise and clean enough to result in state-of-the-art image descriptors. In this paper we carefully investigate this hypothesis and support it with empirical evidence. The proposed LiT works with both supervised and self-supervised pre-trained models. We verify LiT across three image-text datasets, with Vision Transformer [21], ResNet [33], and MLP-Mixer [61] architectures. We also show that with a self-supervised pre-trained model, i.e. DINO [5] or MoCo-v3 [11], LiT achieves better performance compared to from-scratch contrastive-learning. Another contribution of this paper is the proposed recipe for high-performance zero-shot models that can be trained using only modest computational resources and public datasets. By re-using already pre-trained models (e.g. publicly released in the literature), the computational resources used to train the image models can be amortized. Furthermore, we explore publicly available datasets such as YFCC100m [60] and CC12M [6]. Combined with the computational ef\ufb01ciency, we hope to facilitate contributions from a wider audience to research in zero-shot transfer. 2 2. Related work This work is closely related to a vast amount of literature on transfer learning in vision [45, 59]. The main idea of transfer learning is to leverage already pre-trained models to solve a new task better and faster, as opposed to less ef\ufb01cient training from-scratch. This paradigm is usually implemented as a two-step procedure: (1) pre-train (once) an initial model on a large dataset of images that are (weakly)labeled or using self-supervised losses and (2) \ufb01ne-tune the pre-trained model for a task of interest using supervised data. In the context of modern deep learning, many earlier works [20, 33, 34, 48] used supervised pre-training to learn transferrable feature representations, with the Vision Transformer revisiting and improving this approach [21, 69]. It was shown that scaling up model and dataset sizes simultaneously leads to dramatic improvements in transfer effectiveness [21, 33, 69] and robustness [18]. Crucially, large 2Public LiT models available at https : / / github . com / google-research/vision_transformer#lit-models. We provide pre-training code in the big vision codebase [3]. pre-trained models exhibit outstanding capabilities in learning in the low-data (few-shot) regime [9,21,33]. Still, collecting task-speci\ufb01c data and \ufb01ne-tuning large pre-trained models remains time-consuming and potentially costly in many realistic scenarios. Zero-shot transfer is an alternative paradigm that sidesteps the \ufb01ne-tuning stage entirely and performs classi\ufb01cation solely based on a description of the target classes. Early works demonstrated how to train zero-shot classi\ufb01ers based on attributes [36] or numerical descriptors [37]. Another approach, which we adopt in this work, is to learn an alignment between image and text embedding spaces [7, 16, 22, 23, 32, 71]. This approach has demonstrated that with modern architectures, contrastive learning, and large data sources it is possible to obtain performance that is competitive with the classical two-step approach that involves \ufb01ne-tuning on the downstream data [31, 46]. Other efforts in this direction explore image-text alignment or masked language (or image region) modeling [12, 38]. The models have been applied to diverse downstream tasks, including visual question answering [24], visual commonsense reasoning [68] and image captioning [41,42,56]. Contrastive learning techniques are another closelyrelated research direction. The high-level idea of a contrastive loss is to simplify the learning task by requiring the model to select the correct answers out of a \ufb01nite set of carefully designed options. Intuitively, this simpli\ufb01cation of the task may encourage the model to focus on highlevel information in an image instead of generic information, resulting in high quality learned representations. Early works that investigate very speci\ufb01c instances of this idea include [19, 44]. More recently, contrastive learning was formulated and studied in more general settings [8,25,62], leading to very promising results. Finally, [31,46] use contrastive learning for learning from image-text data and derive state-of-the-art zero-shot image classi\ufb01ers. 3. Methods 3.1. Contrastive pre-training Collections of images (potentially noisily) paired with free-form text descriptions have emerged as a powerful resource for training visual models. The key advantage therein is that it is not limited by a \ufb01nite set of prede\ufb01ned categories and instead describes images using open-ended natural language. As a result, models learned from this data can serve as zero-shot learners for a wide range of tasks, e.g. classi\ufb01cation and image/text retrieval. Contrastive pre-training is one particularly effective approach for training models from image-text data, which was recently proven to work well in practice [31, 46]. We take a closer look at this approach and propose a simple, yet highly effective recipe to signi\ufb01cantly enhance contrastive 2 \fu u U u Locked  pre-trained unlocked from-scratch L u Unlocked pre-trained Figure 2. Design choices for contrastive-tuning on image-text data. Two letters are introduced to represent the image tower and text tower setups. L stands for locked variables and initialized from a pre-trained model, U stands for unlocked and initialized from a pre-trained model, u stands for unlocked and randomly initialized. Lu is named as \u201cLocked-image Tuning\u201d (LiT). pre-training from image-text data. The key idea behind the contrastive pre-training approach is to learn two embedding models: an image model and a text model, both of which produce representations of the same dimensionality. These models are trained using a contrastive loss. This loss encourages corresponding image-text pairs to have similar embeddings and, conversely, encourages non-corresponding pairs to have distinct embeddings. See [46, 71] for the detailed discussion of the contrastive loss function. An important detail of this loss function is whether the loss is computed on each accelerator device independently and then accumulated or computed jointly across all devices. We ablate this design choice (Appendix F) and con\ufb01rm that the latter [31,46] consistently results in better performance. We therefore use the global loss in all our experiments and ablations. After image and text towers are trained, they can be readily used for zero-shot classi\ufb01cation: class names or descriptions are embedded with the text model. Then, for a given image the label is selected that has the embedding closest to the embedding of the image. This approach also works for image-text retrieval. 3.2. Contrastive-tuning Contrastive pre-training can be viewed as learning two tasks at the same time: (1) learning an image embedding and (2) learning a text embedding to align with the image embedding space. While contrastive pre-training on imagetext data works well for solving both of these tasks simultaneously, it may be not the optimal approach. When not using contrastive pre-training on image-text data, a standard approach to learning image embeddings is to use a large and relatively clean dataset of (semi)manually labeled images. Large scale and and high quality of such data result in state-of-the-art image embeddings. Some dataset choices for learning powerful image embeddings are ImageNet-21k [15], JFT-300M [57]. However, this common approach has a clear weakness: it is limited to a prede\ufb01ned set of categories and, thus, the resulting models can only reason about these categories. In contrast, image-text data does not have this limitation, as it learns from the free-form text that potentially spans a broad range of real-life concepts. On the other hand, image-text data that is available may be of lower quality (for learning image embeddings) than carefully curated datasets. We propose contrastive-tuning to combine advantages of both sources of data. One speci\ufb01c way of doing this is to initialise the contrastive pre-training with an image model that was already pre-trained using cleaner (semi-)manually labeled data. This way the image-text alignment is learned independently of image embedding, enabling bene\ufb01t from both data sources. Beyond using supervised pre-trained image models, the proposed contrastive-tuning is also \ufb02exible enough to integrate any models that can produce meaningful representations. We verify this in our experiments using selfsupervised pre-trained image models. Similar lines of reasoning can also be applied to the text tower, as there are many powerful pretrained models that use text-speci\ufb01c data sources and learning techniques. 3.3. Design choices and Locked-image Tuning Introducing pre-trained image or text models into the contrastive learning setting involves several design choices. First, each tower (image and text) can independently be initialized randomly or from a pre-trained model. For a pretrained model there are at least two variants: we can lock (freeze) it or allow \ufb01ne-tuning. Note that there are many choices between these two extremes (e.g. partial freezing of selected layers, or custom learning rates), but they are not investigated in this paper. Pre-trained image-text models may have different representation sizes, while the contrastive loss expects representations of the same size. To compensate, we add an optional linear projection (head) to each tower, which maps the representations to a common dimensionality. Preliminary investigations with tried MLP-based heads did not yield signi\ufb01cant improvements over such a simple linear head. We introduce a two-character notation to discuss the potential design choices outlined above (see Figure 2). Each character encodes the setting chosen for the image model and the text model (in this order). We de\ufb01ne three potential settings: L (locked weights, a initialized from pre-trained model), U (unlocked/trainable weights, initialized from a pre-trained model) and u (unlocked/trainable weights, randomly initialized). For example, the notation Lu means locked pre-trained image model, and unlocked (trainable) randomly initialized text model. Previous works training 3 \fmodels from scratch [31,46] are uu. In our experiments we \ufb01nd the Lu setting to work particularly well, so we explicitly name it as Locked-image Tuning (LiT ). 4. Image-text datasets CC12M. The Conceptual Captions dataset [52] extracts, \ufb01lters & transforms image & alt-text pairs from web pages. We use the latest 12 million image-text pair version, i.e. CC12M [6]. Due to expired URLs, only 10 million imagetext pairs were used for our experiments. YFCC100m. The Yahoo Flickr Creative Commons dataset [60] contains 100 million media objects. Of these, 99.2 million are photos that come with rich metadata including camera info, timestamp, title, description, tags, geolocation, and more. [46] de\ufb01nes and uses a subset of 15 million images that have been \ufb01ltered for English text of high quality, which we call YFCC100m-CLIP. A detailed investigation of this dataset and how best to use it, including whether to \ufb01lter it, is presented in Appendix E. Our dataset. We collect 4 billion image and alt-text pairs following the same process as ALIGN [31], with the same image-based \ufb01ltering but simpler text-based \ufb01ltering. Appendix L shows that reducing text \ufb01ltering does not harm performance. To avoid misleading evaluation results, we remove from our dataset near-duplicate images of all splits from all datasets we evaluate on. We do not consider the creation of our dataset a main contribution of this paper; we just simplify the data collection process in ALIGN [31] to demonstrate the ef\ufb01cacy of our methods at scale. 5. Experiments In this section, we \ufb01rst compare LiT to state-of-the-art image-text models. We consider two scenarios: (1) only using public datasets for model training and (2) using privately gathered data. We then present learnings from experimental evaluations of contrastive tuning design choices with various training settings & datasets. We generally perform evaluation on 0-shot ImageNet classi\ufb01cation (\u201c0-shot\u201d) and MSCOCO image (\u201cT\u2192I\u201d) and text (\u201cI\u2192T\u201d) retrieval. 5.1. Comparison to the previous state-of-the-art In this section, we present LiT results on our dataset. The image tower is initialized with a ViT-g/14 model3 pre-trained on JFT-3B [69], which has been de-duplicated against the downstream tasks. We use 32k batch size, and tune for 18 billion image-text pairs seen (roughly 550k steps). See Appendix C for details. 3An earlier version of this paper reported slightly lower numbers with the ViT-g/14 model, e.g. ImageNet accuracy was 84.5% vs 85.2%. We \ufb01xed a model loading bug with ViT-g/14 in this version. Other results are not affected. Dataset INet INet-v2 INet-R INet-A ObjNet ReaL VTAB-N Method Private CLIP [46] 76.2 70.1 88.9 77.2 72.3 ALIGN [31] 76.4 70.1 92.2 75.8 LiT 85.2 79.8 94.9 81.8 82.5 88.6 74.7 Public CLIP [46] 31.3 OpenCLIP [29] 34.8 30.0 LiT 75.7 66.6 60.4 37.8 54.5 82.1 63.1 * ResNet50 [26] 75.8 63.8 36.1 0.5 26.5 82.5 72.6 Table 1. Zero-shot transfer accuracies (%) on ImageNet, \ufb01ve OOD test variants, and seven VTAB-natural tasks. Results are reported on both public datasets and privately gathered data. For reference, we include the ResNet50 model pre-trained on ImageNet, supervised \ufb01ne-tuned on downstream datasets. We use * to denote multiple datasets during supervised \ufb01ne-tuning. We compare the LiT method with the previous state-ofthe-art methods, including CLIP [46] and ALIGN [31]. In Table 1, we report zero-shot classi\ufb01cation results on the ImageNet dataset, \ufb01ve out-of-distribution test variants and seven VTAB-natural tasks [70]. Our model signi\ufb01cantly outperforms the previous state-of-the-art methods at ImageNet zero-shot classi\ufb01cation. The 9% and 8.8% improvement over CLIP and ALIGN, respectively, halves the gap between zero-shot transfer results and supervised \ufb01ne-tuned results [13,69]. Robustness. We evaluate robustness on ImageNetv2 [49], -R [27,64], -A [28], -ReaL [2], and ObjectNet [1], following CLIP and ALIGN. On all of the OOD variants, our model consistently outperforms the previous models. Notably, the LiT model sets a new state-of-the-art 82.5% accuracy on the ObjectNet test set. The pre-trained ViT-g/14 model [69], achieves 70.5% accuracy on the ObjectNet test set when \ufb01ne-tuned on ImageNet. This model gets more than 10% improvement when instead locked-image tuned (LiT) on our image-text dataset. Diverse downstream tasks. We evaluate the LiT models on VTAB, consisting of 19 diverse tasks. We report averaged results on seven VTAB-natural tasks in Table 1. The LiT models achieve promising zero-shot results, comparing to the supervised \ufb01ne-tuned ResNet50 baseline. In Appendix I.2, we present zero-shot transfer details on VTAB, as well as more results and analysis on the specialized tasks and structured tasks. Data & compute ef\ufb01ciency. Figure 1 shows more results when tuning with fewer seen image-text pairs. With LiT the model achieves 81.7% top-1 accuracy on 0-shot ImageNet transfer, with only 300M image-text pairs seen. In comparison, it took the from-scratch method (i.e. CLIP) 4 \fMethod ImgNet ImgNet-v2 Cifar100 Pets Lu 70.1 61.7 70.9 88.1 Uu 57.2 50.2 62.1 74.8 uu 50.6 43.3 47.9 70.3 Table 2. Evaluation of design choices on our large dataset. 12.8B image-text pairs seen, i.e. 40 times more data pairs, to reach 76.2% top-1 accuracy. With a pre-trained image model, the proposed setup converges signi\ufb01cantly faster than the standard from-scratch setups reported in the literature. LiT provides a way to reuse the already pre-trained models in the literature, amortizing the computational resources used to re-generate the image models. Results on public datasets. Given high data ef\ufb01ciency of LiT, we investigate how well it performs when using only smaller, publicly available models and datasets. Speci\ufb01cally, we tune an ImageNet-21k pre-trained ViTL/16 model [55] on the union of the YFCC100m-CLIP and CC12M datasets. More details of the training setup are provided in Appendix D. As a result we achieve unprecedented 75.7% zero-shot transfer on ImageNet, an absolute improvement of 30.9% over the previously reported stateof-the-art result [29] that uses only public data sources. We also obtain strong results on a wide range of robustness datasets and the VTAB-natural tasks, see Table 1. 5.2. Evaluation of design choices Small-scale thorough investigation. We \ufb01rst perform an in-depth study on various combinations of the image and text towers being initialized with pre-trained weights and locked (L) or unlocked (U) or being randomly initialized and unlocked (u). We train each setting many times on the YFCC100m-CLIP dataset, varying the total number of steps from 2 500 to 60 000 in order to understand the setting\u2019s trajectory, and sweeping over learning-rates and weightdecays to avoid being misled. Details can be found in Appendix D. Figure 3 shows the best result for each setting for each duration, i.e. each point on the curves is a separate full run for that duration. It is evident that locking the image tower almost always works best and using a pre-trained image tower signi\ufb01cantly helps across the board, whereas using a pre-trained text tower only marginally improves performance, and locking the text tower does not work well. This still holds in the near-in\ufb01nite data regime. One may hypothesize that locking the pre-trained image tower only helps because the YFCC100m-CLIP dataset is relatively small (15 million images, compared to 400M [46] or 1.8B [31]), and that a randomly initialized image tower will eventually outperform a locked one on much larger imagetext datasets. The trajectory of the Uu and UU settings in 3 6 9 Validation loss 0 20 40 60 LU Lu UU Uu uU uu UL uL LL ImageNet 0-shot 0 20k 40k 60k 0 10 20 30 40 Img Txt recall@1 0 20k 40k 60k 0 10 20 Txt Img recall@1 total training duration [steps] Figure 3. An in-depth study of the possible locking and initialization settings of LiT on the YFCC100m-CLIP dataset. A pretrained image tower works best, while pre-training of the text tower only helps a little. These are not training curves; each point is the \ufb01nal value reached by a training run of that duration. Figure 3 may seem to support this expectation. Maybe surprisingly, experimental results show that this is not the case, and locking the image tower provides bene\ufb01ts even when contrastively tuning on a very large dataset of image-text pairs. Table 2 shows results of contrastive tuning on our dataset of 4 billion images in three settings: Lu, Uu, and uu. Implementation details can be found in Appendix C. The from-scratch method uu unsurprisingly achieves better performance than with smaller datasets such as CC12M and YFCC100m-CLIP. Initializing the image tower from a pre-trained model provides even better performance and is a relatively straightforward extension of CLIP/ALIGN. Perhaps surprisingly, the frozen setup Lu, achieves even better results. While potentially counter-intuitive, another perspective is that LiT simply learns a text tower that extracts knowledge from a strong image embedder. This \ufb02exible & performant setup can turn existing vision backbones into a zero-shot learners, by attaching a text-embedding tower. Why is locked (L) better than unlocked (U)? It is somewhat surprising and counter-intuitive that locking the image tower works better than allowing it to adapt during the contrastive-tuning; Figure 4 gives hints as to why. The \ufb01rst row shows that locking the image tower leads to substantially worse (contrastive) loss on the dataset used for LiT, while the loss of the locked image variant is substan5 \f4 6 8 10 Training loss 4 6 8 10 Validation loss Lu Uu uu 6 8 10 ImageNet classnames val loss 4 6 8 COCO captions val loss 0 20k 40k 0 20 40 60 80 ImageNet linear 10-shot accuracy 0 20k 40k 0 20 40 60 80 CIFAR100 linear 10-shot accuracy Figure 4. Comparing the loss on the dataset used for LiT (top row) to the loss on out-of-distribution (zero-shot) datasets (middle row) and the \u201crepresentation quality\u201d as measured by linear few-shot evaluation on the pre-logits (bottom row). This reveals how the different settings behave, see text for details. tially better on out-of-distribution datasets such as COCO captions (middle row). We also measure the representation quality of the image model (bottom row) via the performance achieved by a few-shot linear regression on its pre-logits, as is commonly done in the self-supervised representation learning literature. Taken together, these \ufb01gures reveal that the image representation of a pre-trained image model generalizes very well, but contrastively \ufb01ne-tuning it worsens the generality of the visual representation, leading it to be better on the contrastive dataset, but worse everywhere else. This indicates that locking the image tower during tuning, i.e. LiT, leads to a text model that is well aligned to an already strong and general image representation, as opposed to an image-text model that is well aligned but specialized to the dataset used for alignment. Intermediate variants, such as \ufb01rst locking and later unlocking the image tower or separating learning-rates are explored in Appendix H; we did not \ufb01nd a strictly better setup than LiT and leave this as an open research question. 5.3. LiT works better for more generally pretrained models One may believe that LiT only works because the image tower is initialized with a backbone that was supervisedly pre-trained for classi\ufb01cation, and hence remains a supervised classi\ufb01er, as opposed to becoming an image-text model. We design a controlled experiment to verify whether that is the case. We \ufb01nd that on the contrary, more generally Model: ViT-B/16 Pre-training LiT Dataset Labels? Full IN 10-shot 0-shot I\u2192T T\u2192I MoCo-v3 [11] IN n 76.7 60.6 55.4 33.5 17.6 DINO [5] IN n 78.2 61.2 55.5 33.4 18.2 AugReg [55] IN21k y 77.4 63.9 55.9 30.3 17.2 AugReg [55] IN y 77.7 77.1 64.3 25.4 13.8 AugReg [55] Places y 22.5 28.5 25.1 12.9 Table 3. The role of pre-training method for the image model: as long as it is general, it does not matter. The background coloring denotes whether a value is similar or far away from the others in that column. pre-trained models are better suited for LiT. We select a set of image models that all use the same ViTB/16 architecture but were pre-trained in various ways: supervised (AugReg [55]) on ImageNet (IN), on the large but narrow Places [39] dataset, on the much broader ImageNet21k (IN21k), or fully unsupervised (DINO and MoCo-v3). All but the Places model achieve similar ImageNet top-1 accuracies of around 77% as reported in their respective publications, and can thus be considered similarly good models. Table 3 shows model performance without LiT (ImageNet 10-shot, and accuracy when fully \ufb01ne-tuned on ImageNet) alongside achieved performance with LiT on YFCC100m-CLIP (zero-shot ImageNet classi\ufb01cation and MS Coco retrieval). From these results, we conclude that models which are pre-trained in a generic way (e.g. on large amounts of data, or in an unsupervised way) and have similar representation quality, become similarly good image-text models after locked-image tuning (LiT). However, this also shows that a narrowly pre-trained model (AugReg-IN and AugRegPlaces) will perform misleadingly well on its narrow task (0-shot IN for AugReg-IN), but signi\ufb01cantly fall behind on more general image-text tasks (MSCOCO captions). These \ufb01ndings highlight the importance of a generally pre-trained model and varied set of evaluation tasks. Is this speci\ufb01c to ViT image models? No. Here we \ufb01xed the architecture to avoid confounders, but Appendix A explores other architectures. 5.4. Which text model to use? While related work has so far focused on the image model, the text model plays an important yet underexplored role in contrastive image-text learning. We consider four possible transformer-based text models [63]\u2014the transformer from ViT-B [21] which also resembles that used in CLIP [46], T5-base [47], mT5-base [67], and the classic 6 \fModel Tok INet 0shot I\u2192T T\u2192I YFCC-CLIP ViT SP 57.2 29.7 16.9 T5 SP 57.8 (+1.4) 29.4 (+1.6) 17.2 (+1.2) mT5 SP 58.1 (+1.2) 28.3 (+0.4) 16.4 (+1.0) BERT WP 58.8 (+0.7) 35.2 (+1.1) 20.0 (+0.7) ViT WP 56.4 28.2 17.3 Ours ViT SP 68.8 43.6 28.5 ViT WP 68.8 45.4 29.7 BERT WP 65.8 43.8 28.6 Table 4. The effect of different text encoders on zero-shot performance. The main numbers show performance achieved when the text tower is randomly initialised; the numbers in brackets are the further improvement achieved when the text tower is initialized with a pre-trained language model. The Tok column indicates whether a SentencePiece or WordPiece tokenizer was used. BERT-base [17]\u2014and whether to initialise them randomly, or from a pre-trained checkpoint. BERT uses a WordPiece (WP) tokenizer [50, 65], and all others use the SentencePiece (SP) tokenizer [35], a component which we also ablate with the ViT model. Table 4 shows the results of LiT using an AugReg-ViTB/32 on YFCC100M-CLIP and our dataset using the base sized variant of these text models. We sweep over various learning-rates and weight-decays separately for each combination to avoid being misled. Our observations differ slightly between the relatively small YFCC100m-CLIP dataset, and our much larger dataset, we \ufb01rst discuss the former. First, we see a small but consistent improvement by initializing the text model with pre-trained weights. Second and somewhat unexpectedly, we \ufb01nd that the BERT model performs signi\ufb01cantly better than others, especially for retrieval. In order to disentangle the contribution of the architecture from the tokenizer, we further apply LiT using a ViT text encoder paired with BERT\u2019s WordPiece tokenizer and see no improvement. We believe that small differences in the architecture, such as initialization and LayerNorm placement, are responsible for the slightly better generalization of BERT that we observe. However, we also found the BERT model to be less stable to train. For the large-scale experiments on our dataset, we do not observe this improvement anymore, and favor sticking with the more stable ViT SentencePiece combination. What about model capacity? Previous works used relatively low-capacity text models. We show in Appendix B that increasing the text tower\u2019s capacity consistently improves performance. The same is true, and more pronounced, for the image tower. Dedup #tune #eval ImgNet I\u2192T T\u2192I 0 0 70.2 43.6 28.4 test 2.6M 76K 70.2 43.3 28.3 train+test 3.6M 220K 69.9 43.7 28.4 Table 5. Results on various de-duplication setups. #tune images are removed from the LiT dataset due to #eval images in the evaluation datasets. We report results averaged across three runs. 5.5. Do duplicate examples matter for LiT? One relevant question in the context of large-scale training is the role of duplicate examples between upstream datasets and downstream datasets. We answer this question by performing experiments on three different upstream deduplication setups: (1) no de-duplication; (2) de-duplicate against downstream test splits only; (3) de-duplicate against downstream train and test splits. We conduct experiments using the Lu setup on our dataset. We use a B/32 image model pre-trained on the JFT-3B dataset [69], which has been de-duplicated against downstream train and test splits. In Table 5, we show the number of duplicate samples found between upstream datasets and downstream datasets during de-duplication. In the de-duplication process, a downstream image may have multiple upstream duplicate examples, e.g. due to image copies on the web. As a result, the number of duplicate examples on the upstream dataset is signi\ufb01cantly larger than the number on the downstream datasets. The downstream number indicates how many downstream images had a duplicate detected, while the upstream number indicates how many images are removed from the image-text dataset. We apply LiT on the three setups, and the zero-shot transfer results vary little. More results with larger backbone can be found in Appendix K, with consistent conclusions. It indicates that the duplication of examples here does not in\ufb02uence the results strongly. This observation is also consistent with previous conclusions [33,46]. A possible interpretation is that with a large upstream dataset, the model may not memorize those duplicate examples. Throughout this paper, we report results using the strictest setup (3) with proper de-duplication against downstream train splits and test splits, to avoid data leakage. 5.6. Technical advantages of locked image models Besides potential modelling advantages previously explored, using a locked image tower has several more bene\ufb01ts. First, the training is signi\ufb01cantly sped-up and memory use reduced as no gradients are computed for the image tower. Second, if no augmentations are used, such as in our large-data experiment, the image model\u2019s embeddings can be precomputed once, further reducing compu7 \fen ru tr es fa fr de ja vi zh ar Prompting language 0% 20% 40% 60% ImageNet 0-shot CLIP subset Lu T5 LU T5 Lu mT5 LU mT5 Figure 5. Including non-English data unlocks multilingual zeroshot models without hurting English performance. In such a regime, multilingual text pre-training can be more useful for lowresource languages. tation time and memory requirements. Appendix G shows concrete measurements. Taken together, these implementation features unlock the use of enormous models at very large batch-sizes. 5.7. Preliminary multilingual experiments It is currently common practice [31, 46] to \ufb01lter imagetext datasets to English language data only. We believe that removing this restriction has the potential to bene\ufb01t a larger part of the world\u2019s population. Concurrent work [30] has relied on additional translated text pairs for training the text encoder. In contrast, we do not require any translations and purely rely on the pre-trained, locked image model to bridge the language barrier. In this section, we report preliminary experiments that show the promise of LiT for multilingual image-text models. We apply LiT on an AugReg-i21k ViT-B/32 with the T5 [47] and mT5 [67] base encoders, both with and without the pre-trained checkpoints. We do this on both the full YFCC100m dataset, and the reduced English-only CLIP subset, and we use all available text as supervision signal (See Appendix E). We evaluate the resulting model\u2019s multilingualism in two ways, both of which have limitations discussed in Appendix J. First, we translate the ImageNet prompts into the most common languages using an online translation service and perform zero-shot classi\ufb01cation in each of them; this evaluation is shown in Figure 5. Second, we use the Wikipedia based Image Text (WIT) dataset [54] to perform T \u2192I retrieval across more than a hundred languages. Figure 6 gives a summary of this evaluation; a more detailed variant is provided in Appendix J. The high-level conclusions are consistent across both evaluations: training on the full dataset improves performance on non-English languages much more than on English, using a multilingual tokenizer (as in mT5) significantly helps languages that do not use the Latin script, and starting from a pre-trained multilingual text model can best   Language, by model   worst 55% 60% 65% 70% 75% Relative mean image rank English Chinese Lu T5 LU T5 Lu mT5 LU mT5 Figure 6. Image retrieval performance over 100 languages reveals that un\ufb01ltered data and a multilingually pre-trained text model can signi\ufb01cantly increase long-tail performance. further help. The combination of all three improvements barely has any effect when evaluated in English, but significantly improves performance on the long tail of languages. This is a promising result for unlocking multimodal models for low-resource languages. 6. Discussion Limitations. This work explores only classi\ufb01cation and retrieval as zero-shot transfer tasks. We leave evaluating zero-shot transfer to a broader set of tasks such as detection, segmentation, visual question answering, and image captioning as future work in order to limit our scope. On cross-modal retrieval tasks, we have not observed as clear a bene\ufb01t of the Lu setup compared to Uu or UU (Figure 3). For very long tuning schedules, Uu or UU sometimes overtake Lu on these tasks. Our results suggest that the proposed Lu setup can still save computational cost within a \ufb01xed budget, but with a large enough budget, it may be useful to also consider the Uu setup if zero-shot classi\ufb01cation is not the primary end goal. Societal impact. This work shows how one can easily add a text-tower to a pre-trained image model. While there are many useful applications, like most research, it is a double-edged sword: the technique also makes it simpler to create malicious, offensive, or obscene text tower pendants to existing image models. Further research is needed on how to best equip open-world image-text models with the behaviour we desire. 7."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2106.04560v2",
+                    "title": "Scaling Vision Transformers",
+                    "abstract": "Attention-based neural networks such as the Vision Transformer (ViT) have\nrecently attained state-of-the-art results on many computer vision benchmarks.\nScale is a primary ingredient in attaining excellent results, therefore,\nunderstanding a model's scaling properties is a key to designing future\ngenerations effectively. While the laws for scaling Transformer language models\nhave been studied, it is unknown how Vision Transformers scale. To address\nthis, we scale ViT models and data, both up and down, and characterize the\nrelationships between error rate, data, and compute. Along the way, we refine\nthe architecture and training of ViT, reducing memory consumption and\nincreasing accuracy of the resulting models. As a result, we successfully train\na ViT model with two billion parameters, which attains a new state-of-the-art\non ImageNet of 90.45% top-1 accuracy. The model also performs well for few-shot\ntransfer, for example, reaching 84.86% top-1 accuracy on ImageNet with only 10\nexamples per class.",
+                    "authors": "Xiaohua Zhai, Alexander Kolesnikov, Neil Houlsby, Lucas Beyer",
+                    "published": "2021-06-08",
+                    "updated": "2022-06-20",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Attention-based Transformer architectures [45] have taken computer vision domain by storm [8,16] and are becoming an increasingly popular choice in research and practice. Previously, Transformers have been widely adopted in the natural language processing (NLP) domain [7,15]. Optimal scaling of Transformers in NLP was carefully studied in [22], with the main conclusion that large models not only perform better, but do use large computational budgets more ef\ufb01ciently. However, it remains unclear to what extent these \ufb01ndings transfer to the vision domain, which has several important differences. For example, the most successful pre-training schemes in vision are supervised, as opposed to unsupervised pre-training in the NLP domain. In this paper we concentrate on scaling laws for transfer performance of ViT models pre-trained on image classi\ufb01ca\u22c6equal contribution 1 10 5 25 Examples per class 60 65 70 75 80 85 ImageNet top-1 accuracy Our ViT-G ViT-H SimCLR v2 BYOL DINO Figure 1. Few-shot transfer results. Our ViT-G model reaches 84.86% top-1 accuracy on ImageNet with 10-shot linear evaluation. tion tasks. In particular, we experiment with models ranging from \ufb01ve million to two billion parameters, datasets ranging from one million to three billion training images and compute budgets ranging from below one TPUv3 core-day to beyond 10 000 core-days. Our main contribution is a characterization of the performance-compute frontier for ViT models, on two datasets. Along the way, we create an improved large-scale training recipe. We investigate training hyper-parameters and discover subtle choices that make drastic improvements in few-shot transfer performance. The few-shot transfer evaluation protocol has also been adopted by previous large-scale pre-training efforts in NLP domain [6]. Speci\ufb01cally, we discover that very strong L2 regularization, applied to the \ufb01nal linear prediction layer only, results in a learned visual representation that has very strong few-shot transfer capabilities. For example, with just a single example per class on the ImageNet dataset (which has 1 000 classes), our best model achieves 69.52% accuracy; and with 10 examples per class it attains 84.86%. In addition, we substantially reduce the memory footprint of the original ViT model proposed in [16]. We achieve this by hardware-speci\ufb01c architecture changes and a different optimizer. As a result, we train a model with two billion parameters and attain a new state-of-the-art 90.45% accuracy on ImageNet. 1 arXiv:2106.04560v2  [cs.CV]  20 Jun 2022 \f100 101 102 103 104 Compute (TPUv3 core days) 10 20 30 40 50 ImageNet finetune error rate [%] E=0.09+0.26(C+0.01) 0.35 30M 300M 1B 3B 100 101 102 103 104 Compute (TPUv3 core days) 20 30 40 50 60 70 80 90 ImageNet 10-shot error rate [%] E=0.12+0.63(C+0.52) 0.32 s/28 S/32 s/16 Ti/16 S/16 B/32 B/28 B/16 L/16 g/14 G/14 101 103 Model Size (Gflops) 102 103 Dataset Size (M) Figure 2. Left/Center: Representation quality, measured as ImageNet \ufb01netune and linear 10-shot error rate, as a function of total training compute. A saturating power-law approximates the Pareto frontier fairly accurately. Note that smaller models (blue shading), or models trained on fewer images (smaller markers), saturate and fall off the frontier when trained for longer. Top right: Representation quality when bottlenecked by model size. For each model size, a large dataset and amount of compute is used, so model capacity is the main bottleneck. Faintly-shaded markers depict sub-optimal runs of each model. Bottom Right: Representation quality by datasets size. For each dataset size, the model with an optimal size and amount of compute is highlighted, so dataset size is the main bottleneck. 2. Core Results We \ufb01rst present our main results on scaling trends, before presenting detailed architecture and training protocol improvements in Section 3. In the following experiments, we train several ViT models on both public ImageNet-21k [14] dataset and privately gathered images, up to three billion weakly-labelled images. We vary the architecture size, number of training images, and training duration. All models are trained on TPUv3, thus total compute is measured in TPUv3 core-days. To evaluate the quality of the representation learned by the models, we measure (i) few-shot transfer via training a linear classi\ufb01er on frozen weights, (ii) transfer via \ufb01ne-tuning the whole model on all data, both to multiple benchmark tasks. 2.1. Scaling up compute, model and data together Figure 2 shows both the 10-shot linear evaluation and \ufb01netuning evaluation on ImageNet [14]. Similar trends on other datasets, Oxford IIIT Pets [28], CIFAR-100 [24], and Caltech-UCSD Birds [47] are presented in the Appendix, Figure 9. For each combination of model size and data size we pre-train for various numbers of steps. In Figure 2, connected points represent the same model trained for a different number of steps. We make the following observations. First, scaling up compute, model and data together improves representation quality. In the left plot and center plot, the lower right point shows the model with the largest size, dataset size and compute achieving the lowest error rate. However, it appears that at the largest size the models starts to saturate, and fall behind the power law frontier (linear relationship on the log-log plot in Figure 2). Second, representation quality can be bottlenecked by model size. The top-right plot shows the best attained performance for each model size. Due to limited capacity, small models are not able to bene\ufb01t from either the largest dataset, or compute resources. Figure 2, left and center, show the Ti/16 model tending towards a high error rate, even when trained on a large number of images. Third, large models bene\ufb01t from additional data, even beyond 1B images. When scaling up the model size, the representation quality can be limited by smaller datasets; even 30-300M images is not suf\ufb01cient to saturate the largest models. In Figure 2, center, the error rate of L/16 model on the the 30M dataset does not improve past 27%. On the larger datasets, this model attains 19%. Further, when increasing the dataset size, we observe a performance boost with big models, but not small ones. The largest models even obtain a performance improvement the training set size grows from 1B to 3B images (Figure 2, bottom right). For small models, however, such as Ti/16 or B/32, increasing the dataset size does not help. For example, in Figure 2, left and center, all of the curves for Ti/16 overlap, showing that this model achieves the same performance irrespective of the dataset size. 2.2. Double-saturating power law Figure 2, left and center, show the Pareto frontier of representation quality versus training compute. The frontier contains the models with the best allocation of compute to model shape and training duration. For over two orders of magnitude of compute, the relation2 \f102 103 104 Images Seen (M) 20 30 40 50 60 70 80 ImageNet 10-shot error rate [%] Ti/16 B/32 L/16 102 103 104 Images Seen (M) 10 20 30 40 ImageNet finetune error rate [%] Ti/16 B/32 L/16 102 103 104 Images Seen (M) 20 30 40 50 ImageNet finetune V2 error rate [%] Ti/16 B/32 L/16 Figure 3. Error rate on ImageNet, with respect to images seen during pre-training. Big models are more sample ef\ufb01cient, which is consistent across diverse setups: few-shot transfer on the frozen representations, \ufb01ne-tune the network on ImageNet, and evaluate the \ufb01ne-tuned models on the v2 test set. ship between compute and performance follows a power-law (E = aCb), resulting in a straight line on the log-log plot. However, we observe \u201csaturation\u201d at both ends of the compute spectrum. At the higher end of compute, the largest models do not tend towards zero error-rate. If we extrapolate from our observations, an in\ufb01nite capacity model will obtain a non-zero error. This effect has also been observed for generative models [19]. The authors of [19] refer to this residual error as the \u201cirreducible entropy\u201d of the task. Since we plot error rate, the information-theoretic interpretation does not apply, but our observations support the notion of fundamental performance ceilings for ImageNet [4]. In terms of the law, this saturation corresponds to an additive constant to the error rate: c in E = aC\u2212b + c. At the lower end of the compute spectrum, we see a saturation for smaller models; the performance of the smallest model is better than that would be predicted by a power-law. This saturation occurs because even trivial solutions can achieve non-zero error. For example, predicting the majority class (almost zero compute) will achieve an accuracy related to its occurence frequency in the test set. This lower bound is not observed in [19], either because their smallest model is large enough to avoid this region, or because log-loss saturates at worse performances than accuracy (it will saturate eventually). This saturation corresponds to a shift in the xaxis: d in E = a(C + d)\u2212b + c. This constant indicates that the zero-compute model will still obtain non-zero accuracy. 2.3. Big models are more sample ef\ufb01cient Figure 3 shows the representation quality with respect to the total number of images \u201cseen\u201d (batch size times number of steps) during pre-training. In addition to ImageNet \ufb01netuning and linear 10-shot results on the public validation set, we also report results of the ImageNet \ufb01ne-tuned model on the ImageNet-v2 test set [33] as an indicator of robust generalization. Three ViT models pre-trained on three billion images are presented in this plot. We observe that bigger models are more sample ef\ufb01cient, reaching the same level of error rate with fewer seen images. For 10-shot, the Ti/16 model needs to see nearly 100 times more images to match the representation quality of the L/16 model. When \ufb01ne-tuning, this factor reduces from 100 to about 20. Our results suggest that with suf\ufb01cient data, training a larger model for fewer steps is preferable. This observation mirrors results in language modelling and machine translation [22,26]. 2.4. Do scaling laws still apply on fewer images? We extend the study to much fewer images, ranging from one million to 13 millions on the public ImageNet-21k 100 101 102 Compute (TPUv3 core days) 30 40 50 60 70 80 ImageNet 10-shot error rate [%] E=0.24+0.96(C+1.57) 0.61 Ti/16 S/16 B/32 B/16 101 Model Size (Gflops) 101 Dataset Size (M) 1.3M 2.6M 5.1M 10.2M 13M Figure 4. Results on the ImageNet-21k dataset. Left: Representation quality, measured as ImageNet linear 10-shot error rate, as a function of total training compute. The double-saturating power law still applies. Right: Representation quality by model sizes and dataset sizes. 3 \fTable 1. The results for ViT-G/14, compared to the previous state-of-the-art models. Benchmark ImageNet INet V2 INet ReaL ObjectNet VTAB (light) NS (Eff.-L2) [49] 88.3 80.2 68.5 MPL (Eff.-L2) [29] 90.2 91.02 CLIP (ViT-L/14) [31] 85.4 75.9 72.3 ALIGN (Eff.-L2) [21] 88.6 70.1 BiT-L (ResNet) [23] 87.54 90.54 58.7 76.29 ViT-H/14 [16] 88.55 90.72 77.63 Our ViT-G/14 90.45\u00b10.03 83.33\u00b10.03 90.81\u00b10.01 70.53\u00b10.52 78.29\u00b10.53 dataset. In Figure 4 left, we found that the double-saturation power law still applies, when varying model sizes, dataset sizes and compute resources. This indicates that the conclusions from the study generalizes well, and can guide future design choices for vision transformer architectures. In Figure 4 right, we observe similar behaviors that the model performance are bottlenecked by the dataset size. When scaling up compute, model and data together, one gets the best representation quality. 2.5. ViT-G/14 results We trained a large Vision Transformer, ViT-G/14, which contains nearly two billion parameters. Section 3.6 details the architecture\u2019s shape. We evaluate the ViT-G/14 model on a range of downstream tasks, and compare it to recent stateof-the-art results. We \ufb01ne-tune on ImaegNet, and report ImageNet [34], ImageNet-v2 [33], ReaL [4], and ObjectNet [2] accuracies. In addition, we report transfer learning result on the VTAB-1k benchmark consisting of 19 tasks [53]. Figure 1 shows the few-shot transfer results on ImageNet. ViT-G/14 outperforms the previous best ViT-H/14 model [16] by a large margin (more than 5%), attaining 84.86% accuracy with 10 examples per class. Ten images per class is less than 1% of ImageNet data (13 examples per class), as commonly used in self-supervised and semisupervised learning [52]. For reference, Figure 1 shows three state-of-the-art self-supervised learning models, SimCLR v2 [10] and BYOL [17], using 1% of ImageNet data, DINO [9] using 20 examples per class. Note, however, that these approaches are quite different: ViT-G/14 uses large source of weakly-supervised data, and is pre-trained only once and transferred to different tasks. Meanwhile, the selfsupervised learning models use unlabeled but in-domain data for pre-training, and target a single task. Table 1 shows the results on the remaining benchmarks. ViT-G/14 achieves 90.45% top-1 accuracy on ImageNet, setting the new state-of-the art. On ImageNet-v2, ViT-G/14 improves 3% over the Noisy Student model [49] based on Ef\ufb01cientNet-L2. For ReaL, ViT-G/14 outperforms ViTH [16] and BiT-L [23] by only a small margin, which indicates again that the ImageNet classi\ufb01cation task is likely reaching its saturation point. For ObjectNet, ViT-G/14 outperforms BiT-L [23] by a large margin, and is 2% better than Noisy Student, but is about 2% behind CLIP [31]. Note that, unlike the other methods, CLIP does not \ufb01ne-tune on ImageNet, and evaluates directly on ObjectNet, this likely improves its robustness. Finally, when transferring the ViTG/14 model to VTAB, it gets consistently better results with just a single hyper parameter across all tasks. The state-ofthe-art on VTAB using a heavyweight per-task hyperparameter sweep is 79.99 [21], we leave running a heavy sweep with ViT-G/14 to future work. 3. Method details We present a number of improvements to the ViT model and training. These improvements are mostly simple to implement, and can signi\ufb01cantly improve memory-utilization and model quality. They allow us to train ViT-G/14 using data-parallelism alone, with the entire model \ufb01tting on a single TPUv3 core. 3.1. Decoupled weight decay for the \u201chead\u201d Weight decay has a drastic effect on model adaptation in the low-data regime. We conduct an study of this phenomena at a mid-size scale. We \ufb01nd that one can bene\ufb01t from decoupling weight decay strength for the \ufb01nal linear layer (\u201chead\u201d), and for the remaining weights (\u201cbody\u201d) in the model. Figure 5 demonstrates this effect: we train a collection ViT-B/32 models on JFT-300M, each cell corresponds to the performance of different head/body weight decay values. The diagonal corresponds to using the same value for both decays. One can observe that the best performance appears off-diagonal (i.e. with a decoupled weight decay for the head and body). Interestingly, we observe that high weight decay in the head decreases performance on the pre-training (upstream) task (not shown), despite improving transfer performance. We do not have a complete explanation of this phenomena. However, we hypothesize that a stronger weight decay in the 4 \f0 0.001 0.01 0.1 0.3 1.0 3.0 10.0 \"Body\" Weight decay 0 0.001 0.01 0.1 0.3 1.0 3.0 10.0 \"Head\" weight decay 58 59 61 60 56 49 37 18 59 60 61 60 56 48 37 20 59 60 61 60 55 48 36 19 59 59 59 58 54 47 34 18 56 57 57 57 53 45 32 16 54 54 55 53 50 43 30 18 53 53 54 53 50 44 30 17 53 54 54 54 50 43 31 17 Linear 5-shot ImageNet accuracy [%] 0 0.001 0.01 0.1 0.3 1.0 3.0 10.0 \"Body\" Weight decay 0 0.001 0.01 0.1 0.3 1.0 3.0 10.0 Upstream performance 0 200k 400k Steps 40% 50% 60% Linear 5-shot ImageNet accuracy Token (ICLR'21) Token (linear) Token + head-wd GAP + head-wd MAP + head-wd Figure 5. Left and middle: The dependence of 5-shot ImageNet accuracy and upstream performance depends on the weight decay strength. Normally, a single weight decay value is applied to all weights (corresponds to the diagonal on the heatmaps). We show that by using weight decay values for the \u201chead\u201d and the rest of the weights one signi\ufb01cantly improves few-shot transfer performance. Right: Few-shot performance on ImageNet for different types of head. A high weight decay on the head works equally well for all of them. head results in representations with larger margin between classes, and thus better few-shot adaptation. This is similar to the main idea behind SVMs [12]. This large decay makes it harder to get high accuracy during upstream pre-training, but our main goal is high quality transfer. 3.2. Saving memory by removing [class] token The largest VIT model from [16] uses 14 \u00d7 14 patches with 224 \u00d7 224 images. This results in 256 visual \u201ctokens\u201d, where each one corresponds to an image patch. On top of this, ViT models have an extra [class] token, which is used to produce the \ufb01nal representation, bringing the total number of tokens to 257. For ViT models, current TPU hardware pads the token dimension to a multiple of 128, which may result in up to a 50% memory overhead. To overcome this issue we investigate alternatives to using the extra [class] token. In particular, we evaluate global average pooling (GAP) and multihead attention pooling (MAP) [25] to aggregate representation from all patch tokens. We set the number of heads in MAP to be equal to the number of attention heads in the rest of the model. To further simplify the head design we remove \ufb01nal non-linear projection before the \ufb01nal prediction layer, which was present in the original ViT paper. To choose the best head, we perform a side-by-side comparison of a [class] token and GAP/MAP heads. Results are summarized in Figure 5 (right). We \ufb01nd that all heads perform similarly, while GAP and MAP are much more memory ef\ufb01cient due to the aforementioned padding considerations. We also observe that non-linear projection can be safely removed. Thus, we opt for the MAP head, since it is the most expressive and results in the most uniform architecture. MAP head has also been explored in [42], in a different context for better quality rather than saving memory. 3.3. Scaling up data For this study, we use the proprietary JFT-3B dataset, a larger version of the JFT-300M dataset used in many previous works on large-scale computer vision models [16,23,37]. This dataset consists of nearly 3 billion images, annotated with a class-hierarchy of around 30k labels via a semiautomatic pipeline. Thus, the data and associated labels are noisy. We ignore the hierarchical aspect of the labels and use only the assigned labels as targets for multi-label classi\ufb01cation via a sigmoid cross-entropy loss, following [16,23]. We have conducted sensitive category association analysis as described in [1]. We measured (per label) the distribution of sensitive categories across the raw data, the cleaned data, the models trained on this data, and labels that were veri\ufb01ed by human raters. Human raters additionally assisted in removing offensive content from the dataset. Figure 6 shows an ablation of the effect of changing from JFT-300M to JFT-3B on model performance, even when scale is not increased. Figure 6, left shows linear 10-shot ImageNet performance evaluated throughout. We observe that JFT-3B results in a better model, even before the model has completely one epoch of JFT-300M. Therefore, over\ufb01tting JFT-300M is not the sole cause of the improvement. This difference can be seen even for the small B/32 model as well as the larger L/16. We \ufb01ne-tune the models to the full ImageNet dataset (right), and con\ufb01rm that these improvements transfer to a full \ufb01ne-tuning setup. Overall, the change in dataset improves transfer to ImageNet by about 1% for both small and large models. Other than the performance improvement, training behavior is similar on JFT-300M and JFT-3B. Most importantly, JFT-3B allows us to scale up further with fewer concerns about over\ufb01tting and regularization. Deduplication. We remove all images from the JFT-3B 5 \f0 1 2 3 4 5 6 7 JFT-300m epochs 25% 50% 75% Linear 10-shot ImageNet accuracy B/32, JFT-300m B/32, JFT-3b L/16, JFT-300m L/16, JFT-3b 0 2.5k 5k 7.5k 10k 12.5k 15k 17.5k 20k Fine-tuning steps 70% 75% 80% 85% 90% ImageNet top-1 accuracy Figure 6. The effect of switching from JFT-300M to JFT-3B, without any further scaling. Both small and large models bene\ufb01t from this change, by an approximately constant factor, both for linear few-shot evaluation (left) and transfer using the full dataset (right). dataset that are near-duplicates of images from both train set and test set of datasets we evaluate on. Overall we identi\ufb01ed and removed 927k duplicate images from JFT-3B. 3.4. Memory-ef\ufb01cient optimizers When training large models, storage required for model parameters becomes a bottleneck. Our largest model, ViT-G, has roughly two billion parameters, which occupies 8 GiB of device memory. To make things much worse, the Adam optimizer that is commonly used for training Transformers, stores two additional \ufb02oating point scalars per each parameter, which results in an additional two-fold overhead (extra 16 GiB). To tackle the overhead introduced by the Adam optimizer we explore two modi\ufb01cations. Adam with half-precision momentum. We empirically observe that storing momentum in half-precision (bfloat16 type) does not affect training dynamics and has no effect on the outcome. This allows to reduce optimizer overhead from 2-fold to 1.5-fold. Notably, storing the second momentum using half-precision resulted in a signi\ufb01cant performance deterioration. Adafactor optimizer. The above optimizer still induces a large memory overhead. Thus, we turn our attention to the Adafactor optimizer [35], which stores second momentum using rank 1 factorization. From practical point of view, this results in the negligible memory overhead. However, the Adafactor optimizer did not work out of the box, so we make the following modi\ufb01cations: \u2022 We re-introduce the \ufb01rst momentum in half-precision, whereas the recommended setting does not use the \ufb01rst momentum at all. \u2022 We disable scaling of learning rate relative to weight norms, a feature that is part of Adafactor. \u2022 Adafactor gradually increases the second momentum from 0.0 to 1.0 throughout the course of training. In our preliminary experiments, we found that clipping the second momentum at 0.999 (Adam\u2019s default value) results in better convergence, so we adopt it. The resulting optimizer introduces only a 50% memory overhead on top the space needed to store model\u2019s parameters. We observe that both proposed optimizers perform on par with or slightly better than the original Adam optimizer. We are aware of other memory-ef\ufb01cient optimizers [32,40], we leave the exploration to future work. 3.5. Learning-rate schedule In our study we want to train each of the models for several different durations in order to measure the tradeoff between model size and training duration. When using linear decay, as in [16], each training duration requires its own training run starting from scratch, which would be an inef\ufb01cient protocol. Inspired by [27], we address this issue by exploring learning-rate schedules that, similar to the warmup phase in the beginning, include a cooldown phase at the end of training, where the learning-rate is linearly annealed toward zero. Between the warmup and the cooldown phases, the learning-rate should not decay too quickly to zero. This can be achieved by using either a constant, or a reciprocal squareroot schedule for the main part of training. Figure 7 (bottom) depicts several of these options, with a cooldown after approximately 200 k, 400 k, and 500 k steps. The upper half of Figure 7 shows the validation score (higher is better) for each of these options and their cooldowns, together with two linear schedules for reference. While the linear schedule is still preferable when one knows the training duration in advance and does not intend to train any longer, all three alternatives come reasonably close, with the advantage of allowing inde\ufb01nite training and evaluating multiple training durations from just one run. For each of the schedules, we optimized the learning-rate and the exact shape. We have also brie\ufb02y 6 \f0 100k 200k 300k 400k 500k Validation score 0 100k 200k 300k 400k 500k Steps Learning rate    ReciprocalTrapezoidal Constant Linear Figure 7. Various \u201cin\ufb01nite\u201d learning-rate schedules, along with the \ufb01nite linear one for reference. tried cyclic learning-rate schedules, however they seemed to perform much worse and we have not investigated further. We therefore opt for the reciprocal square-root schedule. 3.6. Selecting model dimensions ViT models have many parameters that control the model\u2019s shape, and we refer to the original publication for full details. Brie\ufb02y, these include the patch-size, the number of encoder blocks (depth), the dimensionality of patch embeddings and self-attention (width), the number of attention heads, and the hidden dimension of MLP blocks (MLP-width). On top of this, we rely on the XLA compiler to optimize our models for runtime speed and memory footprint. Behind the scenes, XLA uses complex heuristics to compile a model into code for a speci\ufb01c hardware that trades off memory and speed optimally. As a result, it is hard to predict which model con\ufb01gurations will \ufb01t into memory on a single device. Therefore we run an extensive simulation, where we instantiate a large amount of ViTs of various shapes, and attempt to train them for a few steps, without considering the quality. We vary the depth, width, heads, and MLP-width, but keep the patch-size at 14 px. In this way, we measure their speed and whether or not a given model \ufb01ts into the device\u2019s memory. Figure 8 summarizes the result of this simulation. Each block corresponds to one model con\ufb01guration, the shade of the block corresponds to its training speed (brighter is faster). Orange blocks show which original ViT models, without any of our modi\ufb01cations, \ufb01t. Green blocks then further include the memory savings described in Section 3.2 coupled with the half-precision Adam described in Section 3.4. Finally, blue blocks are with our modi\ufb01ed AdaFactor optimizer. The shapes in the white area were not able to \ufb01t into memory in any setting. For space reasons, we show here only the models pertaining to the experiments Table 2. Model architecture details. Name Width Depth MLP Heads Mio. Param GFLOPs 2242 3842 s/28 256 6 1024 8 5.4 0.7 2.0 s/16 256 6 1024 8 5.0 2.2 7.8 S/32 384 12 1536 6 22 2.3 6.9 Ti/16 192 12 768 3 5.5 2.5 9.5 B/32 768 12 3072 12 87 8.7 26.0 S/16 384 12 1536 6 22 9.2 31.2 B/28 768 12 3072 12 87 11.3 30.5 B/16 768 12 3072 12 86 35.1 111.3 L/16 1024 24 4096 16 303 122.9 382.8 g/14 1408 40 6144 16 1011 533.1 1596.4 G/14 1664 48 8192 16 1843 965.3 2859.9 presented, but note that with our modi\ufb01cations we were able to \ufb01t thin ViT models of a depth up to 100 encoder blocks. The original Vision Transformer publication contains a study in Appendix D2 about the trade-offs between scaling the different components, concluding that it is most effective to scale all aspects (depth, width, MLP-width, and patchsize) simultaneously and by a similar amount. We follow this recommendation, and select shapes for ViT-g and ViT-G at the limit of what \ufb01ts in memory accordingly, as shown in Figure 8 and summarized in Table 2. 4. Related Work Smaller Vision Transformers Early work on Transformers for vision focused on small networks for CIFAR10 [11]. The Vision Transformer [16], however, was proposed in the context of state-of-the-art medium and largescale image recognition; the smallest model (ViT-B) containing 86M parameters. [41] present smaller ViT sizes for training from-scratch, down to ViT-Ti, with 5M parameters. New variants of ViT introduce smaller and cheaper architectures. For example, T2T-ViT [51] reduces the number of parameters and compute using a new tokenization and narrower networks. Pyramidal ViTs [46], designed for dense prediction tasks, follow a CNN-like pyramidal structure, that also reduces the size of the model. Hybrids of CNNs and Transformers typically allow smaller models to perform well, such as the ViT-CNN hybrid in [16], BoTNet [36], and HaloNet [44]. However, the other direction, increasing the scale of ViT, is less explored. While language Transformers are still much larger than Vision Transformers, understanding the scaling properties and the improvements introduced in this paper represent a step in this direction. Scaling Laws [22] present a thorough study of the empirical scaling laws of neural language models. The authors 7 \f4096 4608 5120 5632 6144 6656 7168 7680 8192 8704 9216 9728 10240 1024//16 1024//8 1280//20 1280//16 1280//10 1408//22 1408//16 1408//11 1536//24 1536//16 1536//12 1664//26 1664//16 1664//13 1792//28 1792//16 1792//14 1920//30 1920//24 1920//16 1920//15 2048//32 2048//16 SA-width // num heads ViT-L 24 layers 4096 4608 5120 5632 6144 6656 7168 7680 8192 8704 9216 9728 10240 ViT-H 32 layers 4096 4608 5120 5632 6144 6656 7168 7680 8192 8704 9216 9728 10240 MLP-width ViT-g 40 layers 4096 4608 5120 5632 6144 6656 7168 7680 8192 8704 9216 9728 10240 ViT-G 48 layers 4096 4608 5120 5632 6144 6656 7168 7680 8192 8704 9216 9728 10240 Original ViT Savings + Adam-HP Savings + AdaFactor 56 layers Figure 8. Combined results of the \u201cShape\ufb01nder\u201d simulation for the original ViT in orange, our improvements together with half-precision Adam (e.g. ViT-g) in green, and \ufb01nally with our modi\ufb01ed AdaFactor in blue. White areas ran out of memory. The brightness of the dot corresponds to its relative training speed. \ufb01t power laws that describe the relationships between compute, data size, model size, and performance. Following these laws, GPT-3, a 175B parameter language model was successfully trained [7]. [19] presents laws for autoregressive generative modelling in other modalities, including the generation of images. Our paper contains the \ufb01rst study of scaling laws for the discriminative modelling of images. Scaling-up Vision Models Many papers scale up CNNs to attain improved performance. Ef\ufb01cientNets [38, 39] present a scaling strategy that balances compute between depth, width, and resolution and apply it to MobileNets. This strategy is revisited in [3,48] to further improve the performance of ResNets [18]. Large CNNs have attained excellent performance in visual recognition, such as AmoebaNetB(18, 512) (557M parameters) trained using GPipe pipeline parallelism [20], ResNeXt-101 32\u00d748d (829M parameters) pre-trained on weakly-labelled Instagram images [27], Ef\ufb01cientNet-L2 (480M parameters) trained with ImageNet pseudo-labels on JFT-300M [50], and BiT-L-ResNet152x4 (928M parameters) pre-trained on JFT-300M [23]. Recently, [42,54] explore strategies to scale the depth of ViTs. We are the \ufb01rst to scale Vision Transformers to even larger size and reache new state-of-the-art results doing so. The concurrent work [13] focuses on CNN and ViT hybrid architectures. 5. Discussion Limitations. This work uses the proprietary JFT-3B dataset for the scaling laws study. To make our insights more reliable and generalizable, we verify that the scaling laws also apply on the public ImageNet-21k dataset. Societal impact. A potential broader cost of this work is the energy required to perform the experiments in our scaling study, especially in training the largest ViT-G model. However, this cost may be amortized in two ways. First, such studies of scaling laws need only be performed once; We hope future developers of ViT models may use our results to design models that can be trained with fewer compute resources. Second, the models trained are designed primarily for transfer learning. Transfer of pre-trained weights is much less expensive than training from scratch on a downstream task, and typically reaches higher accuracy. Therefore, by transferring our models to many tasks, the pre-training compute is further amortized. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1910.04867v2",
+                    "title": "A Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark",
+                    "abstract": "Representation learning promises to unlock deep learning for the long tail of\nvision tasks without expensive labelled datasets. Yet, the absence of a unified\nevaluation for general visual representations hinders progress. Popular\nprotocols are often too constrained (linear classification), limited in\ndiversity (ImageNet, CIFAR, Pascal-VOC), or only weakly related to\nrepresentation quality (ELBO, reconstruction error). We present the Visual Task\nAdaptation Benchmark (VTAB), which defines good representations as those that\nadapt to diverse, unseen tasks with few examples. With VTAB, we conduct a\nlarge-scale study of many popular publicly-available representation learning\nalgorithms. We carefully control confounders such as architecture and tuning\nbudget. We address questions like: How effective are ImageNet representations\nbeyond standard natural datasets? How do representations trained via generative\nand discriminative models compare? To what extent can self-supervision replace\nlabels? And, how close are we to general visual representations?",
+                    "authors": "Xiaohua Zhai, Joan Puigcerver, Alexander Kolesnikov, Pierre Ruyssen, Carlos Riquelme, Mario Lucic, Josip Djolonga, Andre Susano Pinto, Maxim Neumann, Alexey Dosovitskiy, Lucas Beyer, Olivier Bachem, Michael Tschannen, Marcin Michalski, Olivier Bousquet, Sylvain Gelly, Neil Houlsby",
+                    "published": "2019-10-01",
+                    "updated": "2020-02-21",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.LG",
+                        "stat.ML"
+                    ],
+                    "main_content": "Introduction Distributed representations learned from raw pixels have enabled unprecedented performance on many visual understanding tasks. Hand-crafted features have been replaced with hand-annotated datasets, with thousands to millions of examples (Krizhevsky, 2009; Russakovsky et al., 2015). By contrast, humans learn a wide range vision tasks using just a few examples per task. A key research challenge is to close this gap in sample ef\ufb01ciency, and unlock deep learning for the long tail of problems without many labels. *Equal contribution . Correspondence to: Neil Houlsby <neilhoulsby@google.com>. Preprint. Improving sample ef\ufb01ciency has been approached from many angles: few-shot learning (Fei-Fei et al., 2006), transfer learning (Pan & Yang, 2009), domain adaptation (Wang & Deng, 2018), and representation learning (Bengio et al., 2013). Representation learning is studied in many contexts: supervised pre-training (Sharif Razavian et al., 2014), selfsupervised learning (Doersch et al., 2015), semi-supervised learning (Chapelle et al., 2009), generative modeling (Donahue et al., 2017), and disentanglement learning (Higgins et al., 2017). Each sub-domain has its own evaluation protocol, and the lack of a common benchmark impedes progress. Benchmarks have been critical in other sub-\ufb01elds, such as RL (Mnih et al., 2013), image classi\ufb01cation (Russakovsky et al., 2015), and NLP (Wang et al., 2018). Inspired by these successes, we propose a benchmark with similar principles: (i) minimal constraints to encourage creativity, (ii) a focus on practical considerations, and (iii) make it challenging. We present the Visual Task Adaptation Benchmark (VTAB). Using VTAB, we perform the \ufb01rst extensive cross sub-\ufb01eld study of representation learning. VTAB de\ufb01nes a good representation as one that can be used to solve many diverse previously unseen tasks with the fewest possible labels. We consider low sample complexity to be the key objective of representation learning. Task diversity is also crucial to assess generality. Therefore, VTAB goes beyond standard natural tasks, and includes those related to sensorimotor control, medical imaging, and scene understanding. In our study, we investigate many representation learning algorithms, pre-trained on ImageNet. We carefully control confounding factors such as tuning budget, architecture, and pre-training data. This study quanti\ufb01es existing intuitions and reveals new insights: (i) Supervised ImageNet pretraining yields excellent representations for natural image classi\ufb01cation tasks. Interestingly, it also yields a smaller, but consistent improvement on specialized tasks (e.g. medical imaging). However, these representations are extremely limited for tasks that require structured understanding. (ii) Selfsupervised is less effective than supervised learning overall, but surprisingly, can improve structured understanding. arXiv:1910.04867v2  [cs.CV]  21 Feb 2020 \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark (iii) Combining supervision and self-supervision is effective, and to a large extent, self-supervision can replace, or compliment labels. (iv) Discriminative representations appear more effective than those trained as part of a generative model, with the exception of adversarially trained encoders. (v) GANs perform relatively better on data similar to their pre-training source (here, ImageNet), but worse on other tasks. (vi) Evaluation using a linear classi\ufb01er leads to poorer transfer and different conclusions. 2. The Visual Task Adaptation Benchmark Adapt on DT Test 92.4 96.3 ... 79.7 Upstream  Data Model Adaptation Algorithm \u2130T T1 PT T2 Tk \ud835\udc9c Figure 1. Overview of the VTAB evaluation protocol. We seek algorithms that perform well on a wide variety of unseen visual understanding tasks with few labels per task.1 We \ufb01rst formalize this objective and then specify a practical benchmarking procedure to measure progress. A dataset Dn is a set of n instances {(xi, yi)}n i=1 with observations xi \u2208X and labels yi \u2208Y . A prediction function is any mapping F : X \u2192Y (e.g. a classi\ufb01er). A (learning) algorithm, A, takes as input a dataset and outputs a prediction function. For example, A may be a pre-trained network coupled with a training mechanism. An evaluation procedure, ET , takes F and outputs a scalar measuring F\u2019s performance (e.g. test-set accuracy). We seek the algorithm that maximizes the expected performance over a distribution of tasks PT , where a task T is a tuple containing a taskspeci\ufb01c dataset distribution DT and evaluation procedure. Given only n samples per task we want to maximize: SCOREn(A) = ET \u223cPT ET [A(Dn T )] , (1) This general formulation requires a few clari\ufb01cations. First, the task distribution needs to be appropriate; we desire a spectrum that covers tasks solvable by a vision algorithm 1 In some practical settings additional unlabelled data may be available for unseen tasks in addition to the labelled data. We omit this setting, leaving it to future work. with human-like capabilities. Second, n may be varied to measure an algorithm\u2019s sample complexity. In practice we choose n to be similar to a modest labelling budget (Section 3.1). Third, we assume that PT is known, and we aim to build an algorithm with the best inductive biases for solving samples from it. We desire an \u201copen world\u201d, where evaluation data is extremely diverse and can always be freshly sampled. Unfortunately, in practice the benchmark must contain a \ufb01nite number of test tasks. Therefore, we must ensure that the algorithms are not pre-exposed to speci\ufb01c evaluation samples, as described below. 2.1. A Practical Benchmark We now describe the Visual Task Adaptation Benchmark (VTAB), which is designed to be the best possible proxy for Eq. (1). Fig. 1 present an overview. Task Distribution We aim for universal visual understanding, so we informally de\ufb01ne PT as \u201cTasks that a human can solve, from visual input alone.\u201d. We validate this empirically, see Appendix B. Intuitively, such tasks should bene\ufb01t from visual representations learned by observing and interacting with the natural world. The second clause eliminates tasks that require external knowledge that is (currently) unreasonable for a vision algorithm to acquire \u2013 an extreme example would be to classify objects grouped by their spelling in a natural language. Section 2.2 details the samples. Expectation Over Tasks We approximate the expectation over tasks by an empirical average over a number of hand-picked samples. Ideally, we would sample a new task for each evaluation, as is possible in procedural environments, e.g. (Finn et al., 2017). However, real-world vision tasks are expensive to collect, so we de\ufb01ne a \ufb01xed, representative set of samples from PT . While \ufb01xing the set reduces variance, it introduces a risk of meta-over\ufb01tting. Mitigating Meta-Over\ufb01tting To reduce metaover\ufb01tting, we treat the evaluation tasks like a test set, and consider them unseen. Therefore, algorithms that use pre-training must not pre-train on any of the evaluation tasks (even their unlabelled images). Upstream training should provide useful inductive biases for any draw from PT , so should not use test samples. Fortunately, despite numerous test re-evaluations on popular ML benchmarks, such as ImageNet, progress seems to transfer to new data (Recht et al., 2018; 2019; Kornblith et al., 2019). Uni\ufb01ed Implementation With many diverse tasks, usage could become impractical. As discussed, the algorithms must have no prior knowledge of the downstream tasks; while it is permitted to run a hyperparameter search on each task, the search space cannot be task-dependent. To get meaningful results, we need to de\ufb01ne hyperparameter \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark searches that work well across the benchmark. For this, we convert all tasks into classi\ufb01cation problems. For example, a detection task requiring localization of an object can be mapped to classi\ufb01cation of the (x, y, z) coordinates. With a homogeneous task interface, we may control for possible confounding factors. For example, we may use the same architecture and hyperparameter sweep everywhere. Not all tasks can be ef\ufb01ciently modeled as image-level classi\ufb01cation, such as those requiring per-pixel predictions. Nonetheless, we design the tasks such that success on VTAB requires learning of diverse set of visual features: object identi\ufb01cation, scene classi\ufb01cation, pathology detection, counting, localization, and 3D geometry. 2.2. Tasks VTAB contains 19 tasks which cover a broad spectrum of domains and semantics. Appendix A contains details. These are grouped into three sets: NATURAL, SPECIALIZED, and STRUCTURED. The NATURAL group represents classical vision problems. These tasks contain natural images captured using standard cameras. The classes may represent generic, \ufb01ne-grained, or abstract objects. The group includes: Caltech101, CIFAR100, DTD, Flowers102, Pets, Sun397, and SVHN. The SPECIALIZED group also contains images of the world, but captured through specialist equipment. These images have different invariances to those in the NATURAL tasks. Nonetheless, humans recognize the structures therein, thus generic visual representations should also capture the visual concepts. We have two sub-groups: remote sensing, and medical. Remote sensing includes Resisc45 and EuroSAT: aerial images of the earth captured using satellites or aerial photography. Medical includes Patch Camelyon, metastases detection from microscopy images, and Diabetic Retinopathy, retinopathy classi\ufb01cation from fundus images. The STRUCTURED group assesses comprehension of the structure of a scene, for example, object counting, or 3D depth prediction. Most of these tasks are generated from simulated environments, whose structure is easy for a human to determine, but whose domain differs greatly to datasets like ImageNet. These tasks are intended as a step towards useful representations for perceptual control. We include: Clevr: Simple shapes rendered in a 3D scene, with two tasks: counting and depth prediction. dSprites: Simple black/white shapes rendered in 2D, with two tasks: location and orientation prediction. SmallNORB: Arti\ufb01cial objects viewed under varying conditions, with two tasks: objectazimuth and camera-elevation prediction. DMLab: Frames from a rendered 3D maze. The task involves predicting the time for a pre-trained RL agent to navigate to an object. KITTI: frames captured from a car driver\u2019s perspective. We predict the depth of the nearest vehicle. 2.3. Representation and Transfer Learning Success on VTAB, and ultimately optimizing Eq. (1), requires some knowledge of PT . For example, CNN architectures have a useful inductive bias for vision. While these biases are usually manually designed (e.g. through architecture choice), they can also be learned. For example by learning: architectures Zoph & Le (2017), optimization algorithms Bello et al. (2017), initialization distributions Raghu et al. (2019), or pre-trained data representations. Whilst many strategies merit investigation, we focus on representation learning. Human visual perception is re\ufb01ned through years of observation and interaction with the world, resulting in a system that solves new tasks with few indomain labels. Likewise, we aim to pre-train a network that extracts useful features, or representations, from raw data. Transfer Strategy The representations are not pretrained on the evaluation tasks themselves (which VTAB forbids), so they must be adapted to solve the new tasks (using limited in-domain data). The simplest adaptation strategy in deep representation learning is to \ufb01rst pre-train the network, then freeze the network\u2019s weights and train another \u2013 usually smaller \u2013 model on top. When the upstream and downstream datasets differ signi\ufb01cantly, \ufb01ne-tuning the original weights is more effective (Yosinski et al., 2014; Kornblith et al., 2019). VTAB does not constrain the transfer strategy; here we use \ufb01ne-tuning as it tends to perform best. Upstream Training Representation learning literature often focuses on unsupervised learning which may be applied to any dataset. However, supervised data, where available, can yield good representations. Indeed, the most popular models used in practice are pre-trained on ImageNet labels (Huh et al., 2016, and refs therein). VTAB does not constrain the type of data used for pre-training. 3. Large-Scale Study With VTAB, we evaluate many popular, publicly-available representation learning algorithms across the different types of tasks. Here, we study the pre-training losses, and therefore, control for other factors such as architecture, preprocessing, transfer hyperparameters, and pre-training data. Of course, VTAB may be used subsequently to study and improve these aspects as well. Finally, we provide additional analysis of various of VTAB\u2019s design choices and assess other evaluation protocols. 3.1. Setup Downstream Data Size Section 2.2 describes the tasks. VTAB aims to assess adaptation with limited data, so we \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark evaluate primarily using 1,000 labelled examples per task (called VTAB-1k). We de\ufb01ne train (800 samples) and validation (200 samples) sets for users\u2019 convenience. We emphasize that to avoid meta-over\ufb01tting, one should not use extra images for hyperparameter selection, but may use the 1,000 examples in any manner. We also use the full datasets (VTAB-full). This allows us to assess the value of representation learning as in-domain data increases, and to check performance against prior art. Representation Learning Algorithms We evaluate supervised, semi-supervised, self-supervised, and generative models, selecting popular, public methods from each class. We also compare to \u201cfrom-scratch\u201d models, which use no pre-training, yielding a total of 18 algorithms. All models (except \u201cfrom-scratch\u201d) are pre-trained on ImageNet (1.28M labelled images). Some use the ImageNet labels, others use some or none of the labels. We train two supervised models: one using 100% of the ImageNet labels (SUP-100%), and one using 10% (SUP10%). For self-supervised learning, we include both imagebased and patch-based models. The image-based models include ROTATION (Gidaris et al., 2018) and EXEMPLAR (Dosovitskiy et al., 2014). Patch-based include RELATIVE PATCH LOCATION (Doersch et al., 2015) and JIGSAW (Noroozi & Favaro, 2016). We use the public implementations of Kolesnikov et al. (2019). The patch-based models are converted into image-level classi\ufb01ers by averaging the representations from a 3 \u00d7 3 grid of patches (see Appendix C). The semi-supervised models are trained using 10% of the ImageNet labels with an auxiliary loss on all of the data, see (Zhai et al., 2019). We use either rotation (SUP-ROTATION-10%) or Exemplar (SUPEXEMPLAR-10%) auxiliary losses. These models can also use all of the labels, denoted SUP-ROTATION-100% and SUP-EXEMPLAR-100%. For the generative models, we evaluate GANs and VAEs. As is common, we take the GAN\u2019s image representations from the discriminator, replacing the \ufb01nal linear layer with a classi\ufb01cation layer. We use both the label-conditional and unconditional BigGAN discriminators (Brock et al., 2019) (COND-BIGGAN and UNCOND-BIGGAN) using the implementations of Chen et al. (2019); Lucic et al. (2019). We also evaluate the encoder from BIGBIGAN (Donahue & Simonyan, 2019). For the autoencoders, we use the encoder as the representation. We evaluate VAEs (Kingma & Welling, 2014), and WAEs with three distribution matching losses: GAN, MMD (Tolstikhin et al., 2018), and UKL (Rubenstein et al., 2019). Transfer Algorithm VTAB permits any method of transfer. Here, we \ufb01ne-tuning the entire network on the task data, since that performs best (Kornblith et al., 2019). It is also popular to add a linear model to a frozen network, so we compare to this approach in Section 3.4. Hyperparameters Upstream, we contol the data (ImageNet) and architecture. We \ufb01nd that bigger architectures perform better on VTAB (Appendix L and Kolesnikov et al. (2019)). For comparability, we use ResNet50-v2, or similar, in this study. For the supervised and semi-supervised methods we use the architecture in He et al. (2016). For GANs and auto-encoders we use very similar deep ResNets, but with appropriate modi\ufb01cations to pre-train them successfully as a generative model. Appendix C contains details. Downstream, the transfer hyperparameters in\ufb02uence performance, and different datasets require different settings. We run VTAB in two modes: lightweight and heavyweight. Lightweight mode performs a restricted per-task hyperparameter search. Many hyperparameters are \ufb01xed, including optimizer, batch size, pre-processing, and weight decay. For each task, the lightweight mode sweeps: 2 initial learning rates, and 2 learning rate schedules. See Appendix I for details. We choose short schedules to limit cost, but show in Section 3.5 that these yield near-optimal performance. In heavyweight mode we perform a large random search over learning rate, schedule, optimizers, batch size, train preprocessing functions, evaluation pre-processing, and weight decay. We include longer training schedules and higher resolution images. This mode is used to understand the impact of a larger computational budget on the performance, and establish performance upper bounds on the methods used. Appendix I contains details. We perform our main study in lightweight mode. In Section 3.3 we show that although extensive tuning improves all methods, the relative performances are mostly unaffected. For future study using VTAB, we recommend de\ufb01ning a similar lightweight setup (or using the one here) that facilitates fair comparison without high cost. If ample computational resources are available, a heavyweight mode may be used to improve the state-of-the-art. Tuning and Evaluation Protocol We perform adaptation on the training set and perform model selection for each task on the pre-de\ufb01ned validation sets. We re-train the best model on the union of training and validation sets and evaluate on the test set. Note, for VTAB-1k we de\ufb01ne custom train set with 800 examples and validation set with 200 examples. We run the \ufb01nal evaluation on the test set with three random seeds and report the median score. Metrics We evaluate with top-1 accuracy. We consider other metrics in Appendix D, and \ufb01nd the conclusions are the same. To aggregate scores across tasks, we take the mean accuracy. We investigate more complex aggregation strategies in Section 3.5, but the relative performances are unaffected, so we use mean accuracy for simplicity. \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark Generative From-Scratch Self Supervised Semi Supervised Supervised 0 20 40 60 80 100 Average Top-1 Accuracy Natural Samples 1000 All Generative From-Scratch Self Supervised Semi Supervised Supervised Specialized Generative From-Scratch Self Supervised Semi Supervised Supervised Structured Figure 2. The methods are divided into \ufb01ve groups: Generative, training from-scratch, all methods using 10% labels (Semi Supervised), and all methods using 100% labels (Supervised), For each task group, bars show the mean accuracy across all methods with 3 repeats. Error bars indicate standard deviation across methods. The BigBiGAN outlier causes the high variance in generative models. 3.2. VTAB Results We run all 18 methods on 19 tasks for both VTAB-1k and VTAB-full using the lightweight sweep. Fig. 2 shows aggregate performance of each method group (generative, self-sup, etc.) on each VTAB task group ( NATURAL, etc.). Fig. 3 presents a breakdown of the performance of each algorithm on each taks group. Fig. 4 shows two detailed per-task comparisons: SUP-100% (ImageNet) versus FROM-SCRATCH, and SUP-ROTATION-100% (best overall on VTAB-1k) versus SUP-100%. Appendix G, Table 6 contains the complete results table. Generative Models Overall, these perform worst, Fig. 2. The autoencoders perform more poorly than training from scratch. Despite SOTA generative quality on ImageNet, the GANs\u2019 discriminators do not provide useful representations. The GAN discriminators appear unstable, with large error bars in Fig. 3. The BIGBIGAN encoder stands out, performing much better \u2013 similar to the best self-supervised models, Fig. 3. Interestingly, BIGBIGAN performs relatively well on natural tasks, but less so on structured tasks. UNCONDGAN shows a similar pattern. This indicates that GANs \ufb01t more strongly to ImageNet\u2019s domain (natural images), than self-supervised alternatives. Despite huge advances in generative quality, these models do not yield great representation quality. Similarly, Ravuri & Vinyals (2019) show that models with high sample quality fail to generate data from which an accurate classi\ufb01er can be trained. However, the BigBiGAN result holds promise for adversarially-trained encoders, and consideration of downstream representation quality may lead to further progress. Self-supervised All self-supervised representations outperform from-scratch training. The best, ROTATION, attains 59.6% on VTAB-1k, while FROM-SCRATCH attains 42.1% (Appendix G, Table 6). Methods applied to the entire image (ROTATION, EXEMPLAR) outperform patch-based methods (JIGSAW, REL.PAT.LOC.). However, the patch based methods perform slightly better on DTD (Describable Textures) and Retinopathy (Fundus images), see Appendix G, Table 6. Intuitively, these tasks require sensitivity to local textures, which are indeed captured by patch-based methods. On NATURAL tasks, self-supervision is far behind supervised methods (Fig. 2). On SPECIALIZED, the performances are similar, and most interestingly, on STRUCTURED selfsupervised methods performs slightly better than supervised. This indicates supervised ImageNet models are invariant to useful features required for structured understanding, but self-supervised methods can capture these to some degree. (Semi-)Supervised Overall, supervised models perform best. The bene\ufb01ts are most pronounced on the NATURAL tasks, whose domain and semantics are arguably most similar ImageNet classi\ufb01cation. However, with self-supervision, the bene\ufb01t can be attained with fewer labels. SUP-10% (120k labels) attains 61.6% on VTAB-1k , while SUP100% (1.2M labels) attains 65.6%. With self-supervision, SUP-ROTATION-10% closes 80% of the gap, attaining 64.8%. Recent work has shown in-domain bene\ufb01ts of semisupervised learning (H\u00b4 enaff et al., 2019; Berthelot et al., 2020); our study reveals that semi-supervised methods also learn good representations for transfer to new tasks. Finally, additional self-supervision even improves on top of 100% labelled ImageNet, particularly on STRUCTURED tasks (Fig. 4. SUP-100% attains 65.6%, whereas SUPROTATION-100% attains 67.5%. 3.3. Heavyweight Hyperparameter Sweeps We evaluate FROM-SCRATCH and the best models: SUP100%, SUP-ROTATION-100%, and SUP-EXEMPLAR100% using the heavyweight hyperparameter search: Table 1. Appendix I shows the selected hyperparameters. As expected, all methods improve signi\ufb01cantly. Prior work (He et al., 2018) shows that with suf\ufb01cient data and training time, from-scratch training is competitive for detection. However, we observe that across all task groups, pre-trained representations are better than a tuned from-scratch model. On a couple of tasks FROM-SCRATCH performs competitively \u2014 Clevr-Dist and sNORB-Elev, which require localization and camera elevation respectively \u2014 indicating that even the best ImageNet representations fail to capture these aspects. The ranking of the best methods is similar, but not identical, with a combination supervision and self-supervision \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark WAE-UKL WAE-GAN Cond-BigGAN WAE-MMD VAE From-Scratch Uncond-BigGAN Rel.Pat.Loc Jigsaw Exemplar BigBiGAN Rotation Sup-10% Semi-Exemplar-10% Semi-Rotation-10% Sup-100% Sup-Exemplar-100% Sup-Rotation-100% 10 20 30 40 50 60 70 80 90 100 Top-1 Accuracy 1000-Example Datasets Natural Specialized Structured All WAE-UKL WAE-GAN Cond-BigGAN WAE-MMD VAE Uncond-BigGAN From-Scratch Jigsaw Rel.Pat.Loc BigBiGAN Exemplar Rotation Sup-10% Semi-Rotation-10% Semi-Exemplar-10% Sup-100% Sup-Exemplar-100% Sup-Rotation-100% 10 20 30 40 50 60 70 80 90 100 Top-1 Accuracy Full Datasets Natural Specialized Structured All Figure 3. Average top-1 accuracy across the tasks in each group for VTAB-1k. The x-axis indexes the methods, ordered ordered according to their average accuracy across all tasks (dashed curve). Error bars indicate 95% con\ufb01dence interval using bootstrap resampling from the 3 experiment repeats. Table 1. Top-1 accuracy of the best models and ResNet50 from-scratch using the heavyweight hyperparameter sweep. Caltech101 CIFAR-100 DTD Flowers102 Pets Sun397 SVHN Camelyon EuroSAT Resisc45 Retinopathy Clevr-Count Clevr-Dist DMLab dSpr-Loc dSpr-Ori KITTI-Dist sNORB-Azim sNORB-Elev Mean 1000 From-Scratch 61.2 20.4 46.4 73.4 48.0 13.1 82.5 77.0 91.5 59.5 73.4 66.0 59.2 36.9 88.2 62.6 63.8 22.6 79.1 59.2 Sup-100% 83.6 48.9 65.9 93.0 90.4 31.1 87.0 80.2 95.8 82.0 79.0 78.8 58.3 53.8 89.0 71.2 73.9 34.2 57.8 71.2 Sup-Rot-100% 80.0 42.6 66.5 91.5 88.6 32.7 90.0 78.5 96.4 82.1 78.4 96.2 58.1 55.3 89.3 71.1 69.1 39.3 53.6 71.5 Sup-Ex-100% 83.3 49.3 61.8 94.2 90.2 35.2 87.6 84.2 95.7 81.0 78.7 96.4 56.3 56.1 90.0 69.8 73.3 38.6 59.5 72.7 Full From-Scratch 74.5 77.8 67.1 85.8 70.9 70.1 97.0 91.2 98.8 94.3 82.8 99.8 96.7 76.6 100.0 96.7 68.4 99.9 94.0 86.4 Sup-100% 93.0 83.4 73.7 97.3 92.7 75.6 97.5 87.3 98.8 96.1 83.4 100.0 97.0 78.8 100.0 96.8 81.0 100.0 98.5 91.1 Sup-Rot-100% 93.5 84.0 76.8 97.4 92.6 75.6 97.4 86.5 99.1 96.3 83.7 100.0 96.8 79.6 100.0 96.8 82.4 100.0 96.7 91.3 Sup-Ex-100% 93.8 83.1 76.5 97.8 92.9 75.3 97.5 86.5 99.0 96.3 83.7 99.9 96.8 79.3 100.0 96.7 82.8 100.0 99.1 91.4 (SUP-EXEMPLAR-100%) getting the best performance. We check our results against those recently reported in the literature (Appendix H, Table 8). Our results are comparable; behind on highly popular tasks on which complex architectures have been optimized (e.g. CIFAR), but ahead in others. In Appendix L we show that simply increasing the architecture size improves VTAB performance. 3.4. Frozen Features Extractors Representations are often evaluated by training a linear layer on a frozen model (Kolesnikov et al., 2019). ImageNet is often used for linear evaluation, but this is meaningless for the models we consider because many use ImageNet labels for pre-training. However, linear evaluation is also used in a transfer setting (Goyal et al., 2019a), so we apply this protocol to the VTAB tasks, and contrast it to \ufb01netuning. The full protocol resembles the lightweight \ufb01netuning sweep, see Appendix K. Fig. 5 shows the per-task correlation between linear and \ufb01ne-tuning. Appendix K, Table 9 contains the full table of results. We \ufb01rst note that linear evaluation significantly lowers performance, even when downstream data is limited to 1000 examples. SUP-100% attains 65.6% with \ufb01ne-tuning (lightweight sweep), but 57.3% with linear. Linear transfer would not by used in practice unless infrastructural constraints required it. Second, Fig. 5 shows that on many datasets, particularly SPECIALIZED and STRUCTURED, the correlation is low. Linear evaluation may lead to different conclusions, for example, CONDBIGGAN attains 43.3% (1000-examples) on linear, outperforming both REL.PAT.LOC and JIGSAW. Yet when \ufb01ne-tuned these self-supervised methods signi\ufb01cantly outperform the GAN discriminator. Another discrepancy is between semi-supervised and supervised. SEMI-ROTATION10% and SEMI-EXEMPLAR-10% are 1 \u22122% behind SUP100% with \ufb01ne-tuning, but 4\u22125% behind with linear evaluation. These self-supervised methods extract useful representations, just without linear separability. Previous works (Kornblith et al., 2019; Kolesnikov et al., 2019) claim that linear evaluation results are sensitive to additional factors that we do not vary, such as ResNet version or pre-training regularization parameters. Overall linear evaluation is a poor proxy for overall reduced sample complexity. 3.5. Analysis Metrics We explore alternatives to using top-1 accuracy, and aggregation of scores accross tasks using the mean. Instead of top-1 accuracy, we used mean-per-class accuracy and Cohen\u2019s quadratic kappa. These metrics create only \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark dSpr-Loc Clevr-Dist sNORB-Elev Clevr-Count Camelyon DMLab Retinopathy dSpr-Orient EuroSAT sNORB-Azim SVHN Sun397 KITTI-Dist Resisc45 DTD CIFAR-100 Flowers102 Caltech101 Pets \u221210 0 10 20 30 40 50 60 70 80 Top-1 Accuracy Delta Sup-100% From-Scratch sNORB-Elev sNORB-Azim Sun397 CIFAR-100 Pets Caltech101 EuroSAT Retinopathy Camelyon Flowers102 SVHN Resisc45 dSpr-Orient DTD KITTI-Dist Clevr-Dist DMLab Clevr-Count dSpr-Loc \u22124 \u22122 0 2 4 6 8 10 12 Top-1 Accuracy Delta Sup-Rotation-100% Sup-100% Figure 4. Absolute difference in top-1 accuracy between method pairs for each dataset, using 1k examples. The bar colour indicates the task group: NATURAL, SPECIALIZED, and STRUCTURED. Left: SUP-100% versus FROM-SCRATCH \u2014 supervised pretraining yields a large improvement on the NATURAL datasets and some others. Right: SUP-ROTATION-100% versus SUP-100% \u2014 the additional self-supervised loss on top of supervised loss yields improvements, especially on the STRUCTURED tasks. Note that the y-scales differ. minor ranking differences, leaving the overall picture unchanged. Kendall\u2019s ranking correlation with respect to top-1 accuracy is always > 0.97. Details are in D. For aggregation across tasks, we consider seven alternative strategies: (i) Macro-average across datasets (grouping tasks with the same input data). (ii) Macro-average across groups (merging tasks in the same group). (iii) Geometric mean. (iv) Average rank. (v) Average rank after small perturbation of the scores with noise. (vi) Robust (binned) average rank. (vii) Elimination rank \u2013 equivalent to an \u201cExhaustive Ballot\u201d. All strategies have a high Kendall\u2019s correlation with the vanilla mean across tasks (\u03c4 > 0.87). Appendix E contains details. Based on these results we choose mean top-1 for VTAB since it is simple, interpretable, can be computed for each task independently, and is highly correlated with more complex appraoches. Representative Subset For prototyping, a representative subset of VTAB may be useful. We compute the rank correlation between the mean scores produced by each \u000020 5 \u0001 subsets of \ufb01ve tasks, and the full suite. The top-5 subsets tend to span different domains, but differ to each other. ApSVHN Flowers102 DTD Pets Caltech101 CIFAR-100 Sun397 Retinopathy EuroSAT Camelyon Resisc45 Clevr-Dist sNORB-Azim dSpr-Loc Clevr-Count DMLab sNORB-Elev dSpr-Orient KITTI-Dist \u22120.4 \u22120.2 0.0 0.2 0.4 0.6 0.8 1.0 Kendall Rank Correlation 1000-Example Datasets Figure 5. Kendall\u2019s correlation between \ufb01ne-tuning and linear evaluation on each dataset. 30 100 1000 10 000 50 000 Total training steps 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Accuracy Sup-100% (Full) Sup-100% (1000) Rotation (Full) Rotation (1000) Figure 6. Performance under various time constraints. pendix F contains the results. Although a subset might be useful for screening models, these were computed on our set of models, and may correlate less well in other experiments. Using subsets also increases the risk of meta-over\ufb01tting, which VTAB aims to avoid by having many tasks. Limiting Evaluation Cost The cost is near linear in the schedule length. To determine a minimal, yet meaningful, schedule, we sweep over schedules ranging from 40 to 40,000 steps, using batch size 512. Fig. 6 summarizes the results, details Figs. 19 and 20, Appendix M. Most runs reach their optimal performance within 1,000 steps and do not improve signi\ufb01cantly when trained for longer. Finetuning and evaluating a given ResNet-50 model on a single Nvidia P100 GPU (batch size of 64 images, 0.01 initial learning rate) takes 3 hours for VTAB-1k. To reduce the time and cost we use Google Cloud TPU-v3-16 accelerators and verify that we obtain similar resuts in this setting. 4. Related Work Vision Benchmarks The Visual Decathlon (Rebuf\ufb01et al., 2017) contains ten classi\ufb01cation tasks: Omniglot, and nine using natural images. Four overlap with VTAB, but VTAB includes several other domains and tasks. Importantly, these benchmarks have opposite evaluation protocols: The Visual Decathlon allows direct joint optimization on the tasks, but forbids external data. By contrast, VTAB forbids multi-task learning on the downstream tasks, but permits transfer from \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark any arbitrary dataset \u2014 this simulates one-off training of representations which are then applied to novel tasks. Further, VTAB considers a low-sample regime (1000-examples), which re\ufb02ects performance under a reasonable labelling budget. We conduct experiments showing that the methods ranked according to VTAB are more likely to transfer to new tasks, than those ranked according to the Visual Decathlon (Appendix N). The Facebook AI SSL challenge (Goyal et al., 2019b) was proposed to evaluate self-supervision. It contains four tasks on two datasets of natural images, including classi\ufb01cation, detection, and low-label classi\ufb01cation. The original paper (Goyal et al., 2019a) also includes navigation and surface normal estimation. VTAB uses classi\ufb01cation tasks only to admit task-independent implementations, and attains diversity with many alternative domains and semantics (localization, counting, etc.). Our study includes self-supervision, but are not restricted to it. Indeed, one challenge is to outperform supervised ImageNet on VTAB. Most importantly, Facebook SSL requires transfer via a shallow network stacked on a \ufb01xed CNN, whereas VTAB permits any transfer, and focuses on performance with few samples. Meta-Dataset (Trianta\ufb01llou et al., 2019) contains natural image classi\ufb01cation tasks. The protocol differs to VTAB: most of Meta-Dataset\u2019s evaluation datasets are also in the training set, and the train/test split is made across classes. Meta-Dataset is designed for few-shot learning, rather than 1000 examples, which may entail different solutions. Representation Learning Evaluation Popular evaluations for representation learning are linear/MLP and semisupervised. In linear/MLP evaluation, widely used in for self-supervised representations (Doersch et al., 2015; Zhang et al., 2016; Noroozi & Favaro, 2016; Doersch & Zisserman, 2017), the weights of a pre-trained network are frozen, and a linear layer/MLP is trained on top to predict the labels. Evaluation is often performed on the same dataset that was used to train the representations, but using all labels, defeating the goal of sample ef\ufb01ciency. A semi-supervised protocol is more realistic and performs better (Zhai et al., 2019). This evaluation is performed on a single dataset, by contrast VTAB concerns transfer to unseen tasks. Linear evaluation is sensitive to small upstream training details, such as the ResNet version or regularization, that make little difference when training end-to-end (Kornblith et al., 2019; Kolesnikov et al., 2019). Other intrinsic scores have been proposed to measure the disentanglement of representations (Locatello et al., 2019), or the mutual information between the input and representations (Hjelm et al., 2018). Both were shown to be weakly correlated with representation utility (Locatello et al., 2019; Tschannen et al., 2019). Generative Evaluation Evaluation of generative quality is a dif\ufb01cult goal with numerous proxy metrics, such as reconstruction error, FID (Heusel et al., 2017), IS (Salimans et al., 2016), precision-recall (Sajjadi et al., 2018), log likelihood (for auto-regressive models), or the ELBO (for VAEs). Sometimes generative models are also evaluated using linear classi\ufb01cation (Radford et al., 2016; Donahue et al., 2017; Dumoulin et al., 2017), or semi-supervised learning (Kingma et al., 2014; Narayanaswamy et al., 2017; Tschannen et al., 2018). 5. Discussion Our study answers important and timely questions on representation learning. First, how effective are supervised ImageNet representations? ImageNet labels are indeed effective for natural tasks. Further, although the literature reports mixed results (Liu et al., 2017; Raghu et al., 2019; Neumann et al., 2019), we \ufb01nd that they also work for specialized tasks despite the domain-shift (medical or remote sensing). However, for structured understanding, supervised representations can be poorer than unsupervised ones. As future work, training on diverse data sources, such as video, may enable moving beyond the \u201cImageNet-like\u201d tasks. Second, how do representations trained via generative and discriminative models compare? Generation is a useful task in of itself, but generative losses, in their current form, seem less promising as means towards learning how to represent data. BigBiGAN is a notable recent exception, that is on par with self-supervision and warrants more exploration. Third, to what extent can self-supervision replace labels? Self-supervised learning appears promising, even outperforming supervision on some tasks that require structured understanding. Interestingly, self-supervision can almost (but not quite) replace 90% of ImageNet labels; the gap between pre-training on 10% labels with self-supervison, and 100% labels, is small. Further, self-supervision adds value on top of ImageNet labels on the same data. Overall, it seems we are quite far from general visual representations; simply adding more data on the SPECIALIZED and STRUCTURED tasks is better than the pre-training strategies we evaluated. However, self-supervison can be applied to any dataset of images or videos, so combining large scale open-domain self-supervision with ImageNet or other label sources may be promising future work. VTAB measures representation quality as the ability to adapt with few examples to diverse, unseen tasks. Here, we provide a large study of upstream training losses and control other factors such as hyperparameter sweep, architecture, transfer algorithm, preprocessing, and pre-training data. However, VTAB may be used to analyze and optimze many factors involved in learning generalizable representations. Varying other factors to improve VTAB is valuable future research. To isolate the effect of each factor, confounders \fA Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark such as hyperparameter sweep size should be controlled. The only approach that is out-of-bounds is to condition the algorithm explicitly on the VTAB tasks, since this would compromise representation generalization. Code and data to run VTAB are made available, and progress is monitored at github.com/google-research/ task_adaptation. The models presented here are released. We hope that VTAB drives representation learning towards making deep learning accessible to the many problems without vast labelling budgets. ACKNOWLEDGMENTS We thank Raphael Marinier and Anton Raichuk for help building the DMLab classi\ufb01cation task, and Samy Bengio and Kevin Murphy for useful discussions."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1905.03670v2",
+                    "title": "S4L: Self-Supervised Semi-Supervised Learning",
+                    "abstract": "This work tackles the problem of semi-supervised learning of image\nclassifiers. Our main insight is that the field of semi-supervised learning can\nbenefit from the quickly advancing field of self-supervised visual\nrepresentation learning. Unifying these two approaches, we propose the\nframework of self-supervised semi-supervised learning and use it to derive two\nnovel semi-supervised image classification methods. We demonstrate the\neffectiveness of these methods in comparison to both carefully tuned baselines,\nand existing semi-supervised learning methods. We then show that our approach\nand existing semi-supervised methods can be jointly trained, yielding a new\nstate-of-the-art result on semi-supervised ILSVRC-2012 with 10% of labels.",
+                    "authors": "Xiaohua Zhai, Avital Oliver, Alexander Kolesnikov, Lucas Beyer",
+                    "published": "2019-05-09",
+                    "updated": "2019-07-23",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Modern computer vision systems demonstrate outstanding performance on a variety of challenging computer vision benchmarks, such as image recognition [34], object detection [22], semantic image segmentation [8], etc. Their success relies on the availability of a large amount of annotated data that is time-consuming and expensive to acquire. Moreover, applicability of such systems is typically limited in scope de\ufb01ned by the dataset they were trained on. Many real-world computer vision applications are concerned with visual categories that are not present in standard benchmark datasets, or with applications of dynamic nature where visual categories or their appearance may change over time. Unfortunately, building large labeled datasets for all these scenarios is not practically feasible. Therefore, it is an important research challenge to design a learning approach that can successfully learn to recognize new concepts by leveraging only a small amount of labeled examples. The fact that humans quickly understand new concepts after seeing only a few (labeled) examples suggests that this goal is achievable in principle. Notably, a large research effort is dedicated towards \u2217equal contribution Full dataset Labelled images 0\u00b0 270\u00b0 hat ... ... 270\u00b0 hat 0\u00b0 Figure 1. A schematic illustration of one of the proposed self-supervised semi-supervised techniques: S4L-Rotation. Our model makes use of both labeled and unlabled images. The \ufb01rst step is to create four input images for any image by rotating it by 0\u25e6, 90\u25e6, 180\u25e6and 270\u25e6(inspired by [10]). Then, we train a single network that predicts which rotation was applied to all these images and, additionally, predicts semantic labels of annotated images. This conceptually simple technique is competitive with existing semi-supervised learning methods. learning from unlabeled data that, in many realistic applications, is much less onerous to acquire than labeled data. Within this effort, the \ufb01eld of self-supervised visual representation learning has recently demonstrated the most promising results [17]. Self-supervised learning techniques de\ufb01ne pretext tasks which can be formulated using only unlabeled data, but do require higher-level semantic understanding in order to be solved. As a result, models trained for solving these pretext tasks learn representations that can be used for solving other downstream tasks of interest, such as image recognition. Despite demonstrating encouraging results [17], purely self-supervised techniques learn visual representations that are signi\ufb01cantly inferior to those delivered by fullysupervised techniques. Thus, their practical applicability is 1 arXiv:1905.03670v2  [cs.CV]  23 Jul 2019 \flimited and as of yet, self-supervision alone is insuf\ufb01cient. We hypothesize that self-supervised learning techniques could dramatically bene\ufb01t from a small amount of labeled examples. By investigating various ways of doing so, we bridge self-supervised and semi-supervised learning, and propose a framework of semi-supervised losses arising from self-supervised learning targets. We call this framework self-supervised semi-supervised learning or, in short, S4L. The techniques derived in that way can be seen as new semisupervised learning techniques for natural images. Figure 1 illustrates the idea of the proposed S4L techniques. We thus evaluate our models both in the semi-supervised setup, as well as in the transfer setup commonly used to evaluate self-supervised representations. Moreover, we design strong baselines for benchmarking methods which learn using only 10 % or 1 % of the labels in ILSVRC-2012. We further experimentally investigate whether our S4L methods could further bene\ufb01t from regularizations proposed by the semi-supervised literature, and discover that they are complementary, i.e. combining them leads to improved results. Our main contributions can be summarized as follows: \u2022 We propose a new family of techniques for semisupervised learning with natural images that leverage recent advances in self-supervised representation learning. \u2022 We demonstrate that the proposed self-supervised semi-supervised (S4L) techniques outperform carefully tuned baselines that are trained with no unlabeled data, and achieve performance competitive with previously proposed semi-supervised learning techniques. \u2022 We further demonstrate that by combining our best S4L methods with existing semi-supervised techniques, we achieve new state-of-the-art performance on the semi-supervised ILSVRC-2012 benchmark. 2. Related Work In this work we build on top of the current state-of-theart in both \ufb01elds of semi-supervised and self-supervised learning. Therefore, in this section we review the most relevant developments in these \ufb01elds. 2.1. Semi-supervised Learning Semi-supervised learning describes a class of algorithms that seek to learn from both unlabeled and labeled samples, typically assumed to be sampled from the same or similar distributions. Approaches differ on what information to gain from the structure of the unlabeled data. Given the wide variety of semi-supervised learning techniques proposed in the literature, we refer to [4] for an extensive survey. For more context, we focus on recent developments based on deep neural networks. The standard protocol for evaluating semi-supervised learning algorithms works as such: (1) Start with a standard labeled dataset; (2) Keep only a portion of the labels (say, 10 %) on that dataset; (3) Treat the rest as unlabeled data. While this approach may not re\ufb02ect realistic settings for semi-supervised learning [30], it remains the standard evaluation protocol, which we follow it in this work. Many of the initial results on semi-supervised learning with deep neural networks were based on generative models such as denoising autoencoders [33], variational autoencoders [16] and generative adversarial networks [29, 35]. More recently, a line of research showed improved results on standard baselines by adding consistency regularization losses computed on unlabeled data. These consistency regularization losses measure discrepancy between predictions made on perturbed unlabeled data points. Additional improvements have been shown by smoothing predictions before measuring these perturbations. Approaches of these kind include \u03a0-Model [19], Temporal Ensembling [19], Mean Teacher [41] and Virtual Adversarial Training [24]. Recently, fast-SWA[1] showed improved results by training with cyclic learning rates and measuring discrepancy with an ensemble of predictions from multiple checkpoints. By minimizing consistency losses, these models implicitly push the decision boundary away from high-density parts of the unlabeled data. This may explain their success on typical image classi\ufb01cation datasets, where points in each clusters typically share the same class. Two additional important approaches for semisupervised learning, which have shown success both in the context of deep neural networks and other types of models are Pseudo-Labeling [20], where one imputes approximate classes on unlabeled data by making predictions from a model trained only on labeled data, and conditional entropy minimization [11], where all unlabeled examples are encouraged to make con\ufb01dent predictions on some class. Semi-supervised learning algorithms are typically [30, 24, 2, 42, 1, 23] evaluated on small-scale datasets such as CIFAR-10 [18] and SVHN [25]. We are aware of very few examples in the literature where semi-supervised learning algorithms are evaluated on larger, more challenging datasets such as ILSVRC-2012 [34]. To our knowledge, Mean Teacher [41] currently holds the state-of-the-art result on ILSVRC-2012 when using only 10 % of the labels. Recent concurrent work [43, 13] presents competitive results on ILSVRC-2012. 2.2. Self-supervised Learning Self-supervised learning is a general learning framework that relies on surrogate (pretext) tasks that can be formulated using only unsupervised data. A pretext task is de2 \fsigned in a way that solving it requires learning of a useful image representation. Self-supervised techniques have a variety of applications in a broad range of computer vision topics [15, 37, 7, 31, 36]. In this paper we employ self-supervised learning techniques that are designed to learn useful visual representations from image databases. These techniques achieve stateof-the-art performance among approaches that learn visual representations from unsupervised images only. Below we provided a non-comprehensive summary of the most important developments in this direction. Doersch et al. propose to train a CNN model that predicts relative location of two randomly sampled nonoverlapping image patches [5]. Follow-up papers [26, 28] generalize this idea for predicting a permutation of multiple randomly sampled and permuted patches. Beside the above patch-based methods, there are selfsupervised techniques that employ image-level losses. Among those, in [44] the authors propose to use grayscale image colorization as a pretext task. Another example is a pretext task [10] that predicts an angle of the rotation transformation that was applied to an input image. Some techniques go beyond solving surrogate classi\ufb01cation tasks and enforce constraints on the representation space. A prominent example is the exemplar loss from [6] that encourages the model to learn a representation that is invariant to heavy image augmentations. Another example is [27], that enforces additivity constraint on visual representation: the sum of representations of all image patches should be close to representation of the whole image. Finally, [3] proposes a learning procedure that alternates between clustering images in the representation space and learning a model that assigns images to their clusters. 3. Methods In this section we present our self-supervised semisupervised learning (S4L) techniques. We \ufb01rst provide a general description of our approach. Afterwards, we introduce speci\ufb01c instantiations of our approach. We focus on the semi-supervised image classi\ufb01cation problem. Formally, we assume an (unknown) data generating joint distribution p(X, Y ) over images and labels. The learning algorithm has access to a labeled training set Dl, which is sampled i.i.d. from p(X, Y ) and an unlabeled training set Du, which is sampled i.i.d. from the marginal distribution p(X). The semi-supervised methods we consider in this paper have a learning objective of the following form: min \u03b8 Ll(Dl, \u03b8) + wLu(Du, \u03b8), (1) where Ll is a standard cross-entropy classi\ufb01cation loss of all labeled images in the dataset, Lu is a loss de\ufb01ned on unsupervised images (we discuss its particular instances later in this section), w is a non-negative scalar weight and \u03b8 is the parameters for model f\u03b8(\u00b7). Note that the learning objective can be extended to multiple unsupervised losses. 3.1. Self-supervised Semi-supervised Learning We now describe our self-supervised semi-supervised learning techniques. For simplicity, we present our approach in the context of multiclass image recognition, even though it can be easily generalized to other scenarios, such as dense image segmentation. It is important to note that in practice the objective 1 is optimized using a stochastic gradient descent (or a variant) that uses mini-batches of data to update the parameters \u03b8. In this case the size of a supervised mini-batch xl, yl \u2282Dl and an unsupervised mini-batch xu \u2282Du can be arbitrary chosen. In our experiments we always default to simplest possible option of using minibatches of equal sizes. We also note that we can choose whether to include the minibatch xl into the self-supervised loss, i.e. apply Lself to the union of xu and xl. We experimentally study the effect of this choice in our experiments Section 4.4. We demonstrate our framework on two prominent selfsupervised techniques: predicting image rotation [10] and exemplar [6]. Note, that with our framework, more selfsupervised losses can be explored in the future. S4L-Rotation. The key idea of rotation self-supervision is to rotate an input image then predict which rotation degree was applied to these rotated images. The loss is de\ufb01ned as: Lrot = 1 |R| X r\u2208R X x\u2208Du L(f\u03b8(xr), r) (2) where R is the set of the 4 rotation degrees {0\u25e6, 90\u25e6, 180\u25e6, 270\u25e6}, xr is the image x rotated by r, f\u03b8(\u00b7) is the model with parameters \u03b8,L is the cross-entropy loss. This results in a 4-class classi\ufb01cation problem. We follow a recommendation from [10] and in a single optimization step we always apply and predict all four rotations for every image in a minibatch. We also apply the self-supervised loss to the labeled images in each minibatch. Since we process rotated supervised images in this case, we suggest to also apply a classi\ufb01cation loss to these images. This can be seen as an additional way to regularize a model in a regime when a small amount of labeled image are available. We measure the effect of this choice later in Section 4.4. S4L-Exemplar. The idea of exemplar self-supervision [6] is to learn a visual representation that is invariant to a wide range of image transformations. Speci\ufb01cally, we use \u201cInception\u201d cropping [40], random horizontal mirroring, and HSV-space color randomization as in [6] to produce 8 different instances of each image in a minibatch. Following [17], we implement Lu as the batch hard triplet loss [14] 3 \fwith a soft margin. This encourages transformation of the same image to have similar representations and, conversely, encourages transformations of different images to have diverse representations. Similarly to the rotation self-supervision case, Lu is applied to all eight instances of each image. 3.2. Semi-supervised Baselines In the following section, we compare S4L to several leading semi-supervised learning algorithms that are not based on self-supervised objectives. We now describe the approaches that we compare to. Our proposed objective 1 is applicable for semi supervised learning methods as well, where the loss Lu is the standard semi supervised loss as described below. Virtual Adversarial Training (VAT) [24]: The idea is making the predicted labels robust around input data point against local perturbation. It approximates the maximal change in predictions within an \u03f5vat vicinity of unlabeled data points, where \u03f5vat is a hyperparameter. Concretely, the VAT loss for a model f\u03b8 is: Lvat = 1 |Du| X x\u2208Du KL(f\u03b8(x) \u2225f\u03b8(x + \u2206x)), (3) where \u2206x = arg max \u03b4 s.t. |\u03b4|2=\u03f5 KL(f\u03b8(x) \u2225f\u03b8(x + \u03b4)). (4) While computing \u2206x directly is not tractable, it can be ef\ufb01ciently approximated at the cost of an extra forward and backwards pass for every optimization step. [24]. Conditional Entropy Minimization (EntMin) [11]: This works under the assumption that unlabeled data indeed has one of the classes that we are training on, even when the particular class is not known during training. It adds a loss for unlabeled data that, when minimized, encourages the model to make con\ufb01dent predictions on unlabeled data. Speci\ufb01cally, the conditional entropy minimization loss for a model f\u03b8 (treating f\u03b8 as a conditional distribution of labels over images) is: Lentmin = 1 |Du| X x\u2208Du X y\u2208Y \u2212f\u03b8(y|x) log f\u03b8(y|x) (5) Alone, the EntMin loss is not useful in the context of deep neural networks because the model can easily become extremely con\ufb01dent by increasing the weights of the last layer. One way to resolve this is to encourage the model predictions to be locally-Lipschitz, which VAT does[38]. Therefore, we only consider VAT and EntMin combined, not just EntMin alone, i.e. Lu = wvatLvat + wentminLentmin. Pseudo-Label [20] is a simple approach: Train a model only on labeled data, then make predictions on unlabeled data. Then enlarge your training set with the predicted classes of the unlabeled data points whose predictions are con\ufb01dent past some threshold of con\ufb01dence. Re-train your model with this enlarged labeled dataset. While [30] shows that in a simple \u201dtwo moons\u201d dataset, psuedo-label fails to learn a good model, in many real datasets this approach does show meaningful gains. 4. ILSVRC-2012 Experiments and Results In this section, we present the results of our main experiments. We used the ILSVRC-2012 dataset due to its widespread use in self-supervised learning methods, and to see how well semi-supervised methods scale. Since the test set of ILSVRC-2012 is not available, and numbers from the validation set are usually reported in the literature, we performed all hyperparameter selection for all models that we trained on a custom train/validation split of the public training set. This custom split contains 1 231 121 training and 50 046 validation images. We then retrain the model using the best hyperparameters on the full training set (1 281 167 images), possibly with fewer labels, and report \ufb01nal results obtained on the public validation set (50 000 images). We follow standard practice [41, 32] and perform experiments where class-balanced labels are available for only 10 % of the dataset. Note that 10 % of ILSVRC-2012 still corresponds to roughly 128 000 labeled images, and that previous work uses the full (public) validation set for model selection. While we use a custom validation set extracted from the training set, using such a large validation set does not correspond to a realistic scenario, as already discussed by [33, 41, 30]. We also want to cover more realistic cases in our evaluation. We thus perform experiments on 1 % of labeled examples (roughly 13 000 labeled images), while also using a validation set of only 5000 images. We analyze the impact of validation set size in Section 7. We always de\ufb01ne epochs in terms of the available labeled data, i.e. one epoch corresponds to one full pass through the labeled data, regardless of how many unlabeled examples have been seen. We optimize our models using stochastic gradient descent with momentum on minibatches of size 256 unless speci\ufb01ed otherwise. While we do tune the learning rate, we keep the momentum \ufb01xed at 0.9 across all experiments. Table 1 summarizes our main results. 4.1. Plain Supervised Learning Whenever new methods are introduced, it is crucial to compare them against a solid baseline of existing methods. The simplest baseline to which any semi-supervised learning method should be compared to, is training a plain supervised model on the available labeled data. 4 \fOliver et al. [30] discovered that reported baselines trained on labeled examples alone are unfairly weak, perhaps given that there is not a strong community behind tuning those baselines. They provide strong supervised-only baselines for SVHN and CIFAR-10, and show that the gap shown by the use of unlabeled data is smaller than reported. We observed the same in the case of ILSVRC-2012. Thus, we aim to provide a strong baseline for future research by performing a relatively large search over training hyperparameters for training a model on only 10 % of ILSVRC-2012. Speci\ufb01cally, we try weight-decay values in {1, 3} \u00b7 10{\u22122,\u22123,\u22124}, learning rates in {0.3, 0.1, 0.03}, four different learning rate schedules spanning 30 to 500 epochs, and \ufb01nally we explore various model architectures: ResNet50, ResNet34, ResNet18, in both \u201cregular\u201d (v1) and \u201cpre-activation\u201d (v2) variants, as well as half-, double-, triple-, and quadruple-width variants of these, testing the assumption that smaller or shallower models over\ufb01t less. In total, we trained several thousand models on our custom training/validation split of the public training set of ILSVRC-2012. In summary, it is crucial to tune both weight decay and training duration while, perhaps surprisingly, model architecture, depth, and width only have a small in\ufb02uence on the \ufb01nal results. We thus use a standard, unmodi\ufb01ed ResNet50v2 as model, trained with weight decay of 10\u22123 for 200 epochs, using a standard learning rate of 0.1, ramped up from 0 for the \ufb01rst \ufb01ve epochs, and decayed by a factor of 10 at epochs 140, 160, and 180. We train in total for 200 epochs. The standard augmentation procedure of random cropping and horizontal \ufb02ipping is used during training, and predictions are made using a single central crop keeping aspect ratio. We perform a similar search when training our baseline on 1 % of ILSVRC-2012, but additionally include two choices of data augmentation (whether or not to apply random color augmentation) and two minibatch sizes (256 and 1024) in the hyperparameter search. Perhaps somewhat surprisingly, the results here are similar, in that tuning the weight decay and training duration is crucial, but model architecture does not matter much. Additionally, performing color augmentation becomes important. Here too, we use a standard, unmodi\ufb01ed ResNet50v2 as model, trained with weight decay of 10\u22122 for 1000 epochs, using a learning rate of 0.011, ramped up from 0.0 for the \ufb01rst ten epochs2, and decayed by a factor of 10 at epochs 700, 800, and 900. We train in total for 1000 epochs. A more detailed presentation of the results is provided in the supplementary material. Using this only slightly altered training procedure, our baseline models achieve 80.43 % top5 accuracy (56.35 % top1) on the public ILSVRC-2012 validation set when 1While the standard learning rate of 0.1 worked equally well, learning curves seemed signi\ufb01cantly less stable. 2This was likely not necessary, but kept for consistency. Table 1. Top-5 accuracy [%] obtained by individual methods when training them on ILSVRC-2012 with a subset of labels. All methods use the same standard width ResNet50v2 architecture. ILSVRC-2012 labels: 10 % 1 % (i.e. images per class) (128) (13) Supervised Baseline (Section 4.1) 80.43 48.43 Pseudolabels [20] 82.41 51.56 VAT [24] 82.78 44.05 VAT + Entropy Minimization [11] 83.39 46.96 Self-sup. Rotation [17] + Linear 39.75 25.98 Self-sup. Exemplar [17] + Linear 32.32 21.33 Self-sup. Rotation [17] + Fine-tune 78.53 45.11 Self-sup. Exemplar [17] + Fine-tune 81.01 44.90 S4L-Rotation 83.82 53.37 S4L-Exemplar 83.72 47.02 trained on only 10 % of the full training set. Our 1 % baseline achieves 48.43 % top5 accuracy (25.39 % top1). These results form a solid baseline to compare to, considering that the same standard ResNet50v2 model achieves 92.82 % top5 accuracy (75.89 % top1) on 100 % of the labels. For all further experiments, we reuse the best hyperparameters discovered here, except that we try two additional learning rates: {0.3, 0.1, 0.03} for 10 % and {0.03, 0.01, 0.003} for 1 %, and two additional weight decays: {10\u22124, 3\u00b710\u22124, 10\u22123} for 10 % and {3\u00b710\u22123, 10\u22122, 3\u00b7 10\u22122} for 1 %. We also try two different weights wu for the additionally introduced loss Lu: wu \u2208{0.5, 1.0}. 4.2. Semi-supervised Baselines We train semi-supervised baseline models using (1) Pseudo-Label, (2) VAT, and (3) VAT+EntMin. To the best of our knowledge, we present the \ufb01rst evaluation of these techniques on ILSVRC-2012. Pseudo-Label Using the plain supervised learning models from Section 4.1, we assign pseudo-labels to the full dataset. Then, in a second step, we train a ResNet50v2 from scratch following standard practice, i.e. with learning rate 0.1, weight decay 10\u22124, and 100 epochs on the full (pseudo-labeled) dataset. We try both using all predictions as pseudo-labels, as well as using only those predictions with a con\ufb01dence above 0.5. Both perform closely on our validation set, and we choose no \ufb01ltering for the \ufb01nal model for simplicity. Table 1 shows that a second step training with pseudolabels consistently improves the results on both 10 % and the 1 % labels case. This motivates us to apply the idea to our best semi supervised model, which is discussed in Section 5. 5 \fVAT We \ufb01rst verify our VAT implementation on CIFAR10. With 4000 labels, we are able to achieve 86.41 % top-1 accuracy, which is in line with the 86.13 % reported in [30]. Besides the previously mentioned hyperparameters common to all methods, VAT needs tuning \u03f5vat. Since it corresponds to a distance in pixel space, we use a simple heuristic for de\ufb01ning a range of values to try for \u03f5vat: values should be lower than half the distance between neighbouring images in the dataset. Based on this heuristic, we try values of \u03f5vat \u2208{50, 50 \u00b7 2\u22121/3, 50 \u00b7 2\u22122/3, 25} and found \u03f5vat \u224840 to work best. VAT+EntMin VAT is intended to be used together with an additional entropy minimization (EntMin) loss. EntMin adds a single hyperparameter to our best VAT model: the weight of the EntMin loss, for which we try wentmin \u2208 {0, 0.03, 0.1, 0.3, 1}, without re-tuning \u03f5vat. The results of our best VAT and VAT+EntMin model are shown in Table 1. As can be seen, VAT performs well in the 10 % case, and adding adding entropy minimization consistently improves its performance. In Section 5, we further extend the co-training idea to include the self-supervised rotation loss. 4.3. Self-supervised Baselines Previous work has evaluated features learned via selfsupervision on the unlabeled data in a \u201csemi-supervised\u201d way by either freezing the features and learning a linear classi\ufb01er on top, or by using the self-supervised model as an initialization and \ufb01ne-tuning, using a subset of the labels in both cases. In order to compare our proposed way to do self-supervised semi-supervised learning to these common evaluations, we train a rotation and an exemplar model following the best practice from [17] but with standard width (\u201c4\u00d7\u201d in [17]). Following our established protocol, we tune the weight decay and learning rate for the logistic regression, although interestingly the standard values from [12] of 10\u22124 weight decay and 0.1 learning rate worked best. The results of evaluating these models with both 10 % and 1 % are presented in Table 1 as \u201cSelf-sup. + Linear\u201d and \u201cSelf-sup. + Fine-tune\u201d. Note that while our results for the linear experiment are similar to those reported in [17], they are not directly. This is due to 1) ours being evaluated on the public validation set, while they evaluated on a custom validation set, and 2) they used L-BFGS while we use SGD with standard augmentations. Furthermore, \ufb01ne-tuning approaches or slightly surpasses the supervised baseline. 4.4. Self-supervised Semi-supervised Learning (S4L) For training our full self-supervised semi-supervised models (S4L), we follow the same protocol as for our semisupervised baselines, i.e. we use the best settings of the plain supervised baseline and only tune the learning rate, weight decay, and weight of the newly introduced loss. We found that for both S4L-Rotation and S4L-Exemplar, the self-supervised loss weight w = 1 worked best (though not by much) and the optimal weight decay and learning rate were the same as for the supervised baseline. As described in Section 3.1, we apply the self-supervised loss on both labeled and unlabeled images. Furthermore, both Rotation and Exemplar self-supervision generate augmented copies of each image, and we do apply the supervision loss on all copies of the labeled images. We performed one case study on S4L-Rotation in order to investigate this choice, and found that whether or not the self-supervision loss Lself is also applied on the labeled images does not have signi\ufb01cant impact. On the other hand, applying the supervision loss Lsup on the augmented images generated by self-supervision does indeed improve performance by almost 1 %. Furthermore, this allows to use multiple transformed copies of an image at inference-time (e.g. four rotations) and take the average of their predictions. While this 4-rot prediction is 1 % to 2 % more accurate, the results we report do not make use of this in order to keep comparison fair. The results shown in Table 1 show that our proposed way of doing self-supervised semi-supervised learning is indeed effective for the two self-supervision methods we tried. We hypothesize that such approaches can be designed for other self-supervision objectives. We additionally veri\ufb01ed that our proposed method is not sensitive to the random seed, nor the split of the dataset, see Appendix B for details. Finally, in order to explore the limits of our proposed models and match capacity of the architectures used in concurrent papers (e.g. [13]), we train the S4L-Rotation model with a more powerful architecture, such as ResNet152v2 2\u00d7wider, and also use large computational budget to tune hyperparameters. In this case our model achieves even better results: 86.41 % top-5 accuracy with 10 % labels and 57.50 % with 1 % labels. 5. Semi-supervised Learning is Complementary to S4L Since we found that different types of models perform similarly well, the natural next question is whether they are complementary, in which case a combination would lead to an even better model, or whether they all reach a common \u201cintrinsic\u201d performance plateau. In this section, we thus describe our Mix Of All Models (MOAM). In short: in a \ufb01rst step, we combine S4LRotation and VAT+EntMin to learn a 4\u00d7 wider [17] model. We then use this model in order to generate pseudo labels for a second training step, followed by a \ufb01nal \ufb01ne-tuning step. Results of the \ufb01nal model, as well as the models ob6 \fTable 2. Comparing our MOAM to previous methods in the literature on ILSVRC-2012 with 10 % of the labels. Note that different models use different architectures, larger than those in Table 1. labels Top-5 Top-1 MOAM full (proposed) 10% 91.23 73.21 MOAM + pseudo label (proposed) 10% 89.96 71.56 MOAM (proposed) 10% 88.80 69.73 ResNet50v2 (4\u00d7wider) 10% 81.29 58.15 VAE + Bayesian SVM [32] 10% 64.76 48.41 Mean Teacher [41] 10% 90.89 \u2020UDA [43] 10% 88.52\u2020 68.66\u2020 \u2020CPCv2 [13] 10% 84.88\u2020 64.03\u2020 Training with all labels: ResNet50v2 (4\u00d7wider) 100% 94.10 78.57 MOAM (proposed) 100% 94.97 80.17 \u2020UDA [43] 100% 94.45\u2020 79.04\u2020 \u2020CPCv2 [13] 100% 93.35\u2020 \u2020 marks concurrent work. tained in the two intermediate steps, are reported in Table 2 along with previous results reported in the literature. Step 1: Rotation+VAT+EntMin In the \ufb01rst step, our model jointly optimizes the S4L-Rotation loss and the VAT and EntMin losses. We iterated on hyperparameters for this setup in a less structured way than in our controlled experiments above (always on our custom validation set) and only mention the \ufb01nal values here. Our model was trained with batch size 128, learning rate 0.1, weight decay 2 \u00b7 10\u22124, training for 200 epochs with linear learning rate rampup up to epoch 10, then 10-fold decays at 100, 150, and 190 epochs. We use inception crop augmentation as well as horizontal mirroring. We used the following relative loss weights: wsup = 0.3, wrot = 0.7, wvat = 0.3, wentmin = 0.3. We tried a few heuristics for setting those weights automatically, but found that manually tuning them led to better performance. We also applied Polyak averaging to the model parameters, choosing the decay factor such that parameters decay by 50 % over each epoch. Joint training of these losses consistently improve over the models with a single objective. The model obtained after this \ufb01rst step achieves 88.80% top-5 accuracy on the ILSVRC-2012 dataset. Step 2: Retraining on Pseudo Labels Using the above model, we assign pseudo labels to the full dataset by averaging predictions across \ufb01ve crops and four rotations of each image3. We then train the same network again in the exact same way (i.e. with all the losses) except for the fol3Generating pseudo-labels using 20 crops only slightly improved performance by 0.25 %, but is cheap and simple. 0 100 200 300 400 500 Epochs of Logistic Regression 60 65 70 75 80 Places205 Accuracy [%] Self-supervised Rotation (short) Self-supervised Rotation (long) S4L Rotation (short) S4L Rotation (long) Figure 2. Places205 learning curves of logistic regression on top of the features learned by pre-training a self-supervised versus S4LRotation model on ILSVRC-2012. The signi\ufb01cantly faster convergence (\u201clong\u201d training schedule vs. \u201cshort\u201d one) suggests that more easily separable features are learned. lowing three differences: (1) the network is initialized using the weights obtained in the \ufb01rst step (2) every example has a label: the pseudo label (3) due to this, an epoch now corresponds to the full dataset; we thus train for 18 epochs, decaying the learning rate after 6 and 12 epochs. Step 3: Fine-tuning the model Finally, we \ufb01ne-tune the model obtained in the second step on the original 10 % labels only. This step is trained with weight decay 3\u00b710\u22123 and learning rate 5 \u00b7 10\u22124 for 20 epochs, decaying the learning rate 10-fold every 5 epochs. Remember that all hyper-parameters described here were selected on our custom validation set which is taken from the training set. The \ufb01nal model \u201cMOAM (full)\u201d achieves 91.23 % top-5 accuracy, which sets the new state-of-the-art. We conduct additional experiments and report performance of MOAM (i.e. only Step 1) with 100 % labels in Table 2. Interestingly, MOAM achieves promising results even in the high-data regime with 100 % labels, outperforming the fully supervised baseline: +0.87 % for top-5 accuracy and +1.6 % for top-1 accuracy. 6. Transfer of Learned Representations Self-supervision methods are typically evaluated in terms of how generally useful their learned representation is. This is done by treating the learned model as a \ufb01xed feature extractor, and training a linear logistic regression model on top the features it extracts on a different dataset, usually Places205 [45]. We perform such an evaluation on our S4L models in order to gain some insight into the generality of the learned features, and how they compare to those obtained by pure self-supervision. 7 \f0 10 20 30 40 50 Accuracy [%] on 5k val-set 0 10 20 30 40 50 Accuracy [%] on full val-set r = 0.99; p   0.0 0 10 20 30 40 50 Accuracy [%] on 1k val-set 0 10 20 30 40 50 Accuracy [%] on full val-set r = 0.99; p   0.0 Figure 3. Correlation between validation score on a (custom) validation set of 1000, 5000, and 50 046 images on ILSVRC-2012. Each point corresponds to a trained model during a sweep for plain supervised baseline for the 1 % labeled case. The best model according to the validation set of 1 000 is marked in red. As can be seen, evaluating our models even with only a single validation image per class is robust, and in particular selecting an optimal model with this validation set works as well as with the full validation set. We closely follow the protocol de\ufb01ned by [17]. The representation is extracted from the pre-logits layer. We use stochastic gradient descent (SGD) with momentum for training these linear evaluation models with a minibatch size of 2048 and an initial learning rate of 0.1, warmed up in the \ufb01rst epoch. While Kolesnikov et al. [17] show that a very long training schedule (520 epochs) is required for the linear model to converge using self-supervised representations, we observe dramatically different behaviour when evaluating our self-supervised semi-supervised representations. Figure 2 shows the accuracy curve of the plain self-supervised rotation method [17] and our proposed S4L-Rotation method trained on 10 % of ILSVRC-2012. As can be seen, the logistic regression is able to \ufb01nd a good separating hyperplane in very few epochs and then plateaus, whereas in the self-supervised case it struggles for a very long number of epochs. This indicates that the addition of labeled data leads to much more separable representations, even across datasets. We further investigate the gap between the representation learned by a good S4L model (MOAM) and a corresponding baseline trained on 100 % of the labels (the baseline from Table 2). Surprisingly, we found that the representation learned by \u201cMOAM (full)\u201d transfers slightly better than the baseline, which used ten times more labelled data: 83.3 % accuracy vs. 83.1 % accuracy, respectively. We provide full details of this experiment in the Supplementary Material. 7. Is a Tiny Validation Set Enough? Current standard practice in semi-supervised learning is to use a subset of the labels for training on a large dataset, but still perform model selection using scores obtained on the full validation set.4 But having a large labeled validation set at hand is at odds with the promised practicality of semi-supervised learning, which is all about having only few labeled examples. This fact has been acknowledged by [33], but has been mostly ignored in the semi-supervised literature. Oliver et al. [30] questions the viability of tuning with small validation sets by comparing the estimated model accuracy on small validation sets. They \ufb01nd that the variance of the estimated accuracy gap between two models can be larger than the actual gap between those models, hinting that model selection with small validation sets may not be viable. That said, they did not empirically evaluate whether it\u2019s possible to \ufb01nd the best model with a small validation set, especially when choosing hyperparameters for a particular semi-supervised method. We now describe our analysis of this important question. We look at the many models we trained for the plain supervised baseline on 1 % of ILSVRC-2012. For each model, we compute a validation score on a validation set of 1000 labeled images (i.e. one labeled image per class), 5000 labeled images (i.e. \ufb01ve labeled images per class), and compare these scores to those obtained on a \u201cfull-size\u201d validation set of 50 046 labeled images. The result is shown in Figure 3 and it is striking: there is a very strong correlation between performance on the tiny and the full validation set. Especially, while in parts there is high variability, those hyperparameters which work best do so in either case. Most notably, the best model tuned on a small validation set is also the best model tuned on a large validation set. We thus conclude that for selecting hyperparameters of a model, a tiny validation set is enough. 4To make matters worse, in the case of ILSVRC-2012, this validation set is used both to select hyperparameters as well as to report \ufb01nal performance. Remember that we avoid this by creating a custom validation set from part of the training set for all hyperparameter selections. 8 \f8. Discussion and Future Work In this paper, we have bridged the gap between selfsupervision methods and semi-supervised learning by suggesting a framework (S4L) which can be used to turn any self-supervision method into a semi-supervised learning algorithm. We instantiated two such methods: S4L-Rotation and S4L-Exemplar and have shown that they perform competitively to methods from the semi-supervised literature on the challenging ILSVRC-2012 dataset. We further showed that S4L methods are complementary to existing semisupervision techniques, and MOAM, our proposed combination of those, leads to state-of-the-art performance. While all of the methods we investigated show promising results for learning with 10 % of the labels on ILSVRC2012, the picture is much less clear when using only 1 %. It is possible that in this low data regime, when only 13 labeled examples per class are available, the setting fades into the few-shot scenario, and a very different set of methods would be required for reaching much better performance. Nevertheless, we hope that this work inspires other researchers in the \ufb01eld of self-supervision to consider extending their methods into semi-supervised methods using our S4L framework, as well as researchers in the \ufb01eld of semisupervised learning to take inspiration from the vast amount of recently proposed self-supervision methods. Acknowledgements. We thank the Google Brain Team in Z\u00a8 urich, and especially Sylvain Gelly for discussions."
+                }
+            ],
+            "Lukas Hoyer": [
+                {
+                    "url": "http://arxiv.org/abs/2311.16241v1",
+                    "title": "SemiVL: Semi-Supervised Semantic Segmentation with Vision-Language Guidance",
+                    "abstract": "In semi-supervised semantic segmentation, a model is trained with a limited\nnumber of labeled images along with a large corpus of unlabeled images to\nreduce the high annotation effort. While previous methods are able to learn\ngood segmentation boundaries, they are prone to confuse classes with similar\nvisual appearance due to the limited supervision. On the other hand,\nvision-language models (VLMs) are able to learn diverse semantic knowledge from\nimage-caption datasets but produce noisy segmentation due to the image-level\ntraining. In SemiVL, we propose to integrate rich priors from VLM pre-training\ninto semi-supervised semantic segmentation to learn better semantic decision\nboundaries. To adapt the VLM from global to local reasoning, we introduce a\nspatial fine-tuning strategy for label-efficient learning. Further, we design a\nlanguage-guided decoder to jointly reason over vision and language. Finally, we\npropose to handle inherent ambiguities in class labels by providing the model\nwith language guidance in the form of class definitions. We evaluate SemiVL on\n4 semantic segmentation datasets, where it significantly outperforms previous\nsemi-supervised methods. For instance, SemiVL improves the state-of-the-art by\n+13.5 mIoU on COCO with 232 annotated images and by +6.1 mIoU on Pascal VOC\nwith 92 labels. Project page: https://github.com/google-research/semivl",
+                    "authors": "Lukas Hoyer, David Joseph Tan, Muhammad Ferjad Naeem, Luc Van Gool, Federico Tombari",
+                    "published": "2023-11-27",
+                    "updated": "2023-11-27",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Semantic segmentation models predict pixel-level dense semantic labels for an image. They have important applications in many areas such as autonomous driving, augmented reality, robotics, medical imaging, and remote sensing. However, training such models requires very costly pixel-wise human annotations over a large dataset. To reduce the dependence on large labeled datasets, semi-supervised semantic segmentation aims to effectively learn from a small portion \u2217This work was conducted during an internship at Google. (a) Image (b) G.Truth (c) UniMatch (d) CLIP (e) SemiVL (Ours) Figure 1. While previous SOTA methods such as UniMatch [68] achieve good segmentation even with only 92 labeled images, they struggle to distinguish classes with similar appearance. In contrast, vision-language training such as CLIP [51] learns rich semantic representations but suffers from noisy segmentation (obtained with [73]). Our SemiVL supplements semi-supervised training with vision-language guidance, combining good segmentation with a rich semantic understanding. of labeled images while additionally leveraging a large set of unlabeled images. Typical strategies include adversarial training [44, 56] and self-training [55, 67, 68]. While state-of-the-art methods for semi-supervised semantic segmentation such as UniMatch [68] are able to learn good segmentation masks, segments with similar visual features are prone to misclassification, due to the lack of sufficient labeled examples for learning the exact semantic decision boundaries for these classes. This is particularly problematic for rare instances such as a calve or a boat on land as shown in Fig. 1c. Here, the cow (top, green) is confused with a dog (purple) and the boat (bottom, blue) with a chair (red) even though the segmentation is mostly correct. To better capture semantics, we propose to supplement semi-supervised semantic segmentation with guidance from Vision Language Models (VLM). VLMs such as CLIP [51] are trained on a web-scale image-caption dataset. The diversity of the data and the natural language captions (instead of a fixed set of classes) enables VLMs to capture richer seman1 arXiv:2311.16241v1  [cs.CV]  27 Nov 2023 \ftic representations. For example, they can correctly identify the cow and the boat in Fig. 1d. However, as they are trained on image level, their features do not localize well and their dense predictions [72] are noisy as shown in Fig. 1d. In this work, we study how to combine the good localization of semi-supervised training and the rich semantic understanding of VLMs. Based on the findings, we propose SemiVL, which combines both strengths to achieve a good segmentation quality and a fine-grained semantic discriminability. For example, SemiVL correctly segments and classifies the cow and boat in Fig. 1e. To the best of our knowledge, SemiVL is the first work to utilize vision-language guidance for semi-supervised semantic segmentation to mitigate the issue of limited dense labels. Previous works in VLM semantic segmentation (see Sec. 2.2) either operate without dense labels [5, 63, 73], which significantly limits their performance, or use large-scale annotated segmentation datasets [12, 65, 76], which are expensive to obtain. Instead, SemiVL can learn high-quality semantic segmentation in a semi-supervised setting with only a few labels. Building on consistency regularization [68] for semisupervised training, SemiVL introduces five components to transfer the power of vision-language modeling to semantic segmentation under the constraint of limited segmentation annotations, which are highlighted in green color in Fig. 3: (1) Vision-Language Pre-Training (Sec. 3.2): To utilize the rich semantic prior of VLMs, we initialize the semisupervised training with vision-language pre-training. Compared to the previously used ImageNet pre-training, the VL pre-training on web-scale image-caption pairs provides more diversity and is not limited to a fixed set of classes. (2) Spatial Fine-Tuning (Sec. 3.3): As the VL pre-training is conducted on image level, the vision model needs to be fine-tuned to achieve good feature localization for semantic segmentation. Due to the limited annotations, the fine-tuning is prone to overfitting and forgetting the rich semantics from the pre-training. Therefore, we introduce parameter-efficient spatial fine-tuning. It only fine-tunes network layers that model interactions between different pixels to improve the localization of features while it freezes layers that operate locally to preserve their semantic reasoning capabilities. (3) Language-Guided Decoder (Sec. 3.4): To exploit the alignment of vision and language embeddings from VL pretraining, we integrate it into the segmentation decoder. It processes VL similarity maps using decoupled spatial and semantic reasoning. By sharing parameters across classes or pixels, the limited labels are used effectively. (4) Dense CLIP Guidance (Sec. 3.5): When fine-tuning the VLM towards segmentation, some of the original prior is corrupted and the network drifts to wrong predictions on the unlabeled images. To anchor the semi-supervised training on unlabeled images, we regularize it with the predictions from a frozen VLM, which cannot drift. As these predictions 92 183 366 732 1464 #Labeled Images (of 10K Images in T otal) 60 70 80 90 mIoU (%) +8.8 +8.4 +7.2 +6.8 +6.1 Method CPS (CVPR21) U2PL (CVPR22) PCR (NeurIPS22) UniMatch (CVPR23) SemiVL (Ours) Figure 2. Semi-supervised semantic segmentation on Pascal VOC. Utilizing vision-language guidance, our SemiVL achieves major gains over previous methods, particularly for fewer labels. are noisy, we only use the ones with a high certainty. (5) Class Definitions (Sec. 3.6): Depending on the use case, certain concepts can fall into different classes. For example, a person pushing a bicycle could belong to the class pedestrian or rider. However, a small number of annotations might not be sufficient to learn all relevant dataset-specific decision boundaries between classes. Therefore, we utilize the novel capability of SemiVL to provide language guidance in the form of class definitions to the model. These are often part of the annotation guidelines of datasets or can be created with much less effort than mining images of relevant corner cases and labeling them with semantic segmentation. We evaluate SemiVL on 4 common semantic segmentation datasets, where it significantly outperforms previous semi-supervised methods. For instance, it improves the state of the art by +13.5 mIoU on COCO with 232 annotated images and by +9.7 mIoU on ADE20K with 158 labels. In particular, SemiVL can maintain the performance of previous methods using only a quarter of the segmentation annotations. The significance of the improvements is visualized in Fig. 2, showing that SemiVL achieves major gains over previous works for all labeled subsets. In particular, SemiVL improves the performance for few labels, showing its effectiveness for label-efficient learning. In a comprehensive ablation study on Pascal VOC, we show that all 5 components of SemiVL significantly improve the semi-supervised performance. The gains are particularly large when only a few labels are available, demonstrating SemiVL\u2019s effectiveness to compensate for limited annotations. 2. Related Work 2.1. Semi-Supervised Semantic Segmentation Early works [44, 56] in semi-supervised segmentation use unlabeled images in a GAN framework [18] to match their predictions with the distribution of manual labels. Selftraining [19, 35] methods generate pseudo-labels for unlabeled images and use them for iterative re-training. To mitigate pseudo-label drift and self-confirmation bias [1], strategies such as confidence-weighting [15, 67], curric2 \fula [43, 66, 69], class balancing [21, 22, 25, 31], auxiliary self-supervised tasks [24, 29], contrastive learning [60], symbolic reasoning [38], or soft pseudo-labels [43] can be used. Another popular strategy is consistency regularization [2, 53], which makes predictions invariant to perturbations such as data augmentation [55, 67, 68], mixed samples [16, 31, 49], overlapping crops with different context [26, 27, 34], different model initialization [7], masked images [28], or feature perturbations [68]. Due to its success and simplicity, we base SemiVL on consistency training. 2.2. Semantic Segmentation with VLMs Vision Language Models (VLMs) such as CLIP and its variants [46\u201348, 72, 74, 74] utilize a web-scale image-caption dataset to learn a joint embedding space between image and text. Due to the semantically rich and diverse dataset, these models possess great generalization to a wide range of vision tasks[8, 9, 20, 33, 42, 59]. With prompt engineering, VLMs can be improved by ensembling prompts [51] or generating class descriptions by LLMs [47, 50]. Recently, VLMs also gained attention in semantic segmentation. For instance, several zero-shot methods [5, 54, 63, 73] aim to learn segmentation only from image-caption pairs without any dense annotations. MaskCLIP [73] shows that (noisy) segmentation emerges in the CLIP vision encoder. Further techniques include hierarchical grouping [63], retrieval and co-segmentation [54], or text-grounded masking [5]. However, the produced segmentations are noisy due to the lack of dense supervision. In contrast, open-vocabulary segmentation methods achieve better segmentation as they use a large labeled semantic segmentation dataset. Here, CLIP is often used to deal with unknown classes during test time. OpenSeg [17] pools visual features with learned mask proposals. LSeg [36] learns dense visual embeddings that align with CLIP text embeddings. ZegFormer [12], ZSseg [64], and OVSeg [39] predict class-agnostic segmentation masks and classify their crops with a frozen CLIP. ZegCLIP [76] simplifies this into a one-stage approach, learning an attention decoder between CLIP text embeddings and dense vision embeddings. CAT-Seg [9] learns to aggregate CLIP cost maps. SAN [65] trains a side network to adapt a frozen CLIP to segmentation. To avoid overfitting to the training classes, several methods use parameter-efficient training strategies such as prompt tuning [3, 32, 75, 76], partial finetuning [9, 30], or adapters [23, 57, 65]. Due to its simplicity, SemiVL also follows a one-stage framework [9, 65, 76]. In contrast to these methods, we do not assume access to a large-scale segmentation dataset. 3. Methods In semi-supervised semantic segmentation, the training dataset consists of a set of labeled images Dl = {(xl i, yl i)} and another set of unlabeled images Du = {xu i }. In the folVision-Language  Pre-Training Semi-Supervised  Semantic Segmentation Vision Model (VM)  Language Model (LM)  Contrastive Loss Images Captions 1 Vision-Language  Initialization Unlabeled  Images Labeled  Images \u201ca photo of  a [CLS]\u201d + Class Definitions 5 Dense VLM  Guidance  Loss 4 Supervised  Loss Consistency  Loss Spatial  Fine-Tuning LanguageGuided  Decoder Patch  Similarity 3 VM VM LM 2 Figure 3. Overview of our SemiVL framework. Utilizing the rich semantic representations from vision-language (VL) pre-training (top), we propose 5 strategies (highlighted in green) to guide semisupervised semantic segmentation (bottom): We use the rich VL prior as initialization (1) and regularization on unlabeled images (4). To adapt VL from image-level to dense reasoning, we introduce a label-efficient fine-tuning strategy (2) and a decoder architecture jointly reasoning about VL (3). Finally, we propose to steer the model with text instructions in the form of class definitions (5). lowing, we present our SemiVL framework (see Fig. 3). It is based on the popular consistency training approach to utilize unlabeled images during the semi-supervised training (Sec. 3.1). While this already achieves good segmentation quality, it still struggles to obtain precise semantic decision boundaries from the limited supervision. To address this, we propose five components to guide the semi-supervised training with vision-language guidance, which are highlighted in green color in Fig. 3. First, we initialize the semi-supervised training with a pre-trained VLM as it contains rich semantics learned from web-scale image-text data (Sec. 3.2). Second, we only fine-tune the attention layers of the pre-trained vision model to adapt them from image-level to dense reasoning while we freeze the local MLP layers to preserve their semantic reasoning from the pre-training (Sec. 3.3). Third, we introduce language-based reasoning in the decoder to further exploit the vision-language alignment from the pre-training (Sec. 3.4). Fourth, we regularize the training on unlabeled images with predictions from a frozen VLM to anchor the self-training and avoid drift (Sec. 3.5). And fifth, we provide the model with language-based class definitions (Sec. 3.6). 3.1. Consistency Training While semantic segmentation models are usually trained with a supervised loss Ls such as the pixel-wise crossentropy, this is only possible on the labeled images. In order to additionally utilize the unlabeled images for the semi-supervised training, we resort to consistency train3 \fing [7, 55, 68]. It is based on the idea that the predictions of the same image should be invariant to different data augmentations or model perturbations. Specifically, the consistency loss term C drives the model to produce the same predictions pp under perturbations as the predictions pu without perturbations, also called pseudo-labels: C(pp, pu) = X i,j 1[max(pu ij) \u2265\u03c4]H(pp ij, pu ij) , (1) where i, j are the pixel indices, H is the cross-entropy between pu and pp, and \u03c4 is a fixed confidence threshold to exclude noisy pseudo-labels pu from the consistency training. Consistency training is a variant of self-training as the pseudo-labels pu are generated by the same network as pp. The pseudo-labels pu of model f are obtained from the unlabeled images: pu = f(xu). The perturbed predictions pp are obtained under strong data augmentations A: pp = f(A(xu)). Additionally, perturbations can also achieved by perturbing the features of the model pfp = h(P(g(xw))), where g is the encoder and h the decoder of the model f. In particular, we follow the state-of-the-art approach UniMatch [68] and enforce the consistency over two strong data augmentations (pp1, pp2) and feature perturbation (pfp) Lu = 1 2C(pfp, pu) + 1 4C(pp1, pu) + 1 4C(pp2, pu) . (2) The overall semi-supervised loss is L = 1 2(Ls + Lu). 3.2. Vision-Language Pre-Training While consistency training achieves good segmentation as shown by previous works, it is prone to confusion of visually similar classes as can be seen in Fig. 1. This is caused by the model only having a few labeled images to learn the semantic classes. VLMs like CLIP [51] are trained on web-scale image-text datasets that cover almost all semantic classes a vision agent can ever come across. CLIP uses a vision encoder V to produce embeddings for images and a text encoder T to embed captions. Both are trained jointly to map to an aligned embedding space, where corresponding images and captions have a high similarity. It is trained in a contrastive manner with ground truth image-caption pairs as positive and shuffled image-caption pairs as negatives. In that way, CLIP learns a representation with a rich semantic understanding of images. CLIP can perform zero-shot image classification by computing the similarities of an image embedding V(x) with the text embeddings T (tn) of multiple class prompts tn in the form of \u201ca photo of a [CLSn]\u201d. To address the limitations of consistency training, we propose to initialize the semantic segmentation encoder g with the pre-trained VLM vision encoder V to utilize its rich semantic prior. Our CLIP initialization replaces the prevailing ImageNet initialization. Compared to image classification pre-training on ImageNet, the VLM pre-training does not require a manually annotated dataset but can be trained on web-crawled image-caption pairs. Further, it can learn richer semantic representations due to the versatile captions, which are not restricted to a specific set of classes. 3.3. Spatial Fine-Tuning While the vision-language pre-training of CLIP has learned to distinguish fine-grained semantic concepts, it was only trained to optimize image-level features. Therefore, the semantic features are not necessarily spatially aligned with the image content, resulting in noisy segmentations as shown in Fig. 1d. To mitigate this issue, it is necessary to adapt the backbone towards semantic segmentation. However, due to the limited number of annotated images, the fine-tuning is prone to overfitting and forgetting the rich semantics from the vision-language pre-training. This process can be further reinforced by the self-confirmation bias of the self-training. Inspired by parameter-efficient finetuning [9, 30, 32, 57], we introduce spatial fine-tuning to semi-supervised semantic segmentation. A ViT [13] block consists of a multi-head attention layer and a subsequent MLP. Only the attention layer models interactions between different patches, while the MLP operates locally on each patch (similar to a 1\u00d71 convolution). Spatial fine-tuning only fine-tunes the attention layers, which are responsible for spatial reasoning. In that way, the alignment of semantic features and their corresponding image content can be refined for dense predictions. On the other side, the MLP layers are frozen as they do not perform spatial reasoning to preserve the semantic reasoning capabilities of CLIP pretraining. We further fine-tune the position embeddings to attribute for the shift from global to dense reasoning as well as input size change from pre-training to fine-tuning. 3.4. Language-Guided Decoder A major advantage of the vision-language pre-training is the alignment of the vision and language embeddings, which enables the reasoning about both modalities and their semantic relations. To utilize this capability in the model architecture, we integrate the vision-language alignment in a languageguided decoder architecture, which is visualized in Fig. 4. For that purpose, we obtain a dense vision-language similarity map S \u2208RH\u00d7W \u00d7N with height H, width W and number of classes N as proposed in [73] by computing the patch-wise cosine similarities of the vision embeddings g(x) and the text embeddings T (tn) over all classes n \u2208[0 . . N] with their text prompt tn Sijn = g(x)ij \u00b7 T (tn) \u2225g(x)ij\u2225\u2225T (tn)\u2225. (3) Here, i, j denote the patch indices. The embeddings of g and 4 \fTransformer Spatial Reasoning Similarity Language Model Vision Model \u201ca photo  of a dog\u201d d , Semantic  Segmentation Image N shared  over N Conv shared  over N ASPP Semantic Reasoning shared  over HxW Up x2 shared  over N S S\u2019 S\u2019\u2019 Figure 4. Overview of our vision-language-guided architecture. It is based on dense similarity maps of the vision and text embeddings. These are processed by spatial and semantic reasoning modules and subsequently upsampled. More details are provided in Sec. 3.4. T are aligned as g is initialized from V (see Sec. 3.2). While the original localization of semantics in the similarity map is noisy at the beginning of the training, it is improved during the spatial fine-tuning. The similarity maps are further refined by spatial and semantic reasoning modules. The spatial reasoning module operates on each class similarity map independently (i.e. Sn \u2208RH\u00d7W \u00d71) and models no inter-class relations. Thereby, the learned spatial reasoning is shared across classes. First, each Sn is processed by a 7\u00d77 convolution to learn local spatial structures and embed them to similarity volumes S\u2032 n \u2208RH\u00d7W \u00d7d of d dimensions. Subsequently, a residual ASPP [6] processes the obtained similarity volumes to model long-range context relations, resulting in a combined similarity volume for all classes S\u2032\u2032 \u2208RH\u00d7W \u00d7N\u00d7d. The semantic reasoning module models the relationship between classes. Each pixel in the similarity volumes S\u2032\u2032 ij \u2208 R1\u00d71\u00d7N\u00d7d is processed independently (no spatial interaction) to share the learned relations between spatial locations. It is implemented as two Transformer [58] blocks. Further, the features are supplemented with the N text embeddings of the class names, which act as semantic anchors. By decoupling spatial and semantic reasoning, the learned weights can be shared over different classes for spatial reasoning and shared over different locations for semantic reasoning. In that way, the limited annotations can be utilized more effectively and overfitting is reduced. The common vision transformers operate on a 16 times smaller feature resolution than the input. This limits precise segmentation boundaries. Therefore, we add 2 upsampling blocks [52]. They learn the upsampling using a transpose convolution. Further, skip connections from earlier encoder layers are used. The upsampled features and the skip features are concatenated and fused by two convolutions. For label-efficient learning, the upsampling blocks operate on each class independently. A final convolution maps the d dimensions to one channel to obtain the logits for a class. 3.5. Dense CLIP Guidance While the previous strategies are designed to best transfer the knowledge from vision-language pre-training to semantic segmentation, the self-training on the unlabeled images can cause a drift of the training to erroneous predictions. Incorrect predictions in pu are used for the self-training and result in a self-confirmation bias. To anchor the self-training on the unlabeled images and reduce this issue, we guide the consistency training with predictions from a frozen auxiliary CLIP model, which cannot drift. For this purpose, we extract dense vision-language similarity maps S (see Eq. 3) with a frozen V from the unlabeled images xu. The similarity map is processed by a softmax and can be used as a semantic segmentation pDC pseudolabel. As pDC tends to be noisy (see Fig. 3), we only utilize its high confidence predictions exceeding a threshold \u03b6. To guide the consistency training on the unlabeled images with dense CLIP guidance, we supplement the consistency loss terms C in Eq. 2 by dense CLIP guidance: C(pp, pu, pDC) = C(pp, pu)+ \u03bbDC X 1[max(pDC) \u2265\u03b6]H(pp, pDC) , (4) where \u03bbDC weighs the dense CLIP guidance loss term. As the guidance is picked up by the pseudo-labels pu and spatially refined over the course of the self-training, the guidance is more important at the beginning of the training. Therefore, \u03bbDC is linearly decayed during the training. 3.6. Class Definition Guidance When the number of labeled samples in the semi-supervised setting is small, they might not be sufficient to capture all relevant decision boundaries between classes for the given dataset. This is particularly problematic for edge cases that are not part of the labeled subset or subcategories that could belong to different classes: Does an armchair belong to the chair or sofa class? Does a bench belong to the chair or background class? As similar ambiguities happen when humans annotate a dataset, there are usually annotation guidelines with class definitions for a dataset. For example, Pascal VOC defines: \u201cchair includes armchairs, deckchairs but not stools or benches\u201d. As SemiVL can process arbitrary text, we propose to provide these annotation guidelines to the model to 5 \fImage G. Truth UniMatch [68] SemiVL Image G. Truth UniMatch [68] SemiVL VOC92 ADE158 Figure 5. Example predictions on VOC (92 labels) and ADE20K (158 labels) showing the improved semantic understanding of SemiVL. better capture the decision boundaries for a dataset. Importantly, providing the class descriptions is much less effort than mining images of all relevant corner cases and labeling them with semantic segmentation. Compared to generic class descriptions from LLMs [47, 50] or synonyms [40], this can better capture dataset-specific decision boundaries, e.g. a stool belongs to the background class in Pascal VOC. Even though it would be possible to provide the text encoder T directly with the class definitions, CLIP was trained with image captions that do not match such definitions. In particular, they do not handle negations well. Therefore, we build a set of concepts ba for each class a that contain additional subcategories or descriptions extracted from the class definitions. For example, the concept armchair belongs to the class chair (i.e. armchair \u2208bchair) while stool belongs to background according to the VOC definition. To utilize the concepts, we integrate them into the CLIP guidance training. With the frozen CLIP, we infer the visiontext similarity pconcept over all concepts \u222aaba. We aggregate the concepts back to classes using the concept-class assignment from the class definitions. The concept with the highest score determines the predicted class at position ij pDC ija = max b\u2208ba pconcept ijb . (5) 4. Experiments 4.1. Implementation Details Architecture: SemiVL uses a ViT-B/16 [13] vision encoder and a Transformer [58] text encoder, both pre-trained by CLIP [51]. The dense vision embeddings are extracted following [73]. The language-guided decoder uses an initial 7\u00d77 convolution to map the similarity maps to d=128 channels, a residual ASPP [6] with dilation rates {6,12,18} for spatial reasoning, and 2 Transformer blocks [58] with 4 heads for semantic reasoning. For compute and memory efficiency, the semantic reasoning operates on 4\u00d74 averagepooled features maps. The 2 upsampling blocks use a 2\u00d72 transpose convolution, skip connections with 16/32 channels from the first/fourth ViT blocks, and 2 convolutions with GroupNorm [61] to fuse both into 64/32 channels. For Cityscapes, we use the feature map of the first ResNet block as skip connection to handle particularly small segments. Training: SemiVL is benchmarked on Pascal VOC 2012 [14], COCO [4], ADE20K [71], and Cityscapes [11] using the same labeled subsets as common protocol [7, 60, 68, 77]. It is trained with a batch of 8 labeled and 8 unlabeled images using AdamW [41] for 80/10/40/240 epochs on VOC/COCO/ADE20K/Cityscapes. The inital learning rate is 1\u00d710\u22124/4\u00d710\u22124/4\u00d710\u22124/5\u00d710\u22125 with a 0.9 polynomial decay. Spatial fine-tuning uses a learning rate multiplier of 0.01/0.001/0.001/0.1. We use 801\u00d7801 random crops for Cityscapes following [68] and 512\u00d7512 for the others. For inference, sliding window evaluation is used. Following UniMatch [68], we set \u03c4=0.95, use 50% channel dropout for P, and use random color jitter, grayscale, CutMix, scale, and crop for A. For SemiVL, we set \u03b6=0.9 and use a linear schedule from 0.1 to 0 for \u03bbDC. The class definitions are provided in the supplement. The training is conducted on 4 (for VOC) or 8 (for others) A100 GPUs. 6 \fTable 1. State-of-the-art comparison on Pascal VOC. The mIoU (%) is compared across different splits for the labeled subset Dl. \u2020 denotes re-produced results in the same setting as SemiVL. Method Net 1/115 1/58 1/29 1/14 1/7 (92) (183) (366) (732) (1464) PseudoSeg [77] [ICLR\u201921] R101 57.6 65.5 69.1 72.4 \u2013 CPS [7] [CVPR\u201921] R101 64.1 67.4 71.7 75.9 \u2013 ST++ [67] [CVPR\u201922] R101 65.2 71.0 74.6 77.3 79.1 U2PL [60] [CVPR\u201922] R101 68.0 69.2 73.7 76.2 79.5 PCR [62] [NeurIPS\u201922] R101 70.1 74.7 77.2 78.5 80.7 ESL [43] [ICCV\u201923] R101 71.0 74.0 78.1 79.5 81.8 LogicDiag [38] [ICCV\u201923] R101 73.3 76.7 77.9 79.4 \u2013 UniMatch [68] [CVPR\u201923] R101 75.2 77.2 78.8 79.9 81.2 3-CPS [37] [ICCV\u201923] R101 75.7 77.7 80.1 80.9 82.0 ZegCLIP\u2020 [76] [CVPR\u201923] ViT-B/16 69.3 74.2 78.7 81.0 82.0 ZegCLIP+UniMatch\u2020 \u2014 ViT-B/16 78.0 80.3 80.9 82.8 83.6 UniMatch\u2020 [68] [CVPR\u201923] ViT-B/16 77.9 80.1 82.0 83.3 84.0 Ours \u2014 ViT-B/16 84.0 85.6 86.0 86.7 87.3 (+6.1) (+5.5) (+4.0) (+3.4) (+3.3) Table 2. Class-Wise IoU on Pascal VOC with 92 Labels. Airplane Backgr. Bicycle Bird Boat Bottle Bus Car Cat Chair Cow Din.T able Dog Horse M.Bike Person Plant Sheep Sofa Train Monitor UniMatch(ViT) SemiVL 93 94 69 94 78 76 91 88 94 19 96 56 90 93 82 88 61 94 21 88 71 89 96 78 95 84 84 95 85 97 37 97 72 93 95 90 92 73 94 59 91 68 Table 3. Comparison with state-of-the-art methods on COCO. Method Net 1/512 1/256 1/128 1/64 1/32 (232) (463) (925) (1849) (3697) PseudoSeg [77] [ICLR\u201921] XC65 29.8 37.1 39.1 41.8 43.6 PC2Seg [70] [ICCV\u201921] XC65 29.9 37.5 40.1 43.7 46.1 UniMatch [68] [CVPR\u201923] XC65 31.9 38.9 44.4 48.2 49.8 LogicDiag [38] [ICCV\u201923] XC65 33.1 40.3 45.4 48.8 50.5 UniMatch\u2020 [68] [CVPR\u201923] ViT-B/16 36.6 44.1 49.1 53.5 55.0 Ours \u2014 ViT-B/16 50.1 52.8 53.6 55.4 56.5 (+13.5) (+8.7) (+4.5) (+1.9) (+1.5) 4.2. Comparison with the State of the Art We compare SemiVL with previous semi-supervised methods on four popular semantic segmentation datasets in Tab. 1-5. We additionally reproduce the SOTA method UniMatch [68] with a ViT [13] backbone and the same training parameters as SemiVL for a fair comparison. On all four datasets, SemiVL achieves consistent and significant gains over previous works as detailed in the following. Pascal VOC: The Pascal VOC dataset [14] contains 10582 training images of which 1464 images have segmentation annotations for 21 classes. Tab. 1 compares SemiVL with previous methods for different portions of labeled images ranging from 1/115 of the training set (92 labels) up to 1/7 (1464 labels). SemiVL significantly improves the performance from +3.3 mIoU (1464 labels) up to +6.1 (92 labels) over UniMatch(ViT). The larger improvement for fewer annotations shows the efficiency of SemiVL to learn from limited annotations. SemiVL can reduce the need for labels by a factor Table 4. Comparison with state-of-the-art methods on ADE20K. Method Net 1/128 1/64 1/32 1/16 1/8 (158) (316) (632) (1263) (2526) CutMix [16] [BMVC\u201920] R101 \u2013 \u2013 26.2 29.8 35.6 AEL [31] [NeurIPS\u201921] R101 \u2013 \u2013 28.4 33.2 38.0 UniMatch [68] [CVPR\u201923] R101 15.6 21.6 28.1 31.5 34.6 UniMatch\u2020 [68] [CVPR\u201923] ViT-B/16 18.4 25.3 31.2 34.4 38.0 Ours \u2014 ViT-B/16 28.1 33.7 35.1 37.2 39.4 (+9.7) (+8.4) (+3.9) (+2.8) (+1.4) Table 5. Comparison with state-of-the-art methods on Cityscapes. Method Net 1/30 1/16 1/8 1/4 1/2 (100) (186) (372) (744) (1488) PseudoSeg [77] [ICLR\u201921] R101 61.0 \u2013 69.8 72.4 \u2013 CPS [7] [CVPR\u201921] R101 \u2013 69.8 74.3 74.6 76.8 U2PL [60] [CVPR\u201922] R101 \u2013 74.9 76.5 78.5 79.1 PCR [62] [NeurIPS\u201922] R101 \u2013 73.4 76.3 78.4 79.1 AEL [31] [NeurIPS\u201921] R101 \u2013 75.8 77.9 79.0 80.3 ESL [43] [ICCV\u201923] R101 \u2013 75.1 77.2 78.9 80.5 3-CPS [37] [ICCV\u201923] R101 \u2013 75.7 77.4 78.5 \u2013 UniMatch [68] [CVPR\u201923] R101 73.0 76.6 77.9 79.2 79.5 LogicDiag [38] [ICCV\u201923] R101 \u2013 76.8 78.9 80.2 81.0 UniMatch\u2020 [68] [CVPR\u201923] ViT-B/16 73.8 76.6 78.2 79.1 79.6 Ours \u2014 ViT-B/16 76.2 77.9 79.4 80.3 80.6 (+2.4) (+1.3) (+1.2) (+1.2) (+1.0) of 8 compared to UniMatch(ViT). As a further strong baseline using CLIP, we integrate the open-vocabulary method ZegCLIP [76] into UniMatch. Even though it improves the performance over UniMatch, SemiVL still significantly outperforms it. The class-wise analysis in Tab. 2 shows that SemiVL particularly improves classes, where UniMatch struggles (low IoU), such as chair, dining table, or sofa. These are classes that are often confused with the background class or semantically similar classes. This is also reflected in the example predictions in Fig. 5. COCO: The COCO dataset [4] has 118k training images with segmentation annotations for 81 classes. The higher number of classes makes it more challenging for semisupervised learning. Here, the power of vision-language guidance to distinguish semantic concepts particularly shines. SemiVL achieves major mIoU gains of up to +13.5 (for 232 labels) over UniMatch(ViT) as shown in Tab. 3. ADE20K: The ADE dataset [71] has 20210 training images with segmentation annotations for 150 classes and is even more challenging than COCO. Tab. 4 shows that SemiVL achieves gains of up to +9.7 mIoU (for 158 labels). Cityscapes: The Cityscapes dataset [11] has 2975 training images of street scenes with fine annotations for 19 classes. Tab. 5 shows that SemiVL outperforms UniMatch+ViT by up to +2.4 mIoU (100 labels). We hypothesize that the improvements are smaller than for the other benchmarks due to many challenging small classes in Cityscapes (e.g. distant poles or traffic signs), which are usually not part of the captions used in vision-language pre-training, limiting the 7 \fTable 6. Ablation study of SemiVL\u2019s components on Pascal VOC: vision-language pre-training (VL Pretr.), spatial fine-tuning (SFT), language-guided decoder (Lang.Dec.), dense CLIP guidance (CLIP Guid.), and Class Definition Guidance (Cls.Def.). VL Pretr. SFT Lang.Dec. CLIP Guid. Cls.Def. mIoU92 mIoU1464 \u2013 \u2013 \u2013 \u2013 \u2013 77.9 \u21b1 84.0 \u21b1 \u2713 \u2013 \u2013 \u2013 \u2013 80.6 +2.7 86.0 +2.0 \u2713 \u2713 \u2013 \u2013 \u2013 81.7 +3.8 86.7 +2.7 \u2713 \u2713 \u2713 \u2013 \u2013 82.7 +4.8 87.3 +3.3 \u2713 \u2713 \u2713 \u2713 \u2013 83.2 +5.3 87.4 +3.4 \u2713 \u2713 \u2713 \u2713 \u2713 84.0 +6.1 87.3 +3.3 Table 7. Comparison of fine-tuning strategies on Pascal VOC. Fine-Tuning Method #Updated Params. (Encoder) mIoU92 mIoU1464 Frozen \u2013 75.9 81.7 Deep Prompt Tuning [76] 0.7 M 79.5 84.3 LoRA [30] 1.4 M 80.5 85.8 Full Fine-Tuning 86.8 M 80.6 86.0 Spatial Fine-Tuning 29.1 M 81.7 86.7 Table 8. Decoder ablation study on Pascal VOC. Decoder Variant mIoU92 mIoU1464 Joint Reasoning 64.9 86.3 VL Similarity 68.6 84.5 VL S. + Up 74.1 86.6 VL S. + Up + Spatial 80.3 86.9 VL S. + Up + Spatial + Semantic 82.7 87.3 Table 9. Study on class definition guidance on VOC. Class Definition mIoU92 Dictionary Def. (Text) 82.7 GPT Concepts (Max) 82.9 Class Name 83.2 Guidelines (Text) 83.4 Guidel. (Avg) 83.4 Ours: Guidel. (Max) 84.0 efficiency of vision-language guidance. 4.3. Analysis Ablation Study: Tab. 6 shows the ablation of SemiVL on Pascal VOC. The first row shows the UniMatch baseline with an ImageNet pre-trained ViT encoder and DeepLabv3+ decoder. UniMatch uses consistency training (see Sec. 3.1). All components of SemiVL contribute to the major performance gain over UniMatch. The largest improvement originates from the VL pre-training, improving the semantic understanding. But also the spatial fine-tuning and the language-guided decoder significantly improve the performance for both 92 and 1464 labels. The CLIP guidance and the class definitions improve the mIoU for 92 labels, while they do not affect the mIoU for 1464 labels. Probably, the 16\u00d7 more labels provide sufficient information about the class definitions of the dataset so that no additional CLIP guidance is necessary in this case. All components provide stronger improvements for fewer labels, showing their suitability for training with limited annotations. Fine-Tuning Strategies: Tab. 7 compares different methods to adapt the VLM to semantic segmentation. While a frozen backbone does not overfit to the small labeled subset, it also cannot adapt to segmentation. Deep prompt tuning (DPT) [75, 76] learns additional input tokens to manipulate a frozen ViT. We use the same parameters as [76]. It significantly improves over a frozen backbone. LoRA [30] learns low-rank (r=8) residuals for the attention weights (Wqkvo). It improves over DPT achieving a similar performance as full model fine-tuning. Our spatial fine-tuning can further improve over both. On the one side, it preserves the semantic reasoning of the local MLPs and avoids their overfitting to the limited labels. On the other side, its full rank fine-tuning of the attention layers might be necessary to sufficiently adapt the ViT from image-level to dense reasoning. For all variants, we tuned the backbone learning rate. Decoder Study: SemiVL\u2019s decoder is designed to decouple semantic from spatial reasoning for label-efficient learning. As a baseline for joint reasoning, we input all classes together into the spatial reasoning (N times more convolution channels) instead of processing each class separately (and sharing the parameters across classes). Tab. 8 shows that decoupled reasoning (82.7 mIoU) significantly outperforms joint reasoning (64.9 mIoU) for 92 labels as the joint reasoning overfits to the small labeled subset. Further, Tab. 8 provides an ablation study over SemiVL\u2019s decoder modules. It shows that each module (upsampling, spatial reasoning, and semantic reasoning) is crucial for its performance. Class Definitions: Tab. 9 compares different strategies for text prompts in CLIP inference. It shows that class definitions from the dataset annotation guidelines achieve the best performance. Here, taking the maximum score over concepts (Sec. 3.6) works better than using an averaged text embedding over the concepts or the guidelines as raw text. As baselines, we also benchmark definitions from the Oxford Languages dictionary and class concepts queried from GPT3.5. However, they perform worse, probably because they are dataset agnostic and do not represent the dataset-specific class boundaries as well as the annotation guidelines. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2212.01322v2",
+                    "title": "MIC: Masked Image Consistency for Context-Enhanced Domain Adaptation",
+                    "abstract": "In unsupervised domain adaptation (UDA), a model trained on source data (e.g.\nsynthetic) is adapted to target data (e.g. real-world) without access to target\nannotation. Most previous UDA methods struggle with classes that have a similar\nvisual appearance on the target domain as no ground truth is available to learn\nthe slight appearance differences. To address this problem, we propose a Masked\nImage Consistency (MIC) module to enhance UDA by learning spatial context\nrelations of the target domain as additional clues for robust visual\nrecognition. MIC enforces the consistency between predictions of masked target\nimages, where random patches are withheld, and pseudo-labels that are generated\nbased on the complete image by an exponential moving average teacher. To\nminimize the consistency loss, the network has to learn to infer the\npredictions of the masked regions from their context. Due to its simple and\nuniversal concept, MIC can be integrated into various UDA methods across\ndifferent visual recognition tasks such as image classification, semantic\nsegmentation, and object detection. MIC significantly improves the\nstate-of-the-art performance across the different recognition tasks for\nsynthetic-to-real, day-to-nighttime, and clear-to-adverse-weather UDA. For\ninstance, MIC achieves an unprecedented UDA performance of 75.9 mIoU and 92.8%\non GTA-to-Cityscapes and VisDA-2017, respectively, which corresponds to an\nimprovement of +2.1 and +3.0 percent points over the previous state of the art.\nThe implementation is available at https://github.com/lhoyer/MIC.",
+                    "authors": "Lukas Hoyer, Dengxin Dai, Haoran Wang, Luc Van Gool",
+                    "published": "2022-12-02",
+                    "updated": "2023-03-24",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction In order to train state-of-the-art neural networks for visual recognition tasks, large-scale annotated datasets are necessary. However, the collection and annotation process can be very time-consuming and tedious. For instance, the annotation of a single image for semantic segmentation can take more than one hour [11,77]. Therefore, it would be beneficial to resort to existing or simulated datasets, which are easier to annotate. However, a network trained on such a source     (c) UDA with Masked Image Consistency (MIC) Unsupervised Domain Adaptation (UDA) Method MIC Loss Masking Target Image Masked Image Consistency (MIC) EMA (a) SotA UDA Pred. (b) MIC Prediction Target Image Figure 1. (a) Previous UDA methods such as HRDA [33] struggle with similarly looking classes on the unlabeled target domain. Here, the interior of the sidewalk is wrongly segmented as road, probably, due to the ambiguous local appearance. (b) The proposed Masked Image Consistency (MIC) enhances the learning of context relations to consider additional context clues such as the curb in the foreground. With MIC, the adapted network is able to correctly segment the sidewalk. (c) MIC can be plugged into most existing UDA methods. It enforces the consistency of the predictions of a masked target image with the pseudo-label of the original image. So, the network is trained to better utilize context clues on the target domain. Further details are shown in Fig. 3. dataset usually performs worse when applied to the actual target dataset as neural networks are sensitive to domain gaps. To mitigate this issue, unsupervised domain adaptation (UDA) methods adapt the network to the target domain using unlabeled target images, for instance, with adversarial training [22,29,66,84] or self-training [32,33,83,90,110]. UDA methods have remarkably progressed in the last few years. However, there is still a noticeable performance gap compared to supervised training. A common problem is the confusion of classes with a similar visual appearance on the target domain such as road/sidewalk or pedestrian/rider as there is no ground truth supervision available to learn the 1 arXiv:2212.01322v2  [cs.CV]  24 Mar 2023 \fslight appearance differences. For example, the interior of the sidewalk in Fig. 1 is segmented as road, probably, due to a similar local appearance. To address this problem, we propose to enhance UDA with spatial context relations as additional clues for robust visual recognition. For instance, the curb in the foreground of Fig. 1 a) could be a crucial context clue to correctly recognize the sidewalk despite the ambiguous texture. Although the used network architectures already have the capability to model context relations, previous UDA methods are still not able to reach the full potential of using context dependencies on the target domain as the used unsupervised target losses are not powerful enough to enable effective learning of such information. Therefore, we design a method to explicitly encourage the network to learn comprehensive context relations of the target domain during UDA. In particular, we propose a novel Masked Image Consistency (MIC) plug-in for UDA (see Fig. 1 c), which can be applied to various visual recognition tasks. Considering semantic segmentation for illustration, MIC masks out a random selection of target image patches and trains the network to predict the semantic segmentation result of the entire image including the masked-out parts. In that way, the network has to utilize the context to infer the semantics of the masked regions. As there are no ground truth labels for the target domain, we resort to pseudo-labels, generated by an EMA teacher that uses the original, unmasked target images as input. Therefore, the teacher can utilize both context and local clues to generate robust pseudo-labels. Over the course of the training, different parts of objects are masked out so that the network learns to utilize different context clues, which further increases the robustness. After UDA with MIC, the network is able to better exploit context clues and succeeds in correctly segmenting difficult areas that rely on context clues such as the sidewalk in Fig. 1 b). To the best of our knowledge, MIC is the first UDA approach to exploit masked images to facilitate learning context relations on the target domain. Due to its universality and simplicity, MIC can be straightforwardly integrated into various UDA methods across different visual recognition tasks, making it highly valuable in practice. MIC achieves significant and consistent performance improvements for different UDA methods (including adversarial training, entropy-minimization, and self-training) on multiple visual recognition tasks (image classification, semantic segmentation, and object detection) with different domain gaps (synthetic-to-real, clear-to-adverse-weather, and day-tonight) and different network architectures (CNNs and Transformer). It sets a new state-of-the-art performance on all tested benchmarks with significant improvements over previous methods as shown in Fig. 2. For instance, MIC respectively improves the state-of-the-art performance by +2.1, +4.3, and +3.0 percent points on GTA\u2192Cityscapes(CS), CS\u2192DarkZurich, and VisDA-2017 and achieves an unprece40 60 80 100 Cls. Accuracy, Seg. mIoU, or Det. mAP Det. CS Foggy CS Segm. CS FoggyZ. Segm. CS DarkZ. Segm. Synthia CS Segm. CS ACDC Segm. GTA CS Cls. OfficeHome Cls. VisDA ] +3.6 ] +3.7 ] +4.3 ] +1.5 ] +2.4 ] +2.1 ] +1.9 ] +3.0 w/o MIC w/ MIC Figure 2. MIC significantly improves state-of-the-art UDA methods across different UDA benchmarks and recognition tasks such as image classification (Cls.), semantic segmentation (Segm.), and object detection (Det.). Detailed results can be found in Sec. 4. dented UDA performance of 75.9 mIoU, 60.2 mIoU, and 92.8%, respectively. 2. Related Work 2.1. Unsupervised Domain Adaptation (UDA) In UDA, a model trained on a labeled source domain is adapted to an unlabeled target domain. Due to the ubiquity of domain gaps, UDA methods were designed for all major computer vision problems including image classification [20,55,56,64], semantic segmentation [30,32,84,102], and object detection [7, 8, 49, 52]. The majority of the approaches rely on discrepancy minimization, adversarial training, or self-training. The first group minimizes the discrepancy between domains using a statistical distance function such as maximum mean discrepancy [26, 55, 58], correlation alignment [79,80], or entropy minimization [25,57,87]. In adversarial training, a learned domain discriminator provides supervision in a GAN framework [24] to encourage domain-invariant inputs [23,29], features [20,30,56,84] or outputs [60,71,84,87]. In self-training, pseudo-labels [44] are generated for the target domain based on predictions obtained using confidence thresholds [62,104,110] or pseudolabel prototypes [64,102,103]. To increase the robustness of the self-training, consistency regularization [72,78,82] is often applied to ensure consistency over different data augmentations [1,9,19,63], different crops [33,42], multiple models [101,106,107], or domain-mixup [32\u201334,83,108]. Further UDA strategies utilize pretext tasks [6,34,88,90], follow an adaptation curriculum [12,13,105], exploit the increased domain-robustness of Transformers [32,33,81,98], align the domains with contrastive learning [36,94], use graph matching [4,48,49], or adapt multi-resolution inputs [33]. To facilitate learning domain-robust context dependencies, several UDA methods propose network components 2 \fthat can capture context such as spatial attention pyramids [46], cross-domain attention [99], or context-aware feature fusion [32]. While these network modules provide the ability to capture context, the unsupervised loss on the target domain does not provide sufficient supervision to learn all relevant target context relations. To improve context learning, CrDA [37] aligns local context relations with adversarial training and HRDA [33] uses multi-crop consistency training. However, these mechanisms are not able to capture all relevant context clues as can be seen for HRDA in Fig. 1 a). Due to the random patch masking, MIC is able to learn a larger set of different context clues for robust recognition. 2.2. Masked Image Modeling Predicting withheld tokens of a masked input sequence was shown to be a powerful self-supervised pretraining task in natural language processing [3,15]. Recently, this concept was successfully transferred to self-supervised pretraining in computer vision, where it is known as masked image modeling. Given a partly masked image, the network is trained to reconstruct properties of the masked areas such as VAE features [2, 16, 50], HOG features [91], or color information [27,96]. To sample the mask, block-wise masking [2], random patch masking [27,96], and attention-guided masking [41,53] have been explored. Similarly, our method also uses masked images. However, we pursue a different purpose than previous works. Instead of aiming to learn self-supervised representations, MIC utilizes masked images in a novel way to learn context relations for domain adaptation. Due to this conceptual difference, we do not have to rely on pretext restoration targets such as VAE features but can perform the reconstruction in the actual prediction space of the relevant computer vision task such as semantic segmentation. To the best of our knowledge, MIC is the first method to exploit masked images to enhance context learning for UDA. Particularly, we show that naive masked image modeling on ImageNet does not improve the target domain performance (see Sec. 4.3). 3. Methods 3.1. Unsupervised Domain Adaptation (UDA) A neural network f\u03b8 can be trained on the source domain using images X S = {xS k }NS k=1 and their labels YS = {yS k }NS k=1 with a supervised source loss LS. The specific source loss depends on the computer vision task. For image classification and semantic segmentation, the (pixel-wise) cross-entropy is typically used   \\mathcal   { L}^\\mat h it  { S , c ls/ seg }_ k & =   \\ mat h c al  { H }(f _\\th eta   (x^S _k), y^S_k)\\,,\\\\ \\mathcal {H}(\\hat {y}, y) &= \\sum _{i=1}^{H} \\sum _{j=1}^{W} \\sum _{c=1}^C y_{ijc} \\log \\hat {y}_{ijc}\\,, (2)     Pseudo Label pT Masked Pred. \u0177M Masked Image xM Target Image xT Source Image xS Source Predict. \u0177S MIC Loss LM Masking Network f\u03b8 EMA Teacher g\u03d5 Adaptation Loss LT Supervised Loss LS Masked Image Consistency (MIC) EMA Weight Update Stop Gradient Figure 3. UDA with the proposed Masked Image Consistency (MIC). In UDA, a network is typically trained with a supervised loss on the source domain (blue) and an unsupervised adaptation loss on the target domain (green). MIC enforces the consistency between predictions of masked target images (purple) and pseudolabels that are generated based on the complete image by an exponential moving average (EMA) teacher. To minimize the MIC loss, the network has to learn to infer the predictions of the masked regions from their context. where H=W=1 in case of classification. For object detection, a box regression and a box classification loss are commonly utilized [67]. However, a model trained on the source domain usually experiences a performance drop when applied to another domain. Therefore, unsupervised domain adaptation (UDA) methods use unlabeled images from the target domain X T = {xT k }NT k=1 to adapt the network. For that purpose, an additional unsupervised loss for the target domain LT is added to the optimization problem with a weight \u03bbT   \\ m i n  _{ \\ the ta  }   \\f ra c  {1 }{ N_ S } \\sum _{k=1}^{N_S} \\mathcal {L}^S_k + \\frac {1}{N_T} \\sum _{k=1}^{N_T} \\lambda ^T\\mathcal {L}_k^T\\;.  (3) The target loss LT is defined according to the UDA strategy such as adversarial training [8,20,66,84,85,89] or selftraining [32,62,83,102,103,110]. 3.2. Masked Image Consistency (MIC) To recognize an object (or stuff region), a model can utilize clues from different parts of the image. This can be local information, which originates from the same image patch as the corresponding cell in the feature map, or context information, which comes from surrounding image patches that can belong to different parts of the object or its environment [35]. Many network architectures [17,28] have the capability to integrate both local and context information in their features. While the learning of context clues can be guided by ground 3 \ftruth in supervised learning, there is no ground truth supervision available for the target domain in UDA. Current unsupervised losses are not powerful enough to enable effective learning of context clues as empirically observed such as in Fig. 1 a). Therefore, we propose to specifically encourage the learning of context relations on the target domain to provide additional clues for robust recognition of classes with similar local appearances. In order to facilitate the learning of context relations on the target domain, we introduce a Masked Image Consistency (MIC) module, which can be easily plugged into various existing UDA methods. The domain adaptation process with MIC is illustrated in Fig. 3 and explained below. MIC withholds local information by randomly masking out patches of the target image. For that purpose, a patch mask M is randomly sampled from a uniform distribution   \\mathcal {M }_{\\substac k  { m b+ 1:(m + 1 )b,\\ \\n b+1:(n+1)b}} = [v > r] \\quad \\text {with} \\; v \\sim \\mathcal {U}(0, 1)\\;,  (4) where [\u00b7] denotes the Iverson bracket, b the patch size, r the mask ratio, and m \u2208[0 .. W/b \u22121], n \u2208[0 .. W/b \u22121] the patch indices. The masked target image xM (see Fig. 3) is obtained by element-wise multiplication of mask and image   x ^ { M}  = \\mathcal {M} \\odot x^{T}\\,.  (5) The masked target prediction \u02c6 yM can only use the limited information of the unmasked image regions    \\ h at {y} ^{M} = f_\\theta (x^M)\\,,  (6) making the prediction more difficult. This is also reflected in Fig. 3, where the prediction misses a part of the sidewalk. In order to train the network to use the remaining context clues to still reconstruct the correct label without access to the entire image, the MIC loss LM is introduced   \\ ma thc al {L } ^M = q^T \\mathcal {H}(\\hat {y}^M, p^T)\\,,  (7) where pT denotes a pseudo-label and qT its quality weight. MIC uses pseudo-labels as there is no ground truth available for the target domain. The pseudo-label is the prediction of a teacher network g\u03d5 of the complete target image xT . For image classification and semantic segmentation,   p^\\mathi t  { T, c ls/ seg }_ {ij} = [c =  \\argmax _{c'} g_\\phi (x^T)_{ijc'}]\\,. \\label {eq:pseudo_label}  (8) For object detection pseudo-labels, box predictions from g\u03d5(xT ) are filtered with a confidence threshold \u03b4 and nonmaximum suppression [67]. The teacher network g\u03d5 is implemented as an EMA teacher [82]. Its weights are the exponential moving average of the weights of f\u03b8 with smoothing factor \u03b1   \\p h i _ { t+ 1 } \\l eftarrow \\alpha \\phi _t + (1 \\alpha ) \\theta _t\\,,  (9) where t denotes a training step. The EMA teacher realizes a temporal ensemble of previous student models f\u03b8 [82], which increases the robustness and temporal stability of pseudo-labels. It is a common strategy used in semisupervised learning [18,31,82] and UDA [1,32,33,83]. As the teacher is updated based on the student f\u03b8, it will gradually obtain the enhanced context learning capability from f\u03b8. In contrast to the student f\u03b8, the teacher g\u03d5 has privileged access to the original image xT (see Eq. 8), so that it can use both the context and the intact local appearance information to generate pseudo labels of higher quality. As the pseudo-labels are potentially wrong (especially at the beginning of the training), the loss is weighted by the quality estimate qT . For image classification, we use the maximum softmax probability as certainty estimate [104]   q^\\m a thi t  {T,cl s} = \\max _{c'} g_\\phi (x^T)_{c'}\\,. \\label {eq:q_classification}  (10) For semantic segmentation, we follow [32,33,83] and utilize the ratio of pixels exceeding a threshold \u03c4 of the maximum softmax probability   q^\\m a th it  {T ,seg} = \\ frac {\\sum _ { i= 1 } ^ {H}\\sum _{j=1}^{W} [\\max _{c'} g_\\phi (x^T)_{(ijc')} > \\tau ]}{H \\cdot W}\\,.  (11) And for object detection, we apply the quality estimate from Eq. 10 to each bounding box in the classification branches. The MIC consistency training can be easily integrated into the UDA optimization problem   \\ m i n  _{ \\ the ta  }   \\f ra c  {1 }{N _S }  \\sum  _ {k=1}^{N_S} \\mathcal {L}^S_k + \\frac {1}{N_T} \\sum _{k=1}^{N_T} (\\lambda ^T \\mathcal {L}_k^T + \\lambda ^M \\mathcal {L}_k^M)\\;.  (12) 4. Experiments 4.1. Implementation Details Semantic Segmentation: We study synthetic-to-real, clearto-adverse-weather, and day-to-nighttime adaptation of street scenes. As synthetic datasets, we use GTA [68] containing 24,966 images and Synthia [69] with 9,400 images. As real-world datasets, we use Cityscapes (CS) [11] consisting of 2,975 training and 500 validation images for clear weather, DarkZurich [76] with 2,416 training and 151 test images for nighttime, and ACDC [77] containing 1,600 training, 406 validation, and 2,000 test images for adverse weather (fog, night, rain, and snow). The training resolution follows the used UDA methods (e.g. half resolution for DAFormer [32] or full resolution for HRDA [33]). We evaluate MIC based on a DAFormer network [32] with a MiT-B5 encoder [95], and a DeepLabV2 [5] with a ResNet-101 [28] backbone. All backbones are initialized with ImageNet pretraining. In the default UDA setting, we follow the HRDA [33] multi-resolution self-training strategy and training parameters, i.e. AdamW [59] with a learning 4 \frate of 6\u00d710\u22125 for the encoder and 6\u00d710\u22124 for the decoder, 40k training iterations, a batch size of 2, linear learning rate warmup, a loss weight \u03bbT st=1, an EMA factor \u03b1=0.999, DACS [83] data augmentation, Rare Class Sampling [32], and ImageNet Feature Distance [32]. For adversarial training and entropy minimization, SGD with a learning rate of 0.0025 and \u03bbT adv=\u03bbT ent=0.001 is used. Image Classification: We evaluate MIC on the VisDA-2017 dataset [65], which consists of 280,000 synthetic and real images of 12 classes, as well as the Office-Home dataset [86], which contains 15,500 images from 65 classes for the domains art (A), clipart (C), product (P) and real-world (R). We conduct the experiments with ResNet-101 [28] and ViTB/16 [17]. For UDA training, we follow SDAT [66], which utilizes CDAN [56] with MCC [40] and a smoothness enhancing loss [66]. We use the same training parameters, i.e. SGD with a learning rate of 0.002, a batch size of 32, and a smoothness parameter of 0.02. Object Detection: For object detection UDA, we evaluate MIC on CS [11] to Foggy CS [75]. The experiments are performed based on Faster R-CNN [67] with ResNet-50 [28] and FPN [54]. For UDA, we adopt SADA [8], which utilizes adversarial training on image and instance level. The same parameters as in [8] are used, i.e. 0.0025 initial learning rate, 60k training iterations, \u03bbT adv=0.1, and a batch size of 2. Following previous works [8,70], we report the results in mean Average Precision (mAP) with a 0.5 IoU threshold. MIC Parameters: MIC uses a patch size b=64, a mask ratio r=0.7, a loss weight \u03bbM=1, an EMA weight \u03b1=0.999 following [32,33], and color augmentation (brightness, contrast, saturation, hue, and blur) following the parameters of [32,33,83]. We set the pseudo-label box threshold \u03b4=0.8 following [14, 52] and the quality threshold \u03c4=0.968 following [32,33,83]. If a UDA method trains with half resolution [8,32,83,84,87], the patch size is divided by 2. For image classification and object detection, we use \u03b1=0.9. For object detection, we reduce the mask ratio r=0.5 as the objects of interest are more sparse and a high r increases the risk that they are completely masked out. For target domains with nighttime images (DarkZurich and ACDC), we forgo color augmentation as it can corrupt the content of dark nighttime images due to the locally already low brightness and contrast. The experiments are conducted on an RTX 2080 Ti or a Titan RTX depending on the required memory. 4.2. MIC for Semantic Segmentation First, we combine MIC with different UDA methods and network architectures for semantic segmentation on GTA\u2192CS. Tab. 1 shows that MIC achieves consistent and significant improvements across various UDA methods with different network architectures, ranging from +1.2 up to +4.7 mIoU. Specifically, MIC does not only benefit powerful Transformers such as DAFormer [32] but also CNNs such Table 1. Segmentation performance (mIoU in %) of MIC with different UDA methods on GTA\u2192CS. Network UDA Method w/o MIC w/ MIC Diff. DeepLabV2 [5] Adversarial [84] 44.2 48.2 +4.0 DeepLabV2 [5] Entropy Min. [87] 44.3 49.0 +4.7 DeepLabV2 [5] DACS [83] 53.9 56.0 +2.1 DeepLabV2 [5] DAFormer [32] 56.0 59.4 +3.4 DeepLabV2 [5] HRDA [33] 63.0 64.2 +1.2 DAFormer [32] DAFormer [32] 68.3 70.6 +2.3 DAFormer [32] HRDA [33] 73.8 75.9 +2.1 as DeepLabV2 [5]. Across UDA methods, the performance improvement decreases with a higher UDA performance as expected due to performance saturation. Second, we evaluate the performance of MIC combined with the best-performing UDA method HRDA [33] for further domain adaptation scenarios: synthetic-to-real (GTA\u2192 CS and Synthia\u2192CS), day-to-nighttime (CS\u2192DarkZurich), and clear-to-adverse-weather (CS\u2192ACDC). Tab. 2 shows clear performance improvements on each benchmark. Specifically, MIC improves the state-of-the-art performance by +2.1 mIoU on GTA\u2192CS, by +1.5 mIoU on Synthia\u2192CS, by +4.3 mIoU on CS\u2192DarkZurich, and by +2.4 mIoU on CS\u2192ACDC. Considering the class-wise IoU in Tab. 2, MIC achieves consistent improvements for most classes when compared to the previous state-of-the-art method HRDA. Classes that most profit from MIC are sidewalk, fence, pole, traffic sign, terrain, and rider. These classes have a comparably low UDA performance, meaning that they are difficult to adapt. Here, context clues appear to play an important role in successful adaptation. For some classes such as building or vegetation on synthetic-to-real adaptation, MIC increases the performance by a smaller margin, probably because the target context clues play a smaller role for them. In a few particular cases, the performance of single classes decreases for MIC such as truck on CS\u2192DarkZurich. These are rare classes, which are underrepresented in the data, which might cause MIC to pick up misleading context biases. The observations from Tab. 2 are also reflected in the example predictions in Fig. 4. While previous methods often recognize only parts of ambiguous regions, MIC fixes these issues by using correctly detected parts as context. For instance, the grille of the bus in Fig. 4 resembles a traffic cabinet (building). However, a cabinet between two vehicles is unlikely. Probably using this context prior, MIC can resolve the ambiguity. 4.3. MIC for Image Classification For image classification UDA, we combine MIC with the state-of-the-art method SDAT [66]. On VisDA-2017 (Tab. 3), MIC significantly improves the UDA performance by +2.5 and +3.0 percent points when used with a ResNet and ViT network, respectively The improvement is consistent over almost all classes, where difficult classes generally benefit the most. On Office-Home (Tab. 4), MIC clearly improves 5 \fTable 2. Semantic segmentation performance (IoU in %) on four different UDA benchmarks. Method Road S.walk Build. Wall Fence Pole Tr.Light Sign Veget. Terrain Sky Person Rider Car Truck Bus Train M.bike Bike mIoU Synthetic-to-Real: GTA\u2192Cityscapes (Val.) ADVENT [87] 89.4 33.1 81.0 26.6 26.8 27.2 33.5 24.7 83.9 36.7 78.8 58.7 30.5 84.8 38.5 44.5 1.7 31.6 32.4 45.5 DACS [83] 89.9 39.7 87.9 30.7 39.5 38.5 46.4 52.8 88.0 44.0 88.8 67.2 35.8 84.5 45.7 50.2 0.0 27.3 34.0 52.1 ProDA [102] 87.8 56.0 79.7 46.3 44.8 45.6 53.5 53.5 88.6 45.2 82.1 70.7 39.2 88.8 45.5 59.4 1.0 48.9 56.4 57.5 DAFormer [32] 95.7 70.2 89.4 53.5 48.1 49.6 55.8 59.4 89.9 47.9 92.5 72.2 44.7 92.3 74.5 78.2 65.1 55.9 61.8 68.3 HRDA [33] 96.4 74.4 91.0 61.6 51.5 57.1 63.9 69.3 91.3 48.4 94.2 79.0 52.9 93.9 84.1 85.7 75.9 63.9 67.5 73.8 MIC (HRDA) 97.4 80.1 91.7 61.2 56.9 59.7 66.0 71.3 91.7 51.4 94.3 79.8 56.1 94.6 85.4 90.3 80.4 64.5 68.5 75.9 Synthetic-to-Real: Synthia\u2192Cityscapes (Val.) ADVENT [87] 85.6 42.2 79.7 8.7 0.4 25.9 5.4 8.1 80.4 \u2013 84.1 57.9 23.8 73.3 \u2013 36.4 \u2013 14.2 33.0 41.2 DACS [83] 80.6 25.1 81.9 21.5 2.9 37.2 22.7 24.0 83.7 \u2013 90.8 67.6 38.3 82.9 \u2013 38.9 \u2013 28.5 47.6 48.3 ProDA [102] 87.8 45.7 84.6 37.1 0.6 44.0 54.6 37.0 88.1 \u2013 84.4 74.2 24.3 88.2 \u2013 51.1 \u2013 40.5 45.6 55.5 DAFormer [32] 84.5 40.7 88.4 41.5 6.5 50.0 55.0 54.6 86.0 \u2013 89.8 73.2 48.2 87.2 \u2013 53.2 \u2013 53.9 61.7 60.9 HRDA [33] 85.2 47.7 88.8 49.5 4.8 57.2 65.7 60.9 85.3 \u2013 92.9 79.4 52.8 89.0 \u2013 64.7 \u2013 63.9 64.9 65.8 MIC (HRDA) 86.6 50.5 89.3 47.9 7.8 59.4 66.7 63.4 87.1 \u2013 94.6 81.0 58.9 90.1 \u2013 61.9 \u2013 67.1 64.3 67.3 Day-to-Nighttime: Cityscapes\u2192DarkZurich (Test) ADVENT [87] 85.8 37.9 55.5 27.7 14.5 23.1 14.0 21.1 32.1 8.7 2.0 39.9 16.6 64.0 13.8 0.0 58.8 28.5 20.7 29.7 MGCDA\u2020 [76] 80.3 49.3 66.2 7.8 11.0 41.4 38.9 39.0 64.1 18.0 55.8 52.1 53.5 74.7 66.0 0.0 37.5 29.1 22.7 42.5 DANNet\u2020 [92] 90.0 54.0 74.8 41.0 21.1 25.0 26.8 30.2 72.0 26.2 84.0 47.0 33.9 68.2 19.0 0.3 66.4 38.3 23.6 44.3 DAFormer [32] 93.5 65.5 73.3 39.4 19.2 53.3 44.1 44.0 59.5 34.5 66.6 53.4 52.7 82.1 52.7 9.5 89.3 50.5 38.5 53.8 HRDA [33] 90.4 56.3 72.0 39.5 19.5 57.8 52.7 43.1 59.3 29.1 70.5 60.0 58.6 84.0 75.5 11.2 90.5 51.6 40.9 55.9 MIC (HRDA) 94.8 75.0 84.0 55.1 28.4 62.0 35.5 52.6 59.2 46.8 70.0 65.2 61.7 82.1 64.2 18.5 91.3 52.6 44.0 60.2 Clear-to-Adverse-Weather: Cityscapes\u2192ACDC (Test) ADVENT [87] 72.9 14.3 40.5 16.6 21.2 9.3 17.4 21.2 63.8 23.8 18.3 32.6 19.5 69.5 36.2 34.5 46.2 26.9 36.1 32.7 MGCDA\u2020 [76] 73.4 28.7 69.9 19.3 26.3 36.8 53.0 53.3 75.4 32.0 84.6 51.0 26.1 77.6 43.2 45.9 53.9 32.7 41.5 48.7 DANNet\u2020 [92] 84.3 54.2 77.6 38.0 30.0 18.9 41.6 35.2 71.3 39.4 86.6 48.7 29.2 76.2 41.6 43.0 58.6 32.6 43.9 50.0 DAFormer [32] 58.4 51.3 84.0 42.7 35.1 50.7 30.0 57.0 74.8 52.8 51.3 58.3 32.6 82.7 58.3 54.9 82.4 44.1 50.7 55.4 HRDA [33] 88.3 57.9 88.1 55.2 36.7 56.3 62.9 65.3 74.2 57.7 85.9 68.8 45.7 88.5 76.4 82.4 87.7 52.7 60.4 68.0 MIC (HRDA) 90.8 67.1 89.2 54.5 40.5 57.2 62.0 68.4 76.3 61.8 87.0 71.3 49.4 89.7 75.7 86.8 89.1 56.9 63.0 70.4 \u2020 Method uses additional daytime/clear-weather geographically-aligned reference images. Image DAFormer [32] HRDA [33] MIC (HRDA) Ground Truth road sidew. build. wall fence pole tr. light tr. sign veget. terrain sky person rider car truck bus train m.bike bike n/a. Figure 4. Qualitative comparison of MIC with previous methods on GTA\u2192CS (row 1 and 2), CS\u2192ACDC (row 3), and CS\u2192DarkZurich (row 4). MIC better segments difficult classes such as sidewalk, fence, traffic sign, and bus. Further examples are shown in the supplement. the UDA performance by +1.9. Domains that are difficult to adapt such as A)C or P)C benefit most. Tab. 3 further provides a baseline of SDAT with MAE [27] pretraining, which includes masked image modeling (MIM) and ImageNet supervision. Compared to regular SDAT, additional MIM reduces the performance by -1.4. This demonstrates that naive MIM as additional pretraining is not sufficient to capture the relevant target context dependencies, probably as the learned context is specific to ImageNet and does not transfer well to the target domain. 6 \fTable 3. Image classification accuracy in % on VisDA-2017 for UDA. The last column contains the mean across classes. Method Plane Bcycl Bus Car Horse Knife Mcyle Persn Plant Sktb Train Truck Mean CDAN [56] ResNet 85.2 66.9 83.0 50.8 84.2 74.9 88.1 74.5 83.4 76.0 81.9 38.0 73.9 MCC [40] 88.1 80.3 80.5 71.5 90.1 93.2 85.0 71.6 89.4 73.8 85.0 36.9 78.8 SDAT [66] 95.8 85.5 76.9 69.0 93.5 97.4 88.5 78.2 93.1 91.6 86.3 55.3 84.3 MIC (SDAT) 96.7 88.5 84.2 74.3 96.0 96.3 90.2 81.2 94.3 95.4 88.9 56.6 86.9 TVT [100] ViT 92.9 85.6 77.5 60.5 93.6 98.2 89.3 76.4 93.6 92.0 91.7 55.7 83.9 CDTrans [98] 97.1 90.5 82.4 77.5 96.6 96.1 93.6 88.6 97.9 86.9 90.3 62.8 88.4 SDAT [66] 98.4 90.9 85.4 82.1 98.5 97.6 96.3 86.1 96.2 96.7 92.9 56.8 89.8 SDAT w/ MAE [27] 97.1 88.4 80.9 75.3 95.4 97.9 94.3 85.5 95.8 91.0 93.0 65.4 88.4 MIC (SDAT) 99.0 93.3 86.5 87.6 98.9 99.0 97.2 89.8 98.9 98.9 96.5 68.0 92.8 Table 4. Image classification acc. in % on Office-Home for UDA. Method A )C A )P A )R C )A C )P C )R P )A P )C P )R R )A R )C R )P Avg CDTrans [98] 68.8 85.0 86.9 81.5 87.1 87.3 79.6 63.3 88.2 82.0 66.0 90.6 80.5 TVT [100] 74.9 86.8 89.5 82.8 87.9 88.3 79.8 71.9 90.1 85.5 74.6 90.6 83.6 SDAT [66] 70.8 87.0 90.5 85.2 87.3 89.7 84.1 70.7 90.6 88.3 75.5 92.1 84.3 MIC (SDAT) 80.2 87.3 91.1 87.2 90.0 90.1 83.4 75.6 91.2 88.6 78.7 91.4 86.2 Table 5. Object detection AP in % on CS\u2192Foggy CS. Method Bus Bcycl Car Mcycle Persn Rider Train Truck mAP DAFaster [7] 29.2 40.4 43.4 19.7 38.3 28.5 23.7 32.7 32.0 SW-DA [70] 31.8 44.3 48.9 21.0 43.8 28.0 28.9 35.8 35.3 SC-DA [109] 33.8 42.1 52.1 26.8 42.5 26.5 29.2 34.5 35.9 MTOR [4] 38.6 35.6 44.0 28.3 30.6 41.4 40.6 21.9 35.1 SIGMA [49] 50.4 40.6 60.3 31.7 44.0 43.9 51.5 31.6 44.2 SADA [8] 50.3 45.4 62.1 32.4 48.5 52.6 31.5 29.5 44.0 MIC (SADA) 52.4 47.5 67.0 40.6 50.9 55.3 33.7 33.9 47.6 4.4. MIC for Object Detection For object detection UDA, we combine MIC with the state-of-the-art Scale-aware Domain Adaptive Faster-RCNN (SADA) [8]. On CS\u2192Foggy CS (Tab. 5), MIC obtains consistent improvements over all categories and achieves +3.6 mAP gain compared to the baseline SADA. The classes car, motorcycle, and rider benefit the most. MIC also shows a clear advantage for most categories compared to more recent methods such as SIGMA [49]. 4.5. In-Depth Analysis of MIC Context Utilization: To verify that MIC has learned context priors on the target domain, we mask out an image patch, let the trained model predict the semantics of the patch from the visible context, and calculate the mIoU for the patch. As the patch is masked out, the model can only utilize context information to infer its semantics. We repeat this process for all non-overlapping patches of the size 256\u00d7256 in the CS val. set. MIC(HRDA) achieves a strong context performance of 52.5 mIoU on GTA\u2192CS while HRDA only reaches 22.8 mIoU, showing that MIC indeed enhances context learning. To further illustrate the learned comprehensive context relations, we visualize predictions of masked images in Fig. 5. It shows the learned context priors of helmet and bicycle, which MIC internally exploits to predict the rider\u2019s body. Where to apply MIC? Tab. 6 shows the performance of MIC with HRDA using images from different domains as     a) Only a local patch is visible  \u2192 Rider is confused with \u2018pedestrian\u2019     b) Only context above the patch is visible  \u2192 Rider\u2019s body is predicted from helmet c) Only context below the patch is visible \u2192 Rider\u2019s body is predicted from bicycle d) The entire image is visible \u2192 All local and context clues can be used Figure 5. Predictions of MIC for masked variants of the same image demonstrating the learned context priors of MIC. Table 6. MIC with HRDA [33] for images from different domain. MIC Domain mIoUGTA )CS mIoUCS )ACDC(Val) \u2013 73.8 65.3 Source 71.1 66.5 Target 75.9 66.9 Source+Target 74.5 68.0 masked input: (1) source, (2) target, and (3) both source and target. We observe that: for (1) the performance is -2.7 mIoU worse than HRDA for GTA\u2192CS but it increases by +1.2 mIoU for CS\u2192ACDC, for (2) the performance increases by +2.1 for GTA\u2192CS and +1.6 mIoU for CS\u2192ACDC, and for (3) the mIoU increases by +0.7 for GTA\u2192CS and +2.7 for CS\u2192ACDC. Both benchmarks differ in the domain gap of context relations. While the distributions of context relations can vary between synthetic (GTA) and real data (CS), the context relations of CS and ACDC are very similar as both datasets were recorded in the real world and partly even in the same city. If the context domain gap is large, context relations learned on source images do not transfer well to the target domain and can even hamper the adaptation. However, if the context gap is small, source context relations transfer well to the target domain and can boost the adaptation performance. Therefore, we also apply MIC to the source domain, in addition to the default target domain, for clear-to-adverseweather and day-to-nighttime adaptation. 7 \fTable 7. MIC ablation study with DAFormer [32] on GTA\u2192CS. Masked Img. Color Aug. EMA Teacher Pseudo Lbl. Weight mIoU 1 \u2013 \u2013 \u2013 \u2013 68.3 2 \u2713 \u2713 \u2713 \u2713 70.6 3 \u2013 \u2713 \u2713 \u2713 50.6 4 \u2713 \u2013 \u2713 \u2713 70.3 5 \u2713 \u2713 \u2013 \u2713 69.9 6 \u2713 \u2713 \u2713 \u2013 69.0 Table 8. Parameter study of the patch size b and the mask ratio r of MIC with DAFormer [32] on GTA\u2192CS. The color indicates the difference to the DAFormer performance of 68.3 mIoU. 0.3 0.5 0.7 0.9 Mask Ratio r 32 64 128 256 Patch Size b 69.3 69.9 69.7 69.3 69.2 70.3 70.6 69.7 68.7 70.5 70.4 68.2 66.2 69.5 69.8 68.0 Table 9. Comparison of UDA on GTA\u2192CS and supervised training on CS. \u201cRel.\u201d indicates mIoUUDA/mIoUSuperv.. mIoUUDA mIoUSuperv. Rel. DAFormer [32] 68.3 77.6 88.0% MIC (DAFormer) 70.6 77.9 90.6% Improvement +2.3 +0.3 +2.6% Component Ablation: To gain further insights, we ablate the components of MIC and evaluate the performance with DAFormer [32] (due to the faster training) on GTA\u2192CS in Tab. 7. The complete MIC achieves 70.6 mIoU (row 2), which is +2.3 mIoU better than DAFormer (row 1). First, the masking of the image is ablated, meaning that the consistency training is done with unmasked but still augmented target images (see \u201cMIC Parameters\u201d in Sec. 4.1). Without masking out image patches, the performance heavily decreases by -20.0 mIoU (cf. rows 2 and 3). On the other side, color augmentation is not essential for MIC as its ablation only reduces the performance by -0.3 mIoU (cf. rows 2 and 4). This demonstrates the importance of context learning with masked images. Replacing the EMA predictions with the regular model predictions decreases the performance of MIC by -0.7 mIoU (cf. rows 2 and 5). Without the pseudolabel confidence loss weight, the mIoU drops by -1.6 (cf. rows 2 and 6) showing that it is important to reduce the weight of uncertain samples for MIC training. Patch Size and Mask Ratio: Tab. 8 shows the influence of the mask patch size b and mask ratio r. Compared to DAFormer, MIC achieves significant improvements in a range of b between 64 and 128 and r between 0.5 and 0.7. The best performance is achieved for b=64 and r=0.7. Only for a very large b of 256, which is a quarter of the image height, MIC decreases the performance. Note that b is internally divided by 2 as DAFormer uses half resolution. MIC for Supervised Training: We compare the UDA and the supervised performance of DAFormer with and without Table 10. Runtime and memory consumption during training and inference on an RTX 2080 Ti (row 1-4) or Titan RTX (row 5-6). Training Inference Throughput GPU Memory Throughput GPU Memory Adversarial [84] 1.40 it/s 5.38 GB 11.2 img/s 0.5 GB MIC (Adversarial) 0.81 it/s 5.55 GB 11.2 img/s 0.5 GB DAFormer [32] 0.71 it/s 9.64 GB 8.6 img/s 1.0 GB MIC (DAFormer) 0.57 it/s 9.74 GB 8.6 img/s 1.0 GB HRDA [33] 0.36 it/s 22.46 GB 0.8 img/s 9.4 GB MIC (HRDA) 0.29 it/s 22.55 GB 0.8 img/s 9.4 GB MIC in Tab. 9. Also for supervised training, MIC achieves a slight improvement of +0.3 mIoU. However, the improvement for UDA is much more significant with +2.3 mIoU, showing that MIC is particularly useful for UDA. Therefore, MIC is able to increase the relative UDA performance (column \u201cRel\u201d) by +2.6 percent points, so that UDA with MIC achieves remarkable 90.6% of the performance of a network trained with full supervision on the target domain. Runtime/Memory: Tab. 10 shows the runtime and GPU memory footprint of representative UDA methods with and without MIC. For methods without an EMA teacher such as adversarial training, MIC reduces training speed by 75% due to the additional calculations for MIC and increases the GPU memory consumption by 3% due to the EMA teacher. The memory increase is small as the loss terms LS, LT , and LM are backpropagated separately. For UDA methods that already use an EMA teacher such as DAFormer or HRDA, the teacher and its predictions can be re-used, so that the training speed only increases by 24% and the memory footprint by 1%. Importantly, MIC is only used during training and does not increase the inference time at all. Supplement: The supplement provides results on CS\u2192 FoggyZurich, further results of MIC with DAFormer and HRDADeepLabV2, additional parameter and behavior studies, an extended qualitative analysis, and further discussions. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2111.14887v2",
+                    "title": "DAFormer: Improving Network Architectures and Training Strategies for Domain-Adaptive Semantic Segmentation",
+                    "abstract": "As acquiring pixel-wise annotations of real-world images for semantic\nsegmentation is a costly process, a model can instead be trained with more\naccessible synthetic data and adapted to real images without requiring their\nannotations. This process is studied in unsupervised domain adaptation (UDA).\nEven though a large number of methods propose new adaptation strategies, they\nare mostly based on outdated network architectures. As the influence of recent\nnetwork architectures has not been systematically studied, we first benchmark\ndifferent network architectures for UDA and newly reveal the potential of\nTransformers for UDA semantic segmentation. Based on the findings, we propose a\nnovel UDA method, DAFormer. The network architecture of DAFormer consists of a\nTransformer encoder and a multi-level context-aware feature fusion decoder. It\nis enabled by three simple but crucial training strategies to stabilize the\ntraining and to avoid overfitting to the source domain: While (1) Rare Class\nSampling on the source domain improves the quality of the pseudo-labels by\nmitigating the confirmation bias of self-training toward common classes, (2) a\nThing-Class ImageNet Feature Distance and (3) a learning rate warmup promote\nfeature transfer from ImageNet pretraining. DAFormer represents a major advance\nin UDA. It improves the state of the art by 10.8 mIoU for GTA-to-Cityscapes and\n5.4 mIoU for Synthia-to-Cityscapes and enables learning even difficult classes\nsuch as train, bus, and truck well. The implementation is available at\nhttps://github.com/lhoyer/DAFormer.",
+                    "authors": "Lukas Hoyer, Dengxin Dai, Luc Van Gool",
+                    "published": "2021-11-29",
+                    "updated": "2022-03-29",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction In the last few years, neural networks have achieved overwhelming performance on many computer vision tasks. However, they require a large amount of annotated data in order to be trained properly. For semantic segmentation, annotations are particularly costly as every pixel has to be labeled. For instance, it takes 1.5 hours to annotate a single 2018 2019 2020 2021 Date of Publication 45 50 55 60 65 mIoU AdaptSegNet CBST BDL MRKLD FDA IAST DACS CAG CAMix CTF CorDA ProDA DAFormer Network Architecture DeepLabV2, ResNet-101 DeepLabV2, ResNet-38 DeepLabV3+, ResNet-101 DAFormer, MiT-B5 Figure 1. Progress of UDA over time on GTA\u2192Cityscapes. Most previous UDA methods are evaluated using the outdated DeepLabV2 architecture. We rethink the design of the network architecture as well as its training strategies for UDA and propose DAFormer, signi\ufb01cantly outperforming previous methods. image of Cityscapes [13] while, for adverse weather conditions, it is even 3.3 hours [62]. One idea to circumvent this issue is training with synthetic data [59,61]. However, commonly used CNNs [39] are sensitive to domain shift and generalize poorly from synthetic to real data. This issue is addressed in unsupervised domain adaptation (UDA) by adapting the network trained with source (synthetic) data to target (real) data without access to target labels. Previous UDA methods mostly evaluated their contributions using a DeepLabV2 [6] or FCN8s [48] network architecture with ResNet [25] or VGG [64] backbone in order to be comparable to previously published works. However, even their strongest architecture (DeepLabV2+ResNet101) is outdated in the area of supervised semantic segmentation. For instance, it only achieves a supervised performance of 65 mIoU [72] on Cityscapes while recent networks reach up to 85 mIoU [68,91]. Due to the large performance gap, it stands to question whether using outdated network architectures can limit the overall performance of UDA and can also misguide the benchmark progress of UDA. In order to answer this question, this work studies the in\ufb02uence of the network architecture for UDA, compiles a more sophisticated architecture, and successfully applies it to UDA with a few simple, yet crucial training strategies. Naively using a 1 arXiv:2111.14887v2  [cs.CV]  29 Mar 2022 \fmore powerful network architecture for UDA might be suboptimal as it can be more prone to over\ufb01tting to the source domain. Based on a study of different semantic segmentation architectures evaluated in a UDA setting, we compile DAFormer, a network architecture tailored for UDA (Sec 3.2). It is based on recent Transformers [15,87], which have been shown to be more robust than the predominant CNNs [3]. We combine them with a context-aware multilevel feature fusion, which further enhances the UDA performance. To the best of our knowledge, DAFormer is the \ufb01rst work to reveal the signi\ufb01cant potential of Transformers for UDA semantic segmentation. Since more complex and capable architectures are more prone to adaptation instability and over\ufb01tting to the source domain, in this work, we introduce three training strategies to UDA to address these issues (Sec. 3.3). First, we propose Rare Class Sampling (RCS) to consider the long-tail distribution of the source domain, which hinders the learning of rare classes, especially in UDA due to the con\ufb01rmation bias of self-training toward common classes. By frequently sampling images with rare classes, the network can learn them more stably, which improves the quality of pseudo-labels and reduces the con\ufb01rmation bias. Second, we propose a Thing-Class ImageNet Feature Distance (FD), which distills knowledge from diverse and expressive ImageNet features in order to regularize the source training. This is particularly helpful as the source domain is limited to only a few instances of certain classes (low diversity), which have a different appearance than the target domain (domain shift). Without FD this would result in learning less expressive and source-domain-speci\ufb01c features. As ImageNet features were trained for thing-classes, we restrict the FD to regions of the image that are labeled as a thingclass. And third, we introduce learning rate warmup [23] newly to UDA. By linearly increasing the learning rate up to the intended value in the early training, the learning process is stabilized and features from ImageNet pretraining can be better transferred to semantic segmentation. DAFormer outperforms previous methods by a large margin (see Fig. 1) supporting our hypothesis that the network architecture and appropriate training strategies play an important role for UDA. On GTA\u2192Cityscapes, we improve the mIoU from 57.5 [94] to 68.3 and on Synthia\u2192Cityscapes from 55.5 [94] to 60.9. In particular, DAFormer learns even dif\ufb01cult classes that previous methods struggled with. For instance, we improve the class train from 16 to 65 IoU, truck from 49 to 75 IoU, and bus from 59 to 78 IoU on GTA\u2192Cityscapes. Overall, DAFormer represents a major advance in UDA. Our framework can be trained in one stage on a single consumer RTX 2080 Ti GPU within 16 hours, which simpli\ufb01es its usage compared to previous methods such as ProDA [94], which requires training multiple stages on four V100 GPUs for several days. 2. Related Work Semantic Image Segmentation Since the introduction of Convolutional Neural Networks (CNNs) [39] for semantic segmentation by Long et al. [48], they have been dominating the \ufb01eld. Typically, semantic segmentation networks follow an encoder-decoder design [2, 48, 60]. To overcome the problem of the low spatial resolution at the bottleneck, remedies such as skip connections [60], dilated convolutions [5,90], or resolution preserving architectures [66] were proposed. Further improvements were achieved by harnessing context information, for instance using pyramid pooling [6, 7, 34, 96] or attention modules [18, 35, 82, 91]. Inspired by the success of the attention-based Transformers [75] in natural language processing, they were adapted to image classi\ufb01cation [15, 70] and semantic segmentation [47, 87, 97] achieving state-of-the-art results. For image classi\ufb01cation, CNNs were shown to be sensitive to distribution shifts such as image corruptions [28], adversarial noise [67], or domain shifts [27]. Recent works [3, 55, 57] show that Transformers are more robust than CNNs with respect to these properties. While CNNs focus on textures [20], Transformers put more importance on the object shape [3, 55], which is more similar to human vision [20]. For semantic segmentation, ASPP [7] and skip connections [60] were reported to increase the robustness [36]. Further, Xie et al. [87] showed that their Transformer-based architecture improves the robustness over CNN-based networks. To the best of our knowledge, the in\ufb02uence of recent network architectures on the UDA performance of semantic segmentation has not been systematically studied yet. Unsupervised Domain Adaptation (UDA) UDA methods can be grouped into adversarial training and selftraining approaches. Adversarial training methods aim to align the distributions of source and target domain at input [21, 30], feature [31, 72], output [72, 76], or patch level [73] in a GAN framework [19, 22]. Using multiple scales [9, 72] or category information [16, 50, 84] for the discriminator can re\ufb01ne the alignment. In self-training, the network is trained with pseudo-labels [40] for the target domain. Most of the UDA methods pre-compute the pseudolabels of\ufb02ine, train the model, and repeat the process [14, 89,100,101]. Alternatively, pseudo-labels can be calculated online during the training. In order to avoid training instabilities, pseudo-label prototypes [94] or consistency regularization [65, 69] based on data augmentation [1, 10, 54] or domain-mixup [71, 99] are used. Several methods also combine adversarial and self-training [38,42,78], train with auxiliary tasks [33,77,79], or perform test-time UDA [80]. Datasets are often imbalanced and follow a longtail distribution, which biases models toward common classes [83]. Strategies to address this problem are re2 \fsampling [24,26,85], loss re-weighting [44,63], and transfer learning [37,45]. Also in UDA, re-weighting [53,100] and class-balanced sampling for image classi\ufb01cation [58] were applied. We extend class-balanced sampling from classi\ufb01cation to semantic segmentation and propose Rare Class Sampling, which addresses the co-occurrence of rare and common classes in a single semantic segmentation sample. Further, we demonstrate that re-sampling is particularly effective to train Transformers for UDA. Li et al. [43] have shown that knowledge distillation [29] from an old task can act as a regularizer for a new task. This concept was successfully deployed with ImageNet features to semi-supervised learning [32] and adversarial UDA [9]. We apply this idea to self-training, show that it is particularly bene\ufb01cial for Transformers, and improve it by restricting the Feature Distance to image regions with thingclasses [4] as ImageNet mostly labels thing-classes. 3. Methods 3.1. Self-Training (ST) for UDA First, we will give an overview over our baseline UDA method for evaluating different network architectures. In UDA, a neural network g\u03b8 is trained using source domain images XS = {x(i) S }NS i=1 and one-hot labels YS = {y(i) S }NS i=1 in order to achieve a good performance on target images XT = {x(i) T }NT i=1 without having access to the target labels YT . Naively training the network g\u03b8 with a categorical cross-entropy (CE) loss on the source domain L(i) S = \u2212 H\u00d7W X j=1 C X c=1 y(i,j,c) S log g\u03b8(x(i) S )(j,c) (1) usually results in a low performance on target images as the network does not generalize well to the target domain. To address the domain gap, several strategies have been proposed that can be grouped into adversarial training [31, 72, 78] and self-training (ST) [71, 94, 100] approaches. In this work, we use ST as adversarial training is known to be less stable and is currently outperformed by ST methods [71, 94]. To better transfer the knowledge from the source to the target domain, ST approaches use a teacher network h\u03c6 (which we will describe later) to produce pseudo-labels for the target domain data p(i,j,c) T = [c = arg max c\u2032 h\u03c6(x(i) T )(j,c\u2032)] , (2) where [\u00b7] denotes the Iverson bracket. Note that no gradients will be backpropagated into the teacher network. Additionally, a quality / con\ufb01dence estimate is produced for the pseudo-labels. Here, we use the ratio of pixels exceeding a threshold \u03c4 of the maximum softmax probability [71] q(i) T = PH\u00d7W j=1 [maxc\u2032 h\u03c6(x(i) T )(j,c\u2032) > \u03c4] H \u00b7 W . (3) The pseudo-labels and their quality estimates are used to additionally train the network g\u03b8 on the target domain L(i) T = \u2212 H\u00d7W X j=1 C X c=1 q(i) T p(i,j,c) T log g\u03b8(x(i) T )(j,c) . (4) The pseudo-labels can be generated either online [1, 71, 99] or of\ufb02ine [89, 100, 101]. We opted for online ST due to its less complex setup with only one training stage. This is important as we compare and ablate various network architectures. In online ST, h\u03c6 is updated based on g\u03b8 during the training. Commonly, the weights h\u03c6 are set as the exponentially moving average of the weights of g\u03b8 after each training step t [69] to increase the stability of the predictions \u03c6t+1 \u2190\u03b1\u03c6t + (1 \u2212\u03b1)\u03b8t . (5) ST has been shown to be particularly ef\ufb01cient if the student network g\u03b8 is trained on augmented target data, while the teacher network h\u03c6 generates the pseudo-labels using non-augmented target data for semi-supervised learning [17,65,69] and unsupervised domain adaptation [1,71] In this work, we follow DACS [71] and use color jitter, Gaussian blur, and ClassMix [56] as data augmentations to learn more domain-robust features. 3.2. DAFormer Network Architecture Previous UDA methods mostly evaluate their contributions using a (simpli\ufb01ed) DeepLabV2 network architecture [6, 72], which is considered to be outdated. For that reason, we compile a network architecture that is tailored for UDA to not just achieve good supervised performance but also provide good domain-adaptation capabilities. For the encoder, we aim for a powerful yet robust network architecture. We hypothesize that robustness is an important property in order to achieve good domain adaptation performance as it fosters the learning of domain-invariant features. Based on recent \ufb01ndings [3, 55, 57] and an architecture comparison for UDA, which we will present in Sec. 4.2, Transformers [15,70] are a good choice for UDA as they ful\ufb01ll these criteria. Although the self-attention from Transformers [75] and the convolution both perform a weighted sum, their weights are computed differently: in CNNs, the weights are learned during training but \ufb01xed during testing; in the self-attention mechanism, the weights are dynamically computed based on the similarity or af\ufb01nity between every pair of tokens. As a consequence, the selfsimilarity operation in the self-attention mechanism provides modeling means that are potentially more adaptive and general than convolution operations. In particular, we follow the design of Mix Transformers (MiT) [87], which are tailored for semantic segmentation. The image is divided into small patches of a size of 4\u00d74 (instead of 16\u00d716 as in ViT [15]) in order to preserve 3 \f    Source Data Target Data P(c) c rare common Student Net yS xS xT Teacher Net Augm. ImageNet  Encoder LS LT EMA Update Stop Gradient Hierarchical Transformer Size & Channel Alignment Thing-Class Feat. Dist. Rare Class Sampling Context-Aware Fusion a) b) F4 F3 F2 F1 Stack Figure 2. Overview of our UDA framework with Rare Class Sampling, Thing-Class Feature Distance, and DAFormer network. details for semantic segmentation. To cope with the high feature resolution, sequence reduction [81] is used in the self-attention blocks. The transformer encoder is designed to produce multi-level feature maps Fi \u2208R H 2i+1 \u00d7 W 2i+1 \u00d7Ci. The downsampling of the feature maps is implemented by overlapping patch merging [87] to preserve local continuity. Previous works on semantic segmentation with Transformer backbones usually exploit only local information for the decoder [81, 87, 97]. In contrast, we propose to utilize additional context information in the decoder as this has been shown to increase the robustness of semantic segmentation [36], a helpful property for UDA. Instead of just considering the context information of the bottleneck features [6,7], DAFormer uses the context across features from different encoder levels as the additional earlier features provide valuable low-level concepts for semantic segmentation at a high resolution, which can also provide important context information. The architecture of the DAFormer decoder is shown in Fig. 2 (b). Before the feature fusion, we embed each Fi to the same number of channels Ce by a 1\u00d71 convolution, bilinearly upsample the features to the size of F1, and concatenate them. For the context-aware feature fusion, we use multiple parallel 3\u00d73 depthwise separable convolutions [11] with different dilation rates [90] and a 1\u00d71 convolution to fuse them, similar to ASPP [7] but without global average pooling. In contrast to the original use of ASPP [7], we do not only apply it to the bottleneck features F4 but use it to fuse all stacked multi-level features. Depthwise separable convolutions have the advantage that they have a lower number of parameters than regular convolutions, which can reduce over\ufb01tting to the source domain. 3.3. Training Strategies for UDA One challenge of training a more capable architecture for UDA is over\ufb01tting to the source domain. To circumvent this issue, we introduce three strategies to stabilize and regularize the UDA training: Rare Class Sampling, Thing-Class ImageNet Feature Distance, and learning rate warmup. The overall UDA framework is shown in Fig. 2 (a). Rare Class Sampling (RCS) Even though our more capable DAFormer is able to achieve better performance on dif\ufb01cult classes than other architectures, we observed that the UDA performance for classes that are rare in the source dataset varies signi\ufb01cantly over different runs. Depending on the random seed of the data sampling order, these classes are learned at different iterations of the training or sometimes not at all as we will show in Sec. 4.4. The later a certain class is learned during the training, the worse is its performance at the end of the training. We hypothesize that if relevant samples containing rare classes only appear late in the training due to randomness, the network only starts to learn them later, and more importantly, it is highly possible that the network has already learned a strong bias toward common classes making it dif\ufb01cult to \u2018re-learn\u2019 new concepts with very few samples. This is further reinforced as the bias is con\ufb01rmed by ST with the teacher network. To address this, we propose Rare Class Sampling (RCS). It samples images with rare classes from the source domain more often in order to learn them better and earlier. The frequency fc of each class c in the source dataset can be calculated based on the number of pixels with class c fc = PNS i=1 PH\u00d7W j=1 [y(i,j,c) S ] NS \u00b7 H \u00b7 W . (6) The sampling probability P(c) of a certain class c is de\ufb01ned as a function of its frequency fc P(c) = e(1\u2212fc)/T PC c\u2032=1 e(1\u2212fc\u2032)/T . (7) Therefore, classes with a smaller frequency will have a higher sampling probability. The temperature T controls the smoothness of the distribution. A higher T leads to a more uniform distribution, a lower T to a stronger focus on rare classes with a small fc. For each source sample, a class is sampled from the probability distribution c \u223cP and an image is sampled from the subset of data containing this class xS \u223cuniform(XS,c). Eq. 7 allows to over-sample images containing rare classes (P(c) \u22651/C if fc is small). As a rare class (small fc) usually co-occurs with multiple common classes (large fc) in a single image, it is bene\ufb01cial to sample rare classes more often than common classes (P(crare) > P(ccommon)) to get closer to a balance of the 4 \fre-sampled classes. For example, the common class road co-occurs with rare classes such as bus, train, or motorcycle and is therefore already covered when sampling images with these rare classes. When decreasing T, more pixels of classes with small fc are sampled but also fewer pixels of classes with medium fc. The temperature T is chosen to reach a balance of the number of re-sampled pixels of classes with small and medium fc by maximizing the number of re-sampled pixels of the class with the least. Thing-Class ImageNet Feature Distance (FD) Commonly, the semantic segmentation model g\u03b8 is initialized with weights from ImageNet classi\ufb01cation to start with meaningful generic features. Given that ImageNet also contains real-world images from some of the relevant high-level semantic classes, which UDA often struggles to distinguish such as train or bus, we hypothesize that the ImageNet features can provide useful guidance beyond the usual pretraining. In particular, we observe that the DAFormer network is able to segment some of the classes at the beginning of the training but forgets them after a few hundred training steps as we will show in Sec. 4.5. Therefore, we assume that the useful features from ImageNet pretraining are corrupted by LS and the model over\ufb01ts to the synthetic source data. In order to prevent this issue, we regularize the model based on the Feature Distance (FD) of the bottleneck features F\u03b8 of the semantic segmentation UDA model g\u03b8 and the bottleneck features FImageNet of the ImageNet model d(i,j) = ||FImageNet(x(i) S )(j) \u2212F\u03b8(x(i) S )(j)||2 . (8) However, the ImageNet model is mostly trained on thing-classes (objects with a well-de\ufb01ned shape such as car or zebra) instead of stuff-classes (amorphous background regions such as road or sky) [4]. Therefore, we calculate the FD loss only for image regions containing thing-classes Cthings described by the binary mask Mthings L(i) FD = PHF \u00d7WF j=1 d(i,j) \u00b7 M (i,j) things P j M (i,j) things , (9) This mask is obtained from the downscaled label yS,small M (i,j) things = C X c\u2032=1 yi,j,c\u2032 S,small \u00b7 [c\u2032 \u2208Cthings] . (10) To downsample the label to the bottleneck feature size, average pooling with a patch size H HF \u00d7 W WF is applied to each class channel and a class is kept when it exceeds the ratio r yc S,small = [AvgPool(yc S, H/HF , W/WF ) > r] . (11) This ensures that only bottleneck feature pixels containing a dominant thing-class are considered for the feature distance. The overall UDA loss L is the weighted sum of the presented loss components L = LS + LT + \u03bbFDLFD. Learning Rate Warmup for UDA Linearly warming up the learning rate [23] at the beginning of the training has successfully been used to train both CNNs [25] and Transformers [15,75] as it improves network generalization [23] by avoiding that a large adaptive learning rate variance distorts the gradient distribution at the beginning of the training [46]. We newly introduce learning rate warmup to UDA. We posit that this is particularly important for UDA as distorting the features from ImageNet pretraining would deprive the network of useful guidance toward the real domain. During the warmup period up to iteration twarm, the learning rate at iteration t is set \u03b7t = \u03b7base \u00b7 t/twarm. 4. Experiments 4.1. Implementation Details Datasets For the target domain, we use the Cityscapes street scene dataset [13] containing 2975 training and 500 validation images with resolution 2048\u00d71024. For the source domain, we use either the GTA dataset [59], which contains 24,966 synthetic images with resolution 1914\u00d71052, or the Synthia dataset [61], which consists of 9,400 synthetic images with resolution 1280\u00d7760. As a common practice in UDA [72], we resize the images to 1024\u00d7512 pixels for Cityscapes and to 1280\u00d7720 pixels for GTA. Network Architecture Our implementation is based on the mmsegmentation framework [12]. For the DAFormer architecture, we use the MiT-B5 encoder [87], which produces a feature pyramid with C = [64, 128, 320, 512]. The DAFormer decoder uses Ce = 256 and dilation rates of 1, 6, 12, and 18. All encoders are pretrained on ImageNet-1k. Training In accordance with [47, 87], we train DAFormer with AdamW [49], a learning rate of \u03b7base=6\u00d710\u22125 for the encoder and 6\u00d710\u22124 for the decoder, a weight decay of 0.01, linear learning rate warmup with twarm=1.5k, and linear decay afterwards. It is trained on a batch of two 512\u00d7512 random crops for 40k iterations. Following DACS [71], we use the same data augmentation parameters and set \u03b1=0.99 and \u03c4=0.968. The RCS temperature is set T=0.01 to maximize the sampled pixels of the class with the least pixels. For FD, r=0.75 and \u03bbFD=0.005 to induce a similar gradient magnitude into the encoder as LS. 4.2. Comparison of Network Architectures for UDA First, we compare several semantic segmentation architectures with respect to their UDA performance (see Sec. 3.1) on GTA\u2192Cityscapes in Tab. 1. Additionally, we also provide the performance of the networks trained only with augmented source data (domain generalization) as well as the oracle performance trained with target labels (supervised learning). In all cases, the model is evaluated on the Cityscapes validation set and the performance is provided as mIoU in %. To compare how well a network is suited for 5 \fTable 1. Comparison of the mIoU (%) on the Cityscapes val. set of different segmentation architectures for source-only (GTA), UDA (GTA\u2192Cityscapes), and oracle (Cityscapes) training. Additionally, the relative UDA performance (Rel.) wrt. the oracle mIoU is provided. Mean and SD are calculated over 3 random seeds. Architecture Src-Only UDA Oracle Rel. DeepLabV2 [6] 34.3 \u00b12.2 54.2 \u00b11.7 72.1 \u00b10.5 75.2% DA Net [18] 30.9 \u00b12.1 53.7 \u00b10.2 72.6 \u00b10.2 74.0% ISA Net [35] 32.3 \u00b12.1 53.3 \u00b10.4 72.0 \u00b10.5 74.0% DeepLabV3+ [7] 31.0 \u00b11.4 53.7 \u00b11.0 75.6 \u00b10.9 71.0% SegFormer [87] 45.6 \u00b10.6 58.2 \u00b10.9 76.4 \u00b10.2 76.2% Table 2. Ablation of the SegFormer encoder and decoder. Encoder Decoder UDA Oracle Rel. MiT-B5 [87] SegF. [87] 58.2 \u00b10.9 76.4 \u00b10.2 76.2% MiT-B5 [87] DLv3+ [7] 56.8 \u00b11.8 75.5 \u00b10.5 75.2% R101 [25] SegF. [87] 50.9 \u00b11.1 71.3 \u00b11.3 71.4% R101 [25] DLv3+ [7] 53.7 \u00b11.0 75.6 \u00b10.9 71.0% UDA, we further provide the relative performance (Rel.), which normalizes the UDA mIoU by the oracle mIoU. Note that the oracle mIoU is generally lower than reported in the literature on supervised learning as for UDA the images of Cityscapes are downsampled by a factor of two, which is a necessary common practice in UDA to \ufb01t images from both domains and additional networks into the GPU memory. The majority of works on UDA use DeepLabV2 [6] with ResNet-101 [25] backbone. Interestingly, a higher oracle performance does not necessarily increase the UDA performance as can be seen for DeepLabV3+ [7] in Tab. 1. Generally, the studied more recent CNN architectures, do not provide a UDA performance gain over DeepLabV2. However, we identi\ufb01ed the Transformer-based SegFormer [87] as a powerful architecture for UDA. It increases the mIoU for source-only / UDA / oracle training signi\ufb01cantly from 34.3 / 54.2 / 72.1 to 45.6 / 58.2 / 76.4. We believe that especially the better domain generalization (source-only training) of SegFormer is valuable for the improved UDA performance. To get a better insight into why SegFormer works well for UDA, we swap its encoder and decoder with ResNet101 and DeepLabV3+. As the MiT encoder of SegFormer has an output stride of 32 but the DeepLabV3+ decoder is designed for an output stride of 8, we bilinearly upsample the SegFormer bottleneck features by \u00d74 when combined with the DeepLabv3+ decoder. Tab. 2 shows that the lightweight MLP decoder of SegFormer has a slightly higher relative UDA performance (Rel.) than the heavier DLv3+ decoder (76.2% vs 75.2%). However, the crucial contribution to good UDA performance comes from the Transformer MiT encoder. Replacing it with the ResNet101 encoder leads to a signi\ufb01cant performance drop of the UDA performance. Even though the oracle performance drops as well due to Table 3. In\ufb02uence of the encoder on UDA performance. Enc. Dec. Src-Only UDA Oracle Rel. R50 [25] DLv2 [6] 29.3 52.1 70.8 73.6% R101 [25] DLv2 [6] 36.9 53.3 72.5 73.5% S50 [93] DLv2 [6] 27.9 48.0 67.7 70.9% S101 [93] DLv2 [6] 35.5 53.5 72.2 74.1% S200 [93] DLv2 [6] 35.9 56.9 73.5 77.4% MiT-B3 [87] SegF. [87] 42.2 50.8 76.5 66.4% MiT-B4 [87] SegF. [87] 44.7 57.5 77.1 74.6% MiT-B5 [87] SegF. [87] 46.2 58.8 76.2 77.2% Table 4. In\ufb02uence of learning rate warmup on UDA performance. Architecture LR Warmup UDA Oracle Rel. DeepLabV2 [6] \u2013 49.1 \u00b12.0 67.4 \u00b11.7 72.8% DeepLabV2 [6] \u2713 54.2 \u00b11.7 72.1 \u00b10.5 75.2% SegFormer [87] \u2013 51.8 \u00b10.8 72.9 \u00b11.6 71.1% SegFormer [87] \u2713 58.2 \u00b10.9 76.4 \u00b10.2 76.2% ResNet101 ImageNet Features MiT-B5 ImageNet Features road sidew. build. wall fence pole tr.light sign veget. n/a. terrain sky person rider car truck bus train m.bike bike Figure 3. T-SNE [74] embedding of the bottleneck features after ImageNet pre-training for ResNet101 [25] and MiT-B5 [87] on the Cityscapes val. set, showing a better vehicle separability for MiT. the smaller receptive \ufb01eld of the ResNet encoder [87], the drop for UDA is over-proportional as shown by the relative performance decreasing from 76.2% to 71.4%. Therefore, we further investigate the in\ufb02uence of the encoder architecture on UDA performance. In Tab. 3, we compare different encoder designs and sizes. It can be seen that deeper models achieve a better source-only and relative performance demonstrating that deeper models generalize/adapt better to the new domain. This observation is in line with \ufb01ndings on the robustness of network architectures [3]. Compared to CNN encoders, the MiT encoders generalize better from source-only training to the target domain. Overall, the best UDA mIoU is achieved by the MiT-B5 encoder. To gain insights on the improved generalization, Fig. 3 visualizes the ImageNet features of the target domain. Even though ResNet structures stuff-classes slightly better, MiT shines at separating semantically similar classes (e.g. all vehicle classes), which are usually particularly dif\ufb01cult to adapt. A possible explanation might be the texture-bias of CNNs and the shape-bias of Transform6 \f10k 20k 30k 40k Iteration 0 10 20 30 40 IoU Rider 10k 20k 30k 40k Iteration 0 20 40 60 Bicycle Method w/o RCS w/ RCS Seed 0 1 2 Figure 4. SegFormer UDA performance for the rare classes rider and bicycle without and with Rare Class Sampling (RCS). 2k 4k 6k Iteration 0 20 40 60 IoU Bus 2k 4k 6k Iteration 0 20 40 60 Train Method w/o FD w/ FD Figure 5. SegFormer UDA performance in the beginning of the training with and without ImageNet Feature Distance (FD). Table 5. Ablation of the components of the UDA framework. Network Warmup RCS FD Misc. UDA 1 SegF. [87] \u2013 \u2013 \u2013 \u2013 51.8 \u00b10.8 2 SegF. [87] \u2713 \u2013 \u2013 \u2013 58.2 \u00b10.9 3 SegF. [87] \u2713 \u2713(T=\u221e) \u2013 \u2013 62.0 \u00b11.5 4 SegF. [87] \u2713 \u2713 \u2013 \u2013 64.0 \u00b12.4 5 SegF. [87] \u2713 \u2013 \u2713(all C) \u2013 58.8 \u00b10.4 6 SegF. [87] \u2713 \u2013 \u2713 \u2013 61.7 \u00b12.6 7 SegF. [87] \u2713 \u2713 \u2713 \u2013 66.2 \u00b11.0 8 SegF. [87] \u2713 \u2713 \u2713 Crop PL, \u03b1\u219167.0 \u00b10.4 9 DLv2 [6] \u2013 \u2013 \u2013 \u2013 49.1 \u00b12.0 10 DLv2 [6] \u2713 \u2713 \u2713 Crop PL, \u03b1\u219156.0 \u00b10.5 ers [3]. Before we study the context-aware fusion decoder of DAFormer in Sec. 4.6, we will \ufb01rst discuss how to stabilize the training of MiT with the default SegFormer decoder. 4.3. Learning Rate Warmup Tab. 4 shows that learning rate warmup signi\ufb01cantly improves both UDA and oracle performance. UDA bene\ufb01ts even more than supervised learning from warmup (see column Rel.), showing its particular importance for UDA by stabilizing the beginning of the training, which improves dif\ufb01cult classes (cf. row 1 and 2 in Fig 6). As warmup is essential for a good UDA performance across different architectures, it was already applied in the previous section. 4.4. Rare Class Sampling (RCS) When training SegFormer for UDA, we observe that the performance of some classes depends on the random seed for data sampling as can be seen for the blue IoU curves Road S.walk Build. Wall Fence Pole T.Light T.Sign Veget. Terrain Sky Person Rider Car Truck Bus Train M.bike Bike SegF. UDA +W +RCS +W +FD +W +RCS +FD +W 91 56 87 32 30 40 51 51 89 49 92 62 6 90 60 53 0 20 25 89 50 88 46 44 43 53 55 90 51 93 64 9 91 77 63 0 47 50 95 65 89 50 43 42 54 60 89 47 93 69 30 92 77 70 27 57 63 88 48 88 49 44 41 54 57 90 51 93 66 7 92 73 69 43 53 64 93 62 89 53 44 43 55 61 89 47 93 71 42 92 74 75 62 53 63 Figure 6. Comparison of the class-wise IoU of Warmup (W), RCS and FD. The color visualizes the IoU difference to the baseline. in Fig. 4. The affected classes are underrepresented in the source dataset as shown in the supplement. Interestingly, the IoU for the class bicycle starts increasing at different iterations for different seeds. We hypothesize that this is caused by the sampling order, in particular when the relevant rare classes are sampled. Further, the later the IoU starts improving, the worse is the \ufb01nal IoU of this class, probably due to the con\ufb01rmation bias of self-training that was accumulated over earlier iterations. Therefore, for UDA, it is especially important to learn rare classes early. In order to address this issue, the proposed RCS increases the sampling probability of rare classes. Fig. 4 (orange) shows that RCS results in an earlier increase of the IoU of rider/bicycle and a higher \ufb01nal IoU independent of the data sampling random seed. This con\ufb01rms our hypothesis that an (early) sampling of rare classes is important for learning these classes properly. RCS improves the UDA performance by +5.8 mIoU (cf. row 2 and 4 in Tab. 5). The highest IoU increase is observed for the rare classes rider, train, motorcycle, and bicycle (cf. row 2 and 3 in Fig. 6). RCS also outperforms its special case T = \u221e, which corresponds to \u2018class-balanced sampling\u2019 (cf. row 3 and 4 in Tab. 5), as class-balanced sampling does not consider the co-occurrence of multiple classes in semantic segmentation. 4.5. Thing-Class ImageNet Feature Distance (FD) While RCS gives a performance boost, the performance for thing-classes (e.g. bus and train) could still be further improved as some of the object classes that are fairly well separated in ImageNet features (see Fig. 3 right) are mixed together after the UDA training. When investigating the IoU during the early training (see Fig. 5 orange), we observe an early performance drop for the class train. We assume that the powerful MiT encoder over\ufb01ts to the synthetic domain. When regularizing the training with the proposed FD, the performance drop is avoided (see Fig. 5 green). Also other dif\ufb01cult classes such as bus, motorcycle, and bicycle bene\ufb01t from the regularization (cf. row 2 and 4 in Fig. 6). Overall the UDA performance is improved by +3.5 mIoU (cf. row 2 and 6 in Tab. 5). Note that applying FD only to thingclasses, which the ImageNet features were trained on, is important for its good performance (cf. row 5 and 6). When combining RCS and FD, we observe a further im7 \fTable 6. Comparison with state-of-the-art methods for UDA. The results for DAFormer are averaged over 3 random seeds. Road S.walk Build. Wall Fence Pole Tr.Light Sign Veget. Terrain Sky Person Rider Car Truck Bus Train M.bike Bike mIoU GTA5 \u2192Cityscapes CBST [100] 91.8 53.5 80.5 32.7 21.0 34.0 28.9 20.4 83.9 34.2 80.9 53.1 24.0 82.7 30.3 35.9 16.0 25.9 42.8 45.9 DACS [71] 89.9 39.7 87.9 30.7 39.5 38.5 46.4 52.8 88.0 44.0 88.8 67.2 35.8 84.5 45.7 50.2 0.0 27.3 34.0 52.1 CorDA [79] 94.7 63.1 87.6 30.7 40.6 40.2 47.8 51.6 87.6 47.0 89.7 66.7 35.9 90.2 48.9 57.5 0.0 39.8 56.0 56.6 ProDA [94] 87.8 56.0 79.7 46.3 44.8 45.6 53.5 53.5 88.6 45.2 82.1 70.7 39.2 88.8 45.5 59.4 1.0 48.9 56.4 57.5 DAFormer 95.7 70.2 89.4 53.5 48.1 49.6 55.8 59.4 89.9 47.9 92.5 72.2 44.7 92.3 74.5 78.2 65.1 55.9 61.8 68.3 Synthia \u2192Cityscapes CBST [100] 68.0 29.9 76.3 10.8 1.4 33.9 22.8 29.5 77.6 \u2013 78.3 60.6 28.3 81.6 \u2013 23.5 \u2013 18.8 39.8 42.6 DACS [71] 80.6 25.1 81.9 21.5 2.9 37.2 22.7 24.0 83.7 \u2013 90.8 67.6 38.3 82.9 \u2013 38.9 \u2013 28.5 47.6 48.3 CorDA [79] 93.3 61.6 85.3 19.6 5.1 37.8 36.6 42.8 84.9 \u2013 90.4 69.7 41.8 85.6 \u2013 38.4 \u2013 32.6 53.9 55.0 ProDA [94] 87.8 45.7 84.6 37.1 0.6 44.0 54.6 37.0 88.1 \u2013 84.4 74.2 24.3 88.2 \u2013 51.1 \u2013 40.5 45.6 55.5 DAFormer 84.5 40.7 88.4 41.5 6.5 50.0 55.0 54.6 86.0 \u2013 89.8 73.2 48.2 87.2 \u2013 53.2 \u2013 53.9 61.7 60.9 Table 7. Comparison of decoder architectures with MiT encoder and UDA improvements (DSC: depthwise separable convolution). Decoder Ce #Params UDA Oracle Rel. SegF. [87] 768 3.2M 67.0 \u00b10.4 76.8 \u00b10.3 87.2% SegF. [87] 256 0.5M 67.1 \u00b11.1 76.5 \u00b10.4 87.7% UperNet [86] 512 29.6M 67.4 \u00b11.1 78.0 \u00b10.2 86.4% UperNet [86] 256 8.3M 66.7 \u00b11.2 77.4 \u00b10.3 86.2% ISA [35] Fusion 256 1.1M 66.3 \u00b10.9 76.3 \u00b10.4 86.9% Context only at F4 256 3.2M 67.0 \u00b10.6 76.6 \u00b10.2 87.5% DAFormer w/o DSC 256 10.0M 67.0 \u00b11.5 76.7 \u00b10.6 87.4% DAFormer 256 3.7M 68.3 \u00b10.5 77.6 \u00b10.2 88.0% provement to 66.2 mIoU showing that they complement each other (see row 7 in Tab. 5). We notice pseudo-label drifts originating from image recti\ufb01cation artifacts and the ego car. As these regions are not part of the street scene segmentation task, we argue that it is not meaningful to produce pseudo-labels for them. Therefore, we ignore the top 15 and bottom 120 pixels of the pseudo-label. As Transformers are more expressive, we further increase \u03b1 to 0.999 to introduce a stronger regularization from the teacher. This mitigates the pseudo-label drifts and improves the UDA mIoU (cf. row 7 and 8 in Tab. 5). The overall improvement is +15.2 mIoU for SegFormer (cf. row 1 and 8) and +6.9 mIoU for DeepLabV2 (cf. row 9 and 10). When comparing both architectures, it can be seen that SegFormer bene\ufb01ts noticeably more, supporting our initial hypothesis that the architecture choice can limit the effectiveness of UDA methods. 4.6. DAFormer Decoder After regularizing and stabilizing the UDA training for a MiT encoder and a SegFormer decoder, we come back to the network architecture and investigate our DAFormer decoder with the context-aware feature fusion. Tab. 7 shows that it improves the UDA performance over the SegFormer decoder from 67.0 to 68.3 mIoU (cf. row 1 and 8). Further, DAFormer outperforms a variant without depthwise separable convolutions (cf. last two rows) and a variant with ISA [35] instead of ASPP for feature fusion (cf. row 5 and 8). This shows that a capable but parameter-effective decoder with an inductive bias of the dilated depthwise separable convolutions is bene\ufb01cial for good UDA performance. When the context is only considered for bottleneck features, the UDA performance decreases by -1.3 mIoU (cf. row 6 and 8), revealing that the context clues from different encoder stages used in DAFormer are more domain-robust. We further compare DAFormer to UperNet [86], which iteratively upsamples and fuses the features and was used together with Transformers in [47]. Even though UperNet achieves the best oracle performance, it is noticeably outperformed by DAFormer on UDA, which con\ufb01rms that it is necessary to study and design the decoder architecture, along with the encoder architecture, speci\ufb01cally for UDA. Tab. 6 shows that DAFormer outperforms previous methods by a signi\ufb01cant margin. On GTA\u2192Cityscapes, it improves the performance from 57.5 to 68.3 mIoU and on Synthia\u2192Cityscapes from 55.5 to 60.9 mIoU. In particular, DAFormer learns even dif\ufb01cult classes well, which previous methods struggled with such as train, bus, and truck. Further details are provided in the supplement, including RCS statistics, parameter sensitivity of RCS/FD, ablation of ST, a runtime and GPU memory benchmark, a comprehensive qualitative analysis, and a discussion of limitations. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2012.10782v2",
+                    "title": "Three Ways to Improve Semantic Segmentation with Self-Supervised Depth Estimation",
+                    "abstract": "Training deep networks for semantic segmentation requires large amounts of\nlabeled training data, which presents a major challenge in practice, as\nlabeling segmentation masks is a highly labor-intensive process. To address\nthis issue, we present a framework for semi-supervised semantic segmentation,\nwhich is enhanced by self-supervised monocular depth estimation from unlabeled\nimage sequences. In particular, we propose three key contributions: (1) We\ntransfer knowledge from features learned during self-supervised depth\nestimation to semantic segmentation, (2) we implement a strong data\naugmentation by blending images and labels using the geometry of the scene, and\n(3) we utilize the depth feature diversity as well as the level of difficulty\nof learning depth in a student-teacher framework to select the most useful\nsamples to be annotated for semantic segmentation. We validate the proposed\nmodel on the Cityscapes dataset, where all three modules demonstrate\nsignificant performance gains, and we achieve state-of-the-art results for\nsemi-supervised semantic segmentation. The implementation is available at\nhttps://github.com/lhoyer/improving_segmentation_with_selfsupervised_depth.",
+                    "authors": "Lukas Hoyer, Dengxin Dai, Yuhua Chen, Adrian K\u00f6ring, Suman Saha, Luc Van Gool",
+                    "published": "2020-12-19",
+                    "updated": "2021-04-05",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction Convolutional Neural Networks (CNNs) [35] have achieved state-of-the-art results for various computer vision tasks including semantic segmentation [40, 5]. However, training CNNs typically requires large-scale annotated datasets, due to millions of learnable parameters involved. Collecting such training data relies primarily on manual annotation. For semantic segmentation, the process can be particularly costly, due to the required dense annotations. For example, annotating a single image in the Cityscapes dataset took on average 1.5 hours [9]. Recently, self-supervised learning has shown to be a promising replacement for manually labeled data. It aims to learn representations from the structure of unlabeled data, instead of relying on a supervised loss, which involves manual labels. The principle has been successfully applied in depth estimation for stereo pairs [16] or image sequences [73]. Additionally, semantic segmentation is known to be tightly coupled with depth. Several works have reported that jointly learning segmentation and supervised depth estimation can bene\ufb01t the performance of both tasks [61]. Motivated by these observations, we investigate the question: How can we leverage self-supervised depth estimation to improve semantic segmentation? In this work, we propose a threefold approach to utilize self-supervised monocular depth estimation (SDE) [16, 73, 17] to improve the performance of semantic segmentation and to reduce the amount of annotation needed. Our contributions span across the holistic learning process from data selection, over data augmentation, up to cross-task representation learning, while being uni\ufb01ed by the use of SDE. First, we employ SDE as an auxiliary task for semantic image segmentation under a transfer learning and multitask learning framework and show that it noticeably improves the performance of semantic segmentation, especially when supervision is limited. Previous works only cover full supervision [32], pretraining [26], or improving SDE instead of segmentation [20]. Second, we propose a strong data augmentation strategy, DepthMix, which blends images as well as their labels according to the geometry of the scenes obtained from SDE. In comparison to previous methods [69, 47], DepthMix explicitly respects the geometric structure of the scenes and generates fewer artifacts (see Fig. 1). And third, we propose an Automatic Data Selection for Annotation, which selects the most useful samples to be 1 arXiv:2012.10782v2  [cs.CV]  5 Apr 2021 \fannotated in order to maximize the gain. The selection is iteratively driven by two criteria: diversity and uncertainty. Both of them are conducted by a novel use of SDE as proxy task in this context. While our method follows the active learning cycle (model training \u2192query selection \u2192annotation \u2192model training) [53, 66], it does not require a human in the loop to provide semantic segmentation labels as the human is replaced by a proxy-task SDE oracle. This greatly improves \ufb02exibility, scalability, and ef\ufb01ciency, especially considering crowdsourcing platforms for annotation. The main advantage of our method is that we can learn from a large base of easily accessible unlabeled image sequences and utilize the learned knowledge to improve semantic segmentation performance in various ways. In our experimental evaluation on Cityscapes [9], we demonstrate signi\ufb01cant performance gains of all three components and improve the previous state-of-the-art for semi-supervised segmentation by a considerable margin. Speci\ufb01cally, our method achieves 92% of the full annotation baseline performance with only 1/30 available labels and even slightly outperforms it with only 1/8 labels. Our contributions summarize as follows: (1) To the best of our knowledge, we are the \ufb01rst to utilize SDE as an auxiliary task to exploit unlabeled image sequences and signi\ufb01cantly improve the performance of semi-supervised semantic segmentation. (2) We propose DepthMix, a strong data augmentation strategy, which respects the geometry of the scene and achieves, in combination with (1), state-of-the-art results for semi-supervised semantic segmentation. (3) We propose a novel Automatic Data Selection for Annotation based on SDE to improve the \ufb02exibility of active learning. It replaces the human annotator with an SDE oracle and lifts the requirement of having a human in the loop of data selection. 2. Related Work 2.1. (Semi-Supervised) Semantic Segmentation Since Convolutional Neural Networks (CNNs) [35] were \ufb01rst used by Long et al. [40] for semantic segmentation, they have become the state-of-the-art method for this problem. Most architectures are based on an encoder decoder design such as [40, 51, 6]. Skip connections [51] and dilated convolutions [4, 67] preserve details in the segmentation and spatial pyramid pooling [15, 71, 5] aggregates different scales to exploit spatial context information. Semi-supervised semantic segmentation makes use of additional unlabeled data during training. For that purpose, Souly et al. [59] and Hung et al. [23] utilize generative adversarial networks [18]. Souly et al. [59] use that concept to generate additional training samples, while Hung et al. [23] train the discriminator based on the semantic segmentation probability maps. s4GAN [43] extends this idea by adding a multi-label classi\ufb01cation mean teacher [60]. Another line of work [48, 12, 47] is based on consistency training, where perturbations are applied to unlabeled images or their intermediate features and a loss term enforces consistency of the segmentation. While Ouali et al. [48] study perturbation of encoder features, CutMix [12] mixes crops from the input images and their pseudo-labels to generate additional training data, and ClassMix [47] uses pseudo-label [36] class segments to build the mix mask. Our proposed DepthMix module is inspired by these methods but, in contrast, it also respects the structure of the scene when mixing samples. Commonly, several approaches [43, 12, 47, 11] include selftraining with pseudo-labels [36] and a mean teacher framework [60], which is extended by Feng et al. [11] with a class-balanced curriculum. Another related line of work is learning useful representations for semantic segmentation from self-supervised tasks such as tracking [63], context inpainting [49], colorization [34], depth estimation [26] (see Section 2.3), or optical \ufb02ow prediction [37]. However, all of these approaches are outperformed by ImageNet pretraining and are, therefore, not relevant for semi-supervised semantic segmentation in practice. 2.2. Active Learning Another approach to reduce the number of required annotations is active learning. It iteratively requests the most informative samples to be labeled by a human. On the one side, uncertainty-based approaches select samples with a high uncertainty estimated based on, e.g., entropy [24, 54] or ensemble disagreement [55, 42]. On the other side, diversity-based approaches select samples, which most increase the diversity of the labeled set [44, 52, 58]. For segmentation, active learning is typically based on uncertainty measures such as MC dropout [13, 66, 41], entropy [29, 64], or multi-view consistency [57]. In addition to methods selecting whole images [19, 66, 64], several approaches apply a more \ufb01ne-grained label request at region level [41, 29, 57] and also include a label cost estimate [41, 29]. In contrast to these works, we perform automatic data selection for annotation by replacing the human with SDE as oracle. Therefore, we do not require human-in-theloop annotation during the active learning cycle. Previous works performing unsupervised data selection are restricted to shallow models [68, 70, 45, 22, 56, 39], classi\ufb01cation with low-dimensional inputs [38], or do not perform an iterative data selection [72] to dynamically adapt to the uncertainty of the model trained on the currently labeled set. 2.3. Improving Segmentation with SDE Self-supervised depth estimation (SDE) aims to learn depth estimation from the geometric relations of stereo im2 \fage pairs [14, 16] or monocular videos [73]. Due to the better availability of videos, we use the latter approach, where a neural network estimates depth and camera motion of two subsequent images and a photometric loss is computed after a differentiable warping. The approach has been improved by several follow-up works [17, 8, 74]. The combination of semantic segmentation and SDE was studied in previous works with the goal of improving depth estimation. While [50, 28, 7, 32] learn both tasks jointly, [3, 20, 27] distill knowledge from a teacher semantic segmentation network to guide SDE. To further utilize coherence between semantic segmentation and SDE, [50, 7] proposed additional loss terms that encourage spatial proximity between depth discontinuities and segmentation contours. In contrast to these works, we do not aim to improve SDE but rather semi-supervised semantic segmentation. The closest to our approach are [26], [46], and [32]. Jiang et al. [26] utilizes relative depth computed from optical \ufb02ow to replace ImageNet pretraining for semantic segmentation. In contrast, we additionally study multi-task learning of SDE and semantic segmentation and show that combining SDE with ImageNet features can even further boost performance. Novosel et al. [46] and Klingner et al. [32] improve the semantic segmentation performance by jointly learning SDE. However, they focus on the fully-supervised setting, while our work explicitly addresses the challenges of semisupervised semantic segmentation by using the depth estimates to generate additional training data and an automatic data selection mechanism based on SDE. Another work supporting the usefulness of SDE for semantic segmentation from another viewpoint is [31] demonstrating an improved noise and attack robustness. 3. Methods In this section, we present our three ways to improve the performance of semantic segmentation with self-supervised depth estimation (SDE). They focus on three different aspects of semantic segmentation, covering data selection for annotation, data augmentation, and multi-task learning. Given N images and K image sequences from the same domain, our \ufb01rst method, Automatic Data Selection for Annotation, uses SDE learned on the K (unlabeled) sequences to select NA images out of the N images for human annotation (see Alg. 1). Our second approach, termed DepthMix, leverages the learned SDE to create geometrically-sound \u2018virtual\u2019 training samples from pairs of labeled images and their annotations (see Fig. 1). Our third method learns semantic segmentation with SDE as an auxiliary task under a multi-tasking framework (see Fig. 2). The learning is reinforced by a multi-task pretraining process combining SDE with image classi\ufb01cation. For SDE, we follow the method of Godard et al. [17], which we brie\ufb02y introduce in the following. We \ufb01rst train a depth estimation network fD to predict the depth of a target image and a pose estimation network fT to estimate the camera motion from the target image and the source image. Depth and pose are used to produce a differentiable warping to transform the source image into the target image. The photometric error between the target image and multiple warped source frames is combined by a pixel-wise minimum. Besides, stationary pixels are masked out and an edge-aware depth smoothness term is applied resulting in the \ufb01nal self-supervised depth loss LD. We refer the reader to the original paper [17] for more details. 3.1. Automatic Data Selection for Annotation We use SDE as proxy task for selecting NA samples out of a set of N unlabeled samples for a human to create semantic segmentation labels. The selection is conducted progressively in multiple steps, similar to the standard active learning cycle (model training \u2192query selection \u2192annotation \u2192model training). However, our data selection is fully automatic and does not require a human in the loop as the annotation is done by a proxy-task SDE oracle. Let\u2019s denote by G, GA, and GU, the whole image set, the selected sub-set for annotation, and the un-selected sub-set. Initially, we have GA = \u2205and GU = G. The selection is driven by two criteria: diversity and uncertainty. Diversity sampling encourages that selected images are diverse and cover different scenes. Uncertainty sampling favors adding unlabeled images that are near a decision boundary (with high uncertainties) of the model trained on the current GA. For uncertainty sampling, we need to train and update the model with GA. It is inef\ufb01cient to repeat this every time a new image is added. For the sake of ef\ufb01ciency, we divide the selection into T steps and only train the model T times. In each step t, nt images are selected and moved from GU to GA, so we have PT t=1 nt = NA. After each step t, a model is trained on GA and evaluated on GU to get updated uncertainties for step t + 1. Diversity Sampling: To ensure that the chosen annotated samples are diverse enough to represent the entire dataset well, we use an iterative farthest point sampling based on the L2 distance over features \u03a6SDE computed by an intermediate layer of the SDE network. At step t, for each of the nt samples, we choose the one in GU with the largest distance to the current annotation set GA. The set of selected samples GA is iteratively extended by moving one image at a time from GU to GA until the nt images are collected: GU = GU \\ {Ii} and GA = GA \u222a{Ii}, (1) i = arg max Ii\u2208GU min Ij\u2208GA ||\u03a6SDE i \u2212\u03a6SDE j ||2. (2) Uncertainty Sampling: While Diversity Sampling is able to select diverse new samples, it is unaware of the uncertainties of a semantic segmentation model over these samples. Uncertainty Sampling aims to select dif\ufb01cult samples, 3 \fAlgorithm 1: Automatic Data Selection 1: t = 1 2: i \u2190uniform(1, N) 3: GA = {Ii} and GU = GU \\ {Ii} 4: for k = 2 to NA do 5: if k == Pt t\u2032=1 nt\u2032 then 6: Train depth student \u03a6SIDE on the current GA 7: Calculate E(i) \u2200Ii \u2208GU 8: t = t + 1 9: end if 10: if t == 1 then 11: Obtain index i according to Eq. 2 12: else 13: Obtain index i according to Eq. 4 14: end if 15: GA = GA \u222a{Ii} and GU = GU \\ {Ii} 16: end for i.e., samples in GU that the model trained on the current GA cannot handle well. In order to train this model, active learning typically uses a human-in-the-loop strategy to add annotations for selected samples. In this work, we use a proxy task based on self-supervised annotations, which can run automatically, to make the method more \ufb02exible and ef\ufb01cient. Since our target task is single-image semantic segmentation, we choose to use single-image depth estimation (SIDE) as the proxy task. Importantly, due to our SDE framework, depth pseudo-labels are available for G. Using these pseudo-labels, we train a SIDE method on GA and measure the uncertainty of its depth predictions on GU. Due to the high correlation of single-image semantic segmentation and SIDE, the generated uncertainties are informative and can be used to guide our sampling procedure. As the depth student model is trained only on GA, it can speci\ufb01cally approximate the dif\ufb01culty of candidate samples with respect to the already selected samples in GA. The student is trained from scratch in each step t, instead of being \ufb01netuned from t\u22121, to avoid getting stuck in the previous local minimum. Note that the SDE method is trained on a much larger unlabeled dataset, i.e., the K image sequences, and can provide good guidance for the SIDE method. In particular, the uncertainty is signaled by the disparity error between the student network fSIDE and the teacher network fSDE in the log-scale space under L1 distance: E(i) = || log(1 + fSDE(Ii)) \u2212log(1 + fSIDE(Ii))||1. (3) As the disparity difference of far-away objects is small, the log-scale is used to avoid the loss being dominated by closerange objects. This criterion can be added into Eq. 2 to also select samples with higher uncertainties for the dataset     Random Class Choice Depth Comparison ClassMix (Baseline) DepthMix (Ours) Si Sj MD\u2299Si MCl\u2299Si S\u2018Cl S\u2018D Figure 1. Concept of the proposed DepthMix augmentation (refer to Sec. 3.2) and its baseline ClassMix [47]. By utilizing SDE, DepthMix mitigates geometric artifacts. update in Eq. 1: i = arg max Ii\u2208GU min Ij\u2208GA ||\u03a6SDE i \u2212\u03a6SDE j ||2 + \u03bbEE(i), (4) where \u03bbE is a parameter to balance the contribution of the two terms. For diversity sampling, we still use SDE features instead of SIDE student features as SDE is trained on the entire dataset, which provides better features for diversity estimation. When nt images have been selected according to Eq. 1 and Eq. 4 at step t, a new SIDE model will be trained on the current GA in order to continue further. As presented previously, our selection proceeds progressively in T steps until we collect all NA images. The algorithm of this selection is summarized in Alg. 1, where Pt t\u2032=1 nt\u2032 describes the desired size of GA at the end of step t. 3.2. DepthMix Data Augmentation Inspired by the recent success of data augmentation approaches that mixup pairs of images and their (pseudo) labels to generate more training samples for semantic segmentation [69, 12, 47], we propose an algorithm, termed DepthMix, to utilize self-supervised depth estimates to maintain the integrity of the scene structure during mixing. Given two images Ii and Ij of the same size, we would like to copy some regions from Ii and paste them directly into Ij to get a virtual sample I\u2032. The copied regions are indicated by a mask M, which is a binary image of the same size as the two images. The image creation is done as I\u2032 = M \u2299Ii + (1 \u2212M) \u2299Ij, (5) where \u2299denotes the element-wise product. The label maps of the two images Si and Sj are mixed up with the same mask M to generate S\u2032. The mixing can be applied to labeled data and unlabeled data using human ground truths or pseudo-labels, respectively. Existing methods generate 4 \f    ImageNet Encoder fI Feature Distance Loss LF Seg.  Loss Lce Camera Motion Tt,t+1 SDE Loss LD Pose CNN Shared Encoder fE Depth Decoder fD Semantic Decoder fS Image It Image It+1 Depth Dt Segmentation St Ground Truth St ^ ^ Figure 2. Architecture for learning semantic segmentation with SDE as auxiliary task according to Sec. 3.3. The dashed paths are only used during training and only if image sequences and/or segmentation ground truth are available for a training sample. this mask M in different ways, e.g., randomly sampled rectangular regions [69, 12] or randomly selected object segments [47]. In those methods, the structure of the scene is not considered and foreground and background are not distinguished. We \ufb01nd images synthesized by these methods often violate the geometric relationships between objects. For instance, a distant object can be copied onto a closerange object or only unoccluded parts of mid-range objects are copied onto the other image. Imagine how strange it is to see a pedestrian standing on top of a car or to see sky through a hole in a building (just as shown in Fig. 1 left). Our DepthMix is designed to mitigate this issue. It uses the estimated depth \u02c6 Di and \u02c6 Dj of the two images to generate the mix mask M that respects the notion of geometry. It is implemented by selecting only pixels from Ii whose depth values are smaller than the depth values of the pixels at the same locations in Ij: M(a, b) = \u001a 1 if \u02c6 Di(a, b) < \u02c6 Dj(a, b) + \u03f5 0 otherwise (6) where a and b are pixel indices, and \u03f5 is a small value to avoid con\ufb02icts of objects that are naturally at the same depth plane such as road or sky. By using this M, DepthMix respects the depth of objects in both images, such that only closer objects can occlude further-away objects. We illustrate this advantage of DepthMix with an example in Fig. 1. 3.3. Semi-Supervised Semantic Segmentation In this section, we train a semantic segmentation model utilizing the labeled image dataset GA, the unlabeled image dataset GU, and K unlabeled image sequences. We \ufb01rst discuss how to exploit SDE on the image sequences to improve our semantic segmentation. We then show how to use GU to further improve the performance. Learning with Auxiliary Tasks: For learning semantic segmentation and SDE jointly, we use a network with shared encoder f E \u03b8 and a separate depth f D \u03b8 and segmentation decoder f S \u03b8 (see Fig. 2). The depth branch is trained using the SDE loss LD and the segmentation branch gS \u03b8 = f S \u03b8 \u25e6f E \u03b8 is trained using the pixel-wise cross-entropy Lce. In order to initialize the pose estimation network and the depth decoder properly, the architecture is \ufb01rst trained on K unlabeled image sequences for SDE. As a common practice, we initialize the encoder with ImageNet weights as they provide useful semantic features learned during image classi\ufb01cation. To avoid forgetting semantic features during the SDE pretraining, we utilize a feature distance loss between the current bottleneck features f E \u03b8 and the bottleneck features of the encoder with ImageNet weights f E I : LF = ||f E \u03b8 \u2212f E I ||2. (7) The loss for the depth pretraining is the weighted sum of the SDE loss and the ImageNet feature distance loss: LP = LD + \u03bbF LF . (8) To additionally incorporate transfer learning from depth estimation to semantic segmentation, the weights of f D \u03b8 are used to initialize f S \u03b8 . For effective multi-task learning, we use an attention-guided distillation module [65] to exchange useful intermediate features between both decoders. Learning with Unlabeled Images: In order to further utilize the unlabeled dataset GU, we generate pseudo-labels using the mean teacher algorithm [60], which is commonly used in semi-supervised learning [1, 62, 12, 47]. For that purpose, an exponential moving average is applied to the weights of the semantic segmentation model gS \u03b8 to obtain the weights of the mean teacher \u03b8T : \u03b8\u2032 T = \u03b1\u03b8T + (1 \u2212\u03b1)\u03b8. (9) To generate the pseudo-labels, an argmax over the classes C is applied to the prediction of the mean teacher. SU = arg max c\u2208C (gS \u03b8T (IU)). (10) The mean teacher can be considered as a temporal ensemble, resulting in stable predictions for the pseudo-labels, while the argmax ensures con\ufb01dent predictions [47]. For the semi-supervised setting, the segmentation network is trained with labeled samples (IA, SA) and pseudolabeled samples (IU, SU): LSSL = Lce(gS \u03b8 (IA), SA) + \u03bbP (SU)Lce(gS \u03b8 (IU), SU)) (11) \u03bbP (SU) is chosen to re\ufb02ect the quality of the pseudo-label represented by the fraction of pixels exceeding a threshold \u03c4 for the predicted probability of the most con\ufb01dent 5 \fclass maxc\u2208C(gS \u03b8T (IU)), as suggested in [47]. We incorporate DepthMix samples (I\u2032, S\u2032), which are obtained from the combined labeled and pseudo-labeled data pool Ii, Ij \u2208GA \u222aGU (see Eq. 5), into Eq. 11 to replace the unlabeled samples (SU, LU). Our semi-supervised learning is now changed to: LSSL = Lce(gS \u03b8 (IA), SA) + \u03bbP (S\u2032)Lce(gS \u03b8 (I\u2032), S\u2032)). (12) 4. Experiments 4.1. Implementation Details Dataset: We evaluate our method on the Cityscapes dataset [9], which consists of 2975 training and 500 validation images with semantic segmentation labels from European street scenes. We downsample the images to 1024\u00d7 512 pixels. Besides, random cropping to a size of 512\u00d7512 and random horizontal \ufb02ipping are used in the training. Importantly, Cityscapes provides 20 unlabeled frames before and 10 after the labeled image, which are used for SDE training. During the semi-supervised segmentation, only the originally 2975 labeled training images are used. They are randomly split into a labeled and an unlabeled subset. Network Architecture: Our network consists of a shared ResNet101 [21] encoder with output stride 16 and a separate decoder for segmentation and SDE. The decoder consists of an ASPP [5] block to aggregate features from multiple scales and another four upsampling blocks with skip connections [51]. For SDE, the upsampling blocks have a disparity side output at the respective scale. For effective multi-task learning, we additionally follow PAD-Net [65] and deploy an attention-guided distillation module after the third decoder block. It serves the purpose of exchanging useful features between segmentation and depth estimation. Training: For the SDE pretraining, the depth and pose network are trained using Adam [30], a batch size of 4, and an initial learning rate of 1 \u00d7 10\u22124, which is divided by 10 after 160k iterations. The SDE loss is calculated on four scales with three subsequent images. During the \ufb01rst 300k iterations, only the depth decoder and the pose network are trained. Afterwards, the depth encoder is \ufb01ne-tuned with an ImageNet feature distance \u03bbF = 1 \u00d7 10\u22122 for another 50k iterations. The encoder is initialized with ImageNet weights, either before depth pretraining or before semantic segmentation if depth pretraining is ablated. For the multi-task setting, we train the network using SGD with a learning rate of 1 \u00d7 10\u22123 for the encoder and depth decoder, 1 \u00d7 10\u22122 for the segmentation decoder, and 1 \u00d7 10\u22126 for the pose network. The learning rate is reduced by 10 after 30k iterations and trained for another 10k iterations. A momentum of 0.9, a weight decay of 5\u00d710\u22124, and a gradient norm clipping to 10 are used. The loss for segmentation and SDE are weighted equally. The mean teacher Figure 3. Example semantic segmentations of our method for 100 labeled samples in comparison with ClassMix [47]. has \u03b1 = 0.99 and within an iteration, the network is trained on a clean labeled and an augmented mixed batch with size 2, respectively. The latter uses DepthMix with \u03f5 = 0.03, color jitter, and Gaussian blur. Data Selection for Annotation: In the data selection experiment, we use a slimmed network architecture with a ResNet50 encoder and fewer decoder channels for fSIDE. It is trained using Adam with 1 \u00d7 10\u22124 learning rate and polynomial decay with exponent 0.9 for faster convergence. For calculating the depth feature diversity, we use the output of the second depth decoder block after SDE pretraining. It is downsampled by average pooling to a size of 8x4 pixels and the feature channels are normalized to zero-mean unit-variance over the dataset. The student depth error is weighted by \u03bbE = 1000. The number of the selected samples (Pt t\u2032=1 nt\u2032) is iteratively increased to 25, 50, 100, 200, 372, and 744. For each subset, a student depth network is trained from scratch for 4k, 8k, 12k, 16k, and 20k iterations, respectively, to calculate the student depth error. 4.2. Semi-Supervised Semantic Segmentation First, we compare our approach with several state-of-theart semi-supervised learning approaches. We summarize the results in Tab. 1. The performance (mIoU in %) of the semi-supervised methods and their baselines (only trained on the labeled dataset) are shown for a different number of labeled samples. As the performance of the baselines differs, there are columns showing the absolute improvement for better comparability. As our baseline utilizes a more capable network architecture due to the U-Net decoder with ASPP as opposed to a DeepLabv2 decoder used by most previous works, we also reimplemented the state-of-the-art method, ClassMix [47] with our network architecture and training parameters to ensure a direct comparison. As shown in Tab. 1, our method (without data selection) 6 \fTable 1. Performance on the Cityscapes validation set (mIoU in %, standard deviation over 3 random seeds). Labeled Samples 1/30 (100) 1/8 (372) 1/4 (744) Full (2975) Baseline [23] \u2013 55.50 \u21b1 59.90 \u21b1 66.40 \u21b1 Adversarial [23] \u2013 58.80 +3.30 62.30 +2.40 \u2013 Baseline [43] \u2013 56.20 \u21b1 60.20 \u21b1 66.00 s4GAN [43] \u2013 59.30 +3.10 61.90 +1.70 65.80 \u20130.20 Baseline [12] 44.41 \u00b11.11 \u21b1 55.25 \u00b10.66 \u21b1 60.57 \u00b11.13 \u21b1 67.53 \u00b10.35 \u21b1 CutMix [12] 51.20 \u00b12.29 +6.79 60.34 \u00b11.24 +5.09 63.87 \u00b10.71 +3.30 67.68 \u00b10.37 +0.15 Baseline [11] 45.50 \u21b1 56.70 \u21b1 61.10 \u21b1 66.90 DST\u2013CBC [11] 48.70 +3.20 60.50 +3.80 64.40 +3.30 \u2013 Baseline [47] 43.84 \u00b10.71 \u21b1 54.84 \u00b11.14 \u21b1 60.08 \u00b10.62 \u21b1 66.19 \u00b10.11 ClassMix [47] 54.07 \u00b11.61 +10.23 61.35 \u00b10.62 +6.51 63.63 \u00b10.33 +3.55 \u2013 Baseline 48.75 \u00b11.61 \u21b1 59.14 \u00b11.02 \u21b1 63.46 \u00b10.38 \u21b1 67.77 \u00b10.13 \u21b1 ClassMix [47]1 56.82 \u00b11.65 +8.07 63.86 \u00b10.41 +4.72 65.57 \u00b10.71 +2.11 \u2013 ClassMix [47] (+Video) 56.79 \u00b11.98 +8.04 63.22 \u00b10.84 +4.08 65.72 \u00b10.18 +2.26 68.23 \u00b10.70 +0.46 Ours 58.40 \u00b11.36 +9.65 66.66 \u00b11.05 +7.52 68.43 \u00b10.06 +4.98 71.16 \u00b10.16 +3.40 Ours (+Data Selection) 62.09 \u00b10.39 +13.34 68.01 \u00b10.83 +8.87 69.38 \u00b10.33 +5.92 \u2013 outperforms all other approaches on each labeled subset size for both the absolute performance as well as the improvement to the baseline. The only exception is the absolute improvement of the original results of ClassMix for 100 labeled samples. However, if we consider ClassMix trained in our setting, our method outperforms it also in this case. This can be explained by the considerably higher baseline performance in our setting, which increases the dif\ufb01culty to achieve an high improvement. Adding data selection even further increases the performance by a signi\ufb01cant margin, so that our method, trained with only 1/8 of the labels, even slightly outperforms the fully-supervised baseline. To identify whether the improvement originates from access to more unlabeled data or from the effectiveness of our approach, we compare to another baseline \u201cClassMix (+Video)\u201d. More speci\ufb01cally, we also provide all unlabeled image sequences to ClassMix and see how much it can bene\ufb01t from this additional amount of unlabeled data. Experimental results show no signi\ufb01cant difference. This is probably due to the high correlation of the Cityscapes image dataset and the video dataset (the images are the 20th frames of the video clips). The adequacy of our approach is also re\ufb02ected in the example predictions in Fig. 3. We can observe that the contours of classes are more precise. Moreover, dif\ufb01cult objects such as bus, train, rider, or truck can be better distinguished. This observation is also quantitatively con\ufb01rmed by the class-wise IoU improvement shown in Fig. 4. 4.3. Ablation Study Next, we analyze the individual contribution of each component of the proposed method. For this purpose, we 1 Results of the reimplementation in our experiment setting. Table 2. Ablation of the architecture components (D-T: SDE Transfer Learning, D-M SDE Transfer and Multi-Task Learning, F: ImageNet Feature Distance Loss, P: Pseudo-Labeling, X-C: Mix Class, X-D: Mix Depth, S Data Selection). mIoU in %, standard deviation over 3 seeds. D F P X S 372 Samples 2975 Samples 59.14 \u00b11.02 \u21b1 67.77 \u00b10.13 \u21b1 T 60.46 \u00b10.64 +1.31 69.00 \u00b10.70 +1.23 T \u2713 60.80 \u00b10.69 +1.66 69.47 \u00b10.38 +1.71 M \u2713 61.25 \u00b10.55 +2.10 69.76 \u00b10.39 +1.99 \u2713 62.39 \u00b10.86 +3.24 \u2013 \u2713C 63.16 \u00b10.89 +4.02 69.60 \u00b10.32 +1.83 \u2713D 64.14 \u00b11.34 +5.00 69.83 \u00b10.36 +2.06 M \u2713\u2713D 66.66 \u00b11.05 +7.52 71.16 \u00b10.16 +3.40 \u271364.25 \u00b1 0.18 +5.11 \u2013 M \u2713\u2713D \u271368.01 \u00b10.83 +8.87 \u2013 Road S.walk Building Wall Fence Pole Tr. Light Tr. Sign Veget. Terrain Sky Person Rider Car Truck Bus Train M.cycle Bicycle Baseline DM XD DM+XD DM+XD+S 0.0 0.2 Figure 4. Improvement of the class-wise IoU over the baseline performance for 372 labeled samples (DM: SDE Multi-Task Learning, XD: DepthMix with Pseudo-Labels, S: Data Selection). test several ablated versions of our model for both the cases of 372 and 2975 labeled samples. We summarize the results in Tab. 2. It can be seen that each contribution adds a signi\ufb01cant performance improvement over the baseline. For 372 (2975) annotated samples, transfer and multi-task 7 \f    Image i Image j Depth i Depth j Mixed Image I\u2019 a) b) Figure 5. DepthMix applied to Cityscapes crops. learning improve the performance by +2.10 (+1.99), DepthMix with pseudo-labels by +5.00 (+2.06), and automatic data selection by +5.11 (\u2013) mIoU percentage points. As our components are orthogonal, combining them even further increases performance. SDE Multi-Tasking and DepthMix achieve +7.52 (+3.40) and all three components +8.87 (\u2013) mIoU percentage points improvement. Note that the high variance for few labeled samples is mostly due to the high in\ufb02uence of the randomly selected labeled subset. The chosen subset affects all con\ufb01gurations equally and the reported improvements are consistent for each subset. Furthermore, we compare DepthMix with ClassMix as a standalone. For a fair comparison, we additionally include mixing labeled samples with their ground truth to ClassMix. It can be seen that DepthMix outperforms the ClassMix by 0.98 (0.23) percentage points for 372 (2975) annotated samples, which shows the effect of the geometry aware augmentation. Fig. 5 shows DepthMix examples demonstrating that SDE allows to correctly model occlusions and to produce synthetic samples with a realistic appearance. For more insights into possible reasons for these improvements, we visualize the improvement of the architecture components over the baseline for each class separately in Fig. 4. It can be seen that depth multi-task learning (DM) improves mostly the classes fence, traf\ufb01c light, traf\ufb01c sign, rider, truck, and motorcycle, which is possibly due to their characteristic depth pro\ufb01le learned during SDE. For example, a good depth estimation performance requires correctly segmenting poles or traf\ufb01c signs as missing them can cause large depth errors. This can also be seen in Fig. 3. DepthMix (XD) further improves the performance of wall, truck, bus, and train. This might be caused by the fact the DepthMix presents those rather dif\ufb01cult objects in another context, which might help the network to generalize better. In the suppl. materials, we further show that our method is still applicable if SDE is trained on a different dataset than semantic segmentation within a similar visual domain. 4.4. Automatic Data Selection for Annotation Finally, we evaluate the proposed automatic data selection. Tab. 3 shows a comparison of our method with a baseline and a competing method. The baseline selects the laTable 3. Comparison of data selection methods (DS: Diversity Sampling based on depth features, US: Uncertainty Sampling based on depth student error). mIoU in %, std. dev. over 3 seeds. # Labeled 1/30 (100) 1/8 (372) 1/4 (744) Random 48.75 \u00b11.61 59.14 \u00b11.02 63.46 \u00b10.38 Entropy 53.63 \u00b10.77 63.51 \u00b10.68 66.18 \u00b10.50 Ours (US) 51.75 \u00b11.12 62.77 \u00b10.46 66.76 \u00b10.45 Ours (DS) 53.00 \u00b10.51 63.23 \u00b10.69 66.37 \u00b10.20 Ours (DS+US) 54.37 \u00b10.36 64.25 \u00b10.18 66.94 \u00b10.59 beled samples randomly, while the second, strong competitor uses active learning and iteratively chooses the samples with the highest segmentation entropy. In contrast to our method, this requires a human in the loop to create the semantic labels for iteratively selected images. It can be seen that our method with the combined Diversity Sampling and Uncertainty Sampling (DS+US) outperforms both comparison methods, demonstrating the effectiveness of ensuring diversity and exploiting dif\ufb01cult samples based on depth. It also supports the assumption that depth estimation and semantic segmentation are correlated in terms of sample dif\ufb01culty. The class-wise analysis (see the last row of Fig. 4) shows that data selection signi\ufb01cantly improves the performance of truck, bus, and train, which are usually dif\ufb01cult to distinguish in a semi-supervised setting. We would like to note that our automatic data selection method can be applied to any semantic segmentation method. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1907.13054v2",
+                    "title": "Grid Saliency for Context Explanations of Semantic Segmentation",
+                    "abstract": "Recently, there has been a growing interest in developing saliency methods\nthat provide visual explanations of network predictions. Still, the usability\nof existing methods is limited to image classification models. To overcome this\nlimitation, we extend the existing approaches to generate grid saliencies,\nwhich provide spatially coherent visual explanations for (pixel-level) dense\nprediction networks. As the proposed grid saliency allows to spatially\ndisentangle the object and its context, we specifically explore its potential\nto produce context explanations for semantic segmentation networks, discovering\nwhich context most influences the class predictions inside a target object\narea. We investigate the effectiveness of grid saliency on a synthetic dataset\nwith an artificially induced bias between objects and their context as well as\non the real-world Cityscapes dataset using state-of-the-art segmentation\nnetworks. Our results show that grid saliency can be successfully used to\nprovide easily interpretable context explanations and, moreover, can be\nemployed for detecting and localizing contextual biases present in the data.",
+                    "authors": "Lukas Hoyer, Mauricio Munoz, Prateek Katiyar, Anna Khoreva, Volker Fischer",
+                    "published": "2019-07-30",
+                    "updated": "2019-11-07",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction In many real-world scenarios, the presence of an object, its location and appearance are highly correlated with the contextual information surrounding this object, such as the presence of other nearby objects or more global scene semantics. For example, in the case of an urban street scene, a cyclist is more likely to co-occur on a bicycle and a car to appear on the road below sky and buildings (cf. objects and their context explanations in Fig. 1). These semantic correlations are inherently present in real-world data. A data-driven model, such as a deep neural network, is prone to exploit these statistical biases during training in order to improve its prediction performance. These biases picked up by the network during training can lead to erroneous predictions and impair network generalization (cf. the effect of context on misclassi\ufb01cation in Fig. 2). An effective and safe utilization of deep learning models for real-world applications, e.g. autonomous driving, requires a good understanding of these contextual biases inherent in the data and the extent to which a learned model incorporated them into its decision making process. Saliency methods [4, 5, 6, 7, 8, 9] have become a popular tool to explain predictions of a trained model by highlighting parts of the input that presumably have a high relevance for its predictions. 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada. arXiv:1907.13054v2  [cs.CV]  7 Nov 2019 \f(a) Semantic segment. (b) Context explanations (red) for different segments (outlined in blue) Figure 1: Context explanations by grid saliency for semantic segmentation [1, 2] on Cityscapes [3]. Grid saliency not only can contextually explain correct predictions: in the \ufb01rst row the network looks at the motorbike to correctly predict the class rider (light blue); but can also explain erroneous predictions: in the second row the upper body of the rider is incorrectly predicted as person, but for this prediction the bicycle is not salient in contrast to the correctly predicted legs of the rider. However, to the best of our knowledge the existing saliency methods are mostly focused on image classi\ufb01cation networks and thus are not able to spatially differentiate between prediction explanations. In this work, we propose a way to extend existing saliency methods designed for image classi\ufb01cation towards (pixel-level) dense prediction tasks, which allows to generate spatially coherent explanations by exploiting spatial information in dense predictions. We call our approach grid saliency, which is a perturbation-based saliency method, formulated as an optimization problem of identifying the minimum unperturbed area of the image needed to retain the network predictions inside a target object region. As our grid saliency allows to differentiate between objects and their associated context areas in the saliency map, we speci\ufb01cally explore its potential to produce context explanations for semantic segmentation networks. The contextual information is known to be one of the essential recognition cues [10, 11, 12], thus we aim to investigate which local and global context is the most relevant for the network class predictions inside a target object area (see Fig. 1 and Fig. 2 for examples). In real-world scenarios, context biases are inherently present in the data. To evaluate whether the proposed grid saliency is sensitive to context biases and has the ability to detect them, we introduce a synthetic toy dataset for semantic segmentation, generated by combining MNIST digits [13] with different foreand background textures, for which we arti\ufb01cially induce a context bias between the digit and the background texture. Besides detecting the mere presence of the context bias, we also analyze the ability of our method to localize it in the image (see Sec. 4). We employ this dataset to compare our approach with different baselines, i.e., introduced extensions of gradient-based saliency methods of [4, 14, 15] to produce context explanations. We show that the proposed dataset can serve as a valid benchmark for assessing the quality of saliency methods to detect and localize context biases. We \ufb01nd that gradient-based techniques, in contrast to our grid saliency, are ill-suited for the context bias detection. By design they tend to produce noisy saliency maps which are not faithful to the context bias present in the data, whereas our method has higher sensitivity for context bias and thus can also precisely localize it in the image. We further evaluate grid saliency performance to produce context explanations for semantic segmentation on the real-world Cityscapes dataset [3] and experimentally show that the produced context explanations faithfully re\ufb02ect spatial and semantic correlations present in the data, which were thus picked up by the segmentation network [1, 2]. To the best of our knowledge, we are the \ufb01rst to extend saliency towards dense-prediction tasks and use it to produce context explanations for semantic segmentation. In summary, our contributions are the following: (1) We propose an extension of saliency methods designed for classi\ufb01cation towards dense prediction tasks. (2) We exploit the proposed grid saliency to produce context explanations for semantic segmentation and show its ability to detect and localize context biases. (3) We create a synthetic dataset to benchmark the quality of produced explanations as well as their effectiveness for context bias detection/localization. (4) We investigate the faithfulness of context explanations for semantic segmentation produced by the grid saliency on real-world data. 2 Related Work Explanations. Many methods attempt to interpret the network decision making process by producing explanations via bounding boxes [16, 17] or attributes [18], providing textual justi\ufb01cations [19, 20, 21] or generating low-level visual explanations [4, 22, 5, 6]. Our work builds on top of 2 \f(a) Context explanation (b) Semantic segm. (c) Image w/o context (d) Segm. w/o context Figure 2: Effect of context on semantic segmentation and the context explanation provided by grid saliency for an erroneous prediction, the image is taken from MS COCO [23]. The grid saliency (a) shows the responsible context for misclassifying the cow (green) as horse (purple) in the semantic segmentation (b). It shows the training bias that horses are more likely on road than cows. Removing the context (c) yields a correctly classi\ufb01ed cow (d). the latter approaches, also known as saliency methods, which try to identify the image pixels that contribute the most to the network prediction. These methods mostly focus on the task of image classi\ufb01cation and can be divided into two categories: gradient-based and perturbation-based methods. Gradient-based methods [24, 4, 25] compute a saliency map that visualizes the sensitivity of each image pixel to a speci\ufb01c class prediction, which is obtained by backpropagating the gradient for the prediction with respect to the image and estimating how moving along the gradient in\ufb02uences the class output. To circumvent noise and visual diffusion in saliency maps, [14] proposes to sum up the gradients over the intensity-scaled input versions, while [15] averages over many noisy samples of the input. Other methods [5, 26, 27] explore integrating network activations into their saliency maps. Gradient-based methods mostly rely on heuristics for backpropagation and as has been shown by [7] may provide explanations which are not faithful to the model or data. Perturbation-based methods [6, 28, 9, 8] evaluate the class prediction change with respect to a perturbed image, e.g. in which speci\ufb01c regions of the image are either replaced with the mean image values or removed by applying blur or Gaussian noise. The approach in [6] formulates this output change as an optimization problem of the original and perturbed image, while [28] estimates the perturbation mask by training an auxiliary network. To account for image discontinuities, [9, 8] compute the saliency of the masked region by marginalizing it out either over neighboring image regions or by conditioning the trained generative model on the non-masked region and then estimating the classi\ufb01cation change. Perturbation approaches might be vulnerable to network artifacts resulting in arbitrary saliency regions [8]. To overcome this, [6, 28] resort to perturbe large image regions. The above methods are limited to explanations of image classi\ufb01cation. In this work, we propose a way to extend them to produce grid saliency maps for dense prediction networks as well (see Sec. 3). We showcase the usability of grid saliency for context explanations of semantic segmentation (see Sec. 4 and 5). The closest related work to produce context explanations is [29], which proposes a network that jointly learns to predict and contextually explain its decisions. In contrast to [29], we focus on the post hoc model explanations and are not limited to the image classi\ufb01cation task. Semantic segmentation. CNNs have become a default technique for semantic segmentation [30, 31, 32]. [33] \ufb01rst showcased the use of CNNs for segmentation. Since then, multiple techniques have been proposed, from utilizing dilated convolutions [34, 35] and post-processing smoothing operations [36, 37, 38, 39] to employing spatial pyramid pooling [40, 30, 41, 2] and encoder/decoder architectures [42, 43, 44, 45]. As the accuracy of these methods comes at a high computational cost, there has been an increasing interest in developing real-time semantic segmentation networks with low memory needs [46, 1, 47, 48, 49]. In this work, we aim to produce context explanations for semantic segmentation. To investigate the effectiveness of our approach and show its generalization across architectures, we employ DeepLabv3+ [2] and U-Net [44] with different backbones [1, 50, 51]. 3 Method In Sec. 3.1 we introduce the grid saliency method, which allows to produce spatially coherent explanations for dense predictions, and present a way to use it for context explanations of semantic segmentation. Next, in Sec. 3.2 we extend the popular gradient-based saliency methods [4, 14, 15] to produce spatial explanations as well, which we later compare with grid saliency in Sec. 4. 3 \f3.1 Grid Saliency via Perturbation Let f : I \u2192O denote the prediction function, e.g. a deep neural network, which maps a grid input space I = RHI\u00d7WI\u00d7CI to a grid output space O = RHO\u00d7WO\u00d7CO, where W and H are the respective width and height of the input and output, and CI and CO are the number of input channels (e.g. 3 or 1 for images) and output prediction channels (e.g. number of classes for semantic segmentation). To keep the discussion concrete, we consider only images as input, x \u2208I, and per-pixel dense predictions of the network f(x) \u2208O as output. The goal is to \ufb01nd the smallest saliency map M \u2208[0, 1]HI\u00d7WI that must retain in the image x in order to preserve the network prediction in the request mask area R \u2208{0, 1}HO\u00d7WO for class (channel) c \u2208{1, ..., CO}. Further on, for simplicity we assume that the input and output spatial dimensions are the same. Our method builds on top of perturbation saliency methods [6, 28, 9, 8] designed for image classi\ufb01cation. They aim to \ufb01nd salient image regions most responsible for a classi\ufb01er decision by replacing parts of the image with uninformative pixel values, i.e. perturbing the image, and evaluating the corresponding class prediction change. We follow the same image perturbation strategy as [6]. Let p denote a perturbation function that removes information from an image x outside of the saliency M. For example, such perturbation function can be the interpolation between x and a \u2208I, where a can be a constant color image, gaussian blur, or random noise. In this case, p(x, M) = x \u25e6M + a \u25e6(1 \u2212M). Note, that in practice M operates on a lower resolution to avoid adversarial artifacts [6, 28] and is later upsampled to the original image resolution. In addition, the pixel values of the perturbed image p(x, M) are clipped to preserve the range of the original image space. With this notation in hand, we can formulate the problem of \ufb01nding the saliency map M for the prediction of class c as the following optimization problem: M \u2217(x, c) = argmin M \u03bb \u00b7 \u2225M\u22251 + \u2225max(fc(x) \u2212fc(p(x, M)), 0)\u22251, (1) where \u2225\u00b7 \u22251 denotes the l1 norm and fc(x) is the network prediction for class c . The \ufb01rst term can be considered as a mask loss that minimizes the salient image area and perturbs the original image as much as possible. The second term serves as a preservation loss which ensures that the network prediction fc(p(x, M)) for class c on the perturbed image p(x, M) reaches at least the con\ufb01dence of the network prediction fc(x) on the original unperturbed image. Thus, the second loss term can be considered as a penalty for not meeting the constraint fc(p(x, M)) \u2a7efc(x), hence the use of max(\u00b7, 0) in Eq. 1. The parameter \u03bb controls the sparsity of M. We then can spatially disentangle explanations given in the saliency map M for the network predictions in the requested area of interest R from the explanations for the other predictions, by restricting the preservation loss to the request mask R in Eq. (1): M \u2217 grid(x, R, c) = argmin M \u03bb \u00b7 \u2225M\u22251 + \u2225R \u25e6max(fc(x) \u2212fc(p(x, M)), 0)\u22251 \u2225R\u22251 . (2) Further on, we will refer to M \u2217 grid in Eq. (2) as a grid saliency map. Context explanations for semantic segmentation. We now adapt the grid saliency formulation from Eq. (2) to speci\ufb01cally provide context explanations for the requested area of interest R. Context explanations are of particular interest for semantic segmentation, as context often serves as one of the main cues for semantic segmentation networks (see Fig. 2). Thus, here we focus on context explanations for semantic labelling predictions and assume that R is the area covering the object of interest in the image x. To optimize for salient parts of the object context, we integrate the object request mask R in the perturbation function. For the request mask R, the perturbed image p(x, R) \u2208I contains only the object information inside R and all the context information outside R is removed (with a constant color image a). For optimization, we will now use this new perturbed image p(x, R) instead of the maximally perturbed image p(x, M = 0) = a and denote the context perturbation function as pcontext(x, R, M) = x \u25e6M + p(x, R) \u25e6(1 \u2212M). The context saliency map for class c and request object R can be obtained via optimization of M \u2217 context(x, R, c) = argmin M \u03bb \u00b7 \u2225M\u22251 + \u2225R \u25e6max(fc(x) \u2212fc(pcontext(x, R, M)), 0)\u22251 \u2225R\u22251 , (3) where the saliency map is optimized to select the minimal context necessary to at least yield the original prediction for class c inside the request mask R. Note that, M \u2217 context can be an empty mask if no context information is needed to recover the original prediction inside R. 4 \f3.2 Gradient-Based Variants Another way to produce spatially coherent saliency maps is to make use of the popular gradient-based saliency methods. Thus, we additionally consider the Vanilla Gradient (VG) [24], Integrated Gradient (IG) [14], and SmoothGrad (SG) [15] saliency methods. Let G(x, c) = \u2202gc(x)/\u2202x \u2208RHI\u00d7WI\u00d7CI denote the gradient of the classi\ufb01cation network prediction gc(x) \u2208R for class c with respect to the input image x \u2208I. For the classi\ufb01cation task, the saliency maps of VG, IG and SG are computed as: M VG(x, c) = X c\u2208CI \f \fG(x, c) \f \f, M SG(x, c) = X c\u2208CI \f \f \f \f 1 n n X k=1 G \u0000x + N(0, \u03c32), c \u0001 \f \f \f \f, M IG(x, c) = X c\u2208CI \f \f \f \f 1 n n X k=1 G \u0012k nx, c \u0013 \f \f \f \f, (4) where n is the number of approximation steps for IG or the number of samples for SG, and N(0, \u03c32) represents Gaussian noise with standard deviation \u03c3. Following Sec. 3.1, we next extend the above approaches to produce the saliency M for dense predictions fc(x) \u2208O and to spatially disentangle explanations given in the saliency M for the network predictions in the request area R from other predictions. For a given input x and a binary request mask R, we denote the normalized network prediction score for class c in the request area R as S(x, R, c) = \u2225R \u25e6fc(x)\u22251/\u2225R\u22251, S(x, R, c) \u2208R. Similarly to G(x, c), we de\ufb01ne Ggrid(x, R, c) . .= \u2202S(x, R, c)/\u2202x \u2208RHI\u00d7WI\u00d7CI which directly yields M VG/SG/IG grid (x, R, c) by replacing G(x, c) in Eq. 4 with Ggrid(x, R, c). For the gradient-based context saliency, as in Sec. 3.1 only salient pixels outside of the object area are considered, i.e. M VG/IG/SG context (x, R, c) . .= (1 \u2212R) \u25e6M VG/IG/SG grid (x, R, c). (5) Gradient-based saliency maps are prone to be noisy. Thus, to circumvent this and also make them more comparable to the lower resolution perturbation-based grid saliency, in our experiments we apply the spatial mean \ufb01lter on top of the saliency map with a (WI/WS) \u00d7 (HI/HS) kernel and stride, where WS \u00d7 HS is the resolution of the perturbation-based saliency map. 4 Context Bias Detection on Synthetic Data Although context biases are inherently present in real-world data, in practice it is hard to measure and alter these correlations and thus to employ this data for benchmarking context bias detection. In order to evaluate whether the grid saliency is sensitive to context biases and has the ability to detect them, in Sec. 4.1 we introduce a synthetic dataset for semantic segmentation with arti\ufb01cially induced context biases.1 We use this dataset to compare the grid saliency methods proposed in Sec. 3 and show that this dataset can serve as a valid benchmark for assessing the quality of saliency methods for context bias detection and localization, see Sec. 4.2. 4.1 Benchmark for Context Bias Detection Dataset. The proposed synthetic toy dataset consists of gray scale images of size 64 \u00d7 64 pixels, generated by combining upscaled digits from MNIST [13] with foreground and background textures from [52, 53], as can be seen in Fig. 3 (a). In order to introduce different context information for the digit, two background textures are used for the upper and lower half of the image. For each synthetic image a corresponding segmentation ground truth is generated, where the MNIST digit de\ufb01nes the mask and the semantic class (including a background class). To evaluate the ability of saliency methods to explain context biases, we propose to generate biased and unbiased versions of the dataset. For the unbiased version (DSno-bias), all foreand background textures appear with equal probability for all digits. For the biased version, a single digit class is coupled with a speci\ufb01c background texture, located randomly either in the upper or lower half of the background. We consider two variants of it, with a weakly DSw-bias and strongly induced bias 1The code for generating the proposed synthetic dataset with induced context biases can be found here: https://github.com/boschresearch/GridSaliency-ToyDatasetGen. 5 \fToy data examples a) Input VG Ours b) Unbiased Biased IG SG Figure 3: a) Synthetic dataset images, see Sec. 4.1 for details. b) Context saliency maps of different methods. For the image on the left, the different context saliency maps Mcontext are shown for the networks [1] trained on the unbiased (top row) and biased (bottom row) dataset versions. For the biased version, the digit 2 is biased with the background texture in the top half of the image. Our perturbation-based grid saliency is able to precisely detect this bias, in contrast to other methods. DSs-bias. For the dataset with a strongly induced bias DSs-bias, a speci\ufb01c texture appears if and only if a certain digit class is present. For a weakly induced bias DSw-bias, a speci\ufb01c texture always appears along with the biased digit but also uniformly appears with other digits. From the pool of 25 textures, 5 textures are chosen randomly to induce context bias for one of 10 digits. For all 50 texture/digit combinations, a weakly and strongly biased dataset variant with train/test splits is generated. Evaluation metrics. To evaluate the extent to which a network is able to pick up a context bias present in the training data, we propose to measure on the unbiased DSno\u2212bias test set the performance of the network trained on the biased DSw/s-bias training set, using the standard Intersection over Union (IoU, see [33]) metric for semantic segmentation. If the network has picked up the bias, we expect to see a signi\ufb01cant drop in IoU for the biased digit segmentation. To benchmark how well different saliency methods (perturbationor gradient-based) can detect the context bias of semantic segmentation networks, we propose to evaluate to which extent a context saliency map Mcontext(x, R, c) (cf. Eq. 3 and 5 in Sec. 3) for the request object mask R and its corresponding class c is concentrated on the ground truth context area C = 1 \u2212RGT , where RGT is the ground truth mask of the object R, by using a context bias detection metric (CBD): CBD(x, R, c) = \u2225C \u25e6Mcontext(x, R, c)\u22251/\u2225C\u22251. (6) To benchmark the ability of different saliency methods to localize a context bias, we propose to measure how much of the context saliency Mcontext(x, R, c) falls into the ground truth biased context area Cbias, the upper or lower half of the image x by the design of DSw/s-bias, where Cbias is a binary mask of the biased context area. We refer to this metric as a context bias localization (CBL): CBL(x, R, c) = \u2225Cbias \u25e6Mcontext(x, R, c)\u22251/\u2225C \u25e6Mcontext(x, R, c)\u22251. (7) In our experiments we report mIoU and the mean CBD and CBL measures (mCBD, mCBL) per biased digit class c, averaging the results across all images in the DSw/s-bias test set, 5 randomly selected bias textures and 5 different initial random seeds for the training set generation. 4.2 Experimental Results Implementation details. We use the U-Net [44] architecture with a VGG16 [51] backbone. As request mask, the segmentation prediction of the target digit is used. The saliency maps with a size of 4\u00d74 are optimized using SGD with momentum of 0.5 and a learning rate of 0.2 for 100 steps starting with a 0.5 initialized mask. A weighting factor of \u03bb = 0.05 is used (see Eq. 3). A constant color image is used for perturbation. Further implementation details are provided in the supp. material. Bias in the trained network. We \ufb01rst investigate if the networks trained on DSw-bias and DSs-bias have picked up the induced weak and strong context biases in the training data. For this purpose, we evaluate their performance in terms of mIoU on the unbiased dataset DSno-bias and report the results in Fig. 4 (a), which visualizes the digit-wise mIoU with respect to the biased digit. The \ufb01rst row of the heat map (labeled as N) in Fig. 4 (a) shows the performance of the networks trained on DSno-bias. We observe a clear drop in performance for biased digits (diagonal elements) in comparison to the \ufb01rst row. As expected, the performance drop is higher for the stronger bias. Moreover, the mIoU of the unbiased digits (non-diagonal elements) is also affected by the introduced bias. For example, 6 \fN 9 8 7 6 5 4 3 2 1 0 Strong bias mIoU 0 1 2 3 4 5 6 7 8 9 Digit N 9 8 7 6 5 4 3 2 1 0 Weak bias VG SG IG Ours 0 1 2 3 4 5 6 7 8 9 Digit 0 1 2 3 4 5 6 7 8 9 Digit 0 1 2 3 4 5 6 7 8 9 Digit 0 1 2 3 4 5 6 7 8 9 Digit 0.0 0.5 1.0 mCBL (strong) VG SG IG Ours 0.0 0.5 1.0 mCBL (weak) (a) Segmentation (b) Bias detection (c) Bias localization Figure 4: Context bias detection/localization of different saliency methods. (a) shows the segmentation mIoU and (b) & (c) report the context bias detection and localization results, for the strongly and weakly biased datasets, respectively. In (a)-(b), the y axis indicates the network bias towards the digit (0-9), with N denoting the unbiased setting. The x axis indicates the corresponding digit result. In contrast to VG [24], IG [14], and SG [15], our grid saliency is able to accurately detect (see diagonal elements in (b)) and localize (c) the induced context bias, see Sec. 4.2 and suppl. material for details. inducing a bias for the digit nine leads to a decreased performance for the digit four (see second row in Fig. 4 (a)). We observe that this effect mostly occurs for the similar looking digits and, most likely, is caused by the fact that on the unbiased dataset the bias textures also occur with the unbiased digits, resulting in the confusion of similar looking digits for the network. From the observed mIoU drop for biased digits we can conclude that the networks have picked up the introduced bias. However, in real world it is often impossible to collect fully unbiased data. Thus, we next evaluate the ability of our grid saliency to detect context bias only using the biased data. Context bias detection with grid saliency. In the last column of Fig. 4 (b), we report the context bias detection results for our perturbation-based grid saliency method, described in Sec. 3.1, using the CBD metric (see Eq. 6). The mCBD values are visualized with respect to networks trained on data biased to different digits DSs/w-bias (y-axis) and for the different digit classes (x-axis) in the biased test set. The only exception is the \ufb01rst row (labeled as N), where for comparison we show the results with no bias, for the network trained on DSno-bias. We observe that our grid saliency shows substantial evidence of context bias for digits with induced bias (diagonal elements), both strong and weak. Even for the weak bias in Fig. 4 (b) the grid saliency still clearly differentiates between biased and unbiased digits (diagonal vs. non-diagonal elements). Note that the bias detection using the grid saliency does not require an unbiased test set. In the suppl. material, we also study the in\ufb02uence of the bias texture as well as the choice of hyperparameters. Comparison across different saliency methods. In Fig. 4 (b) and (c) we compare our perturbationbased grid saliency with the context saliency extensions of gradient-based methods, i.e. VG, SG, and IG (see Eq. 4), using mCBD and mCBL metrics proposed in Sec. 4.1. From Fig. 4 (b) we see that VG and SG are not able to reliably detect the context bias, while IG achieves a comparable performance to our grid saliency. However, in contrast to the perturbation saliency, IG has also high mCBD values for unbiased digits, complicating its use in practice, as one would need to tune a reliable detection threshold, which is particularly challenging for weakly biased data. Context bias localization. In addition to the detection performance, we evaluate how well the saliency methods localize the context bias using the mCBL measure. Fig. 4 (c) shows that VG, IG, and SG have low localization performance, comparable to random guessing (\u223c0.5 mCBL), while our grid saliency is the only method which is able to accurately localize the context bias (mCBL above 0.9) on both strongly and weakly biased data. We evaluate the ability of our perturbation grid saliency to detect and localize context bias across different semantic segmentation networks. For U-Net [44] with the VGG16 [51], ResNet18 [50] and MobileNetv2 [1] backbones, we observe similar bias detection and localization performance (see also supp. material). 7 \fSemantic segment. [40] Person (corr) Rider (err) Rider (err) Rider (corr) Person (corr) Pole (corr) Traf\ufb01c sign (corr) Figure 5: Context explanations by grid saliency for erroneous (err) and correct (corr) semantic segmentation predictions on Cityscapes [3]. Salient red regions visualize the most important context Mcontext for the requested segment prediction R outlined in blue. See Sec. 5 for discussion. 5 Cityscapes Experiments Motivated by the success of our perturbation-based grid saliency at detecting context biases on the synthetic dataset described in Sec. 4, we next apply it to Cityscapes [3] in order to produce and analyze context explanations for semantic segmentation in real-world scenes. Experimental setup. We use 500 \ufb01nely annotated images of the Cityscapes validation set, considering only a subset of classes for the analysis. For our experiments we use the Deeplabv3+ [2] network with a Mobilenetv2 backbone [1]. Our optimization setup largely carries over from Sec. 4.2, with the exception that we optimize a coarse 16 by 32 pixel mask using SGD with a learning rate of 1 for 80 steps and use \u03bb = 0.01. Additional implementation details are provided in the supp. material. Analysis of context explanations. In order to gain a global understanding of the model context bias, we aggregate statistics from the produced grid saliency maps for several requested classes across all validation images. These statistics are summarized in Fig. 6. For each image and class, the context saliency is computed for all suf\ufb01ciently large instances (at least 10k pixels), as can be seen in the second row of Fig. 5. Next, for each image and semantic class we compute the weighted label distribution of context salient pixels. The weight of each pixel is given by the computed saliency intensity at said pixel and its context class label is taken from the ground truth segmentation. For each requested object class, we group by label across all images and sum the saliency values within each group. This yields an accumulated saliency value for each label. We normalize the results by the sum of all saliency values across all classes and images to yield a probability distribution across labels. To show that context explanations produced by our grid saliency are meaningful, we additionally compare its accumulated context class statistics with their baseline class distributions. These baselines are computed in the same manner as above, but instead of relying on the optimized saliency map we use a \ufb01xed dilation of the contour of all predicted object segments considered in the image, thereby capturing the immediate context around each object in a uniform set of directions. Fig. 6 compares these accumulated statistics of context explanations for the requested object classes, given on the y axis. We observe that the context explanations are focused across reasonable and somewhat expected class subsets, which vary per class. For instance, in comparison to the baseline, context saliency for the rider class shifts attention from its spatially co-occurring classes such as road, vegetation and building, to mainly the bicycle class. Car context saliency attention is decreased on building and vegetation and mainly focused on road, sidewalk and other cars. Bicycle context saliency mostly attends to sidewalk rather than the road class. Overall, our grid saliency is able to provide sensible and coherent explanations for network decision making, which re\ufb02ect semantic dependencies present in street scenes. These quantitative \ufb01ndings are validated by the qualitative results in Fig. 5. The second row of Fig. 5 gives several representative examples of context saliencies. Please refer to the supp. material for additional examples and a high-resolution version of Fig. 6 as well as a qualitative model comparison on MS COCO. Context explanations of erroneous predictions. One of the motivations for this work is also to explain unexpected model behavior. Speci\ufb01cally, in case of erroneous predictions (i.e., Fig. 1 and 2), we wish to understand to which extent context bias contributes to the failure. Table 1 shows class statistics of context explanations for correct and erroneous predictions, where the intersection of the 8 \froad building vegetation sidewalk car bicycle motorcycle person rider pole traffic_sign Context Class car bicycle motorcycle person rider pole traffic_sign Request Class a) Grid Saliency road building vegetation sidewalk car bicycle motorcycle person rider pole traffic_sign Context Class b) Baseline 0.00 0.08 0.16 0.24 0.32 0.40 road building vegetation sidewalk car bicycle motorcycle person rider pole traffic_sign Context Class Difference of (a) and (b) 0.16 0.08 0.00 0.08 0.16 Figure 6: Class statistics of context explanations on Cityscapes, see Sec. 5 for discussion. Table 1: Class statistics of context explanations of correct and erroneous predictions on Cityscapes. GT Prediction Context class road bicycle motorcycle car vegetation building sidewalk rider person pole rider rider 0.17 0.22 0.03 0.07 0.09 0.11 0.10 0.07 0.02 0.02 person 0.13 0.10 0.01 0.05 0.10 0.24 0.05 0.08 0.11 0.04 person person 0.12 0.03 0.00 0.09 0.15 0.22 0.13 0.01 0.05 0.05 rider 0.09 0.18 0.02 0.04 0.10 0.13 0.07 0.08 0.15 0.03 car 0.19 0.01 0.00 0.32 0.05 0.11 0.03 0.01 0.19 0.01 bicycle bicycle 0.18 0.06 0.01 0.05 0.09 0.15 0.19 0.05 0.04 0.04 motorcycle 0.19 0.27 0.13 0.01 0.02 0.11 0.13 0.03 0.01 0.04 car car 0.39 0.02 0.00 0.06 0.11 0.10 0.08 0.01 0.04 0.04 motorcycle 0.07 0.06 0.22 0.17 0.14 0.07 0.02 0.01 0.00 0.04 person 0.14 0.01 0.00 0.28 0.03 0.14 0.07 0.01 0.22 0.02 motorcycle motorcycle 0.18 0.04 0.10 0.04 0.07 0.14 0.18 0.05 0.03 0.06 bicycle 0.13 0.10 0.07 0.10 0.10 0.14 0.11 0.01 0.05 0.03 car 0.16 0.01 0.18 0.10 0.10 0.10 0.11 0.05 0.04 0.06 pole pole 0.04 0.02 0.00 0.05 0.28 0.23 0.10 0.00 0.03 0.04 building 0.04 0.01 0.00 0.04 0.07 0.57 0.02 0.00 0.01 0.05 vegetation 0.03 0.00 0.00 0.03 0.63 0.08 0.03 0.00 0.00 0.07 ground truth mask and the correct or misclassi\ufb01ed prediction region are used as the request mask R, and similarly to Fig. 6 semantic class statistics are computed inside the salient context for all suf\ufb01ciently large request masks (at least 625 pixels). We observe that the saliency severely changes for the error cases in comparison to the correct predictions. E.g. for the correctly classi\ufb01ed rider class, context saliency mostly focuses on bicycle (22%), but is much less present (10%) when rider is mistaken as person and in this case the context saliency shifts from bicycle to building (24%). The opposite effect is observed when person is misclassi\ufb01ed as rider. Sidewalk saliency is decreased when bicycle is misclassi\ufb01ed as motorcycle (19% vs. 13%) and when pole is misclassi\ufb01ed as building (10% vs. 2%). In general, grid saliency is able to provide reasonable explanations of erroneous predictions. Fig. 1 (second row) and Fig. 5 (\ufb01rst row) showcase how applying grid saliency on the same input image but different prediction outputs is useful for isolating failures caused by context bias. Both of them show examples of context explanations for erroneous segmentations of a single object. In Fig. 5, the arms, head and shoulders of a pedestrian are classi\ufb01ed as rider, while the rest of the body is correctly identi\ufb01ed as person. The context saliency for the former body parts activates highly on the arms and stroller handles, whereas activations for person does not highlight this support. Thus, a reasonable conclusion is that the misclassi\ufb01cation may be attributed to the arm pose and potentially also to the context bias given by the similar appearance of the stroller and bicycle handles. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1903.08960v2",
+                    "title": "Short-Term Prediction and Multi-Camera Fusion on Semantic Grids",
+                    "abstract": "An environment representation (ER) is a substantial part of every autonomous\nsystem. It introduces a common interface between perception and other system\ncomponents, such as decision making, and allows downstream algorithms to deal\nwith abstracted data without knowledge of the used sensor. In this work, we\npropose and evaluate a novel architecture that generates an egocentric,\ngrid-based, predictive, and semantically-interpretable ER. In particular, we\nprovide a proof of concept for the spatio-temporal fusion of multiple camera\nsequences and short-term prediction in such an ER. Our design utilizes a strong\nsemantic segmentation network together with depth and egomotion estimates to\nfirst extract semantic information from multiple camera streams and then\ntransform these separately into egocentric temporally-aligned bird's-eye view\ngrids. A deep encoder-decoder network is trained to fuse a stack of these grids\ninto a unified semantic grid representation and to predict the dynamics of its\nsurrounding. We evaluate this representation on real-world sequences of the\nCityscapes dataset and show that our architecture can make accurate predictions\nin complex sensor fusion scenarios and significantly outperforms a model-driven\nbaseline in a category-based evaluation.",
+                    "authors": "Lukas Hoyer, Patrick Kesper, Anna Khoreva, Volker Fischer",
+                    "published": "2019-03-21",
+                    "updated": "2019-07-26",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.AI",
+                        "cs.LG",
+                        "cs.RO"
+                    ],
+                    "main_content": "Introduction In recent years, deep learning methods have been investigated to control autonomous systems, such as self-driving cars or robots. An important property of such systems is their capability to perceive complex situations using multiple sensors and to act accordingly in a fast and reliable way. To enable intelligent decision making, a common environment representation (ER) as interface between different sensors and the downstream control has to be provided. Target in Image Space Target Frame 1 Frame 2 Prediction Unknown Road Sidewalk Vegetation Building Pole+Sign Bus Pedestrian Loss Mask Figure 1. (best viewed in color) Semantic grid prediction: Our framework is based on semantic segmentations in image space (top left). These are transformed into the proposed bird\u2019s eye view semantic grid representation (top right). We feed our architecture with a sequence of input frames (bottom left and bottom center) to predict the subsequent frame (bottom right), which is compared with the target frame (top right). In the shown example, a bus (dark turquoise) is entering the grid at the top (bottom row). One sees that our architecture is able to predict its movement further into the grid even though the vehicle became just visible in frame 2. For details on the loss mask and prediction artifacts, see Sec. 5. Such an ER should have certain properties. First, it should unify different sensor representations to support sensor fusion. In that way, downstream algorithms can work with this abstract representation without knowledge of the used sensors. Second, the ER should be interpretable to modularize the system and enhance human accessibility. So, the system can be debugged and understood more easarXiv:1903.08960v2  [cs.CV]  26 Jul 2019 \fily, improving its reliability. And third, it should be predictive to compensate for system-inherent latencies caused by sensor measurements or signal processing, which is particularly important in the context of recent computationally expensive computer vision algorithms. Due to these reasons, we present and evaluate a proof of concept for generating an egocentric, interpretable, predictive, and grid-based ER, most importantly including the spatio-temporal fusion of semantic information of multiple cameras and short-term prediction. We call this ER semantic grid. It is a bird\u2019s eye view 2D projection of semantic features of the environment (see Fig. 1 top row). Our concrete architecture for producing semantic grids can be disentangled into two major parts (see I and II in Fig. 2): First, semantic information is extracted from each camera signal using deep neural networks for semantic segmentation (see S in Fig. 2). The semantic information is then spatially transformed into a top-down, bird\u2019s eye view semantic grid, using depth information provided by the stereo camera (see P in Fig 2). Grids from different cameras and past time frames are temporally aligned to a certain future time \u03c4 using the agent\u2019s egomotion (see T in Fig. 2). Note that in particular independently moving objects will be transformed incorrectly, as there is no motion model of other objects than the agent itself, yet. Second, these spatio-temporally aligned representations are combined into a single grid using a deep encoderdecoder (ED) neural network. This network contains all trainable parameters of our architecture and has the nontrivial task to fuse the grids from different cameras and past times, making assumptions about the environment, as well as predicting the motion of potentially multiple dynamic objects. Such a prediction is shown in Fig. 1. Even though in this work we focus on multi-camera fusion, our architecture can be extended to other modalities such as LiDAR given an appropriate algorithm for extracting spatial semantic features from the sensor. We have evaluated our architecture with respect to different semantic categories on the camera-based Cityscapes dataset of driving scenarios. Our approach signi\ufb01cantly outperforms solely model-driven baselines for single camera and multi-camera prediction, considering missing egomotion information, different sequence length, and varying prediction horizons. To the best of our knowledge, we are the \ufb01rst to investigate a deep convolutional neural network for short-term prediction and multi-camera sequence fusion on semantic grids. We evaluate the proposed architecture on real-world data. Our approach enables autonomous systems to perceive the environment with multiple cameras and helps to mitigate the problem of long runtimes of recent computer vision algorithms. After giving an overview of related work in Section 2 camera 1 at C1 camera 2 at C2 m(ti) tB 1 tB 2 S P T \u03b3(ti) g(ti \u2192\u03c4) stack encoder decoder prediction \u02c6 g(\u03c4) I II Figure 2. (best viewed in color) Schematic overview of our semantic grid fusion and prediction architecture. The framework is fed with image sequences m(ti) of different cameras from past times ti. These are semantically segmented (S), projected (P) into a agent-centric top-down view 2D point cloud \u03b3(ti) and transformed (T ) to the same time \u03c4 resulting in spatio-temporally aligned grids g(ti \u2192\u03c4). The encoder-decoder deep neural network (ED) fuses these grids and predicts environment dynamics. The two-step description in the Methods Section is indicated as I and II on the right. Note that the point cloud \u03b3(ti) is represented as grid for visualization purpose. and a general description of the presented approach in Section 3, we lay out experimental setups in Section 4, present the results in Section 5, followed by a discussion and conclusion in Sections 6 and 7 respectively. 2. Related Work Due to the recent discovery of certain cells dedicated to spatial referencing in brains [11], grid-based representations seem biologically plausible. One well known example of interpretable grid-based ER are singleand multi-sensor occupancy grids (e.g., [4, 18]), which have also gained interest in recent deep learning studies (e.g., [6, 10, 17]). Due to the lack of suitable feature extractors before the raise of 2 \fdeep learning, these grids normally only consisted of one semantic feature encoding the occupancy property. However, now semantic grid representations have become a recent subject of interest. For instance, [8, 9, 14] deal with the generation of single-camera, non-predictive semantic grids, while [1] utilize them for decision-making in autonomous driving as semantic grids are easier to simulate than raw sensor data. Other approaches generate sparse bird\u2019s eye view representations of the environment by detecting objects in image space and transforming them to the top-down view (e.g., [21, 25]). We want to differentiate and hence disentangle two different but both desired predictive properties of autonomous agents. First, the agent should be able to reason about future events, e.g., for planning, and in this sense should be able to form midor long term predictions. For instance, [1, 12, 22] perform explicit long-term vehicle trajectory prediction using RNNs. From this, we differentiate a second form of prediction, intrinsic to perception. Sensors and their downstream signal processing (e.g. semantic segmentation) induce a temporal delay between the actual present state of the world and the agent\u2019s belief about this state. Our aim is to design an environment representation that can compensate these short-term system-inherent latencies and synchronize data from different time horizons. For short-term prediction of camera data, there are approaches predicting extracted image features such as object bounding boxes [2], semantic segmentation [16], or instance segmentation [15]. However, all of these work are in the sensor-dependent image space, which unnecessarily complicates the task considering downstream sensor fusion. In contrast, [6, 7, 19, 20] and [24] employ an end-to-end trainable recurrent architecture to directly predict an unoccluded occupancy grid from laser data, capable to track multiple objects. We distinguish ourselves by investigating the capacity of our architecture in the context of sensor fusion and further, by using camera data instead of laser data, enabling semantically richer representations. Moreover, we combine semantic predictions with a moving sensor instead of analyzing both scenarios separately. 3. Methods In this section, we describe our architecture that generates semantic grids from semantic segmentations of camera images (see Fig. 2). A semantic grid is a g \u2208RWG\u00d7HG\u00d7F , with WG, HG, F \u2208N denoting the grid\u2019s width, height, and number of semantic features (e.g. different object classes) respectively. Each grid cell represents a certain area in space and hence the spatial coordinate is determined by the cell\u2019s indices. The semantic grid is egocentric in the sense that the agent is constantly located at the same grid cell. Generating synchronized semantic grids (Fig. 2, I): Given a RGB image m \u2208RWI\u00d7HI\u00d73 and a depth map d \u2208RWI\u00d7HI, where WI, HI \u2208N denote the pixel resolution, we \ufb01rst use some semantic segmentation network S to generate semantic information in image space s = S(m) \u2208 [0, 1]WI\u00d7HI\u00d7F . Second, this sensor-dependent representation (in image space) is transformed into a spatially aligned (w.r.t. the agent) continuous 2D point cloud \u03b3 \u2208RWI\u00d7HI\u00d7(2+F ), using the depth map d to project the semantic information from pixel space to the \ufb02oor plane of the camera coordinate system at recording time ti: \u03b3(ti) = P(s(ti), d(ti)) (1) For that purpose, each segmentation pixel suv \u2208RF with u \u2208{1, . . . , WI) and v \u2208{1, . . . , HI} is transformed to a 2D point with semantic information \u03b3uv = (\u03b3x uv, \u03b3z uv, \u03b3F uv)T \u2208R2 \u00d7RF in the agent coordinate system A. This transformation is achieved using the intrinsic camera matrix K and the extrinsic parameters RC\u2192A and tC\u2192A describing the camera pose C with respect to the agent (see supplementary material for details on those matrices). \uf8eb \uf8ed \u03b3x uv \u03b3y uv \u03b3z uv \uf8f6 \uf8f8= (RC\u2192A, tC\u2192A) \u00b7 K\u22121 \u00b7 duv \u00b7 \uf8eb \uf8ed u v 1 \uf8f6 \uf8f8 (2) Using these agent-centric 3D coordinates, the semantic features s are projected onto the ground \ufb02oor. For simplicity, we de\ufb01ne P as well as the following operators on a single point while they are applied to the whole point cloud. P(suv, duv) := (\u03b3x uv, \u03b3z uv, suv)T (3) Third, the semantic grids are spatially aligned according to the agent\u2019s current orientation \u03b1ti t0 = Pi j=1 \u02d9 \u03b1tj \u00b7 (tj \u2212tj\u22121), which is the integrated angular component of the egomotion, and the point cloud is discretized to the grid representation. g(ti) = D(T1(\u03b3(ti), \u03b1ti t0)) (4) T1(\u03b3uv, \u03b1) := \uf8ee \uf8f0 cos \u03b1 \u2212sin \u03b1 0 sin \u03b1 cos \u03b1 0 IdF \uf8f9 \uf8fb\u00b7 \uf8eb \uf8ed \u03b3x uv \u03b3z uv \u03b3F uv \uf8f6 \uf8f8 (5) Thereby, rotations of the agent do not cause rotation of the entire grid, but only result in a change of the agent\u2019s internal orientation \u03b1ti t0 and a rotation of potential new data, reducing quantization errors. The discretization D assigns the 2D points \u03b3uv to grid cells gkl. If two points are allocated to the same grid cell, they are prioritized preferring small and dynamic classes. In that way, we ensure that no important information is occluded by another class. If no point is assigned to a grid cell, it is classi\ufb01ed as unknown (black). 3 \fConv2D + Batch Normalization + Activation Max Pooling Dropout Upsampling + Conv2D Softmax \ud835\udc54(\ud835\udc61\ud835\udc56\u2192\ud835\udf0f)   \ud835\udc54(\ud835\udf0f) \ud835\udc56= 1 \ud835\udc56= 2 \ud835\udc56= 3 Figure 3. (best viewed in color) Encoder-decoder CNN (ED). The ED combines synchronized semantic grids into one prediction including dynamic object transformation, temporal \ufb01ltering, and multi-camera fusion. And fourth, the integrated translational component of egomotion q\u03c4=tk ti = Pk\u22121 j=i \u02d9 qtj \u00b7 (tj+1 \u2212tj) is used for the parameter-free temporal extrapolation (grid translation) into the future time \u03c4: g(ti \u2192\u03c4) = T2(g(ti), q\u03c4 ti). (6) T2(gkl, q) := gk\u2212qx,l\u2212qz (7) assuming that q is represented in pixel coordinates. If k + qx >= WG or l + qz >= HG, gkl is assigned the class unknown (black). Let {m(t1), . . . , m(tn)} denote an input sequence of length n of past images at times t1, . . . , tn from one sensors, then {g(t1 \u2192\u03c4), . . . , g(tn \u2192\u03c4)} are accordingly synchronized to the same future time \u03c4 using the parameterfree (and hence not trained) transformations P and T = T2 \u25e6D \u25e6T1. Predictive fusion architecture (Fig. 2, II): These, now synchronized single-sensor semantic grids, are stacked and fed into an encoder-decoder deep neural network (ED) (see Fig. 2 stack). Its task is to fuse the grids from multiple sensors and different original time steps and to incorporate object motion. The ED network yields the predicted semantic grid \u02c6 g(\u03c4) at time \u03c4. In case \u03c4 \u2212tn covers the system-inherent latency, \u02c6 g(\u03c4) represents the agents belief about the actual present situation. The ED is trained using self-supervised learning. It is provided with grid input sequences {g(t1 \u2192\u03c4), . . . , g(tn \u2192\u03c4)} for multiple sensors, while withholding the grid g(\u03c4), measured at the time \u03c4 > tn, as ground truth. Loss and validation metrics: For training and validation, we use the categorical cross-entropy loss function and the intersection over union (IoU) [13], respectively. For both, the area M that the sensors have already seen in past frames, but where no ground truth information is available for the target frame, is ignored. For this purpose, we calculate a masked loss Lmasked = f(g(\u03c4), (1 \u2212M) \u00b7 \u02c6 g(\u03c4) + M \u00b7 g(\u03c4)), (8) where g(\u03c4) and \u02c6 g(\u03c4) are the ground truth and predicted semantic grids respectively, f is some distance measure (e.g. categorical cross entropy), and M is the mask of the covered area without ground truth information M = Mcovered \u2212 Mtarget. Here, Mtarget denotes the area with ground truth information Mtarget = known(g(\u03c4)) and Mcovered the area the sensors have seen during the entire synchronized sequence (including the target frame g(\u03c4)) Mcovered = known(g(\u03c4) + n P i=1 g(ti \u2192\u03c4)). The function \u201dknown\u201d determines the area within a grid where the classi\ufb01cation is not the class \u201dunknown\u201d. In the area determined through M, the network is not punished for any predictions. In that way, we want to encourage the network to remember areas, which are newly occluded in the target frame (e.g. the waiting car in Fig. 7b that is newly occluded by another car in the target frame; M visualized in white color), instead of predicting the class \u201dunknown\u201d (white/black). 4. Experiment Setup Dataset: For our experiments, we have chosen the Cityscapes dataset of driving scenarios [5], which provides real-world RGB image sequences (30 frames, 17 Hz) including the preprocessed disparity of the stereo camera, the egomotion of the vehicle, as well as training data for semantic segmentation. In that way, Cityscapes allows the generation of semantic grids with various and diverse classes. As some classes are comparatively rare due to their small 2D projection during the semantic grid generation for training, we have combined large vehicles (bus, truck, and train) as well as small static objects (poles, traf\ufb01c signs, and traf\ufb01c lights), which have a similar semantic meaning and behave similarly in the semantic grid representation. Therefore, the 4 \fFigure 4. (best viewed in color) Sensor Split 2 for simulating multiple sensors. The semantic segmentation of the camera image (top left) is cropped to simulate two separate sensors (top center and right). In the lower row the associated egocentric semantic grids are visualized. The white dashed lines mark the blind spot. color coding for the classes in a semantic grid (see Tab. A1) does not exactly match with the Cityscapes colors. To evaluate the performance of the semantic grid prediction with respect to multi-camera fusion, we split the camera frames into separate image sections. While the Split 1 scenario just divides the images into a lower and an upper part, the Split 2 scenario additionally has a left and right black margin, as well as a blind area between the lower and upper image section (see Fig. 4). These splits are especially challenging due to the vertical split direction, as there are ambiguities when merging the resulting semantic grids. Moreover, the image is split close below the horizon, which is the most crucial part of the image. In that way, we want to underline the capabilities of the ED to solve ambiguities, track objects between both cameras, and conclude information about blind spots. Semantic Grid Generation: We generate the semantic segmentation of the Cityscapes sequences using the state-of-the-art deep neural network DeepLabv3 [3], which achieves 80.31 % IoU on Cityscapes. The semantic labels are transformed to the semantic grid using the ego motion of the vehicle and the disparity of the stereo camera (see Section 3). For the semantic grid, we have chosen a size of 128 \u00d7 128 pixels, which is the equivalent of 100 \u00d7 100 meters. Dataset of Grid Sequences: For training and evaluation of our semantic grid prediction framework, the frames o, o+ s, o+2s, . . . , o+(n\u22121)s, where o is the sequence offset and s the step size, are used as input frames, while frame o+n\u00b7s represents the target frame. Here, we have used the step size s = 5. The training sequences are overlappingly sampled (o is chosen arbitrarily), whereas the validation sequences are disjoint and their ground truth frames are aligned for the different experiments. In case of n = 2, we work with 59500 training sequences and 500 validation sequences. All results are reported on the validation dataset. Encoder-Decoder CNN: In this paper, we have used an encoder-decoder Convolutional Neural Network (ED CNN) similar to U-Net [23] (see Fig. 3). The encoder with depth d contains d down sampling blocks. Each of them reduces the spatial resolution by half and doubles the initial number of features f. The decoder consists of d\u22121 up sampling blocks and \ufb01nally a softmax layer. Between each corresponding downand upsampling block are skip connections to enable dense predictions [13]. Further information on the implementation and training can be found in the supplementary material. 5. Experiments In this section, the behavior of the proposed framework is studied. For this purpose, we have designed several experiments, each analyzing certain aspects of our architecture. For all experiments, the results were analyzed according to categories that contain similarly behaving semantic classes: static (unknown, building, road, sidewalk, vegetation), vehicles (car, bus, truck, train), small static (pole, traf\ufb01c light, traf\ufb01c sign) and small dynamic (person, bicycle). Note that bus, truck, and train as well as pole, traf\ufb01c light, and traf\ufb01c sign were already combined into one class in the dataset. The class-wise mean intersection over union (IoU) for each of those categories is plotted in Fig. 5. The experiment labels (DC, NT, SF,...) are associated with the following paragraphs. DC Default con\ufb01g: We compared our framework utilizing a single sensor (ED-DC) with a simple baseline (BLDC), which transforms the last sensor frame into the target time. NT No explicit translation: We trained the ED to estimate and apply the translational ego motion q\u03c4 tn without additional external sensor input or the explicit translation step T2 (ED-NT). SF Sensor fusion: We analyze the performance of the framework to fuse grids from multiple simulated cameras (ED-SplX). SL Sequence length: We varied the sequence length to study its relation with the prediction performance (EDSLX). PH Prediction horizon: We evaluated how far the network is able to predict into the future (ED-PHX). Baselines: To assess the performance of the proposed semantic grid prediction, we designed several baselines that leave out the ED. The simplest baseline BL-NT just replicates the last input grid as prediction \u02c6 g(\u03c4) = g(tn), while the baseline BL-DC transforms the last input frame using the vehicle\u2019s egomotion into the time of the prediction \u02c6 g(\u03c4) = T2(g(tn), q\u03c4 tn). To have baselines that consider the dif\ufb01culties of the sensor split (Sp), we also designed BLSp1 and BL-Sp2, which overlay the semantic grid of the lower sensor with the grid of the upper sensor. 5 \fBL-DC ED-DC BL-NT ED-NT BL-Sp1 ED-Sp1 BL-Sp2 ED-Sp2 ED-SL2 ED-SL3 ED-SL4 ED-SL5 BL-PH1 ED-PH1 BL-PH2 ED-PH2 BL-PH3 ED-PH3 0.55 0.60 0.65 0.70 0.75 0.80 IoU DC NT SF SL PH (a) static BL-DC ED-DC BL-NT ED-NT BL-Sp1 ED-Sp1 BL-Sp2 ED-Sp2 ED-SL2 ED-SL3 ED-SL4 ED-SL5 BL-PH1 ED-PH1 BL-PH2 ED-PH2 BL-PH3 ED-PH3 0.20 0.25 0.30 IoU DC NT SF SL PH (b) small static BL-DC ED-DC BL-NT ED-NT BL-Sp1 ED-Sp1 BL-Sp2 ED-Sp2 ED-SL2 ED-SL3 ED-SL4 ED-SL5 BL-PH1 ED-PH1 BL-PH2 ED-PH2 BL-PH3 ED-PH3 0.3 0.4 0.5 0.6 IoU DC NT SF SL PH (c) vehicles BL-DC ED-DC BL-NT ED-NT BL-Sp1 ED-Sp1 BL-Sp2 ED-Sp2 ED-SL2 ED-SL3 ED-SL4 ED-SL5 BL-PH1 ED-PH1 BL-PH2 ED-PH2 BL-PH3 ED-PH3 0.25 0.30 0.35 0.40 IoU DC NT SF SL PH (d) small dynamic Figure 5. Category-wise mean IoU reported on the validation set with respect to different framework con\ufb01gurations. Further details on the experiments (columns within the plots) are provided in the paragraphs associated with the corresponding references (DC, NT, SF,...). (a) 1.0 Frame 1 Frame 2 Target (b) Frame 1 Frame 2 Prediction Target Certainty 1.0 0.0 Figure 6. (best viewed in color) Example prediction sequence using ED-DC. (a) shows the sequence of semantic segmentations while (b) presents the associated semantic grids. The framework is fed with frame 1 and 2. The prediction is compared with the target frame. The maximum softmax activation of the prediction is visualized on the right. In this example, a car (blue) is moving to the right. Black areas represent unknown parts of the grid and the white color in the target frame visualizes the loss mask M. Note that the left car in the target frame cannot be predicted as it was not visible in frame 1 and 2. Default con\ufb01g (DC): We compared our framework utilizing a single sensor (ED-DC) with the baseline (BL-DC). As default ED con\ufb01guration, we have used n = 2, d = 3, and f = 64, as this con\ufb01guration provides a good trade-off between performance and runtime (see supplementary material). In Fig. 5a and 5c column DC, it can be seen that EDDC signi\ufb01cantly outperforms its baseline BL-DC for large static objects and vehicles. On the one hand, the higher performance for large static objects demonstrates the ED\u2019s ability to combine the information of the input frames and predict an improved combined grid, which also may contain information about newly occluded areas (see white area of target frame in Fig. 7b). On the other hand, the high performance for vehicles indicates that the ED is able to predict 6 \f(a) Frame 1 Frame 2 Prediction Target Certainty 1.0 (b) Frame 1 Frame 2 Prediction Target Certainty 1.0 (c) Frame 1 Frame 2 Prediction Target Certainty 1.0 Figure 7. (best viewed in color) Series of sequences and their associated predictions. (a) The left car (blue) is approaching while the right one drives away (ED-DC). (b) The ED is able to remember the waiting car, which is occluded in the white area of the target frame (ED-DC) (c) A bicycle (maroon) is moving to the right (ED-Sp2). In contrast to the other examples, it uses the Split 2 dataset. Frame 1 and frame 2 are visualized as combination of both sensors. Moreover, the curvature of the left sidewalk towards the blind spot is recognized and correctly completed. Also, the CNN concludes that there is sidewalk around the pole on the right side. the motion of dynamic objects and use it for the prediction (see Fig. 6b). Even though classes covering a small area are supposedly quite challenging for ED architectures, our framework is able to outperform the baseline for small dynamic (Fig. 5d) and small static (Fig. 5b) objects (compare ED-DC with BL-DC). However, the absolute IoU is relatively low for both ED-DC and BL-DC. During qualitative analysis of the data, we found that the effect may be due to temporal noise caused by the alignment of RGB and depth image as well as the grid discretization. Small objects sometimes vanish and appear again making it dif\ufb01cult to predict them. This effect can be mitigated by excluding segmentation borders and regions with a minimum depth difference as well as by increasing the resolution of the semantic grid. To underline the abilities of the ED CNN as well as the straightforward human-interpretability of the semantic grid, Fig. 1, 6, and 7 visualize selected predictions from the validation dataset. In Fig. 6, a car (blue) is moving perpendicular to the camera. The ED is provided with frame 1 and frame 2. It can be seen, that it translates the car according to its velocity to the right, which is quite accurate compared with the target frame. On the right side of Fig. 6b, the maximum softmax activation of the ED is visualized. It can be interpreted as certainty of the CNN\u2019s predictions. While the network is certain (red) about already seen static objects such as roads (purple), the maximum softmax activation indicates its uncertainty (green) in the area behind the car and at the borders between classes. The ED is even able to differentiate between instances of the same class and translate them according to their different velocities. This can be seen in Fig. 7a, in which the left car (blue) is approaching while the right one is driving away. Moreover, the ED can correctly predict areas that would have been occluded in the target frame while they were visible in the frames before (see waiting car in Fig 7b and compare with white area M in target frame). While the network is able to predict correct classes in most of M, it often fails at the bottom of the grid (see Fig. 1) as there is never ground truth information available during the training process. For a deployable system, we would remove this region from M. Further examples are given in the supplementary material. No explicit translation (NT): To study the ED\u2019s ability to estimate the translational egomotion and the motion of dynamic objects simultaneously and superimpose both, we disabled the explicit translation T2 of our framework. However, the ED is still provided with the orientationally aligned semantic grids g(ti). The framework with disabled translation (ED-NT) reaches the performance of the framework with explicit transformation (ED-DC) for large static areas and vehicles (see Fig. 5a and 5c column DC and NT). Hence, the network is able to solve both tasks in general. 7 \fHowever, it struggles with small objects (see Fig. 5b and 5c), probably, as a more precise estimate of the egomotion is needed to predict them correctly due to their small size. Sensor fusion (SF): In this experiment, the ability of the framework to fuse grids from multiple simulated cameras is evaluated. When using the Split 1 dataset, the framework (ED-Sp1) achieves as good results as without a camera split (ED-DC), as can be seen in Fig. 5 in the corresponding columns. This shows, that the ED is able to fuse multiple sensors and solve ambiguities. However, a CNN trained on Split 2 (ED-Sp2) performs worse than the other con\ufb01gurations so far, as there is information missing in the input data due to the margins left and right of the sensor and the blind area between both sensors (see Fig. 4 and 7c). This fact can also clearly be seen in the baseline of Split 2 (see Fig. 5 BL-Sp2). Even though the missing information decreases the absolute performance, the ED is still able to maintain the difference to its associated baseline compared to other models (see BL-DC/ED-DC and BL-Sp2/ED-Sp2 in Fig. 5). For static objects, the CNN is even able to increase the difference as it is able to make assumptions about the environment. For instance, ED-Sp2 concludes that there is sidewalk around the pole on the right side of the grid in Fig. 7c even though it has never seen the sidewalk before. Note that during training, the semantic grid generated of the whole camera image, instead of the limited image sections for the virtual cameras, was provided as target frame. Sequence length (SL): To analyze the in\ufb02uence of the sequence length, we varied n. It can be seen in Fig. 5 column SL that one additional input frame (ED-SL3) improves the performance of all categories in comparison with ED-SL2 as it probably allows better denoising and speed estimation. However, a longer sequence length also decreases the number of sequences that can be sampled from the whole training dataset, leading to a drop of performance for ED-SL4/5. Prediction horizon (PH): In the last experiment, we have analyzed, how far our framework is able to predict into the future. We have maintained the same step size of about 300 ms between the input frames and evaluated target frames one, two, and three steps after the last input frame (ED-PHX). As expected, the performance of the ED drops with increasing prediction horizon (see Fig. 5 column PH). Still, the CNN is able to outperform its associated baselines (BL-PHX) for large and medium objects. However, using a long prediction horizon, the ED struggles with small objects, which is probably caused by the increased sensitivity towards spatial mismatches. 6. Discussion Grid-based representations are potentially bene\ufb01cial in applications where very fast reaction times are required and where the duration to compute and update the ER should be independent of the number of objects. However, one limitation of grid-based representations is their linear scalability with respect to the number of semantic features. In the context of autonomous agents, this is especially true for objects carrying a high variety of semantic information, such as road signs. However, the grid could provide an attention mechanism to identify relevant areas for further processing (e.g. road sign recognition). In contrast to other representations, which are not interpretably modularized, our approach provides direct interfaces to verify and test the trained encoder-decoder fusion architecture. Normally, for camera or laser input signals, it is hard to synthesize new, especially critical, scenarios. Whereas for our approach, it is comparably easy to generate such sequences of semantic input and according output grid representations. Further, the interpretable interface functions as a human-readable monitor which can for example be used for debugging or determining corner cases. Even though we have concentrated on semantic features in this work, the semantic grid can be extended to support a variety of other information. As the semantic grid already provides an abstract, scale-invariant, and low resolution representation of the environment, down-stream algorithms can easily extract further information such as correspondence information or object tracks. If more detailed information is necessary, additional semantic maps encoding local features provided by preprocessing steps can be added to the semantic grid (e.g. local velocities, pedestrian pose, instance segmentation, or uncertainty). Other interesting subjects for future work include: multiscale or dynamic-resolution grid representations, multimodal sensor inputs (e.g., LiDAR or radar), sensor signals with different dynamics (e.g., frame rates, offsets), and alternative ED architectures for fusion. In this work, we omitted the estimation of the actual system-inherent latency. The latency depends on the used hardware and other choices, such as the architecture to extract the semantic information from sensors or the chosen encoder-decoder architecture. Possible solutions are estimating the delay on the \ufb01nal real-world system and using this estimate for the temporal synchronization of the grid representation before fusion or integrating an additional adaptive component that estimates the current latency. The presented architecture should be thought of as part of a larger system. For example, the provided environment representation could be combined with a recurrent representation to store a mid-term ER, a SLAM approach for global mapping, or with a decision making module. In this context, the semantic grid can act as interpretable interface, which can be used for debugging or determining corner cases. Moreover, this interface simpli\ufb01es synthesizing arti\ufb01cial training data for new, especially critical, scenarios in comparison to the simulation of raw sensor signals. 8 \f7."
+                }
+            ]
+        },
+        "edge_feat": {}
+    }
+}
\ No newline at end of file