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+{
+    "url": "http://arxiv.org/abs/2405.04700v1",
+    "title": "Robust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures",
+    "abstract": "Large Language Models (LLMs) deployed on edge devices learn through\nfine-tuning and updating a certain portion of their parameters. Although such\nlearning methods can be optimized to reduce resource utilization, the overall\nrequired resources remain a heavy burden on edge devices. Instead,\nRetrieval-Augmented Generation (RAG), a resource-efficient LLM learning method,\ncan improve the quality of the LLM-generated content without updating model\nparameters. However, the RAG-based LLM may involve repetitive searches on the\nprofile data in every user-LLM interaction. This search can lead to significant\nlatency along with the accumulation of user data. Conventional efforts to\ndecrease latency result in restricting the size of saved user data, thus\nreducing the scalability of RAG as user data continuously grows. It remains an\nopen question: how to free RAG from the constraints of latency and scalability\non edge devices? In this paper, we propose a novel framework to accelerate RAG\nvia Computing-in-Memory (CiM) architectures. It accelerates matrix\nmultiplications by performing in-situ computation inside the memory while\navoiding the expensive data transfer between the computing unit and memory. Our\nframework, Robust CiM-backed RAG (RoCR), utilizing a novel contrastive\nlearning-based training method and noise-aware training, can enable RAG to\nefficiently search profile data with CiM. To the best of our knowledge, this is\nthe first work utilizing CiM to accelerate RAG.",
+    "authors": "Ruiyang Qin, Zheyu Yan, Dewen Zeng, Zhenge Jia, Dancheng Liu, Jianbo Liu, Zhi Zheng, Ningyuan Cao, Kai Ni, Jinjun Xiong, Yiyu Shi",
+    "published": "2024-05-07",
+    "updated": "2024-05-07",
+    "primary_cat": "cs.LG",
+    "cats": [
+        "cs.LG",
+        "cs.AI",
+        "cs.DC",
+        "cs.IR"
+    ],
+    "label": "Original Paper",
+    "paper_cat": "Retrieval AND Augmented AND Generation AND RAG",
+    "gt": "Large Language Models (LLMs) deployed on edge devices learn through\nfine-tuning and updating a certain portion of their parameters. Although such\nlearning methods can be optimized to reduce resource utilization, the overall\nrequired resources remain a heavy burden on edge devices. Instead,\nRetrieval-Augmented Generation (RAG), a resource-efficient LLM learning method,\ncan improve the quality of the LLM-generated content without updating model\nparameters. However, the RAG-based LLM may involve repetitive searches on the\nprofile data in every user-LLM interaction. This search can lead to significant\nlatency along with the accumulation of user data. Conventional efforts to\ndecrease latency result in restricting the size of saved user data, thus\nreducing the scalability of RAG as user data continuously grows. It remains an\nopen question: how to free RAG from the constraints of latency and scalability\non edge devices? In this paper, we propose a novel framework to accelerate RAG\nvia Computing-in-Memory (CiM) architectures. It accelerates matrix\nmultiplications by performing in-situ computation inside the memory while\navoiding the expensive data transfer between the computing unit and memory. Our\nframework, Robust CiM-backed RAG (RoCR), utilizing a novel contrastive\nlearning-based training method and noise-aware training, can enable RAG to\nefficiently search profile data with CiM. To the best of our knowledge, this is\nthe first work utilizing CiM to accelerate RAG.",
+    "main_content": "INTRODUCTION The emerging Large Language Models (LLMs) are deployed primarily on centralized cloud platforms [1, 2] (Cloud LLMs), raising concerns about user privacy and trustworthy issues [3]. These issues become even more prominent in areas such as healthcare [4], companionship [5], and personal assistance [6], where the user privacy and trustworthiness of LLMs are crucial. To address these issues, the cloud LLMs will eventually transform into personalized LLMs, capable of generating personalized responses, deployed on edge devices (Edge LLMs), where users can keep all their private data and the model learns from those data locally. To better suit the needs of individual users, Edge LLMs must learn from user interactions. However, their capability of learning is constrained by their limited RAM and computational power. Similar to Cloud LLMs, the Edge LLMs primarily learn by finetuning their model parameters. Yet, given that these models often contain over 3 billion parameters, updates can be challenging, even with numerous efforts to accelerate them [7\u20139]. For example, using the experimental high-performance embedded system like NVIDIAAGX, the pockengine method [9] can still take 90 hours to learn from a middle-sized dataset Alpaca with only 52k documents, making this option impractical for normal users. E x : \u201cI  am sick?\u201d Sentence  Embedding  Model E(x) User Query Profile Data Embedding Space S \u2026 P(x, d83) P(x, d29) P(x, d37) Top k (k = 1) DAC ADC \u2026 CiM E(d83) Data: d83, d29, d37 User Query LLM Output NVM Digital Logic \ud835\udc6c(\ud835\udc99) \u2219\ud835\udc6c(\ud835\udc85\ud835\udc8a) = \ud835\udc0f(\ud835\udc31, \ud835\udc85\ud835\udc8a) E(d1) E(d2) E(d3) E(d4) E(dn) Document 2 Document 1 Document n \u2026 Figure 1: The workflow of RAG on edge-based CiM. CiM performs max inner product search (MIPS) to retrieve the top-ranked documents, concatenating them with user query to allow the LLM to generate personalized responses. Retrieval-augmented generation (RAG), on the other hand, is a more resource-efficient choice [10], and hence becoming the de facto learning method for Edge LLMs. In a typical RAG system, it consists of a retriever and a generator. The retriever is commonly backed by max inner product search (MIPS). When the retriever receives a user query, it will retrieve the most relevant document from profile data, as shown in Figure 1. The profile data has many documents, and each document \ud835\udc51\ud835\udc56contains specific information that may be relevant to user queries. The generator can be seen as a LLM, which takes the user query \ud835\udc65and retriever-obtained documents as a prompt and generates a corresponding response. For every document\ud835\udc51\ud835\udc56and the user query \ud835\udc65, RAG utilizes a sentence embedding model shown in Figure 1 to convert them into vectors (i.e., \ud835\udc38(\ud835\udc51\ud835\udc56) and \ud835\udc38(\ud835\udc65), respectively). The vectors for documents can be named as document embeddings and stored as a matrix as shown in Figure 1. The vector for user query, named query embedding \ud835\udc38(\ud835\udc65), will be used in MIPS to perform inner product with every document embedding. The larger the product \ud835\udc43(\ud835\udc65,\ud835\udc51\ud835\udc56), the more semantic similar it will be between the user query and the document. Using RAG, Edge LLMs can provide user-preferred responses by retrieving relevant documents from profile data, and the profile data can be incrementally updated with new documents. This is an efficient learning process without costly updating the model parameters via fine-tuning [11]. Other than the inevitable LLM inference cost, the primary computational cost of RAG is about retrieval, which is more than ten times less than the cost of updating model parameters. While the computational cost of RAG is more edge-friendly, there still exist two issues impeding RAG from being deployed for real-time user interaction on Edge LLMs. Firstly, the growing profile data as stored cannot be unlimited without affecting the access time. If the size of the profile data exceeds the RAM capacity, arXiv:2405.04700v1  [cs.LG]  7 May 2024 \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 it will need to be offloaded into the storage, such as a hard disk drive (HDD) or solid-state drive (SSD). Accessing data from HDD or SSD will significantly increase the data transfer latency [12], rendering real-time user interaction impractical. Secondly, the core retrieval method of RAG, MIPS, may experience decreased efficiency as profile data grows, and it can become potentially prohibitive when dealing with overwhelmingly large datasets. For example, on Raspberry Pi 4B, MIPS can take 5 minutes to find one appropriate profile data among 21M documents [10], which is even longer than the 2-minute inference time of an Edge LLM. Unfortunately, few efforts have been made to optimize RAG towards Edge LLMs. Thus, we propose to utilize the Computing-in-Memory (CiM) architecture to address this issue. As shown in Figure 1, CiM architectures using memory arrays have shown substantial promise in accelerating matrix-vector multiplication [13], which is the key operation of MIPS. The CiM architectures often utilize massive parallel processing to perform computations directly within the memory array where the data is stored, such that they can minimize the data movement through in-situ data access and significantly increase the throughput [14]. Given the same amount of documents, CiM can finish computation within 50ms [15], which is negligible compared to the computation latency on normal edge devices. Furthermore, by incorporating non-volatile memory (NVM) devices, such as phase-change memories (PCMs), resistive random-access memories (RRAMs), and ferroelectric field-effect transistors (FeFETs), CiM can outperform conventional MOSFET-based designs in terms of energy efficiency [16]. 0.00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Level of noise ( ) 0.0 0.2 0.4 0.6 0.8 1.0 Accuracy Citation Movie Rating News DBLP Figure 2: The impact on MIPS accuracy when the RAG\u2019s document embedding is perturbed by various levels of Gaussian noise caused by the device variations. An accurate retrieval means the document retrieved under the impact of the noise is the same as that retrieved without noise. Unfortunately, simply changing the underlying hardware is not enough, as the non-idealities of the NVM devices in CiM array could greatly deteriorate the RAG performance. First, the operations performed in CiM architectures are susceptible to various sources of noise, including electronic noise (thermal, shot, and flicker), deviceto-device variability, and line noise from the supporting circuitry [17]. These noise sources can corrupt the computations, especially when the signal levels are close to the noise floor, which is a common scenario in high-precision tasks. Such noise issues are critical in RAG applications where the accuracy and quality of the generated content heavily rely on the precision of the underlying computations. Additionally, the CiM architecture is primarily designed and optimized for low-resolution computation [18]. Moreover, CiM arrays are typically sized at a fixed dimension, such as 64x64 [19], which is different from the documents\u2019 embedding dimension (e.g., 128). Therefore, both RAG\u2019s data precision (typically FP32) and its embedding dimension need to be reduced to fit in the size of CiM\u2019s crossbar arrays. To illustrate the impact of these on RAG, as an example, we present a preliminary study on MIPS performance in Figure 2, where we use a simple yet representative Gaussian noise to simulate the noise from the device variations in CiM. As shown in Figure 2, as the noise level increases, MIPS accuracy (specified in section 4.1.3) drops dramatically, approaching random guessing. To address these issues, we further propose a novel optimization framework for CiM-backed RAG, called Robust CiM-backed RAG (RoCR). The framework consists of three parts. The first part is a contrastive learning method. We use it to optimize the document embedding model. The second part is a novel data construction method to generate both positively and negatively labeled data pairs for contrastive learning. For the profile data, they can be either labeled to indicate the explicit user-preferred response to certain input, or simply statements without explicit labels that only implicitly indicate user preferences. Our data construction method is capable of dealing with both types of profile data. The third part is a noise-aware training method. It goes in tandem with contrastive learning to obtain a sentence embedding model that can generate document and user query embeddings with high noise-resilient capability, while such embeddings can fit into CiM architectures under different designs and configurations. Our major contributions can be summarized as: \u2022 We propose the first work to harvest CiM advantages for RAG acceleration on the edge. We provide a pathway to utilize emerging CiM devices to expand the Edge LLMs\u2019 capability in terms of storing a high volume of profile data with fast MIPS computing. \u2022 We introduce noise-aware training to enhance the noiseresilient capabilities of RAG\u2019s document embedding. The resulting noise-resilient embeddings can be reused robustly, saving resources needed to calibrate and regenerate embeddings. \u2022 Our experiments on various datasets show that our proposed framework can improve the RAG performance on multiple CiM devices up to 35%, approaching to the theoretical RAG performance. Across a wide device variation (noise) range on a single CiM device, our proposed framework can still improve the RAG performance. 2 RELATED WORK 2.1 CiM Architectures and their NVMs As shown in the middle part of Figure 1, memory arrays are the key component for vector-matrix multiplication. In this array, matrix values are stored at NVM cells, such as emerging NVM technologies like PCMs, RRAMs, and FeFETs, at the cross-points of vertical and horizontal lines. Simultaneously, vector values flow along the horizontal lines of the array. Operations within the memory array take place in the analog domain by exploiting law of physics directly. However, for other essential functions like shift-and-add for multiple bits and sorting to find the top-k ranked values would be done in the digital domain. Thus, digital-to-analog and analog-to-digital \fRobust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures profile  data Data Construction  Module positive  examples anchor  examples \u2026 Reshape  Module embeddings Contrastive  Learning close far Device Variation NVMs Sentence  Embedding  Model negative  examples Flexible Noise-aware  Training Module optimize constraints Figure 3: Overview of the proposed Robust CiM-backed RAG framework (RoCR). It optimizes the sentence embedding model to adapt different types of NVMs utilized by CiM. converters (DACs and ADCs) are used to connect these different components. CiM arrays suffer from various sources of variations and noises. Two major ones include spatial variations and temporal variations. Spatial variations result from fabrication defects and have both local and global correlations. FeFET devices also suffer from temporal variations due to the stochasticity in memory switching and also aging, which causes fluctuations in conductance when programmed at different times. Temporal variations are typically independent from device to device and are irrelevant to the value to be programmed [20]. In this work, as a proof of concept, we focus on the impact of temporal variations in the programming process on DNN performance. Temporal variation makes the programmed resistance of a device deviate from what is expected. The proposed framework can also be extended to other sources of variations with modification. Measurement results [21, 22] show that the noise on DNN weights caused by device variations can be safely modeled as a Gaussian noise with zero mean, each with a standard deviation associated with the weight value. A detailed representation is given by: v = v0 + \u0394v, \u0394v \u223cN (0, \ud835\udf0e\ud835\udc63) (1) where v is the actual embedding deployed on the accelerators, v0 is the target embedding value, and \ud835\udf0e\ud835\udc63is a value measured by the experiments. We collect the measurement results from RRAM and FeFET devices and the specific value will be discussed in Section 4.1. 2.2 Past Noise Mitigation Methods Several strategies have been introduced to tackle the challenge of device variations in CiM accelerators. These methods can be separated into software and hardware-based techniques. The software-based techniques are generally developed to obtain more robust DNN models [19, 22\u201324] or recommendation systems [25], and are thus not suitable for generating more robust MIPS solutions. For the hardware techniques, the write-verify procedure [26, 27] is one of the most commonly used approach during programming. Initially, a NVM device is programmed to a set state via a designated pulse pattern. Subsequent to this, the device\u2019s value is verified to ascertain if its conductance aligns with a stipulated range of the desired value, essentially assessing its accuracy. If discrepancies arise, a supplemental update pulse is initiated to reset the device conductance nearer to the target. This loop persists until the disparity between the programmed device value and the target value diminishes to a satisfactory margin, typically taking a handful of cycles. Cutting-edge research suggests that by selectively applying write-verify to a subset of pivotal devices, one can uphold the average accuracy of a DNN [21]. Additionally, a variety of circuit design initiatives [18, 28] have been put forth to counteract device variations. 3 PROPOSED WORK 3.1 Framework Overview As shown in Figure 3, our proposed framework, Robust CiM-backed RAG (RoCR), consists of three stages. First, we apply contrastive learning to utilize the training data to optimize the training module. To do that, in the second stage, we take the profile data and construct via a data construction module to obtain contrastive training data pairs, which are then used in the flexible noise-aware training module. In the third stage, we obtain the constraints of NVMs in CiM via profiling. These constraints will be encoded into the flexible noise-aware training module and used to train the sentence embedding model so that it can generate embedding that are robust against device variation of the target NVMs. After training, the training module can be turned into a new sentence embedding model and generate CiM-friendly embeddings. 3.2 Contrastive Learning: Triplet Loss Function When we apply RAG using CiM, we first need to store embeddings into NVMs as shown in Figure 1. Such embeddings are generated by the sentence embedding model, and they are the numerical representations of profile data. Each single document in the profile data can have its unique embedding, which is a vector. The embeddings stored on NVMs can consist of a matrix as the orange blocks shown in Figure 1. Given a user query, which will also be converted into an embedding, CiM can operate MIPS between this user query embedding and all profile embeddings simultaneously via vector-matrix multiplication. The top-ranked values in the product will be used as the index to retrieve the corresponding document data, as the pink block shown in Figure 1. This retrieved user-relevant document is the output of MIPS. However, as we have explained in Section 2.1, writing the document embeddings into NVMs can cause them to suffer from temporal variations (device variations). Then, the NVM-stored embeddings will be different from the original sentence embedding model generated embeddings. As shown in Figure 4, the vanilla embedding model generates desired embedding, which will deviate to the noise embedding under device variation, such that the irrelevant embedding is ranked higher than desired embedding due to its larger inner product. Contrastive learning can learn the representations via push away dissimilar examples and pull close similar examples [29]. In particular, the contrastive loss function can be used to increase the distance between dissimilar examples. In our work, we propose to improve the noise-resilient capability by contrastive learning. By increasing the distance between \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 noise  embedding irrelevant  embedding query retrieve the wrong data irrelevant  embedding query NVMs Device Variation lead to Vanilla  CiM-backed  RAG Robust  CiM-backed  RAG Our  embedding  model noise-resilient  embeddings user  profile  data NVMs Device Variation desired  embedding vanilla  embedding  model user  profile  data retrieve the desired data embeddings Figure 4: Improvement by our Robust CiM-backed RAG. Our framework generates noise-resilient embeddings, as shown the orange and blue point in right subfigure dissimilar examples, as shown the right subfigure in Figure 4, deviated desired embedding will still have a larger inner product with the query compared to the irrelevant embedding. Our contrastive learning loss function is based on Weinberger et al. [30]. For each example \ud835\udc65\ud835\udc56in a mini-batch of N anchor examples, our data construction method will construct \ud835\udc3epositive and \ud835\udc3enegative examples corresponding to \ud835\udc65\ud835\udc56. We can have {{(\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56,\ud835\udc65+ \ud835\udc56)\ud835\udc58}\ud835\udc56=1,...,\ud835\udc41}\ud835\udc58=1,...,\ud835\udc3e, in which \ud835\udc65\u2212and \ud835\udc65+ are negative and positive examples corresponding to \ud835\udc65\ud835\udc56, where \ud835\udc65\ud835\udc56is closer to \ud835\udc65+ \ud835\udc56compared to \ud835\udc65\u2212 \ud835\udc56. Also, \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56) represents the learned embedding of \ud835\udc65\ud835\udc56. Then the loss function L can be defined as: L = \ud835\udc41 \u2211\ufe01 \ud835\udc56=1 1 \ud835\udc3e \ud835\udc3e \u2211\ufe01 \ud835\udc58=1 max \u0010 0, d(\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56(\ud835\udc58)) \u2212d(\ud835\udc65\ud835\udc56,\ud835\udc65+ \ud835\udc56(\ud835\udc58)) + \ud835\udc5a \u0011 , d(\ud835\udc65\ud835\udc4e,\ud835\udc65\ud835\udc4f) = sim(emb(\ud835\udc65\ud835\udc4e), emb(\ud835\udc65\ud835\udc4f)) (2) The distance \ud835\udc51(\ud835\udc65\ud835\udc4e,\ud835\udc65\ud835\udc4f) is calculated by the Euclidean distance between embeddings of two data \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc4e) and \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc4f). The function \ud835\udc60\ud835\udc56\ud835\udc5a() calculate the semantic similarity. 3.3 Data Construction To train the sentence embedding model via contrastive learning, it is critical to construct pairs of examples where the positive examples and negative examples need to be distinct from each other [31]. In our work, since we use triplet contrastive loss, instead of pairs of examples, we will construct trios of examples where each triplet contains an anchor, positive, and negative example. We use profile data to construct triplets of examples. For the profile data, it is generated by the user during the user-LLM interaction and contains the user preference information. There exists two situations for such data. First, the profile data can contain explicit labels indicating the user preferred response to the corresponding content. Second, the profile data also can be statements containing the user-related information but without explicit user preferences As shown in Figure 5, to deal with the two situations, we come up with two data construction methods: Construction Data with Explicit labels (CDE) and Construction Data with Implicit labels (CDI). \u201cJake Blues, just released from prison,  puts his old band back together to save  the Catholic home where he and his  brother Elwood were raised.\u201d is  \u201cdystopia\u201d negative r = 0.1 r = 0 r = 0 \u201cFresh out of prison, Jake Blues  rallies his old band to save their  childhood Catholic home\u201d is  \u201cclassic\u201d positive example (embedding) \u201cJake Blues, just released\u2026\u201d is  \u201cclassic\u201d anchor example (embedding) \u201cJake Blues, just released \u2026\u201d is  \u201cdystopia\u201d negative example (embedding) r = 0.1 \u201cVictims of traumatized \u2026\u201d r = 0 r = 0.9 CDE CDI E anchor/positive example negative example \u201cTwo victims of traumatized  childhoods become lovers and serial  murderers irresponsibly glorified by the  mass media.\u201d anchor/positive/negative example \u201cTwo people with traumatic pasts  turn into a couple on a crime spree,  mistakenly idolized by the media.\u201d \u201cIndividuals, mired in traumas, unite *()  crime-ridden bond, enthrall\u2606\u2609\u00a7ing the  media's distorted spotlight.\" \u201cJake Blues, just released from prison,  puts his old band back together to save  the Catholic home where he and his  brother Elwood were raised.\u201d is \u201cclassic\u201d explicit label Statement/implicit label Figure 5: Examples of the two data construction methods. For data with explicit labels, CDE is used to construct the training data. For data without explicit labels (implicit labeled data), CDI is used to construct the training data. 3.3.1 Construction Trios via Data with Explicit Labels (CDE). For the data with explicit labels, each of the data consists of a textual content c and its corresponding label l which indicates the user preferred response regarding to the content c. As shown in the CDE part in Figure 5, there exists explicit label circled by dashed line. Using the profile data, we will construct triplet examples in the format of (\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56,\ud835\udc65+ \ud835\udc56). Given a dataset D with size of \ud835\udc5bprofile documents, each piece of data consists of a content \ud835\udc50\ud835\udc56and the corresponding label \ud835\udc59\ud835\udc56where \ud835\udc56\u2208{1, 2, ...,\ud835\udc5b}. The anchor example \ud835\udc65\ud835\udc56can be constructed as: \ud835\udc65\ud835\udc56= \ud835\udc50\ud835\udc56\u2295\ud835\udc59\ud835\udc56, for \ud835\udc56= 1, 2, . . . ,\ud835\udc5b (3) where \u2295denotes a concatenation operation, specifically used here to combine label and content. Negative examples \ud835\udc65\u2212 \ud835\udc56can be constructed by concatenating \ud835\udc50\ud835\udc56with a random label \ud835\udc59\ud835\udc57that is different from \ud835\udc59\ud835\udc56as follows: \ud835\udc65\u2212 \ud835\udc56= \ud835\udc50\ud835\udc56\u2295\ud835\udc59\ud835\udc57, where \ud835\udc59\ud835\udc56\u2260\ud835\udc59\ud835\udc57. (4) Randomly assigning a different label ensures diversity in the negative examples while maintaining the same content from the anchor. Different from constructing anchor and its negative examples, it is challenging to construct positive examples corresponding to the anchor examples since it is more difficult to formalize semantically similar data than to formalize semantically dissimilar data. To construct positive examples, we follow the SimCSE method [32] to add a dropout rate \ud835\udc5finto the sentence embedding model M. The process for constructing positive examples involves two main steps. First, the textual positive example is formalized as: \ud835\udc65+ \ud835\udc56= \ud835\udc65\ud835\udc56, for \ud835\udc56= 1, 2, ...,\ud835\udc5b (5) where we align each anchor with the corresponding positive example. This step effectively duplicates the anchor data as a starting point for generating the embeddings. \fRobust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures Second, the embedding generation process varies based on the dropout rate applied within the model M. When model M is utilized to generate embeddings for anchor and negative examples, the dropout rate is set to 0. In contrast, for generating embeddings for positive examples, a non-zero dropout rate \ud835\udc5fis used. The anchor, negative, positive examples, as shown in Figure 5, can be constructed as: \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56) = M(\ud835\udc65\ud835\udc56,\ud835\udc51\ud835\udc5f\ud835\udc5c\ud835\udc5d\ud835\udc5c\ud835\udc62\ud835\udc61= 0) \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\u2212 \ud835\udc56) = M(\ud835\udc65\u2212 \ud835\udc56,\ud835\udc51\ud835\udc5f\ud835\udc5c\ud835\udc5d\ud835\udc5c\ud835\udc62\ud835\udc61= 0) \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65+ \ud835\udc56) = M(\ud835\udc65+ \ud835\udc56,\ud835\udc51\ud835\udc5f\ud835\udc5c\ud835\udc5d\ud835\udc5c\ud835\udc62\ud835\udc61= \ud835\udc5f) (6) The condition of \ud835\udc5f\u22600 can induce variation in the embeddings, enhancing the model\u2019s ability to recognize semantically similar yet variably expressed content. Given the construction factor \ud835\udc3e, we can construct the triplet data examples as: D\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc5d\ud835\udc59\ud835\udc52\ud835\udc61= \ud835\udc41 \u00d8 \ud835\udc56=1 n (\ud835\udc65\ud835\udc56(\ud835\udc58),\ud835\udc65\u2212 \ud835\udc56(\ud835\udc58),\ud835\udc65+ \ud835\udc56(\ud835\udc58)) : \ud835\udc58= 1, 2, . . . , \ud835\udc3e o (7) For the triplet data examples D\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc5d\ud835\udc59\ud835\udc52\ud835\udc61, their embeddings for each augmentation \ud835\udc58are given by: E = \ud835\udc41 \u00d8 \ud835\udc56=1 n (\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56(\ud835\udc58)),\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\u2212 \ud835\udc56(\ud835\udc58)),\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65+ \ud835\udc56(\ud835\udc58)) : \ud835\udc58= 1, 2, . . . , \ud835\udc3e o (8) As shown in Figure 5, for data with explicit labels, a content\ud835\udc50can concatenate with its corresponding label \ud835\udc59to formalize the positive and anchor example. That content \ud835\udc50can also concatenate with other labels \ud835\udc59\u2032 to formalize the negative example. The positive example can be finally obtained from the sentence embedding model with dropout rate \ud835\udc5f. The anchor and negative example can be finally obtained from the sentnece embedding model with \ud835\udc5f= 0. 3.3.2 Construction Trios via Data with Implicit Labels (CDI). For data with implicit labels, each of the data consists of solely textual content c. As shown of the CDI part in Figure 5, there is no explicit label to indicate user preferences. Instead, the data can be seen as a statement containing some user-related information. To construct the anchor examples and positive examples, we can use the exact same method in EDC. Given a dataset D with size of n profile data, each piece of data consits of a content \ud835\udc50\ud835\udc56. The anchor data \ud835\udc65\ud835\udc56can be constructed as: \ud835\udc65\ud835\udc56= \ud835\udc50\ud835\udc56, for \ud835\udc56= 1, 2, . . . ,\ud835\udc5b (9) For each anchor data \ud835\udc65\ud835\udc56, constructing its corresponding negative example is not as simple as merely concatenating the content\ud835\udc50\ud835\udc56with a non-corresponding label \ud835\udc59\ud835\udc58. To construct negative examples, we employ a reciprocal approach with the positive examples, applying a similar method to both. We first initialize the negative example and positive example following the equation 5: \ud835\udc65\u2212 \ud835\udc56= \ud835\udc65+ \ud835\udc56= \ud835\udc65\ud835\udc56, for \ud835\udc56= 1, 2, . . . ,\ud835\udc5b (10) For the positive example \ud835\udc65+ \ud835\udc56, it can be finalized by incorporating a dropout rate \ud835\udc5finto the sentence embedding model M, where a rate of 0 < \ud835\udc5f\u22640.2 can generate a sentence embedding with a semantic representation similar to \ud835\udc65\ud835\udc56and ensure good model training performance [32]. Increasing the dropout rate to a higher value, such as 0.5, can distort the semantic representation of \ud835\udc65+ \ud835\udc56, making it dissimilar to that of \ud835\udc65\ud835\udc56. Training the model with such positive examples can result in poorer performance. For positive examples in training the sentence embedding model, the higher dropout rate performs more like a noise rather than a data augmentation method. In our work, we train the sentence embedding model to generate embeddings that maintain their integrity under noisy conditions, such as during writing into Compute-in-Memory (CiM). The noise can alter or fragment the original semantic representations. For instance, as illustrated in Figure 5, using a high dropout rate \ud835\udc5f= 0.9 can lead to a negative example with a corrupted representation. Although it may lack certain informative content, this negative example becomes semantically distinct from both the anchor and positive examples, effectively simulating the effect of CiM corruption. This approach not only differentiates the negative examples semantically but also aligns them with the corrupted data scenarios for noise-aware training. Given the triple examples (\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56,\ud835\udc65+ \ud835\udc56), for \ud835\udc56= 1, 2, ...,\ud835\udc5bas shown in equation 10, we have the dropout rate \ud835\udc5ffor formalizing the positive examples where 0 < \ud835\udc5f\u22640.2. Correspondingly, the dropout rate for formailzing the negative examples can be 1 \u2212\ud835\udc5f. Given the sentence embedding model M, the anchor example, positive example, and negative example can be constructed as: emb(\ud835\udc65\ud835\udc56) = M(\ud835\udc65\ud835\udc56, dropout = 0) emb(\ud835\udc65\u2212 \ud835\udc56) = M(\ud835\udc65\u2212 \ud835\udc56, dropout = 1 \u2212\ud835\udc5f) emb(\ud835\udc65+ \ud835\udc56) = M(\ud835\udc65+ \ud835\udc56, dropout = \ud835\udc5f) (11) 3.4 Flexible Noise-aware Training In the previous two stages, we construct the data to train the sentence embedding model based on contrastive learning. Meanwhile, the training can be more effective when injecting the simulated device variation [33] so that the model can be optimized with consideration of the device variation. Additionally, the sentence embedding model needs to produce embeddings that can fit with the different CiMs, which might have various NVM designs. To do that, we need the sentence embedding model reshapes its output embeddings into certain dimensions and precision. Hence, we propose a flexible noise-aware training method, which can generate the noise-resilient embedding, fitting to various CiMs. As shown in Figure 3, in the flexible noise-aware training module, the embedding generated by sentence embedding model will be shaped based on the CiM\u2019s NVMs constraints where required dimension is \ud835\udc51and required precision is \ud835\udc5d, and being injected device variation to formalize the embeddings. The reshape module, shown in Figure 3, seen as an autoencoder to reconstruct its input embedding [34], can be expressed as \ud835\udc60\u210e\ud835\udc5d(), initialized by \ud835\udc51 and \ud835\udc5d, takes the anchor embedding \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56) as input. We can have \ud835\udc60\u210e\ud835\udc5d(\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)) = \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d. Based on the device variation shown as Table 2, we can have: \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d \ud835\udf0e = (\ud835\udc52\u2032 \u2217\ud835\udc3f0 + \ud835\udc52\u2032 \u2217\ud835\udc3f1 + \ud835\udc52\u2032 \u2217\ud835\udc3f2 + \ud835\udc52\u2032 \u2217\ud835\udc3f3) \u2217\ud835\udf0e, (12) \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 Table 1: Performance comparison between our framework and four baselines on five CiM devices with device variation specified in Table 2 across five datasets. Evaluate the performance of our framework using EDC (RoCR-EDC) and using IDC (RoCR-IDC) to optimize the performance of RAG, which utilizes Gemma-2 as its LLM. Dataset Citation Movie Rating News DBLP CiM Method Acc \u2191 F1 \u2191 Acc \u2191 F1 \u2191 MAE \u2193 RMSE \u2193 ROUGE-1 \u2191 ROUGE-L \u2191 ROUGE-1 \u2191 ROUGE-L \u2191 Device-1 SWV 0.4208 0.3339 0.1305 0.1974 0.3850 0.8093 0.0754 0.0731 0.1709 0.1590 CxDNN 0.4223 0.3576 0.1516 0.1762 0.4404 0.9135 0.0640 0.0632 0.1646 0.1449 CorrectNet 0.4155 0.3791 0.0996 0.1305 0.3609 0.7071 0.0512 0.0764 0.1603 0.1538 Vanilla RAG 0.4401 0.3476 0.1017 0.0838 0.3903 0.8944 0.0754 0.0731 0.1731 0.1473 RoCR-CDE 0.5536 0.3956 0.2242 0.2303 0.3108 0.6856 0.1041 0.0987 0.2066 0.1924 RoCR-CDI 0.5409 0.5117 0.2273 0.2487 0.2767 0.6083 0.0831 0.0808 0.2317 0.2176 Device-2 SWV 0.1831 0.1552 0.1992 0.1957 0.4205 0.8775 0.0296 0.0289 0.1968 0.1874 CxDNN 0.4013 0.3557 0.2167 0.2019 0.4423 0.8367 0.0604 0.0791 0.1517 0.1401 CorrectNet 0.3827 0.3209 0.1625 0.1909 0.3762 0.8062 0.0513 0.0505 0.2042 0.1945 Vanilla RAG 0.4801 0.3462 0.1576 0.2079 0.4153 0.9354 0.0296 0.0289 0.1618 0.1353 RoCR-CDE 0.5407 0.4396 0.2924 0.2509 0.2553 0.5385 0.1209 0.0946 0.2025 0.1906 RoCR-CDI 0.5299 0.4591 0.2971 0.2386 0.2124 0.5763 0.0884 0.0853 0.2240 0.2098 Device-3 SWV 0.2450 0.2564 0.1695 0.1641 0.3460 0.7416 0.0725 0.069 0.1018 0.0954 CxDNN 0.4811 0.4006 0.2367 0.2113 0.2851 0.6928 0.0761 0.0707 0.1425 0.1111 CorrectNet 0.4510 0.3918 0.0792 0.1029 0.3704 0.7937 0.0585 0.0555 0.1715 0.1346 Vanilla RAG 0.4852 0.3618 0.1614 0.1636 0.3255 0.7649 0.0725 0.0690 0.1647 0.1437 RoCR-CDE 0.5139 0.4116 0.2242 0.2215 0.3208 0.6481 0.0825 0.0805 0.1893 0.1754 RoCR-CDI 0.5515 0.4984 0.2152 0.2131 0.2916 0.6245 0.1099 0.1049 0.2294 0.2140 Device-4 SWV 0.5135 0.4260 0.1271 0.1178 0.3610 0.8196 0.0259 0.0256 0.1871 0.1786 CxDNN 0.4733 0.3964 0.1267 0.2158 0.3468 0.7616 0.0646 0.0634 0.1603 0.1538 CorrectNet 0.4628 0.4019 0.1592 0.1847 0.4013 0.9274 0.0705 0.0750 0.1628 0.1292 Vanilla RAG 0.2101 0.2401 0.1219 0.2019 0.4015 0.8544 0.0505 0.0489 0.1929 0.1814 RoCR-CDE 0.5836 0.5555 0.1706 0.2817 0.3139 0.6856 0.0873 0.0851 0.1984 0.1882 RoCR-CDI 0.5352 0.4289 0.1642 0.2445 0.2706 0.5916 0.1154 0.1128 0.2148 0.1978 Device-5 SWV 0.4320 0.3541 0.1250 0.1076 0.3652 0.7616 0.0434 0.0427 0.0985 0.0923 CxDNN 0.4301 0.0538 0.0751 0.0458 0.3503 0.8185 0.0707 0.0682 0.2042 0.1945 CorrectNet 0.4145 0.3926 0.1083 0.1395 0.5526 0.8185 0.0735 0.0776 0.2096 0.1879 Vanilla RAG 0.4256 0.3522 0.0847 0.0863 0.3951 0.8515 0.0676 0.0653 0.2018 0.1846 RoCR-CDE 0.5698 0.5223 0.2152 0.1669 0.2959 0.6245 0.0936 0.0891 0.1946 0.1844 RoCR-CDI 0.5254 0.4504 0.2394 0.2458 0.2624 0.6325 0.0799 0.0764 0.2238 0.2095 where \ud835\udc52\u2032 = \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d. The device variation, as noise, is injected into embeddings to formalize \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d \ud835\udf0e , which will be used in contrastive learning to train the sentence embedding model, as shown in Figure 3. 4 EXPERIMENTAL EVALUATION 4.1 Experimental Setup 4.1.1 Datasets. To demonstrate our robust CiM-backed RAG, we employ five datasets with different tasks and domains, including Citation Identification [35] (Citation), Movie Tagging [36] (Movie), Product Rating [37] (Rationg), News Headline Generation [38] (News), and DBLP-Citation-network V14 [39] (DBLP) to evaluate the proposed framework. The data in each dataset consists of query data and profile data. In our evaluation, the profile data will be used to formalize user history, and the profile corresponding query data will be used as the user input. The first three datasets contain binary, five-class, and fifteen-class classification tasks respectively. The last two datasets contain text generation tasks. In the Citation Identification dataset, every piece of query data consists of a paper title and two references, and the correct reference is provided. RAG uses the profile data corresponding to the paper titles with their detailed contents to choose the appropriate reference. In the Movie Tagging dataset, each query data contains a description of a movie, and RAG uses a similar description and its corresponding tag in the profile data to tag the query data. The Product Rating dataset has a similar structure as the Movie Tagging dataset. In News Headline Generation and DBLP datasets, each query data contains an abstract, which can be summarized into a title. RAG uses a similar abstract and its corresponding title in profile data to generate the title for query data. All five datasets have labels in their query data. 4.1.2 Default Experimental Setting. Our framework chooses all-MiniLM-L6-v2 [40] as the sentence embedding model. For each dataset, we randomly select 2000 documents from profile data as the anchor examples. To examine the data construction method of CDE, we set the augmentation factor \ud835\udc58= 5 to obtain 10000 negative and positive examples. We set dropout rate as 0.1 to obtain the positive examples while maintain it as 0 when process anchor and negative examples. To examine the data construction method CDI, we set dropout rate for positive examples as 0.1 and dropout rate for negative examples as 0.9. To align with experiments for CDE, we also set \ud835\udc58= 5 in the experiments for CDI. For the experimental results, we run five times and get the average. In experiments, we set the device variation \ud835\udf0e= 0.1 and shape embeddings into dimension of 64 with precision of \ud835\udc56\ud835\udc5b\ud835\udc618. The learning rate is 2\ud835\udc52\u22125. In all experiments, we adhere to the device variation model previously described. The specific parameters are abstracted and then simplified from three representative NVM devices, two of them \fRobust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (a) Citation on Gemma-2B D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (b) Citation on Phi-2 D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (c) Citation on Mistral-7B D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (d) Citation on Llama-2-3B D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (e) Movie on Gemma-2B D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (f) Movie on Phi-2 D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (g) Movie on Mistral-7B D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (h) Movie on Llama-2-3B Figure 6: Performance comparison between our framework and four baselines on RAG utilizing the LLMs including Gemma-2B, Phi-2, Mistral-7B, and Llama-2-3B with device variation specified in Table 2, given dataset \ud835\udc36\ud835\udc56\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5band \ud835\udc40\ud835\udc5c\ud835\udc63\ud835\udc56\ud835\udc52. Table 2: Device non-ideality modeling for different real and synthesized devices. For devices with more than two levels, the device variation for each level is depicted as \ud835\udc3f\ud835\udc65. Name # of Levels Device Variations \ud835\udf0e\ud835\udc63 \ud835\udc3f0 \ud835\udc3f1 \ud835\udc3f2 \ud835\udc3f3 \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc401 (Device-1) 1 0.0100 0.0100 0.0100 0.0100 \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc472 (Device-2) 4 0.0067 0.0135 0.0135 0.0067 \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc473 (Device-3) 4 0.0049 0.0146 0.0146 0.0049 \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc404 (Device-4) 4 0.0038 0.0151 0.0151 0.0038 \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc476 (Device-5) 4 0.0026 0.0155 0.0155 0.0026 are resistive random-access memory (RRAM) devices extracted from [27, 41] and the other is a ferroelectric field effect transistor (FeFET) device extracted from [42]. We name them \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc401, \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc404 and \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc472, respectively. We also extrapolate the modeling data to obtain two synthesized \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc473 and \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc476 devices. Detailed device modeling results are demonstrated in Table 2. A \ud835\udc65-level device means this device can represent \ud835\udc65distinct values and \ud835\udf0e\ud835\udc3f2 = 0.01 means the variation of this device is 0.01 when it is representing the level value 2. Using the device variations obtained from real CiM devices, we perform our experiments on a single Nvidia A10 GPU. Document embeddings are shaped based on different CiM devices and stored as parallel arrays, similar to how they would be mapped to multiple NVM devices in practical scenarios. For example, if an embedding is shaped to contain all uint8 values, when it is mapped to 4-level (2-bit) devices such as \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc472, each element of the vector is represented by four devices. 4.1.3 Evaluation Methods. Our first three datasets examine the model classification capability, and the rest of two datasets examine the text generation capability. In particular, dataset \ud835\udc36\ud835\udc56\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5band \ud835\udc40\ud835\udc5c\ud835\udc63\ud835\udc56\ud835\udc52has two and fifteen labels respectively. We can examine the binary and multiclass classification capabilities of the LLMs enhanced by our framework. In this way, we use accuracy to examine the ability of the models to correctly classify instances across different classes, and we use F1 score to examine the balance between precision and recall in classification tasks. For dataset \ud835\udc45\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5b\ud835\udc54, while it has five labels and also examine the multiclass classification, we use mean absolute error (MAE) and root mean square error (RMSE) to evaluate from from a regression perspective [43]. For MAE, it measures the average magnitude of errors in the predictions, providing a straightforward assessment of the model\u2019s overall accuracy in predicting the rating values. For RMSE, it captures the square root of the average squared differences between predicted and actual ratings, offering a metric sensitive to larger errors, which can highlight significant discrepancies between the model\u2019s predictions and true values. For dataset \ud835\udc41\ud835\udc52\ud835\udc64\ud835\udc60and \ud835\udc37\ud835\udc35\ud835\udc3f\ud835\udc43, their labels are sentences. Such datasets examine the text generation capabilities. We use ROUGE1 and ROUGE-L to evaluate the overlap between generated texts and reference texts [44], capturing both the precision and recall of individual words (ROUGE-1) and the longest matching sequence (ROUGE-L), ensuring a comprehensive evaluation of the text generation quality. For accuracy, F1, ROUGE-1 and ROUGE-L, their higher values reflect the better performance. For MAE and RMSE, their lower value represent the better performance. Additionally, we use accuracy to measure the MIPS performance (MIPS accuracy), representing the ratio of MIPS results under device variation and MIPS results without device variation (references). 4.1.4 Baselines. As this is the first work to improve the RAG robustness on Edge-based CiM, we do not have state-of-the-art for comparison. As such, we construct baselines from the past noise mitigation methods originally designed to boost DNN robustness. The first baseline is selective write verify [21] (SWV). While it originally utilizes the second derivation to evaluate the device variation impact on neural network weights, we use the second derivation to measure the embedding deviation between the ground truth embedding and the embedding under device variation. The second baseline is (CxDNN) [45]. While they use compensation factor to improve the robustness of vector-matrix multiplication, we use the compensation factor the calibrate the embedding impacted by device variation. The third baseline is CorrectNet [46], where it utilizes the cross entropy loss and regularization to improve the robustness of neural networks in CiM. To use it as a baseline, we also use the cross entropy loss the regularization as the loss function to \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 calibrate the device output embedding. Additionally, we examine the Vanilla RAG, which contains no noise mitigation methods, as our fourth baseline. The baselines use the same experimental setting as our framework does. 4.2 Results For RAG, it can be simplified as the combination of MIPS and LLM, where the MIPS as a retriever searches the appropriate information and the LLM as a generator processes the searched results. Hence, in our experiments, we first evaluate the performance of MIPS under the device variation of device-1. We take the MIPS results obtained without device variation as the references (i.e., ground truth). Using the metric of MIPS accuracy, we examine how many MIPS results under device variation will match the references. Since the quality of retrieved content largely depends on the base sentence embedding model, and we focus on mitigating the device variation impact on the embedding model, we do not assess the quality of references. As shown in Table 3, our framework using the two data construction methods outperforms the four baselines across five datasets. It shows that our framework can mitigate the embedding perturbation due to device variation. These results can also correspond to the preliminary study shown in Figure 2, where the increment of \ud835\udf0e in naive Gaussian noise will jeopardize the MIPS performance. Table 3: Performance (MIPS accuracy) comparison between our framework and baselines. Accuracy is computed based on MIPS-retrieved documents under device variation of device-1 and the these retrieved without device variation. Dataset Citation Movie Rating News DBLP SWV 0.4200 0.1728 0.1050 0.0855 0.2295 CxDNN 0.4401 0.2017 0.0503 0.0754 0.1681 CorrectNet 0.4013 0.0699 0.0509 0.0533 0.1609 Vanilla RAG 0.4547 0.1694 0.0933 0.0649 0.1747 RoCR-CDE 0.9231 0.4639 0.1583 0.1921 0.2750 RoCR-CDI 0.9344 0.4355 0.1266 0.1708 0.2905 After we compare the MIPS performance of our framework and baselines, we further present a comprehensive evaluation to show the RAG performance of them. We use Gemma-2B as the LLM in RAG. Additionally, with Gemma-2B, we run RAG without device variation to obverse its ideal performance, where we get 0.5200 of accuracy for Citation, 0.3728 of accuracy for Movie, 0.3150 of MAE for Rating, 0.0855 of ROUGE-1 for News, and 0.2295 of ROUGE-1 for DBLP. On five CiM devices, whose device variations have been shown in Table 2, we examine RAG with five datasets. As shown in Table 1, given the same datasets, it is clear that each device variation significantly compromises the RAG robustness, whereas our framework can mitigate the different device variation. For example, the RAG performance for Citation dataset on Device-2 can range from 0.18 to 0.48, while our framework can boost the accuracy performance of Citation dataset above 0.5 for all five devices. Compared to the four baselines whose performances are relatively worse than the ideal performance, our framework significantly approaches and sometimes outperforms the ideal performance via generating better sentence embeddings. This is because RoCR also serves as a regularization to improve the model\u2019s generalization. In addition, we evaluate the impact of different LLMs on the performance of our framework. As Figure 1 shown, the LLM takes the concatenation of MIPS searched data and user query as the input and generates the response regarding the user query. Since different LLMs may have different response given the same query, we select four emerging edge-friendly medium-size LLMs in our experiments to examine the performance of our framework. Gemma-2B [47] is a new SOTA open model introduced by Google, with 4.95G model weights. According to Google, Gemma can outperform the same sized Llama-2 in reasoning capabilities. Hence, we also use Llama2-3B [48], one of the earliest open LLMs introduced by Meta, with 6.85G model weights. Similarly, Phi-2 [49] released by Microsoft, is a powerful small LLM with 5G model weights. Additionally, Mistral7B-GPTQ [50] made by Mistral AI, is a well-performed LLM after Llama model. We select dataset \ud835\udc36\ud835\udc56\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5band dataset \ud835\udc40\ud835\udc5c\ud835\udc56\ud835\udc63\ud835\udc52. We use the default experimental setting with \ud835\udf0e= 0.1 and use CiM Device-1 as the experimental environment. The results are shown on Figure 6. It is evident that our framework outperforms each baseline across five CiM devices. Besides, the performance of each baseline on the same dataset can be largely different given different device, while our framework can produce a more robust performance. 0.000 0.025 0.050 0.075 0.100 0.125 0.150 Device Variation ( ) 0.10 0.15 0.20 0.25 0.30 ROUGE-1 SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI Figure 7: Performance comparison between our framework and four baselines on CiM device-1 with different device variation \ud835\udf0e, given dataset DBLP. By default, we use \ud835\udf0e= 0.1 to calculate the device variation of the five CiM devices. We also conduct an additional study to evaluate our framework given different \ud835\udf0evalues. Since we have already use dataset Citation and dataset Movie to study the performance of our frameworks seen in Figure 6, we choose a different dataset DBLP, using ROUGE-1 as the metric. For the LLM in RAG, we choose Mistral-7B. We examine the \ud835\udf0evalues higher and lower than 0.1, including 0, 0.025, 0.05, 0.075, 0.125, and 0.15. The case of \ud835\udf0e = 0 reflects the ideal performance. For the CiM device, we use CiM device-1. As shown in Figure 7, our framework outperforms baselines across different device variation values. Finally, RoCR is a training method that generates more robust weights for the sentence embedding model. It does not change the model structure. Thus, there is no hardware (e.g., energy and latency) overhead during inference. 5",
+    "additional_graph_info": {
+        "graph": [
+            [
+                "Ruiyang Qin",
+                "Zheyu Yan"
+            ],
+            [
+                "Ruiyang Qin",
+                "Zhenge Jia"
+            ],
+            [
+                "Ruiyang Qin",
+                "Ahmed Abbasi"
+            ],
+            [
+                "Zheyu Yan",
+                "Weiwen Jiang"
+            ],
+            [
+                "Zhenge Jia",
+                "Dawei Li"
+            ],
+            [
+                "Zhenge Jia",
+                "Cong Liu"
+            ],
+            [
+                "Ahmed Abbasi",
+                "Abiy Tasissa"
+            ],
+            [
+                "Ahmed Abbasi",
+                "Shuchin Aeron"
+            ]
+        ],
+        "node_feat": {
+            "Ruiyang Qin": [
+                {
+                    "url": "http://arxiv.org/abs/2405.04700v1",
+                    "title": "Robust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures",
+                    "abstract": "Large Language Models (LLMs) deployed on edge devices learn through\nfine-tuning and updating a certain portion of their parameters. Although such\nlearning methods can be optimized to reduce resource utilization, the overall\nrequired resources remain a heavy burden on edge devices. Instead,\nRetrieval-Augmented Generation (RAG), a resource-efficient LLM learning method,\ncan improve the quality of the LLM-generated content without updating model\nparameters. However, the RAG-based LLM may involve repetitive searches on the\nprofile data in every user-LLM interaction. This search can lead to significant\nlatency along with the accumulation of user data. Conventional efforts to\ndecrease latency result in restricting the size of saved user data, thus\nreducing the scalability of RAG as user data continuously grows. It remains an\nopen question: how to free RAG from the constraints of latency and scalability\non edge devices? In this paper, we propose a novel framework to accelerate RAG\nvia Computing-in-Memory (CiM) architectures. It accelerates matrix\nmultiplications by performing in-situ computation inside the memory while\navoiding the expensive data transfer between the computing unit and memory. Our\nframework, Robust CiM-backed RAG (RoCR), utilizing a novel contrastive\nlearning-based training method and noise-aware training, can enable RAG to\nefficiently search profile data with CiM. To the best of our knowledge, this is\nthe first work utilizing CiM to accelerate RAG.",
+                    "authors": "Ruiyang Qin, Zheyu Yan, Dewen Zeng, Zhenge Jia, Dancheng Liu, Jianbo Liu, Zhi Zheng, Ningyuan Cao, Kai Ni, Jinjun Xiong, Yiyu Shi",
+                    "published": "2024-05-07",
+                    "updated": "2024-05-07",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI",
+                        "cs.DC",
+                        "cs.IR"
+                    ],
+                    "main_content": "INTRODUCTION The emerging Large Language Models (LLMs) are deployed primarily on centralized cloud platforms [1, 2] (Cloud LLMs), raising concerns about user privacy and trustworthy issues [3]. These issues become even more prominent in areas such as healthcare [4], companionship [5], and personal assistance [6], where the user privacy and trustworthiness of LLMs are crucial. To address these issues, the cloud LLMs will eventually transform into personalized LLMs, capable of generating personalized responses, deployed on edge devices (Edge LLMs), where users can keep all their private data and the model learns from those data locally. To better suit the needs of individual users, Edge LLMs must learn from user interactions. However, their capability of learning is constrained by their limited RAM and computational power. Similar to Cloud LLMs, the Edge LLMs primarily learn by finetuning their model parameters. Yet, given that these models often contain over 3 billion parameters, updates can be challenging, even with numerous efforts to accelerate them [7\u20139]. For example, using the experimental high-performance embedded system like NVIDIAAGX, the pockengine method [9] can still take 90 hours to learn from a middle-sized dataset Alpaca with only 52k documents, making this option impractical for normal users. E x : \u201cI  am sick?\u201d Sentence  Embedding  Model E(x) User Query Profile Data Embedding Space S \u2026 P(x, d83) P(x, d29) P(x, d37) Top k (k = 1) DAC ADC \u2026 CiM E(d83) Data: d83, d29, d37 User Query LLM Output NVM Digital Logic \ud835\udc6c(\ud835\udc99) \u2219\ud835\udc6c(\ud835\udc85\ud835\udc8a) = \ud835\udc0f(\ud835\udc31, \ud835\udc85\ud835\udc8a) E(d1) E(d2) E(d3) E(d4) E(dn) Document 2 Document 1 Document n \u2026 Figure 1: The workflow of RAG on edge-based CiM. CiM performs max inner product search (MIPS) to retrieve the top-ranked documents, concatenating them with user query to allow the LLM to generate personalized responses. Retrieval-augmented generation (RAG), on the other hand, is a more resource-efficient choice [10], and hence becoming the de facto learning method for Edge LLMs. In a typical RAG system, it consists of a retriever and a generator. The retriever is commonly backed by max inner product search (MIPS). When the retriever receives a user query, it will retrieve the most relevant document from profile data, as shown in Figure 1. The profile data has many documents, and each document \ud835\udc51\ud835\udc56contains specific information that may be relevant to user queries. The generator can be seen as a LLM, which takes the user query \ud835\udc65and retriever-obtained documents as a prompt and generates a corresponding response. For every document\ud835\udc51\ud835\udc56and the user query \ud835\udc65, RAG utilizes a sentence embedding model shown in Figure 1 to convert them into vectors (i.e., \ud835\udc38(\ud835\udc51\ud835\udc56) and \ud835\udc38(\ud835\udc65), respectively). The vectors for documents can be named as document embeddings and stored as a matrix as shown in Figure 1. The vector for user query, named query embedding \ud835\udc38(\ud835\udc65), will be used in MIPS to perform inner product with every document embedding. The larger the product \ud835\udc43(\ud835\udc65,\ud835\udc51\ud835\udc56), the more semantic similar it will be between the user query and the document. Using RAG, Edge LLMs can provide user-preferred responses by retrieving relevant documents from profile data, and the profile data can be incrementally updated with new documents. This is an efficient learning process without costly updating the model parameters via fine-tuning [11]. Other than the inevitable LLM inference cost, the primary computational cost of RAG is about retrieval, which is more than ten times less than the cost of updating model parameters. While the computational cost of RAG is more edge-friendly, there still exist two issues impeding RAG from being deployed for real-time user interaction on Edge LLMs. Firstly, the growing profile data as stored cannot be unlimited without affecting the access time. If the size of the profile data exceeds the RAM capacity, arXiv:2405.04700v1  [cs.LG]  7 May 2024 \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 it will need to be offloaded into the storage, such as a hard disk drive (HDD) or solid-state drive (SSD). Accessing data from HDD or SSD will significantly increase the data transfer latency [12], rendering real-time user interaction impractical. Secondly, the core retrieval method of RAG, MIPS, may experience decreased efficiency as profile data grows, and it can become potentially prohibitive when dealing with overwhelmingly large datasets. For example, on Raspberry Pi 4B, MIPS can take 5 minutes to find one appropriate profile data among 21M documents [10], which is even longer than the 2-minute inference time of an Edge LLM. Unfortunately, few efforts have been made to optimize RAG towards Edge LLMs. Thus, we propose to utilize the Computing-in-Memory (CiM) architecture to address this issue. As shown in Figure 1, CiM architectures using memory arrays have shown substantial promise in accelerating matrix-vector multiplication [13], which is the key operation of MIPS. The CiM architectures often utilize massive parallel processing to perform computations directly within the memory array where the data is stored, such that they can minimize the data movement through in-situ data access and significantly increase the throughput [14]. Given the same amount of documents, CiM can finish computation within 50ms [15], which is negligible compared to the computation latency on normal edge devices. Furthermore, by incorporating non-volatile memory (NVM) devices, such as phase-change memories (PCMs), resistive random-access memories (RRAMs), and ferroelectric field-effect transistors (FeFETs), CiM can outperform conventional MOSFET-based designs in terms of energy efficiency [16]. 0.00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Level of noise ( ) 0.0 0.2 0.4 0.6 0.8 1.0 Accuracy Citation Movie Rating News DBLP Figure 2: The impact on MIPS accuracy when the RAG\u2019s document embedding is perturbed by various levels of Gaussian noise caused by the device variations. An accurate retrieval means the document retrieved under the impact of the noise is the same as that retrieved without noise. Unfortunately, simply changing the underlying hardware is not enough, as the non-idealities of the NVM devices in CiM array could greatly deteriorate the RAG performance. First, the operations performed in CiM architectures are susceptible to various sources of noise, including electronic noise (thermal, shot, and flicker), deviceto-device variability, and line noise from the supporting circuitry [17]. These noise sources can corrupt the computations, especially when the signal levels are close to the noise floor, which is a common scenario in high-precision tasks. Such noise issues are critical in RAG applications where the accuracy and quality of the generated content heavily rely on the precision of the underlying computations. Additionally, the CiM architecture is primarily designed and optimized for low-resolution computation [18]. Moreover, CiM arrays are typically sized at a fixed dimension, such as 64x64 [19], which is different from the documents\u2019 embedding dimension (e.g., 128). Therefore, both RAG\u2019s data precision (typically FP32) and its embedding dimension need to be reduced to fit in the size of CiM\u2019s crossbar arrays. To illustrate the impact of these on RAG, as an example, we present a preliminary study on MIPS performance in Figure 2, where we use a simple yet representative Gaussian noise to simulate the noise from the device variations in CiM. As shown in Figure 2, as the noise level increases, MIPS accuracy (specified in section 4.1.3) drops dramatically, approaching random guessing. To address these issues, we further propose a novel optimization framework for CiM-backed RAG, called Robust CiM-backed RAG (RoCR). The framework consists of three parts. The first part is a contrastive learning method. We use it to optimize the document embedding model. The second part is a novel data construction method to generate both positively and negatively labeled data pairs for contrastive learning. For the profile data, they can be either labeled to indicate the explicit user-preferred response to certain input, or simply statements without explicit labels that only implicitly indicate user preferences. Our data construction method is capable of dealing with both types of profile data. The third part is a noise-aware training method. It goes in tandem with contrastive learning to obtain a sentence embedding model that can generate document and user query embeddings with high noise-resilient capability, while such embeddings can fit into CiM architectures under different designs and configurations. Our major contributions can be summarized as: \u2022 We propose the first work to harvest CiM advantages for RAG acceleration on the edge. We provide a pathway to utilize emerging CiM devices to expand the Edge LLMs\u2019 capability in terms of storing a high volume of profile data with fast MIPS computing. \u2022 We introduce noise-aware training to enhance the noiseresilient capabilities of RAG\u2019s document embedding. The resulting noise-resilient embeddings can be reused robustly, saving resources needed to calibrate and regenerate embeddings. \u2022 Our experiments on various datasets show that our proposed framework can improve the RAG performance on multiple CiM devices up to 35%, approaching to the theoretical RAG performance. Across a wide device variation (noise) range on a single CiM device, our proposed framework can still improve the RAG performance. 2 RELATED WORK 2.1 CiM Architectures and their NVMs As shown in the middle part of Figure 1, memory arrays are the key component for vector-matrix multiplication. In this array, matrix values are stored at NVM cells, such as emerging NVM technologies like PCMs, RRAMs, and FeFETs, at the cross-points of vertical and horizontal lines. Simultaneously, vector values flow along the horizontal lines of the array. Operations within the memory array take place in the analog domain by exploiting law of physics directly. However, for other essential functions like shift-and-add for multiple bits and sorting to find the top-k ranked values would be done in the digital domain. Thus, digital-to-analog and analog-to-digital \fRobust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures profile  data Data Construction  Module positive  examples anchor  examples \u2026 Reshape  Module embeddings Contrastive  Learning close far Device Variation NVMs Sentence  Embedding  Model negative  examples Flexible Noise-aware  Training Module optimize constraints Figure 3: Overview of the proposed Robust CiM-backed RAG framework (RoCR). It optimizes the sentence embedding model to adapt different types of NVMs utilized by CiM. converters (DACs and ADCs) are used to connect these different components. CiM arrays suffer from various sources of variations and noises. Two major ones include spatial variations and temporal variations. Spatial variations result from fabrication defects and have both local and global correlations. FeFET devices also suffer from temporal variations due to the stochasticity in memory switching and also aging, which causes fluctuations in conductance when programmed at different times. Temporal variations are typically independent from device to device and are irrelevant to the value to be programmed [20]. In this work, as a proof of concept, we focus on the impact of temporal variations in the programming process on DNN performance. Temporal variation makes the programmed resistance of a device deviate from what is expected. The proposed framework can also be extended to other sources of variations with modification. Measurement results [21, 22] show that the noise on DNN weights caused by device variations can be safely modeled as a Gaussian noise with zero mean, each with a standard deviation associated with the weight value. A detailed representation is given by: v = v0 + \u0394v, \u0394v \u223cN (0, \ud835\udf0e\ud835\udc63) (1) where v is the actual embedding deployed on the accelerators, v0 is the target embedding value, and \ud835\udf0e\ud835\udc63is a value measured by the experiments. We collect the measurement results from RRAM and FeFET devices and the specific value will be discussed in Section 4.1. 2.2 Past Noise Mitigation Methods Several strategies have been introduced to tackle the challenge of device variations in CiM accelerators. These methods can be separated into software and hardware-based techniques. The software-based techniques are generally developed to obtain more robust DNN models [19, 22\u201324] or recommendation systems [25], and are thus not suitable for generating more robust MIPS solutions. For the hardware techniques, the write-verify procedure [26, 27] is one of the most commonly used approach during programming. Initially, a NVM device is programmed to a set state via a designated pulse pattern. Subsequent to this, the device\u2019s value is verified to ascertain if its conductance aligns with a stipulated range of the desired value, essentially assessing its accuracy. If discrepancies arise, a supplemental update pulse is initiated to reset the device conductance nearer to the target. This loop persists until the disparity between the programmed device value and the target value diminishes to a satisfactory margin, typically taking a handful of cycles. Cutting-edge research suggests that by selectively applying write-verify to a subset of pivotal devices, one can uphold the average accuracy of a DNN [21]. Additionally, a variety of circuit design initiatives [18, 28] have been put forth to counteract device variations. 3 PROPOSED WORK 3.1 Framework Overview As shown in Figure 3, our proposed framework, Robust CiM-backed RAG (RoCR), consists of three stages. First, we apply contrastive learning to utilize the training data to optimize the training module. To do that, in the second stage, we take the profile data and construct via a data construction module to obtain contrastive training data pairs, which are then used in the flexible noise-aware training module. In the third stage, we obtain the constraints of NVMs in CiM via profiling. These constraints will be encoded into the flexible noise-aware training module and used to train the sentence embedding model so that it can generate embedding that are robust against device variation of the target NVMs. After training, the training module can be turned into a new sentence embedding model and generate CiM-friendly embeddings. 3.2 Contrastive Learning: Triplet Loss Function When we apply RAG using CiM, we first need to store embeddings into NVMs as shown in Figure 1. Such embeddings are generated by the sentence embedding model, and they are the numerical representations of profile data. Each single document in the profile data can have its unique embedding, which is a vector. The embeddings stored on NVMs can consist of a matrix as the orange blocks shown in Figure 1. Given a user query, which will also be converted into an embedding, CiM can operate MIPS between this user query embedding and all profile embeddings simultaneously via vector-matrix multiplication. The top-ranked values in the product will be used as the index to retrieve the corresponding document data, as the pink block shown in Figure 1. This retrieved user-relevant document is the output of MIPS. However, as we have explained in Section 2.1, writing the document embeddings into NVMs can cause them to suffer from temporal variations (device variations). Then, the NVM-stored embeddings will be different from the original sentence embedding model generated embeddings. As shown in Figure 4, the vanilla embedding model generates desired embedding, which will deviate to the noise embedding under device variation, such that the irrelevant embedding is ranked higher than desired embedding due to its larger inner product. Contrastive learning can learn the representations via push away dissimilar examples and pull close similar examples [29]. In particular, the contrastive loss function can be used to increase the distance between dissimilar examples. In our work, we propose to improve the noise-resilient capability by contrastive learning. By increasing the distance between \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 noise  embedding irrelevant  embedding query retrieve the wrong data irrelevant  embedding query NVMs Device Variation lead to Vanilla  CiM-backed  RAG Robust  CiM-backed  RAG Our  embedding  model noise-resilient  embeddings user  profile  data NVMs Device Variation desired  embedding vanilla  embedding  model user  profile  data retrieve the desired data embeddings Figure 4: Improvement by our Robust CiM-backed RAG. Our framework generates noise-resilient embeddings, as shown the orange and blue point in right subfigure dissimilar examples, as shown the right subfigure in Figure 4, deviated desired embedding will still have a larger inner product with the query compared to the irrelevant embedding. Our contrastive learning loss function is based on Weinberger et al. [30]. For each example \ud835\udc65\ud835\udc56in a mini-batch of N anchor examples, our data construction method will construct \ud835\udc3epositive and \ud835\udc3enegative examples corresponding to \ud835\udc65\ud835\udc56. We can have {{(\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56,\ud835\udc65+ \ud835\udc56)\ud835\udc58}\ud835\udc56=1,...,\ud835\udc41}\ud835\udc58=1,...,\ud835\udc3e, in which \ud835\udc65\u2212and \ud835\udc65+ are negative and positive examples corresponding to \ud835\udc65\ud835\udc56, where \ud835\udc65\ud835\udc56is closer to \ud835\udc65+ \ud835\udc56compared to \ud835\udc65\u2212 \ud835\udc56. Also, \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56) represents the learned embedding of \ud835\udc65\ud835\udc56. Then the loss function L can be defined as: L = \ud835\udc41 \u2211\ufe01 \ud835\udc56=1 1 \ud835\udc3e \ud835\udc3e \u2211\ufe01 \ud835\udc58=1 max \u0010 0, d(\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56(\ud835\udc58)) \u2212d(\ud835\udc65\ud835\udc56,\ud835\udc65+ \ud835\udc56(\ud835\udc58)) + \ud835\udc5a \u0011 , d(\ud835\udc65\ud835\udc4e,\ud835\udc65\ud835\udc4f) = sim(emb(\ud835\udc65\ud835\udc4e), emb(\ud835\udc65\ud835\udc4f)) (2) The distance \ud835\udc51(\ud835\udc65\ud835\udc4e,\ud835\udc65\ud835\udc4f) is calculated by the Euclidean distance between embeddings of two data \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc4e) and \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc4f). The function \ud835\udc60\ud835\udc56\ud835\udc5a() calculate the semantic similarity. 3.3 Data Construction To train the sentence embedding model via contrastive learning, it is critical to construct pairs of examples where the positive examples and negative examples need to be distinct from each other [31]. In our work, since we use triplet contrastive loss, instead of pairs of examples, we will construct trios of examples where each triplet contains an anchor, positive, and negative example. We use profile data to construct triplets of examples. For the profile data, it is generated by the user during the user-LLM interaction and contains the user preference information. There exists two situations for such data. First, the profile data can contain explicit labels indicating the user preferred response to the corresponding content. Second, the profile data also can be statements containing the user-related information but without explicit user preferences As shown in Figure 5, to deal with the two situations, we come up with two data construction methods: Construction Data with Explicit labels (CDE) and Construction Data with Implicit labels (CDI). \u201cJake Blues, just released from prison,  puts his old band back together to save  the Catholic home where he and his  brother Elwood were raised.\u201d is  \u201cdystopia\u201d negative r = 0.1 r = 0 r = 0 \u201cFresh out of prison, Jake Blues  rallies his old band to save their  childhood Catholic home\u201d is  \u201cclassic\u201d positive example (embedding) \u201cJake Blues, just released\u2026\u201d is  \u201cclassic\u201d anchor example (embedding) \u201cJake Blues, just released \u2026\u201d is  \u201cdystopia\u201d negative example (embedding) r = 0.1 \u201cVictims of traumatized \u2026\u201d r = 0 r = 0.9 CDE CDI E anchor/positive example negative example \u201cTwo victims of traumatized  childhoods become lovers and serial  murderers irresponsibly glorified by the  mass media.\u201d anchor/positive/negative example \u201cTwo people with traumatic pasts  turn into a couple on a crime spree,  mistakenly idolized by the media.\u201d \u201cIndividuals, mired in traumas, unite *()  crime-ridden bond, enthrall\u2606\u2609\u00a7ing the  media's distorted spotlight.\" \u201cJake Blues, just released from prison,  puts his old band back together to save  the Catholic home where he and his  brother Elwood were raised.\u201d is \u201cclassic\u201d explicit label Statement/implicit label Figure 5: Examples of the two data construction methods. For data with explicit labels, CDE is used to construct the training data. For data without explicit labels (implicit labeled data), CDI is used to construct the training data. 3.3.1 Construction Trios via Data with Explicit Labels (CDE). For the data with explicit labels, each of the data consists of a textual content c and its corresponding label l which indicates the user preferred response regarding to the content c. As shown in the CDE part in Figure 5, there exists explicit label circled by dashed line. Using the profile data, we will construct triplet examples in the format of (\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56,\ud835\udc65+ \ud835\udc56). Given a dataset D with size of \ud835\udc5bprofile documents, each piece of data consists of a content \ud835\udc50\ud835\udc56and the corresponding label \ud835\udc59\ud835\udc56where \ud835\udc56\u2208{1, 2, ...,\ud835\udc5b}. The anchor example \ud835\udc65\ud835\udc56can be constructed as: \ud835\udc65\ud835\udc56= \ud835\udc50\ud835\udc56\u2295\ud835\udc59\ud835\udc56, for \ud835\udc56= 1, 2, . . . ,\ud835\udc5b (3) where \u2295denotes a concatenation operation, specifically used here to combine label and content. Negative examples \ud835\udc65\u2212 \ud835\udc56can be constructed by concatenating \ud835\udc50\ud835\udc56with a random label \ud835\udc59\ud835\udc57that is different from \ud835\udc59\ud835\udc56as follows: \ud835\udc65\u2212 \ud835\udc56= \ud835\udc50\ud835\udc56\u2295\ud835\udc59\ud835\udc57, where \ud835\udc59\ud835\udc56\u2260\ud835\udc59\ud835\udc57. (4) Randomly assigning a different label ensures diversity in the negative examples while maintaining the same content from the anchor. Different from constructing anchor and its negative examples, it is challenging to construct positive examples corresponding to the anchor examples since it is more difficult to formalize semantically similar data than to formalize semantically dissimilar data. To construct positive examples, we follow the SimCSE method [32] to add a dropout rate \ud835\udc5finto the sentence embedding model M. The process for constructing positive examples involves two main steps. First, the textual positive example is formalized as: \ud835\udc65+ \ud835\udc56= \ud835\udc65\ud835\udc56, for \ud835\udc56= 1, 2, ...,\ud835\udc5b (5) where we align each anchor with the corresponding positive example. This step effectively duplicates the anchor data as a starting point for generating the embeddings. \fRobust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures Second, the embedding generation process varies based on the dropout rate applied within the model M. When model M is utilized to generate embeddings for anchor and negative examples, the dropout rate is set to 0. In contrast, for generating embeddings for positive examples, a non-zero dropout rate \ud835\udc5fis used. The anchor, negative, positive examples, as shown in Figure 5, can be constructed as: \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56) = M(\ud835\udc65\ud835\udc56,\ud835\udc51\ud835\udc5f\ud835\udc5c\ud835\udc5d\ud835\udc5c\ud835\udc62\ud835\udc61= 0) \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\u2212 \ud835\udc56) = M(\ud835\udc65\u2212 \ud835\udc56,\ud835\udc51\ud835\udc5f\ud835\udc5c\ud835\udc5d\ud835\udc5c\ud835\udc62\ud835\udc61= 0) \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65+ \ud835\udc56) = M(\ud835\udc65+ \ud835\udc56,\ud835\udc51\ud835\udc5f\ud835\udc5c\ud835\udc5d\ud835\udc5c\ud835\udc62\ud835\udc61= \ud835\udc5f) (6) The condition of \ud835\udc5f\u22600 can induce variation in the embeddings, enhancing the model\u2019s ability to recognize semantically similar yet variably expressed content. Given the construction factor \ud835\udc3e, we can construct the triplet data examples as: D\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc5d\ud835\udc59\ud835\udc52\ud835\udc61= \ud835\udc41 \u00d8 \ud835\udc56=1 n (\ud835\udc65\ud835\udc56(\ud835\udc58),\ud835\udc65\u2212 \ud835\udc56(\ud835\udc58),\ud835\udc65+ \ud835\udc56(\ud835\udc58)) : \ud835\udc58= 1, 2, . . . , \ud835\udc3e o (7) For the triplet data examples D\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc5d\ud835\udc59\ud835\udc52\ud835\udc61, their embeddings for each augmentation \ud835\udc58are given by: E = \ud835\udc41 \u00d8 \ud835\udc56=1 n (\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56(\ud835\udc58)),\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\u2212 \ud835\udc56(\ud835\udc58)),\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65+ \ud835\udc56(\ud835\udc58)) : \ud835\udc58= 1, 2, . . . , \ud835\udc3e o (8) As shown in Figure 5, for data with explicit labels, a content\ud835\udc50can concatenate with its corresponding label \ud835\udc59to formalize the positive and anchor example. That content \ud835\udc50can also concatenate with other labels \ud835\udc59\u2032 to formalize the negative example. The positive example can be finally obtained from the sentence embedding model with dropout rate \ud835\udc5f. The anchor and negative example can be finally obtained from the sentnece embedding model with \ud835\udc5f= 0. 3.3.2 Construction Trios via Data with Implicit Labels (CDI). For data with implicit labels, each of the data consists of solely textual content c. As shown of the CDI part in Figure 5, there is no explicit label to indicate user preferences. Instead, the data can be seen as a statement containing some user-related information. To construct the anchor examples and positive examples, we can use the exact same method in EDC. Given a dataset D with size of n profile data, each piece of data consits of a content \ud835\udc50\ud835\udc56. The anchor data \ud835\udc65\ud835\udc56can be constructed as: \ud835\udc65\ud835\udc56= \ud835\udc50\ud835\udc56, for \ud835\udc56= 1, 2, . . . ,\ud835\udc5b (9) For each anchor data \ud835\udc65\ud835\udc56, constructing its corresponding negative example is not as simple as merely concatenating the content\ud835\udc50\ud835\udc56with a non-corresponding label \ud835\udc59\ud835\udc58. To construct negative examples, we employ a reciprocal approach with the positive examples, applying a similar method to both. We first initialize the negative example and positive example following the equation 5: \ud835\udc65\u2212 \ud835\udc56= \ud835\udc65+ \ud835\udc56= \ud835\udc65\ud835\udc56, for \ud835\udc56= 1, 2, . . . ,\ud835\udc5b (10) For the positive example \ud835\udc65+ \ud835\udc56, it can be finalized by incorporating a dropout rate \ud835\udc5finto the sentence embedding model M, where a rate of 0 < \ud835\udc5f\u22640.2 can generate a sentence embedding with a semantic representation similar to \ud835\udc65\ud835\udc56and ensure good model training performance [32]. Increasing the dropout rate to a higher value, such as 0.5, can distort the semantic representation of \ud835\udc65+ \ud835\udc56, making it dissimilar to that of \ud835\udc65\ud835\udc56. Training the model with such positive examples can result in poorer performance. For positive examples in training the sentence embedding model, the higher dropout rate performs more like a noise rather than a data augmentation method. In our work, we train the sentence embedding model to generate embeddings that maintain their integrity under noisy conditions, such as during writing into Compute-in-Memory (CiM). The noise can alter or fragment the original semantic representations. For instance, as illustrated in Figure 5, using a high dropout rate \ud835\udc5f= 0.9 can lead to a negative example with a corrupted representation. Although it may lack certain informative content, this negative example becomes semantically distinct from both the anchor and positive examples, effectively simulating the effect of CiM corruption. This approach not only differentiates the negative examples semantically but also aligns them with the corrupted data scenarios for noise-aware training. Given the triple examples (\ud835\udc65\ud835\udc56,\ud835\udc65\u2212 \ud835\udc56,\ud835\udc65+ \ud835\udc56), for \ud835\udc56= 1, 2, ...,\ud835\udc5bas shown in equation 10, we have the dropout rate \ud835\udc5ffor formalizing the positive examples where 0 < \ud835\udc5f\u22640.2. Correspondingly, the dropout rate for formailzing the negative examples can be 1 \u2212\ud835\udc5f. Given the sentence embedding model M, the anchor example, positive example, and negative example can be constructed as: emb(\ud835\udc65\ud835\udc56) = M(\ud835\udc65\ud835\udc56, dropout = 0) emb(\ud835\udc65\u2212 \ud835\udc56) = M(\ud835\udc65\u2212 \ud835\udc56, dropout = 1 \u2212\ud835\udc5f) emb(\ud835\udc65+ \ud835\udc56) = M(\ud835\udc65+ \ud835\udc56, dropout = \ud835\udc5f) (11) 3.4 Flexible Noise-aware Training In the previous two stages, we construct the data to train the sentence embedding model based on contrastive learning. Meanwhile, the training can be more effective when injecting the simulated device variation [33] so that the model can be optimized with consideration of the device variation. Additionally, the sentence embedding model needs to produce embeddings that can fit with the different CiMs, which might have various NVM designs. To do that, we need the sentence embedding model reshapes its output embeddings into certain dimensions and precision. Hence, we propose a flexible noise-aware training method, which can generate the noise-resilient embedding, fitting to various CiMs. As shown in Figure 3, in the flexible noise-aware training module, the embedding generated by sentence embedding model will be shaped based on the CiM\u2019s NVMs constraints where required dimension is \ud835\udc51and required precision is \ud835\udc5d, and being injected device variation to formalize the embeddings. The reshape module, shown in Figure 3, seen as an autoencoder to reconstruct its input embedding [34], can be expressed as \ud835\udc60\u210e\ud835\udc5d(), initialized by \ud835\udc51 and \ud835\udc5d, takes the anchor embedding \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56) as input. We can have \ud835\udc60\u210e\ud835\udc5d(\ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)) = \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d. Based on the device variation shown as Table 2, we can have: \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d \ud835\udf0e = (\ud835\udc52\u2032 \u2217\ud835\udc3f0 + \ud835\udc52\u2032 \u2217\ud835\udc3f1 + \ud835\udc52\u2032 \u2217\ud835\udc3f2 + \ud835\udc52\u2032 \u2217\ud835\udc3f3) \u2217\ud835\udf0e, (12) \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 Table 1: Performance comparison between our framework and four baselines on five CiM devices with device variation specified in Table 2 across five datasets. Evaluate the performance of our framework using EDC (RoCR-EDC) and using IDC (RoCR-IDC) to optimize the performance of RAG, which utilizes Gemma-2 as its LLM. Dataset Citation Movie Rating News DBLP CiM Method Acc \u2191 F1 \u2191 Acc \u2191 F1 \u2191 MAE \u2193 RMSE \u2193 ROUGE-1 \u2191 ROUGE-L \u2191 ROUGE-1 \u2191 ROUGE-L \u2191 Device-1 SWV 0.4208 0.3339 0.1305 0.1974 0.3850 0.8093 0.0754 0.0731 0.1709 0.1590 CxDNN 0.4223 0.3576 0.1516 0.1762 0.4404 0.9135 0.0640 0.0632 0.1646 0.1449 CorrectNet 0.4155 0.3791 0.0996 0.1305 0.3609 0.7071 0.0512 0.0764 0.1603 0.1538 Vanilla RAG 0.4401 0.3476 0.1017 0.0838 0.3903 0.8944 0.0754 0.0731 0.1731 0.1473 RoCR-CDE 0.5536 0.3956 0.2242 0.2303 0.3108 0.6856 0.1041 0.0987 0.2066 0.1924 RoCR-CDI 0.5409 0.5117 0.2273 0.2487 0.2767 0.6083 0.0831 0.0808 0.2317 0.2176 Device-2 SWV 0.1831 0.1552 0.1992 0.1957 0.4205 0.8775 0.0296 0.0289 0.1968 0.1874 CxDNN 0.4013 0.3557 0.2167 0.2019 0.4423 0.8367 0.0604 0.0791 0.1517 0.1401 CorrectNet 0.3827 0.3209 0.1625 0.1909 0.3762 0.8062 0.0513 0.0505 0.2042 0.1945 Vanilla RAG 0.4801 0.3462 0.1576 0.2079 0.4153 0.9354 0.0296 0.0289 0.1618 0.1353 RoCR-CDE 0.5407 0.4396 0.2924 0.2509 0.2553 0.5385 0.1209 0.0946 0.2025 0.1906 RoCR-CDI 0.5299 0.4591 0.2971 0.2386 0.2124 0.5763 0.0884 0.0853 0.2240 0.2098 Device-3 SWV 0.2450 0.2564 0.1695 0.1641 0.3460 0.7416 0.0725 0.069 0.1018 0.0954 CxDNN 0.4811 0.4006 0.2367 0.2113 0.2851 0.6928 0.0761 0.0707 0.1425 0.1111 CorrectNet 0.4510 0.3918 0.0792 0.1029 0.3704 0.7937 0.0585 0.0555 0.1715 0.1346 Vanilla RAG 0.4852 0.3618 0.1614 0.1636 0.3255 0.7649 0.0725 0.0690 0.1647 0.1437 RoCR-CDE 0.5139 0.4116 0.2242 0.2215 0.3208 0.6481 0.0825 0.0805 0.1893 0.1754 RoCR-CDI 0.5515 0.4984 0.2152 0.2131 0.2916 0.6245 0.1099 0.1049 0.2294 0.2140 Device-4 SWV 0.5135 0.4260 0.1271 0.1178 0.3610 0.8196 0.0259 0.0256 0.1871 0.1786 CxDNN 0.4733 0.3964 0.1267 0.2158 0.3468 0.7616 0.0646 0.0634 0.1603 0.1538 CorrectNet 0.4628 0.4019 0.1592 0.1847 0.4013 0.9274 0.0705 0.0750 0.1628 0.1292 Vanilla RAG 0.2101 0.2401 0.1219 0.2019 0.4015 0.8544 0.0505 0.0489 0.1929 0.1814 RoCR-CDE 0.5836 0.5555 0.1706 0.2817 0.3139 0.6856 0.0873 0.0851 0.1984 0.1882 RoCR-CDI 0.5352 0.4289 0.1642 0.2445 0.2706 0.5916 0.1154 0.1128 0.2148 0.1978 Device-5 SWV 0.4320 0.3541 0.1250 0.1076 0.3652 0.7616 0.0434 0.0427 0.0985 0.0923 CxDNN 0.4301 0.0538 0.0751 0.0458 0.3503 0.8185 0.0707 0.0682 0.2042 0.1945 CorrectNet 0.4145 0.3926 0.1083 0.1395 0.5526 0.8185 0.0735 0.0776 0.2096 0.1879 Vanilla RAG 0.4256 0.3522 0.0847 0.0863 0.3951 0.8515 0.0676 0.0653 0.2018 0.1846 RoCR-CDE 0.5698 0.5223 0.2152 0.1669 0.2959 0.6245 0.0936 0.0891 0.1946 0.1844 RoCR-CDI 0.5254 0.4504 0.2394 0.2458 0.2624 0.6325 0.0799 0.0764 0.2238 0.2095 where \ud835\udc52\u2032 = \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d. The device variation, as noise, is injected into embeddings to formalize \ud835\udc52\ud835\udc5a\ud835\udc4f(\ud835\udc65\ud835\udc56)\ud835\udc51\u2217\ud835\udc5d \ud835\udf0e , which will be used in contrastive learning to train the sentence embedding model, as shown in Figure 3. 4 EXPERIMENTAL EVALUATION 4.1 Experimental Setup 4.1.1 Datasets. To demonstrate our robust CiM-backed RAG, we employ five datasets with different tasks and domains, including Citation Identification [35] (Citation), Movie Tagging [36] (Movie), Product Rating [37] (Rationg), News Headline Generation [38] (News), and DBLP-Citation-network V14 [39] (DBLP) to evaluate the proposed framework. The data in each dataset consists of query data and profile data. In our evaluation, the profile data will be used to formalize user history, and the profile corresponding query data will be used as the user input. The first three datasets contain binary, five-class, and fifteen-class classification tasks respectively. The last two datasets contain text generation tasks. In the Citation Identification dataset, every piece of query data consists of a paper title and two references, and the correct reference is provided. RAG uses the profile data corresponding to the paper titles with their detailed contents to choose the appropriate reference. In the Movie Tagging dataset, each query data contains a description of a movie, and RAG uses a similar description and its corresponding tag in the profile data to tag the query data. The Product Rating dataset has a similar structure as the Movie Tagging dataset. In News Headline Generation and DBLP datasets, each query data contains an abstract, which can be summarized into a title. RAG uses a similar abstract and its corresponding title in profile data to generate the title for query data. All five datasets have labels in their query data. 4.1.2 Default Experimental Setting. Our framework chooses all-MiniLM-L6-v2 [40] as the sentence embedding model. For each dataset, we randomly select 2000 documents from profile data as the anchor examples. To examine the data construction method of CDE, we set the augmentation factor \ud835\udc58= 5 to obtain 10000 negative and positive examples. We set dropout rate as 0.1 to obtain the positive examples while maintain it as 0 when process anchor and negative examples. To examine the data construction method CDI, we set dropout rate for positive examples as 0.1 and dropout rate for negative examples as 0.9. To align with experiments for CDE, we also set \ud835\udc58= 5 in the experiments for CDI. For the experimental results, we run five times and get the average. In experiments, we set the device variation \ud835\udf0e= 0.1 and shape embeddings into dimension of 64 with precision of \ud835\udc56\ud835\udc5b\ud835\udc618. The learning rate is 2\ud835\udc52\u22125. In all experiments, we adhere to the device variation model previously described. The specific parameters are abstracted and then simplified from three representative NVM devices, two of them \fRobust Implementation of Retrieval-Augmented Generation on Edge-based Computing-in-Memory Architectures D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (a) Citation on Gemma-2B D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (b) Citation on Phi-2 D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (c) Citation on Mistral-7B D-1 D-2 D-3 D-4 D-5 0.2 0.4 0.6 0.8 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (d) Citation on Llama-2-3B D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (e) Movie on Gemma-2B D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (f) Movie on Phi-2 D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (g) Movie on Mistral-7B D-1 D-2 D-3 D-4 D-5 0.0 0.1 0.2 0.3 0.4 0.5 Accuracy SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI (h) Movie on Llama-2-3B Figure 6: Performance comparison between our framework and four baselines on RAG utilizing the LLMs including Gemma-2B, Phi-2, Mistral-7B, and Llama-2-3B with device variation specified in Table 2, given dataset \ud835\udc36\ud835\udc56\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5band \ud835\udc40\ud835\udc5c\ud835\udc63\ud835\udc56\ud835\udc52. Table 2: Device non-ideality modeling for different real and synthesized devices. For devices with more than two levels, the device variation for each level is depicted as \ud835\udc3f\ud835\udc65. Name # of Levels Device Variations \ud835\udf0e\ud835\udc63 \ud835\udc3f0 \ud835\udc3f1 \ud835\udc3f2 \ud835\udc3f3 \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc401 (Device-1) 1 0.0100 0.0100 0.0100 0.0100 \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc472 (Device-2) 4 0.0067 0.0135 0.0135 0.0067 \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc473 (Device-3) 4 0.0049 0.0146 0.0146 0.0049 \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc404 (Device-4) 4 0.0038 0.0151 0.0151 0.0038 \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc476 (Device-5) 4 0.0026 0.0155 0.0155 0.0026 are resistive random-access memory (RRAM) devices extracted from [27, 41] and the other is a ferroelectric field effect transistor (FeFET) device extracted from [42]. We name them \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc401, \ud835\udc45\ud835\udc45\ud835\udc34\ud835\udc404 and \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc472, respectively. We also extrapolate the modeling data to obtain two synthesized \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc473 and \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc476 devices. Detailed device modeling results are demonstrated in Table 2. A \ud835\udc65-level device means this device can represent \ud835\udc65distinct values and \ud835\udf0e\ud835\udc3f2 = 0.01 means the variation of this device is 0.01 when it is representing the level value 2. Using the device variations obtained from real CiM devices, we perform our experiments on a single Nvidia A10 GPU. Document embeddings are shaped based on different CiM devices and stored as parallel arrays, similar to how they would be mapped to multiple NVM devices in practical scenarios. For example, if an embedding is shaped to contain all uint8 values, when it is mapped to 4-level (2-bit) devices such as \ud835\udc39\ud835\udc52\ud835\udc39\ud835\udc38\ud835\udc472, each element of the vector is represented by four devices. 4.1.3 Evaluation Methods. Our first three datasets examine the model classification capability, and the rest of two datasets examine the text generation capability. In particular, dataset \ud835\udc36\ud835\udc56\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5band \ud835\udc40\ud835\udc5c\ud835\udc63\ud835\udc56\ud835\udc52has two and fifteen labels respectively. We can examine the binary and multiclass classification capabilities of the LLMs enhanced by our framework. In this way, we use accuracy to examine the ability of the models to correctly classify instances across different classes, and we use F1 score to examine the balance between precision and recall in classification tasks. For dataset \ud835\udc45\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5b\ud835\udc54, while it has five labels and also examine the multiclass classification, we use mean absolute error (MAE) and root mean square error (RMSE) to evaluate from from a regression perspective [43]. For MAE, it measures the average magnitude of errors in the predictions, providing a straightforward assessment of the model\u2019s overall accuracy in predicting the rating values. For RMSE, it captures the square root of the average squared differences between predicted and actual ratings, offering a metric sensitive to larger errors, which can highlight significant discrepancies between the model\u2019s predictions and true values. For dataset \ud835\udc41\ud835\udc52\ud835\udc64\ud835\udc60and \ud835\udc37\ud835\udc35\ud835\udc3f\ud835\udc43, their labels are sentences. Such datasets examine the text generation capabilities. We use ROUGE1 and ROUGE-L to evaluate the overlap between generated texts and reference texts [44], capturing both the precision and recall of individual words (ROUGE-1) and the longest matching sequence (ROUGE-L), ensuring a comprehensive evaluation of the text generation quality. For accuracy, F1, ROUGE-1 and ROUGE-L, their higher values reflect the better performance. For MAE and RMSE, their lower value represent the better performance. Additionally, we use accuracy to measure the MIPS performance (MIPS accuracy), representing the ratio of MIPS results under device variation and MIPS results without device variation (references). 4.1.4 Baselines. As this is the first work to improve the RAG robustness on Edge-based CiM, we do not have state-of-the-art for comparison. As such, we construct baselines from the past noise mitigation methods originally designed to boost DNN robustness. The first baseline is selective write verify [21] (SWV). While it originally utilizes the second derivation to evaluate the device variation impact on neural network weights, we use the second derivation to measure the embedding deviation between the ground truth embedding and the embedding under device variation. The second baseline is (CxDNN) [45]. While they use compensation factor to improve the robustness of vector-matrix multiplication, we use the compensation factor the calibrate the embedding impacted by device variation. The third baseline is CorrectNet [46], where it utilizes the cross entropy loss and regularization to improve the robustness of neural networks in CiM. To use it as a baseline, we also use the cross entropy loss the regularization as the loss function to \fRuiyang Qin1, Zheyu Yan1, Dewen Zeng1, Zhenge Jia1, Dancheng Liu2, Jianbo Liu1, Ahmed Abbasi1,Zhi Zheng1, Ningyuan Cao1, Kai Ni1, Jinjun Xiong2, Yiyu Shi1 calibrate the device output embedding. Additionally, we examine the Vanilla RAG, which contains no noise mitigation methods, as our fourth baseline. The baselines use the same experimental setting as our framework does. 4.2 Results For RAG, it can be simplified as the combination of MIPS and LLM, where the MIPS as a retriever searches the appropriate information and the LLM as a generator processes the searched results. Hence, in our experiments, we first evaluate the performance of MIPS under the device variation of device-1. We take the MIPS results obtained without device variation as the references (i.e., ground truth). Using the metric of MIPS accuracy, we examine how many MIPS results under device variation will match the references. Since the quality of retrieved content largely depends on the base sentence embedding model, and we focus on mitigating the device variation impact on the embedding model, we do not assess the quality of references. As shown in Table 3, our framework using the two data construction methods outperforms the four baselines across five datasets. It shows that our framework can mitigate the embedding perturbation due to device variation. These results can also correspond to the preliminary study shown in Figure 2, where the increment of \ud835\udf0e in naive Gaussian noise will jeopardize the MIPS performance. Table 3: Performance (MIPS accuracy) comparison between our framework and baselines. Accuracy is computed based on MIPS-retrieved documents under device variation of device-1 and the these retrieved without device variation. Dataset Citation Movie Rating News DBLP SWV 0.4200 0.1728 0.1050 0.0855 0.2295 CxDNN 0.4401 0.2017 0.0503 0.0754 0.1681 CorrectNet 0.4013 0.0699 0.0509 0.0533 0.1609 Vanilla RAG 0.4547 0.1694 0.0933 0.0649 0.1747 RoCR-CDE 0.9231 0.4639 0.1583 0.1921 0.2750 RoCR-CDI 0.9344 0.4355 0.1266 0.1708 0.2905 After we compare the MIPS performance of our framework and baselines, we further present a comprehensive evaluation to show the RAG performance of them. We use Gemma-2B as the LLM in RAG. Additionally, with Gemma-2B, we run RAG without device variation to obverse its ideal performance, where we get 0.5200 of accuracy for Citation, 0.3728 of accuracy for Movie, 0.3150 of MAE for Rating, 0.0855 of ROUGE-1 for News, and 0.2295 of ROUGE-1 for DBLP. On five CiM devices, whose device variations have been shown in Table 2, we examine RAG with five datasets. As shown in Table 1, given the same datasets, it is clear that each device variation significantly compromises the RAG robustness, whereas our framework can mitigate the different device variation. For example, the RAG performance for Citation dataset on Device-2 can range from 0.18 to 0.48, while our framework can boost the accuracy performance of Citation dataset above 0.5 for all five devices. Compared to the four baselines whose performances are relatively worse than the ideal performance, our framework significantly approaches and sometimes outperforms the ideal performance via generating better sentence embeddings. This is because RoCR also serves as a regularization to improve the model\u2019s generalization. In addition, we evaluate the impact of different LLMs on the performance of our framework. As Figure 1 shown, the LLM takes the concatenation of MIPS searched data and user query as the input and generates the response regarding the user query. Since different LLMs may have different response given the same query, we select four emerging edge-friendly medium-size LLMs in our experiments to examine the performance of our framework. Gemma-2B [47] is a new SOTA open model introduced by Google, with 4.95G model weights. According to Google, Gemma can outperform the same sized Llama-2 in reasoning capabilities. Hence, we also use Llama2-3B [48], one of the earliest open LLMs introduced by Meta, with 6.85G model weights. Similarly, Phi-2 [49] released by Microsoft, is a powerful small LLM with 5G model weights. Additionally, Mistral7B-GPTQ [50] made by Mistral AI, is a well-performed LLM after Llama model. We select dataset \ud835\udc36\ud835\udc56\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5band dataset \ud835\udc40\ud835\udc5c\ud835\udc56\ud835\udc63\ud835\udc52. We use the default experimental setting with \ud835\udf0e= 0.1 and use CiM Device-1 as the experimental environment. The results are shown on Figure 6. It is evident that our framework outperforms each baseline across five CiM devices. Besides, the performance of each baseline on the same dataset can be largely different given different device, while our framework can produce a more robust performance. 0.000 0.025 0.050 0.075 0.100 0.125 0.150 Device Variation ( ) 0.10 0.15 0.20 0.25 0.30 ROUGE-1 SWV CxDNN CorrectNet Vanilla RAG RoCR-CDE RoCR-CDI Figure 7: Performance comparison between our framework and four baselines on CiM device-1 with different device variation \ud835\udf0e, given dataset DBLP. By default, we use \ud835\udf0e= 0.1 to calculate the device variation of the five CiM devices. We also conduct an additional study to evaluate our framework given different \ud835\udf0evalues. Since we have already use dataset Citation and dataset Movie to study the performance of our frameworks seen in Figure 6, we choose a different dataset DBLP, using ROUGE-1 as the metric. For the LLM in RAG, we choose Mistral-7B. We examine the \ud835\udf0evalues higher and lower than 0.1, including 0, 0.025, 0.05, 0.075, 0.125, and 0.15. The case of \ud835\udf0e = 0 reflects the ideal performance. For the CiM device, we use CiM device-1. As shown in Figure 7, our framework outperforms baselines across different device variation values. Finally, RoCR is a training method that generates more robust weights for the sentence embedding model. It does not change the model structure. Thus, there is no hardware (e.g., energy and latency) overhead during inference. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2311.12275v4",
+                    "title": "Enabling On-Device Large Language Model Personalization with Self-Supervised Data Selection and Synthesis",
+                    "abstract": "After a large language model (LLM) is deployed on edge devices, it is\ndesirable for these devices to learn from user-generated conversation data to\ngenerate user-specific and personalized responses in real-time. However,\nuser-generated data usually contains sensitive and private information, and\nuploading such data to the cloud for annotation is not preferred if not\nprohibited. While it is possible to obtain annotation locally by directly\nasking users to provide preferred responses, such annotations have to be sparse\nto not affect user experience. In addition, the storage of edge devices is\nusually too limited to enable large-scale fine-tuning with full user-generated\ndata. It remains an open question how to enable on-device LLM personalization,\nconsidering sparse annotation and limited on-device storage. In this paper, we\npropose a novel framework to select and store the most representative data\nonline in a self-supervised way. Such data has a small memory footprint and\nallows infrequent requests of user annotations for further fine-tuning. To\nenhance fine-tuning quality, multiple semantically similar pairs of question\ntexts and expected responses are generated using the LLM. Our experiments show\nthat the proposed framework achieves the best user-specific content-generating\ncapability (accuracy) and fine-tuning speed (performance) compared with vanilla\nbaselines. To the best of our knowledge, this is the very first on-device LLM\npersonalization framework.",
+                    "authors": "Ruiyang Qin, Jun Xia, Zhenge Jia, Meng Jiang, Ahmed Abbasi, Peipei Zhou, Jingtong Hu, Yiyu Shi",
+                    "published": "2023-11-21",
+                    "updated": "2024-04-16",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL"
+                    ],
+                    "main_content": "INTRODUCTION While most Large Language Models (LLMs) are still deployed in cloud servers, more and more LLMs (e.g. Llama-3B with parameter size of 6GB) are being deployed on edge and mobile devices such as prompt-driven robots to assist people in daily life [3] and provide personalized companionship [10] while preserving users\u2019 privacy. Traditionally, most LLMs are pretrained in high-performance servers and then deployed in these devices without further training. However, such a generic model usually falls behind in adapting to each individual user\u2019s unique needs and habits. It is often desirable for the deployed LLMs to further learn from real-world input data (e.g. userand LLM-generated texts in their interaction), so that This work was supported in part by NSF 2324864, NSF 2122320, NSF 2324937, and NIH R01EB033387. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. DAC \u201924, June 23\u201327, 2024, San Francisco, CA, USA \u00a9 2024 Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 979-8-4007-0601-1/24/06. https://doi.org/10.1145/3649329.3655665 the LLMs can be personalized and adapted to the user\u2019s immediate context in real-time. This allows more accurate and context-aware responses, improving overall effectiveness of the LLMs. Although LLMs are pre-trained in a self-supervised way through next-token prediction, existing work has demonstrated that their fine-tuning must be supervised, where human-written outputs for task instructions or annotated outputs in a specific task dataset are given. For on-device personalization, it is usually impractical to send new data to the cloud for annotation due to data privacy and security concerns [19]. As such, any annotation has to be done locally by directly asking users to provide preferred responses during the user-LLM interaction. Such annotations need to be sparse because frequent inquires impede the user experience of LLM. Thus, for on-device LLM personalization, it is desirable to learn from new streaming data in-situ with as few annotations as possible. In addition, for on-device personalization, considering limited hardware resources on the edge, it is necessary to learn from usergenerated data streams without accumulating a large dataset. In other words, a small data buffer should be used to form each minibatch for training. Existing LLM training methods assume that each mini-batch is independent and identically distributed (iid) by sampling uniformly at random from each semantic domain [6]. However, it is challenging to maintain the most representative data in the buffer so that learning from this buffer efficiently derives a model that is as effective as if the entire data is used. This is due to two reasons. First, the streaming data collected on edge devices are usually temporally correlated [14] and result in a correlation within each mini-batch. There can be a few rounds of uncontroversial dialogue sets before switching to those that contain useful information. Second, there is no easy way to select representative data for each domain topic such that the data contain rich information in each domain topic from non-iid streaming data, due to the fact that the streaming data are unlabeled. If annotations were available for all the data, we could easily select representative data based on all the annotations even if the streaming data were non-iid. Without addressing these challenges, directly learning from temporally correlated non-iid mini-batches would result in poor representations and inefficient personalization. To tackle the challenges of sparse local annotations and limited buffer size for on-device LLM personalization, in this paper, we propose to utilize embedding entropy, domain-related information, and embedding similarity to measure data quality from different perspectives in an unsupervised way. For each dialogue set in the data, the scores measured by the three metrics reflect the quality of the data regarding the information it contains as well as the domain it belongs to. Based on the three metrics, we propose a data replacement policy for the buffer, which always replaces the data in the buffer that has the lowest scores in these metrics if the arXiv:2311.12275v4  [cs.CL]  16 Apr 2024 \fDAC \u201924, June 23\u201327, 2024, San Francisco, CA, USA Qin et al. buffer is full and the new data have higher scores. To provide annotation needed in the fine-tuning, we ask users to provide preferred responses as annotations for all the data in the buffer. Finally, multiple semantically similar question-answer pairs can lead to better model fine-tuning [11]. Therefore, for each dialogue set selected to store in the buffer, we utilize the LLM to synthesize semantically similar pairs, also without user supervision. In summary, the main contributions of the paper include: \u2022 On-device LLM personalization framework. We propose a framework to form mini-batches of training data for fine-tuning LLM on the fly from the unlabeled input stream generated from user-LLM interactions. It only uses a small data buffer and eliminates the necessity of storing all the streaming data in the device. \u2022 Quality metrics for data selection. We propose a data replacement policy guided by three quality metrics to maintain the most representative data in the buffer for on-device LLM fine-tuning. Annotation is not needed in the data replacement process. \u2022 Data synthesis for labeled pairs. We propose to use the LLM model to generate additional data that are semantically similar to the selected data to enhance fine-tuning quality. As this is the first work for on-device LLM personalization, no state-of-the-art is available, and we constructed a few vanilla baselines for comparison. Experimental results on multiple datasets of varying temporal correlation including ALPACA [16], DOLLY [5], MedDialog [4], Prosocial-Dialog [12], OPENORCA [18], and Empathetic-Dialog [9] show that the proposed framework achieves up to 38% higher ROUGE-1 than the baselines and at the same time greatly improves the learning speed. Table 1: Three example domains and their lexicons. Domain Example Lexicons medical Admin dose vial inhale inject ml pills ingredient Anatomy Pelvis arm sinus breast chest lymph tonsil Drug ACOVA ACTONEL CARTIA EMGEL emotion Fear bunker cartridge cautionary chasm cleave Surprise amazingly hilarious lucky merriment Trust advocate alliance canons cohesion GloVe GloVeTW26 extreme potential activity impact movement GloVeCC41 symptomatic thrombosis fibrillation GloVeTW75 nyquil benadryl midol pepto midol ritalin 2 BACKGROUND AND RELATED WORK 2.1 Background 2.1.1 Text Domain. Text domain usually refers to either the text topic like medical conversation or the embedding lexicon dictionary such as GloVe embedding dictionary. The lexicons related to certain text domains are organized as TABLE 1 shown. The medical, emotion and GloVe are three domains. In each domain, high-level lexicons such as fear and drug are used to index the detailed lexicons shown in Example Lexicons in TABLE 1. 2.1.2 Text Embedding. Text embedding involves converting text data into machine-readable numeric data. The quality of this embedding can influence subsequent text learning tasks and is also intrinsically linked to alignment\u2014a crucial aspect of NLP [20]. A prevalent embedding method assigns unique indices to words based on their position within a comprehensive vocabulary dictionary. Consequently, the same word, regardless of its occurrence in different text data, can be represented consistently using its unique index. In this work, we adopt an embedding technique using a pretrained transformer model. This model not only captures semantic information but also offers superior alignment capabilities. 2.2 Related Work 2.2.1 LLM Personalization. LLM personalization employs fine-tune the model to enhance its capability of text-understanding and textgenerating in specific domains. While existing works concentrate more on scaling up the LLM to enable its comprehensive capabilities, some efforts [15] have been made to fine-tune LLM using relative small dataset with high quality. However, all these works still involve large-scale computation and high-intensive neural network training with the overwhelming dataset size regarding on-device learning, and they assume that each mini-batch can be formed by sampling from the dataset. But when learning from streaming data, data is collected sequentially as it is. Random sampling from the entire input stream to create iid mini-batches is infeasible since it requires to store all the data, which is unavailable for device storage and computationally intractable for device computational resources. Therefore, an approach to form mini-batches on-the-fly while including the most representative data in each mini-batch is needed to enable efficient on-device LLM learning. 2.2.2 Data Selection in Streaming and Continual Learning. There are several supervised streaming and continual learning models that can learn from a stream of data [1]. To overcome the problem of catastrophic forgetting of previously seen data, a data buffer is usually needed to store previous data for rehearsal [17]. However these works cannot handle text input with user-specific semantic domains due to the lack of semantic level data processing and evaluation of these works. Efficiently evaluating input text data and selecting the most representative text which can shape the LLM towards user-specific text generation on devices have not been explored and studied. 3 PROPOSED WORK In this section, we first provide an overview of our framework. We then delve into the details, starting from the three metrics we have found to benefit most for the data selection in LLM personalization, and demonstrate how they collaborate with the data buffer to select data. After that, we demonstrate the data synthesis method we use to augment the selected data and explain the reason to use that. 3.1 Framework Overview In our framework, we assume the atomic unit of data selection is a dialogue set, which contains a pair of question and answer during user-LLM interaction. As shown in Figure 1, the proposed framework has three stages. The first stage selects data to store in the data buffer based on \fEnabling On-Device Large Language Model Personalization with Self-Supervised Data Selection and Synthesis DAC \u201924, June 23\u201327, 2024, San Francisco, CA, USA Figure 1: Overview of the framework. Fine-tune LLMs using data from data selection and following data generating. certain quality metrics, and the selected data will be annotated by the user. The second stage takes the selected (and annotated) data and synthesizes additional data using the LLM. And finally, the selected and synthesized data together will be used for fine-tuning. In the discussion below, we focus on the first two stages where our framework resides. Specifically, with details discussed in Section 3.2, in the first stage, the proposed framework takes each dialogue set in the input streaming data from user-LLM interaction on-the-fly, calculate the quality metrics, and discard the data or update the data buffer based on the metrics. Considering the resource limitation, only a small data buffer is used to maintain the highest quality data. We will inquire user about the expected response as annotation for each selected dialogue set. With details discussed in Section 3.3, in the second stage, each selected dialogue set in the buffer is sent to the LLM for generation of additional dialogue sets that are semantically similar. We use the user annotation to replace the LLM generated response in the selected dialogue set. A pre-stored and fixed prompt is given to instruct the LLM for data generation. For the generated dialogue sets, a sanity check is made to make sure that their semantic similarity with the original dialogue set is above a user-specified threshold. 3.2 Data Selection by Quality Scores Each dialogue set\u2019s quality is captured by scores from three metrics. Each of them measures the quality of data from different perspectives as detailed below. Metric 1: Entropy of Embedding. Entropy of embedding (EOE) comes from the idea of Shannon\u2019s Entropy [22], where higher entropy means more features to learn. For each input \ud835\udc47, EOE aims to qualify and measure the information of embedding vector \u00ae E = \ud835\udc53(\ud835\udc47) generated by an end-to-end embedding function \ud835\udc53(\u00b7). The embedding vector \u00ae E = [\ud835\udc521,\ud835\udc522, . . . ,\ud835\udc52\ud835\udc5e] where \ud835\udc52\ud835\udc56is the embedding of the \ud835\udc56\ud835\udc61\u210etoken in the input and \ud835\udc5eis the length of the embedding. \ud835\udc38\ud835\udc42\ud835\udc38(\u00b7) can then be defined as: EOE(\u00ae E\ud835\udc56) = \u2212\u00cd \ud835\udc52\ud835\udc56\u2208\u00ae E \ud835\udc5d(\ud835\udc52\ud835\udc56) log\ud835\udc5d(\ud835\udc52\ud835\udc56) log(\ud835\udc5b) (1) where \ud835\udc5d(\ud835\udc52\ud835\udc56) represents the probability distribution of \ud835\udc52\ud835\udc56, and \ud835\udc5b is the number of tokens in \ud835\udc47. The term \ud835\udc5d(\ud835\udc52\ud835\udc56) log\ud835\udc5d(\ud835\udc52\ud835\udc56) represents the contribution of each token\u2019s embedding to the overall entropy. The normalization by log(\ud835\udc5b) adjusts for the effect of the sequence length, ensuring that entropy is comparable across sequences of different lengths. Metric 2: Domain Specific Score. While EOE measures the amount of information the input data contains, it cannot provide assessment regarding how much the information can be related to certain domains. As shown in TABLE 1, the domain of medical, emotion, and GloVe embedding can include distinct lexicons. The value of a dialogue set with respect to a particular domain can then be indicated by the token overlapping between the dialogue set and the common lexicons in each domain. Note that this would require a pre-stored dictionary containing common lexicons of domains of interests in the device, which can be easily constructed. Given a dialogue set \ud835\udc47containing \ud835\udc5btokens, and a collection of lexicon set \ud835\udc3f= {\ud835\udc591,\ud835\udc592, . . . ,\ud835\udc59\ud835\udc5a} from \ud835\udc5adifferent domains, the Domain Specific Score (DSS) can be calculated as: DSS(\ud835\udc47, \ud835\udc3f) = 1 \ud835\udc5a \ud835\udc5a \u2211\ufe01 \ud835\udc56=1 |\ud835\udc47\u2229\ud835\udc59\ud835\udc56| \ud835\udc5b (2) where it measures the ratio of tokens in \ud835\udc47belonging to every domain lexicons and output the mean of all ratios across all the domains. As domain can be highly important when adapting LLM to different tasks, the texts in different domains should not be compared to each other purely using EOE, and the text in the same domain should be evaluated together. Metric 3: In-Domain Dissimilarity. While DSS calculates the general overlapping between \ud835\udc47and all domain lexicons, it is important to evaluate how much value \ud835\udc47brings to the domain it overlaps most with, i.e., the dominant domain. The dominant domain can be obtained as: \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51= arg max \ud835\udc59\ud835\udc56\u2208\ud835\udc3f |\ud835\udc47\u2229\ud835\udc59\ud835\udc56| (3) When a dialogue set is stored in the buffer, we also store its dominant domain and its embedding. When a new dialogue set is considered, we identify all the dialogue sets already in the buffer that have the same dominant domain as the new set, and compute the dissimilarity between them, which will reflect the amount of new information that can be brought by the new set to the dominant domain. Specifically, the In-Domain Dissimilarity (IDD) can be calculated by cosine embedding similarity: IDD(\u00ae E, \ud835\udc35) = 1 \ud835\udc45 \ud835\udc45 \u2211\ufe01 \ud835\udc56=1 (1 \u2212cos(\u00ae E, \u00ae E\ud835\udc56 \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51)) (4) where E\ud835\udc56 \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51is the embedding vector of the \ud835\udc56\ud835\udc61\u210edialogue set in the buffer \ud835\udc35that has the same dominant domain as \ud835\udc47, and \ud835\udc45is the total number of such dialogue sets in \ud835\udc35. cos(\u00ae E, \u00ae E\ud835\udc56 \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51) is the cosine \fDAC \u201924, June 23\u201327, 2024, San Francisco, CA, USA Qin et al. similarity between \u00ae E and \u00ae E\ud835\udc56 \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51, calculated as: cos(\u00ae E, \u00ae E\ud835\udc56 \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51) = \u00ae E \u00b7 \u00ae E\ud835\udc56 \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51 \u2225\u00ae E\u2225\u2225\u00ae E\ud835\udc56 \ud835\udc37\ud835\udc5c\ud835\udc5a\ud835\udc51\u2225 (5) Note that we store the embedding of all the selected dialogue sets in the buffer, so that they do not need to be re-computed each time a new dialogue set is being evaluated. Quality Score Based Data Selection. When a new dialogue set arrives and the buffer is full, we need to decide whether this new set needs to be discarded, or to replace a dialogue set already in the buffer. If the latter, we also need to decide which set in the buffer needs to be replaced. In our framework, for each new input dialogue set\ud835\udc47, its EOE, DSS, and IDD scores will be computed and compared with these scores of all the data in the buffer. If all the three metrics of \ud835\udc47are higher than a dialogue set already in the buffer, then we use \ud835\udc47to replace it. Note that if there are more than one options to replace, we will randomly select one. Users will then be asked to provide annotation to this new dialogue set, for example, by asking \u201cDo you think my response is acceptable and if not what would be an ideal response?\u201d If users provided an alternative response that is preferred, the dialog set will be updated using the user provided content before being placed into the buffer. Finally, from the definition of the three metrics and the replacement policy, it is easy to see that for each new dialogue set, our data selection policy has a linear complexity with respect to the size of the buffer. 3.3 Data Synthesis The selected data in the buffer can capture features unique to the user. However, when such data are used in LLM fine-tuning, the limited size can confine the effectiveness. To address this problem, inspired by the observation that multiple semantically similar question-answer pairs can lead to better model fine-tuning[11], we deploy a self-generated instruction strategy to generate additional data. Specifically, each dialogue set (i.e., \u201coriginal\u201d dialogue set) in the buffer will be sent to the LLM to generate similar ones, by giving the following prompt \u201cPlease refine and generate a text semantically similar to the following text block, no need to answer it, no need to explain, use [ ] to hold your generated response: \u201d followed by the original dialogue set. We run this multiple times to generate several additional sets for each original one. To avoid complicating the data replacement, the data synthesis process will only occur right before the fine-tuning starts each time. However, sometimes we find that the dialogue sets generated by LLM, even though the prompt instructs it to generate semantically similar ones, still differ from the original dialogue set significantly, if measured by ROUGE-1. As such, we add a sanity check for each generated dialogue set, and if ROUGE-1 between it and original set is above a threshold, it will be discarded. 4 EXPERIMENTAL EVALUATION 4.1 Experimental Setup We first explain the datasets used in the experiments, the settings under different experiments, and baselines. Datasets. To show the generalization capability of our framework, we use multiple and diverse datasets, including ALPACA [16], DOLLY [5], MedDialog [4], Prosocial-Dialog [12], OPENORCA [18], and Empathetic-Dialog [9], to evaluate the proposed framework. These datasets reflect different temporal correlation scenarios in the input data stream: ALPACA, DOLLY and OPENORCA contain diversified dialogue sets not bounded to a single domain, and the input data streams formed on them have little temporal correlation. While the other three ones are domain-specific, and thus the data streams are highly temporal correlated. All these datasets are fully annotated. However, our framework only uses annotations for the data selected to finetune the LLM; and the fully annotated dataset is used in the evaluation. Default Experimental Setting. We use a pre-trained Llama-3B [21], one of the most popular on-device LLM, as the model embedded on devices. For each dataset, we randomly choose 10% of the data to simulate input data stream and run our framework on it for model fine-tuning, and the remaining 90% is reserved for evaluation of the fine-tuned mode. For every 800 dialogue sets received in the input stream, we will start the fine-tuning process with 100 epochs using optimizer AdamW. The buffer will not be cleared after the fine-tuning, and the data selection continues after the fine-tuning is done. We obtain input text embedding from Llama-3B last hidden layer during its inference. Unless otherwise mentioned, in data synthesis each dialogue set in the buffer is sent to LLM to generate three additional sets. With the selected and synthesized data, we fine-tune Llama-3B using Low-Rank Adaptation (LoRA) [8], a parameter efficient fine-tune technique. Unless otherwise specified, the batch size is 128 with fixed learning rate of 0.0003. For LoRA settings, the trainable layers are the QKV layers (q_proj, k_proj, v_proj) and attention output layer (o_proj), max sequence length is 512, LoRA rank r is 8, loRA metrics scaling facotr alpha is 16, and LoRA dropout rate is 0.05. For consistency, when we use the fine-tuned model to generate text for evaluation, the temperature \ud835\udf0f is set to 0.5 for all experiments. As for the data selection buffer design, for efficient memory operations, we divide it into bins of equal size and each bin is able to hold the text of one dialog set, its domain as well as its embedding. Considering that the maximum dialogue set is of length 1,024 tokens (512 tokens x2) and the embedding is a floating point vector of length 4,096 for Llama-32B, the bin size is set to 22KB. In the experiments, we will explore the impact of the buffer size. To make sure that our framework can be applied to the various edge devices, we explore buffer sizes from 32 bins (704KB) to 512 bins (11MB). To efficiently evaluate our framework, we use compact, 150 watt, single-slot A10 GPU, which is much smaller than 300 watt double-width A100 GPU. A10 is compatible to fit into robotics applications. ROUGE-1 as Evaluation Metric. After the LLM model is finetuned using our framework, for each dialogue set in the test set, we feed the same user question to the model and collect the response generated. The quality of the data can then be evaluated by measuring the overlapping between the generated responses and the responses in the test dialogue set under the same question, which can be captured by ROUGE-1. ROUGE-1 is commonly used in natural language processing to measure the overlap of unigrams \fEnabling On-Device Large Language Model Personalization with Self-Supervised Data Selection and Synthesis DAC \u201924, June 23\u201327, 2024, San Francisco, CA, USA (a) ALPACA (b) DOLLY (c) Prosocial-Dialog (d) Empathetic-Dialog (e) OPENORCA (f) MedDialog Figure 2: The learning curve of our proposed framework, Random Replace, FIFO Replace, and K-Center with buffer size 281KB on datasets (a) ALPACA (b) DOLLY (c) Prosocial-Dialog (d) Empathetic-Dialog (e) OPENORCA (f) MedDialog. (single words) between the machine-generated summary and a reference summary in terms of F-1 score [13]. A higher ROUGE-1 score suggests that the generated text is more similar to the reference text. Table 2: ROUGE-1 of different methods on six datasets with data buffer 2816KB Random FIFO K-Center Ours ALPACA 0.2457 0.2013 0.2384 0.3736 DOLLY 0.2417 0.1976 0.2403 0.3465 Prosocial 0.2375 0.2190 0.2147 0.3062 Empathetic 0.2352 0.1902 0.2098 0.3260 OPENORCA 0.2286 0.1833 0.2048 0.2813 MedDialog 0.2465 0.2074 0.2204 0.3429 Baselines. As this is the first work on on-device LLM personalization, we do not have state-of-the-art for comparison. As such, we construct a few vanilla baselines. Random Replace is recently used for continual learning [7]. It selects data uniformly at random from new data to replace the ones already in the buffer. FIFO Replace is also recently employed for continual learning [7]. It replaces the oldest data in the buffer with new data. K-Center is a SOTA active learning approach [23] which selects the most representative data by performing k-center clustering in the features space. While not directly used in LLM personalization, these works also do not require labeling information and seemingly simple, and have demonstrated superior performance in maintaining image data for continual learning. In addition, to demonstrate the importance to consider all the three metrics EOE, DSS and IDD, we will perform ablation study on additional three baselines, each only using one of the three for data selection. For fair comparison, for all of these methods we used the same data synthesis based on the selected data as used in our framework. 4.2 Results We start with comparing the ROUGE-1 of Random Replace (Random), FIFO Replace (FIFO), and K-Center on all the datasets using buffer size 128 bins (2816 KB). The results are presented in TABLE 2. From the table we can see that our method outperforms all the baselines by a significant margin, indicating that its superiority in both weak and strong temporal correlation settings. The results also show that the most competitive baseline is the seemingly simple, yet surprisingly effective approach random replace. These results match the results in [2] for image classification tasks, where a random replacement policy outperforms elaborately designed approaches. Table 3: ROUGE-1 based on MedDialog with different buffer sizes. Buffer Size (KB) Ours Random FIFO K-Center 176 0.3040 0.2281 0.2383 0.2160 352 0.3447 0.2455 0.2304 0.2175 704 0.3353 0.2536 0.2389 0.2080 1408 0.3353 0.2791 0.2417 0.2204 2816 0.3940 0.2638 0.2309 0.2073 5632 0.3944 0.2748 0.2381 0.2167 11264 0.4215 0.2834 0.2315 0.2122 Next, as a very important profiling tool for on-device learning, we evaluate the learning curve of the proposed framework and the baselines on these datasets. The learning curve represents how well the LLM can be fine-tuned to generate user-specific text with respect to the number of input dialogue sets seen as the data streams in. The same buffer size is used. The results are depicted in Figure 2 (a)-(f), respectively. From all the figures, we can clearly see that the ROUGE-1 of the proposed framework consistently increases with the increase of seen data, while the ROUGE-1 of the baselines only demonstrate minor improvement. In addition, we evaluate the impact of buffer size on the performance of the proposed framework. The model is trained on the \fDAC \u201924, June 23\u201327, 2024, San Francisco, CA, USA Qin et al. MedDialog dataset. The number of bins in the buffer is in {8, 16, 32, 64, 128, 256, 512} corresponding to a buffer size of {176KB, 353KB, 704KB, 1408KB, 2816KB, 5632KB, 11264KB} respectively. The corresponding learning rate is scaled to {2, 3, 4, 5, 7, 10, 14} X 10\u22125, roughly following a learning rate \u221d \u221a batch size scaling scheme. The proposed framework consistently outperforms the baselines under different buffer sizes. As shown in TABLE 3, under the different buffer sizes, the ROUGE-1 by the proposed framework maintains a clear margin over the baselines. Besides, the margin becomes larger as the buffer size increases. This is because a larger buffer size provides the framework a better opportunity to select more high quality data, and the framework can leverage this opportunity to maintain richer quality data for learning, while the baselines cannot. Also, the proposed framework achieves higher ROUGE-1 when the buffer size becomes larger. This is because a larger buffer size provides a larger batch size, which naturally benefits the LLM fine-tuning. Table 4: ROUGE-1 of our framework and the baselines using only one of the three metrics EOE, DSS or IDD on six datasets with buffer size 2816KB. EOE DSS IDD Ours ALPACA 0.2821 0.2726 0.2950 0.3736 DOLLY 0.2782 0.2633 0.2247 0.3465 Prosocial 0.2617 0.2441 0.2324 0.3062 Empathetic 0.2661 0.2726 0.2707 0.3260 OPENORCA 0.2468 0.2362 0.2468 0.2813 MedDialog 0.2608 0.2726 0.2931 0.3429 Finally, we perform two ablation studies. The first one is to demonstrate the advantage of simultaneously considering all the three quality metrics EOE, DSS and IDD, we modify our framework to use only one of them for data replacement, those only considering one of them. The results on all six datasets are presented in TABLE 4. From the table we can see that simultaneously considering all the metrics always achieves the highest ROUGE-1. Figure 3: ROUGE-1/training time on MedDialog dataset with different number of dialogue sets generated from each original set in the buffer. The second study shows the relationship between the number of additional sets generated during data synthesis for each original dialogue set in the buffer and ROUGE-1/training time per epoch. From Figure 3 we can see that the maximum gain in ROUGE-1 can be attained when six additional sets are generated, while the training time consistently increases. Generating more then six dialogue sets will not further boost the performance, but would cost more time to fine-tune the model. For the sake of balanced efficiency and preference , as mentioned in the experimental setup, in all the experiments we generated three additional dialogue sets. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2112.02994v1",
+                    "title": "IBERT: Idiom Cloze-style reading comprehension with Attention",
+                    "abstract": "Idioms are special fixed phrases usually derived from stories. They are\ncommonly used in casual conversations and literary writings. Their meanings are\nusually highly non-compositional. The idiom cloze task is a challenge problem\nin Natural Language Processing (NLP) research problem. Previous approaches to\nthis task are built on sequence-to-sequence (Seq2Seq) models and achieved\nreasonably well performance on existing datasets. However, they fall short in\nunderstanding the highly non-compositional meaning of idiomatic expressions.\nThey also do not consider both the local and global context at the same time.\nIn this paper, we proposed a BERT-based embedding Seq2Seq model that encodes\nidiomatic expressions and considers them in both global and local context. Our\nmodel uses XLNET as the encoder and RoBERTa for choosing the most probable\nidiom for a given context. Experiments on the EPIE Static Corpus dataset show\nthat our model performs better than existing state-of-the-arts.",
+                    "authors": "Ruiyang Qin, Haozheng Luo, Zheheng Fan, Ziang Ren",
+                    "published": "2021-11-05",
+                    "updated": "2021-11-05",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.AI"
+                    ],
+                    "main_content": "INTRODUCTION The cloze test is a test in which the participant is asked to supply words that have been removed from a passage. Cloze tests are generally used to evaluate a participant\u2019s ability to comprehend the given text. One signi\uffffcant di\ufffference between the cloze task from other Natural Language Processing (NLP) problems is that the former requires a considerably larger long-term memory to make decisions and the ability to comprehend this content. Solving the cloze tests will provide insight into existing NLP tasks in text comprehension. In this paper, we employ a BERT-based sequence-to-sequence (Seq2Seq) model to solve the cloze task. Given a speci\uffffc context in the form of a passage, the solution of a cloze problem based on this context is generated in two steps: understanding the meaning of the idiom and choosing the correct idiom for each \u201cblank\u201d \u2013 where the original word is removed \u2013 in the passage. Previous research approaches NLP problems similar to the cloze task by applying neural network models such as the Knowledgeable Reader [9] and Entity Tracking [5]. These aforementioned approaches treat idioms as regular phrases. However, idiomatic expressions usually have meanings that are highly non-compositional and should not be \u2217All authors contributed equally to this research. understood from a literal standpoint. For example, if we look at the idiom word by word literally, the expression \"it\u2019s raining cats and dogs\" describes cats and dogs falling form the sky. This understanding is apparently incorrect as we generally use this expression to describe a heavy rain. Failure to understand the correct meaning of idiomatic expressions is detrimental to the decision-making process of the models, as even if a model is good enough to comprehend the passage itself, it cannot associate an expression that, from the literal perspective, is completely unrelated to the context. This calls for di\ufffferent approaches to the cloze task as our model is required to not only provide a grammatically sound candidate, but also ensure that the semantical meanings are coherent as well. Previous work on contextual relation understanding has yielded solutions with excellent performance. The BERT [3] performs well at understanding the contextual meaning of a given context. Pretrained BERT and other similar models can be adapted to perform contextual comprehension in other tasks. In this paper, we use pretrained models to solve the context comprehension step of the cloze task. To understand the meaning of idiomatic expressions correctly, we employ another pretrained model XLNET [20] and \uffffne-tune it to solve the cloze task. We train XLNET to learn not only the local context of the removed word (the sentence from which an original word is removed), but also the global context (the entire passage). This knowledge is represented by the last layer output of the hidden layers in XLNET. The contextual embeddings generated by our BERT-based pretrained models are combined with the idiom embeddings from the XLNET model. Softmax [8] is applied to decide on the best word from the combined embeddings. Our models are \uffffne-tuned on the EPIE [14] Static Corpus dataset. The remainder of the paper is organized as follows. Section 2 introduces some related work in cloze-style reading comprehension and modern NLP and Deep Learning technologies. Section 3 describes the method we developed to solve the challenge our problem. Section 4 describes our experiment to examine the performance of the models. Finally, Section 5 and 6 present the result and conclusion of our experiment and research work. 2 RELATED WORK 2.1 Cloze-style reading comprehension Cloze-style reading comprehension uses a passage of word tokens G1:=, with one token G9 masked; the task is to \uffffll in the masked word y, which was originally at position j. There already have a lot of works achieve a great performance in the cloze work [5, 9, 15]. \fRuiyang Qin, Haozheng Luo, Zheheng Fan, and Ziang Ren And Researchers created many large-scale cloze-style reading comprehension datasets yet, such as RACE [6], Children\u2019s Book Test (CBT)[4]. However,these research work only works on the normal words to \uffffnished the Cloze-style reading comprehension, and the idiom phrases are oftentimes non-contextual in the paragraphs. In this research e\uffffort, we want to utilize the most advanced technologies in our tasks to get the state of the art performance in the Idiom Cloze-style reading comprehension. The dataset we use EPIE used in this paper is also a large scale cloze-style dataset but focuses on English idiom prediction. 2.2 Pre-trained Language Models During daily usage, enormous sources of unilateral context can in\uffffuence the model\u2019s accuracy. In practice, it\u2019s highly possible to have the issue that they do not have symbols after sentences and polysemous condition in the sentence. There are previous researches on improving word embedding [10, 12], but it still cannot help us solve the challenge in our tasks. With the milestone of the appearance of transformer [19], there were several pre-trained models proposed in the NLP area like BERT and XLNet. With the various research, Language model pre-training has been proven to be e\uffffective over a list of natural language tasks at both sentence level [2] and token level [18] 2.3 Idiom detection A lot of research work are beginning forces on the idiom clozestyle reading comprehension. The \uffffrst popular neural network reading comprehension models were the Dual Embedding Model with bert [17]. In this previous work from Tan et al. [17], they use the BERT model to encode the contextual sentence as well as candidate phrases. However, the normal BERT model have the shortcoming to handle the long text sequence and lack of con\uffffdence in commonsense pragmatic inference. Also, the BERT model show mediocre performance in negation situation in the sentence. For some Cloze-style reading comprehension, it consist a huge context size and the related keys of cloze choice are far from the masks. Also, we could not ignore the condition of negation and commonsense in the challenge of the Cloze-style reading comprehension. 3 METHOD 3.1 Task De\uffffnition The main idea about our project is the idiom cloze test. We choose idiom as candidate and take o\uffffthe candidate from each sentence. For those candidates, they need to be \ufffflled into each blank spot. The \uffffrst thing is to padding each candidate. Then we need to \uffffnd the most appropriate candidate each time we hit the blank spot. 3.2 Padding Sequence In this part, we begin with processing the candidates. In our project, the \uffffrst step is to grasp idioms or candidates. We will search for the most appropriate candidate in the following steps. To do search, we have to process those candidates. The reason we have to do that is because every candidate has di\ufffferent length and it\u2019s hard to make any operations on those di\ufffferent-length candidates The default padding and probably the most usual padding is zero-padding, in which we add numbers of zeros in the back of each candidate. For the padding length, we will take the value of the longest candidate and use it as our padding length. This task can be achieved by calling speci\uffffc functions. We will need to search for longest candidate \uffffrst. Figure 1: Our Method Diagram about how our Attention Baselines works 3.3 Attention Baselines Previous methods applied to the BERT-based Dual Embedding [17] are only based on the BERT architecture. Because of the shortcoming in BERT and success of XLNET for many NLP task, especially reading comprehension, we proposed to present new methods (shown as Figure 1) based on XLNET and BERT to resolve the Idiom Cloze-style reading comprehension. The \uffffrst one, we treats a English idiom as a sequence of characters and we used the BERT to embedding the passage to analysis the original relationship in each contextual words. We combine the passges with each candidate idiom into a sequence and processes with multiple sequences, one for each candidate. The second baseline we treat the each idiom as a single token which has its own embedding vector and use XLNET to process the passage and then matches the encoded passage with each candidate idiom\u2019s embedding. \fIBERT: Idiom Cloze-style reading comprehension with A\uffffention Attention Baseline with Idioms as Character Sequence Attention baseline with idiom as candidate sequence is work to apply the BERT model for idiom cloze-style reading comprehension. Given a passage P = (?1, ?2, ?3, \u00b7 \u00b7 \u00b7 , [\"\ud434( ], \u00b7 \u00b7 \u00b7 , p=) and a candidate 3: 2 \u21e1, we \uffffrst concatenate them into a single sequence ([CLS], ?1, ?2, ?3, \u00b7 \u00b7 \u00b7 ,d=1,3=2, \u00b7 \u00b7 \u00b7 ,d=: , \u00b7 \u00b7 \u00b7 ,p=,[SEQ]), where3=1 to3=: are the characters and padding of the idiom3=. We can directly use the BERT to process this sequence and obtain the hidden representation for [CLS] in the last hidden layer, denoted by \u2318! :,0 2 R3. In order to detect the candidate idiom 3: among all the candidate, we use the linear layer to process \u2318! :,0 for : = 1, 2, \u00b7 \u00b7 \u00b7 ,   and use the softmax function with the each probability value of the candidates in D. After that, we will choose the best one candidate as the \uffffnal one for our cloze-style reading comprehension. Attention Baseline with Idiom Embedding Many idiom are non-compositional and therefore their meaning should not be directly derived from the its full individual characters. For example, \"It\u2019s a piece of cake\" literally means it is a number of cake, but it usually used to describe the task is pretty easy. Therefore, if we only embedding with its each characters meaning will cause a lot of confusion and a single embedding vector for the entire idiom can help the model understands the contextual relationship in the reading comprehension. In this baseline, instead of concatenating the passage and a candidate answer into a single sequence of BERT model, we separate them. We used the XLNET to process the passage sequence ([CLS], ?1,?2,?3, \u00b7 \u00b7 \u00b7 , [\"\ud434( ], \u00b7 \u00b7 \u00b7 ,p=,[SEP]). After that, we use the hidden representation of [MASK] at the decode layer, denoted as \u2318! 1, to match each candidate answer with BERT. In this way, no matter how much candidate we are, we only embedding the whole passage once with XLNET model. It can help our work prevent the problem of idiom\u2019s non-composition. We use 3: to denote the embedding vector for candidate 3: 2 \u21e1, the hidden representation \u2318! 1 is work with each candidate idiom via element-wise multiplication. Then the probability of selecting 3: among all the candidate is de\uffffned as equation 1. ?: = 4G?(F \u00b7 (3: \u2326\u2318! 1) + 1) \u00d5\u21e1 30=1 4G?(F \u00b7 (3: \u2326\u2318! 1) + 1) (1) whereF 2 R3 and1 2 R are model parameters, and \u2326is elementwise multiplication. To train the model, we use cross entropy loss as the loss function. 3.4 Context-aware Pooling The attention model baseline have potential problem. We can easily observe the idiom are always non-compositional. As a result, in order the idiom candidate can be \ufffft in the passage well, it not only the grammar should \ufffft with the contextual surrounding, also its meaning should also greatly \ufffft the whole passages. As those reason, how to get the candidate not just reasonable in grammar, such as verbs or noun, is importance for us to solve the cloze-style reading comprehension. That is why we need do some development on our baseline in order to context-aware. As the milestone of the transformer, the more and more pre-train model support the function of the context-aware with the contextual surrounding. Also, some models support the global contextaware, like the BERT work in SQuAD dataset [13]. As a result, we decide to evaluate whether a idiom candidate is suitable in a passage, we need not only understand its neighbour contextual details, also need to know the semantic meaning in the entire passage. That is why we need not only match the idiom candidate to place its context place on, also we must match it meaning with the entire passage. Recall that \ud43b! = (\u2318! 0,\u2318! 1,\u2318! 2, \u00b7 \u00b7 \u00b7 ,\u2318! =) represent the hidden states of the last hidden layer of baseline after it process the sequence. Our model with context-aware pooling can be work as equation 2. ?: = 4G?(3: \u00b7 \u2318! 1 + <0G= 8=0(3: \u00b7 \u2318! 8 )) \u00d5\u21e1 30=1 4G?(3: \u00b7 \u2318! 1 + <0G= 8=0(3: \u00b7 \u2318! 8 )) (2) 3.5 Dual Pretrain Attention Model In our method (Figure 1), we proposed use the linear interpolations [22] to make our solution of idiom detection for the cloze-style reading comprehension become more smooth. We perform linear interpolations in \uffffnal texture hidden space between both training output, one from the output of classi\uffffed with the softmax function in Attention Baseline with Idioms as Character Sequence, and the other is the possibility idiom candidate from the Attention Baseline with Idiom Embedding. We use the _ as a parameter of the weight to make sure both baseline can work smoothly on the \uffffnal output. In our model, The _ is a sample from beta distribution. We ensure the lambda larger than 0.7, ensure the Attention Baseline with Idioms as Character Sequence dominate the combination. With the linear interpolation, we can easily get a new possibility of the candidate and we choose the highest one as our \uffffnal candidate of the cloze-style reading comprehension. 4 DATA We evaluate the accuracy of the classi\uffffcation model with one standard dataset EPIE consisting of 359 kinds of candidate (listed as table 1) , and we make a program to create the classi\uffffcation what idiom should be used in cloze-type reading comprehension sentences. In our program we developed program to use self-attention for the task. After that, We use the training, validation, and test splits as de\uffffned in Lee and Dernoncourt [7] to analyse our loss score of the classi\uffffcation. Table 1 shows the statistics for dataset. There are many number of sentences to classify what kind of the idiom are the most reasonable in the sentence. All of idiom are normally used in the English speaking society and all types of the sentence are the condition of the idioms using. As table 1 shows, we have 21890 candidate as total. For each candidate, we have a corresponding tag (shown as 2). Both O, BIDIOM, and I-IDIOM represent the one position of words or special symbol. Given 1 sentence, we need to take o\uffffthe part which start from the \"B-IDIOM\" and end at the last position of \"I-IDIOM\". We got the 21890 sentences after deleting the candidate from the original sentences. \fRuiyang Qin, Haozheng Luo, Zheheng Fan, and Ziang Ren We use 3 types of labels for our data. For all 21890 candidate, we add a [\u2019CLS\u2019] at the beginning of the candidate and add [\u2019SEP\u2019] at end of the candidate in order to label our data. For all sentences which delete the candidate, we add a [\u2019CLS\u2019] at the beginning add [\u2019SEP\u2019] at end in order to label our deleting sentences, in addition to that, we add a [\u2019UNK\u2019] label at the position of the deleting candidate in the 21890 sentences. By the above labeling, we could encode all 21891 candidate and deleting sentences. The example is showing at table 3. Table 1: Number of Sentences in the Dataset Dataset Train Validation Test T N EPIE 15k 5k 2k 359 21890 |T| represents the number of idiom candidate and |N| represents the sentence data size Table 2: Type of Tags in the Dataset Type1 Type2 Type3 O B-IDIOM I-IDIOM Table 3: Example of Encoding and Labeling Original Sentence:Anyway , thanks MKM and keep up the good work ! Candidate:keep up the good work Label Candidate:\u2019[CLS]\u2019, \u2019keep\u2019, \u2019up\u2019, \u2019the\u2019, \u2019good\u2019, \u2019work\u2019, \u2019[SEP]\u2019 encode:11815, 17, 19, 3466, 414, 536, 692, 21, 435, 76, 18, 195, 154, 17, 136, 4, 3 deleting sentence:\u2019Anyway\u2019, \u2019,\u2019, \u2019thanks\u2019, \u2019MKM\u2019, \u2019and\u2019, \u2019! 5 RESULT We have compared the classi\uffffcation accuracy of our method with several other models (Table 4). We make the Dual BERT Embedding Model and Entity Tracking as the baseline. For methods using attention and deep contextualization word representation in some approaches to classify the idiom candidate of cloze-style reading comprehension, some of them use the self-attention for the task. However, they did not perform as well as our model. All models and their variables were trained eight times, making an average for the performance as a result. From our experiment, as shown as Figure 2, we can get our morel loss reduce with the epochs increase. And we make the cooperation with other two kinds of baselines and our method using the accuracy (Shown as table 4). As we can see, our method have 7.21% higher than the Dual BERT Embedding Model and 14.66% higher than the Entity Tracking in our challenge. That is reasonable that the Dual BERT Embedding Model is designed to solve the challenge of Chinese Idiom Cloze-style Reading comprehension and the Entity Tracking is only designed for the normal Cloze-style Figure 2: Epoch number and the loss Table 4: Accuracy of task in Idiom Cloze-style Reading comprehension Performance with baselines Model EPIE(%) Tan et al. [17] 71.02 Our Method 78.23 Entity Tracking [5] 63.57 Reading comprehension. The Idiom is pretty complex and Idiom always are non-compositional. It is hard to guarantee out methods can work on other kinds of Idiom yet, and from the work of Tan et al, we can see their Dual BERT Embedding Model have a better performance on Chinese Idiom Cloze-style Reading comprehension, they already reach the accuracy of 84.43% in Chinese conditions. 6"
+                }
+            ],
+            "Zheyu Yan": [
+                {
+                    "url": "http://arxiv.org/abs/2107.06871v1",
+                    "title": "Uncertainty Modeling of Emerging Device-based Computing-in-Memory Neural Accelerators with Application to Neural Architecture Search",
+                    "abstract": "Emerging device-based Computing-in-memory (CiM) has been proved to be a\npromising candidate for high-energy efficiency deep neural network (DNN)\ncomputations. However, most emerging devices suffer uncertainty issues,\nresulting in a difference between actual data stored and the weight value it is\ndesigned to be. This leads to an accuracy drop from trained models to actually\ndeployed platforms. In this work, we offer a thorough analysis of the effect of\nsuch uncertainties-induced changes in DNN models. To reduce the impact of\ndevice uncertainties, we propose UAE, an uncertainty-aware Neural Architecture\nSearch scheme to identify a DNN model that is both accurate and robust against\ndevice uncertainties.",
+                    "authors": "Zheyu Yan, Da-Cheng Juan, Xiaobo Sharon Hu, Yiyu Shi",
+                    "published": "2021-07-06",
+                    "updated": "2021-07-06",
+                    "primary_cat": "cs.AR",
+                    "cats": [
+                        "cs.AR",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION Deep Neural Networks (DNNs) have achieved superhuman performance in various perception tasks and have become one of the most popular solutions for these applications. Thus, there is an obvious trend in deploying DNNs on edge devices such as automobiles, smartphones, and smart sensors. However, implementing computational intense DNNs directly on edge devices is a significant challenge due to the limited computation resource and constrained power budget of these devices. Moreover, most of the DNN accelerator designs are confined in a design space where the researchers only consider conventional von-Neumann architectures (e.g., GPUs, mobile CPUs, or FPGAs) as candidate platforms. In von-Neumann architectures, data movement inevitably becomes the bottleneck Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. ASP-DAC 2021, Tokyo, Japan \u00a9 2021 Association for Computing Machinery. ACM ISBN 978-1-4503-7999-1/21/01...$15.00 https://doi.org/10.1145/3394885.3431635 0 20 40 60 80 100 FC Net LeNet ResNet56 VGG19 Accuracy (%) Model Original Deployed Figure 1: Accuracy differences between models trained in data center and deployed on CiM simulations for different neural architectures. FC Net and LeNet target MNIST, ResNet 56 targets CIFAR-10 and VGG-19 targets ImageNet. An accuracy drop close to 10% can be observed. for system efficiency, due to the well-known memory wall where the computational unit must fetch and store data from the memory hierarchy. Emerging device-based Compute-in-Memory (CiM) neural accelerators [5] offer a great opportunity to break the memory wall with special architectural advantages. CiM architectures offer reduced data movement by in-situ weight data access [17]. Highly efficient emerging devices (e.g. RRAMs, STT-RAMS, and FeFETs) can be devised to offer higher energy efficiency and higher memory density compared with traditional MOSFET [16] based designs. However, such accelerators suffer greatly from design limitations. Non-ideal conditions of emerging devices due to their non-ideal manufacturing process induce uncertainties on emerging devices. These uncertainties, such as device-to-device (D2D) variations, thermal noise, and retrieval limitations, cause value changes that, the weights in the actually deployed accelerators may be different from the desired weight value trained offline in data centers. This weight value change leads to performance degradation in actual accelerator implementations. As an illustration, we train the four models, multilayer perceptron (MLP) and LeNet for MNIST, ResNet 56 for CIFAR-10, and VGG-19 for ImageNet, to state-of-the-art accuracy and deploy them on CiM simulation tools [4]. As shown in Fig. 1, an accuracy degradation of close to 10% is observed in each model implementation. arXiv:2107.06871v1  [cs.AR]  6 Jul 2021 \fASP-DAC 2021, Tokyo, Japan Yan, Juan, Hu and Shi The device uncertainty-induced performance degradation has been studied from different perspectives, including device-level observations [25], architecture level analysis [6], and behavioral level explorations [21]. Finding suitable pairs of DNN models and hardware designs that can together offer both desirable hardware reliability and high inference accuracy requires great effort. Neural Architecture Search (NAS) [24, 26, 27] is one of the most successful efforts to address this issue. NAS liberates human labor from endlessly exploring optimal handcrafted DNN models by automatically identifying neural architectures that can offer desired performances from a pre-determined search space. Co-exploration of neural architecture and hardware design [7\u20139] pushes this concept further by incorporating hardware design specifications into NAS search spaces, so as to offer neural architecture-hardware design pairs that are accurate, efficient, and robust against hardware uncertainties. In this work, we adopt a statistical analysis perspective to study the effect of device uncertainties on the performance of DNNs. We model the emerging device uncertainty as a whole into Gaussian noise on weights and thoroughly investigate the behavior of different DNN models under such uncertainties. We conduct a Monte-Carlo simulation-based analysis on the statistical behavior of the models under the influence of device uncertainties. We then abstract our analysis results to offer supports for NAS applications. The detailed contributions of this work are: \u2022 We propose a Monte-Carlo simulation-based experimental flow to measure the device uncertainty-induced perturbations to DNN models. \u2022 We then thoroughly investigate the behaviors of different DNN models under such perturbations and show that the value changes of their output vectors follow Gaussian distribution. \u2022 To alleviate this effect, we then propose UAE, a device uncertaintyaware NAS framework, to search for architectures that are more robust to device uncertainties. Experimental results show that UAE offers a 2.49% higher accuracy than NACIM [6] with 1.2x of time consumption. By further increasing search complexity, UAE reaches 6.39% higher accuracy than NACIM with 2.5x of search time. 2 BACKGROUND 2.1 CiM DNN Accelerators Researchers have proposed different crossbar-based CiM DNN accelerator architectures [3, 16] for efficient DNN inference. We assume an ISAAC-like architecture [16], and the architecture of the system is organized into multiple tiles, with crossbar arrays as the heart of each tile. The crossbar not only stores the synaptic weights but also performs dot-product computations. One certain crossbar is dedicated to processing a set of neurons in a given DNN layer. The outputs of that layer are fed to other crossbars that are dedicated to processing the next layer. The computation in the crossbar is performed inanalog domain. However, ADC and DAC are used to convert the signal from and to the analog domain dot-product computation and other digital domain operations needed in DNN computation. Crossbar is the key component of CiM DNN accelerators. As shown in Fig. 2, a crossbar can be considered as a processing element for matrix-vector multiplication where matrix value (i.e. weights for NNs) are stored at the cross point of each vertical and horizontal line with resistive emerging devices such as RRAMs and FeFETs, and each vector value is propagated through horizontal data lines. In this work, we assume an RRAM-based design. The calculation in crossbar is performed in analog domain but additional peripheral digital circuits are needed for other key NN operations (e.g., nonlinear activation), so DAC and ADCs are adopted between different components. Device-level limitations confine the application of crossbars. The precision of ADC and DACs limits the precision of DNN activations for each layer and the non-ideal characteristics of emerging devices impose noises on the weights of deployed DNNs. DAC DAC DAC DAC ADC ADC ADC Mux Synapse Figure 2: Illustration on crossbar architecture. 2.2 Device Variation In this work, we assume an RRAM-based crossbar design. RRAM devices suffer various types of faults due to manufacturing and runtime non-idealities. Noise sources that are directly relevant to crossbar-based designs include thermal noise, shot noise, random telegraph noise (RTN), and programming errors [4]. When the circuitry is used for inference, programming errors due to deviceto-device variations could be the dominant error source. Write-and-verify [1, 15, 19] is a simple, accurate, and widely used programming scheme for RRAMs. The key operation is to iteratively apply a series of short pulses and check the difference between current and target resistance, converging progressively on the target resistance. In deploying accelerators for Neural Network inference, this time-consuming progress is tolerable because once programmed, no more modifications to the resistance are needed during the entire life span of the accelerator. Although this scheme pulls down the D2D variation-induced error to less than 1%, a significant error drop can still be observed in conditions shown in Fig. 1. 2.3 Neural Architecture Search Neural Architecture Search (NAS) has achieved state-of-the-art performance in various perceptual tasks, such as image classifications [22, 23], inference security [2] and image segmentation [20]. \fUncertainty Modeling of Emerging Device based Computing-in-Memory Neural Accelerators with Application to Neural Architecture Search ASP-DAC 2021, Tokyo, Japan NAS is becoming increasingly successful mainly because it liberates human designing labors by automatically identifying highperformance neural architectures. Co-exploration of neural architecture and hardware design [7\u20139] push this concept further by incorporating hardware design specifications into NAS search spaces, so as to offer neural architecture-hardware design pairs that are accurate, efficient, and robust against hardware uncertainties. Formally speaking, NAS deals with a problem that, given a perceptual task \ud835\udc47, a human-defined design space S, and a set of figures of merit (FOM) P, what is the best neural architecture in S that can offer optimal performance (in terms of FOM in P) on task \ud835\udc47. A typical reinforcement learning (RL)-based NAS that solves this issue, such as the framework proposed in [26], is composed of three key components, a controller, a trainer and an evaluator. In one iteration (named episode) of RL-based NAS, (1) the controller generates a neural architecture from the design space; (2) the trainer builds the generated neural architecture into a DNN model, named child network, and trains the child network on a held-out training dataset; (3) the evaluator collects the figures of merit (FOM), e.g., test accuracy of the trained child network on test dataset, its latency and/or energy consumption; and (4) the controller use a user-defined reward function to calculate a \ud835\udc5f\ud835\udc52\ud835\udc64\ud835\udc4e\ud835\udc5f\ud835\udc51data from FOM collected by the evaluator and use the \ud835\udc5f\ud835\udc52\ud835\udc64\ud835\udc4e\ud835\udc5f\ud835\udc51to update itself so that it can predict neural architectures with higher FOM. This iterative method terminates under two circumstances: (1) the controller repeatedly predicts the same child network; and (2) the number of predicted architectures exceeds a predefined threshold (episode limit). The child network that offers the highest reward among all the generated neural architectures is presented as the search result. The chosen neural architecture is then re-trained on the training dataset for a longer training time to offer optimal performance. More recently, differentiable NAS [12, 13, 18] has achieved stateof-the-art performance with a much-reduced search time by transforming the search process into training an over-parameterized neural network. However, those approaches suffer from flexibility. More specifically, in the field of research considered in this paper, differentiable NAS struggles in handling large search spaces with multiple different hardware design parameters and complex designs where the number of channels varies for each layer. Thus, in this work, we adopt RL based NAS as our search algorithm. 3 UNCERTAINTY MODELING 3.1 Uncertainty Model In this work, we model device uncertainties as a whole and use a Gaussian distribution to represent them [4, 6]. We set the mean of the uncertainty distribution to be zero, its variation to be 0.04, and for each device, its uncertainty is independently distributed. which is referred from [25], where the uncertainties are measured from actual physical devices. For an easier representation of the latter part of this paper, the uncertainty model is depicted as: \ud835\udc4a\ud835\udc37\ud835\udc52\ud835\udc5d= \ud835\udc4a\ud835\udc38\ud835\udc65\ud835\udc5d+ N(\ud835\udf07, \ud835\udf0e) (1) where N is a Gaussian variable which, on each individual element of the weights, is independent and identically distributed. \ud835\udc4a\ud835\udc38\ud835\udc65\ud835\udc5d and \ud835\udc4a\ud835\udc37\ud835\udc52\ud835\udc5dare the expected weights trained in the data center and the actual weights deployed on the accelerators, respectively. 3.2 Effects on DNN Outputs In this work, we focus on the impact of device uncertainty on classification tasks, and the reasons go as follows: (1) most emerging device-based DNN accelerators target classification tasks, analyzing the effect of device uncertainties on these tasks helps the majority of the researchers to improve their work; (2) DNNs for classification tasks are typically composed of convolution layers and fully connected layers, which are also the basic components of DNNs targeting other application. The effect of device uncertainties on these two components is essential for all types of DNNs. We start by understanding the effect of device uncertainties on the output of a DNN model. Formally speaking, a DNN model \ud835\udc40 can be defined as a combination of its neural architecture and its trained weights. Thus, the inference process of a DNN model with input I that generate an output O can be defined as: O = \ud835\udc39(\ud835\udc4a, I) (2) where \ud835\udc39is the neural architecture of \ud835\udc40, \ud835\udc4ais its weights, I is this input vector and O is the output vector. During training, O is then passed through a loss function, where a version of O after SoftMax is compared with the ground truth classification label \ud835\udc3a\ud835\udc47to generate a loss for the backpropagation process. During inference, the final predicted class of I can be calculated by \ud835\udc4e\ud835\udc5f\ud835\udc54\ud835\udc5a\ud835\udc4e\ud835\udc65(O), which is the index of the item in O that has the maximum value. Although in inference, classification result is the final outcome of a DNN model, the output vector O serves as a better representative of the behavior of this model. The classification result is only an index of the maximum value of O and is thus only a simplified discrete proxy of O. The continuous, multi-dimension vector O contains more information than the classification result. In order to understand how uncertainties in weights may affect the network, it is of crucial importance to understand how it affects O. As defined in 1 and 2, a deployed neural network under the effect of device uncertainties can be depicted as: O\ud835\udc37\ud835\udc52\ud835\udc5d= \ud835\udc39(\ud835\udc4a\ud835\udc37\ud835\udc52\ud835\udc5d, \ud835\udc3c) = \ud835\udc39(\ud835\udc4a\ud835\udc38\ud835\udc65\ud835\udc5d+ \ud835\udc41\ud835\udc57, \ud835\udc3c) (3) where \ud835\udc4a\ud835\udc38\ud835\udc65\ud835\udc5dis the trained value of the neural network to be deployed, \ud835\udc41\ud835\udc57is one sample from the noise distribution, and O\ud835\udc37\ud835\udc52\ud835\udc5dis the affected output. We analyze the distributional behavior of the effect of device uncertainties on the output vector of a DNN model. To conduct this analysis, we first (1) train a DNN model \ud835\udc39to converge and collects its trained weight \ud835\udc4a\ud835\udc38\ud835\udc65\ud835\udc5d. We then (2) fix one input image \ud835\udc3cand collect its output on the trained weight \ud835\udc4a\ud835\udc38\ud835\udc65\ud835\udc5d. We denote this output as the original output O\ud835\udc42\ud835\udc5f\ud835\udc56. After that, we (3) sample \ud835\udc3edifferent instances of noises \ud835\udc411, \ud835\udc412, ...\ud835\udc41\ud835\udc3eand then feed them to Eq. 3, collecting \ud835\udc3edifferent output vectors. Finally, we calculate output change using Eq. 4 O\ud835\udc36\ud835\udc3a= O\ud835\udc37\ud835\udc52\ud835\udc5d\u2212O\ud835\udc42\ud835\udc5f\ud835\udc56 (4) where the output change is the element-wise subtraction of the perturbed and original output. \fASP-DAC 2021, Tokyo, Japan Yan, Juan, Hu and Shi 3.3 Experimental Results In order to get a glance at the statistical behavior of output change, according to the workflow introduced in Sect. 3.2, we train a LeNet model for the MNIST dataset [11] to state-of-the-art accuracy. We then randomly choose one input image in the test dataset and sampled 10k different instances of noise. with this setup, we gathered 10k different output change vectors. The output change is a vector of 10, with each element representing the confidence of classifying the input image into one certain category. Because a high-dimensional vector is not a good choice for analytical study and visualization, we analyze each element of these vectors. We analyze the statistical behavior of each element across different vectors and gathered 10 instances of distribution data. Surprisingly, each element of the output change follows Gaussian distribution. To visualize this finding, we plot the histogram of the distribution of each element of output change vector and the corresponding Gaussian distribution that fits it. The visualization result for the first element of output change is shown in Fig. 3 that the first element of output change vector is a Gaussian variable. Figure 3: Output change distribution of LeNet for MNIST. 10k output change vectors are gathered from one trained LeNet model affected by 10k different instances of noise sampled from N (0, 0.04). This figure shows the distribution of the fist item of the gathered output change vectors. To verify this observation, We tested various networks in various datasets. With the MNIST dataset, we analyze both LeNet and multilayer perceptrons (MLP) using ReLU and Sigmoid activation with 2 layers. With the CIFAR-10 dataset [10], we test a conventional floating-point CNN, a quantized CNN, and two ResNets, ResNet56 and ResNet-110. We also train these models with 3 different initializations to get different trained weights. We evaluate how these output change vectors fit into Gaussian variables by two widely used standards: mean square error (MSE) and Chi-square (\ud835\udf122) test. MSE can be described as: \ud835\udc40\ud835\udc46\ud835\udc38= 1 \ud835\udc41 \ud835\udc41 \u2211\ufe01 \ud835\udc56=1 (\ud835\udc42\ud835\udc56\u2212\ud835\udc38\ud835\udc56)2 (5) and \ud835\udf122 test can be depicted as: \ud835\udf122 = \ud835\udc41 \u2211\ufe01 \ud835\udc56=1 (\ud835\udc42\ud835\udc56\u2212\ud835\udc38\ud835\udc56)2 \ud835\udc38\ud835\udc56 (6) where \ud835\udc42\ud835\udc56and \ud835\udc38\ud835\udc56are the observed (output change) and estimated (Gaussian) value of probability and \ud835\udc41is a user-defined granularity. We define \ud835\udc41= 100 because it is precise enough when we have a total of 10k instances of data. Model Dataset \ud835\udf122 (10\u22122) MSE (10\u22124) MLP-ReLU MNIST 5.22 3.20 MLP-Sigmoid MNIST 5.81 2.20 LeNet MNIST 4.59 2.67 Float-Conv CIFAR-10 7.01 3.03 Fixed-Conv CIFAR-10 6.79 2.74 ResNet-56 CIFAR-10 4.56 1.79 ResNet-110 CIFAR-10 4.81 2.01 Table 1: Gaussian fit for different models. The \ud835\udf122 test result and MSE between the output change and Gaussian distribution is presented. Both tests show that the output change follows Gaussian distribution w.r.t different instances of noise. The evaluation result is shown in Table 1. Note that we have test 3 different initializations for each model and for each model, \ud835\udc5c\ud835\udc62\ud835\udc61\ud835\udc5d\ud835\udc62\ud835\udc61\ud835\udc50\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52is a vector of 10. The result shown in Table 1 is an average of them. For each model tested, \ud835\udf122 test results are all below 0.1 and MSE are all below 1 \u00d7 10\u22123, which indicates that they are well fit into Gaussian distributions. Moreover, both errors do not increase when the model is extremely shallow (e.g. 2-layer MLP) and very deep (RestNet-110), so this observation generalizes across different DNN models. The study on each of the models supports the previous observation that their output vectors values follow Gaussian distribution. Based on these studies, we can claim that, with any independent and identically distributed Gaussian noise on weight, the output vector of the same input image follows a multi-dimensional Gaussian distribution1 over different samples of noise. This is a very strong claim but is not counter-intuitive. The output of the first convolution layer is the summation of the multiplication result of deterministic inputs and Gaussianly distributed weights and is thus a summation of Gaussian distributions. The summation of Gaussian variables is also a Gaussian variable, so the output of the first layer is a Gaussian variable. After activation, the input of the second layer is a transformed Gaussian variable and after propagating through this layer, with enough number of operands, the accumulated variable can be approximated by Gaussian variables. Thus, although the final output may not strictly be a Gaussian variable, a Gaussian approximation can be observed. 4 UNCERTAINTY AWARE SEARCH 4.1 Methodology In addition to understanding the effect of device uncertainties, we propose a remedy method to reduce the effect of this issue by adopting NAS. In this work, we propose Uncertainty Aware sEarch (UAE), a more comprehensive uncertainty aware NAS for better exploration of neural architectures in non-ideal emerging devices based CiM NN accelerators. 1Note that each element of the output are deeply co-related, not independent \fUncertainty Modeling of Emerging Device based Computing-in-Memory Neural Accelerators with Application to Neural Architecture Search ASP-DAC 2021, Tokyo, Japan Similar to the state-of-the-art Reinforcement Learning based NAS framework NACIM [6], UAE works iteratively and in each iteration: (1) an LSTM-based controller is used to learn and generate neural architectures; (2) an uncertainty aware trainer is used to train each generated neural architecture to get a model to deploy; (3) an uncertainty aware evaluator is adopted to evaluate the actual performance of the deployed model; (4) the evaluated performance is used as a reward to update the controller so that it can generate neural architectures with higher rewards. The detailed implementation of the uncertainty aware trainer and evaluator are described below. 4.2 Uncertainty-Aware Training & Evaluation We adopt an uncertainty-aware training scheme similar to the one used in NACIM [6]. The training process is organized the same as traditional DNN training that, in each iteration, a subset (batch) of the training data is used to train the model, and after the whole training dataset has been used to train the model, and an epoch of training is finished and another epoch is started. The trainer trains the model for multiple epochs to get a trained model. The uncertainty-aware training augments the training process for each batch to learn a DNN model that is more robust against device variations. In each training batch, before feeding the input into the model, the trainer (1) save the original weight \ud835\udc4a\ud835\udc42\ud835\udc5f\ud835\udc56of the model; (2) sample a noise from the uncertainty distribution and add the noise to the weight of the model to form a \ud835\udc4a\ud835\udc37\ud835\udc52\ud835\udc5d; (3) perform forward inference and back propagation in the perturbed model and collect gradient data for each weight; (4) load the saved \ud835\udc4a\ud835\udc42\ud835\udc5f\ud835\udc56 back to the model and update\ud835\udc4a\ud835\udc42\ud835\udc5f\ud835\udc56with the collected gradient data via stochastic gradient descent. Uncertainty-aware training simulates the process of training DNNs directly on CiM-based accelerators. Experiments in NACIM show that uncertainty-aware training learns DNN models that are robust against device uncertainties. The uncertainty-aware evaluation is performed similarly to the training process. Before evaluation, the evaluator samples an instance of noise from the uncertainty distribution and add the noise to the trained weight \ud835\udc4a\ud835\udc42\ud835\udc5f\ud835\udc56to get a \ud835\udc4a\ud835\udc37\ud835\udc52\ud835\udc5d. The evaluator then evaluates the classification accuracy of the perturbed model on a test dataset. This process is performed for \ud835\udc3etimes and \ud835\udc3edifferent accuracy data are gathered. The evaluator then report one distributional property (e.g., mean, maximum value, 95% minimum value) of the \ud835\udc3eaccuracy data to form a reward. The distributional property to be used is specified by the user. 4.3 Experimental Results We demonstrate the effectiveness of UAE by searching for an optimal quantized CNN for CIFAR-10. The fixed design parameters and hyper-parameters included in the search space are shown in Table 2. For device uncertainty specifications, we assume an ISAAClike [16] neural accelerator architecture and a four-bit RRAM device, whose behavioral model is extracted from [25]. The search process is conducted in a GPU server machine with an Nvidia GTX 1080ti accelerator. As described in Sect. 4.2, there are two major search parameters: the instances (\ud835\udc3e) of noise sampled for each architecture and the Hyper-Parameters Value choices Dataset CIFAR-10 Type Quantized CNN # of Conv Layers 6 # of FC Layers 2 FC Hidden size 1024 # channels (24, 36, 48, 64) Filter Height/Width (1, 3, 5, 7) # of integer bits (0, 1, 2, 3) # of fraction bits (0, 1, 2, 3, 4, 5, 6) Table 2: Quantized CNN for CIFAR-10 search setups. Upperhalf: configurations fixed to be the same among all searched architectures; lower half: hyper-parameters to be searched. distributional properties used to form the accuracy data collected by the evaluator into a reward. We test two different values of \ud835\udc3e, 5, and 100 samples, for the reason that will be explained afterward. We also use two different distributional properties, one is the mean value of all accuracy data (mean) and the other is 95% minimum of the accuracy data (95). The mean value indicates how a model performs under the effect of device uncertainty in average circumstances and the 95% minimum shows the models\u2019 behavior in worst-case scenarios. We offer a comparison for different specifications of UAE and two baseline methods, quantNAS [14], a state-of-the-art NAS framework to search for the optimal quantized CNN and NACIM [6], another uncertainty aware searching framework for CiM-based accelerators. In each experiment, the NAS controller searches for 2000 different architectures (episodes) and the trainer trains each generated architecture for 15 epochs. The DNN models finally presented by each search framework are also evaluated by mean and 95% minimum value with their accuracy data collected by 10k Monte-Carol simulation. The experimental result is shown in Table. 3.2 Method \ud835\udc3e w/o noise mean 95 Time (h) QuantNAS [14] 0 84.92% 08.48% N/A 53 NACIM [6] 1 73.88% 73.45% N/A 98 UAE-M 5 77.48% 75.94% 75.55% 118 UAE-M 100 82.99% 79.84% 77.82% 255 UAE-95 100 80.64% 78.39% 77.98% 255 Table 3: Comparison for different specifications of UAE and two other baselines. Different methods sample different instances of noise (\ud835\udc3e) in uncertainty-aware evaluation. UAEM uses mean value of the accuracy data collected by the evaluator to form a reward and UAE-95 uses the 95% minimum data. Accuracy without noise shows the model accuracy with an ideal device and the mean and 95% minimum (95) value shows the behavior of the searched DNN models under the effect of device uncertainty, evaluated by MonteCarol simulation. 2Because the data for quantNAS and NACIM are collected from published work, we do not have the 95% minimum accuracy result for them. \fASP-DAC 2021, Tokyo, Japan Yan, Juan, Hu and Shi Experimental results show that, without uncertainty-aware training, QuantNAS can identify an optimal DNN model that can offer close to 85% of test accuracy, but struggles in finding proper neural architectures that are robust to device uncertainties, as the test accuracy of the DNN model identified by quantNAS is down to 8.5%, even worse than random guessing (10%). With the help of uncertainty-aware training, NACIM can identify DNN models that are robust against the impact of device uncertainties. However, as NACIM only evaluates the performance of the searched architecture once, the randomness of the device uncertainty hinders the performance of NACIM, resulting in a model with only 73.45% of average accuracy. UAE, on the contrary, is able to find DNN models that are both accurate and robust against the impact of device uncertainties. When collecting only 5 samples in uncertainty-aware evaluation, UAE achieves 2.49% higher accuracy than NACIM with a time overhead of only 20%. When collecting 100 samples in uncertaintyaware evaluation, UAE achieves 6.39% higher accuracy than NACIM a search time overhead of 2.5x. Though further increasing the number of samples is possible, the search time overhead will be too large to endure. The adoption of a 95% minimum value standard is also effective. UAE-95 offers 0.16% higher worst-case accuracy than its UAE-M counterpart with a 1.45% lower average accuracy. 5"
+                }
+            ],
+            "Zhenge Jia": [
+                {
+                    "url": "http://arxiv.org/abs/2212.08624v2",
+                    "title": "Development of A Real-time POCUS Image Quality Assessment and Acquisition Guidance System",
+                    "abstract": "Point-of-care ultrasound (POCUS) is one of the most commonly applied tools\nfor cardiac function imaging in the clinical routine of the emergency\ndepartment and pediatric intensive care unit. The prior studies demonstrate\nthat AI-assisted software can guide nurses or novices without prior sonography\nexperience to acquire POCUS by recognizing the interest region, assessing image\nquality, and providing instructions. However, these AI algorithms cannot simply\nreplace the role of skilled sonographers in acquiring diagnostic-quality POCUS.\nUnlike chest X-ray, CT, and MRI, which have standardized imaging protocols,\nPOCUS can be acquired with high inter-observer variability. Though being with\nvariability, they are usually all clinically acceptable and interpretable. In\nchallenging clinical environments, sonographers employ novel heuristics to\nacquire POCUS in complex scenarios. To help novice learners to expedite the\ntraining process while reducing the dependency on experienced sonographers in\nthe curriculum implementation, We will develop a framework to perform real-time\nAI-assisted quality assessment and probe position guidance to provide training\nprocess for novice learners with less manual intervention.",
+                    "authors": "Zhenge Jia, Yiyu Shi, Jingtong Hu, Lei Yang, Benjamin Nti",
+                    "published": "2022-12-16",
+                    "updated": "2022-12-19",
+                    "primary_cat": "eess.IV",
+                    "cats": [
+                        "eess.IV",
+                        "cs.CV",
+                        "cs.HC",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Automated acquisition guidance can help novice learners to study how to manipulate probes to acquire high-quality POCUS images. However, the di\ufb00erence between human vision and machine vision may confuse novice learners, because the guidance from sonographers and the DNN model may be inconsistent due to the di\ufb00erent interpretations of images between the DNN-based models and humans [6]. For example, DNNs rely more on textures and high-frequency features rather than shapes and low-frequency features. Our previous works show that machine learning based detection methods on medical image can achieve signi\ufb01cantly high accuracy [7\u201315]. This means images considered highquality for humans may not be high-quality for DNN-based machine vision. The discrepancy becomes more signi\ufb01cant due to the large inter-observer variability. As a result, if the acquired images are of high quality for both human and DNN-based segmentation models, the segmentation accuracy can be improved, and accordingly, the human e\ufb00orts needed to correct or identify the bad segmentation can be reduced. As an example in a closely-related area, previous works for medical image compression demonstrated that image compressed with machine vision considered achieved signi\ufb01cantly higher segmentation accuracy than conventional methods that only consider human vision at the same compression rate [16]. Therefore, we will explore methods to optimize the image quality assessment and acquisition guidance to be preferred by both human and machine models. 2 Methods The overall framework proposed is shown in Figure 1. During data acquisition, the scan is checked by sonographers to ensure high visual quality for humans. After that, the segmentation model will 1 \fMulti-Task DNN Image Quality Assessment 6D Distance (feedback together with 6D Distance) Figure 1: Framework of simultaneously optimizing image quality assessment and acquisition guidance by a multi-task DNN. produce segmentation results. Then a multi-task DNN will predict the quality of the input scan image according to the segmentation results, and meanwhile, the model will generate 6D (probe position plus orientation) geometric distances, which can be feedback to novice learners for guidance on how to move from the current probe location and the probe location anticipated to optimize the image. Here, such a DNN model needs to be conducted in a real-time manner, because the learners need immediate feedback from the model to learn how to move the probe. If the predicted quality is lower than the prede\ufb01ned threshold, the novice learner will be prompted to do a re-scan with guidance. The study/exam process is done once the predicted image quality exceeds the threshold (a max number of re-scans can be set to avoid too many trials). Because the prediction model helps learners to select the data that the model can segment well, the quality of the segmentation results and the subsequent medical analysis will be improved. Box 1. Equations C = cc R \u03b1f(\u03b1)d\u03b1; Eq.1 \u22c4C: Original average cost \u22c4cc: Average cost of correcting a failed segmentation \u22c4\u03b1: Probability of segmentation fails \u22c4f(\u03b1): Probability density function of \u03b1 C\u2032 \u03b1 = \u03b1(1 \u2212r)cc + \u03b1r p (cs + C\u2032 \u03b1); Eq.2 \u22c4C\u2032: New average cost \u22c4cs: Average cost of an additional scan \u22c4p: Precision of the quality prediction model \u22c4r: Recall of the quality prediction model C\u2032 \u03b1 = p\u03b1cc\u2212p\u03b1rcc+\u03b1rcs p\u2212\u03b1r ; Eq.3 h(\u03b1) = C\u2032 \u03b1 C\u03b1 = p\u2212pr+r cs cc p\u2212\u03b1r ; Eq.4 \u22c4h(\u03b1): Ratio of new cost to original cost for \u03b1 C\u2032 C = R \u03b1f(\u03b1)h(\u03b1)d\u03b1 R \u03b1f(\u03b1)d\u03b1 ; Eq.5 The key component of the framework is the multi-task DNN model for both image quality prediction and 6D distance generation. We will develop a DNN with a shared encoder (i.e., the front layers of DNN) to extract common features, and then it will be split into two paths for image quality assessment and acquisition guidance tasks, respectively. For the image quality assessment task, we will use a modi\ufb01cation of DNN-based segmentation model as the image quality prediction model. It can take advantage of existing techniques such as model architecture for image segmentation, uncertainty estimation of neural networks, and the attention mechanism. We will compare the quality prediction models that are trained to predict conventional quality metrics and the new quality metric proposed in Aim 1. In this way, we can additionally validate whether the new metric also helps in this framework. On the other hand, for the 6D distance generation task, we will add a tail after the encoder, and each output neuron will correspond to one distance in the 6D distance. We will track the image quality after the learners move the probe according to the predicted 6D distance to evaluate the e\ufb00ectiveness of the model. A threshold applied to the predicted image quality will control the learning loop. A low threshold means a re-scan is suggested only if the predicted image quality is quite low and thus the feedback information to novice learners is minimal. A high threshold means the data is satisfying only if the predicted quality is quite high which leads to higher image quality, and in turn, it indicates a strict requirement for learners. In practice, the threshold can be set based on the progress of the learner. The learning quality of a learner followed the proposed framework can be analyzed via a theoretical cost model. Because the cost for the initial scan and veri\ufb01cation are not a\ufb00ected by the proposed method, we focus on the cost of correcting segmentation and the cost of doing a re-scan in the analysis below. We consider the segmentation on the video obtained fails if correction is needed, and denote \u03b1 as the failure probability. A \ufb01xed \u03b1 for all subjects may be inaccurate because each subject may have a unique condition that makes it easier or more di\ufb03cult to obtain images that can be segmented well. Therefore, we consider each subject has a unique \u03b1 and repetitive scans for the same subject are independent. The original average per-subject cost is given in Eq. 1 (Box 1). The quality prediction model is used to identify failed scans. The new average cost for a certain \u03b1 is given in Eq. 2. The \ufb01rst 2 \fterm is for predicted passed segmentation and the second term is for predicted failed segmentation. Solving Eq. 2 we get Eq. 3. It can be derived that, for a certain \u03b1, the new average cost is lower than the original average cost if and only if p > \u03b1+ cs cc . This indicates a lower bound of precision can reduce the cost. Finally the ratio of the new cost C\u2032 to the original cost C is given in Eq. 4 and Eq. 5. 3 Results Table 1: Estimated cost reduction. \u03b1 is the probability of a segmentation fails. cs cc is the ratio of the scan cost to the correction cost. Cost reduction is 1 \u2212C\u2032 C . \u03b1 0.2 0.3 0.2 0.2 0.2 0.2 cs cc 0.1 0.1 0.2 0.1 0.1 0.1 Precision 0.8 0.8 0.8 0.6 0.9 0.7 Recall 0.8 0.8 0.8 0.6 0.7 0.9 Cost reduction 64% 57% 50% 37% 55% 69% This cost model re\ufb02ects how various factors a\ufb00ect the cost-saving of the proposed framework. f(\u03b1) accounts for the data distribution in the speci\ufb01c applications which can be estimated by examining the segmentation results of interest. Bigger \u03b1 leads to lower cost reduction because it takes more scans to obtain good segmentation. h(\u03b1) is determined by cs cc , p, and r. cs cc is the ratio of the re-scan cost to the correction cost which can be easily estimated by sonographers and radiologists. Smaller cs cc means smaller C\u2032 C and thus more cost reduction. An accurate prediction model is crucial for the cost reduction as re\ufb02ected by p and r. In Table 1, we provide a few numerical examples of the cost reduction when variables are set to typical values. The cost model shows a substantial possible cost reduction. 4"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2008.08060v1",
+                    "title": "Personalized Deep Learning for Ventricular Arrhythmias Detection on Medical IoT Systems",
+                    "abstract": "Life-threatening ventricular arrhythmias (VA) are the leading cause of sudden\ncardiac death (SCD), which is the most significant cause of natural death in\nthe US. The implantable cardioverter defibrillator (ICD) is a small device\nimplanted to patients under high risk of SCD as a preventive treatment. The ICD\ncontinuously monitors the intracardiac rhythm and delivers shock when detecting\nthe life-threatening VA. Traditional methods detect VA by setting criteria on\nthe detected rhythm. However, those methods suffer from a high inappropriate\nshock rate and require a regular follow-up to optimize criteria parameters for\neach ICD recipient. To ameliorate the challenges, we propose the personalized\ncomputing framework for deep learning based VA detection on medical IoT\nsystems. The system consists of intracardiac and surface rhythm monitors, and\nthe cloud platform for data uploading, diagnosis, and CNN model\npersonalization. We equip the system with real-time inference on both\nintracardiac and surface rhythm monitors. To improve the detection accuracy, we\nenable the monitors to detect VA collaboratively by proposing the cooperative\ninference. We also introduce the CNN personalization for each patient based on\nthe computing framework to tackle the unlabeled and limited rhythm data\nproblem. When compared with the traditional detection algorithm, the proposed\nmethod achieves comparable accuracy on VA rhythm detection and 6.6% reduction\nin inappropriate shock rate, while the average inference latency is kept at\n71ms.",
+                    "authors": "Zhenge Jia, Zhepeng Wang, Feng Hong, Lichuan Ping, Yiyu Shi, Jingtong Hu",
+                    "published": "2020-08-18",
+                    "updated": "2020-08-18",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "eess.SP",
+                        "stat.ML"
+                    ],
+                    "main_content": "INTRODUCTION Sudden cardiac death (SCD) is one of the most signi\ufb01cant causes of natural death in the US, which occurs when the heart electrical system malfunctions and leads to 325,000 deaths per year [5]. There are two main causes of SCD, namely, ventricular tachycardia (VT) and ventricular \ufb01brillation (VF), which are the life-threatening ventricular arrhythmias (VA). For people under a great risk of SCD, implantable cardioverter de\ufb01brillator (ICD) is served as a preventive treatment. ICD is a small device implanted under the skin with wires inserted in the heart to detect the intracardiac rhythm [30]. This battery-powered device continuously monitors the intracardiac rhythm and is programmed to deliver a shock when VT or VF is detected, and bring the rhythm back to normal. While ICDs reduce the risk of SCD and increase survival rate, the ICD recipients might experience inappropriate shocks, which are the delivery of shock on the rhythm other than VT or VF. Inappropriate shocks have been associated with proarrhythmias, intolerable pain, and depression [16]. They have been reported to occur in 12% to 23% ICD recipients and constitute 30% to 50% of all shocks [8, 11, 13]. Current VT/VF detection methods count on a wide variety of criteria on intracardiac rhythm, such as heart rate, the number of intervals to detection (NID), and probability counter [3, 28, 31], where there are hundreds of programmable parameters a\ufb00ecting the detection performance. However, it is complicated to obtain the best parameters setting for each patient as it requires massive clinician experience and frequent manual intervention. Moreover, such optimization cannot be conducted in time since the ICD programming update could only be accomplished every 3-6 months [4]. Recently, deep learning technique has been widely applied on electrocardiogram (ECG) arrhythmias classi\ufb01cation using Convolutional Neural Network (CNN) [1, 10, 25]. The CNN-based arrhythmias classi\ufb01cation could eliminate the cumbersome criteria selections and parameters setting in traditional arrhythmia detection methods, while achieves decent detection accuracy. The objective of this paper is to propose a framework for deep learning based VT/VF detection. The framework addresses the following challenges: (1) Existing CNN-based detection models cannot be directly deployed on the implantable devices due to the hardware resources constraints, such as limited memory capacity and computational power; (2) The only intracardiac rhythm sensed by ICDs is the bottleneck for VT/VF detection accuracy improvement. Performing inference on surface and intracardiac rhythm simultaneously could improve the accuracy but the memory and energy overhead would signi\ufb01cantly increase; (3) The personalization of detection model for each patient su\ufb00ers from a long delay and frequent manual intervention on current ICDs platform. To tackle the above challenges, we propose P-VA, a Personalized computing framework and architecture for deep learning based VA detection. The overview of P-VA is demonstrated in Fig. 1. P-VA consists of two parts: the Detection Nodes and the Cloud. The Detection Nodes, which constitute an implantable node and a wearable node, detect VT/VF on intracardiac electrograms (IEGMs) and surface electrocardiograms (ECG) simultaneously. A CNN is designed and deployed on the both nodes to perform a real-time inference. To further improve the detection accuracy, we propose \fthe cooperative inference with a con\ufb01dence score thresholding policy, which enables two nodes to detect VT/VF collaboratively and signi\ufb01cantly reduce the memory and energy overhead by only transmitting the hardly-decidable rhythm segments. On the Cloud, we develop an in-time personalization using deep domain adaptation and a policy network to personalize CNNs for the speci\ufb01c patient. Evaluation results show that our detection nodes could e\ufb00ectively make each inference within 71ms on average. Furthermore, when compared with traditional detection algorithm, our method achieves a comparable VT/VF detection rate and 6.6% reduction in inappropriate shock rate. The main contributions of this paper are summarized as follows: \u2022 We present P-VA, a personalized computing framework for deep learning based VA detection, to improve the accuracy of VA detection. \u2022 To the best of our knowledge, this paper is the \ufb01rst to deploy a CNN on a resources-constrained and ultra-low power processor with real-time VT/VF detection. \u2022 The cooperative inference reduces the inappropriate shock rate by enabling the implantable and the wearable nodes to detect VT/VF collaboratively. \u2022 We develop an in-time personalization to personalize CNNbased detection model for each speci\ufb01c patient. The rest of the paper is organized as follows. In Section 2, background and motivation are introduced. Section 3 \ufb01rst provides an overview of the P-VA framework, followed by the construction of CNN-based detection model, the cooperative inference and the model personalization. Evaluation results of the proposed detection methods and its comparisons with the state-of-the-arts are shown in Section 4. Finally, Section 5 concludes the paper. 2 BACKGROUND AND MOTIVATION In this section, we \ufb01rst introduce the background of ICDs and the impedance of improving VT/VF detection accuracy on ICDs. Next we demonstrate the motivation of proposing deep learning based detection and the challenges in application and deployment. 2.1 VT/VF Detection in ICDs The ICD is implanted to deliver de\ufb01brillation on VT/VF, which are life-threatening VA leading to SCD. Until now, ICDs are still the best preventive treatment for the population under high risk of SCD. As aforementioned, patients with ICDs, however, might receive inappropriate shocks, which severely a\ufb00ect the patients\u2019 quality of life. According to the surveys, 12% to 23% of ICD recipients receive inappropriate shocks in the following 20 months to 11 years after implantation [11, 17, 19, 21]. The study [11] shows that the main cause of inappropriate shock comes from supraventricular arrhythmia misdiagnosed as VT (74.8% of inappropriate shocks). The optimization goal of ICDs is to improve the detection accuracy on VT/VF. However, there are three main obstacles to improve VT/VF detection performance. Complex detection criteria and parameter settings. Current VT/VF detection algorithms count on a combination of various criteria. In the detection programming, there are hundreds of programmable parameters a\ufb00ecting shock delivery decision. The modi\ufb01cation of non-nominal parameter settings (i.e., changing the ICD out-of-the-box factory default settings for the recipient) and innovation of new detection criteria are necessary for the improvement of detection accuracy. However, they require extensive clinician proactivity and engagement. Long delay in personalized detection. The criteria parameters are supposed to be optimized to accommodate the unique rhythm feature of each ICD recipient. After implanting an ICD, the recipient should be followed-up every 3-6 months [4]. During the follow-up, the detection criteria parameters would be \ufb01ne-tuned by a group of doctors based on historical records. However, the detection algorithms could not be updated in time due to the 3-6 months follow-up interval. The in-time personalized detection is unrealistic on the current ICDs platform since it requires massive manual intervention and expertise to \ufb01ne tune criteria parameters, which severely restrict the personalization frequency. Limited rhythm sources. In the current data sources acquisition, there is an intrinsic bottleneck in detection accuracy improvement, that is, ICDs detect VT/VF purely based on the sensed intracardiac rhythm. The involvement of surface rhythm would improve detection accuracy since there would be more rhythm information in the detection. However, due to the constraints in physical size and lifetime, the computational power and memory capacity of ICDs are severely restricted. The processing unit on ICDs must run with few hundreds of kilobyte of on-chip memory and an active power consumption below the 10 mW mark. Those constraints are the impedance to enable ICDs to detect surface rhythm simultaneously. 2.2 Deep Learning in Arrhythmia Detection Recently, deep learning has been applied to classify arrhythmias on ECGs by using convolutional neural networks (CNNs) and achieved outstanding performance in terms of accuracy [1, 7, 10, 12]. Different from existing detection algorithms, the deep learning based methods eliminates the process of criteria design and optimizations. It signi\ufb01cantly reduces the magnitude of cardiological expertise required in the detection parameters \ufb01ne-tuning. CNN could detect the arrhythmias with high accuracy if the training rhythm data is properly and accurately labeled by doctors. However, in ICDs scenarios, existing CNN models cannot be directly deployed. The limited memory capacity and real-time inference requirement are the impedance to deploy CNN models on ICDs. Further e\ufb00ort on the CNN architecture design should be applied to \ufb01t those hardware constraints. Moreover, only one welltrained CNN as a generalized detection model cannot accommodate all patients due to the unique rhythm features of each patient. The in-time personalization of each speci\ufb01c patient\u2019s CNN based detection model remains unsolved. Nevertheless, the improvement of the CNN model detection accuracy is also restricted by the limited rhythm sources. The inference mechanism on both surface and intracardiac rhythm could generate a more accurate prediction. 3 PERSONALIZED DEEP LEARNING FOR VA DETECTION In this section, we \ufb01rst provide an overview of the proposed VT/VF detection framework. Then we introduce the CNN-based detection 2 \fC1 C2 C3 C4 C5 F6 F7 C1 C2 C3 C4 C5 F6 F7 VT/VF Non VT/VF IEGM Segments CONV FC Implantable Node Detection Nodes C1' C2' C3' C4' C5' F6' F7' C1 C2 C3 C4 C5 F6 F7 C1 C2 C3 C4 C5 F6 F7 1. Unsupervised  Domain Adaptation IEGM Domain ECG Domain 2. Policy Network  Training Policy Network C1 C2 C3 C4 C5 F6 F7 Reward IEGM  Segments Policy Prediction Gradient Domain  Adaptation Policy Freeze Or Fine-Tuning Cloud All ECG Segments Hardly-decidable  IEGM Segments Policy Network Detection Networks \u0102\u0102  All ECG Segments Hardly-decidable  IEGM Segments Policy Network Detection Networks \u0102\u0102  C1' C2' C3' C4' C5' F6' F7' ECG Segments VT/VF Non VT/VF CONV FC C1' C2' C3' C4' C5' F6' F7' ECG Segments VT/VF Non VT/VF CONV FC Wearable Node Cooperative  Inference Figure 1: An Overview of P-VA Framework. model. After that, we illustrate the cooperative inference mechanism. Finally, we present the in-time CNN model personalization to accommodate each patient\u2019s unique rhythm. 3.1 An Overview of P-VA Framework. In this subsection, we present an overview of the personalized deep learning based VT/VF detection framework. Fig. 1 illustrates our P-VA framework, which consists of the Detection Nodes and the Cloud. There are two main functions provided by the framework, the VT/VF detection on Detection Nodes and the detection model personalization on the Cloud. Detection Nodes. The detection nodes consist of the implantable node and the wearable node, and both nodes perform arrhythmias detection on the sensed rhythm. As shown in the Detection Nodes of Fig. 1, the implantable node continuously monitors intracardiac rhythm while the wearable node continuously monitors surface rhythm. Due to the constraints caused by implantation, the implantable node is equipped with an ultra-low power processing unit with few hundreds of KB on-chip memory and an active power consumption below 10 mW. The wearable node is equipped with a more powerful processing unit due to the relatively loose physicalsize restrictions and easy access to power supply. As illustrated in Fig. 1, both nodes conduct VT/VF detection on intracardiac and surface rhythm simultaneously using CNN with the same architecture. The CNN model is designed to satisfy the hardware constraints and perform real-time inference. To further improve the detection accuracy, the cooperative inference is proposed to enable both nodes to detect VT/VF rhythm collaboratively. However, due to the energy constraints, not all IEGMs could be uploaded from the implantable node to the wearable node. Hence a con\ufb01dence thresholding policy is implemented on the implantable \u0102\u0102  Input 250x1 10 10 16 16 7 5 3 Conv1 BN, ReLU Conv2 BN, ReLU Conv3 BN, ReLU Conv4 BN, ReLU Conv5 BN, ReLU FC1 ReLU FC2 Softmax 3 5 10 10 10 VT/VF Non-VT/VF Stride 2 Stride 2 Stride 1 Stride 1 Stride 1 10 Figure 2: CNN Architecture and Detection Process. node to identify the limited number of hardly-decidable IEGM segments and trigger the cooperative inference. Moreover, the wearable node would upload the received hardly-decidable IEGMs along with its fully-recorded ECGs to the Cloud for further personalization. Cloud. Once the fully-recorded ECGs and the hardly-decidable IEGMs are uploaded to the Cloud, the fully-recorded ECGs would \ufb01rst be labeled by doctors. However, the uploaded IEGMs could not be accurately labeled due to its non-continuity, and only the CNN on ECG domain could be correctly personalized by retraining. To tackle the challenge, as illustrated in Fig. 1, we \ufb01rst apply Maximum Mean Discrepancy (MMD) distance to perform unsupervised domain adaptation on the ECG-domain speci\ufb01c CNN to accommodate the IEGM domain [15]. Here, all possible IEGM-domain speci\ufb01c CNNs with di\ufb00erent CONV layers being frozen or \ufb01netuned would be generated as candidate CNNs. In the step 2 on the Cloud in Fig. 1, the policy network is utilized to determine the best-\ufb01t CNN from the candidate pool for incoming IEGM segment. Once the domain adaptation and the policy network training completes, the personalized CNNs on ECG and IEGM domains along with the policy network would be propagated back to the wearable and the implantable nodes. Moreover, our framework provides the feasibility that the model could be personalized once the uploaded rhythm is manually labeled by the doctors. Di\ufb00erent from traditional upgrading process, our personalization could be completed in a more \ufb02exible and timely way, where the patients do not need to wait for 3-6 months to upgrade the detection model. 3.2 CNN Detection Model In this subsection, we introduce the CNN-based VT/VF detection model. Then, we will introduce the mechanism to determine the shockable rhythm on the implantable node. Since CNN model detects VT/VF on resources-constrained platforms, the network is designed to be relatively small when compared with the existing CNN-based detection models [1, 7, 10, 12]. Fig. 2 illustrates the CNN architecture and the detection process. It consists of \ufb01ve 1D-convolutions (CONV) followed by ReLU function and two fully connected (FC) layers. This CNN-based model requires 26.5 KB to store and the intermediate results during layerby-layer calculation require at most 4.54 KB. As demonstrated in Fig. 2, the input of CNN are 2-second (2s) ECG or IEGM segments (i.e., sampling points within the 2s segment) in a time series and the output would be a series of inference 3 \fC1 C2 C3 C4 C5 F6 F7 C1 C2 C3 C4 C5 F6 F7 Predictions IEGM Segment CONV FC Implantable Node C1' C2' C3' C4' C5' F6' F7' ECG Segment Predictions CONV FC CSIN < T Policy Network Chosen CNN CSIN > T Evaluator Wearable Node Hardly-decidable  IEGM Segment Hardly-decidable  IEGM Segment CS Comparator Figure 3: Cooperative Inference with Con\ufb01dence Thresholding Policy. results (i.e. VT/VF or non-VT/VF) on each input segment. Based on the series of inference results, on the implantable node, an evaluator would determine the shockable rhythm through a simple but e\ufb00ective decision criteria. This criteria is that the rhythm would be determined as shockable if there are four consecutive VT/VF segments. The detection period on VT/VF is 8 seconds since the detection delay of traditional VT/VF detection algorithm is about 5 to 9 seconds [16]. In other words, if there are four consecutive VT/VF segments in the rhythm, the shock therapy would be triggered. 3.3 Cooperative Inference The CNN-based VT/VF detection performed purely on the implantable node is e\ufb03cient but it does not achieve the perfect detection due to the limited data sources. A way to improve accuracy is to conduct the inference on intracardiac and surface rhythm through the implantable and the wearable node simultaneously, and to choose the inference result with higher con\ufb01dence. However, if such detection is performed, both nodes would frequently communicate with each other to send inference results. The transmission overhead and the corresponding energy consumption would greatly reduce the lifetime of the implantable node and increase the response time to the shockable rhythm. Hence, there is a trade-o\ufb00 between accuracy and lifetime on the device. To address the problem, we propose the cooperative inference along with an inference con\ufb01dence thresholding policy. The cooperative inference is designed to be simple but e\ufb03cient to \ufb01t the resources constraints. It is based on the concept Con\ufb01dence Score (CS), which indicates the con\ufb01dence of the inference on each possible class. As shown in Fig. 3, on the implantable node, its CNN would perform inference on the input IEGM segment. The CS for both classes (i.e., VT/VF and non-VT/VF) on the segment would be calculated and fed into the CS comparator. The CS on the implantable node is de\ufb01ned as follows: \ud436\ud446\ud43c\ud441= |\ud443\ud43c\ud441(\ud466\ud463\ud461\ud463\ud453|\ud465) \u2212\ud443\ud43c\ud441(\ud466\ud45b\ud45c\ud45b\u2212\ud463\ud461\ud463\ud453|\ud465)|, (1) where \ud443\ud43c\ud441(\ud466|\ud465) is the probability of being classi\ufb01ed as VT/VF segment (\ud466\ud463\ud461\ud463\ud453) or not (\ud466\ud45b\ud45c\ud45b\u2212\ud463\ud461\ud463\ud453) given the input IEGM segment \ud465. This probability is obtained by utilizing \ud446\ud45c\ud453\ud461\ud45a\ud44e\ud465on the outputs of the last FC layer. Here, a higher \ud436\ud446\ud43c\ud441represents a more reliable inference result on the input IEGM segment. When \ud436\ud446\ud43c\ud441is low, it reveals the fact that there is a certain level of uncertainty of the inference result, which indicates that the CNN on the implantable could not clearly discriminate the segment. In this case, the CNN on the wearable node should be involved in the VT/VF detection. A con\ufb01dence thresholding policy is proposed to utilize a threshold \ud447on CS to determine the participation of the wearable node in VT/VF detection. As shown in Fig. 3, when \ud436\ud446\ud43c\ud441< \ud447, the corresponding input IEGM segment is de\ufb01ned as the hardly-decidable segment and would be transmitted to the wearable node. On the wearable node, the ECG segment on the same timestamp of the IEGM segment would be the \ufb01rst to feed into CNN and generate the inference result along with the con\ufb01dence score \ud436\ud446\ud44a\ud441, and the \ud436\ud446\ud44a\ud441on the wearable node is de\ufb01ned as follows: \ud436\ud446\ud44a\ud441= |\ud443\ud44a\ud441(\ud466\ud463\ud461\ud463\ud453|\ud465) \u2212\ud443\ud44a\ud441(\ud466\ud45b\ud45c\ud45b\u2212\ud463\ud461\ud463\ud453|\ud465)|, (2) where \ud443\ud44a\ud441(\ud466|\ud465) is the probability of being classi\ufb01ed as VT/VF segment (\ud466\ud463\ud461\ud463\ud453) or not (\ud466\ud45b\ud45c\ud45b\u2212\ud463\ud461\ud463\ud453) given the input ECG segment \ud465. Moreover, the uploaded hardly-decidable IEGM segment is fed into the policy network (which would be introduced in Section 3.4). Here, the policy network would choose one CNN to make inference on the given IEGM segment and the corresponding CS is de\ufb01ned as \ud436\ud446\ud443\ud43f. As shown in Fig. 3, by comparing the value of \ud436\ud446\ud443\ud43fwith \ud436\ud446\ud44a\ud441, the \ufb01nal inference result (i.e., VT/VF or non-VT/VF) with higher CS value would be transmitted back to the implantable node and fed into the decision evaluator on the implantable node. On the other hand, if \ud436\ud446\ud43c\ud441\u2265\ud447, the inference result generated by CNN on the implantable node would be fed into the evaluator directly without the cooperative inference. In the cooperative inference, the threshold\ud447a\ufb00ects the chance of data transmission between the implantable node and the wearable node. The determination of the value of \ud447is based on the tradeo\ufb00among the corresponding performance of accuracy, inference latency and energy consumption. The detailed experimental results and the determination of \ud447will be demonstrated in Section 4.2 3.4 In-time CNN Personalization To cater for the unique features of each patient\u2019s rhythm, the CNNbased detection model needs to be personalized with the sensed rhythm. In the P-VA framework, once the data uploaded to the Cloud is accurately labeled, the CNN model personalization would be automatically conducted by \ufb01ne-tuning CNN with the newly labeled data. This process signi\ufb01cantly reduces the degree of manual intervention and the personalization period. However, in our scenarios, only the arrhythmias on the fullyrecorded ECGs could be accurately labeled by doctors since the hardly-decidable IEGM segments are non-continuous and the ventricular arrhythmias on IEGMs could not be correctly diagnosed. Thus, only the ECG-domain speci\ufb01c CNN could be personalized by \ufb01ne-tuning the CNN on the fully-recorded ECGs. To obtain the personalized CNN model on IEGM domain, we invoke Deep Adaptation Networks (DAN) proposed in [15] and conduct further optimizations by utilizing the policy network. The core of DAN is to integrate the Maximum Mean Discrepancy (MMD) in the loss function [15]. The invocation of MMD explicitly reduces the domain discrepancy on the FC layers. By minimizing the MMD between the source and the target domain, the mean embedding of distributions across domains can be explicitly matched. In this work, the source domain is set as labeled ECGs and the target domain is set as unlabeled IEGMs. For the generalized CNN trained on the rhythm from databases, DAN is applied to \ufb01netune some CONV layers and all FC layers on the ECG domain (i.e., 4 \fReward IEGM Segment Inference Gradient Policy Freeze Freeze Fine-tune Fine-tune Fine-tune C1 C2 C3 C4 C5 F6 F7 C1 C2 C3 C4 C5 F6 F7 C1 C2 C3 C4 C5 F6 F7 C1 C2 C3 C4 C5 F6 F7 \u0102\u0102  C1 C2 C3 C4 C5 F6 F7 C1 C2 C3 C4 C5 F6 F7 \u0102\u0102  Policy Network Candidate CNNs Conv, 5x1 BN, LeakyRelu Conv, 5x1 BN FC3 LeakyReLu FC2 LeakyReLu Policy Network Conv, 5x1 BN, Relu FC1 LeakyReLu LeakyRelu, Dropout max  pool x 3 Conv, 5x1 BN, LeakyRelu Conv, 5x1 BN FC3 LeakyReLu FC2 LeakyReLu Policy Network Conv, 5x1 BN, Relu FC1 LeakyReLu LeakyRelu, Dropout max  pool x 3 Policy Figure 4: The Illustration of Policy Generation and Training on Policy Network. personalize the CNN on ECG domain for the speci\ufb01c patient), and use MMD to tailor the FC layers to \ufb01t the un-labelled IEGMs domain (i.e., personalize the CNN on IEGM domain for the speci\ufb01c patient). In this way, the model could be personalized once the uploaded ECG is manually labeled by the doctors. The personalization could be completed with less manual intervention. DAN employs a global \ufb01ne-tuning strategy, which \ufb01xes the CONV layers to freeze or \ufb01ne-tune for all input. That is, DAN freezes the low-level CONV layers and \ufb01ne-tunes the high-level CONV layers on the source domain. It assumes that the low-level CONV layers can learn generic features, while high-level CONV layers are slightly domain-biased. However, the assumption are not met when the target training data is not su\ufb03cient or unbalanced. For example, the speci\ufb01c person\u2019s IEGM segments might have higher similarity with the ECG segments from rhythm databases, and routing those segments through some the generalized CNN\u2019s CONV layers could generate more inference results. Inspired by the idea from BlockDrop [29], which uses a policy network to dynamically select which layer of a Residual Network to execute during an inference, we utilize the policy network to route each IEGM segment to a speci\ufb01c CNN. As shown in Fig. 4, we \ufb01rst obtain IEGM-domain speci\ufb01c CNNs under all possible freezing or \ufb01ne-tuning strategy with DAN as the candidate CNNs. Then, the policy network is used to route each incoming IEGM segment for its best-\ufb01t IEGM-domain speci\ufb01c CNN. Fig. 4 demonstrates the architecture of the policy network. It consists of one CONV layer, four dense blocks and three FC layers. Inside each dense block, there are two CONV layers with skip connection. As shown in Fig. 4, there are 32 IEGM-domain speci\ufb01c CNNs (i.e., freeze or \ufb01ne-tune on each CONV layer out of 5) with the application of DAN. Given an IEGM segment s, the policy of deciding which candidate CNN to choose is de\ufb01ned as a 5-dimensional Bernoulli distribution, which takes the form: \ud70b(a|s) = P[\ud434= a|\ud446= s] = 5 \u00d6 \ud456=1 \ud465\ud44e\ud456 \ud456(1 \u2212\ud465\ud456)1\u2212\ud44e\ud456, (3) where s is the input IEGM segment and a is the policy that chooses a certain CNN, as shown in Fig. 4. Here, \ud465is the output vector of the policy network after the Sigmoid function. The i-th element of the vector \ud465, \ud465\ud456\u2208[0, 1], represents the likelihood of the corresponding CONV layer of the ECG-domain speci\ufb01c CNN being frozen or \ufb01netuned. The policy a is also a vector with binary number, which are selected based on \ud465, where \ud44e\ud456= 0 represents frozen CONV layer \ud456 and \ud44e\ud456= 1 represents \ufb01ne-tuned CONV layer \ud456. To encourage the highest accuracy on all segments, as illustrated in Fig. 4, the reward is set based on its prediction correctness of the input segment. This prediction is generated by the IEGM-domain speci\ufb01c CNN selected by the policy network. The reward function associated with the action is de\ufb01ned as \ud445(a) = \u001a \ud6fd \ud456\ud453\ud450\ud45c\ud45f\ud45f\ud452\ud450\ud461 \u2212\ud6fd \ud45c\ud461\u210e\ud452\ud45f\ud464\ud456\ud460\ud452, (4) where \ud6fdis a positive constant. Here, we treat all candidate CNNs equally and reward the correct prediction on the selected CNN with \ud6fd. Similarly, incorrect prediction is penalized by \u2212\ud6fd. Finally, to obtain the optimal choice of the IEGM-domain speci\ufb01c CNN, we maximize the expected reward associated with the action \ud438\ud445= E\ud70b[\ud445(a)]. To maximize the expected value, we utilize the policy gradient to compute the gradients of \ud43d, which is calculated as follow: \u2207\ud438\ud445= E\ud70b[\ud445(a)\u2207log \ud70b(a|s)] = E[\ud445(a)\u2207log 5 \u00d6 \ud456=1 \ud465\ud44e\ud456 \ud456(1 \u2212\ud465\ud456)1\u2212\ud44e\ud456]. (5) Here, we can further optimize the expected gradient to \u2207\ud438\ud445= E\ud70b[\ud445(a)\u2207 5 \u00d5 \ud456=1 log[\ud465\ud456\ud44e\ud456+ (1 \u2212\ud465\ud456)(1 \u2212\ud44e\ud456)]], (6) where the value of \ud44e\ud456is either 0 or 1. Based on the calculated expected gradient on the policy network, we could generate the choice of which IEGM-domain speci\ufb01c CNNs to be applied to the incoming IEGM segment. After that, the \ufb01ne-tuned ECG-domain speci\ufb01c CNN, the policy network, and all candidate IEGM-domain speci\ufb01c CNNs would be propagated back from the Cloud to the wearable node. Due to the hardware constraints, the policy network and all candidate CNNs could only be deployed on the wearable node. One IEGM-domain CNN which achieves the highest accuracy during policy network training would be transmitted from the wearable node and deployed on the implantable node. 4 EVALUATIONS We evaluate the performance of the proposed CNN-based VT/VF detection method with a series of experiments. In this section, 5 \fTable 1: Database Summary. Database # of Recordings Rhythm Source Sampling Rate (Hz) Recording Duration (min) MITDB 48 Lead I 360 30 VFDB 22 Lead I 250 30 CUDB 35 Lead I 250 8 AAEL 304 Lead I, RVA-Bi 1,000 0.5-5 Table 2: Training and Testing Dataset for Generalized CNN Training and Personalization. Dataset # of VT/VF Segments # of Non-VT/VF Segments Data Source Generalization-TrainSet 6,093 25,524 MITDB, CUDB, VFDB Personalization-G1 780 2,274 AAEL Personalization-G2 781 2,275 AAEL Personalization-G3 781 2,275 AAEL we \ufb01rst illustrate the experiments setup and then introduce the evaluation results. 4.1 Experiments Setup 4.1.1 Implementations. In our experiments, the board Apollo3 Blue [27] serves as the implantable node and a Raspberry Pi 3B+ [26] serves as the wearable node. The size of our CNN model is 26.2 KB and requires at most 4.54 KB for intermediate data storage. Such storage requirements could be met by the Apollo3 Blue. Moreover, the total size of the 32 candidate CNN models is 838.4 KB, which is not a large memory overhead for the wearable node. We adopt PyTorch for the generalized CNN model and the policy network training. For the generalized CNN, we set learning rate to 1\ud452\u22124 and use a batch size of 64 during training. We use stochastic gradient descent (SGD) with 0.9 momentum and the loss function is cross-entropy loss function. The total number of epochs is 100. For the policy network, we set learning rate to 1\ud452\u22124 and use a batch size of 4 during training. We use SGD with 0.9 momentum and the loss function is cross-entropy loss function. The total number of epochs is 50. All those training experiments run on the PC as the Cloud with 8 core of Intel Xeon E5-2620 v5 CPU, 512 GB memory, and an NVIDIA GeForce GTX 2080Ti GPU. 4.1.2 Data Preprocessing. The data used in the experiments is obtained from four databases, namely the MIT-BIH arrhythmia database (MITDB) [18] [23], the MIT-BIH malignant ventricular arrhythmia database (VFDB) [9] [22], the Creighton Univeristy ventricular tachyarrhythmia database (CUDB) [20] [24], and the Ann Arbor Electrogram Libraries (AAEL) [2]. The attributes of the recordings acquired from the four databases are summarized in Table 1. As demonstrated in Table 1, the databases MITDB, CUDB and VFDB only contain the ECG recordings (from Lead I) whereas the recordings in the AAEL are recorded as ECG (from lead I) and IEGM (i.e., from the lead placed in right ventricle, named RVA-Bi) simultaneously. Moreover, the sampling rate of the four databases is di\ufb00erent. To ensure the standardization across the databases and reduce inference time, all recordings are downsampled to 125Hz. The ECG recordings in the MITDB, CUDB and VFDB are utilized to train the CNN on ECG domain as the generalized CNN model. The recordings from those three databases are segmented into 2s non-overlapping segments and each segment is labeled as VT/VF or non-VT/VF using ground truth annotations provided in those three databases. The 2s segments from those three databases constitute the training set for the generalized CNN model. As shown in Table 2, there are 6,093 VT/VF segments and 25,524 non-VT/VF segments in the training set, denoted as Generalization-TrainSet. The ECG and IEGM recordings in the AAEL are utilized to perform personalization on generalized CNN model and to evaluate detection performance. The data preprocessing procedure for the AAEL is recording level \u2212 \u2192event level \u2212 \u2192segment level. On the recording level, there are 304 recordings as demonstrated in Table 1. Each recording consists of two channels of data from lead I and RVA-Bi. Then, the recording periods diagnosed as VT and VF in the AAEL are labeled as VT/VF events while other are labeled as non-VT/VF events. The reason for the existence of an event level is that the shock therapy is supposed to be delivered during a VT/VF event and the detection accuracy should be calculated based on the correct detection on the events. After labeling and participating in 304 recordings, there are 198 VT/VF and 273 non-VT/VF events in total. Each event is then segmented into non-overlapping 2s segments labeled with VT/VF or non-VT/VF. There are 2,342 VT/VF segments and 6,824 non-VT/VF segments. For the segments in a time series from the same event, we participate those segments into three equally partitioned set and place those set into three groups, denoted as Personalization-G1, G2, and G3. The detailed statistics of each group are demonstrated in Table 2. The rationale behind such grouping is that the segments (i.e., the patient\u2019s rhythm from G1 and G2) would be utilized to personalize the model \ufb01rst. The data from the same patient (i.e., the same patient\u2019s rhythm from G3) should be applied to evaluate the performance of the personalized CNN model. 4.1.3 Our Methods and Baseline Methods. We \ufb01rst obtain the generalized CNN trained on the ECG segments from GeneralizationTrainSet. Next, we utilize Personalization-G1 and G2 to personalize the generalized CNN model and utilize Personalization-G3 to evaluate performance. We denote the detection model which is personalized with Personalization-G1 and G2 as CNN-2G. The detection model which is personalized with only PersonalizationG1 is denoted as CNN-1G. The generalized CNN which is directly deployed on both nodes are de\ufb01ned as CNN-0G. The cooperative inference and the policy network are activated during the inference. We further implement the CNN-2G without policy network involved. This detection model is used to evaluate the e\ufb00ect of the policy network to detection accuracy. This detection method is denoted as CNN-NoPolicy. 6 \fTable 3: Performance metrics for baseline methods and CNN with cooperative inference and policy network. Methods F1 Score Se/Sp BAC/Acc PPV/NPV Classic .925 .970/.908 .939/.934 .885/.976 SVM-ML .883 .874/.923 .898/.902 .892/.910 CNN-DL .901 .944/.890 .917/.913 .862/.957 CNN-NoPolicy .953 .970/.952 .961/.960 .937/.977 CNN-0G .831 .909/.799 .854/.845 .766/.924 CNN-1G .950 .960/.956 .958/.958 .941/.970 CNN-2G .975 .984/.974 .980/.979 .965/.989 80% 90% 100% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Detection Accuracy  Confidence Score Threshold (T) CNN-2G CNN-1G CNN-0G CNN-NoPolicy Figure 5: Detection Accuracy vs Con\ufb01dence Score Threshold. For the baseline methods, we \ufb01rst simulate a VT/VF discrimination method used in single-chamber ICDs [30], denoted as Classic. In the Classic, the VT zone is set to be 160-200 bpm. The VF zone is set to be the heart rate faster than 200 bpm. The detection window is set as 10 most recent ventricular R-R intervals. The event is determined as the shockable event if the Classic methods decides to deliver the shock. The heart rate boundary of VT/VF zone is also \ufb01ne-tuned for each testing patient to simulate the intervention such that the best discrimination performance could be achieved. We then implement an existing machine learning based detection method using support vector machine (SVM) [14], denoted as SVM-ML. The features extracted in SVM-ML are Count2 and Leakage. The training set is Personalization-G1 and G2, and testing set is Personalization-G3. We do not implement SVM-ML on the implantable node due to its complex feature extraction process, which consumes extensive computational resources and cannot be completed within 1 second for 2s segment input. Existing machine learning based detection methods are not designed to \ufb01t real-time requirements. We also implement a conventional deep learning detection method using CNN. We utilize the IEGM segments from Personalization-G1 and G2 to train the generalized CNN model. The detection performance is evaluated by Personalization-G3 on the implantable node. This detection is denoted as CNN-DL. 4.2 Performance Assessment In this section, we evaluate the performance of CNN-based detection on the P-VA framework with the performance metrics in terms of detection accuracy, latency and energy consumption. 4.2.1 Detection Accuracy. The detection accuracy performance is \ufb01rst reported in terms of sensitivity (Se), speci\ufb01city (Sp), positive and negative predictive accuracy (PPV and NPV), total accuracy (Acc), balanced accuracy (BAC), and F1 score. Those statistics are calculated based on the resulted event detections. We conduct performance evaluations using the aforementioned detection methods. As shown in Table 3, when compared with the classic VT/VF discrimination algorithm, CNN-DL achieves a slight decrease on performance in terms of VA detection (Se) and inappropriate shock rate (Sp). When compared with SVM-ML, CNNDL achieves a 1.1% increment from a baseline of 90.2% in Acc and a 6.9% increment on VT/VF event detection rate represented by Se. In other words, the performance of a single CNN on IEGM domain shows that the deep learning based VT/VF detection could achieve decent detection accuracy. With the assistance of the P-VA framework, the cooperative inference enables CNN model to achieve higher detection accuracy. In the experiment, the con\ufb01dence score threshold \ud447is set to be 0.5 due to the fact that the accuracy becomes relatively stable when \ud447> 0.4. We will demonstrate the accuracy trend along with \ud447in Fig. 5. Here, CNN-2G could achieve the accurate detection, speci\ufb01cally 97.9% accuracy on all events. When compared with the classic detection algorithm, CNN-2G improves by 1.4% on VT/VF event detection accuracy represented by Se and 6.6% on non-VT/VF event detection accuracy represented by Sp. CNN-2G also achieves the best performance in terms of F1 score among all methods. In addition, the personalization process is shown to be necessary for the CNN-based model. As shown in Table 3, CNN-1G experiences a slight degradation in F1 score when compared with CNN-2G. CNN-0G has the worst detection performance among all methods, which we believe is due to the fact that it has no personalization done on the CNN models, so that both node could only make the inference based on the learned features from other patients in rhythm database (i.e., Generalization-TrainSet). Moreover, when compared with CNN-2G, the performance of CNN-NoPolicy illustrates that the policy network could further improve detection accuracy by selecting the best-\ufb01t IEGM-domain CNN. We further investigate the e\ufb00ect of the con\ufb01dence score threshold\ud447on the detection accuracy. The line plot in Fig. 5 demonstrates the accuracy achieved by CNN-0G, CNN-1G, CNN-2G, and CNNNoPolicy. These four CNN-based detection methods show to follow the same trend: the accuracy become relatively stable when\ud447> 0.4 and the cooperative inference could be more accurate for a larger T. It indicates that the appropriate setting of threshold\ud447could achieve the higher detection accuracy through cooperative inference between the wearable and the implantable node. 4.2.2 Detection Delay. The experiments aim at assessing the inference latency on the implantable node. On the implantable node, 7 \f30 50 70 90 110 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Average Inference Latency (ms) Confidence Score Threshold (T) CNN-2G CNN-NoPolicy Figure 6: Detection Latency vs Con\ufb01dence Score Threshold. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 CNN-DL CNN-2G Average Energy Consumption (mJ) Methods Figure 7: Average Energy Consumption of Inference over One Segment for CNN-DL and CNN-2G. inference on a 2s IEGM segment could be completed within 31ms without cooperative inference. Here, we do not consider the latency caused by data collection on the implantable node. Data collection is the process of sensing heart rhythm and sending the sensed data to processing unit. Since our focus is on the calculation in inference, we do not consider the latency caused by data collection for all methods. With cooperative inference involved, the actual inference latency on the implantable node should include the transmission delay and the inference latency caused by the wearable node. Here, we investigate the e\ufb00ect of the con\ufb01dence score threshold \ud447to the inference latency on the implantable node. The line plot in Fig. 6 illustrates the changes of average inference latency on one IEGM segment with an increasing \ud447. The performance of two methods, CNN-2G and CNN-NoPolicy, is evaluated in this experiment. Here, both methods show the same trend: the inference latency would increase as the\ud447increases. It is straightforward since the implantable node would be more likely to upload its IEGM segment for cooperative inference as \ud447increases to 1. Moreover, the average inference time of CNN-2G is longer than that of CNN-NoPolicy because the policy network also takes time to execute on the wearable node. One interesting observation is that there is a plateau in the inference latency for both methods, where the \ud447ranges from 0.5 to 0.7. Therefore, \ud447\u2286[0.5, 0.7] is a suitable value in terms of accuracy and latency. 4.2.3 Energy Consumption. In this subsection, we compare average energy consumption of detection for various methods on the 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Average Energy Consumption (mJ) Confidence Score Threshold (T) CNN-2G CNN-NoPolicy Figure 8: Average Energy Consumption of Inference vs Con\ufb01dence Score Threshold. implantable node. We also evaluate the e\ufb00ect of the con\ufb01dence score threshold \ud447on the energy consumption for the CNN-based methods. The energy consumption of the implantable node comes from several operations, including inference and data transmission. Fig. 7 demonstrates the average energy consumption of inference on a 2s segment for methods CNN-DL and CNN-2G (where \ud447= 0.5) deployed on the implantable node. The performance indicates that the deep learning based methods could achieve a high energy e\ufb03ciency. As shown in Fig. 7, the energy consumption of CNN-DL is less than that of CNN-2G since the data transmission between the implantable and the wearable node consumes extra energy. Fig. 8 illustrates the changes of the average energy consumption of inference over a 2s segment along with the con\ufb01dence score threshold \ud447on the implantable node. The performances of two methods, CNN-2G and CNN-NoPolicy, are evaluated in the experiments. As shown in Fig. 8, both methods would consume more energy with an increasing \ud447since there is a higher probability for the implantable node to transmit data to the wearable node. And CNN-2G would consume more energy because the implantable node should wait for a longer time to receive the results from the wearable node during the cooperative inference. 5"
+                }
+            ],
+            "Ahmed Abbasi": [
+                {
+                    "url": "http://arxiv.org/abs/2211.07621v1",
+                    "title": "Alternating minimization algorithm with initialization analysis for r-local and k-sparse unlabeled sensing",
+                    "abstract": "The unlabeled sensing problem is to recover an unknown signal from permuted\nlinear measurements. We propose an alternating minimization algorithm with a\nsuitable initialization for the widely considered k-sparse permutation model.\nAssuming either a Gaussian measurement matrix or a sub-Gaussian signal, we\nupper bound the initialization error for the r-local and k-sparse permutation\nmodels in terms of the block size $r$ and the number of shuffles k,\nrespectively. Our algorithm is computationally scalable and, compared to\nbaseline methods, achieves superior performance on real and synthetic datasets.",
+                    "authors": "Ahmed Abbasi, Abiy Tasissa, Shuchin Aeron",
+                    "published": "2022-11-14",
+                    "updated": "2022-11-14",
+                    "primary_cat": "eess.SP",
+                    "cats": [
+                        "eess.SP",
+                        "cs.IT",
+                        "cs.LG",
+                        "math.IT",
+                        "math.PR"
+                    ],
+                    "main_content": "Introduction For the linear inverse problem, Y = BX\u2217+ W, where B is the measurement matrix and Y \u2208Rn\u00d7m are linear measurements of unknown X\u2217\u2208Rd\u00d7m. In the unlabeled sensing problem, the measurements are scrambled by an unknown permutation P\u2217: Y = P\u2217BX\u2217+ W, (1) where P\u2217is the unknown permutation matrix and W is additive Gaussian noise with Wij = N(0, \u03c32). Given scrambled measurements Y and the measurement matrix B, the unlabeled sensing problem is to estimate the signal X\u2217. This problem is called the single-view (multi-view) unlabeled sensing problem for m = 1(m > 1). The work in [1] introduced the unlabeled sensing problem and establish that n = 2d measurements are necessary and su\ufb03cient for recovery of x\u2217. Subsequent works [2, 3] have generalized this result and developed information theoretic inachievability analysis. Because estimating the unknown permutation matrix P\u2217is a challenging problem, several works [4, 5, 6, 7, 8, 9] assume a k-sparse, or partially shu\ufb04ed, permutation model. [10, 11] consider an r-local, or block diagonal, permutation model. A permutation matrix P\u2217 k is k-sparse if it has k o\ufb00-diagonal elements, i.e., \u27e8I, P\u2217 k\u27e9= n \u2212k. A permutation matrix P\u2217 r is r-local with s blocks if P\u2217 r = blkidag (P1, \u00b7 \u00b7 \u00b7 , Ps). Fig. 1 illustrates the two models. The two models are compared by information-theoretic inachiveability in [11]. For single-view unlabeled sensing, algorithms based on branch and bound and expectation maximization are proposed in [12, 13, 14]. These algorithms are applicable to small problem sizes. A stochastic alternating minimization approach is proposed in [15]. For multi-view unlabeled sensing, the Levsort subspace matching algorithm was proposed in [16]. A subspace clustering approach was proposed in [7]. The works [4, 5] propose methods based on bi-convex optimization and robust regression, respectively. A spectral initialization based method was proposed recently in [17]. 1.1 Contributions We extend our alternating minimization algorithm for the r-local unlabeled sensing model [10], [11] to the widely considered k-sparse model, Section II. We propose and analyze the initialization to our algorithm, 1 arXiv:2211.07621v1  [eess.SP]  14 Nov 2022 \f0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Figure 1: Left. Sparse (or partially shu\ufb04ed) permutation considered in [4, 5, 6, 7], with number of shu\ufb04es k = 10. Right. The r-local permutation structure considered in [10], [11], with block size r = 10. In this paper, we propose and analyze a general algorithm that is applicable to both permutation models. Section III. Speci\ufb01cally, assuming random B or X\u2217, we derive upper bounds on the initialization error for both r-local and k-sparse models. Numerical results and comparisons to baseline methods showing superior performance are given in Section IV. A theoretical contribution is upper-bounding the error in the minimum norm solution to an underdetermined linear inverse problem. Under-determined systems have also been studied as the sketching problem [18], but the results, derived assuming a random sub-sampling strategy and \ufb01xed x\u2217, are not applicable as our sub-sampling strategy is deterministic because of the unknown permutation P\u2217in (1) and random x\u2217. 1.2 Applications of unlabeled sensing The linked linear regression problem [19, 20, 21], or linear regression under blocking, is to \ufb01t a linear model by minimizing over x and P the objective \u2225y \u2212PBx\u22252 2, where P is an unknown r-local permutation. For example, consider a simple statistical model where the weight of an individual depends linearly on age and height. The vector y \u2208Rn contains the weight of n = 10 individuals, the matrix B \u2208Rn\u00d7d, d = 2 contains the values of the age, height of the individuals. The 10 individuals comprise 5 males, 5 females and each record (weight, age, height) is known to belong to a male or a female individual. Letting the \ufb01rst (second) block of 5 rows of y, B contain the records of male (female) individuals, the unknown r-local permutation, r = 5, assigns each weight value (row of y) to its corresponding age, height (row of B). Experimental results and comparisons with baseline methods for the linked linear regression problem are given in Section 4. For the pose and correspondence problem [22], the rows of B \u2208Rn\u00d7d contain the d-dimensional coordinates of the landmark points of an object. The object is transformed by an unknown linear transformation X\u2217and the rows of Y = P\u2217BX\u2217contain the coordinates of the points of the transformed object. Given coordinates B, Y, the problem is to assign a point in B to the corresponding point in Y and estimate the transformation X\u2217. Another application of unlabeled sensing is signal amplitude estimation from binary quantized unlabeled samples [23]. The unknown scalar quantity x is the signal amplitude and the sensing matrix B is replaced by a threshold function that models bit quantization. The image correspondence problem [24] and the 1-d unassigned distance geometry problem (uDGP) were recently formulated as unlabeled sensing problems in [11]. 1.3 Notation and de\ufb01nitions P\u2217\u2208\u03a0n denotes a permutation matrix, where \u03a0n = {Z: Z \u2208{0, 1}n\u00d7n, Z1n = 1n, Z\u22ba1n = 1n} is the set of n \u00d7 n permutation matrices and 1n \u2208Rn is the vector of all ones. P\u2217 r denotes r-local permutations with block size r. P\u2217 k denotes k-sparse permutations with k o\ufb00-diagonal elements. The Hamming distortion dH = \u03a3i1(b P(i) \u0338= P(i)) is the number of mismatches in permutation matrices b P, P\u2217. A\u2020 denotes the Moore-penrose pseudoinverse of matrix A. [A; B] denotes the vertical concatenation of matrices A and B. \u27e8A, B\u27e9= trace(A\u22baB) denotes the matrix inner product. \u2225\u00b7\u2225p denotes the vector p-norm and \u2225\u00b7\u2225F denotes the matrix Frobenius norm. \u2225X\u2225\u03d51(\u2225X\u2225\u03d52) denotes the sub-exponential (sub-Gaussian) norm of the sub-exponential (sub-Gaussian) random variable X. c\u2032 \u2208R denotes an absolute constant. 2 \f2 Algorithm We propose to estimate the permutation P\u2217and the signal X\u2217by minimizing the forward error in (1), (b P, b X) = argmin P\u2208\u03a0n,X F(X, P) = \u2225Y \u2212PBX\u22252 F . (2) Alternating minimization (alt-min) updates for (2) are P(t) = argmin P\u2208\u03a0n \u27e8\u2212YX(t)\u22baB\u22ba, P\u27e9, (3) X(t+1) = argmin X F(X, P(t)) = (P(t)B) \u2020Y. (4) When the unknown permutation P\u2217is r-local, the P update in (3) decouples along the blocks of P\u2217and reduces to the simpler updates in line 9 of Algorithm 1. (2) is a non-convex optimization problem because the set of permutations is a discrete (non-convex) set. Therefore, the initialization to the updates (3), (4) has to be chosen carefully. We propose and analyze initializations for both r-local and k-sparse models in the following sections. A key contribution of our algorithm is that the model assumption is incorporated in the initialization. We note that Algorithm 1 for r-local P\u2217 r without initilization analysis was \ufb01rst proposed in our earlier work [11]. 2.1 Initialization for the r-local model When the unknown permutation P\u2217 r is r-local, we propose to initialize b X by the collapsed initialization (7). Let Pi denote each r \u00d7 r block of P\u2217 r = diag(P1, \u00b7 \u00b7 \u00b7 , Ps). Let Bi \u2208Rr\u00d7d denote blocks of the measurement matrix B = [B1; \u00b7 \u00b7 \u00b7 ; Bs]. Since the permutation P\u2217 r is block-diagonal, the r permuted measurements in each block [Yi : PiBi] are summed to extract one labelled measurement on X\u2217\u2208Rd\u00d7m j=r X j=1 [PiBi(j)]\u22baX\u2217= j=r X j=1 Yi(j) \u2200i \u2208[s], (5) where PiBi(j) \u2208Rd\u00d71 denotes row j of PiBi. Assuming all blocks have the same size r, the number of blocks is s = n/r. The s labelled measurements in (5) are represented in the s \u00d7 d collapsed linear system of equations e Y = e BX\u2217, (6) where the rows of the collapsed measurement matrix e B \u2208Rs\u00d7d and measurements e Y \u2208Rs\u00d7m are e B(i) = j=r X j=1 Bi(j), e Y(i) = j=r X j=1 Yi(j) \u2200i \u2208[s]. Note that e B(i) = Pj=r j=1 PiBi(j) = Pj=r j=1 Bi(j). The proposed initialization b X is the minimum-norm solution to (6), b X = e B\u2020 e Y. (7) Substituting the singular value decomposition form of e B\u2020 = e VS\u22121 e U\u22ba, where e B = e US e V\u22ba, shows that b X is the projection of X\u2217onto the the row space of e B. Formally, \u2225X\u2217\u2212b X\u22252 = \u2225(I \u2212e V e V\u22ba)X\u2217\u22252. (8) For s \u2265d, b X = X\u2217. In section 3.1, we upper bound the error in initialization b X for the under-determined s < d case. 2.2 Initialization for the k-sparse model When the unknown permutation P\u2217 k is k-sparse , we propose to initialize b P(0) = I, which sets b Y(0) = Y = P\u2217 kBX\u2217. Since the permutation matrix P\u2217 k has k o\ufb00-diagonal elements, the initialization b P(0) = I is close to the true solution P\u2217 k as dH(b P(0), P\u2217 k) = k. The initialization error is upper bounded in section 3.2. 3 \fAlgorithm 1 alt-min Input: mode \u2208{r-local, k-sparse}, convergence threshold \u03f5 1: if mode is r-local then 2: b X \u2190collapsed initialization in (7) 3: b Y \u2190B b X 4: else 5: b Y \u2190Y 6: while relative change > \u03f5 do 7: if mode is r-local then 8: for i \u22081 \u00b7 \u00b7 \u00b7 s do //s is the number of blocks 9: b Pi \u2190argmin Pi\u2208\u03a0r \u2212\u27e8Yi b Y\u22ba i , Pi\u27e9 10: b P \u2190diag(b P1, \u00b7 \u00b7 \u00b7 , b Ps) 11: else 12: b P = argmin P\u2208\u03a0n \u2212\u27e8Y b Y\u22ba, P\u27e9 13: b X \u2190B\u2020 b P\u22baY 14: b Y \u2190B b X 15: Return b P, b X 3 Initialization analysis 3.1 r-local model Assuming random models on either B or X\u2217, the results in this section upper bound the error in the proposed initialization (7). Without any assumptions, the worst case error \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2264\u2225y\u22252 2 \u2212\u03c32 min(B)\u2225\u02c6 x\u22252 2 \u03c32 min(B) (9) is achieved for x\u2217aligned with the direction of the singular vector corresponding to the smallest singular value \u03c3min(B). For x\u2217with zero-mean, independent sub-Gaussian random coordinates, Theorem 2.1 in [25] shows that the error \u2225x\u2217\u2212\u02c6 x\u22252 is sub-Gaussian. In contrast, Theorem 1 in this work shows a sub-Gaussian error distribution for \ufb01xed x\u2217and Gaussian B. Theorem 2 shows that the error is sub-Gaussian assuming \ufb01xed B and sub-Gaussian x\u2217. Particularly, our result in Theorem 2 does not assume that the entries of x\u2217are independent. Lemma 1. Let \u02c6 x be as de\ufb01ned in (7). Let x\u2217\u2208Rd be the \ufb01xed unknown vector and let B be a Gaussian measurement matrix. For t > 0, Pr B h\u2225x\u2217\u2212\u02c6 x\u22252 \u2225x\u2217\u22252 \u2265(1 + t) r d \u2212s d i \u22642 exp(\u2212c\u2032t2(d \u2212s)). (10) The result in (10) con\ufb01rms that the relative error decreases with increasing number of measurements s in the under-determined system. For example, let s = 3d/4, the probability that the relative error exceeds 1/2 decays with a sub-Gaussian tail. The bound in (10) is also sharp because it is the upper tail of the following two-sided inequality Pr B h (1 \u2212t) r d \u2212s d \u2264\u2225x\u2217\u2212\u02c6 x\u22252 \u2225x\u2217\u22252 \u2264(1 + t) r d \u2212s d i \u22651 \u22122 exp(\u2212c\u2032t2(d \u2212s)). Proof of Lemma 1. Since B \u2208Rn\u00d7d is assumed Gaussian, the scaled collapsed matrix e B/ \u221a \u2018r (6) is also a Gaussian matrix. The singular vectors of a rectangular Gaussian matrix M \u2208Rp\u00d7q, p > q, span a qdimensional uniformly random subspace of Rp, see Section 5.2.6 in [26]. Therefore, \u02c6 x = (I \u2212e V e V\u22ba)x\u2217is the projection of x\u2217\u2208Rd onto a uniformly random (d\u2212s)-dimensional subspace. The result in (10) follows from invoking the Johonson-Lindenstrauss (JL) Lemma (66) with p = d and q = d \u2212s. 4 \fTheorem 1. Let b X be as de\ufb01ned in (7). Let X\u2217\u2208Rd\u00d7m be the \ufb01xed unknown matrix and let B be a Gaussian measurement matrix. For log m \u2264c\u2032t2(d \u2212s)/2, Pr B h\u2225X\u2217\u2212b X\u2225F \u2225X\u2217\u2225F \u2265(1 + t) r d \u2212s d i \u22642 exp(\u2212c\u2032t2(d \u2212s)/2). (11) Proof of Theorem 1. Let event E = Si=m i=1 Ei be de\ufb01ned as the union of events Ei, where Ei = {x\u2217| \u2225x\u2217 i \u2212\u02c6 xi\u22252 2 \u2265(1 + t)2(d \u2212s) d \u2225x\u2217 i \u22252 2}. (12) Then, Pr h i=m X i=1 \u2225x\u2217 i \u2212\u02c6 xi\u22252 2 \u2264(1 + t)2(d \u2212s) d i=m X i=1 \u2225x\u2217 i \u22252 2 i (13) \u22651 \u2212 i=m X i=1 Pr[Ei] (14) \u22651 \u22122 exp(\u2212c\u2032t2(d \u2212s)/2). (15) (14) follows from noting that Pr[Ec] = 1 \u2212Pr[E] and applying the union bound to E. (15) is by substituting (10) and bounding m exp(\u2212c\u2032t2(d \u2212s)/2) \u22641 for log m \u2264c\u2032t2(d \u2212s)/2. Lemma 2. Let \u02c6 x be as de\ufb01ned in (7). Let B be a \ufb01xed measurement matrix and let x\u2217be a zero-mean sub-Gaussian random vector (31). For t \u22650, Pr x\u2217 \u0002 \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2265c1 + 2K2( \u221a d \u2212s + 1)t] \u2264exp(\u2212t), (16) where c1 = K2(d \u2212s + 1 2 \u221a d \u2212s) and K is as given in (31). Proof of Lemma 2. The result in (16) follows from applying the Hanson-Wright inequality (65) to (8). To derive the result in Theorem 2, it is necessary to show that \u2225x\u2217\u2212\u02c6 x\u22252 2 is a sub-exponential random variable. A random variable X is sub-exponential if for t \u22650, K1 > 0 X \u223csubExp \u21d0 \u21d2Pr[|X| \u2265t] \u2264c\u2032 exp(\u2212t/K1), (17) where c\u2032 \u22651. It cannot be concluded that the shifted random variable \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 is sub-exponential because the lower tail probability Pr[\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u2264\u2212t], t > 0 is not upper bounded in (16). However, our goal is to upper bound the probability that the error exceeds a certain value. We therefore de\ufb01ne the non-negative random variable \u02dc e \u2212c1, which upper bounds the lower tail as {\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u2264\u2212t, t > 0} \u22640, see (18). The de\ufb01nition of \u02dc e \u2212c1 is given in (18), (19). Fig. 2 is a visual description of the de\ufb01nition. We show that \u02dc e \u2212c1 is a sub-exponential random variable in Lemma 3. Lemma 3. The random variable \u02dc e \u2212c1, de\ufb01ned in (18), (19), is a sub-exponential random variable with sub-exponential norm \u2225\u02dc e \u2212c1\u2225\u03d51 = c\u2032K1, where \u02dc e \u2212c1 | E1 = 0, (18) \u02dc e \u2212c1 | E2 = \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c, (19) c = K2(d \u2212s + 1 2 \u221a d \u2212s), K1 = 2K2( \u221a d \u2212s + 1), c\u2032 is an absolute constant and events E1, E2 are E1 = {x\u2217| \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u22640}, (20) E2 = Ec 1 = {x\u2217| \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 > 0}. (21) 5 \f!c1 0 jjx$ ! ^ xjj2 2 ! c1 0 ~ e ! c1 ~ e ! c1 against jjx$ ! ^ xjj2 2 ! c1 -1 0 1 X ! c1 9 Unif[!1; 1] 0 0.5 1 Example of de-nition. c1 = 1 FX FY Figure 2: Left. The derived random variable (rv) \u02dc e \u2212c1 is plotted against the error rv \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1. Note that i) \u02dc e \u2212c1 \u22650, ii) \u02dc e \u2212c1 \u2265\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1, iii) \u02dc e \u2212c1 | E1 = 0 and iv) \u02dc e \u2212c1 | E2 = \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1. Right. Example. The cdfs of Y (\u02dc e \u2212c1) and X(\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1) \u223cUnif[\u22121, 1] are plotted. Proof of Lemma 3. Pr[\u02dc e \u2212c1 \u2265t] = Pr[\u02dc e \u2212c1 \u2265t | E1] \u00b7 Pr[E1] + Pr[\u02dc e \u2212c1 \u2265t | E2] \u00b7 Pr[E2] = Pr[\u02dc e \u2212c1 \u2265t | E2] \u00b7 Pr[E2] (22) = Pr[\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u2265t | E2] \u00b7 Pr[E2] (23) = Pr[\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u2265t], (24) where (22) follows from (18); speci\ufb01cally, Pr[\u02dc e\u2212c1 \u2265t | E1] = 0\u2200t > 0. (23) follows from (19) and (24) is by Pr[\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u2265t | E2] Pr[E2] = Pr[\u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u2265t \u2229E2] and that the event {x\u2217| \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 \u2265t, t > 0} is a subset of E2. From (16), for t \u22650 Pr h \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 2K2( \u221a d \u2212s + 1) \u2265t i \u2264exp(\u2212t). (25) From (24), for t > 0 Pr h \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 2K2( \u221a d \u2212s + 1) \u2265t i = Pr h \u02dc e \u2212c1 2K2( \u221a d \u2212s + 1) \u2265t i , (26) where \u221a d \u2212s + 1 > 0 is a positive number since s < d, see (8). From (18) and (19), \u02dc e \u2212c1 is non-negative so that \u02dc e \u2212c1 = |\u02dc e \u2212c1| and Pr h \u02dc e \u2212c1 2K2( \u221a d \u2212s + 1) \u2265t i = Pr h |\u02dc e \u2212c1| 2K2( \u221a d \u2212s + 1) \u2265t i . (27) Combining (26) and (27), for t > 0 Pr h |\u02dc e \u2212c1| 2K2( \u221a d \u2212s + 1) \u2265t i = Pr h \u2225x\u2217\u2212\u02c6 x\u22252 2 \u2212c1 2K2( \u221a d \u2212s + 1) \u2265t i . (28) Substituting (25) in (28), for t > 0, Pr h |\u02dc e \u2212c1| 2K2( \u221a d \u2212s + 1) \u2265t i \u2264exp(\u2212t). (29) Since exp(0) = 1, (29) also holds at t = 0 Pr h |\u02dc e \u2212c1| 2K2( \u221a d \u2212s + 1) \u2265t i \u2264exp(\u2212t) \u2200t \u22650. (30) In (30), we have veri\ufb01ed the de\ufb01nition in (17) for \u02dc e \u2212c1 with K1 = 2K2( \u221a d \u2212s + 1). By proposition 2.7.1 (a), (d) de\ufb01nition 2.7.5 in [26], and (30), the sub-exponential norm is \u2225\u02dc e \u2212c1\u2225\u03d51 = c\u2032K1. 6 \fTheorem 2. Let X\u2217\u2208Rd\u00d7m be the unknown matrix such that the columns x\u2217 i are identical, independent, zero-mean sub-Gaussian random vectors, i.e., for \u03b1 \u2208Rd, K \u22650 E[exp(\u03b1\u22bax\u2217 i )] \u2264exp(\u2225\u03b1\u22252 2K2/2). (31) Let b X be as de\ufb01ned in (7). Let B be a \ufb01xed measurement matrix. For t \u22650, Pr \u0002 i=m X i=1 \u2225x\u2217 i \u2212\u02c6 xi\u22252 \u2212mc2 \u2265t \u0003 \u22642 exp \u0000\u2212c\u2032t2 mK1 \u0001 , (32) where c2 = K(d \u2212s + 5 2 \u221a d \u2212s + 2) 1 2 , K1 = 2K2( \u221a d \u2212s + 1) and c\u2032 is an absolute constant. Proof of Theorem 2. Note that Pr[|\u02dc e \u2212E[\u02dc e]| \u2265t] = Pr[|\u02dc e \u2212c1 \u2212E[\u02dc e \u2212c1]| \u2265t] \u22642 exp(\u2212c\u2032t/\u2225\u02dc e \u2212c1 \u2212E[\u02dc e \u2212c1]\u2225\u03d51). (33) Since \u2225\u02dc e \u2212c1 \u2212E[\u02dc e \u2212c1]\u2225\u03d51 = c\u2032\u2225\u02dc e \u2212c1\u2225\u03d51 (by 2.7.10 in [26]) and \u2225\u02dc e \u2212c1\u2225\u03d51 = c\u2032K1 (Lemma 3), Pr[|\u02dc e \u2212E[\u02dc e]| \u2265t] \u22642 exp(\u2212c\u2032t/K1), (34) where K1 is as in (30). For z \u22650, |z \u22121| \u2265\u03b4 = \u21d2|z2 \u22121| \u2265max(\u03b4, \u03b42). Therefore, Pr h\f \f \f \u221a \u02dc e \u2212 p E[\u02dc e] p E[\u02dc e] \f \f \f \u2265\u03b4 i \u2264Pr h\f \f \f\u02dc e \u2212E[\u02dc e] E[\u02dc e] \f \f \f \u2265max(\u03b4, \u03b42) i \u2264Pr h\f \f \f\u02dc e \u2212E[\u02dc e] E[\u02dc e] \f \f \f \u2265\u03b42i (35) \u22642 exp(\u2212c\u2032E[\u02dc e]\u03b42/K1), (36) where (36) follows by substituting t = \u03b42 in (34). Changing variables t = \u03b4 p E[\u02dc e] in (36), Pr \u0002\f \f\u221a \u02dc e \u2212 p E[\u02dc e] \f \f \u2265t \u0003 \u22642 exp(\u2212c\u2032t2/K1). (37) From (37) and proposition 2.5.2 (i), (iv) and de\ufb01nition 2.5.6 in [26], the sub-Gaussian norm squared is bounded as \u2225 \u221a \u02dc e \u2212 p E[\u02dc e]\u22252 \u03d52 = c\u2032K1. (38) Applying Hoe\ufb00ding\u2019s inequality (67) to \u221a \u02dc e \u2212 p E[\u02dc e], Pr \u0002 i=m X i=1 \u0000p \u02dc ei \u2212 p E[\u02dc ei] \u2212E[ p \u02dc ei \u2212 p E[\u02dc ei]] \u0001 \u2265t \u0003 \u22642 exp \u0000\u2212c\u2032t2 mK1 \u0001 . (39) Substituting E[ \u221a \u02dc e] \u2212 p E[\u02dc e] \u22640 (Jensen\u2019s inequality) in (39), Pr \u0002 i=m X i=1 p \u02dc ei \u2212m p E[\u02dc e] \u2265t \u0003 \u22642 exp \u0000\u2212c\u2032t2 mK1 \u0001 . (40) For non-negative random variable X, E[X] = R \u221e 0 Pr[X > t]dt. Applied to \u02dc e \u2212c1/2K2( \u221a d \u2212s + 1), E h \u02dc e \u2212c1 2K2( \u221a d \u2212s + 1) i \u2264 Z \u221e 0 exp(\u2212t)dt = 1 (41) In (41), we have substituted the tail probability upper bounds from (30). Substituting c1 from (16) in (41), E[\u02dc e] \u2264K2(d \u2212s + 5 2 \u221a d \u2212s + 2). (42) 7 \fBy de\ufb01nition in (18), (19) (also see Fig. 2), \u2225x\u2217 i \u2212\u02c6 xi\u22252 \u2264\u221a\u02dc ei \u2200i \u2208[m]. Substituting (42) and using \u2225x\u2217\u2212\u02c6 x\u22252 \u2264 \u221a \u02dc e in (40) completes the proof. Remark. Assuming Gaussian B, Theorem 1 shows that the probability that the relative error in (7) exceeds p (d \u2212s)/d decays at a sub-Gaussian rate. Assuming sub-Gaussian x\u2217instead, Theorem 2 shows that the probability that the error exceeds mc2, where c2 as in (32), also decays at a sub-Gaussian rate. Both results agree that the estimate (7) improves as the number of measurements s in the under-determined system (6) increases. The quantity K in (32) depends linearly on the sub-Gaussian norm of x\u2217and can be estimated in practice. For x\u2217\u223cN(0, I), K = 1. For x\u2217uniformly distributed on the Euclidean ball, K = c\u2032, where c\u2032 is an absolute constant. 3.2 k-sparse model For the standard (un-permuted) linear inverse problem y = Bx\u2217+ w, the error in the solution \u2225x\u2217\u2212\u02c6 x\u22252 is upper bounded in terms of the noise \u2225w\u22252 and \u03c3min(B), the smallest singular value of B. In this section, we bound the error in the proposed initialization for the k-sparse permutation model in terms of the value of the objective function F, \u03c3min(B) and the number of shu\ufb04es k. Lemma 4. Let P\u2217 k be the \ufb01xed unknown k-sparse permutation matrix with \u27e8I, P\u2217 k\u27e9= n \u2212k. Let x\u2217be the \ufb01xed unknown vector. Assuming Gaussian B, let y\u2217= Bx\u2217, \u02c6 y(0) = P\u2217 kBx\u2217. For k = 1, \u00b7 \u00b7 \u00b7 , n \u22121 and t \u22650, Pr \u0002 \u2225y\u2217\u2212\u02c6 y(0)\u22252 2 \u22652\u2225y\u2217\u22252 2 \u22122\u2225x\u2217\u22252 2 \u0000n \u2212k \u2212c3 \u221a t \u22123t \u0001\u0003 \u22647 exp(\u2212t), (43) where c3 = 2 \u221a n \u2212k + 2 \u221a 3k. Proof of Lemma 4. Under the k-sparse assumption on P\u2217, the known vector y has k shu\ufb04ed entries. For b P(0) = I, i.e., \u02c6 y(0) = y, the forward error is \u2225y\u2217\u2212\u02c6 y(0)\u22252 2 = 2\u2225y\u2217\u22252 2 \u22122\u27e8y\u2217, \u02c6 y(0)\u27e9, (44) where y\u2217= Bx\u2217is the unknown vector of unshu\ufb04ed measurements. Since \u2225y\u2217\u22252 = \u2225\u02c6 y(0)\u22252 and initialization \u02c6 y(0) = P\u2217 kBx\u2217is known, the only unknown term in (44) is the inner product \u27e8y\u2217, \u02c6 y(0)\u27e9. The scaled inner product \u27e8y\u2217, \u02c6 y(0)\u27e9/\u2225x\u2217\u22252 2 is expanded as \u27e8y\u2217, \u02c6 y(0)\u27e9 \u2225x\u2217\u22252 2 = 1 \u2225x\u2217\u22252 2 i=n X i=1 b\u22ba i x\u2217b\u22ba P\u2217(i)x\u2217 (45) = 1 \u2225x\u2217\u22252 2 i=n\u2212k X i=1 (b\u22ba i x\u2217)2 | {z } \u225cT1 + 1 \u2225x\u2217\u22252 2 n X i=n\u2212k+1 b\u22ba i x\u2217b\u22ba P\u2217(i)x\u2217 | {z } \u225cT2 . We have assumed in (45), wlog, that the last k rows of P\u2217 k are shu\ufb04ed. Assuming Gaussian B, (b\u22ba i x\u2217/\u2225x\u2217\u22252)2 \u223c \u03c72 is Chi-squared distributed with 1 degree of freedom. T1, de\ufb01ned in (45), is the sum of n \u2212k independent Chi-square random variables and is bounded, using (69), as Pr \u0002 T1 \u2264n \u2212k \u22122 p (n \u2212k)t \u0003 \u2264exp(\u2212t). (46) The product random variables in T2 are distributed as the di\ufb00erence of two independent \u03c72 random variables. To see this, let X = b\u22ba i x\u2217/\u2225x\u2217\u22252 , Y = b\u22ba P(i)x\u2217/\u2225x\u2217\u22252. (47) The random variables X and Y in (47) are independent normal random variables. The product random variable XY is expressed as XY = 1 2 \u0000X + Y \u221a 2 \u00012 \u22121 2 \u0000X \u2212Y \u221a 2 \u00012. (48) 8 \fThe random variables (X + Y )/ \u221a 2 \u223cN(0, 1) and (X \u2212Y )/ \u221a 2 \u223cN(0, 1) are jointly Gaussian and uncorrelated. Therefore, (X + Y )/ \u221a 2 is independent of (X \u2212Y )/ \u221a 2 and the product XY is distributed as the di\ufb00erence of two independent \u03c72 distributed random variables Z1 i , Z2 i b\u22ba i x\u2217b\u22ba P(i)x\u2217 \u2225x\u2217\u22252 2 \u223c1 2Z1 i \u22121 2Z2 i \u2200i \u2208n \u2212k + 1, \u00b7 \u00b7 \u00b7 , n. (49) The random variables (rv) in (49) are not mutually independent, but each rv depends on, at most, two other rvs. To see this, let permutation P such that P(i) 7\u2192j, then b\u22ba i x\u2217b\u22ba j x\u2217is not independent of b\u22ba j x\u2217b\u22ba P(j)x\u2217, b\u22ba P\u22ba(i)x\u2217b\u22ba i x\u2217. (50) The k rvs in (49) can therefore be partitioned into three sets P, Q, R such that the rvs within each set are independent. Let k1 be the number of rvs in set P. The sum TP , where TP \u225c 1 \u2225x\u2217\u22252 2 X i\u2208P b\u22ba i x\u2217b\u22ba P(i)x\u2217= 1 2 X i\u2208P Z1 i \u22121 2 X i\u2208P Z2 i , (51) is upper bounded in probability as Pr[TP \u2264\u22122 p k1t \u2212t] \u22642 exp(\u2212t). (52) (52) follows from applying the union bound to probabilities p1, p2 p1 = Pr \u0002 X i\u2208P Z1 i \u2264k1 \u22122 p k1t \u0003 \u2264exp(\u2212t), (53) p2 = Pr \u0002 X i\u2208P Z2 i \u2265k1 + 2 p k1t + 2t \u0003 \u2264exp(\u2212t), (54) and bounding p1, p2 using tail inequalities (69), (68), respectively. De\ufb01ning TQ, TR similarly to (51), with cardinalities k2, k3 and applying the union bound as in (53), (54) gives Pr \u0002 T2 \u2264\u22122( p k1t + p k2t + p k3t) \u22123t \u0003 \u22646 exp(\u2212t), (55) where T2 = TP + TQ + TR. Since \u2225[\u221ak1 \u221ak2 \u221ak3]\u22ba\u22252 1 \u22643\u2225\u00b7\u22252 2 = 3k, \u221ak1 + \u221ak2 + \u221ak3 \u2264 \u221a 3k. Substituting in (55), Pr \u0002 T2 \u2264\u22122 \u221a 3kt \u22123t \u0003 \u22646 exp(\u2212t). (56) Applying the union bound to (46), (56) gives the result in (43). Lemma 5. For the same assumptions as in Lemma 4, k = 1, \u00b7 \u00b7 \u00b7 , n \u22121 and t \u22650, Pr \u0002 \u03c32 min(B)\u2225x\u2217\u2212\u02c6 x(1)\u22252 2 \u22652\u2225y\u2217\u22252 2 \u22122\u2225x\u2217\u22252 2 \u0000n \u2212k \u2212c3 \u221a t \u22123t \u0001 \u2212F (1)\u0003 \u22647 exp(\u2212t), (57) where \u02c6 x(1) = B\u2020\u02c6 y(0), F (1) = \u2225y \u2212B\u02c6 x(1)\u22252 2 and c3 = 2 \u221a n \u2212k + 2 \u221a 3k. Proof of Lemma 5. Let e denote the error term such that \u02c6 y(0) = B\u02c6 x(1) + e, (58) where \u02c6 x(1) = B\u2020\u02c6 y(0), \u02c6 y(0) = P\u2217 kBx\u2217and e \u22a5R(B) is orthogonal to the range space of B because \u02c6 x(1) = argmin x\u2225\u02c6 y(0) \u2212Bx\u22252. Substituting y\u2217= Bx\u2217and (58) below, \u2225y\u2217\u2212\u02c6 y(0)\u22252 2 = \u2225B(x\u2217\u2212\u02c6 x(1)) \u2212e\u22252 2. (59) Using Pythogras\u2019 theorem, \u2225B(x\u2217\u2212\u02c6 x(1)) \u2212e\u22252 2 = \u2225B(x\u2217\u2212\u02c6 x(1))\u22252 2 + \u2225e\u22252 2. (60) Substituting the lower bound \u03c32 min(B)\u2225x\u2217\u2212\u02c6 x(1)\u22252 2 \u2264\u2225B(x\u2217\u2212\u02c6 x(1))\u22252 2 in (60), \u03c32 min(B)\u2225x\u2217\u2212\u02c6 x(1)\u22252 2 \u2264\u2225y\u2217\u2212\u02c6 y(0)\u22252 2 \u2212\u2225e\u22252 2. (61) (62) follows by substituting the probability upper bound for \u2225y\u2217\u2212\u02c6 y(0)\u22252 2 from (43) into (61) and noting that \u2225e\u22252 2 = F (1) = F(b P(0), \u02c6 x(1)). 9 \f10  20  50  100 125 200 250 block size r 0 0.2 0.4 0.6 0.8 1 dH=n P$ r n = 1000 m = 50 d = 100 B 9 N(0; 1) Spectral Biconvex RLUS `2-regularized Proposed (a) 600 650 700 750 800 850 900 number of shu0es k 0 0.2 0.4 0.6 0.8 1 P$ k n = 1000 m = 50 d = 100B 9 N(0; 1) Spectral RLUS `2-regularized Proposed (b) 600 650 700 750 800 825 850 875 900 number of shu0es k 0 0.2 0.4 0.6 0.8 1 P$ k n = 1000 m = 50 d = 100 B 9 Unif Spectral RLUS `2-regularized Proposed (c) 100 105 110 115 120 125 130 135 140 145 150 number of shu0es k 0 0.2 0.4 0.6 0.8 1 P$ k n = 200 m = 10 d = 20B 9 N(0; 1) Spectral RLUS `2-regularized ds+ Proposed (d) 200 250 300 350 400 500 550 600 650 700 750 800 number of shu0es k 2 4 6 8 10 12 14 16 estimation error jjx$ ! ^ xjj2 P$ k n = 1000 m = 1 d = 100B 9 N(0; 1) Alt-min mulitple init Stochastic Alt-min Proposed (e) Figure 3: Synthetic simulations. Y = P\u2217Bn\u00d7dX\u2217 d\u00d7m + W. The entries of X\u2217are drawn from the normal distribution. The permutation matrix P\u2217 r(P\u2217 k) is sampled uniformly from the set of r-local (k-sparse) permutations. The results are averaged over 15 Monte-Carlo runs. (a). Figure plots the fractional Hamming distortion dH/n against block size r. (b), (c), (d). Figure plots dH/n against number of shu\ufb04es k. (e). Figure plots the estimation error against k. Theorem 3. Let P\u2217 k be the \ufb01xed unknown permutation matrix with \u27e8I, P\u2217 k\u27e9= n \u2212k. Let X\u2217be the \ufb01xed unknown matrix. Assuming Gaussian B, let Y\u2217= BX\u2217and b Y(0) = P\u2217 kBX\u2217. For k = 1, \u00b7 \u00b7 \u00b7 , n \u22121 and t \u2265log m2, Pr \u0002 \u03c32 min(B)\u2225X\u2217\u2212b X(1)\u22252 F \u22652\u2225Y\u2217\u22252 F \u22122\u2225X\u2217\u22252 F \u0000n \u2212k \u2212c3 \u221a t \u22123t \u0001 \u2212F (1)\u0003 \u22647 exp(\u2212t), (62) where b X(1) = B\u2020 b Y(0), F (1) = \u2225Y \u2212B b X(1)\u22252 F and c3 = 2 \u221a n \u2212k + 2 \u221a 3k. Proof of Theorem 3. The proof follows by applying the union bound to (57). An immediate upper bound that does not depend on the number of diagonal entries of P\u2217 k is \u03c32 min(B)\u2225X\u2217\u2212b X(1)\u22252 F \u22644\u2225Y\u2217\u22252 F \u2212F (1). (63) (62) shows that the probability that the error exceeds \u2225Y\u2217\u22252 F \u2212c, c = \u2126(n \u2212k) and n \u2212k is the number of diagonal entries in P\u2217 k, decays exponentially. Furthermore, the term \u2225Y\u2217\u22252 F is known and \u2225X\u2217\u2225F can be lower bounded in terms of known quantities as \u2225X\u2217\u22252 F \u2264\u2225Y\u2217\u22252 F /\u03c32 max(B). 4 Results MATLAB code and datasets are at the \ufb01rst author\u2019s GitHub 1. Baselines. We compare against six baseline methods. We refer to these methods as \u2018\u21132-regularized\u2019 [5],\u2018ds+\u2019 [5], \u2018Spectral\u2019 [17], \u2018Biconvex\u2019 [4], \u2018RLUS\u2019 [10], and \u2018Stochastic alt-min\u2019 [15]. The \u2018\u21132-regularized\u2019 1https://github.com/aabbas02/ksparse-and-rlocal 10 \fDataset n d m rmax s Oracle Naive RLUS \u21132-reg alt-min-k alt-min-r ftp 335 30 6 43 46 (0.88, 0) (0.44, 0.70) (0.62, 0.01) (0.57, 0.59) (0.73, 0.41) (0.85, 0.17) scm 8966 35 16 424 1137 (0.58, 0) (0.21, 0.79) (0.52, 0.29) (0.46, 0.49) (0.54, 0.22) (0.55, 0.22) scs 44484 10 6 3553 356 (0.81, 0) (0.01, 0.99) (0.74, 0.24) (0.02, 0.98) (0.73, 0.33) (0.77, 0.21) air-qty 27605 27 16 95 366 (0.69, 0) (0.29, 0.78) (0.65, 0.20) (0.64, 0.21) (0.66, 0.17) (0.65, 0.19) air-qty 27605 27 16 46 744 (0.69, 0) (0.10, 0.94) (0.60, 0.34) (0.26, 0.79) (0.57, 0.42) (0.62, 0.32) Table 1: rmax denotes the largest block size of P\u2217 r and s denotes the number of blocks. The methods are compared based on the coe\ufb03cient of determination R2 and relative error (R2, relative error). The relative error is \u2225X\u2217\u2212 b X\u2225F /\u2225X\u2217\u2225F , where X\u2217= B\u2020Y\u2217is the \u2018oracle\u2019 regression matrix given unpermuted data Y\u2217. The \u2018naive\u2019 estimate from permuted data Y is b X = B\u2020Y. The coe\ufb03cient R2( b X) = 1 \u2212(\u2225Y\u2217\u2212B b X\u2225F /\u2225Y\u2217\u2225F ) measures the goodness of \ufb01t for the unpermuted data. method considers the k-sparse permutation model and incorporates the model assumption by imposing a row-wise group sparse penalty on \u2225b Yi,:\u22252, where b Y = B b X. Other methods are discussed in the following paragraphs. We set \u03f5 = 0.01 in Algorithm 1. Figs. 3a, 3b. We compare our algorithm for the r-local and k-sparse permutation models to baselines. To adapt the \u2018Spectral\u2019, \u2018\u21132-regularized\u2019 and \u2018Biconvex\u2019 methods to the r-local model, we add a constraint enforcing the permutation estimates to be block-diagonal. The results show that our algorithm recovers P\u2217 with decreasing block size r and number of shu\ufb04es k. This observation con\ufb01rms the conclusions of Theorems 1,2,3 as the initialization to our algorithm improves with lower values of r and k. The results also show that our algorithm is applicable to both models, unlike baselines. Our algorithm is also computationally scalable. For r = 125 in Fig. 3a and k = 850 in Fig. 3b, the MATLAB run-times, with 16 Gb RAM, are less than one second. Fig. 3c. The entries of the measurement matrix B are sampled i.i.d. from the uniform [0, 1] distribution. Compared to the case for Gaussian B (Fig. 3b), the performance of the \u2018Spectral\u2019 and \u2018RLUS\u2019 methods deteriorates signi\ufb01cantly. This is because both algorithms consider quadratic measurements YY\u22ba. Speci\ufb01cally, the \u2018Spectral\u2019 method is based on the spectral initialization technique [27], which assumes that the measurement matrix is Gaussian. In contrast, the performance of the proposed algorithm and the \u2018\u21132-regularized\u2019 method does not deteriorate. Fig. 3d. The \u2018ds+\u2019 algorithm [5] considers the convex relaxation of (2) by minimizing the objective over the set of doubly stochastic matrices. Assuming an upper bound on the number of shu\ufb04es k is known, \u2018ds+\u2019 also constrains \u27e8I, P\u27e9\u2265n \u2212k. Each iteration of \u2018ds+\u2019 minimizes a linear program, greatly increasing the run-time of the algorithm. The results show that the proposed algorithm outperforms \u2018ds+\u2019, possibly because, for our algorithm, the objective is optimized directly over the set of permutations. Fig. 3e. We compare to the method in [15] which considers the m = 1 single-view setup and proposes stochastic alternating minimization (S.alt-min) to optimize (2). The run-time for S.alt-min is 50 times the run-time of the proposed algorithm because S.alt-min updates P multiple times in each iteration and retains the best update. [15] also proposes alt-min with multiple initializations for b P(0). The results in Fig. 3e show that our algorithm (alt-min with b P(0) = I initialization) outperforms both S.alt-min and alt-min with multiple initializations. Real data. For the linked linear regression problem introduced in Section 1.2, we report results in Table 1 on the four datasets from [5]. The measurement matrix and the regression model are also de\ufb01ned as in [5]. For \u2018ftp\u2019,\u2018scm\u2019 and \u2018scs\u2019, the values of a feature (or column of the measurement matrix) are rounded and data points with the same feature value are assigned to the same block and permuted. The \u2018air-qty\u2019 dataset comprises time-stamped readings with year, month, day and hour information. In row 4 (5) of Table 1, readings with the same month and day (day and hour) are assigned to the same block and permuted, as also done in [5]. The results show that the proposed alt-min algorithm outperforms the competing baselines. \u2018alt-min-k\u2019, i.e. alt-min initialized to b P(0) = I, is also competitive, possibly because permuting correlated rows may not greatly corrupt the measurement matrix B. 11 \f5"
+                }
+            ],
+            "Weiwen Jiang": [
+                {
+                    "url": "http://arxiv.org/abs/2311.12333v1",
+                    "title": "QuGeo: An End-to-end Quantum Learning Framework for Geoscience -- A Case Study on Full-Waveform Inversion",
+                    "abstract": "The rapid advancement of quantum computing has generated considerable\nanticipation for its transformative potential. However, harnessing its full\npotential relies on identifying \"killer applications\". In this regard, QuGeo\nemerges as a groundbreaking quantum learning framework, poised to become a key\napplication in geoscience, particularly for Full-Waveform Inversion (FWI). This\nframework integrates variational quantum circuits with geoscience, representing\na novel fusion of quantum computing and geophysical analysis. This synergy\nunlocks quantum computing's potential within geoscience. It addresses the\ncritical need for physics-guided data scaling, ensuring high-performance\ngeoscientific analyses aligned with core physical principles. Furthermore,\nQuGeo's introduction of a quantum circuit custom-designed for FWI highlights\nthe critical importance of application-specific circuit design for quantum\ncomputing. In the OpenFWI's FlatVelA dataset experiments, the variational\nquantum circuit from QuGeo, with only 576 parameters, achieved significant\nimprovement in performance. It reached a Structural Similarity Image Metric\n(SSIM) score of 0.905 between the ground truth and the output velocity map.\nThis is a notable enhancement from the baseline design's SSIM score of 0.800,\nwhich was achieved without the incorporation of physics knowledge.",
+                    "authors": "Weiwen Jiang, Youzuo Lin",
+                    "published": "2023-11-21",
+                    "updated": "2023-11-21",
+                    "primary_cat": "quant-ph",
+                    "cats": [
+                        "quant-ph"
+                    ],
+                    "main_content": "INTRODUCTION Quantum computing\u2019s integration with machine learning is a rapidly expanding \ufb01eld, though it is still in search of groundbreaking applications to fully showcase its capabilities. In geophysical inversion, critical for areas like civil infrastructure and energy exploration, recent advancements have applied quantum annealing to seismic inversion problems. These e\ufb00orts, however, primarily tackle linearized versions of the problem. For instance, Souza\u2019s approach reformulates seismic inversion into linear equations and QUBO formulations [1], and Greer develops a QUBO-compatible linearized inversion for dual velocity values, bothbased on traditional seismic techniques [2]. In contrast, our work introduces the \ufb01rst learningbased seismic inversion technique on general-purpose quantum computing platforms, marking a novel direction in this domain. Characterizing subsurface geology is essential for a range of applications, from earthquake research and civil infrastructure to new energy exploration and environmental studies. Seismic inversion, which reconstructs subsurface images from seismic waves, uses full-waveform inversion (FWI) to consider complete waveform data, including amplitude and phase. FWI, addressing nonlinear challenges, o\ufb00ers enhanced accuracy and resolution compared to linear methods. Currently, FWI is approached through physics-based and machine learning-based methods. While physicsbased methods are precise, they face computational challenges and issues like ill-posedness and cycle skipping. Conversely, machine learning-based methods, designed to address these issues, are promising but heavily dependent on extensive training data, which can result in high computational demands. What\u2019s more, seismic waves continuously received by sensors from di\ufb00erent locations have a high correlation on both spatial and temporal. Our study introduces quantum computing as a novel solution to reduce the computational burden in data-driven FWI and leverage fundamental quantummechanics, particularly the entanglement, to extract highly correlated features, showcasing its potential to enhance both the e\ufb03ciency and e\ufb00ectiveness of machine learning in seismic inversion. Unique characteristics and structures of geophysical problems, however, set challenges in designing quantum learning for FWI. Firstly, there is a scarcity of standard datasets tailored to current quantumcapabilities; careless design could result in data misaligned with physical realities, a\ufb00ecting performance. Secondly, a dedicated quantum design framework is lacking; straightforwardly using the quantum learning algorithm (a.k.a., Variational Quantum Circuit or VQC) can easily be suboptimal or ine\ufb03cient because it cannot exploit the speci\ufb01c properties inherent in geophysics to optimize the design. Finally, the most e\ufb00ective ways to harness the full potential of quantum computing in this context remain unclear, requiring further exploration and innovation in the \ufb01eld. In our work, we introduce QuGeo, an innovative quantum learning framework tailoredto address seismic FWI challenges. To tackle the mentioned hurdles, \ufb01rstly, we have created a physics-informed dataset via a governing wave equation, on top of which we further developed a classical machine learning-based data converter to perform data scaling according to the quantum resource constraints e\ufb03ciently. Secondly, we developed an application-speci\ufb01c VQC, namely QuGeoVQC, to incorporate domain-speci\ufb01c knowledge in optimizing the design. QuGeoVQC explores the designs of a data encoder and VQC computing structure to extract spatial and temporal features; it also exploits characteristics of FWI problem to simplify and optimize the encoder to boost the performance. Furthermore, we present a novel data batching technique adapted for quantum computing, which enables quantum computers to process \ud441batches of data in parallel with only \ud459\ud45c\ud454\ud441additional qubits, further catering to the high computational demands from learningbased FWI and advancing the potential of QuGeo in seismic inversion. The main contributions of this paper are as follows: \fConference\u201917, July 2017, Washington, DC, USA W. Jiang and Y. Lin km/s depth (km) o\ufb00set (m) 0 0 surface surface receivers r1 r1 r2 r2 r3 r3 2 4 6 700 4000 3500 time o\ufb00set (m) 0 0 1024 700 o\ufb00set (m) 0 700 o\ufb00set (m) 0 700 (a) Flat rock layers w/ seismic receivers (c) Subsurface velocity map (b) Received seismic data Figure 1: Illustration of the target geophysics application: (a) the photo of a \ufb02at rock layer; (b) the seismic waveform data obtained from the receivers placed on the surface; and (c) the velocity map used to characterize the subsurface structure. \u2022 To the best of our knowledge, this is the very \ufb01rst pilot work in exploring the capacity of general-purpose quantum computing for seismic inversion problems, showcasing the potential of quantum computing for geoscienti\ufb01c analyses. \u2022 We unveil the essential need for a physics-informed dataset to ensure data is aligned with physical realities while simultaneously satisfying the resource constraint in current quantum computing platforms. \u2022 We develop an application-speci\ufb01c variational quantum circuit to leverage domain-speci\ufb01c knowledge in optimizing the design, which is equipped with a novel data batch engine to underpin the high computational demand to process extensive data for learning-based seismic inversion. To evaluate the e\ufb00ectiveness of ourproposedQuGeo framework, we implemented the proposed quantum learning innovations on TorchQuantum platform [3] and conducted tests using the widely recognized OpenFWI dataset from the geophysics community [4]. These tests aimed to validate the integration of physics knowledge and assess the performance of our custom-designed QuGeoVQC module. Our \ufb01ndings demonstrate that the incorporationof physics knowledge in QuGeo not only leads to high prediction accuracy, as re\ufb02ected in the SSIM values, but also results in a remarkably e\ufb03cient learning model utilizing just 576 parameters. This paper is structured as follows: Section 2 presents the background. In Section 3, we detail the QuGeo Framework. Evaluation results and concluding remarks are presented in Sections 4-5. 2 BACKGROUND AND RELATED WORK Data-driven Quantum Learning. The core component of a typical quantum learning framework is a parameterized quantum circuit (a.k.a., variational quantum circuit or ansatz circuit). Like classical machine learning, there are two steps in quantum learning: forward propagation and backward propagation. The forward propagation only involves quantum computing, where data will be encoded, processed, and measured in the quantum circuit. The backward propagation further involves classical computing to calculate the loss according to the measurement results and label, which will then be used to update the parameters in VQC. The optimization framework is to \ufb01nd the optimal parameters in quantum circuits so that the cost function of a certain task can be minimized. Quantum learning has a wide of applications, such as Quantum Approximate Optimization Algorithm (QAOA) for combinatorial optimization problems [5], Variational Quantum Eigensolver (VQE) for quantum chemistry and quantum simulations [6], and Quantum Neural Networks (QNN) for machine learning tasks [3, 7\u201311]. Compared with classical computing, quantum learning gets bene\ufb01ts from entanglement, which has been proven to be scalable to learn from the exponentially increasing size of data [12]. Geoscience application \u2014 Full-Waveform Inversion (FWI). Subsurface structures are typically layer-structured; an example is the \ufb02at rock layers, as illustrated in Figure 1(a). Understanding subsurface structures is a multidisciplinary e\ufb00ort with broad implications. In the \ufb01eld of new and renewable energy exploration, accurately characterizing subsurface structures is crucial for e\ufb00ectively locating energy sources below the Earth\u2019s surface. Note that it not only contributes to resource exploration and extraction but also plays a vital role in addressing environmental challenges, ensuring infrastructure stability, and advancing our knowledge of Earth\u2019s geological processes. Seismic data and velocity maps are two key elements to describe subsurface structures, discussed below. Seismic data, or seismic waveforms, are composed of waves that travel through the Earth\u2019s subsurface. These waves are often generated by controlled explosions or by striking the ground with heavy weights. Captured by geophones or receivers, these waves re\ufb02ect varying subsurface layers. The collected data is then processed to create detailed waveforms in a shot gather, as illustrated in Figure 1b. From this shot-gather waveform data, a velocity map can be derived, indicating the speed at which seismic waves travel through di\ufb00erent subsurface media, as shown in Figure 1c. Since rock properties a\ufb00ect wave speed, the velocity map is essential in identifying and characterizing subsurface geological structures, such as rock layers, faults, and reservoirs. For instance, the example in Figure 1c displays a subsurface structure with \ufb01ve distinct layers, each representing di\ufb00erent rock properties and depths. Building on those concepts, we introduce Full-Waveform Inversion (FWI), a sophisticated seismic imaging technique widely used in geophysics for creating detailed and accurate subsurface models. FWI aims to minimize the discrepancy between observed and simulated seismic waveforms. Applied extensively in both academic and engineering [13], FWI essentially predicts the subsurface velocity map from seismic waveforms recorded at surface receivers. Traditionally, FWI relies on physics-based simulations that are computationally intensive. The concept of data-driven FWI, introduced in recent years [14], has led to its growing success. In this work, we pilot a data-driven quantum learning for Geoscience FWI, harnessing the power of quantum computing to enhance the FWI. 3 QuGeo FRAMEWORK Figure 2 shows the proposed QuGeo framework, which is composed of three components:\u2460QuGeoDatawill consider the quantum device\u2019s capacity and scale the seismic wave data appropriately; \u2461QuGeoVQC is a computation engine which is key to achieving practical usage of near-term noisy quantum computers; and \u2462QuBatch is a performance booster that provides the ability to process data in parallel to unleash the power of quantum computing. \fQGeo: Qantum Learning for Geoscience Conference\u201917, July 2017, Washington, DC, USA depth (km) time o\ufb00set (m) 0 0 2 4 6 700 350 o\ufb00set (m) num of qubits 0 0 200 400 600 800 700 350 time o\ufb00set (m) 0 0 200 400 600 800 700 350 Training data: seismic wave & velocity map QuGeoData 1 Test seismic data 1000 \u00b4 700 standard phy-guide 32 \u00b4 8 8 \u00b4 8 8 \u00b4 8 QuGeoVQC 2 Encoder Decoder VQC # groups # layers pixel vs. layer # batches 3 QuBatch: Data Batch on Quantum Circuits Figure 2: Illustration of the proposed QuGeo framework. 3.1 QuGeoData: Physics-Guided Data Scaling 3.1.1 Scaling data with velocity map. We have pairs of seismic wave and velocity maps in the training dataset. Instead of directly scaling seismic data, we propose to employ Forward Modeling to generate the seismic wave data from downsampled velocity maps. The governing equation of the acoustic wave propagation in a 2dimension isotropic medium with a constant density is as follows: \u22072\ud45d\u22121 \ud4502 \ud7152\ud45d \ud715\ud4612 = \ud460 (1) where \u22072 = \ud7152 \ud715\ud4652 + \ud7152 \ud715\ud4662 , \ud450is velocity map, \ud45dis pressure \ufb01eld and \ud460is source term. Velocity map \ud450depends on the spatial location (\ud465,\ud466) while the pressure \ufb01eld \ud45dand the source term \ud460depend on the spatial location and time (\ud465,\ud466) and \ud461. We will follow the Forward Modeling algorithm [15]. The seismic data is simulated using \ufb01nite di\ufb00erence methods with the absorbing boundary condition. 3.1.2 Scaling data without velocity map. When the QuGeo is applied to real-world applications, we do not have the subsurface velocity map, which requires the scaling of data without using the velocity map. In addition, it has the requirement of high e\ufb03ciency in performing data scaling. To address these issues, we propose an ML-based data scaling approach to train a convolution neural network (CNN) for data compression. Our hypnosis is that physicsrelated features can be learned from the feature extraction by CNN. The key step is to build the dataset. Here, we have the original seismic wave data (\ud437), which has the dimension of 1000 \u00d7 700 as in the example from Figure 2. We also have the physics-guided scaled data (\ud45d\u210e\ud466\ud437), which has the dimension of 32 in the example. The pair of data \u27e8\ud437, \ud45d\u210e\ud466\ud437\u27e9will be built as the training dataset. On top of the dataset, we designed a LeNet-like CNN to perform data compression, which contains two convolutional layers (including a ReLU function after the convolution operation) and a fully connected (FC) layer. Although simple, it is quite e\ufb00ective to perform the data compression, as will be shown in the experimental results. 3.2 QuGeoVQC: Quantum Circuit Design 3.2.1 Encoderof Multi-source Seismic Data. The encoding of seismic wave data to qubits is based on a state-of-the-art quantum encoder from [16], called spatial-temporal encoder \u201cST-Encoder\u201d. It was originally designed for natural images, encoding a group of data (i.e., spatial-close \ud441data) to amplitudes of \ud459\ud45c\ud4542\ud441qubits. To use ST-Encoder in QuGeoVQC, we need to identify how to map data to the amplitudes of a set of qubits. Seismic data has three dimensions, including (1) sources, which generate waves from the surface at di\ufb00erent locations; (2) receivers, which receive waves on the surface; and (3) elapsed time, which shows the wave pressure received over time. As shown in Eq. 1, one source represents an independent event (say vibration from vibroseis trucks), and each of them can be calculated by the di\ufb00erential of pressure on time and location. Therefore, to better extract features from di\ufb00erent sources, we group and encode data from the same source into qubits. 3.2.2 Variational Qantum Circuit. We will follow ST-VQC [16] to create sub-VQCs to independently process data in each group, and gradually commute between groups by using multi-qubit gates across di\ufb00erent sub-VQCs. There are a couple of design hyperparameters that need to be determined. In each sub-VQC, it is necessary to determine how many layers of VQC are needed, which re\ufb02ects the number of trainable parameters. In addition, the order of entanglement between di\ufb00erent groups needs to be identi\ufb01ed. 3.2.3 Decoder of Velocity Maps. In carrying out the waveform inversion task, we observe the simpli\ufb01cation of the decoding circuit can signi\ufb01cantly improve performance, which takes bene\ufb01ts of the task\u2019s property. The decoder design is related to the de\ufb01nition of the loss function used in training VQC. Let \ud43abe the groundtruth velocity map, and \ud43a\ud456,\ud457the velocity at location \u27e8\ud456, \ud457\u27e9, where \ud456, \ud457\u2208[0, \ud441) represents the coordinate. A straightforward design is to conduct a pixel-wise comparison between ground truth and the forward results. In this case, the decoder needs \ud4412 velocities from the quantum circuit, denoted as \ud437. For example, the mean squared error between ground truth and VQC outputs can be de\ufb01ned as: \ud459\ud45c\ud460\ud460\ud45d\ud456\ud465\ud452\ud459= \u00d5 \ud456\u2208[0,\ud441) \u00d5 \ud457\u2208[0,\ud441) {(\ud43a\ud456,\ud457\u2212\ud437\ud456,\ud457)2} (2) As most subsurface has a \ufb02at structure, we can simplify the regression output data. Speci\ufb01cally, instead of pixel-wise comparison, we can predict one velocity of each row. As a result, we only predict \ud441velocities, denoted as \ud437\u2032, leading to the following MSE: \ud459\ud45c\ud460\ud460\ud459\ud44e\ud466\ud452\ud45f= \u00d5 \ud456\u2208[0,\ud441) \u00d5 \ud457\u2208[0,\ud441) {(\ud43a\ud456,\ud457\u2212\ud437\u2032 \ud457)2} (3) The above method can be generalized for the non-\ufb02at subsurface, such as curve structures. Because the subsurface mediums between curves have the same material, indicating a similar velocity. In this case, we can still use one velocity for each row but need to predict a multi-variable function to describe the curve. The row velocity will be used for all locations between two curves. \fConference\u201917, July 2017, Washington, DC, USA W. Jiang and Y. Lin q\u007f q\u0081 |0\u00f1 s0 (a)  (b)  (c)  s1 s2 |0\u00f1 Ue U(q) I \u00c4 U(q) =  |j\u00f1 = I \u00c4 U(q) \u00d7 Ue \u00d7|0\u00f1 =  =  =  \u00c4 U(q) s1\u00aes2 I Figure 3: Illustration of the proposed QuBatch with 2 qubits. G\u007f G\u0081 (d) No batch for 2 groups  (a) No batch  (b) Batch of 2  (c) Batch of 4 U\u007f(q) U!(q) U\u0081(q) G\u007f G\u0081 q_batch\u007f q_batch\u0081 (e) Batch of 2 for 2 groups   U\u007f(q) U\u0081(q) I I U!(q) D D U(q) q_batch\u007f U(q) I D q_batch\u0081 U(q) I q_batch\u007f I Figure 4: Illustration of integrating QuBatch to QuGeoVQC with di\ufb00erent batch sizes and group sizes. 3.3 QuBatch: Data Batch on Quantum Circuits 3.3.1 Fundamentals. Before introducing details, we use an example to illustrate the ability of quantum computing to process data batches in parallel in Figure 3. We have two vectors \ud4371 and \ud4372, each of which contains two features. The weight matrix is \ud448(\ud703). There are two computations to be performed: \ud448(\ud703) \u00b7 \ud4371 and \ud448(\ud703) \u00b7 \ud4372. We construct a quantum circuit with three steps (\ud4460 to \ud4462) to perform these computations. As shown in Figure 3(a), an encoder \ud448\ud452 entangle these qubits from \ud4460 to \ud4461: |\ud713\u27e9= \ud448\ud452\u00b7 |0\u27e9= [\ud4371, \ud4372]\ud447. The subcircuit from step \ud4461 to \ud4462 constructs the computation operator, where \u201cno gate\u201d is placed on qubit \ud45e0, indicated by a virtual \ud43c, and the operator \ud448(\ud703) is placed on qubit \ud45e1. As qubits are entangled at step \ud4461, we need to combine these two operators for computation by tensor product, shown in Figure 3(b). Now, we have two \ud448(\ud703) operators located on the diagonal of the matrix. Kindly note that this indicates that we can duplicate the computation operator without any cost, providing the fundamentals to e\ufb03ciently support SIMD operation. Then, as shown in Figure 3(c), we can obtain the results of \ud448(\ud703) \u00b7 \ud4371 and \ud448(\ud703) \u00b7 \ud4372 on the output amplitudes. 3.3.2 \ud444\ud462\ud435\ud44e\ud461\ud450\u210eintegration. To integrate \ud444\ud462\ud44f\ud44e\ud461\ud450\u210eto \ud444\ud462\ud43a\ud452\ud45c\ud449\ud444\ud436, we extend the approach in Figure 3 from two aspects: (1) a larger number of batches and (2) the circuit with multiple groups. For the \ufb01rst extension, \ud444\ud462\ud44f\ud44e\ud461\ud450\u210ecan process batch size of 2\ud441by adding only \ud441qubits (\ud441\u22650), as shown in Figures 4(a)-(c). The second extension is to support multiple groups. This can be achieved by introducing a swap gate before communicating between two groups, as shown in Figures 4(d)-(e). We omit detailed analysis here to save space. 3.3.3 Complexity analysis. Before detailed analysis, we \ufb01rst discuss the overhead introduced by \ud444\ud462\ud435\ud44e\ud461\ud450\u210e. As multiple operators can be produced by using tensor products between one operator with an identity matrix, we do not need to pay overhead for the computation. Kindly note that we need to pay overhead on the data encoding. First, the amplitude encoding will lead to a longer circuit. Fortunately, as shown in [16], the circuit length grows linearly with the increase of qubits. Second, because the sum of the squared magnitudes of the amplitudes of all possible states of a qubit must equal 1, if more data are encoded to a batch, we need to normalize the data, indicating that the data precision will be decreased. Kindly note although the data precision will be decreased, the relevant relationship between data can be maintained. In addition, we can control the group size and batch size to make a tradeo\ufb00 among circuit depth, number of qubits, and data precision. Now, we discuss the complexity. Let \ud435be the number of batches implemented in QuBatch, \ud43abe the number of groups in the encoder, and \ud442(\ud44b) be the original time-space complexity (i.e., qubits times circuit depth). By introducing \ud435bathes to the system, we will have overhead on both qubits number and depth: (1) the additional number of qubits will be \ud442(\ud43a\u00b7 log\ud435); (2) for each group, the circuit length increased along with the additional number of qubits; therefore, it has an overhead of \ud442(log\ud435). As the encoding of di\ufb00erent groups is conducted in parallel, we \ufb01nally have the time-space complexity of \ud442(\ud43a\u00b7 log2 \ud435\u00b7 \ud44b). Compared to the implementation of \ud435batches independently, it will lead to a complexity of \ud442(\ud435\u00b7\ud44b). If \ud435>> \ud43a, QuBatch can exponentially decrease the complexity. 4 EXPERIMENTS This section reports experimental setups, followed by results. Dataset. Evaluations are performed on the FlatVelA dataset in OpenFWI [4]. QuGeoData is applied to adjust the data scale to \ufb01t the target quantum backends. More speci\ufb01cally, the original data in OpenFWI has the dimension of 350, 000 = 5 \u00d7 1000 \u00d7 70 (#source \u00d7 time steps \u00d7 #receiver) for seismic data and 70 \u00d7 70 (depth \u00d7 width) for velocity maps. We set the constraint on the number of the qubits to be less than 16, which \ufb01ts most of today\u2019s superconducting-based or ion-trap-based quantum computers. To this end, we will scale the dimension of seismic data to 256, and the velocity map to 8 \u00d7 8. For comparison, we employ a standard nearest neighbor resampling algorithm to downsample both waveform and velocity map as the baseline, denoted as \u201cD-Sample\u201d. Two methods in QuGeoData will be used for comparison. We denote \u201cQ-D-FW\u201d to indicate the data (D) scaling using forward modeling (FW), which is proposed in Section 3.1.1, and denote \u201cQ-D-CNN\u201d to indicate the data (D) scaling using CNN-based data compression proposed in Section 3.1.2. VQCs. We will compare two QuGeoVQC designs with di\ufb00erences in the decoder. The \ufb01rst one is designed to use the pixelwise loss in Eq. 2, where 8 \u00d7 8 velocities will be decoded as the magnitude of 64 amplitudes from the quantum system. We denote the design as \u201cQ-M-PX\u201d, indicating the VQC model (M) based on the pixel-wise loss. The second one is application-speci\ufb01c design, which leverages the knowledge that the subsurface has a \ufb02at layerwise (LY) structure, denoted as \u201cQ-M-LY\u201d. It uses the loss function in Eq. 3, and the velocities will be decoded using Z-measurement of independent qubits. Q-M-PX and Q-M-LY use the ansatz with 12 blocks, each of which is a \u2018U3+CU3\u2019 block as proposed in [3]. To compare quantum learning and classical learning, we implemented two classical convolutional neural networks (CNN) with \fQGeo: Qantum Learning for Geoscience Conference\u201917, July 2017, Washington, DC, USA 0.72 0.0 0 0.002 0.004 0.006 0.008 0.010 0 100 200 300 400 500 0 100 200 300 400 500 0.4 0.8 0.75 0.78 0.81 0.84 0.87 0.0004 0.0007 0.001 D-Sample Q-D-FW Q-D-CNN D-Sample Q-D-FW Q-D-CNN D-Sample Q-D-FW Q-D-CNN SSIM SSIM MSE MSE (a) (b) (c) Epoch Figure 5: Performance of Q-M-PX VQC on dataset scaled by di\ufb00erent approaches: (a) SSIM vs. MSE, the best solution is on the most left-top corner; (b) convergence comparison of SSIM in training; (c) comparison of MSE in training. pixel-wise and layer-wise decoding, denoted as \u201cCNN-PX\u201d and \u201cCNNLY\u201d. For a fair comparison, we use the same training and testing data and control the number of parameters of all models at the same level. Environment. We carry out the concept-proof evaluation on quantum learning design based on the Torchquantum [3] framework. To enable the proposed encoder and QuBatch, we added two components in TorchQuantum. For all VQC model training, we employ the Adam optimizer with 500 epochs where the initial learning rate is set to 0.1, followed by a cosine annealing schedule. To support training, we split the FlatVelA dataset with 500 samples into a training set (size of 400) and a test set (size of 100). The classical CNNs used in Q-D-CNN, CNN-PX, and CNN-LY are trained from scratch in Pytorch. We used the same training setting (i.e., 500 epoch, Adam optimizer, and 0.1 initial learning rate, etc.). For Q-D-CNN, we applied 500 other samples from the FlatVelA dataset to train the CNN. Then, the trained CNN model is used to generate data for the test on quantum computing in QD-CNN. 4.1 QuGeoData: Physics Guidance is Needed Figure 5 reports the comparison results of data scaling approaches using Q-M-PX VQC. Each point in Figure 5(a) is a VQC model obtained in training, where the x-axis and y-axis stand for Structural Similarity Image Metric (SSIM) and Mean Squared Error (MSE), respectively. Figures 5(b)-(c) give the convergence of model training. We have a couple of observations from these results. First, we can clearly observe that the model trained based on the physicsguided data scaling (i.e., Q-D-FW) signi\ufb01cantly outperformsD-Sample (i.e., green diamond) in both SSIM and MSE. Second, Q-D-FW and Q-D-CNN have similar performance in two metrics. Speci\ufb01cally, the SSIM of Q-D-CNN is 0.8619, which is even slightly higher than Q-D-FW with 0.8591, while the MSE of Q-FM is 0.000461, slightly outperforming Q-D-CNN with 0.000460. The above results emphasize the importance of the incorporation of physics knowledge in data scaling. Furthermore, it re\ufb02ects that the learning-based approach can e\ufb00ectively scale data to keep physics features, which will be very useful in processing the raw seismic data. Time Steps Reciver Reciver Reciver Reciver Reciver Reciver Time Steps (a) Seismic data obtained by Q-D-FM, D-Sampling, Q-D-CNN Q-D-FM (b) Normalized seismic data by quantum encoder SSIM: 1.0 (baseline) SSIM: 0.0597 SSIM: 0.9255 SSIM: 1.0 (baseline) SSIM: 0.5253 SSIM: 0.9989 D-Sample Q-D-CNN Q-D-FM D-Sample Q-D-CNN Figure 6: Visualization of seismic data by di\ufb00erent methods: (a) scaled classical data; (b) normalized quantum data. depth (km) velocity (km/s) 0 0.2 0.4 0.6 0.05 0 0.10 0.05 0 0.10 0.05 0 0.10 0.7 0.06 0.04 km/s D-Sample (0.9613) Ground truth Q-D-CNN (0.9742) Q-D-FM (0.9772) depth (km) (a) Output visualization of Q-M-PX on different datasets (b) Comparison of vertical velocity profiles of inversion results at x = 400 m 0 0.2 0.4 0.6 0.7 depth (km) 0 0.2 0.4 0.6 0.7 depth (km) 0 0.2 0.4 0.6 0.7 horizon distance (m) 0 200 400 600 0 200 400 600 0 200 400 600 0 200 400 600 D-Sample B Ground truth Ground truth Ground truth Q-D-FW Q-D-CNN C D F G E A C Figure 7: Visualization of the predicted velocity map: (a) outputs from di\ufb00erent data scaling approaches; (b) vertical velocity pro\ufb01ling for physics information analysis. Figure 6 gives the visualization of seismic waveform data to better show the causes of performance gain. First, in comparing Q-DFM and D-Sample, we can see a larger wavelength in Q-D-FM. The adjustment of the sampling rate and source wavelet causes this. By down-scaling the time dimension in seismic waves, the sampling rate is decreased. Consequently, we lower the source wavelet frequency from 15Hz to 8Hz to prevent the loss of information from a physics standpoint, resulting in an increased wavelength. Conversely, D-Sample\u2019s approach of directly downsampling the waveform data leads to the inevitable loss of vital physical information, evident in the waveform measurements\u2019 incoherence, which will eventually degrade the resulting imaging accuracy. Secondly, the waveform data from Q-D-FM and Q-D-CNN, with an initial SSIM of 0.9255, see this value rise to 0.9989 following data normalization within quantum state constraints. These \ufb01ndings underscore the signi\ufb01cance of physics-guided data scaling and the e\ufb03ciency of CNN-based data compression methods. Finally, in Figure 7, we present the visualization of results. From Figure 7(a), we can again see the improvements obtained by QD-FM and Q-D-CNN over D-Sample. More interesting results are shown in Figure 7(b), where we compare the vertical velocity pro\ufb01les at horizon distance \ud465= 400. The brown lines are the velocity changes along depth for the ground truth, where the y-value stands the velocity, and the layer boundary (say point B) indicates an interface between two layers. From these \ufb01gures, we can see that D-Sample leads to a larger di\ufb00erence in velocity values, and it cannot predict the interface. Speci\ufb01cally, for all seven in\ufb02ection points, there are \ufb01ve wrong predictions (points A, B, C, E, and G) and only two correct predictions (points D and F). On the other \fConference\u201917, July 2017, Washington, DC, USA W. Jiang and Y. Lin 0.72 0.78 0.84 0.9 0.96 0.0009 0.0006 0.0003 0 Q-D-FM (a) (b) D-Sample Q-D-CNN Q-D-FM D-Sample Q-D-CNN SSIM MSE Q-M-LY Q-M-PX Q-M-LY Q-M-PX Figure 8: Comparison results of Q-M-PX and Q-M-LY. depth (km) velocity (km/s) 0 0.2 0.4 0.6 0.05 0 0.10 0.05 0 0.10 0.7 0.06 0.04 0.02 km/s Q-M-PX Q-D-FW (0.9492) Ground truth Q-M-LY Q-D-FW  (0.9854) Q-M-LY depth (km) (a) Output visualization of differne models on different datasets (b) Comparison of vertical velocity profiles of inversion results at x = 400 m 0 0.2 0.4 0.6 0.7 0 0.2 0.4 0.6 0.7 depth (km) depth (km) 0 0.2 0.4 0.6 0.7 horizon distance (m) 0 200 400 600 0 200 400 600 0 200 400 600 D-Sample (0.9606) 0 200 400 600 Q-D-FW &  Q-M-PX Ground truth Ground truth Q-D-FW &  Q-M-LY 0.05 0 0.10 D-Sample &  Q-M-LY Ground truth A C D E B Figure 9: Visualization of the predicted velocity map for QM-LY: (a) ground truth and prediction results; (b) vertical velocity pro\ufb01ling for physics information analysis. hand, there are three correct interface predictions for both Q-DFW and Q-D-CNN. Clearly, the aforementioned results indicate that creating a dataset in accordance with physical principles can enhance performance. However, to fully leverage quantum computing for FWI problems, further improvements in performance are necessary. 4.2 QuGeoVQC Further Boost Performance Figure 8 reports the comparison results on two di\ufb00erent VQC designs: Q-M-PX and Q-M-LY, using both SSIM and MSE metrics. As shown in the \ufb01gures, for all three data scaling methods, results obtained by Q-M-LY signi\ufb01cantly outperform those by Q-M-PX. Speci\ufb01cally, for SSIM using data from Q-D-CNN data, the SSIM improves from 0.862 to 0.905. These \ufb01gures are from 0.859 to 0.892 for Q-D-FW, and 0.800 to 0.842 for D-Sample. Overall, Q-M-LY achieves a 4.5% improvement over Q-M-PX. For MSE, the average improvement reaches 33.23%. These results show the e\ufb00ectiveness of layer-wise VQC design. We have one more interesting observation from the results. Considering that the Q-M-PX using D-Sample is the straightforward implementation of FWI on quantum computing, the proposedQuGeo design can improve the SSIM from 0.800 to 0.905 and reduce MSE from 0.000855 to 0.000328, achieving 11.6% and 61.69% improvements on SSIM and MSE, respectively. This result shows the huge optimization room for implementing FWI on quantum computing, and the e\ufb00ectiveness of our proposed methods. Figure 9 presents the visualization results for Q-M-PX and QM-LY using Q-D-FW, as well as Q-M-LY using D-Sample. In the Model Dataset Batch Extra Qubits SSIM vs. BL Q-M-LY Q-D-FW 0 0 0.8926 BL 2 1 0.8864 0.69% 4 2 0.8678 2.77% Table 1: Evaluation of QuBatch with di\ufb00erent batch sizes on Q-D-FW dataset using Q-M-LY VQC. Table 2: Comparison between quantum and classical learning Model Par. Q-D-FW Q-D-CNN SSIM vs. MSE vs. SSIM vs. MSE vs. CNN-PX 634 0.870 BL 4.34E-04 BL 0.87 BL 4.38E-04 BL CNN-LY 616 0.871 0.04% 4.36E-04 -0.43% 0.87 0.00% 4.36E-04 0.38% Q-M-PX 576 0.859 -1.28% 4.61E-04 -6.10% 0.86 -0.98% 4.62E-04 -5.45% Q-M-LY 576 0.893 2.50% 3.48E-04 19.84% 0.91 3.87% 3.28E-04 25.17% vertical velocity analysis depicted in Figure 9(b), we observe that Q-M-PX inaccurately predicts two interfaces at Points A and B. Although Q-M-LY using D-Sample accurately predicts all interfaces, it incorrectly interprets the relative positioning between two layers at three interfaces, as highlighted at Points C, D, and E. In contrast, Q-M-LY using Q-D-FW, successfully predicts all interfaces while maintaining the correct relative relationships between the layers. These \ufb01ndings further emphasize the e\ufb00ectiveness of layerwise VQC design and highlight the promising potential of quantum learning in addressing the challenges of FWI. 4.3 QuBatch Achieves Competitive Results Next, Table 1 shows the proposed QuBatch can be successfully used in the model training, where we can perform 2\ud441batches in parallel with only \ud441more qubits. In addition, we observe the SSIM has slight degradation compared with the results obtained without using data batch. The root cause might be the decrease in data precision caused by the normalization, which is required by the constraints of amplitudes. As discussed in Section 3.3.3, such effects can be mitigated by making tradeo\ufb00s among data precision and qubits, which will be the future work. 4.4 QuGeo Outperforms Classical ML Table 2 reports the comparison of our proposed quantum learning for FWI with classical learning. We constrain the number of parameters to the same level. As shown in Table 2, CNN-PX and CNN-LY have 58 and 40 more parameters. Whereas, Q-M-LY outperforms both classical results on a dataset obtained by both Q-DFW and Q-D-CNN. Speci\ufb01cally, using CNN-PX as a baseline, Q-MLY achieves 19.84% and 25.17% MSE improvements on two datasets, respectively. The potential reason that quantum learning can outperform classical learning at the same scale is because of the complicated entanglement in quantum computing, which has the potential to extract high-correlation features among data. Achieving better performance even at the small scale is meaningful since realworld applications commonly have hard real-time requirements, in particular, for monitoring tasks in geoscience. \fQGeo: Qantum Learning for Geoscience Conference\u201917, July 2017, Washington, DC, USA 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2012.10360v1",
+                    "title": "When Machine Learning Meets Quantum Computers: A Case Study",
+                    "abstract": "Along with the development of AI democratization, the machine learning\napproach, in particular neural networks, has been applied to wide-range\napplications. In different application scenarios, the neural network will be\naccelerated on the tailored computing platform. The acceleration of neural\nnetworks on classical computing platforms, such as CPU, GPU, FPGA, ASIC, has\nbeen widely studied; however, when the scale of the application consistently\ngrows up, the memory bottleneck becomes obvious, widely known as memory-wall.\nIn response to such a challenge, advanced quantum computing, which can\nrepresent 2^N states with N quantum bits (qubits), is regarded as a promising\nsolution. It is imminent to know how to design the quantum circuit for\naccelerating neural networks. Most recently, there are initial works studying\nhow to map neural networks to actual quantum processors. To better understand\nthe state-of-the-art design and inspire new design methodology, this paper\ncarries out a case study to demonstrate an end-to-end implementation. On the\nneural network side, we employ the multilayer perceptron to complete image\nclassification tasks using the standard and widely used MNIST dataset. On the\nquantum computing side, we target IBM Quantum processors, which can be\nprogrammed and simulated by using IBM Qiskit. This work targets the\nacceleration of the inference phase of a trained neural network on the quantum\nprocessor. Along with the case study, we will demonstrate the typical procedure\nfor mapping neural networks to quantum circuits.",
+                    "authors": "Weiwen Jiang, Jinjun Xiong, Yiyu Shi",
+                    "published": "2020-12-18",
+                    "updated": "2020-12-18",
+                    "primary_cat": "quant-ph",
+                    "cats": [
+                        "quant-ph",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION In the past few years, we have witnessed many breakthroughs in both machine learning and quantum computing research fields. On machine learning, the automated machine learning (AutoML) [47, 48] significantly reduces the cost of designing neural networks to achieve AI democratization. On quantum computing, the scale of the actual quantum computers has been rapidly evolving (e.g., IBM [13] recently announced to debut quantum computer with 1,121 quantum bits (qubits) in 2023). Such two research fields, however, have met the bottlenecks when applying the theoretical knowledge in practice. With the large-size inputs, the size of machine learning models (i.e., neural networks) significantly exceed the resource provided by the classical computing platform (e.g., GPU and FPGA); on the other hand, the development of quantum applications is far behind the development of quantum hardware, that is, it lacks killer applications to take full advantage of high-parallelism provided by a quantum computer. As a result, it is natural to see the emerging of a new research field, quantum machine learning. Like applying machine learning to the classical hardware accelerators, when machine learning meets quantum computers, there will be tons of opportunities along with the challenges. The development of machine leering on the classical hardware accelerator experienced two phases: (1) the design of neural network tailored hardware [15, 16, 25, 26, 40, 45], and (2) the co-design of neural network and hardware accelerator [3, 5, 7, 11, 12, 14, 17, 21, 34\u2013 37, 39]. To best exploit the power of the quantum computer, it would be essential to conduct the co-design of neural network and quantum circuits design; however, with the different basic logic gates between quantum circuit and classical circuit designs, it is still unclear how to design a quantum accelerator for the neural network. In this work, we aim to fix such a missing link by providing an open-source design framework. In general, the full acceleration system will be divided into three parts, the data pre-processing and data post-processing on a classical computer, and the neural network accelerator on the quantum circuit. In the quantum circuit, it will further include the quantum state preparation and the quantum computing-based neural computation. In the following of this paper, we will introduce all the above components in detail and demonstrate the implementation using IBM Qiskit for quantum circuit design and Pytorch for the machine learning model process. The remainder of the paper is organized as follows. Section 2 presents an overview of the full system. Section 3 presents the case study on the MNIST dataset. Insights are discussed in Section 4. Finally, concluding remarks are given in Section 5. arXiv:2012.10360v1  [quant-ph]  18 Dec 2020 \fASPDAC \u201921, January 18\u201321, 2021, Tokyo, Japan W. Jiang, et al. Classical Data Pre-Pro Compute (Q) UP UN Classical Data Post-Pro Classical Data Post-Pro Classical Data Pre-Pro Classical (a) Classical Comp. Accelerator (b) Quantum Comp. Accelerator (c) Hybrid Quantum-Classical Comp. Accelerator Data Pre-Pro Compute (C) Compute (hybrid Q+C) * + + + + + + * * * * * * * w w w w w w w w I0 O0 ON I1 I2 I3 C C Q Q Q C Classical Data Post-Pro |0\u232a |0\u232a |0\u232a \u2026 Figure 1: Illustration of three different types of computing schemes: (a) classical computing \u201cC\u201d based neural computation, where\ud835\udc4ais weights; (b) quantum computing \u201cQ\u201d based neural computation, where\ud835\udc48\ud835\udc5dis the quantum-state preparation and \ud835\udc48\ud835\udc41is the neural computation; (c) hybrid quantumclassical computing \u201cQ+C\u201d based neural computation. 2 OVERVIEW Figure 1 demonstrates three types of neural network design: (1) the classical hardware accelerator; (2) the pure quantum computing based accelerator; (3) the hybrid quantum and classical accelerator. All of these accelerators follow the same flow that the data will be first pre-processed, then the neural computation is accelerated, and finally, the output data will go through the post-processing to obtain the final results. 2.1 Classical acceleration After the success of deep neural networks (e.g., Alexnet [23] and VGGNet [32]) in achieving high accuracy, designing hardware accelerator became the hot topic in accelerating the execution of deep neural networks. On the application-specific integrated circuit (ASIC), works [6, 8, 41\u201343] studied how to design neural network accelerator using different dataflows, including weight stationery, output stationery, etc. By selecting dataflow for a dedicated neural computation, it can maximize the data reuse to reduce the data movement and accelerate the process, which derived the co-design of neural network and ASICs [38]. On the FPGA, work [40] first proposed the tiling based design to accelerate the neural computation, and works [15, 16, 24, 45] gave different designs and extended the implementation to multiple FPGAs. Driven by the AutoML, work [21] proposed the first co-design framework to involve the FPGA implementation into the search loop, so that both software accuracy and hardware efficiency can be maximized. The co-design philosophy also applied in other designs [11, 12, 20, 44] and in this direction, there exist many research works in further integrating the model compression into consideration [19, 28], accelerating the search process [27, 46], 2.2 Pure quantum computing Most recently, the emerging works in using the quantum circuit to accelerate neural computation. The typical work include [9, 18, 33], among which the work [18] first demonstrates the potential quantum advantage that can be achieved by using a co-design philosophy. These works encode data to either qubits [9] or qubit states [18] and use superconducting-based quantum computers to run neural networks. These methods have the following limitations: Due to the short decoherence times in the superconducting-based quantum computers, the condition logic is not supported in the computing process. This makes it hard to implement a function that is not differentiable at all points, like the commonly used Rectified Linear Unit (ReLU) in machine learning models. However, it also has advantages, such as the design can be directly evaluated on an actual quantum computer, and there is no communication between the quantum-classical interface during the computation. In the quantum circuit design, it includes two components: \ud835\udc48\ud835\udc43 for quantum states preparation and \ud835\udc48\ud835\udc41for neural computation, as shown in Figure 1(b). After the component \ud835\udc48\ud835\udc41, it will measure the quantum qubits to extract the output data, which will be further sent to the data post-processing unit to obtain the final results. 2.3 Hybrid quantum-classical computing To overcome the disadvantage of pure quantum computing and take full use of classical computing, the hybrid quantum-classical computing for machine learning tasks is proposed [4]. It establishes a computing paradigm where different neurons can be implemented on either quantum or classical computers, as demonstrated in Figure 1(c). This brings flexibility in implementing functions (e.g., ReLU). However, at the same time, it will lead to massive data transfer between quantum and classical computers. 2.4 Our Focus in The Case Study This work focus on providing a full workflow, starting from the data pre-processing, going through quantum computing acceleration, and ending with the data post-processing. We will apply the MNIST data set as an example to carry out a case study. Computing architecture and neural operation can affect the design. In this work, for the computing architecture, we focus on the pure quantum computing design, since it can be easily extended to the hybrid quantum-classical design by connecting the inputs and output of the quantum acceleration to the traditional classical accelerator; for the neural network, we focus on the multi-layer perceptron, which is the basic operation for a large number of neural computation, like the convolution. 3 CASE STUDY ON MNIST DATASET In this section, we will demonstrate the detailed implementation of four components in the pure quantum computing based neural computation as shown in Figure 1(b): data pre-processing, quantum state preparation (\ud835\udc48\ud835\udc43), neural computation (\ud835\udc48\ud835\udc41), and data postprocessing. 3.1 Data Pre-Processing The first step of the whole procedure is to prepare the quantum data to be encoded to the quantum states. Kindly note in order \fWhen Machine Learning Meets Quantum Computers: A Case Study ASPDAC \u201921, January 18\u201321, 2021, Tokyo, Japan to utilize \ud835\udc41qubits to represent 2\ud835\udc41data, it has constraints on the numbers; more specifically, if a vector\ud835\udc480 of 2\ud835\udc41data can be arranged in the first column of a unitary matrix \ud835\udc48, then for the initial state of |\ud835\udf13\u27e9= 1 \u00b7 |0\u27e9\u2297\ud835\udc41, we can obtain \ud835\udc480 by conducting \ud835\udc48|\ud835\udf13\u27e9= \ud835\udc480, where |0\u27e9\u2297\ud835\udc41represents the zero state with \ud835\udc41qubits. 1 import torch 2 import numpy as np 3 import torchvision.transforms as transforms 4 # Input: img_size =4 to represent the resolution of 4*4 5 class ToQuantumData(object): 6 def __call__(self , tensor): 7 device = torch.device(\"cuda\" if torch.cuda. is_available () else \"cpu\") 8 data = tensor.to(device) 9 input_vec = data.view(-1) 10 vec_len = input_vec.size()[0] 11 input_matrix = torch.zeros(vec_len , vec_len) 12 input_matrix [0] = input_vec 13 input_matrix = np.float64(input_matrix. transpose (0,1)) 14 u, s, v = np.linalg.svd(input_matrix) 15 output_matrix = torch.tensor(np.dot(u, v)) 16 output_data = output_matrix [:, 0]. view(1, img_size ,img_size) 17 return output_data 18 # Similarly , we have \"class ToQuantumMatrix(object)\" which return the output_matrix 19 transform = transforms.Compose ([ transforms.Resize (( img_size , img_size)), transforms.ToTensor (), ToQuantumData ()]) 20 # transform = transforms.Compose ([ transforms.Resize (( img_size ,img_size)),transforms.ToTensor (), transforms.Normalize ((0.1307 ,), (0.3081 ,)), ToQuantumData ()]) Listing 1: Converting classical data to qautnum data Listing 1 demonstrates the data conversion from the classical data to quantum data. We utilize the transforms in torchvision to complete the data conversation. More specifically, we create the ToQuantumData class in Line 5. It will receive a tensor (the original data) as input (Line 6). We apply Singular Value Decomposition (svd) provided by np.linalg to obtain the unitary matrix output_matrix (Line 14), then we extract the first vector from output_matrix as the output_data (Line 16), where the output_matrix represents \ud835\udc48and the output_data represents \ud835\udc480. After we build the ToQuantumData class, we will integrate it into one \u201ctransform\u201d variable, which can further include the data pre-processing functions, such as image resize (Line 20) and data normalization (Line 21). In creating the data loader, we can apply the \u201ctransform\u201d to the dataset (e.g., we can obtain train data by using \u201ctrain_data=datasets.MNIST(root=datapath, train=True,download=True, transform=transform)\u201d). 3.2 \ud835\udc48\ud835\udc43: Quantum State Preparation Theoretically, with the \ud835\udc5b\u00d7 \ud835\udc5bunitary matrix \ud835\udc48, we can directly operate the oracle on the quantum circuit to change 2\ud835\udc41states from the zero state |0\u27e9\u2297\ud835\udc41to \ud835\udc480. This process is widely known as quantum-state preparation. The efficiency of quantum-state preparation can significantly affect the complexity of the whole circuit, and therefore, it is quite important to improve the efficiency of such a process. In general, there are two typical ways to perform the quantum-state preparation: (1) quantum random access memory (qRAM) [29] based approach [1, 22] and (2) computing based approach [2, 10, 30]. Let\u2019s first see the qRAM-based approach, where the vector in \ud835\udc480 will be stored in a binary-tree based structure in qRAM, which can be queried in quantum superposition and can generate the states efficiently. In IBM Qiskit, it provides the initialization function to perform quantum-state preparation, which is based on the method in [31]. 1 from qiskit import QuantumRegister , QuantumCircuit , ClassicalRegister 2 from qiskit.extensions import XGate , UnitaryGate 3 from qiskit import Aer , execute 4 import qiskit 5 # Input: a 4*4 matrix (data) holding 16 input data 6 inp = QuantumRegister (4,\"in_qbit\") 7 circ = QuantumCircuit(inp) 8 data_matrix = Q_InputMatrix = ToQuantumMatrix ()(data. flatten ()) 9 circ.append(UnitaryGate(data_matrix , label=\"Input\"), inp [0:4]) 10 # Using StatevectorSimulator from the Aer provider 11 simulator = Aer.get_backend('statevector_simulator ') 12 result = execute(circ , simulator).result () 13 statevector = result.get_statevector(circ) 14 print(statevector) Listing 2: Quantum-State Preparation in IBM Qiskit In Listing 2, we give the codes to initialize the quantum states, using the unitary matrix \ud835\udc48which is converted from the original data in Listing 1(see Line 18). In this code snippet, we first create a 4-qubit QuantumRegister \u201cinp\u201d (line 6) and the quantum circuit (line 7). Then, we convert the input data to data_matrix, which is then employed to initialize the circuit using function UnitaryGate from qiskit.extensions. Finally, from line 10 to line 14, we output the states of all qubits to verify the correctness. 3.3 \ud835\udc48\ud835\udc41: Neural Computation Now, we have encoded the image data (16 inputs) onto 4 qubits. The next step is to perform the neural computation, that is, the weighted sum with quadratic function using the given binary weights \ud835\udc4a. Neural computation is the key component in quantum machine learning implementation. To clearly introduce this component, we first consider the computation of the hidden layer, which can be further divided into two stages: (1) multiplying inputs and weights, and (2) applying the quadratic function on the weighted sum. Then, we will present the computation of the output layer to obtain the final results. Computation of one neural in the hidden layer Stage 1: multiplying inputs and weights. Since the weight \ud835\udc4ais given, it is pre-determined. We use the quantum gate to operate the weights with the inputs. The quantum gates applied here include the \ud835\udc4bgate and the 3-controlled-Z gate with 3 trigger qubits. The function of such a 3-controlled-Z is to flip the sign of state |1111\u27e9, and the function of \ud835\udc4bgate is to swap one state to another state. For example, if the weight for state |0011\u27e9is \u22121. We operate it on the input follows three steps. First, we swap the amplitude of state |0011\u27e9to state |1111\u27e9using two \ud835\udc4bgates on the first two qubits. Then, in the second step, we apply controlled-Z gate to flip the sign of the state |1111\u27e9. Finally, in the third step, we swap the amplitude of state |1111\u27e9back to state |0011\u27e9using two \ud835\udc4bgates on the first two qubits. Therefore, we can transverse all weights and apply the above three steps to flip the sign of corresponding states. Kindly note that since the non-linear function is a quadratic function, if \fASPDAC \u201921, January 18\u201321, 2021, Tokyo, Japan W. Jiang, et al. the number of \u22121 is larger than +1, we can flip all signs of weights to minimize the number of gates to be put in the circuit. 1 def cccz(circ , q1, q2, q3, q4, aux1 , aux2): 2 # Apply Z-gate to a state controlled by 4 qubits 3 circ.ccx(q1 , q2, aux1) 4 circ.ccx(q3 , aux1 , aux2) 5 circ.cz(aux2 , q4) 6 # cleaning the aux bits 7 circ.ccx(q3 , aux1 , aux2) 8 circ.ccx(q1 , q2, aux1) 9 return circ 10 def neg_weight_gate(circ ,qubits ,aux ,state): 11 for idx in range(len(state)): 12 if state[idx]=='0': 13 circ.x(qubits[idx]) 14 cccz(circ ,qubits [0], qubits [1], qubits [2], qubits [3], aux[0],aux [1]) 15 for idx in range(len(state)): 16 if state[idx]=='0': 17 circ.x(qubits[idx]) 18 # input: weight vector , weight_1_1 19 aux = QuantumRegister (2,\"aux_qbit\") 20 circ.add_register(aux) 21 if weight_1_1.sum() <0: 22 weight_1_1 = weight_1_1 *-1 23 for idx in range(weight_1_1.flatten ().size()[0]): 24 if weight_1_1[idx]== -1: 25 state = \"{0:b}\".format(idx).zfill (4) 26 neg_weight_gate(circ ,inp ,aux ,state) 27 circ.barrier () 28 print(circ) Listing 3: Multiplying inputs and weights on quantum Listing 3 demonstrates the procedure of multiplying inputs and weights. In the list, the function cccz utilizing the basic quantum logic gates to realize the 3-controlled-Z gate with 3 control qubits. The involved basic gates include Toffoli gate (i.e., CCX) and controlled-Z gate (i.e., CZ). Since such a function needs auxiliary (a.k.a., ancilla) qubits, we include 2 additional qubits (i.e., \ud835\udc4e\ud835\udc62\ud835\udc65) in the quantum circuit (i.e., \ud835\udc50\ud835\udc56\ud835\udc5f\ud835\udc50), as shown in Lines 19-20. The function neg_weights_gate flips the sign of the given state, applying the 3-step process. Lines 11-13 complete the first step to swap the amplitude of the given state to the state of |1\u27e9\u22974. Then, the cccz gate is applied to complete the second step. Finally, from line 15 to line 17, the amplitude is swap back to the given state. With the above two functions, we traverse the weights to assign the sign to each state from Lines 21-27. Kindly note that, after this operation, the states vector changed from the initial state |\ud835\udf13\u27e9= \ud835\udc480 to |\ud835\udf13\u2032\u27e9= \ud835\udc48\u2032 0 where the states have the weights. Stage 2: applying a quadratic function on the weighted sum. In this stage, it also follows 3 steps to complete the function. In the first step, we apply the Hadamard (H) gates on all qubits to accumulates all states to the zero states. Then, the second step swap the amplitude of zero state |0\u27e9\u2297\ud835\udc41and the one-state |1\u27e9\u2297\ud835\udc41. Finally, the last step applies the N-control-X gate to extract the amplitude to one output qubit \ud835\udc42, in which the probability of \ud835\udc42= |1\u27e9is equal to the square of the weighted sum. In the first step, the H gates can be applied to accumulate the amplitude of states, because the first row of \ud835\udc3b\u22974 is 1 4 \u00d7 [1, 1, 1, 1] and the \ud835\udc3b\u22974|\ud835\udf13\u2032\u27e9performs the multiplication between the 4 \u00d7 4 matrix and the state vector \ud835\udc48\u2032 0. As a result, the amplitude of |0\u27e9\u2297\ud835\udc41 will be the weighted sum with the coefficient of 1 \u221a \ud835\udc41. 1 def ccccx(circ , q1, q2, q3, q4, q5, aux1 , aux2): 2 circ.ccx(q1 , q2 , aux1) 3 circ.ccx(q3 , q4 , aux2) 4 circ.ccx(aux2 , aux1 , q5) 5 # cleaning the aux bits 6 circ.ccx(q3 , q4 , aux2) 7 circ.ccx(q1 , q2 , aux1) 8 return circ 9 # input: circ after stage 1 10 hidden_neuron = QuantumRegister (1,\"out_qbit\") 11 circ.add_register(hidden_neuron) 12 circ.h(inp) 13 circ.x(inp) 14 ccccx(circ ,inp[0],inp[1],inp[2],inp[3], hidden_neuron , aux[0],aux [1]) Listing 4: Applying quadratic function on the weighted sum Listing 4 demonstrates the implementation of the quadratic function on the weighted sum on Qiskit. In the list, function ccccx is based on the basic Toffoli gate (i.e., CCX) to implement a 4-controlX gate to swap the amplitude between the zero state |0\u27e9\u22974 and the one-state |1\u27e9\u22974. In Line 14, \u210e\ud835\udc56\ud835\udc51\ud835\udc51\ud835\udc52\ud835\udc5b_\ud835\udc5b\ud835\udc52\ud835\udc62\ud835\udc5f\ud835\udc5c\ud835\udc5bis an additional output qubit in the quantum circuit (i.e., \ud835\udc50\ud835\udc56\ud835\udc5f\ud835\udc50) to hold the result for the neural computation, which is added in Lines 10-11. For a neural network with \ud835\udc41neurons in the hidden layer, it has \ud835\udc41sets of weights. We can apply the above neural computation on \ud835\udc41set of weights to obtain \ud835\udc41output qubits. Computation of one neuron in the output layer With these \ud835\udc41output qubits, we have two choices: (1) go to the classical computer and then encode the output of these \ud835\udc41outputs to log2 \ud835\udc41qubits and then repeat these computations for the hidden layer to obtain the final results; (2) continuously use these qubits to directly compute the outputs, but the fundamental computation needs to be changed to the multiplication between random variables because the data associated with a qubit represents the probability of the qubit to be |0\u27e9state. In the following, we demonstrate the implementation of the second choices (fundamental details please refer to [18, 33]). In this example, we follow the network structure with 2 neurons in the hidden layer. In addition, we consider there is only one parameter for the normalization function using one additional qubit for each output neuron. Let \u210e\ud835\udc56\ud835\udc51\ud835\udc51\ud835\udc52\ud835\udc5b_\ud835\udc5b\ud835\udc52\ud835\udc62\ud835\udc5f\ud835\udc5c\ud835\udc5b\ud835\udc60be the outputs of 2 neurons in the hidden layer; let \ud835\udc64\ud835\udc52\ud835\udc56\ud835\udc54\u210e\ud835\udc61_2_1 be the weights for the 1\ud835\udc60\ud835\udc61output neuron in the 2\ud835\udc5b\ud835\udc51layer; let norm_flag_1 and norm_para_1 be the normalization related parameters for the 1\ud835\udc60\ud835\udc61output neuron. Then, we have the following implementation. 1 # Additional registers 2 inter_q_1 = QuantumRegister (1,\"inter_q_1_qbits\") 3 norm_q_1 = QuantumRegister (1,\"norm_q_1_qbits\") 4 out_q_1 = QuantumRegister (1,\"out_q_1_qbits\") 5 circ.add_register(inter_q_1 ,norm_q_1 ,out_q_1) 6 circ.barrier () 7 # Input and weight multiplication 8 if weight_2_1.sum() <0: 9 weight_2_1 = weight_2_1 *-1 10 idx = 0 11 for idx in range(weight_2_1.flatten ().size()[0]): 12 if weight_2_1[idx ]== -1: 13 circ.x(hidden_neurons[idx]) 14 circ.h(inter_q_1) 15 circ.cz(hidden_neurons [0], inter_q_1) 16 circ.x(inter_q_1) 17 circ.cz(hidden_neurons [1], inter_q_1) 18 circ.x(inter_q_1) 19 # quadratic function on weighted sum 20 circ.h(inter_q_1) \fWhen Machine Learning Meets Quantum Computers: A Case Study ASPDAC \u201921, January 18\u201321, 2021, Tokyo, Japan 21 circ.x(inter_q_1) 22 circ.barrier () 23 # normalization for two cases 24 norm_init_rad = float(norm_para_1.sqrt().arcsin ()*2) 25 circ.ry(norm_init_rad ,norm_q_1) 26 if norm_flag_1: 27 circ.cx(inter_q_1 ,out_q_1) 28 circ.x(inter_q_1) 29 circ.ccx(inter_q_1 ,norm_q_1 ,out_q_1) 30 else: 31 circ.ccx(inter_q_1 ,norm_q_1 ,out_q_1) 32 # Recove the inputs for the next neuron computation 33 for idx in range(weight_2_1.flatten ().size()[0]): 34 if weight_2_1[idx]== -1: 35 circ.x(hidden_neurons[idx]) Listing 5: Implementation of the second layer neural computation without measurement after the first layer In the above list, it follows the 2-stage pattern for the computation in the hidden layer. If we modify all sub-index _1 to _2, then we can obtain the quantum circuit for the second output neuron. 3.4 Data Post-Processing After all outputs are computed and stored in the out_q_1 and out_q_2 qubits, we can then measure the output qubits, run a simulation or execute on the IBM Q processors, and finally obtain the classification as follows. 1 from qiskit.tools.monitor import job_monitor 2 3 def fire_ibmq(circuit ,shots ,Simulation = False , backend_name='ibmq_essex '): 4 count_set = [] 5 if not Simulation: 6 provider = IBMQ.get_provider('ibm -q-academic ') 7 backend = provider.get_backend(backend_name) 8 else: 9 backend = Aer.get_backend('qasm_simulator ') 10 job_ibm_q = execute(circuit , backend , shots=shots) 11 job_monitor(job_ibm_q) 12 result_ibm_q = job_ibm_q.result () 13 counts = result_ibm_q.get_counts () 14 return counts 15 16 def analyze(counts): 17 mycount = {} 18 for i in range (2): 19 mycount[i] = 0 20 for k,v in counts.items(): 21 bits = len(k) 22 for i in range(bits): 23 if k[bits -1-i] == \"1\": 24 if i in mycount.keys(): 25 mycount[i] += v 26 else: 27 mycount[i] = v 28 return mycount ,bits 29 30 qc_shots =8192 31 counts = fire_ibmq(circ ,qc_shots ,True) 32 (mycount ,bits) = analyze(counts) 33 class_prob =[] 34 for b in range(bits): 35 class_prob.append(float(mycount[b])/qc_shots) 36 class_prob.index(max(class_prob)) Listing 6: Extract the classification results Listing 6 demonstrate the above three tasks. The fire_ibmq function can execute the constructed circuit in either simulation or a given IBM Q processor backend. The parameter \u201cshots\u201d defines the number of execution to be executed. Finally, the counts for each state will be returned. On the implementation, the probability of each qubit (instead of each state) gives the probability to choose the corresponding class. Therefore, we create the \u201canalyze\u201d function to get the probability for each qubits. Finally, we obtain the classification results by extracting the index of the max probability in the \u201cclass_prob\u201d set. Kindly note that the Listing 6 can also be applied for the hybrid quantum-classical computing. 4 INSIGHTS From the study of implementing neural networks onto the quantum circuits, there are several insights in terms of achieving quantum advantages, listed as follows. \u2022 Data encoding: this case study encodes 2\ud835\udc41data to \ud835\udc41quantum qubits, which provides the opportunity to achieve quantum advantage for conducting inference for each input. An alternative way is to encode \ud835\udc41data to \ud835\udc41qubits, however, with the consideration that each data needs to be operated in the neural computation, such an encoding approach can hardly achieve the quantum advantage. \u2022 Quantum-state preparation: by encoding 2\ud835\udc41data to \ud835\udc41 quantum qubits, we can achieve quantum advantage only if the quantum-state preparation can be efficiently conducted with complexity at \ud835\udc42(\ud835\udc41). \u2022 Quantum computing-based neural computation: Neural computation can also become the performance bottleneck, using the design in Listing 3 to flip one sign at each time, it requires \ud835\udc42(2\ud835\udc41) gates in the worst case. To overcome this, [18] proposed a co-design approach to reduce the number of gates to \ud835\udc42(\ud835\udc412). 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2006.14815v2",
+                    "title": "A Co-Design Framework of Neural Networks and Quantum Circuits Towards Quantum Advantage",
+                    "abstract": "Despite the pursuit of quantum advantages in various applications, the power\nof quantum computers in neural network computations has mostly remained\nunknown, primarily due to a missing link that effectively designs a neural\nnetwork model suitable for quantum circuit implementation. In this article, we\npresent the co-design framework, namely QuantumFlow, to provide such a missing\nlink. QuantumFlow consists of novel quantum-friendly neural networks (QF-Nets),\na mapping tool (QF-Map) to generate the quantum circuit (QF-Circ) for QF-Nets,\nand an execution engine (QF-FB). We discover that, in order to make full use of\nthe strength of quantum representation, it is best to represent data in a\nneural network as either random variables or numbers in unitary matrices, such\nthat they can be directly operated by the basic quantum logical gates. Based on\nthese data representations, we propose two quantum friendly neural networks,\nQF-pNet and QF-hNet in QuantumFlow. QF-pNet using random variables has better\nflexibility, and can seamlessly connect two layers without measurement with\nmore qbits and logical gates than QF-hNet. On the other hand, QF-hNet with\nunitary matrices can encode 2^k data into k qbits, and a novel algorithm can\nguarantee the cost complexity to be O(k^2). Compared to the cost of O(2^k)in\nclassical computing, QF-hNet demonstrates the quantum advantages. Evaluation\nresults show that QF-pNet and QF-hNet can achieve 97.10% and 98.27% accuracy,\nrespectively. Results further show that for input sizes of neural computation\ngrow from 16 to 2,048, the cost reduction of QuantumFlow increased from 2.4x to\n64x. Furthermore, on MNIST dataset, QF-hNet can achieve accuracy of 94.09%,\nwhile the cost reduction against the classical computer reaches 10.85x. To the\nbest of our knowledge, QuantumFlow is the first work to demonstrate the\npotential quantum advantage on neural network computation.",
+                    "authors": "Weiwen Jiang, Jinjun Xiong, Yiyu Shi",
+                    "published": "2020-06-26",
+                    "updated": "2020-09-09",
+                    "primary_cat": "quant-ph",
+                    "cats": [
+                        "quant-ph",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Although quantum computers are expected to dramatically outperform classical computers, so far quantum advantages have only been shown in a limited number of applications, such as prime factorization1 and sampling the output of random quantum circuits2. In this work, we will demonstrate that quantum computers can achieve potential quantum advantage on neural network computation, a very common task in the prevalence of arti\ufb01cial intelligence (AI)1. In the past decade, neural networks3\u20135 have become the mainstream machine learning models, and have achieved consistent success in numerous Arti\ufb01cial Intelligence (AI) applications, such as image classi\ufb01cation6\u20139, object detection10\u201313, and natural language processing14\u201316. When the neural networks are applied to a speci\ufb01c \ufb01eld (e.g., AI in medical or AI in astronomy), the high-resolution input images bring new challenges. For example, one 3D-MRI image contains 224 \u00d7 224 \u00d7 10 \u22485 \u00d7 106 pixels17 while one Square Kilometre Array (SKA) science data contains 32,768 \u00d7 32,768 \u22481 \u00d7 109 pixels18,19. The large inputs greatly increase the computa1Quirk demos at https://wjiang.nd.edu/categories/qf/ tion in neural network20, which gradually becomes the performance bottleneck. Among all computing platforms, the quantum computer is a most promising one to address such challenges2,21 as a quantum accelerator for neural networks22\u201324. Unlike classical computers with N digit bits to represent 1 Nbit number at one time, quantum computers with K qbits can represent 2K numbers and manipulate them at the same time25. Recently, a quantum machine learning programming framework, TensorFlow Quantum, has been proposed26; however, how to exploit the power of quantum computing for neural networks is still remained unknown. One of the most challenging obstacles to implementing neural network computation on a quantum computer is the missing link between the design of neural networks and that of quantum circuits. The existing works separately design them from two directions. The \ufb01rst direction is to map the existing neural networks designed for classical computers to quantum circuits; for instance, recent works27\u201330 map McCulloch-Pitts (MCP) neurons31 onto quantum circuits. Such an approach has dif\ufb01culties in consistently mapping the trained model to quantum circuits. For example, it needs a large number of 1 \fQuantumFlow Co-Design Machine Leanring Models (QF-pNet, QF-hNet) Classic Computer Quantum Computer Quantum Circuit Design and Optimization (QF-Circ) Efficient Forward/Backward Propagation (QF-FB) Datasets QF-Map QF-FB(C) QF-FB(Q) P-LYR U-LYR N-LYR Figure 1. QuantumFlow, an end-to-end co-design framework, provides a missing link between neural network and quantum circuit designs, which is composed of QF-Nets, QFhNet, QF-FB, QF-Circ, QF-Map that work collaboratively to design neural networks and their quantum implementations. qbits to realize the multiplication of real numbers. To overcome this problem, some existing works27\u201330 assume binary representation (i.e., \u201c-1\u201d and \u201c+1\u201d) of activation, which cannot well represent data as seen in modern machine learning applications. This has also been demonstrated in work32, where data in the interval of (0,2\u03c0] instead of binary representation are mapped onto the Bloch sphere to achieve high accuracy for support vector machines (SVMs). In addition, some typical operations in neural networks cannot be implemented on quantum circuits, leading to inconsistency. For example, to enable deep learning, batch normalization is a key step in a deep neural network to improve the training speed, model performance, and stability; however, directly conducting normalization on the output qbit (say normalizing the qbit with maximum probability to probability of 100%) is equivalent to reset a qbit without measurement, which is simply impossible. In consequence, batch normalization is not applied in the existing multi-layer network implementation28. The other direction is to design neural networks dedicated to quantum computers, like the tree tensor network (TTN)33,34. Such an approach suffers from scalability problems. More speci\ufb01cally, the effectiveness of neural networks is based on a trained model via the forward and backward propagation on large training sets. However, it is too costly to directly train one network by applying thousands of times forward and backward propagation on quantum computers; in particular, there are limited available quantum computers for public access at the current stage. An alternative way is to run a quantum simulator on a classical computer to train models, but the time complexity of quantum simulation is O(2m), where m is the number of qbits. This signi\ufb01cantly restricts the trainable network size for quantum circuits. To address all the above obstacles, it demands to take quantum circuit implementation into consideration when designing neural networks. This paper proposes the \ufb01rst codesign framework, namely QuantumFlow, where \ufb01ve subcomponents (QF-pNet, QF-hNet, QF-FB, QF-Circ, and QFMap) work collaboratively to design neural networks and implement them to quantum computers, as shown in Figure 1. In QuantumFlow, the start point is the co-design of networks and quantum circuits. We \ufb01rst propose QF-pNet, which contains multiple neural computation layer, namely P-LYR. In the design of P-LYR, we take full advantage of the ability of quantum logic gates to operate random variables represented by qbits. Speci\ufb01cally, data in P-LYR are modeled as random variables following a two-point distribution, which is consistent to the expression of a qbit; computations in P-LYR can be easily implemented by the basic quantum logic gates. Kindly note that P-LYR can model both inputs and weights to be random variables. But because binary weights can achieve comparable high accuracy for deep neural network applications35 and signi\ufb01cantly reduce circuit complexity, we employ random variables for inputs only and binary values for weights in P-LYR. Bene\ufb01ting from the quantum-aware data interpretation for inputs, P-LYR can be attached to the output qbits of previous layers without measurement; however, it utilizes 2k qbits to represent 2k input data items, and the computation needs at least one quantum gate for each qbit. Therefore, it has high cost complexity. Towards achieving the quantum advantage, we propose a hybrid network, namely QF-hNet, which is composed of two types of neural computation layers: P-LYR and U-LYR. ULYR is based on the unitary matrix, where 2k input data are converted to a vector in the unitary matrix, such that all data can be represented by the amplitudes of states in a quantum circuit with k qbits. The reduction in input qbits provides the possibility to achieve quantum advantage; however, the stateof-the-art implementation27 using hypergraph state for computation still has the cost complexity of O(2k). In this work, we devise a novel optimization algorithm to guarantee the cost complexity of U-LYR to be O(k2), which takes full use of the properties of neural networks and quantum logic gates. Compared with the complexity of O(2k) on classical computing platforms, U-LYR demonstrates the quantum advantages of executing neural network computations. In addition to neural computation, QF-Nets also integrates a quantum-friendly batch normalization N-LYR, which can be plugged into both QF-pNet and QF-hNet. It includes additional parameters to normalize the output of a neuron, which are tuned during the training phase. To support both the inference and training of QF-Nets, we further develop QF-FB, a forward/backward propagation engine. When QF-FB is integrated into PyTorch to conduct inference and training of QF-Nets on classical computers, we denote it as QF-FB(C). QF-FB can also be executed on a quantum computer or a quantum simulator. Based on Qiskit Aer simulator, we implement QF-FB(Q) for inference with or without error models. For each operation in QF-Nets (e.g., neural computations and batch normalization), a corresponding quantum circuit is designed in QF-Circ. In neural computation, an encoder is involved to encode the inputs and weights. The output will be sent to the batch normalization which involves additional control qbits to adjust the probability of a given qbit to be ranged 2/14 \f                  MLP(C) w/o BN {1,5} 0.2 0.4 0.6 0.8 1.0 0.0 {3,6} {0,3,6} {0,1,3,6,9} {0,1,2,3,4} {0,3,6,9} {1,3,6} {3,8} {3,9} QF-pNet w/o BN FFNN(Q) w/o BN MLP(C) w/ BN binMLP(C) w/ BN binMLP(C) w/o BN QF-pNet w/ BN QF-hNet w/o BN QF-hNet w/ BN FFNN(Q) w/ BN Figure 2. QF-hNet achieves state-of-the-art accuracy in image classi\ufb01cations on different sub-datasets of MNIST. from 0 to 1. Based on QF-Nets and QF-Circ, QF-Map is an automatic tool to conduct (1) network-to-circuit mapping (from QF-Nets to QF-Circ); (2) virtual-to-physic mapping (from virtual qbits in QF-Circ to physic qbits in quantum processors). Network-to-circuit mapping guarantees the consistency between QF-Nets and QF-Circ with or without internal measurement; while virtual-to-physic mapping is based on Qiskit with the consideration of error rates. As a whole, given a dataset, QuantumFlow can design and train a quantum-friendly neural network and automatically generate the corresponding quantum circuit. The proposed co-design framework is evaluated on the IBM Qikist Aer simulator and IBM Quantum Processors. Results This section presents the evaluation results of all \ufb01ve subcomponents in QuantumFlow. We \ufb01rst evaluate the effectiveness of QF-Nets (i.e., QF-pNet and QF-hNet) on the commonly used MNIST dataset36 for the classi\ufb01cation task. Then, we show the consistency between QF-FB(C) on classical computers and QF-FB(Q) on the Qiskit Aer simulator. Next, we show that QF-Map is a key to achieve quantum advantage. We \ufb01nally conduct an end-to-end case study on a binary classi\ufb01cation test case on IBM quantum processors to test QF-Circ. QF-Nets Achieve High Accuracy on MNIST Figure 2 reports the results of different approaches for the classi\ufb01cation of handwritten digits on the commonly used MNIST dataset36. Results clearly show that with the same network structure (i.e., the same number of layers and the same number of neurons in each layer), the proposed QFhNet can achieve the highest accuracy than the existing models: (i) multi-level perceptron (MLP) with binary weights for the classical computer, denoted as MLP(C); (ii) MLP with binary inputs and weights designed for the classical computer, denoted as binMLP(C); and (iii) a state-of-the-art quantumaware neural network with binary inputs and weights28, denoted as FFNN(Q). Before reporting the detailed results, we \ufb01rst discuss the experimental setting. In this experiment, we extract sub-datasets from MNIST, which originally include 10 classes. For instance, {3,6} indicates the sub-datasets with two classes (i.e., digits 3 and 6), which are commonly used in quantum machine learning (e.g., Tensor\ufb02ow-Quantum37). To evaluate the advantages of the proposed QF-Nets, we further include more complicated sub-datasets, {3,8}, {3,9}, {1,5} for two classes. In addition, we show that QF-Nets can work well on larger datasets, including {0,3,6} and {1,3,6} for three classes, and {0,3,6,9}, {0,1,3,6,9}, {0,1,2,3,4} for four and \ufb01ve classes. For the datasets with two or three classes, the original image is downsampled from the resolution of 28\u00d728 to 4\u00d74, while it is downsampled to 8 \u00d7 8 for datasets with four or \ufb01ve classes. All original images in MNIST and the downsampled images are with grey levels. For all involved datasets, we employ a two-layer neural network, where the \ufb01rst layer contains 4 neurons for two-class datasets, 8 neurons for three-class datasets, and 16 neurons for fourand \ufb01ve-class datasets. The second layer contains the same number of neurons as the number of classes in datasets. Kindly note that theses architectures are manually tuned for higher accuracy, the neural architecture search (NAS) will be our future work. In the experiments, for each network, we have two implementations: one with batch normalization (w/ BN) and one without batch normalization (w/o BN). Kindly note that FFNN28 does not consider batch normalization between layers. To show the bene\ufb01ts and generality of our newly proposed BN for improving the quantum circuits\u2019 accuracy, we add that same functionality to FFNN for comparison. From the results in Figure 2, we can see that the proposed \u201cQFhNet w/ BN\u201d (abbr. QF-hNet_BN) achieves the highest accuracy among all networks (even higher than MLP running on classical computers). Speci\ufb01cally, for the dataset of {3,6}, the accuracy of QF-hNet_BN is 98.27%, achieving 3.01% and 15.27% accuracy gain against MLP(C) and FFNN(Q), respectively. It even achieves a 1.17% accuracy gain compared to QF-pNet_BN. An interesting observation attained from this result is that with the increasing number of classes in the dataset, QF-hNet_BN can maintain the accuracy to be larger than 90%, while other competitors suffer an accuracy loss. Speci\ufb01cally, for dataset {0,3,6} (input resolution of 4 \u00d7 4), {0,3,6,9} (input resolution of 8 \u00d7 8), {0,1,3,6,9} (input resolution of 8 \u00d7 8), the accuracy of QF-hNet_BN are 90.40%, 93.63% and 92.62%; however, for MLP(C), these \ufb01gures are 75.37%, 82.89%, and 70.19%. This is achieved by the hybrid use of two types of neural computation in QF-hNet to better extract features in images. The above results validate that the proposed QF-hNet has a great potential in solving machine learning problems and our co-design framework is effective to design a quantum neural network with high accuracy. Furthermore, we have an observation for our proposed 3/14 \fTable 1. Inference accuracy and ef\ufb01ciency comparison between QF-FB(C) and QF-FB(Q) on both QF-pNet and QF-hNet using MNIST dataset to show the consistency of implementations of QF-Nets on classical computers and quantum computers. QF-pNet QF-hNet Qbits (Neurons) Accuracy Elapsed CPU Time Qbits (Neurons) Accuracy Elapsed CPU Time dataset L1 L2 QF-FB(C) QF-FB(Q) Diff. QF-FB(C) QF-FB(Q) L1 L2 QF-FB(C) QF-FB(Q) Diff. QF-FB(C) QF-FB(Q) {3,6} 28(4) 12(2) 97.10% 95.53% -1.57% 5.13S 2,555H 7(4) 5(2) 98.27% 97.46% -0.81% 4.30S 16.57H {3,8} 28(4) 12(2) 86.84% 83.59% -3.25% 5.59S 2,631H 7(4) 5(2) 87.40% 88.06% +0.54% 4.05S 16.56H {1,3,6} 28(8) 18(3) 87.91% 81.99% -5.92% 15.89S 14,650H 7(8) 8(3) 88.53% 88.14% -0.39% 6.96S 47.98H batch normalization (BN). For almost all test cases, BN helps to improve the accuracy of QF-pNet and QF-hNet, and the most signi\ufb01cant improvement is observed at dataset {1,5}, from less than 70% to 84.56% for QF-pNet and 90.33% to 96.60% for QF-hNet. Interestingly, BN also helps to improve MLP(C) accuracy signi\ufb01cantly for dataset {1,3,6} (from less than 60% to 81.99%), with a slight accuracy improvement for dataset {3,6} and a slight accuracy drop for dataset {3,8}. This shows that the importance of batch normalization in improving model performance and the proposed BN is de\ufb01nitely useful for quantum neural networks. QF-FB(C) and QF-FB(Q) are Consistent Next, we evaluate the results of QF-FB(C) for both QFpNet and QF-hNet on classical computers, and that of QFFB(Q) simulation on classical computers for the quantum circuits QF-Circ built upon QF-Nets. Table 1 reports the comparison results in the usage of qbits in QF-Circ, inference accuracy and elapsed time, where results under Column QF-FB(C) are the golden results. Because of the limitation of Qiskit Aer (whose backend is \u201cibmq_qasm_simulator\u201d) used in QFFB(Q) that can maximally support 32 qbits, we measure the results after each neuron. We select three datasets, including {3,6}, {3,8}, and {1,3,6}, for evaluation. Datasets with more classes (e.g., {0,3,6,9}) are based on larger inputs, which will lead to the usage of qbits in QF-pNet to exceed the limitation (i.e., 32 qbits). Speci\ufb01cally, for 4\u00d74 input image in QF-pNet, in the \ufb01rst hidden layer, it needs 23 qbits (16 input qbits, 4 encoding qbits, and 3 auxiliary qbits) for neural computation and 4 qbits for batch normalization, and 1 output qbit; as a result, it requires 28 qbits in total. On the contrary, since QF-hNet is designed in terms of the quantum circuit implementation, which takes full use of all states of k qbits to represent 2k data. In consequence, the number of required qbits can be signi\ufb01cantly reduced. In detail, for the 4 \u00d7 4 input, it needs 4 qbits to represent the data, 1 output qbit, and 2 auxiliary qbits; as a result, it only needs 7 qbits in total. The number of qbits used for each hidden layer (\u201cL1\u201d and \u201cL2\u201d) is reported in column \u201cQbits\u201d, where numbers in parenthesis indicate the number of neurons in a hidden layer. Column \u201cAccuracy\u201d in Table 1 reports the accuracy comparison. For QF-FB(C), there will be no difference in accuracy among different executions. For QF-FB(Q), we implement the obtained QF-Circ from QF-Nets on Qiskit Aer simulation with 8,192 shots. We have the following two observations from these results: (1) There exist accuracy differences bedeviation [0.0,1.0] [0.1,0.9] [0.2,0.8] [0.3,0.7] [0.4,0.6] [0.6,0.4] [0.7,0.3] [0.8,0.2] [0.9,0.1] [1.0,0.0] [0.5,0.5] inputs -0.08 -0.06 -0.04 -0.02 -0.00 0.02 0.04 ibmq_armonk QF-FB(C) QF-FB(Q)-ideal QF-FB(Q)-noise Figure 3. Output probability comparison on QF-FB(C), QFFB(Q)-ideal assuming perfect qbits, QF-FB(Q)-noise applying noise model for \u201cibm_armonk\u201d backend, and results of circuit design (\u201cdesign 4\u201d) in Figure 5(d) on \u201cibm_armonk\u201d backend on IBM quantum processor. tween QF-FB(C) and QF-FB(Q). This is because Qiskit Aer simulation used in QF-FB(Q) is based on the Monte Carlo method, leading to the variation. In addition, since the output probability of different neurons may quite close in some cases, it will easily result in different classi\ufb01cation results for small variations. (2) Such accuracy differences for QF-hNet is much less than that of QF-pNet, because QF-pNet utilizes much more qbits, which leads to the accumulation of errors. In QF-hNet, we can see that there is a small difference between QF-FB(C) and QF-FB(Q). For the dataset {3,8}, QFFB(Q) can even achieve higher accuracy. The above results demonstrate both QF-pNet and QF-hNet can be consistently implemented on classical and quantum computers. Column \u201cElapsed Time\u201d in Table 1 demonstrates the ef\ufb01ciency of QF-FB. The elapsed time is the inference time (i.e., forward propagation), used for executing all images in the test datasets, including 1968, 1983, and 3102 images for {3,6}, {3,8}, and {1,3,6}, respectively. As we can see from the table, QF-FB(Q) for QF-pNet takes over 2,500 Hours for classifying 2 digits and 14,000 Hours for classifying 3 digits, and these \ufb01gures are 16 Hours and 48 Hours for QF-hNet. On the other hand, QF-FB(C) only takes less than 16 seconds for both QF-Nets on all datasets. The speedup of QF-FB(C) over QFFB(Q) is more than six orders of magnitude larger (i.e., 106\u00d7) for QF-pNet, and more than four orders of magnitude larger (i.e., 104\u00d7) for QF-hNet. This veri\ufb01es that QF-FB(C) can provide an ef\ufb01cient forward propagation procedure to support the lengthy training of QF-pNet. In Figure 3, we further verify the accuracy of QF-FB by conducting a comparison for design 4 in Figure 5(d) on IBM quantum processor with \u201cibm_armonk\u201d backend. Kindly note that the quantum processor backend is selected by QF-Map. In this experiment, the result of QF-FB(C) is taken as a base4/14 \f101 16 32 64 128 256 512 1024 2048 102 103 Cost (# of operators) Input Sizes of Neural Computation 104 U-LYR 50 trails U-LYR Average FC(Q) 50 trails FC(Q) Average FC(C) FC(C) v.s. U-LYR 64\u00b4 32\u00b4 20\u00b4 13\u00b4 7.6\u00b4 4.8\u00b4 3.3\u00b4 2.4\u00b4 Figure 4. Demonstration of Quantum Advantage Achieved by U-LYR in QuantumFlow: comparison is conducted by using 50 random generated weights for each input size. line. In the \ufb01gure, the x-axis and y-axis represent the inputs and deviation, respectively. The deviation indicates the difference between the baseline and the results obtained by Qiskit Aer simulation or that by executing on IBM quantum processor. For comparison, we involve two con\ufb01gurations for QFFB(Q): (1) QF-FB(Q)-ideal assuming perfect qbits; (2) QFFB(Q)-noise with error models derived from \u201cibm_armonk\u201d. We launch either simulation or execution for respective approaches for 10 times, each of which is represented by a dot in Figure 3. We observe that the results of QF-FB(Q)-ideal are distributed around that generated by QF-FB(C) within 1% deviation; while QF-FB(Q)-noise obtains similar results of that on the IBM quantum processor. These results verify that the QF-Nets on the classical computer can achieve consistent results with that of QF-Circ deployed on a quantum computer with perfect qbits. QF-Map is the Key to Achieve Quantum Advantage Two sets of experiments are conducted to demonstrate the quantum advantage achieved by QuantumFlow. First, we conduct an ablation study to compare the operator/gate usage of the core computation component, neural computation layer. Then, the comparison on gate usage is further conducted on the trained neural networks for different subdatasets from MNIST. In these experiments, we compare QuantumFlow to MLP(C) and FFNN(Q)28. For MLP(C), we consider the adder/multiplier as the basic operators, while for FFNN(Q) and QuantumFlow, we take the quantum logic gate (e.g., Pauli-X, Controlled Not, Toffoli) as the operators. The operator usage re\ufb02ects the total cycles for neural computation. Kindly note that the results of QuantumFlow are obtained by using QF-Map on neural computation U-LYR; and that of FFNN(Q) are based on the state-of-the-art hypergraph state approach proposed in27. For a fair comparison, QuantumFlow and FFNN(Q) are based on the same weights. Figure 4 reports the comparison results for the core component in neural network, the neural computation layer. The x-axis represents the input size of the neural computation, and the y-axis stands for the cost, that is, the number of operators used in the corresponding design. For quantum implementation (both FC(Q)27 in FFNN(Q)28 and U-LYR in QuantumTable 2. QuantumFlow demonstrates quantum advantages on neural networks for MNIST datasets with the increasing model sizes: comparison on the number of used gates. Dataset Structure MLP(C) FFNN(Q) QF-hNet(Q) In L1 L2 L1 L2 Tot. L1 L2 Tot. Red. L1 L2 Tot. Red. {1,5} 16 4 2 132 18 150 80 38 118 1.27\u00d7 74 38 112 1.34\u00d7 {3,6} 16 4 2 96 38 134 1.12\u00d7 58 38 96 1.56\u00d7 {3,8} 16 4 2 76 34 110 1.36\u00d7 58 34 92 1.63\u00d7 {3,9} 16 4 2 98 42 140 1.07\u00d7 68 42 110 1.36\u00d7 {0,3,6} 16 8 3 264 51 315 173 175 348 0.91\u00d7 106 175 281 1.12\u00d7 {1,3,6} 16 8 3 209 161 370 0.85\u00d7 139 161 300 1.05\u00d7 {0,3,6,9} 64 16 4 2064 132 2196 1893 572 2465 0.89\u00d7 434 572 1006 2.18\u00d7 {0,1,3,6,9} 64 16 5 2064 165 2229 1809 645 2454 0.91\u00d7 437 645 1082 2.06\u00d7 {0,1,2,3,4} 64 16 5 1677 669 2346 0.95\u00d7 445 669 1114 2.00\u00d7 {0,1,3,6,9}\u2217256 8 5 4104 85 4189 5030 251 5281 0.79\u00d7 135 251 386 10.85\u00d7 \u2217: Model with 16\u00d716 resolution input for dataset {0,1,3,6,9} to test scalability, whose accuracy is 94.09%, which is higher than 8\u00d78 input with accuracy of 92.62%. Flow), the value of weights will affect the gate usage, so we generate 50 sets of weights for each scale of input, and the dots on the lines in this \ufb01gure represent average cost. From this \ufb01gure, it clearly shows that the cost of FC(C) in MLP(C) on classical computing platforms grows exponentially along with the increase of inputs. The state-of-the-art quantum implementation FC(Q) has the similar exponentially growing trend. On the other hand, we can see that the growing trend of U-LYR is much slower. As a result, the cost reduction continuously increases along with the growth of the input size of neural computation. For the input size of 16 and 32, the average cost reductions are 2.4\u00d7 and 3.3\u00d7, compared with the implementations on classical computers. When the input size grows to 2,048, the cost reduction increased to 64\u00d7 on average. The cost reduction trends in this \ufb01gure clearly demonstrate the quantum advantage achieved by U-LYR. In the Methods section, for the neural computation with an input size of 2k, we will show that the complexity for quantum implementation is O(k2), while it is O(2k) for classical computers. Table 2 reports the comparison results for the whole network. The neural network models for MNIST in Figure 2 are deployed to quantum circuits to get the cost. In addition, to demonstrate the scalability, we further include a new model for dataset \u201c{0,1,3,6,9}\u2217\u201d, which takes the larger sized inputs but less neurons in the \ufb01rst layer L1 and having higher accuracy over \u201c{0,1,3,6,9}\u201d. In this table, columns L1, L2, and Tot. under three approaches report the number of gates used in the \ufb01rst and second layers, and in the whole network. Columns \u201cRed.\u201d represent the comparison with baseline MLP(C). From the table, it is clear to see that all cases implemented by QF-hNet can achieve cost reduction over MLP(C), while for datasets with more than 3 classes, FFNN(Q) needs more gates than MLP(C). A further observation made in the results is that QF-hNet can achieve higher cost reduction with the increase of input size. Speci\ufb01cally, for input size is 16, the reduction ranges from 1.05\u00d7 to 1.63\u00d7. The reduction increases 5/14 \fto 2.18\u00d7 for input size is 64, and it continuously increases to 10.85\u00d7 when the input size grows to 256. The above results are consistent with the results shown in Figure 4. It further indicates that even the second layer in QF-hNet uses the PLYR which requires more gates for implementation, the quantum advantage can still be achieved for the whole network because the \ufb01rst layer using U-LYR can signi\ufb01cantly reduce the number of gates. Above all, QuantumFlow demonstrates the quantum advantages on MNIST dataset. QF-Circ on IBM Quantum Processor This subsection further evaluates the ef\ufb01cacy of QuantumFlow on IBM Quantum Processors. We \ufb01rst show the importance of quantum circuit optimization in QF-Circ to minimize the number of required qbits. Based on the optimized circuit design, we then deploy a 2-input binary classi\ufb01er on IBM quantum processors. Figure 5 demonstrates the optimization of a 2-input neuron step by step. All quantum circuits in Figures 5(a)-(d) achieve the same functionality, but with a different number of required qbits. The equivalency of all designs will be demonstrated in the Supplementary Information. Design 1 in Figure 5(a) is directly derived from the design methodology presented in Methods section. To optimize the circuit using fewer qbits, we \ufb01rst convert it to the circuit in Figure 5(b), denoted as design 2. Since there is only one controlled-Z gate from qbit I0 to qbit E/O, we can merge these two qbits, and obtain an optimized design in Figure 5(c) with 2 qbits, denoted as design 3. The circuit can be further optimized to use 1 qbit, as shown in Figure 5(d), denoted as design 4. The function f in design 4 is de\ufb01ned as follows: f(\u03b1,\u03b2) = 2 \u00b7arcsin( p x+ y\u22122 \u00b7x\u00b7y), (1) where x = sin2 \u03b1 2 , y = sin2 \u03b2 2 , representing input probabilities. To compare these designs, we deploy them onto IBM Quantum Processors, where \u201cibm_velencia\u201d backend is selected by QF-Map. In the experiments, we use the results from QFFB(C) as the golden results. Figure 5(e) reports the deviations of design 1 and design 4 against the golden results. The results clearly show that design 4 is more robust because it uses fewer qbits in the circuit. Speci\ufb01cally, the deviation of design 4 against golden results is always less than 5%, while reaching up to 13% for design 1. In the following experiments, design 4 is applied in QF-Circ. Next, we are ready to introduce the case study on an end-toend binary classi\ufb01cation problem as shown in Figure 6. In this case study, we train the QF-pNet based on QF-FB(C). Then, the tuned parameters are applied to generate QF-Circ. Finally, QF-Map optimizes the deployment of QF-Circ to IBM quantum processor, selecting the \u201cibmq_essex\u201d backend. The classi\ufb01cation problem is illustrated in Figure 6(a), which is a binary classi\ufb01cation problem (two classes) with two inputs: x and y. For instance, if x = 0.2 and y = 0.6, it indicates class 0. The QF-pNet, QF-Circ, and QF-Map are demonstrated in Figure 6(b)-(d). First, Figure 6(b) shows that E/O I0 I1 |0\u00f1 |0\u00f1 |0\u00f1 Ry(b) Ry(a) X X H H E/O I0 I1 |0\u00f1 |0\u00f1 |0\u00f1 Ry(b) Ry(a) X X H H \u00c5 I0/E/O I1 |0\u00f1 |0\u00f1 Ry(b) Ry(a) X X \u00c5 I0/I1 E/O |0\u00f1 Ry(f(a,b)) (a) (b) (c) (d) [0.0,1.0] [0.1,0.9] [0.2,0.8] [0.3,0.7] [0.4,0.6] [0.6,0.4] [0.7,0.3] [0.8,0.2] [0.9,0.1] [1.0,0.0] [0.5,0.5] deviation inputs 0.00 0.03 0.06 0.09 0.12 0.15 (e) design 1 design 4 Figure 5. Evaluation of the quantum circuits for a two-input neural computation, where weights are {-1,+1}: (a) design 1: original neural computation design; (b-d) three optimized designs (design 2-4), based on design 1; (e) the deviation of design 1 and design 4 obtained from \u201cibm_velencia\u201d backend IBM quantum processor, using QF-FB(C) as golden results. QF-pNet consists of one hidden layer with one 2-input neuron and batch normalization. The output is the probability p0 of class 0. Speci\ufb01cally, an input is recognized as class 0 if p0 \u22650.5; otherwise it is identi\ufb01ed as class 1. The quantum circuit QF-Circ of the above QF-pNet is shown in Figure 6(c). The circuit is composed of three parts, (1) neural computation, (2) batch_adj in batch normalization, and (3) indiv_adj in batch normalization. The neural computation is based on design 4 as shown in Figure 5(d). The parameter of Ry gate in neural computation at qbit q0 is determined by the inputs x and y. Speci\ufb01cally, f(x,y) = 2 \u00b7 arcsin(\u221ax+ y\u22122 \u00b7x\u00b7y), as shown in Formula 1. Then, batch normalization is implemented in two steps, where qbits q2 and q4 are initialized according to the trained BN parameters. During the process, q1 holds the intermediate results after batch_adj, and q3 holds the \ufb01nal results after indiv_adj. Finally, we measure the output on qbit q32. After building QF-Circ, the next step is to map qbits from the designed circuit to the physic qbits on the quantum processor, and this is achieved through our QF-Map. In this experiment, QF-Map selects \u201cibm_essex\u201d as backend with its physical properties shown in Figure 6(d), where error rates of each qbit and each connection are illustrated by different colors. By following the rules as de\ufb01ned by QF-Map (see Method section), we obtain the physically mapped QF-Circ shown in Figure 6(d). For example, the input q0 is mapped to the physical qbit labeled as 4. After QuantumFlow goes through all the steps from input data to the physic quantum processor, we can perform inference on the quantum computer. In this experiments, we 2A Quirk-based example of inputs 0.2 and 0.6 leading to f(x,y) = 1.6910 can be accessed by https://wjiang.nd.edu/quirk_0_2_0_6.html, which is accessible at 06-19-2020. The output probability of 60.3% is larger than 50%, implying the inputs belong to class 0. 6/14 \f\u0013 \u0014 \u0015 q2 q3 q4 1.0 1.0 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.4 0.2 1.0 0.5 0.0 1.0 0.5 0.0 1.0 0.5 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.5 1.0 0.8 0.6 0.4 0.2 \u0016 \u0017 7.516e-3 1.788e-2 CNOT error rate 3.474e-4 6.272e-4 single qbit error rate |0\u00f1 |0\u00f1 |0\u00f1 x x y y q1 q0 (a) (b) (d) (e) (f) (g) (h) (c) q2 \u00c5 |0\u00f1 q3 |0\u00f1 q4 Ry(0.765) X \u00c5 \u00c5 q0 q1 x=0.2; y=0.6 input output class 0 class 0 0.8 0.6 0.8 0.6 x=0.8; y=0.8 49.53% 49.44% class 1 Batch Normalization batch_adj (t==0) indiv_adj neural comp. weights input prob of 0 \u22640.5 0.765 +1 -1 2.793 class 1 class 0 class 1 x y p x y p x y p x y p Ry(f(x,y)) Ry(2.793) Figure 6. Results of a binary classi\ufb01cation case study on IBM quantum processor of \u201cibmq_essex\u201d backend: (a) binary classi\ufb01cation with two inputs \u201cx\u201d and \u201cy\u201d; (b) QF-Nets with trained parameters; (c) QF-Circ derived from the trained QF-Nets; (d) the virtual-to-physic mapping obtained by QF-Map upon \u201cibmq_essex\u201d quantum processor; (e) QF-FB(C) achieves 100% accuracy; (f) QF-FB(Q) achieves 98% accuracy where 2 marked error cases having probability deviation within 0.6% ; (g) results on \u201cibmq_essex\u201d using the default mapping, achieving 68% accuracy; (h) results obtained by \u201cibmq_essex\u201d with the mapping in (d), achieving 82% accuracy; shots number in all tests is set as 8,192. test 100 combinations of inputs from \u27e8x,y\u27e9= \u27e80.1,0.1\u27e9to \u27e8x,y\u27e9= \u27e81.0,1.0\u27e9. First, we obtain the results using QF-FB(C) as golden results and QF-FB(Q) as quantum simulation assuming perfect qbits, which are reported in Figure 6(e) and (f), achieving 100% and 98% prediction accuracy. The results verify the correctness of the proposed QF-pNet. Second, the results obtained on quantum processors are shown in Figure 6(h), which achieves 82% accuracy in prediction. For comparison, in Figure 6(g), we also show the results obtained by using the default mapping algorithm in IBM Qiskit, whose accuracy is only 68%. This result demonstrates the value of QF-Map in further improving the physically achievable accuracy on a physical quantum processor with errors. Discussion In summary, we propose a holistic QuantumFlow framework to co-design the neural networks and quantum circuits. Novel quantum-aware QF-Nets are \ufb01rst designed. Then, an accurate and ef\ufb01cient inference engine, QF-FB, is proposed to enable the training of QF-Nets on classical computers. Based on QFNets and the training results, the QF-Map can automatically generate and optimize a corresponding quantum circuit, QFCirc. Finally, QF-Map can further map QF-Circ to a quantum processor in terms of qbits\u2019 error rates. The neural computation layer is one key component in QuantumFlow to achieve state-of-the-art accuracy and quanTable 3. Comparison of the implementation of Neural Computation with m = 2k input neurons. Layers FC(C)38 FC(Q)28 P-LYR U-LYR Complexity # Bits/Qbits O(2k) O(k) O(2k) O(k) # Operators O(2k) O(2k) O(k \u00b72k) O(k2) Data Representation Input Data F32 Bin R.V. F32 Weights Bin (F32) Bin Bin (R.V.) Bin Connect Layers w/o Measurement \u2713 \u2713 \u00d7 Summary Flexibility \u00d7 \u2713 \u00d7 Qu. Adv. \u00d7 \u00d7 \u2713 tum advantage. We have shown in Figure 2 that the existing quantum-aware neural network28 that interprets inputs as the binary form will degrade the network accuracy. To address this problem, in QF-pNet, we \ufb01rst propose a probability-based neural computation layer, denoted as P-LYR, which interprets real number inputs as random variables following a two-point distribution. As shown in Table 3, P-LYR can represent both input and weight data using random variables, and it can directly connect layers without measurement. In summary, PLYR provides better \ufb02exibility to perform neural computation than others; however, it suffers high complexity, i.e., O(2k) for the usage of qbits and O(k\u00d72k) for the usage of operators (basic quantum gates). In order to acquire quantum advantages, we further propose a unitary matrix based neural computation layer, called U7/14 \fLYR. As illustrated in Table 3, U-LYR sacri\ufb01ces some degree of \ufb02exibility on data representation and non-linear function but can signi\ufb01cantly reduce the circuit complexity. Specifically, with the help of QF-Map, the number of basic operators used by U-LYR can be reduced from O(2k) to O(k2), compared to FC(C) and FC(Q). Kindly note that this work does not take the cost of inputs encoding into consideration in demonstrating quantum advantage; instead, we focus on the speedup of the commonly used computation component layer, that is, the neural computation layer. The cost of encoding inputs can be reduced to O(1) by preprocessing data and storing them into quantum memory, or approximating the quantum states by using basic gate (e.g., Ry). For neural computation, we demonstrated that U-LYR can successfully achieve quantum advantage in the next section. Batch normalization is another key technique in improving accuracy, since the backbone of the quantum-friendly neuron computation layers (P-LYR and U-LYR) is similar to that in classical computers, using both linear and non-linear functions. This can be seen from the results in Figure 2. Batch normalization can achieve better model accuracy, mainly because the data passing a nonlinear function y2 will lead to outputs to be signi\ufb01cantly shrunken to a small range around 0 for real number representation and 1/m for a two-point distribution representation, where m is the number of inputs. Unlike straightforwardly doing normalization on classical computers, it is non-trivial to normalize a set of qbits. Innovations are made in QuantumFlow for a quantum-friendly normalization. The philosophy of co-design is demonstrated in the design of P-LYR, U-LYR, and N-LYR. From the neural network design, we take the known operations as the backbones in PLYR, U-LYR, and N-LYR; while from the quantum circuit design, we take full use of its ability in processing probabilistic computation and unitary matrix based computations to make P-LYR, U-LYR, and N-LYR quantum-friendly. In addition, as will be shown in the next section, the key to achieve quantum advantage for U-LYR is that QF-Map fully considers the \ufb02exibility of the neural networks (i.e., the order of inputs can be changed), while the requirement of continuously executing machine learning algorithms on the quantum computer leads to a hybrid neural network, QF-hNet, with both neural computation operations: P-LYR and U-LYR. Without the co-design, the previous works did not exploit quantum advantages in implementing neural networks on quantum computers, which re\ufb02ects the importance of conducting co-design. We have experimentally tested QuantumFlow on a 32-qbit Qiskit Aer simulator and a 5-qbit IBM quantum processor based on superconducting technology. We show that the proposed quantum oriented neural networks QF-Nets can obtain state-of-the-art accuracy on the MNIST dataset. It can even outperform the conventional model on a similar scale for the classical computer. For the experiments on IBM quantum processors, we demonstrate that, even with the high error rates of the current quantum processor, QF-Nets can be applied to classi\ufb01cation tasks with high accuracy. In order to accelerate the QF-FB on classical computers to \u2026 \u2026 \u2026 \u2026 \u2026 m Swixi w0 w1 wm-1 I1 I0 O=E(y2) Im-1 y= y2 (c) R R E C A R R (p0) (x0) (x1) (xm-1) (p1) (pm-1) 28\u00b428 downsample grey level 4\u00b44 (a) (b) \u2026 \u2026 x0 r.v. -1 1.0 0.9 1.0 0.0 0.1 0.0 +1 x1 x15 m Swiui w0 w1 wm-1 I1 I0 O=y2 Im-1 y= y2 U Cu Au (p0) (u0) (u1) (um-1) (p1) (pm-1) 0.0 0.9 1.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.9 1.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 (d) U [0, 0.9, 0, 0, 0, 0.1, 0, 0, 1.0, 0.5, 0.5, 0, 0, 0, 0]T [0, 0.59, 0, 0, 0, 0.07, 0, 0, 0.66, 0.33, 0.33, 0, 0, 0, 0]T Figure 7. Neural Computation: (a) prepossessing of inputs by i) down-sampling the original 28 \u00d7 28 image in MNIST to 4 \u00d7 4 image and ii) get the 4 \u00d7 4 matrix with grey level normalized to [0,1]. (b-c) P-LYR: (b) input data are converted from real number to a random variable following a two-point distribution; (c) four operations in P-LYR, i) R: converting a real number ranging from 0 to 1 to a random variable, ii) C: average sum of weighted inputs, iii) A: non-linear activation function, iv) E: converting random variable to a real number. (d-e) U-LYR: (d) m input data are converted to a vector in the \ufb01rst column of a m\u00d7m unitary matrix; (e) three operations in U-LYR, i) U: unitary matrix converter, ii) Cu: average sum of weighted inputs; iii) Au: non-linear activation function. support training, we make the assumptions that the perfect qbits are used. This enables us to apply theoretic formulations to accelerate the simulation process; however, it leads to some error in predicting the outputs of its corresponding deployment on a physical quantum processor with high error rates (such as the current IBM quantum processor with error rates in the range of 10\u22122). However, we do not deem this as a drawback of our approach, rather this is an inherent problem of the current physical implementation of quantum processors. As the error rates get smaller in the future, it will help to narrow the gap between what QF-Nets predicts and what quantum processor delivers. With the innovations on reducing the error rate of physic qbits, QF-Nets will achieve better results. Methods We are going to introduce QuantumFlow in this section. Neural computation and batch normalization are two key components in a neural network, and we will present the design and implementation of these two components in QF-Nets, QF-FB, QF-Circ, and QF-Map, respectively. QF-pNet and QF-hNet Figure 7 demonstrates two different neural computation components in QuantumFlow: P-LYR and U-LYR. As stated in the Discussion section, P-LYR and U-LYR have their different features. Before introducing these two components, we demonstrate the common prepossessing step in Figure 7(a), 8/14 \fz = E(y2) Batch Normalization (a) (b) batch_adj(t, q) if t == 0:      z =  else:       z =  indiv_adj(g) batch_adj(t, q) indiv_adj(g) (1-z) \u00b4 (sin     ) + z q 2 2 q 2 2 z \u00b4 (sin     )  z =  ^ ^ ^ ~ g 2 2 z \u00b4 (sin     )  z  ~ A E C Cu Au Figure 8. Quantum implementation aware batch normalization: (a) connected to P-LYR; (b) connected to U-LYR. which goes through the downsampling and grey level normalization to obtain a matrix with values in the range of 0 to 1. With the prepossessed data, we will discuss the details of each component in the following texts. Neural Computation P-LYR: An m-input neural computation component is illustrated in 7(d), where m input data I0,I1,\u00b7\u00b7\u00b7 ,Im\u22121 and m corresponding weights w0,w1,\u00b7\u00b7\u00b7 ,wm\u22121 are given. Input data Ii is a real number ranging from 0 to 1, while weight wi is a {\u22121,+1} binary number. Neural computation in P-LYR is composed of 4 operations: i) R: this operation converts a real number pk of input Ik to a two-point distributed random variable xk, where P{xk = \u22121} = pk and P{xk = +1} = 1\u2212pk, as shown in 7(b). For example, we treat the input I0\u2019s real value of p0 as the probability of x0 that outcomes \u22121 while q0 = 1 \u2212p0 as the probability that outcomes +1. ii) C: this operation calculates y as the average sum of weighted inputs, where the weighted input is the product of a converted input (say xk) and its corresponding weight (i.e., wk). Since xk is a two-point random variable, whose values are \u22121 and +1 and the weights are binary values of \u22121 and +1, if wk = \u22121, wk \u00b7 xk will lead to the swap of probabilities P{xk = \u22121} and P{xk = +1} in xk. iii) A: we consider the quadratic function as the non-linear activation function in this work, and A operation outputs y2 where y is a random variable. iv) E: this operation converts the random variable y2 to 0-1 real number by taking its expectation. It will be passed to batch normalization to be further used as the input to the next layer. Neural Computation U-LYR: Unlike P-LYR taking advantage of the probabilistic properties of qbits to provide the maximum \ufb02exibility, U-LYR aims to minimize the gates for quantum advantage using the property of the unitary matrix. The 2k input data are \ufb01rst converted to 2k corresponding data that can be the \ufb01rst column of a unitary matrix, as shown in Figure 7(d). Then the linear function Cu and activation quadratic function Au are conducted. U-LYR has the potential to signi\ufb01cantly reduce the quantum gates for computation, since the 2k inputs are the \ufb01rst column in a unitary matrix and can be encoded to k qbits. But the state-of-the-art hypergraph based approach27 needs O(2k) basic quantum gates to encode 2k corresponding weights to k qbits, which is the same with that of classical computer needing O(2k) operators (i.e., adder/multiplier). In the later section of QF-Map, we propose an algorithm to guarantee that the number of used basic quantum gates to be O(k2), achieving quantum advantages. Multiple Layers: P-LYR and U-LYR are the fundamental components in QF-Nets, which may have multiple layers. In terms of how a network is composed using these two components, we present two kinds of neural networks: QF-pNet and QF-hNet. QF-pNet is composed of multiple layers of P-LYR. For its quantum implementation, operations on random variables can be directly operated on qbits. Therefore, R operation is only conducted in the \ufb01rst layer. Then, C and A operations will be repeated without measurement. Finally, at the last layer, we measure the probability for output qbits, which is corresponding to the E operation. On the other hand, QFhNet is composed of both U-LYR and P-LYR, where the \ufb01rst layer applies U-LYR with the converted inputs. The output of U-LYR is directly represented by the probability form on a qbit, and it can seamlessly connect to C in P-LYR used in later layers. Batch Normalization: Figure 8 illustrates the proposed batch normalization (N-LYR) component. It can take the output of either P-LYR or U-LYR as input. N-LYR is composed of two sub-components: batch adjustment (\u201cbatch_adj\u201d) and individual adjustment (\u201cindiv_adj\u201d). Basically, batch_adj is proposed to avoid data to be continuously shrunken to a small range (as stated in Discussion section). This is achieved by normalizing the probability mean of a batch of outputs to 0.5 at the training phase, as shown in Figure 9(c)-(d). In the inference phase, the output \u02c6 z can be computed as follows: \u02c6 z = (1 \u2212z)\u00d7 (sin2\u03b8 2 )+ z, if t = 0 \u02c6 z = z\u00d7 (sin2 \u03b8 2 ), if t = 1 (2) After batch_adj, the outputs of all neurons are normalized around 0.5. In order to increase the variety of different neurons\u2019 output for better classi\ufb01cation, indiv_adj is proposed. It contains a trainable parameter \u03bb and a parameter \u03b3 (see Figure 9(e)). It is performed in two steps: (1) we get a start point of an output pz according to \u03bb, and then moves it back to p=0.5 to obtain parameter \u03b3; (2) we move pz the angle of \u03b3 to obtain the \ufb01nal output. Since different neurons have different values of \u03bb, the variation of outputs can be obtained. In the inference phase, its output \u02dc z can be calculated as follows. \u02dc z = \u02c6 z\u00d7 (sin2 \u03b3 2) (3) The determination of parameters t, \u03b8, and \u03b3 is conducted in the training phase, which will be introduced later in QF-FB. QF-FB QF-FB involves both forward propagation and backward propagation. In forward propagation, all weights and parameters are determined, and we can conduct neural computation and batch normalization layer by layer. For -LYp, the neural computation will compute y = \u2211\u2200i{xi\u00d7wi} m and y2, where xi is a twopoint random variable. The distributions of y and y2 are illustrated in Figure 9(a)-(b). It is straightforward to get the 9/14 \f\u00d5pi + \u00d5qi pm-1...p1q0 pm-1...q1p0 + + qm-1...p1p0 \u2026 + qm-1...q1p0 qm-1...p1q0 + + pm-1...q1q0 \u2026 \u2026 \u2026 \u2026 -1 1 ( ) 0 -m+2 y2 2 y m m-2 m \u00d5pi + \u00d5qi pm-1...p1q0 pm-1...q1p0 + + + qm-1...p1p0 \u2026 + + qm-1...q1p0 qm-1...p1q0 + + pm-1...q1q0 \u2026 \u2026 \u2026 1 0 m-2 m (a) (b) pmean (c) (d) (e) t=0; q=2\u00b4arcsin(                   )   pmean 1-pmean 0.5-pmean downward p = 0 |0\u00f1 |1\u00f1 p = 1 p = 0.5 p = 0.5 p = 0.5 upward pz pz (pz / n+0.5)\u00b4l g g ^ t=1; q=2 \u00b4 arcsin(               ) pmean ^ ^ pmean g=2\u00b4arcsin(                             ) (pz / n+0.5)\u00b4l \u00d6 \u00d6 pmean 0.5 0.5 \u00d6 Figure 9. QF-FB: (a-b) distribution of random variable y and y2 in neural computation component of QF-pNet; (c-e) determination of parameters, t, \u03b8, and \u03b3, in batch normalization component of QF-Nets. expectation of y2 by using the distribution; however, for m inputs, it involves 2m terms (e.g., \u220fqi is one term), and leads to the time complexity to be O(2m). To reduce the time complexity, QF-FB takes advantage of independence of inputs to calculate the expectation as follows: E([\u2211\u2200i wixi]2) = E(\u2211\u2200i[wixi]2 + 2 \u00d7\u2211\u2200i\u2211\u2200j>i[wixiwjxj]) = m+ 2 \u00d7\u2211\u2200i\u2211\u2200j>i E(wixi)\u00d7 E(wjxj) (4) where E(\u2211\u2200i[wixi]2) = m, since [wixi]2 = 1 and there are m inputs in total. The above formula derives the following algorithm with time complexity of O(m2) to simulate the neural computation P-LYR. Algorithm 1: QF-FB: simulating P-LYR Input: (1) number of inputs m; (2) m probabilities \u27e8p0,\u00b7\u00b7\u00b7 , pm\u22121\u27e9; (3) m weights \u27e8w0,\u00b7\u00b7\u00b7 ,wm\u22121\u27e9. Output: expectation of y2 1. Expectation of random variable xi: ei = E(xi) = 1\u22122\u00d7 pi; 2. Expectation of wi \u00d7xi: E(wi \u00d7xi) = wi \u00d7ei; 3. Sum of pair product sumpp = \u2211\u2200i \u2211\u2200j>i{E(wi \u00d7xi)\u00d7E(wj \u00d7xj)}; 4. Expectation of y2: E(y2) = m+2\u00d7sumpp m2 ; 5. Return E(y2); For -LYu, the neural computation will \ufb01rst convert inputs I = {i0,i1,\u00b7\u00b7\u00b7 ,im\u22121} to a vector U = {u0,u1,\u00b7\u00b7\u00b7 ,um\u22121} who can be the \ufb01rst column of a unitary matrix MATu. By operating MATu on K = log2m qbits with initial state (i.e., |0\u27e9), we can encode U to 2K = m states. The generating of unitary matrix MATu is equivalent to the problem of identifying the nearest orthogonal matrix given a square matrix A. Here, matrix A is created by using I as the \ufb01rst column, and 0 for all other elements. Then, we apply Singular Value Decomposition (SVD) to obtain B\u2211C\u2217= SVD(A), and we can obtain MATu = BC\u2217. Based on the obtained vector U in MATu, -LYu computes y = \u2211\u2200i{ui\u00d7wi} m and y2, as shown in the following algorithm. Algorithm 2: QF-FB: simulating U-LYR Input: (1) number of inputs m; (2) m input values \u27e8p0,\u00b7\u00b7\u00b7 , pm\u22121\u27e9; (3) m weights \u27e8w0,\u00b7\u00b7\u00b7 ,wm\u22121\u27e9. Output: [ \u2211\u2200i{ui\u00d7wi} m ]2 1. Generating square matrix A and compute B\u2211C\u2217= SVD(A) 2. Calculating MATu = BC\u2217and extract vector U from MATu ; 3. Compute y = \u2211\u2200i{ui\u00d7wi} m ; 4. Return y2; The forward propagation for batch normalization can be ef\ufb01ciently implemented based on the output of the neural computation. A code snippet is given as follows. Algorithm 3: QF-FB: simulating N-LYR Input: (1) E(y2) from neural computation; (2) parameters t, \u03b8, \u03b3 determined by training procedure. Output: normalized output \u02dc z 1. Initialize z: z = E(y2); 2. Calculate \u02c6 z according to Formula 2; 3. Calculate \u02dc z according to Formula 3; 4. Return \u02dc z; For the backward propagation, we need to determine weights and parameters (e.g., \u03b8 in N-LYR). The typically used optimization method (e.g., stochastic gradient descent39) is applied to determine weights. In the following, we will discuss the determination of N-LYRparameters t, \u03b8, \u03b3. The batch_adj sub-component involves two parameters, t and \u03b8. During the training phase, a batch of outputs are generated for each neuron. Details are demonstrated in Figure 9(c)-(d) with 6 outputs. In terms of the mean of outputs in a batch pmean, there are two possible cases: (1) pmean \u22640.5 and (2) pmean > 0.5. For the \ufb01rst case, t is set to 0 and \u03b8 = 2 \u00d7 arcsin( q 0.5\u2212pmean 1\u2212pmean ) can be derived from Formula 2 by setting \u02c6 z to 0.5; similarly, for the second case, t is set to 1 and \u03b8 = 2 \u00d7 arcsin( q 0.5 pmean ). Kindly note that the training procedure will be conducted in multiple iterations of batches. As with the method for batch normalization in the conventional neural network, we employ moving average to record parameters. Let xi be the parameter of x (e.g., \u03b8) at the ith iteration, and xcur be the value obtained in the current iteration. For xi, it can be calculated as xi = m \u00d7 xi\u22121 + (1 \u2212m) \u00d7 xcur, where m is the momentum which is set to 0.1 by default in the experiments. In forward propagation, the sub-module indiv_adj is almost the same with batch_adj for t = 0; however, the determination of its parameter \u03b3 is slightly different from \u03b8 for batch_adj. As shown in Figure 9(e), the initial probability of \u02c6 z after batch_adj is pz. The basic idea of indiv_adj is to move \u02c6 z by an angle, \u03b3. It will be conducted in three steps: (1) we move start point at pz to point A with the probability of (pz/n+0.5)\u00d7\u03bb, where n is the batch size and \u03bb is a trainable variable; (2) we obtain \u03b3 by moving point A to p = 0.5; (3) we \ufb01nally move solution at pz by the angle of \u03b3 to obtain the \ufb01nal result. By replacing P mean by (pz/n + 0.5)\u00d7 \u03bb in batch_adj when t = 1, we can calculate \u03b3. For each batch, we calculate the mean of \u03b3, and we also employ the moving average to record \u03b3. 10/14 \fIm-2 I0 Im-1 |0\u00f1 E0 |0\u00f1 E1 H H |0\u00f1 |0\u00f1 Ek-1 |0\u00f1 O Ek-2 H H H H H \u2026 |0\u00f1 |0\u00f1 |0\u00f1 \u2026 \u2026 ... ... ... H \u00c5 (a) R M (c) (b) (e) O I P \u00c5 |j\u00f1 |0\u00f1 |0\u00f1 Ry(q) Ry(q1) Ry(qm-2) Ry(qm-1) X \u00c5 P(O=|1\u00f1)= P(O = |1\u00f1) =  |j\u00f1 |0\u00f1 |0\u00f1 Ry(g) \u00c5 g(q,g)=2\u00b4arcsin(         \u00b4         )  (1-z)\u00b4(sin     )+z q 2 2 ^ g 2 2 z \u00b4 (sin     )  q 2 sin      g 2 sin      W O I P (d) P(O = |1\u00f1) =  |j\u00f1 |0\u00f1 |0\u00f1 Ry(q) \u00c5 q 2 2 z \u00b4 (sin     )  O I P (f) |j\u00f1 |0\u00f1 |0\u00f1 Ry(g(q,g)) \u00c5 O I P A C |0\u00f1 E0 |0\u00f1 E1 |0\u00f1 |0\u00f1 Ek-1 |0\u00f1 O Ek-2 \u2026 MATu \u00c5 U M Au Cu W QF-Map Figure 10. QF-Circ: (a) quantum circuit designs for QF-pNet; (b) quantum circuit design for QF-hNet; (c-f): batch normalization, quantum circuit designs for different cases; (c) design of \u201cbatch_adj\u201d for the case of t = 0; (d) design of \u201cbatch_adj\u201d for the case of t = 1; (e) design of \u201cindiv_adj\u201d; (f) optimized design for a speci\ufb01c case when t = 1 in \u201cbatch_adj\u201d. QF-Circ We now discuss the corresponding circuit design for components in QF-Nets, including P-LYR, U-LYR, and N-LYR. Figures 10(a)-(b) demonstrate the circuit design for P-LYR (see Figure 7(c)) and U-LYR (see Figure 7(e)), respectively; Figures 10(c)-(f) demonstrate the N-LYR in Figure 8. Implementing P-LYR on quantum circuit: For an minput neural computation, the quantum circuit for P-LYR is composed of m input qbits (I), and k = log2m encoding qbits (E), and 1 output qbit (O). In accordance with the operations in P-LYR, the circuit is composed of four parts. In the \ufb01rst part, the circuit is initialized to perform R operation. For qbits I, we apply m Ry gates with parameter \u03b8 = 2\u00d7arcsin(\u221apk) to initialize the input qbit Ik in terms of the input real value pk, such that the state of Ik is changed from |0\u27e9to \u221aqk|0\u27e9+ \u221apk|1\u27e9. For encoding qbits E and output qbit O, they are initialized as |0\u27e9. The second part completes the average sum function, i.e., C operation. It further includes three steps: (1) dot product of inputs and weights on qibits I, (2) make encoding qbits E into superposition, (3) encode m probabilities in qbits I to 2k = m states in qbits E. The third part implements the quadratic activation function, that is the A operation. It applies the control gate to extract the amplitudes in states |I0I1\u00b7\u00b7\u00b7Im\u22121\u27e9\u2297|00\u00b7\u00b7\u00b70\u27e9to qbit O. As we know that the probability is the square of the amplitude, the quadratic activation function can be naturally implemented. Finally, E operations corresponds to the fourth part that measures qbit O to obtain the output real number E(y2), where the state of O is |O\u27e9= p 1 \u2212E(y2)|0\u27e9+ p E(y2)|1\u27e9. A detailed demonstration of the equivalency between QF-Circ and P-LYR can be found in the Supplementary Information. Kindly note that for a multi-layer network composed of PLYR, namely QF-pNet, there is no need to have a measurement at interfaces, because the converting operation R initializes a qbit to the state exactly the same with |O\u27e9. In addition, the batch normalization can also take |O\u27e9as input. Implementing U-LYR on quantum circuit: For an minput neural computation, the quantum circuit for U-LYR contains k = log2m encoding qbits E and 1 output qbit O. According to U-LYR, the circuit in turn performs U, Cu, Au operations, and \ufb01nally obtains the result by a measurement. In the \ufb01rst operation, unlike the circuit for P-LYR using R gate to initialize circuits using m qbits; for U-LYR, we using the matrix MATu to initialize circuits on k = log2m qbits E. Recalling that the \ufb01rst column of MATu is vector V, after this step, m elements in vector V will be encoded to 2k = m states represented by qbits E. The second operation is to perform the dot product between all states in qbits E and weights W, which is implemented by control Z gates and will be introduced in QF-Map. Finally, like the circuit for P-LYR, the quadratic activation and measurement are implemented. Kindly note that, in addition to quadratic activation, we can also implement higher orders of non-linearity by duplicating the circuit to perform U, Cu, and Au to achieve multiple outputs. Then, we can use control NOT gate on the outputs to achieve higher orders of non-linearity. For example, using a Toffoli gate on two outputs can realize y4. Let the non-linear function be yk and the cost complexity of U-LYR using quadratic activation be O(N), then the cost complexity of U-LYR using yk as the non-linear function will be O(kN). For neural networks with given inputs, we can preprocess the U operation and store the states in quantum memory40. Thus, the key for quantum advantage is to exponentially reduce the number of gates used in neural computation, compared with the number of basic operators used in classical computing. We will present an algorithm in QF-Map for ULYR to achieve this goal. Implementing N-LYR on quantum circuit: Now, we discuss the implementation of N-LYR in quantum circuits. In these circuits, three qbits are involved: (1) qbit I for input, which can be the output of qbit O in circuit without measurement, or initialized using a Ry gate according to the measurement of qbit O in circuit; (2) qbit P conveys the parameter, which is obtained via training procedure, see details in QFFB; (3) output qbits O, which can be directly used for the next layer or be measured to convert to a real number. Figures 10(b)-(c) show the circuit design for two cases in batch_adj. Since parameters in batch_adj are determined in the inference phase, if t = 0, we will adopt the circuit in Figure 10(b), otherwise, we adopt that in Figure 10(c). Then, Figure 10(d) shows the circuit for indiv_adj. We can see that circuits in Figures 10(c) and (d) are the same except the ini11/14 \f(a) (b) |0\u00f1 q3 |0\u00f1 q2 |0\u00f1 |0\u00f1 q0 q1 |0\u00f1 q3 |0\u00f1 q2 |0\u00f1 |0\u00f1 q0 q1 (c) |0\u00f1 q3 |0\u00f1 q2 |0\u00f1 |0\u00f1 q0 q1 Z Figure 11. Illustration of state |6\u27e9= |0110\u27e9and |4\u27e9= |0100\u27e9 in a k = 4 computation system: (a) FG6; (b) PG6; (c) PG4. tialization of parameters, \u03b8 and \u03b3. For circuit optimization, we can merge the above two circuits into one by changing the input parameters to g(\u03b8,\u03b3), as shown in Figure 10(e). In this circuit, \u02dc z\u2032 = z \u00d7 sin2 g(\u03b8,\u03b3) 2 , while for applying circuits in Figures 10(c) and (d), we will have \u02dc z = z\u00d7 sin2 \u03b8 2 \u00d7 sin2 \u03b3 2. To guarantee the consistent function, we can derive that g(\u03b8,\u03b3) = 2 \u00d7 arcsin(sin \u03b8 2 \u00d7 sin \u03b3 2). QF-Map QF-Map is an automatic tool to map QF-Nets to the quantum processor through two steps: network-to-circuit mapping, which maps QF-Nets to QF-Circ; and virtual-to-physic mapping, which maps QF-Circ to physic qbits. Mapping QF-Nets to QF-Circ: The \ufb01rst step of QFMap is to map three kinds of layers (i.e., P-LYR, U-LYR, and N-LYR) to QF-Circ. The mappings of P-LYR and N-LYR are straightforward. Speci\ufb01cally, for P-LYR in Figure 10(a), the circuit for weight W is determined using the following rule: for a qbit Ik, an X gate is placed if and only if Wk = \u22121. Let the probability P(xk = \u22121) = P(Ik = |0\u27e9) = qk, after the X gate, the probability becomes 1 \u2212qk. Since the values of random variable xk are \u22121 and +1, such an operation computes \u2212xk. For N-LYR, let O1 be the output qbit of the \ufb01rst layer. It can be directly connected to the qbit I in Figure 10(c)-(f), according to the type of batch normalization, which is determined by the training phase. The mapping of U-LYR to quantum circuits is the key to achieve quantum advantages. In the following texts, we will \ufb01rst formulate the problem, and then introduce the proposed algorithm to guarantee the cost for a neural computation with 2k inputs to be O(k2). Before formally introducing the problem, we \ufb01rst give some fundamental de\ufb01nitions that will be used. We \ufb01rst de\ufb01ne the quantum state and the relationship between states as follows. Let the computational basis be composed of k qbits, as in Figure 10(b). De\ufb01ne |xi\u27e9= |bi k\u22121,\u00b7\u00b7\u00b7 ,bi j,\u00b7\u00b7\u00b7 ,bi 0\u27e9to be the ith state, where b j is a binary number and xi = \u2211\u2200j{bi j \u00b72 j}. For two states |xp\u27e9and |xq\u27e9, we de\ufb01ne |xp\u27e9\u2286|xq\u27e9if \u2200bp j = 1, we have bq j = 1. We de\ufb01ne sign(xi) to be the sign of xi. Next, we de\ufb01ne the gates to \ufb02ip the sign of states. The controlled Z operation among K qbits (e.g., CKZ) is a quantum gate to \ufb02ip the sign of states25,27. De\ufb01ne FGxi to be a CKZ gate to \ufb02ip the state xi only. It can be implemented as follows: if bi j = 1, the control signal of the jth qbit is enabled by |1\u27e9, otherwise if bi j = 0, it is enabled by |0\u27e9. De\ufb01ne PGxi to be a controlled Z gate to \ufb02ip all states \u2200xm if xm \u2286xi. It can be implemented as follows: if bi j = 1, there is a control signal of the jth qbit, enabled by |1\u27e9, otherwise, it is not a control qbit. Speci\ufb01cally, if there is only bi m = 1 for all bi k \u2208xi, we put a Z gate on the mth qbit. Figure 11 illustrates FG6, PG6 and PG4. We de\ufb01ne cost function C to be the number of basic gates (e.g., Pauli-X, Toffoli) used in a control Z gate. Now, we formally de\ufb01ne the weight mapping problem as follows: Given (1) a vector W = {wm\u22121,\u00b7\u00b7\u00b7 ,w0} with m binary weights (i.e., -1 or +1) and (2) a computational basis of k = log2m qbits that include m states X = {xm\u22121,\u00b7\u00b7\u00b7 ,x0} and \u2200xj \u2208X, sign(xj) is +, the problem is to determine a set of gates G in either FG or PG to be applied, such that the circuit cost is minimized, while the sign of each state is the same with the corresponding weight; i.e., \u2200xj \u2208X, signG(xj) = sign(wj) and min = \u2211g\u2208G(C(g)), where signG(xj) is the sign of state xj under the computing conducted by a sequence of quantum gates in G. A straightforward way to satisfy the sign \ufb02ip requirement without considering cost is to apply FG for all states whose corresponding weights are \u22121. A better solution for cost minimization is to use hypergraph states27, which starts from the states with less |1\u27e9, and apply PG to reduce the cost. However, as shown in the previous work, both methods have the cost complexity of O(2k), which is the same as classical computers and no quantum advantage can be achieved. Toward the quantum advantage, we made the following important observation: the order of weights can be adjusted, since matrix MAT \u2032 u obtained by switching two rows in the unitary matrix MATu will still be a unitary matrix. Based on this property, we can simplify the weight mapping problem to determine a set of gates, such that the cost is minimized while the number of states with sign \ufb02ip is the same as the number of \u22121 in weight W. On top of this, we propose an algorithm to guarantee the cost complexity to be O(k2). Compared to O(2k) needed for classical computers, we can achieve quantum advantages. To demonstrate how to guarantee the cost complexity to be O(k2), we \ufb01rst have the following theorem. Theorem 1. For an integer number R where R > 0 and R \u2264 2k\u22121, the number R can be expressed by a sequence of addition (+) or subtraction (\u2212) of a subset of Sk = {2i|0 \u2264i < k}; when the terms of Sk are sorted in a descending order, the sign of the expression (addition and subtraction) are alternative with the leading sign being addition (+). Proof. The above theorem can be proved by induction. First, for k = 2, the possible values of R are {1,2}, the set S2 = {2,1}. The theorem is obviously true. Second, for k = 3, the possible values of R are {1,2,3,4}, and the set S3 = {4,2,1}. In this case, only R = 3 needs to involve 2 numbers from S3 using the expression 3 = 4\u22121; other numbers can be directly expressed by themselves. So, the theorem is true. Third, assuming the theorem is true for k = n \u22121, we can prove that for k = n the theorem is true for the following three cases. Case 1: For R < 2n\u22122, since the theorem is true for k = n\u22121, based on the assumption, all numbers less than 2n\u22122 can be expressed by using set Sn\u22121 and thus we can also express them by using set Sn because Sn\u22121 \u2286Sn; Case 2: For R = 2n\u22122, 12/14 \fitself is in set Sn; Case 3: For R > 2n\u22122, we can express R = 2n\u22121 \u2212T, where T = 2n\u22121 \u2212R < 2n\u22122. Since the theorem is true for k = n\u22121, we can express T by using set Sn \u22122n\u22121 = {2i|0 \u2264i < n \u22121} = Sn\u22121; hence, R can be expressed using set Sn. Above all, the theorem is correct. We propose to only use PG gate in a set {PGx(j) | x(j) = 2 j \u22121 & j \u2208[1,k]} for any required number R \u2208(0,2k) of sign \ufb02ips on states; for instance, if k = 4, the gate set is {PG0001,PG0011,PG0111,PG1111} and R \u2208(0,16). This can be demonstrated using the above theorem and properties of the problem: (1) the problem has symmetric property due to the quadratic activation function. Therefore, the weight mapping problem can be reduced to \ufb01nd a set of gates leading to the number of \u22121 no larger than 2k\u22121; i.e., R \u2208(0,2k\u22121]. (2) for PGx(j), it will \ufb02ip the sign of 2k\u2212j states; since j \u2208[1,k], the numbers of the \ufb02ipped sign by these gates belong to a set Sk = {2i|0 \u2264i < k}; These two properties make the problem in accordance with that in Theorem 1. The weight mapping problem is also consistent with three rules in the theorem. (1) A gate can be selected or not, indicating the \ufb01nally determined gate set is the subset of Sk; (2) all states at the beginning have the positive sign, and therefore, the \ufb01rst gate will increase sign \ufb02ips, indicating the leading sign is addition (+); (3) \u2200p < q, x(q) \u2286x(p); it indicates that among the 2k\u2212p states whose signs are \ufb02ipped by x(p), there are 2k\u2212q states signs are \ufb02ipped back; this is in accordance to alternatively use + and \u2212in the expression in Theorem 1. Followed by the proof procedure, we devise the following recursive algorithm to decide which gates to be employed. Algorithm 4: QF-Map: weight mapping algorithm Input: (1) An integer R \u2208(0,2k\u22121]; (2) number of qbits k; Output: A set of applied gate G void recursive(G,R,k){ if (R < 2k\u22122){ recursive(G,R,k\u22121); // Case 1 in the third step } else if (R == 2k\u22121){ G.append(PG2k\u22121); // Case 2 in the third step return; }else{ G.append(PG2k\u22121); recursive(G,2k\u22121 \u2212R,k\u22121); // Case 3 in the third step } } // Entry of weight mapping algorithm set main(R,k){ Initialize empty set G; recursive(G,R,k); return G } In the above algorithm, the worst case for the cost is that we apply all gates in {PGx(j) | x(j) = 2 j \u22121 & j \u2208[1,k]}. Let the state x(j) has y |1 > states, if y > 2 the PGx(j) can be implemented using 2y\u22121 basic gates, including y\u22121 Toffoli gates for controlling, 1 control Z gate, and y\u22121 Toffoli gates for resetting; otherwise, it uses 1 basic gates. Based on these understandings, we can calculate the cost complexity in the worst case, which is 1 + 1 + 3 + \u00b7\u00b7\u00b7 + (2 \u00d7 k \u22121) = k2 + 1. Therefore, the cost complexity of linear function computation is O(k2). The quadratic activation function is implemented by a CkZ gate, whose cost is O(k). Thus, the cost complexity for neural computation U-LYR is O(k2). Finally, to make the functional correctness, in generating the inputs unitary matrix, we swap rows in it in terms of the weights, and store the generated results in quantum memory. Mapping QF-Circ to physic qbits: After QF-Circ is generated, the second step is to map QF-Circ to quantum processors, called virtual-to-physic mapping. In this paper, we deploy QF-Circ to various IBM quantum processors. Virtual-tophysic mapping in QF-Map has two tasks: (1) select a suitable quantum processor backend, and (2) map qbits in QF-Nets to physic qbits in the selected backend. For the \ufb01rst task, QFMap will i) check the number of qbits needed; ii) \ufb01nd the backend with the smallest number of qbit to accommodate QF-Circ; iii) for the backends with the same number of qbits, QF-Map will select a backend for the minimum average error rate. The second task in QF-Map is to map qbits in QF-Nets to physic qbits. The mapping follows two rules: (1) the qbit in QF-Nets with more gates is mapped to the physic qbit with a lower error rate; and (2) qbits in QF-Nets with connections are mapped to the physic qbits with the smallest distance. Data availability The authors declare that all data supporting the \ufb01ndings of this study are available within the article and its Supplementary Information \ufb01les. Source data can be accessed via https://wjiang.nd.edu/categories/qf/. Code availability All relevant codes will be open in github upon the acceptance of the manuscript or be available from the corresponding authors upon reasonable request."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1907.08985v2",
+                    "title": "Achieving Super-Linear Speedup across Multi-FPGA for Real-Time DNN Inference",
+                    "abstract": "Real-time Deep Neural Network (DNN) inference with low-latency requirement\nhas become increasingly important for numerous applications in both cloud\ncomputing (e.g., Apple's Siri) and edge computing (e.g., Google/Waymo's\ndriverless car). FPGA-based DNN accelerators have demonstrated both superior\nflexibility and performance; in addition, for real-time inference with low\nbatch size, FPGA is expected to achieve further performance improvement.\nHowever, the performance gain from the single-FPGA design is obstructed by the\nlimited on-chip resource. In this paper, we employ multiple FPGAs to\ncooperatively run DNNs with the objective of achieving super-linear speed-up\nagainst single-FPGA design. In implementing such systems, we found two barriers\nthat hinder us from achieving the design goal: (1) the lack of a clear\npartition scheme for each DNN layer to fully exploit parallelism, and (2) the\ninsufficient bandwidth between the off-chip memory and the accelerator due to\nthe growing size of DNNs. To tackle these issues, we propose a general\nframework, \"Super-LIP\", which can support different kinds of DNNs. In this\npaper, we take Convolutional Neural Network (CNN) as a vehicle to illustrate\nSuper-LIP. We first formulate an accurate system-level model to support the\nexploration of best partition schemes. Then, we develop a novel design\nmethodology to effectively alleviate the heavy loads on memory bandwidth by\nmoving traffic from memory bus to inter-FPGA links. We implement Super-LIP\nbased on ZCU102 FPGA boards. Results demonstrate that Super-LIP with 2 FPGAs\ncan achieve 3.48x speedup, compared to the state-of-the-art single-FPGA design.\nWhat is more, as the number of FPGAs scales up, the system latency can be\nfurther reduced while maintaining high energy efficiency.",
+                    "authors": "Weiwen Jiang, Edwin H. -M. Sha, Xinyi Zhang, Lei Yang, Qingfeng Zhuge, Yiyu Shi, Jingtong Hu",
+                    "published": "2019-07-21",
+                    "updated": "2020-02-08",
+                    "primary_cat": "cs.DC",
+                    "cats": [
+                        "cs.DC"
+                    ],
+                    "main_content": "INTRODUCTION Deep Neural Networks (DNNs) have been continuously achieving breakthroughs in many challenging AI domains, such as image recognition [1], object detection [2], and natural language processing [3]. More recently, DNNs have been applied to process live data in interactive services. For instance, a trained DNN can be employed to process the live video stream for traffic surveillance and emergency response [4], and to analyze medical scans to help doctors during surgery [5]. Thus, the design of systems for real-time DNN inference becomes an imminent challenge, which attracts increasing studies from both industry [6, 7] and academia [2, 5, 8]. Out of all leading computation platforms for DNNs, FPGAs stand out due to their flexibility and versatility over ASICs and their efficiency over CPUs and GPUs. In addition, FPGAs are expected to be more suitable for real-time DNN inference [6, 7] for the following reasons. First, even though ASICs can achieve better latency and energy efficiency, it is prohibitive to design ASIC for each different application or upgrade the design when needed. Second, the realtime inference has rigorous requirements of guaranteed latency to ensure user experience, reliability, and even safety. To meet the hard deadlines, uncertainties in CPUand GPU-based designs (caused by accessing caches) force them to apply the worst-case analysis with a safety margin [9], which leads to inferior design. In contrast, designers are able to customize FPGAs to make the accelerators process the deterministic timing characteristics, which can avoid the overhead caused by the safety margin. Third, realtime inference commonly needs to process data with low batch size, which renders the batch throughput optimization inefficient. For instance, Google\u2019s TPU [10] requires the batch size of at least 16 for energy efficiency while FPGAs can extract parallelism from individual execution instance to reduce latency for the low or even no batching. While most FPGA-based DNN acceleration has been focusing on single-FPGA platform [11\u201314], the growth of resource requirement in DNNs has far exceeded the growth of the resource integrated into one FPGA. As a result, the limited on-chip resources will hinder the exploitation of model parallelism from further boosting time performance. To overcome this challenge, [15] proposed to employ multiple FPGAs, on which DNN layers can be processed in a pipelined fashion. Pipelining designs can achieve high throughput; however, the latency cannot be reduced. With the objective of minimizing latency for real-time AI applications, we exploit the parallelisms in DNN layers and concurrently process each DNN layer across multiple FPGAs. However, we observe that with the growing size of models and volume of input data, a straightforward partition of DNNs to multiple FPGAs leads the severe performance degradation due to insufficient communication bandwidth. In this paper, we propose a new framework, namely \u201cSuper-LIP\u201d, to address the performance bottlenecks in DNNs. To illustrate the framework, we take Convolutional Neural Networks (CNNs) as a vehicle, since CNNs have large amount of intermediate data and more complicated data reuse patterns than Recurrent Neural Networks (RNNs), which results in the acceleration for CNNs on multi-FPGA more challenging. Given a CNN, we first formulate an accurate performance model to detect performance bottleneck in an early stage for the latter optimization of accelerator design. Compared with the existing model [14], the proposed one is more accurate since we investigate the fine-grained data accesses/communication patterns. Based on the accurate model, we identify that the communication bottleneck is commonly at accessing off-chip memory. We propose a novel design, \u201cXFER\u201d, to take advantages of the high-bandwidth inter-FPGA links by offloading part of the data traffic from the memory bus to these links. As a result, performance bottleneck on memory bandwidth can be significantly alleviated. arXiv:1907.08985v2  [cs.DC]  8 Feb 2020 \fCODES+ISSS, 2019, New York, NY Layer Model 1 Input  Application Accelerator Design 2 Performace Model and Bottleneck Analysis 3 Computation Bound Communication Bound Analytic Model design parameters Workload Balance 4 Communication Balance 5 PE PE IFM OFM XFER FPGA1 FPGA2 WEI IFM OFM WEI SW/HW scale-up 6 Output Multi-FPGA design XFER XFER XFER XFER XFER Design Super-LIP Framework one FPGA FPGA Accelerator Design Figure 1: Overview of Super-LIP Framework. The main contributions made in this paper are threefold: \u2022 Super-LIP Framework. We build a framework, Super-LIP, to control the exploration of FPGA-based designs for realtime DNN inference to achieve Super-Linear speedup across multiple FPGAs. Inside Super-LIP, we have further made two contributions listed as follows. \u2022 Accurate Model. First, we formulate an accurate analytic model to quantify the performance-resource trade-off in terms of data access/communication patterns, which can guide designers to better design DNN accelerator and provide insights on how to partition DNNs onto multiple-FPGAs. \u2022 XFER Design. Second, we propose a novel design, XFER, to partition and map DNNs onto multiple FPGAs to exploit high parallelism in DNN layers. XFER can further alleviate the performance bottleneck on memory bandwidth by transferring part of the traffic to inter-FPGA links. Evaluations are conducted on Xilinx ZCU102 FPGA boards connected via optical fiber cables using SFP+ transceiver. Results show that Super-LIP can achieve 2.82\u00d7 and 5.81\u00d7 reduction in latency, while achieving 8.61\u00d7 and 1.81\u00d7 improvement in energy efficiency, compared to GPU and mobile GPU, respectively. Compared with the state-of-the-art single-FPGA design, Super-LIP with 2 FPGAs achieves 3.48\u00d7 speedup and 39.86% improvement in energy efficiency. In addition, when the size of the FPGA cluster scales up to 16, the latency can be consistently reduced. For instance, the latency of YOLO [16] on one FPGA is 126.6ms, which can be reduced to 4.53ms, achieving 27.93\u00d7 reduction. This confirms the applicability and scalability of Super-LIP. The remainder of the paper is organized as follows. Section 2 presents the Super-LIP framework and design challenges. Section 3 and Section 4 present the accurate analytic model and novel XFER design in Super-LIP. Experimental results are shown in Section 5. Section 6 discusses related work. Finally, concluding remarks are given in Section 7. 40 60 80 100 20 0 0 10 20 30 40 50 60 70 80 Computation roof Bandwidth roof (2.23GB/s) Peformance (GFLOPS) Computation to Communication Ratio (FLOP/DRAM byte access) Model Performance Real Performance Model Performance A B Real Performance Figure 2: Model performance vs. real performance. 2 SUPER-LIP FRAMEWORK & CHALLENGES Figure 1 demonstrates the overview of the proposed Super-LIP framework. Super-LIP takes a Deep Neural Network (DNN) as input. And it outputs a multi-FPGA design, onto which the given DNN is partitioned and mapped through a novel technique proposed in this paper. The design objective of Super-LIP is to achieve the minimum latency for real-time AI with ultra-low batch size. Super-LIP, in the middle of Figure 1, sequentially explores two design spaces: (1) accelerator design space on the left-hand side (\u2780-\u2782), which determines the hardware-level parallelism (e.g., how many DSPs to compute multiply-accumulate operations, and how many channels to move data between off-chip and on-chip memory); (2) multi-FPGA design space on the right-hand side (\u278d-\u278f), which determines task-level parallelism (e.g., how to partition each DNN layer and how to communicate between FPGAs). In the exploration of both spaces, there exist several challenges needing to be addressed, which are demonstrated as follows. Challenge 1: Inaccurate performance models hinder designers to optimize accelerators. Figure 2 shows the design space exploration of Layer 5 in AlexNet [1] using the model proposed by [14] which is based on roof-line model. In this figure, each point represents a design. Computation roof is determined by the computation resource in FPGA, while bandwidth roof is the theoretic peak memory bandwidth to access off-chip memory in terms of designs. Design points under these roofs are regarded as attainable. We implement designs A and B in Figure 2 on the ZCU102 FPGA board to capture its real performance of on-board execution. We observe that even though design points A and B are under both roofs, their real performance cannot reach the estimated model performance. This is because the model in [14] assumes the uninterrupted memory access, which is basically impossible due to the synchronization of operations (i.e, the completed operations need to wait for the slower ones, see Figure 6). In addition, design A with the best model performance is inferior to design B in real performance. Therefore, it is imperative to develop an accurate performance model which can explore the optimal designs. Challenge 2: How to alleviate communication bottleneck without costly modifications on hardware? As shown in Figure 1 \u2782, there are potentially two kinds of performance bottlenecks: computation bottleneck bounded by the computation resource and communication bottleneck bounded by \fCODES+ISSS, 2019, New York, NY PE PE IFM OFM XFER WEI IFM OFM WEI FPGA1 FPGA2 Off-chip  memory On-chip buffer Accelerator IFM OFM WEI PE Cycles 1412 234 2953 1782 Ops IFM OFM W&X PE Cycles 1412 234 1695 1782 Ops (a) w/o XFER w/ XFER Lat Lat 2953 1782 (b) (c) Figure 3: The XFER design and the performance gain. the off-chip memory bandwidth. For computation bottleneck, we can enlarge the accelerator to involve more computation resource to alleviate it. However, it is hard to alleviate communication bottlenecks without costly modifications on the FPGA hardware (e.g., increase the size of on-chip memory). Thanks to the inter-FPGA links in an FPGA cluster, memory bandwidth bottleneck can be alleviated by offloading traffic from the memory bus to inter-FPGA links, which can be realized from a high-level implementation without any modifications on hardware. This idea is inspired by the observation from the results in comparing data transmission time between accessing off-chip memory and switching between FPGAs. Experimental results on two connected ZCU102 FPGAs via SFP+ cables show that the speed of inter-FPGA communication is competitive with accessing offchip memory. Specifically, inter-FPGA communication is 3 times faster than accessing off-chip memory when the packet size is 1KB. The figure is 1.6 times when the packet size increases to 64KB and 128KB. The obtained speedup is mainly because platforms provide high-speed serial communication, while the speed of memory accesses is bounded by the accelerator designs (details in Section 3 \u2781-1). Motivated by the above results, we propose a novel design methodology in the Super-LIP framework, namely \u201cXFER\u201d, to address two implementation problems in exploring multi-FPGA designs: (1) how to partition computations in DNN layers for higher model parallelism; and (2) what type of data can be off-loaded to inter-FPGA links. Figure 1, from \u278dto \u278f, demonstrates three steps in XFER to optimize the accelerators on multiple FPGAs: first, it determines the partitions of DNN layers to balance computation workloads (\u278d); second, it identifies the traffic to be off-loaded to inter-FPGA links to balance communication (\u278e); last, it scales up the number of FPGAs to further speedup the whole DNN network (\u278f). Figure 3 (a) illustrates an example of XFER employed between two FPGAs. As shown in the figure, FPGA1 and FPGA2 share the same set of weights and have different sets of input/output feature maps (IFMs/OFMs). Traditionally, each FPGA will load the weight and compute by itself. In XFER, each FPGA only loads half of the shared weight from the off-chip memory. Then, they will send the loaded half weight to each other through inter-FPGA links. In this way, each FPGA only loads parts of the weights from off-chip memory, which significantly reduces the traffic loads on the memory bus. As a result, the overall latency regarding to the pipeline cycle time (Lat2 in Figure 6) can be reduced from 2,953 to 1,782, achieving 39.65% improvement, as shown in Figure 3 (b)-(c). Kindly note that the pipeline cycle time is determined by the slowest operation (details can be found in Formulas 12 and 13). In the following sections, we will address the first challenge by formulating an accurate performance analytic model in Section3, which is the base of optimizing DNN accelerators on a multi-FPGA L1 L2 L3 L4 M R C N K K B \u2026 \u2026 IFM OFM L=\u2329B,M,N,R,C,K\u232a Figure 4: Super-LIP \u2780: CNN layer model. platform. Second, in the design of a computing platform with multiple FPGAs where communication channels can be established between two FPGAs, we present the novel XFER design in Section 4 to address the second challenge. 3 ACCURATE ANALYTIC MODELS Figures 4-6 shows the details of accelerator optimization (left-hand side) in Super-LIP. We first formulate the model for one CNN layer in \u2780; then, the accelerator design for both off-chip optimization and on-chip implementations are depicted in \u2781-1 and \u2781-2, respectively. At the end of this section, we present the performance model and bottleneck detection (component \u2782in Figure 1). \u2780Layer Model. Layer model describes the properties of a CNN layer. In Figure 4, we show the details of the second layer L2 in a CNN with 4 layers. A CNN layer is defined as L = \u27e8B, M, N,R,C,K\u27e9, where B is the batch size; M and N represent the number of channels in output/input feature maps (OFM/IFM); R and C represent the number of rows and columns in OFM; K refers to kernel size. For example, L = \u27e82, 128, 192, 13, 13, 3\u27e9describes Layer 5 in AlexNet with the batch size of 2. Based on the proposed layer model, we will introduce how to design CNN accelerators on an FPGA (component \u2781in Figure 1). \u2781Accelerator Design. The core of FPGA-based accelerator design is the on-chip computation engine (as shown in right-hand part of Figure 5(b)), which will conduct a set of multiplicationand-accumulation in parallel. Since the memory and computation requirement of one layer significantly exceeds the on-chip resource, the computation engine cannot process all operations in one CNN layer at once; instead, it will be invoked repeatedly. To match the speed between the data consumed by the computation engine and the data produced by accessing off-chip memory, we use the on-chip memory (i.e., BRAM) to be a cache, which constructs a two-level computing model, as shown in Figure 5(b). In the first level, we move data between off-chip memory and on-chip memory, denoted as \u2781-1 off-chip design. In the second level, we move data between on-chip memory and computation engine, denoted as \u2781-2 on-chip design. \u2781-1 Off-Chip Design. The off-chip design needs to control the sequence of data to be uploaded to the on-chip memory. To ensure the functional correctness, we need to determine the size of data and the order of data to be uploaded, which correspond to loop tiling and loop ordering. A data tile of each data type is the basic unit to be moved between off-chip and on-chip memory. Let \u27e8Tm,Tn,Tr ,Tc\u27e9be tiling parameters on OFM channel, IFM channel, row, column. Then, we can get the size of data tile for IFM (to be Tn \u00b7 Tr \u00b7 Tc), OFM (to be Tm \u00b7Tr \u00b7Tc), and weight (to be Tm \u00b7Tn \u00b7 k \u00b7 k, where k is the kernel \fCODES+ISSS, 2019, New York, NY A A 1 C E D D E Tn Tm Tr Tc Tn Tm k Tr Tc \u2329Tm,Tn,Tr,Tc\u232a -1: off-chip optimization: 2 Off-Chip Memory (DRAM) Tn Tm Ip Wp Op Tn Tm . Tm F \u2026 (a) B B = \u00d7 + + \u00d7 = \u00d7 + + \u00d7 + + \u2026 \u2026 \u2026 Tm Tn On-Chip Memory and Data Flow IFM WEI OFM words On-Chip Computation Engine -2: on-chip design: 2 (b) \u2329Ip,Wp,Op\u232a \u2026 Figure 5: Super-LIP \u2781: (a) \u2781-1 the off-chip optimization; (b) \u2781-2 the on-chip accelerator design. size). In Figure 5(a), the colored data demonstrate the data tiles. Note that these tiling parameters will be constrained by on-chip resource, which will be introduced later in this section. Next, the loop order will determine the sequence of data to be moved between off-chip and on-chip memory. The convolution operation involves 4-level of nested loops (details please refer to Fig. 5 in [14]). These loops traverse along IFM channel, OFM channel, row/column, and batch, which correspond to directions C, D, E, F in Figure 5(a), respectively. Based on the above loop order, we can get the trip count, which will be used in modeling the computation latency. We first introduce the trip count for loops along IFM channel (C). According to the tiling parameters, in the loop each step will involve Tn channels, and there are N IFM channels in total. Therefore, the trip count for direction C is \u2308N Tn \u2309. Similarly, we can obtain the trip count for direction D as \u2308M Tm \u2309. Then, for direction E, it will move along R rows and C columns, and each step along row and column is Tr and Tc, so the trip count is \u2308C Tc \u2309\u00d7 \u2308R Tr \u2309. Finally, for batch size of B, we need to traverse each batch and the trip count for F is B. \u2781-2 On-Chip Design. Based on the loop optimization parameters, we model the on-chip computation and buffer allocation. Then, we model the off-chip/on-chip communication based on the data width related parameters \u27e8Ip,Wp,Op\u27e9. For the on-chip computation model, as shown in Figure 5(b), there areTm\u00d7Tn Multiply-Accumulate (MAC) operations conducted in parallel. In this paper, we consider different data types: For the 16bits fixed point, each MAC utilizes 1 DSP, while for the 32bits floating point, each MAC utilizes 5 DSPs. Let D be the number of DSPs provided by the platform, we have the following constraints for 32bits float-point and 16bits fix-point, respectively: 5 \u00d7Tm \u00d7Tn \u2264D (1) Tm \u00d7Tn \u2264D (2) ... { \uf8eeN/Tn\uf8f9 Lat2 O I W COMP prologue { Lat ... { one load Lat1 ITER epilogue Performance  Bottleneck Figure 6: Super-LIP \u2782: Performance model and bottleneck detection. As to the on-chip buffer design, there are three kinds of buffers: IFM, OFM and Weight (WEI) buffers. The size of IFM buffer is determined by the loop tiling on IFM. For each iteration, Tn \u00d7 Tr \u00d7 Tc pixels in IFM are loaded to the on-chip buffer, as shown in Figure 5(a). Hence, the IFM buffer is declared as a 3-dimension array I[Tn][Tr ][Tc]. Similarly, OFM and WEI buffers are declared as O[Tm][Tr ][Tc] and W [Tm][Tn][K][K], respectively. In order to match the speed of computation, these arrays should be partitioned into different on-chip memories (i.e. BRAM) which can be accessed in parallel. As shown in Figure 5(b), each computation involves Tn, Tm, and Tm \u00d7Tn pixels/weights in IFM, OFM, and WEI buffers. Accordingly, we completely partition IFM and OFM along their first dimension, and WEI along its first two dimensions. Then, we calculate the usage of BRAMs for IFM (bI), OFM (bO), WEI (bW ): bI = 2 \u00d7Tn \u00d7 \u2308Tr \u00b7 Tc \u00b7 BITs/18K\u2309 (3) bO = 2 \u00d7Tm \u00d7 \u2308Tr \u00b7 Tc \u00b7 BITs/18K\u2309 (4) bW = 2 \u00d7Tm \u00d7Tn \u00d7 \u2308K \u00b7 K \u00b7 BITs/18K\u2309 (5) where 2 represents the double-buffer technique adopted in the design. For the ease of illustration, we do not put the double buffer in Figure 5(b). Let B be the number of BRAMs in the platform, we have the following constraint. bI + bO + bW \u2264B (6) Finally, we determine the data width parameters \u27e8Ip,Wp,Op\u27e9, which are used to model the off-chip/on-chip communication bandwidth. Ip,Wp,Op represent the number of AXI_STREAMs employed in transmitting IFM, WEI, and OFM, and in turn determine the width of AXI bus. In a given platform, the data width of the memory bus is limited, denoted as W. We have the following constraint. BITs \u00d7 (Ip +Wp + Op) \u2264W (7) where notation BITs is the data bit-width adopted in designs. \u2782Performance Model and Analysis. We first model the offchip/on-chip communication latency for transferring IFM, OFM, and WEI, based on parameters \u27e8Ip,Wp,Op\u27e9. For instance, the size of IFM buffer is Tn \u00b7 Tr \u00b7 Tc, and we employ Ip AXI_STREAMs for data transfer, indicating that Ip pixels can be loaded to IFM buffer within 1 clock cycle in the pipelined fashion. Hence, the latency of loading IFM (tImem) can be formulated. tImem = Tn \u00b7 Tr \u00b7 Tc/Ip (8) Similarly, we model the latency of loading WEI buffer (tWmem) and offloading OFM buffer (tOmem) as follows. tWmem = Tm \u00b7 Tn \u00b7 K \u00b7 K/Wp (9) tOmem = Tm \u00b7 Tr \u00b7 Tc/Op (10) \fCODES+ISSS, 2019, New York, NY Then, after filling up the on-chip buffers, data in them can support K \u00d7 K \u00d7Tr \u00d7Tc \u00d7Tm \u00d7Tn MACs. As stated in \u2781-2 On-Chip Design, the Processing Element (PE) can conduct Tm \u00d7 Tn MAC operations in parallel. Therefore, the latency of one execution of PE (tComp) can be modeled as follows. tComp = K \u00b7 K \u00b7 Tr \u00b7 Tc (11) Next, we are going to model the system latency. Benefiting from the double buffer technique, loading IFM, loading WEI, and executing PE can be conducted in parallel. In addition, loading OFM can be overlapped with \u2308N Tn \u2309executions, as shown in Figure 6. With the known trip counts in \u2781-1, we model the latency as follows. Lat1 = max{tComp,tImem,tWmem} (12) Lat2 = max{\u2308N Tn \u2309\u00b7 Lat1,tOmem} (13) Lat = B \u00d7 \u2308R Tr \u2309\u00d7 \u2308C Tc \u2309\u00d7 \u2308M Tm \u2309\u00d7 Lat2 + (tOmem + Lat1) (14) where Lat1 and Lat2 are the latencies of one trip for loop C and D, respectively, and Lat is the overall system latency. Based on the above performance and resource usage models, we can formulate the optimization problem of minimizing latency as integer non-linear programming problem, which incorporates constraints from Formula 1 to 14. The objective of this problem is to minimize latency, as follows. Objective : min = Lat (15) Performance Bottleneck Detection. The above analytic model can help designer to detect the performance bottleneck. Specifically, we have the following corollary. Corollary 1. Given a CNN layer and the design parameters, we can detect the performance bottlenecks by considering Lat1 and Lat2 as follows: \u2022 if Lat2 is dominated by tOmem, the performance bottleneck is on transmitting OFM data, otherwise, \u2022 if Lat1 is dominated by tImem, the performance bottleneck is on transmitting IFM data, \u2022 if Lat1 is dominated by tWmem, the performance bottleneck is on transmitting weights, \u2022 if Lat1 is dominated by tComp, we have fully utilized the involved computation resource. 4 XFER DESIGN In this section, we will present the XFER design for CNNs on multiple FPGAs, which can effectively alleviate the performance bottleneck on off-chip memory bandwidth and in turn achieve the super-linear performance. 4.1 Design Principles and Objective Here, we present three design principles and our ultimate objective for implementing CNNs on multi-FPGA clusters. P1. Maximizing the utilization of computation resources. The design should fully utilize the computation resources. In order to achieve this goal, the workloads should be balanced so that FPGAs will not be idle. Meanwhile, we need to avoid memory access stalls, hence DSPs don\u2019t have to wait for the data loading from off-chip memory. P2. Balancing the traffic loads across multiple FPGAs. Different from the single-FPGA design where the memory bus is the only communication channel to access off-chip data, in multi-FPGA clusters, inter-FPGA links provide extra communication bandwidth, which can considerably improve the efficiency of data transfer. However, it is essential to balance the traffic on memory buses and inter-FPGA links for performance improvement. P3. Minimizing the exchange of data in off-chip memories. The movements of data stored in off-chip memory are controlled by CPUs, which incurs high latency. Thus, we should avoid such movement as much as possible. Objective. Achieving super-linear speedup. The design objective is to achieve super-linear performance, such that the latency can be minimized without compromising on throughput or energy efficiency, which leads to the ultimate realization of real-time DNN inference. Following the above principles and objective, XFER consists of the following three steps: First, XFER achieves linear speedup through balancing computation workloads. Then, to achieve further speedup, XFER identifies the shared data among different partitions, and distributes these data across FPGAs to reduce the traffic loads on each FPGA\u2019s memory bus. During run time, FPGAs transfer a part of the data stored in their local off-chip memory via inter-FPGA links. 4.2 Layer Partition and Workload Balance A CNN layer can be partitioned into different parts to be computed in parallel. The most common partition is the batch partition, where the IFM and OFM are divided along batching direction, as shown in Figure 7(a). The computation of a batch of OFM only relies on the corresponding batch of IFM and the whole weights. In consequence these batches can be computed in parallel in multiple processing elements (PEs) if weights are duplicated to PEs. Similarly, we can partition CNN layers along rows (R) and columns (C), as shown in Figure 7(b)-(c). In addition, we can partition a CNN layer as follows: dividing the OFM into multiple parts along channel direction, and dividing weights correspondingly, as shown in Figure 7(d). In this case, we use the whole IFM and part of weights to compute part of OFM, and we call such partition as OFM channel partition. Similarly, we can partition IFM along channel direction and weights correspondingly, as shown in Figure 7(e). For each kind of partition, we define a partition factor to indicate the number of parts generated by the partition. We use notations Pb, Pr , Pc, Pm, Pn to represent factors for partitions along with batch (B), rows (R), columns (C), OFM channels (M), and IFM channels (N). For instance, Pr = 2 indicates that we partition IFM/OFM along rows into 2 parts, as shown in Figure 7(b). Kindly note that we use factors of 2 in Figure 7 for the simplicity of illustration; however, these factors can be other positive integers except 2, which will be restricted by the number of available FPGAs in a system. According to the types of shared data, we can classify partitions into 3 categories: first, \u201cweight shared\u201d case, where the computations of different partitions use the same weights, such as row or column partitions in Figure 7(a)-(c); second, \u201cIFM shared\u201d case, where the computations under the OFM channel partition use the same IFM, as shown in Figure 7(d); third, \u201cOFM shared\u201d case, where \fCODES+ISSS, 2019, New York, NY (b) Row Partition (d) OFM Channel Partition (e) IFM Channel Partition (a) Batch Partition PE PE PE PE I W O I W O I W O I W O (f) Weight Shared (g) IFM Shared (c) Column Partition PE PE I W O I W O (h) OFM Shared Summation Processing System (DRAM) Programmable Logic FPGA f1  f2  f1  f2  f1  f2 batch #1 batch #2 Figure 7: Five kinds of partitions and the base-line designs: (a) batch partition Pb = 2; (b) row partition Pr = 2; (c) column partition Pc = 2; (d) OFM channel partition Pm = 2; (e) IFM channel partition Pn = 2; (f) design for batch/row/column partitions; (g) design for OFM channel partition; (h) design for IFM channel partition. PE PE PE PE I W O I O W O W O (a) Pr = 2 (c) Pm = 2 W XFER I I XFER void XFER_load_weight(float bW[SIZE],         stream<float> &mem_in,                               stream<float> &b2b_in,         stream<float> &b2b_out){     for(int i=0; i<SIZE/2; i++){ #pragma HLS PIPELINE II=1 #pragma HLS dependence variable=bW intra false         bW[i] = mem_in.read();         bW[i+SIZE/2] = b2b_in.read();         b2b_out.write(bW[i]); }}// end of for loop and function (b) HLS codes for FPGA f2 in (a) void XFER_load_ifm(float bI[SIZE],    stream<float> &mem_in,                          stream<float> &b2b_in,    stream<float> &b2b_out){     for(int i=0; i<SIZE/2; i++){ #pragma HLS PIPELINE II=1 #pragma HLS dependence variable=bI intra false         bI[i] = mem_in.read();         bI[i+SIZE/2] = b2b_in.read();         b2b_out.write(bW[i]);     }}// end of for loop and function f1 f2 f1 f2 (d) HLS codes for FPGA f2 in (c) Figure 8: XFER design: (a) weight shared case; (b) HLS codes (a); (c) IFM shared case; (d) HLS codes for (c). the computations under the IFM channel partition share the OFM. Kindly note that \u201cOFM shared\u201d will cause the transmission of intermediate data as shown in Figure 7(h), which violates design principle P3. Hence, we do not consider it in the designs. For different types of partitions, we present a straightforward design that follows design principle P1 to balance workloads among FPGAs. This design will be the base of XFER. The main idea of this design is to map each partition to one FPGA and replicate the shared data to each FPGA. Figure 7(f) shows the design for weight shared partitions, where IFM and OFM are independent on two FPGAs and the whole weights are replicated to these FPGAs. Similarly, Figure 7(g) shows the design of IFM shared partition. In the above design, since the shared data are replicated to each FPGA, the computations in any two FPGAs have no dependency. It implies that all FPGAs can be executed in parallel, enabling linear speedup in the DNN inference. 4.3 Share Data through inter-FPGA links XFER can further boost the performance by transferring parts of the traffic loads from the memory bus to inter-FPGA links. Since XFER modifies the data communication subsystem, the matched system model needs to be revised. In the following text, we will introduce the detailed XFER design for weight shared partition and IFM shared partition. Weight Shared Partition. Figure 8(a) demonstrates the XFER design for weight shared partition, where each FPGA load a part of weights from its local off-chip memory, and obtain the remaining parts from its neighbors via inter-FPGA links. In other words, a copy of the whole weights are distributed in the off-chip memory across FPGAs. During the execution, each FPGA conducts three operations: (1) loading half of the data from its off-chip memory to on-chip buffer, (2) sending the loaded data to other FPGAs through the inter-FPGA links, and (3) receiving the remaining data from other FPGAs through the inter-FPGA links. HLS design. To efficiently conduct these operations, data are streamed in and out from the on-chip weight buffers (e.g., utilizing the AXI streams for Xilinx FPGAs). The detailed HLS codes are given in Figure 8(b). Kindly note that in order to transmit data in parallel, we make the intra dependency on bW to be false. In addition, we use the pipeline pragma to load and send the weight in the pipelined fashion. Model modification. XFER can reduce the latency of loading weightstWmem. With partition factors \u27e8Pb, Pr , Pc\u27e9,tWmem in XFER will be Pb \u00b7 Pr \u00b7 Pc times smaller. Thus, we replace Formula 9 as follows. tWmem = Tm \u00b7 Tn \u00b7 K \u00b7 K/(Wp \u00b7 Pb \u00b7 Pr \u00b7 Pc) (16) Then, we will formulate the newly incurred inter-FPGA communications. In XFER, each FPGA only holds 1 Pb \u00b7Pr \u00b7Pc part of the whole weights. It requires Pb \u00b7 Pr \u00b7 Pc \u22121 communication channels. For each channel, the latency is the same and we model the latency of the ith channel as follows. tW i b2b = Tm \u00b7 Tn \u00b7 K \u00b7 K/(W b2b p \u00b7 Pb \u00b7 Pr \u00b7 Pc) (17) where W b2b p is the number of ports in one channel. Finally, the latency Lat1 should be modified, since XFER incurs the inter-FPGA communication in the inner-most loop C in Figure 5(a). Formula 12 is revised as follows. Lat1 = max{tComp,tImem,tWmem, max i \u2208[1,..,P\u22121]{tW i b2b}} (18) where P \u22121 is the number of communication channels. IFM Shared Partition. Similar to the design of the weight shared case, XFER will transfer the movements of IFM from the memory bus to inter-FPGA links, as shown in Figure 8(c)-(d). We \fCODES+ISSS, 2019, New York, NY XFER PE PE O O I I XFER PE PE O I I XFER O W W XFER W W PE O I W (b) XFER Design (a) Workload-Balance Design PE O f1 I W I PE O W f2 I PE O W f4 f3 f1 f2 f3 f4 A B B A Figure 9: Workload-balance design and XFER design for the partitions with factors Pr = 2 and Pm = 2. add Formula 19 to model the latency on the ith inter-FPGA link, and replace formulations for modeling the latency of loading IFM and the latency Lat1. tIi b2b = Tm \u00b7 Tn \u00b7 K \u00b7 K/(Ib2b p \u00b7 Pm) (19) tImem = Tm \u00b7 Tn \u00b7 K \u00b7 K/(Ip \u00b7 Pm) (20) Lat1 = max{tComp,tImem,tWmem, max i \u2208[1,..,Q\u22121]{tIi b2b}} (21) where Ib2b p is the number of data transmitted via inter-FPGA links in parallel and Q \u22121 = Pm \u22121 is the number of links. 4.4 Extension to Hybrid Partitions We continue to extend XFER to support hybrid partitions where both weights and IFMs are shared. For partition \u27e8Pb, Pr , Pc, Pm\u27e9, XFER involves Pb \u00b7 Pr \u00b7 Pc \u00b7 Pm FPGAs. In the following texts, we solve three problems in constructing a network of FPGAs: (1) how to organize FPGAs, (2) what\u2019s the topology of the network, (3) how to formulate the inter-FPGA bandwidth constraints. Organization. We organize FPGAs in a two-dimensional array (2D-array) with Pm columns and Pb \u00b7 Pr \u00b7 Pc rows by the following two steps. First, for the IFM shared partition (e.g., partition A in Figure 9), the weights and OFM are partitioned into independent parts and allocated to a column of FPGAs, while the whole IFM is replicated to all FPGAs. Second, when considering the OFM shared partition (e.g., partition B), for all FPGAs in one column, the IFM and OFM are split into independent partitions. Then, the design with workload balanced can be obtained, as shown in Figure 9(a). The design has one property as follows. property 2. All FPGAs in one column share a part of weights, while all FPGAs in one row share a part of IFMs. Based on the above property, we extend XFER to support hybrid partitions. Specifically, for a column of FPGAs, a corresponding part of the weights are distributed among these FPGAs, and exchange the weights during execution like that in the 2-FPGA system. For a row of FPGAs, they share a part of the IFMs. Figure 9(b) demonstrates XFER for the hybrid partitions with factors Pr = 2 and Pm = 2, where the traffic loads on the memory bus can be significantly reduced. 1 4 3 2 1 4 3 2 1 4 3 2 1 2 3 4 1 2 3 1 4 3 2 1 4 3 2 1 4 3 2 1 2 3 1 2 3 1 2 3 1 1 2 1 3 1 3 2 1 3 2 fa fb fc fd fe ff fg fh fi fj fk fl FPGA FPGA FPGA data from local off-chip memory data from inter-FPGA links shared 1/4 weight in column 1 shared 1/3 IFM in row 1 shared 1/4 weight in column 2 shared 1/4 weight in column 3 shared 1/4 weight in column 4 1 2 2 2 3 1 1 3 2 shared 1/3 IFM in row 2 shared 1/3 IFM in row 3 1 3 # of columns = Pm # of columns = Pb*Pr*Pc bI bI bI bW bW bW bW Figure 10: 2D-torus topology and data movement. Tn Tn Tn \u2026 Tn \u2026 FPGA1 FPGA2 (a) XFER based design needing data transfers for intermediate data (b) XFER based design without transferring intermediate data Figure 11: Intermediate data placement across layers. Topology. Benefiting from the regular 2D-array organization, we have a wide range of choices for the networking topology, such as mesh, torus, folded torus, etc. In this work, we employ the 2Dtorus topology to build the connections among FPGAs [17, 18]. One reason to select 2D-torus is that we can apply a uniform design for each FPGA. As shown in Figure 10, where Pm = 4 and Pb \u00b7Pr \u00b7Pc = 3, each FPGA has two incoming links and two outgoing links; meanwhile, the amount of data received and transmitted by each FPGA are the same. In addition, with such a 2D-torus topology, the traffic loads on columns and rows are balanced, which overrides our design principle P2. Bandwidth Constraints. Based on 2D-torus topology, we formulate the bandwidth constraints. As shown in Figure 10, the loads on incoming links and outgoing links are the same; therefore, we build the constraints on one direction. We first consider the communication on one row for IFM sharing. The size of shared IFM is bI, which is shared by Pm FPGAs. Take FPGA fa in Figure 10 as an example, the total amount of data transmitted on outgoing link will be Drow = (Pm\u22121)\u00b7 bI Pm . Similarly, for one column, the total amount of data on outgoing link will be Dcol = (Pb \u00b7 Pr \u00b7 Pc \u22121) \u00b7 bW Pb \u00b7Pr \u00b7Pc . Kindly note that these data transmissions need to be completed in the time of Lat1 to avoid worsening the whole latency (see Equation 12). Let NB be the maximum bandwidth of the inter-FPGA communication on one direction. We have the following constraint. Drow + Dcol \u2264NB \u00b7 Lat1 (22) 4.5 Extension to Multiple CNN Layers We have discussed the optimizations for single CNN layer on multiple FPGAs. However, due to CNNs have many layers, how to \fCODES+ISSS, 2019, New York, NY ZYNQ Accelerator AXI DMA Wp Ip \u2026 \u2026 Op\u2026 pl_clk HP0 HP3 \u2026 AXI4 Asyn FIFO Aurora user_clk clock domain 1 clock domain 2 on-chip Main Memory off-chip ZYNQ Accelerator AXI DMA Wp Ip \u2026 \u2026 Op \u2026 pl_clk HP0 HP3 \u2026 AXI4 Asyn FIFO Aurora user_clk on-chip Main Memory off-chip b2b comm SFP+ Asyn FIFO Asyn FIFO Figure 12: Implementation overview of a multi-FPGA cluster with two FPGAs connected by SFP+. seamlessly execute multiple layers on multiple FPGAs is important. This subsection will follow our design principle P3 to extend XFER to support multiple CNN layers. To make data remain in-situ to minimize the intermediate data movements, the partition schemes for different layers should be coordinated according to XFER. Figure 11 shows two cases for the IFM shared partitions. The partition in Figure 11(a) will lead the exchange of intermediate data between FPGAs f1 and f2. Because in XFER (Figure 8(d)), the first of Tn/2 channels of IFM are loaded to f1, and the remained Tn/2 channels are loaded to f2. This will be applied to all of the channels until the end of the IFM. With the partition shown in Figure 11(a), all first Tn channels in OFM are produced at f1. In order to make it work for the convolution operation in the next layer, we need to exchange half of the OFM between f1 and f2. In contrast, if the OFM channels are partitioned in an interleaving way as shown in Figure 11(b), no data movements are required across layers. Based on the above observations, we investigate different partition methods for consecutive layers. For batch partition, the required data for the next layer on an FPGA is totally produced by itself, and no data movement is required. For row/column partition, only borders need to be transferred, where the small number of data can be transmitted during the execution via inter-FPGA links without passing CPUs. For OFM channel partition, we can avoid data movements by employing the interleaving partition, as shown in Figure 11(b). Furthermore, we found that if the consecutive layers employ different partition methods, data movement is unavoidable. In consequence, we will deploy CNNs on multi-FPGA clusters with uniform partition factors across CNN layers. 4.6 Analysis of Performance In the following, we analyze the performance in both latency and elapsed time to explore the design space. Single Layer. XFER can achieve super-linear speedup due to the following two reasons. First, the trip counts of loops D, E, F can be linearly reduced. Second, by alleviating the performance bottleneck on accessing off-chip memory, the latency Lat1 can be further reduced. In consequence, XFER is able to achieve superlinear speedup. Multiple Layers. The accelerator with uniform design parameters is simple to design and can avoid the costly inter-layer communications and FPGA reconfiguration, but it may be sub-optimal for some layers. Table 1 gives an example of the uniform design against the layer-customized design. From this table, we can see that the overall latency of the uniform design is 2,239 clock cycles, which is Table 1: Layer-specific and cross-layer optimization AlexNet Design Cycles(\u00d71000) Elap. Tm Tn Tr Tc Pb Pr Pc Pm Comp.[+Comm.] (sec.) Layer1 96 3 1 55 4 1 1 1 375 [+290] 0.5 Layer2 10 48 14 27 2 2 1 1 514 [+186] 9.7 Layer3 55 9 13 13 4 1 1 1 314 [+0] 162.3 Layer4 28 18 13 13 4 1 1 1 242 [+64] 60.7 Layer5 32 15 13 13 2 1 1 2 167 [+0] 40.4 Total neglect reprogramming overhead 2,152 Cross-Layer 64 7 7 14 2 1 2 1 2,239 797.2 slightly (within 5%) slower than the layer-customized design. Kindly note that for the layer-customized design, we consider the interlayer communication latency (in brackets), but ignore the FPGA reprogramming overhead, which will lead the layer-customized design inefficient against the uniform design. In consequence, we use uniform design in experiments. Finally, the column \u201cElap.\u201d shows the elapsed time to obtain the design for layer-specific or cross-layer optimization. As shown in this table, for the layer-specific optimization, all explorations can be finished in 3 minutes; while for the cross-layer optimization, it takes 13 minutes to obtain the design. These results demonstrate the efficiency of the proposed formulation. 5 EXPERIMENTAL RESULTS This section reports the evaluation designs synthesized by SuperLIP framework with XFER technique on Xilinx FPGAs. Results demonstrate the significant improvements on performance and energy efficiency achieved by Super-LIP against GPUs and the existing designs. We also demonstrate the scalability of Super-LIP and validate the accuracy of the presented system-level model. A. Implementation System Implementation. Figure 12 shows the implementation of a cluster with two Xilinx ZCU102 FPGAs. FPGAs are connected by SFP+ using the Xilinx Aurora IP. In this way, data in two FPGAs can be directly moved between their on-chip buffers. The implementation of each FPGA utilizes the ZYNQ architecture, which controls the startup of CNN accelerator, the off-chip/on-chip communications, etc. As shown in this figure, each FPGA has two clock domains: one for accelerator and the other for board-to-board communication. We employ asynchronous FIFOs to coordinate data movements in different clock domains. The accelerator on FPGA is implemented with Vivado HLS, which generates design\u2019s IP core from C language. In HLS, we apply HLS-defined pragma to implement loop optimization. Then, \fCODES+ISSS, 2019, New York, NY Table 2: Experimental results of Super-LIP with comparisons to GPUs and the existing FPGA designs Design mGPU GPU FPGA15 ISCA17 ISLPED16 Super-LIP Precision 32bits float 32bits float 32bits float 32bits float 16bits fixed 32bits float 16bits fixed Device Jetson TX2 Titan X VX485T VX485T 4\u00d7VX690t 2\u00d7ZCU102 2\u00d7ZCU102 Freq (MHz) 1300MHz 1139MHz 100MHz 100MHz 150MHz 100MHz 200MHz Power (Watt) 16.00 162.00 18.61 126.00 52.40 54.40 DSP Uti. 80% 80% 90.79% 55.87% BRAM Uti. 49.71% 43.25% 72.92% 92.43% Overall Perf. Lat. Thr. Lat. Thr. Lat. Thr. Lat. Thr. Lat. Thr. Lat. Thr. Lat. Thr. ms GOPS ms GOPS ms GOPS ms GOPS ms GOPS ms GOPS ms GOPS 11.1 13.2 110.75 5.1 6.4 235.55 21.62 69.09 60.13 85.47 30.6 128.8 10.13 149.54 2.27 679.04 E.-E. (GOPS/W) 6.88 1.45 3.71 1.02 2.85 12.48 the obtained IP cores are connected, synthesized and implemented in Vivado (v2017.4). In Vivado, we employ Xilinx Aurora IP core to control inter-FPGA communication and add an axi-timer to capture the exact elapsed time. FPGA boards are connected through SFP+ cables, as shown in Figure 13. Finally, we employ Xilinx SDK to program MPSoC on ZCU102, which controls the start-up of the accelerator and off-chip/on-chip communication. Design Parameters. The accelerator contains two sub-systems: computation subsystem and communication subsystem. In computation subsystem, the working frequency and computation parallelism significantly affect the performance. We use frequency of 100MHz for floating points, while 200MHz for fixed points. Then, the tiling parameters \u27e8Tm,Tn,Tr ,Tc\u27e9determines the computation parallelism. These parameters can be obtained via our proposed accelerator design methodology in Super-LIP. The communication subsystem includes the off-chip/on-chip memory communication and board-to-board communication. The design parameters \u27e8Ip,Wp,Op\u27e9are pre-set according to the bandwidth requirement captured from our model. Specifically, for floating points, we set Ip = 2,Wp = 2, and Op = 2, indicating peak bandwidth is 6\u00d732/8 100M = 2.4GB/s. For fixed points, we set Ip = 4, Wp = 8 and Op = 4, indicating peak bandwidth of 24\u00d716/8 100M = 2.4GB/s. The data width used for board-to-board communication is set according to Ip and Wp. For 16bits fixed point, Wp = 8 indicates the transmission data width is 16 \u00d7 8 = 128. Kindly note that the ZCU102 board can provide the maximum data width of 256bits for bi-direction board-to-board communication. In a cluster, the number of FPGAs N is determined by all partitioning parameters, i.e., N = Pb \u00b7 Pr \u00b7 Pc \u00b7 Pm. The infrastructure of the network in the FPGA cluster applies the 2D-torus topology, as illustrated in Figure 10 of Section 4.4, where each FPGA has two incoming and two outgoing inter-FPGA links. Super-LIP will also control the data flow among FPGAs. B. Low Latency and Energy Efficiency Table 2 reports the comparison results in latency, throughput and energy efficiency of AlexNet with a batch size of 1 on different platforms and designs. The competitors of Super-LIP include mobile GPU (Jetson TX2) and GPU (Titan X), single-FPGA design (FPGA15 [14], ISCA17 [12]), and multi-FPGA design (ISLPED16 [15]). The power consumption of our implementation is measured by a power meter as demonstrated in Figure 13. Note that notation \u201c-\u201d indicates that data is not reported in references or inapplicable. Board-to-Board SFP+ Links Total Power:  54.9W Plug plate Power:  0.5W ZCU102 Idle Power:  \u223c20W Figure 13: Power measurement of on-board executions. On-Board Measurement. Before presenting the detailed results, we first introduce the on-board measurement, where latency and power consumption are two main metrics. For all platforms and implementations, they will consistently process a set of input images, say 1,000 images. For a fair comparison, we record the latency and power consumption when the system enters the stable state (i.e., after the process of the first image). By recording the latency of different images, we observe that the elapsed time of GPUs is varied for different executions, while it is stable in FPGAs. For example, the latency on mGPU ranges from 11.1ms to 13.2ms, as shown in Table 2. It implies that the GPU implementations need to apply the worst-case execution time with an extra safety margin to satisfy the hard real-time constraint, which will drastically degrade the resource utilization and energy efficiency. Latency. Real-time DNN inference requires ultra-low latency to avoid missing deadline. For 32bits float-point, Super-LIP achieves latency of 10.13ms, which is 23.26%, 2.13\u00d7, 5.94\u00d7 less than that of mGPU, FPGA15, and ISCA17. However, Super-LIP with 32bits float-point is slower than Titan X GPU, whose latency is 6.4ms. This is because such GPU is much more powerful, with the penalty of consuming more than 3\u00d7 power over the FPGA implementation in Super-LIP. Benefiting from the flexibility of FPGAs to apply different data types for computation, it is possible to reduce latency by using lower-precision data type. As shown in this table, by applying 16bits fix-point, Super-LIP can achieve the lowest latency among all competitors, i.e., 2.27ms. Throughput and Energy Efficiency. Super-LIP makes better trade-offs between latency and throughput than the existing \fCODES+ISSS, 2019, New York, NY Table 3: Comparison results on ZCU102 Design 32bits float 16bits fixed FPGA15 Super-LIP FPGA15 Super-LIP \u27e8Tm, Tn \u27e9 \u27e864, 7\u27e9 \u27e864, 7\u27e9 \u27e864, 24\u27e9 \u27e8128, 10\u27e9 Power (W) 25.70 52.40 26.00 54.40 (1 FPGA) (2 FPGAs) (1 FPGA) (2 FPGAs) Perf. Lat. Thr. Lat. Thr. Lat. Thr. Lat. Thr. ms GOPS ms GOPS ms GOPS ms GOPS conv1 7.36 28.6 3.66 57.6 3.74 56.5 0.94 224.5 conv2 5.20 86.1 2.55 175.5 1.48 302.6 0.48 933.1 conv3 4.50 66.4 1.73 172.7 1.20 249.6 0.33 906.2 conv4 3.41 65.7 1.31 171.0 0.89 252.6 0.35 640.8 conv5 2.28 66.0 0.88 170.9 0.59 251.7 0.17 879.5 overall 22.75 66.6 10.13 149.5 7.90 195.1 2.27 679.0 Perf. Impr. 1.00\u00d7 2.25\u00d7 1.00\u00d7 3.48\u00d7 E.-E. 2.59 2.85 7.51 12.48 (GOPS/W) E.-E. Impr. 9.21% 39.86% pipeline-based FPGA competitors (ISCA17 and ISLPED16). Compared with ISCA17 with 32bits float-point, Super-LIP achieves 5.94\u00d7 lower latency together with 1.75\u00d7 higher throughput. The improvement in throughput is less than that on latency is because ISCA17 aims to improve throughput, but its throughput is still less than Super-LIP\u2019s. Similarly, compare with ISLPED16 with 16bits fix-point, Super-LIP achieves 13.48\u00d7 lower latency together with 5.27\u00d7 higher throughput. Benefiting from the higher throughput, Super-LIP achieves the highest energy efficiency than competitors. Utilization. From Table 2, we observe that the DSP resource is not fully utilized in Super-LIP with 16bits. This is because there is not enough BRAM resource to support parallel data access (Equations 1-6 for details). By adding BRAM in the FPGA, we expect to further reduce latency and achieve higher energy efficiency. C. Results on ZCU102 We notice that, in Table 2, the energy efficiency of Super-LIP with 32bits is less than FPGA15. This is because the power consumed by ZCU102 in idle state (i.e., \u223c20W ) is already larger than that of FPGA15 at run-time (18.61W). For a fair comparison, we reimplement FPGA15 on ZCU102, and the results are reported in Table 3. 32bits float-point. We have several conclusions. First, the design parameters (Tm and Tn) of FPGA15 and Super-LIP are the same; therefore, the only difference is that Super-LIP has extra inter-FPGA communication. Second, the power in FPGA15 design (25.70W) is less than half of that in Super-LIP (52.40W). The gap 52.40 \u221225.70 \u00d7 2 = 1.0W on power consumption is caused by the inter-FPGA communication (including IP core, communication links, etc.), which only occupies 1.91% of the total power. Third, Super-LIP with 2 FPGAs achieves 2.25\u00d7 speedup over FPGA15, indicating that we have achieved super-linear speedup. Furthermore, benefiting from the significant performance achievement, SuperLIP obtains 9.21% improvement in energy efficiency. 16bits fix-point. Unlike 32bits float-point, the optimal design parameters (\u27e8Tm,Tn\u27e9) explored by FPGA15 and Super-LIP are different. This is because the design \u27e8128, 10\u27e9alleviates bottleneck on 0 2 4 6 model in [13] our model on-board exec cannot model \u232912,16\u232a \u232910,22\u232a \u23298,32\u232a \u23298,32\u232a Clock Cycles Design Parameters: \u2329Tm,Tn\u232a  2 FPGAs 1 FPGA 1 FPGA 1 FPGA \u00d7105 Figure 14: Comparisons of predictable models and onboard executions on latency: employing different designs on single-FPGA and 2-FPGA systems. memory bandwidth, which results in severe performance degradation in the overall assessment for FPGA15 design, while Super-LIP can resolve such bottlenecks to achieve better performance. Specifically, our Super-LIP design has achieved 3.48\u00d7 speedup and 39.86% improvements in energy efficiency, when compared with FPGA15. D. Model Accuracy and Effectiveness Now, we are going to validate the accuracy and effectiveness of the proposed system-level model. We will conduct two sets of experiments: (1) we compare the proposed model with the existing one in predict latency; (2) we compare the proposed model with the final implementation results from Vivado in memory resource, computation resource, and on-board execution latency. First, Figure 14 reports the comparison results among different models and on-board execution latency. The x-axis and y-axis represent different designs and latency in clock cycles, respectively. In the first three designs, we employ one FPGA for implementation; while for the fourth one, we employ 2 FPGAs. Results in Figure 14 clearly show that the latency predicted by our proposed model is always close to the on-board execution latency, where the average deviation is only 2.53%. In contrast, the existing model in [14] has larger deviations on designs of \u27e810, 22\u27e9 and \u27e88, 32\u27e9, which are 18.49% and 45.47%. In addition, the existing model cannot predict for multiple FPGAs, but ours can. We have another observation from Figure 14. For the design of \u27e812, 16\u27e9, model in [14] predicts the same latency with ours. This is because the computation latency dominates the whole system. In this case, the inaccurate estimation of communication will not affect prediction accuracy. However, when we employ more computation resource (by increasingTm \u00d7Tn), the performance bottleneck moves to communication which leads to the large latency deviations between the existing model and the on-board execution. The above results verify the accuracy and effectiveness of the proposed system-level model in predicting system latency. With such an accurate model, it can help designers to get the accurate system performance to make better design decisions. Next, Table 4 reports the comparison between the proposed model and the final implementation results from Vivado in BRAMs and DSPs. It is clear that the deviations on BRAM and DSP usages are less than 7.5% and 3.9%, respectively. These deviations are mainly caused by the overhead on extra operations besides the accelerator itself, such as DSPs used for address calculation. The above results further verify the accuracy of the proposed model. \fCODES+ISSS, 2019, New York, NY Table 4: Experimental results on model validation, performance bottleneck detection and alleviation Design Precision \u27e8Tm, T n\u27e9 Partition Our Model On-Board Deviation Speedup Cycles BRAM DSPs Bound Cycles BRAM DSPs Cycles BRAM DSPs A (Single) 32b float \u27e88, 32\u27e9 519168 592 1280 IFM 535530 624 1326 3.06% 5.13% 3.47% baseline B (XFER) \u27e88, 32\u27e9 Pm=2 158880 592 1280 Comp. 162114 640 1331 1.99% 7.50% 3.83% 3.30X C (Single) 16b fixed \u27e864, 20\u27e9 115200 1448 1280 Weight 118688 1516 1324 2.94% 4.49% 3.32% baseline D (XFER) \u27e864, 20\u27e9 Pr=2 32760 1448 1280 Comp. 34622 1530 1330 5.38% 5.36% 3.76% 3.43X Results in Table 4 further verify the effectiveness of the proposed techniques in detecting and alleviating system performance bottlenecks. For the single-FPGA designs A and C, we employ Corollary 1 to detect their performance bottlenecks, as shown in Column \u201cBound\u201d under Column \u201cOur Model\u201d. It indicates that the performance of design A is bounded by loading IFM data, while that of design C is bounded by loading weights data. For design A, we apply XFER technique by setting Pm = 2 to share IFM data on interFPGA links, and it outputs the design B. As shown in this table, the performance bottleneck on design B has been successfully moved to computation, and therefore, it achieves 3.3\u00d7 speedup. Similarly, we apply XFER technique on design C to generate design D with 3.43\u00d7 speedup. The above results validate the accuracy of the proposed model in modeling system resources. In addition, the proposed system-level model can be applied to effectively detect performance bottleneck. And the proposed XFER technique can be applied to alleviate different kinds of bottlenecks. As a result, Super-LIP can achieve super-linear speedups with multiple FPGAs. E. Design Space Exploration Finally, we explore the design space to demonstrate the scalability of Super-LIP. Specifically, we scale up the number of FPGAs with the same design parameters (the optimal ones in single-FPGA design) but different partitions. Figure 15 reports the experimental results of four widely used CNNs with 16bits fixed point on the clusters with up to 16 FPGAs, including AlexNet, SqueezeNet, VGG, and YOLO. In the figure, the x-axis and y-axis represent the number of involved FPGAs and the total clock cycles. Each point corresponds to a design with specific loop tiling and partition factors. We give the tiling value (Tm and Tn) of CNNs in each sub-figure; for instance, in AlexNet, Tm = 128 and Tn = 10. For the system with no more than 2 FPGAs, we have implemented the accelerators on the testbed in Figure 13, and obtain the on-board execution latency. While for larger FPGA cluster, we obtain the latency using the following method. First, according to these tiling factors, we implement the accelerator on FPGA to obtain its on-board execution latency. Then, according to the partition factors for each design point, we can obtain how many times the accelerator will be invoked in each layer. Based on the above two kinds of information, we can get the computation latency. In addition, according to the determined partition factors, we can get the communication load on intra-FPGA and inter-FPGA links to calculate the communication latency. Finally, the overall latency can be derived from Formula 14. As shown in Figures 15(a)-(d), with the increasing number of FPGAs, Super-LIP can consistently reduce the overall latency. We observe that the speedup of SqueezeNet is relatively small, mainly because the sizes of weight and IFM are small owing to the squeeze operations. However, its latency can still be drastically reduced, from 6.69ms to 0.45ms. For AlexNet, VGG, and YOLO, super-linear 1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 0 2 4 6 10 12 14 16 x 105 0 2 4 6 8 10 12 1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 (a) AlexNet \u2329128,10\u232a 2FPGAs 2.54\u00d7 3FPGAs 3.71\u00d7 4FPGAs 4.86\u00d7 16FPGAs 17.95\u00d7 Single FPGA Clock Cycles # of FPGAs # of FPGAs 8 0.31ms 5.63ms 2FPGAs 2.52\u00d7 4FPGAs 5.30\u00d7 16FPGAs 21.19\u00d7 Single FPGA 3.37ms 3FPGAs 4.02\u00d7 71.46ms (c) VGG \u232964,26\u232a x 106 1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 # of FPGAs 0 5 10 15 20 30 25 x 106 2FPGAs 2.64\u00d7 4FPGAs 7.31\u00d7 16FPGAs 27.93\u00d7 Single FPGA 3FPGAs 5.69\u00d7 4.53ms 126.6ms Clock Cycles (d) YOLO \u232964,25\u232a 0 2 10 Clock Cycles 4 6 8 12 14 1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 # of FPGAs (b) SqueezeNet \u232996,16\u232a x 105 2FPGAs 2.01\u00d7 4FPGAs 3.92\u00d7 16FPGAs 14.75\u00d7 Single FPGA 0.45ms 3FPGAs 2.92\u00d7 6.69ms Clock Cycles Figure 15: Design space exploration of Super-LIP with the increasing number of FPGAs using different CNNs. performance can be consistently achieved with 2-16 FPGAs. Specifically, for YOLO, the latency is reduced from 126.6ms to 4.53ms with 16 FPGAs, achieving 27.93\u00d7 speedup. We have another observation from Figure 15(b): for SqueezeNet, when the number of FPGAs increases to 3, the speedup is only 3.92\u00d7, which indicates the failure of achieving super-linear speedup. This is because SqueezeNet contains many convolution operations with the kernel size of 1, which leads the performance bottleneck mainly on computation. In contrast, we can consistently achieve super-linear speedup for the other three CNNs when the size of FPGA cluster scales up to 16. However, when the size of FPGA cluster scales up, the linear performance will be terminated since the number of channels (or row, column) in a CNN layer is fixed, and we cannot further improve parallelism by adding partition factor when it reaches the maximum number. Benefiting from the super-linear performance, the system energy efficiency can be improved. Compared with the single FPGA design, for AlexNet, VGG, and YOLO on 4 FPGAs, the energy efficiency improvements are 11.29%, 20.65%, and 41.02%, respectively. \fCODES+ISSS, 2019, New York, NY When the size of FPGA cluster scales up to 16, these figures are 3.93%, 18.61%, and 36.25%. We observe that the improvements are decreased because the overheads on communication are increased as the cluster size increases. Overall, these results verify that by applying the proposed techniques on multiple FPPGAs, we can achieve super-linear speedup, and in turn, the energy efficiency can be improved against the single-FPGA design. Finally, we analyze the bandwidth requirement of clusters. From Figures 15(a)-(d), we can observe that the performance improvement is converged when the number of FPGAs approaches 16 with a 4\u00d74 torus topology. In this scale, the inter-FPGA bandwidth provided by ZCU102 is sufficient. Specifically, for each FPGA, there are 3 weights and 3 IFMs needing to be transmitted simultaneously. Thus, the BW requirement for each FPGA is (3 + 3) \u00d7 (16bits/cycle) = 144bits/cycle; while ZCU102 provides a bandwidth of 256bits/cycle (4 SPF+ ports with 64bits wide each). Furthermore, we can add 4 QSFP ports for additional bandwidth of 4 \u00d7 256 = 1024bits/cycle for even larger clusters. The above results verify the scalability of Super-LIP. In addition, our techniques are effective to explore the design space to provide more options for different timing constraints. 6 RELATED WORK The development of FPGA-based DNNs accelerator evolves in three stages. At the early stage [14, 19\u201323], the whole FPGA is designed as one accelerator, and a controller iteratively moves data from off-chip DRAM to the accelerator to be executed. In the second stage, it is observed that the computation resource cannot be fully utilized with the one-size accelerator due to the varied computation and memory requirements in DNN layers. To overcome this shortage, multiple accelerators are integrated into one FPGA [12, 24, 25]. However, the restrict resource on one board still limits the performance boosting of DNNs on FPGAs. Most recently, with the growing demand in time performance, it is a trend to employ a cluster of FPGAs to execute DNNs [15, 26\u201332]. In [15, 28], authors construct multiple FPGAs as a pipeline to execute a set of input images in a pipeline fashion. In [26], authors split the CNN layers to balance pipeline stages for higher throughput and lower cost. Authors in [27] employ multiple FPGAs for the training phase. In [29, 30], multi-FPGA platforms are utilized to accelerate the lung nodule segmentation. All the above works target on improving throughput by using a pipeline of FPGAs, which can achieve high throughput but make sacrifices on latency. To satisfy the low latency requirement for real-time DNN inference, Microsoft in Brainwave [6, 7] devise techniques to pin weights on different FPGAs. Such an approach can work well for RNNs with small intermediate data, but awkward for CNN implementations due to the large intermediate data and complicated data reuse pattern. Kindly note that in [6, 7], authors use only one FPGA for CNNs, whose input image has low resolution that hides the bandwidth bottleneck issue. However, for more realistic CNN applications with high resolution, like medical images, it is still unknown how to achieve real-time inference with ultra-low latency using multiple FPGAs. Super-LIP is proposed to fill this gap. Another branch of related work is to deploy CNNs on multi-core mobile devices or multi-processor system on-chip (MPSoC) [33\u2013 38]. Unlike FPGA-based implementation that requires designers to determine the designs of communication and computation subsystems, processing elements in these systems use fixed designs (e.g., CPUs, GPUs). In consequence, the optimization problem on such systems is how to run tasks to computation components in parallel, without considering how to tailor hardware designs. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1901.11211v1",
+                    "title": "Accuracy vs. Efficiency: Achieving Both through FPGA-Implementation Aware Neural Architecture Search",
+                    "abstract": "A fundamental question lies in almost every application of deep neural\nnetworks: what is the optimal neural architecture given a specific dataset?\nRecently, several Neural Architecture Search (NAS) frameworks have been\ndeveloped that use reinforcement learning and evolutionary algorithm to search\nfor the solution. However, most of them take a long time to find the optimal\narchitecture due to the huge search space and the lengthy training process\nneeded to evaluate each candidate. In addition, most of them aim at accuracy\nonly and do not take into consideration the hardware that will be used to\nimplement the architecture. This will potentially lead to excessive latencies\nbeyond specifications, rendering the resulting architectures useless. To\naddress both issues, in this paper we use Field Programmable Gate Arrays\n(FPGAs) as a vehicle to present a novel hardware-aware NAS framework, namely\nFNAS, which will provide an optimal neural architecture with latency guaranteed\nto meet the specification. In addition, with a performance abstraction model to\nanalyze the latency of neural architectures without training, our framework can\nquickly prune architectures that do not satisfy the specification, leading to\nhigher efficiency. Experimental results on common data set such as ImageNet\nshow that in the cases where the state-of-the-art generates architectures with\nlatencies 7.81x longer than the specification, those from FNAS can meet the\nspecs with less than 1% accuracy loss. Moreover, FNAS also achieves up to\n11.13x speedup for the search process. To the best of the authors' knowledge,\nthis is the very first hardware aware NAS.",
+                    "authors": "Weiwen Jiang, Xinyi Zhang, Edwin H. -M. Sha, Lei Yang, Qingfeng Zhuge, Yiyu Shi, Jingtong Hu",
+                    "published": "2019-01-31",
+                    "updated": "2019-01-31",
+                    "primary_cat": "cs.DC",
+                    "cats": [
+                        "cs.DC",
+                        "cs.LG"
+                    ],
+                    "main_content": "INTRODUCTION The performance of a Deep Neural Network (DNN) is mostly decided by its architecture. Yet the design of DNN architecture had signi\ufb01cantly relied on human expertise and labor until the recent development of Neural Architecture Search (NAS) that can automatically explore the optimal architecture for a particular application. Existing research e\ufb00orts [6, 16] have demonstrated that NAS can generate DNNs of competitive or even better accuracy against the human-invented ones (e.g., AlexNet, VGGNet, GoogleNet and ResNet). However, the prevalence of NAS is obstructed by its ef\ufb01ciency. As reported in [16], the search process can take several days even with hundreds of GPUs. The issue mainly comes from the fact that the search space can be huge, and for each candidate architecture, lengthy training process is needed to evaluate it. In addition, a mainstay for any existing NAS framework is that accuracy is the mono-objective to guide the search [16]. If the resulting architecture is to be deployed in the cloud or latency is not a critical factor, they will still work. However, if the architecture is to be implemented on hardware with latency speci\ufb01cation, then there is no guarantee that the speci\ufb01cation will be met. In these scenarios, the optimal architecture found by NAS is simply useless. In this paper, we propose a novel hardware-aware NAS framework to address the above issues. To illustrate our framework, we choose to use Field programmable gate array (FPGA) as a vehicle, as it has gradually become one of the most popular platforms to implement DNNs due to its high performance and energy e\ufb03ciency, in particular for low-batch real-time applications [2]. To introduce hardware awareness, it seems to be straightforward to simply include an additional metric in existing NAS frameworks that describe the latency of a neural architecture on an FPGA. However, the evaluation of the metric can be challenging. First, unlike the regular path-based structures in most human-invented DNNs, the architecture obtained by NAS can be irregular. Many existing design \ufb02ows dedicated for human-invented DNNs are not suitable for such complicated structures [2\u20134, 8, 9, 13\u201315]. Second, the architectures from NAS commonly have large sizes, which may require multi-FPGAs to collaborate for implementation. Consequently, the scheduling of tasks on multiple FPGAs should be taken into consideration. As such, a more elegant way to evaluate the metric is warranted. Towards this, we put forward an abstraction model that builds a bridge between the software (neural architecture) and hardware (FPGA designs) for e\ufb03cient latency estimation. Speci\ufb01cally, a tilebased graph model is presented to describe a given DNN under an FPGA design. In the model, we determine the granularity of tasks, the task dependencies, and the data accesses according to the DNN architecture and the tiling parameters in the design. Then, the complicated dependencies among DNN layers can be captured by adding extra edges among tasks. As for the scheduling on multiple FPGAs, a limited number of works exist in the literature [4, 14], all of which still follow the scheduler design for a single FPGA [13] (see Figure 5(a)). Such schedule paradigm, however, cannot fully exploit the parallelism among FPGAs. In this work, we propose a more \ufb02exible schedule mechanism (see Figure 5(b)). We \ufb01rst study the design principle for schedulers, based on which we present the mechanism to schedule \fNN Architectures \u201cHyperparameters\u201d Number of Filters Filter Size Stride Para. Filter Size Layer N Layer N+1 Layer N-1 \u2026 \u2026 The controller (RNN) Trainning Accuracy \u201cA\u201d. Implement NN     Get performacne. R=A Target FPGAs     DSP number ... Not satisfy  req. perf. Failed Succ. Satisfy  req. perf. Number of Filters Layer N-1 \u2026 Number of Filters Generated NNs (a) NAS Framework (b) implementation for generated NN after convergence before convergence Figure 1: NAS framework [16] with its implementation. tasks in the abstraction model. Furthermore, we theoretically analyze the latency of executions and pipeline stalls to estimate the overall latency. Kindly note that the proposed schedule paradigm can also be widely applicable in the design of multi-FPGA systems beyond the scope of this work. The main contributions of this paper are as follows. \u2022 Framework. We build an FPGA-implementation aware neural architecture search framework, namely FNAS, which can generate optimal DNN architectures with guaranteed latency on target FPGAs. \u2022 Abstraction Model. We propose a graph model to describe neural architectures based on FPGA implementations, which provides the fundamental support for latency analysis. In addition, it can model di\ufb00erent kinds of architectures. \u2022 Schedule Paradigm. We present a novel schedule paradigm, which can fully exploit the parallelism among multiple FPGAs. Experimental results on common data set such as ImageNet show that in the cases where the state-of-the-art [16] generates architectures with latencies 7.81\u00d7 longer than the speci\ufb01cations, those from FNAS can meet them with less than 1% accuracy loss; meanwhile FNAS also achieves 11.13\u00d7 speedup for the search process. The remainder of the paper is organized as follows. Section 2 reviews related background and Section 3 demonstrates our motivation and problem formulation. The detailed FNAS algorithm is presented in Section 4. Experimental results are shown in Section 5 and concluding remarks are given in Section 6. 2 BACKGROUND In this section, we will present the background on the neural architecture search and the FPGA-based DNN implementations. Searching Neural Network Architecture. Although the research on automatically predicting neural network architectures can trace back to the 1980s [7], after deep neural networks have achieved great success in AI domains, there has been growing interests in generating good neural architectures recently. With the fact that the architectures are growing deeper, the search space grows exponentially, which makes the search process di\ufb03cult. In existing works, there are two main directions in searching an architecture: (1) employing reinforcement learning [1, 16, 17], and (2) applying the evolutionary algorithms [6, 10]. Figure 1 shows the NAS framework presented in [16]. In NAS, it iteratively generates a new child network, and obtains its accuracy A by training it on a held-out data set. Then, accuracy A will be used as the reward signal for the next iteration. The search process will be stopped if the Table 1: FNAS uses less time to generate architectureshaving lower latency on PYNQ with small accuracy degradations. Methods TC Elasp. Lat. Acc. ms (m,s) Imp. (ms) Imp. (%) Deg. NAS [17] 190m33s 19.70 99.42% FNAS 10 74m29s 2.55\u00d7 8.67 2.27\u00d7 99.34% -0.08% 5 59m19s 3.21\u00d7 4.77 4.13\u00d7 99.18% -0.24% 2 17m07s 11.13\u00d7 1.80 10.94\u00d7 98.61% -0.81% controller is converged for the maximum accuracy, or the accuracy of child network satis\ufb01es the required accuracy rA. The generated \ufb01nal design will then be implemented into FPGAs. Existing work has demonstrated that the automatically searched network architectures can achieve close accuracy to the best human-invented architectures [16, 17]. However, there are two important challenges that need to be addressed. First, the searching process is ine\ufb03cent. [16] reported that 20,000 networks were trained across 500 P100 GPUs over 4 days to \ufb01nd the descired network. Second, the generated neural architectures achieve high accuracy with the sacri\ufb01ce of inference speed. The resultant network are usually complex and slow, which frequently fails to satisfy the required timing speci\ufb01cation with available computing resources for real-time AI applications. Table 1 reports our results of NAS [16] for image classi\ufb01cation using MNIST data set targeting on PYNQ board [5]. We can observe that it takes 190m 33s to complete the search by NAS, with 19.70 ms of the latency and 99.42% of the accuracy for the generated network. FPGA Implementation. FPGA has demonstrated its excellent abilities to achieve high performance and energy e\ufb03ciency for lowbatch real-time inferences. With such vision, a series of works for implementing neural networks on FPGAs have been carried out: (1) a single accelerator is implemented on a single FPGA [11, 13]; (2) multiple accelerators are integrated into a single FPGA [8, 9, 15]; (3) multiple accelerators are deployed on multiple FPGAs [2\u20134, 14]. Existing NAS did not take implementation into consideration at all. In order to make NAS process implementation-aware, the inference latency in FPGA has to be obtained for each child network. Existing research e\ufb00orts commonly analyze the latency of DNNs on FPGAs by generating the HLSor RTL-level code [8, 13, 15], which may involve human intervention and vast amount of time. Therefore, if we simply obtain inference latency using existing techniques and naively integrating it into the reward for architecture selection, the NAS search process will take even longer time. In this work, one of the main contributions is to accurately and quickly estimate FPGA inference latency so that both search process and network generating are e\ufb03cient. Table 1 also reports results of the proposed FNAS. From this table, we can see that for the same dataset, by setting the inference timing speci\ufb01cations (TS) to 2ms, FNAS can reduce the search time from 190 minutes to 17 minutes, achieving 11.13\u00d7 reduction. What is more, the latency of the resultant architecture on PYNQ is 10.94\u00d7 shorter against that explored by NAS. Meanwhile, the accuracy degradation is within 1%. For other two cases, FNAS signi\ufb01cantly achieves more than 2\u00d7 speedup with only 0.08% and 0.24% penalty on accuracy. These verify that the implementation of aware-FNAS can signi\ufb01cantly reduce the search time and guarantee the generated architecture on \f      FNAS-Design  \u201cDesign on Program Logic\u201d       FNAS-GG \u201cTile-based Task Graph Generator\u201d 2       FNAS-Sched \u201cScheduler on Processing System\u201d 3       FNAS-Analyzer Performance \u201cL\u201d 4 1 Target FPGAs     DSP number ... Trainning Accuracy \u201cA\u201d. FNAS Tool FNAS Framework NN Architectures \u201cHyperparameters\u201d R=f(A,L) Number of Filters Filter Size Stride Para. Filter Size Layer N Layer N+1 Layer N-1 \u2026 \u2026 The controller (RNN) Number of Filters Layer N-1 \u2026 Number of Filters Figure 2: Overview of the proposed FNAS framework. target FPGAs to satisfy the timing speci\ufb01cation in the inference phase while maintaining the accuracy. 3 FNAS FRAMEWORK 3.1 Problem De\ufb01nition and FNAS Overview In this paper, we aim to develop an FPGA-implementation aware Neural Architecture Search. The problem is formally de\ufb01ned as follows: Given a speci\ufb01c data set, a target FPGA platform and a required inference latency rL, our objective is to automatically generate a neural network, such that its inference latency on the given FPGA platform is less than rL, while achieving the maximum accuracy for the machine learning task on the given data set. Figure 2 shows the overview of FPGA-implementation aware Neural Architecture Search (FNAS) framwork. In FNAS, it takes the FPGA-based inference performance into consideration during child network searching. Speci\ufb01cally, instead of directly applying accuracy A as reward, FNAS employs a reward function f to calculate the reward in terms of accuracy A and performance/latency L. In order to e\ufb03ciently and accurately estimate the inference latency L for a given NN architecture on target FPGAs, we develop the \u201cFNAS tool\u201d. There are 4 components in FNAS tool, including \u2780FNAS-Design, \u2781FNAS-GG, \u2782FNAS-Sched, and \u2783FNASAnalyzer. In the following texts, we will \ufb01rst present the reward function, then introduce these components one-by-one. 3.2 Reward function Reward function takes the accuracy A, latency L, and the required latency rL to calculate the reward signal. The function to calculate the reward R is de\ufb01ned as follows. R = \u001a r L\u2212L r L \u22121 L > rL (A \u2212b) + L r L L \u2264rL (1) In the above function, there are two cases. First, if L > rL, it indicates that the performance of the resultant system cannot satisfy the timing speci\ufb01cation. In this case, we do not train the child network and directly return a negative reward to the controller. In the second case, we sum up the reward of performance and accuracy, where the performance reward is set as L r L , which indicates a solution has higher performance reward if its latency approaches the required level. Here, b is a baseline function, which is an exponential moving average of the previous architecture accuracies [16]. 3.3 Tiling Parameters: \u2780FNAS-Design weights \u2026 Tm \u2026 layer 1 layer 2 Tm N Tn Tn (a) N/Tn=2 Tr R Tc (b) R/Tr=2, C/Tc=2 (c) M/Tm=3, N/Tn=2, R/Tr=1, C/Tc=2 C 1 2 3 4 1 2 layer 1 layer 2 layer 3 T1,1,1 ifm T1,1,2 ifm T1,2,1 ifm T1,2,2 ifm T2,1,1 ofm T2,1,2 ofm T2,2,1 ofm T2,2,2 ofm ofm T2,3,1 ofm T2,3,2 ofm T2,1,1 ifm T2,1,2 ifm T2,2,1 ifm T2,2,2 ifm T3,1,1 ofm T3,1,2 ofm T3,2,1 ofm T3,2,2 ofm ofm T3,3,1 ofm T3,3,2 ofm (d) Tm!=Tn T1,1,1 ifm T1,1,2 ifm T1,2,1 ifm T1,2,2 ifm T2,1,1 ofm T2,1,2 ofm T2,2,1 ofm T2,2,2 ofm T2,3,1 ofm T2,3,2 ofm v1,1,1,1 v1,1,2,1 v1,1,3,1 v1,1,1,2 v1,1,2,2 v1,1,3,2 v1,2,1,1 v1,2,2,1 v1,2,3,1 v1,2,1,2 v1,2,2,2 v1,2,3,2 T2,1,1 ifm T2,1,2 ifm T2,2,1 ifm T2,2,2 ifm T3,1,1 ofm T3,1,2 ofm T3,2,1 ofm T3,2,2 ofm T3,3,1 ofm T3,3,2 ofm v2,1,1,1 v2,1,2,1 v2,1,3,1 v2,1,1,2 v2,1,2,2 v2,1,3,2 v2,2,1,1 v2,2,2,1 v2,2,3,1 v2,2,1,2 v2,2,2,2 v2,2,3,2 layer 1 layer 2 layer 3 conv 1 conv 2 (e) Tile-based Task Graphs with convolutional tasks derived from (d) inter-layer dependencies intra-layer dependencies inter-layer dependencies Figure 3: FNAS-Design and FNAS-GG: (a) channel tiles; (b) row/col tiles; (c) tile-based convolution; (d) encoding of tiles; (e) tile-based task graph. Due to the limited resource on FPGA, it may be di\ufb03cult to place a whole convolutional layer on FPGA. In consequence, it is common to apply tiling technique to split convolutional operations into multiple small tasks [8, 12\u201315]. FNAS-Design is to determine the tiling parameters for a given NN architecture on target FPGAs. Take one convolutional operation as an example, it involves four parameters \u27e8Tm,Tn,Tr,Tc\u27e9, related to the input/output feature maps (IFM/OFM). Here, the number of IFM channel is N . The size of corresponding tiles isTn (channels). IFM is then partitioned into \u2308N Tn \u2309tiles, as shown in Figure 3(a). Similarly, OFM with M channels is partitioned to \u2308M Tm \u2309tiles. In addition, the numbers of row/column of OFM are R and C, respectively. They are tiled according to Tr and Tc as shown in Figure 3(b). After tiling the IFM/OFM/row/col, one convolutional operation is divided to smaller tasks, as shown in Figure 3(c). Each task corresponds to a pair of IFM/OFM tiles. Tasks in one layer will be continuously loaded to a Processing Element (PE) on FPGA for execution (the load sequence is determined by \u2782FNAS-Sched). For a task, it involves Kh \u00d7Kw \u00d7Tr \u00d7Tc \u00d7Tm \u00d7Tn Multiply-Accumulate (MAC) operations, where Kh and Kw are the height and weight of \ufb01lter determined by the controller. A PE composed ofTm\u00d7Tn DSPs can execute Tm \u00d7 Tn MAC operations (16bit \ufb01x-point) in parallel [13]. Then the latency of a task is Kh \u00d7 Kw \u00d7Tr \u00d7Tc. In FNAS, each layer is allocated to a dedicated PE, and PEs are performed in the pipeline fashion. Such architecture can be implemented on one FPGA as in [8, 15] or multiple FPGAs as in [4, 14]. The resource (e.g., DSP and memory bandwidth) for each layer can be obtained by considering the load balance. And then the best parameters \u27e8Tm,Tn,Tr ,Tc\u27e9can be obtained according to [8, 13]. 3.4 Tile-based task graph generator: \u2781FNAS-GG FNAS-GG is a graph generator that takes the design parameters and NN architecture to generate the dependency graph between \fT1,1,1 ifm T1,1,2 ifm T1,2,1 ifm T1,2,2 ifm T2,1,1 ofm T2,1,2 ofm T2,2,1 ofm T2,2,2 ofm T2,3,1 ofm T2,3,2 ofm v1,1,1,1 v1,1,2,1 v1,1,3,1 v1,1,1,2 v1,1,2,2 v1,1,3,2 v1,2,1,1 v1,2,2,1 v1,2,3,1 v1,2,1,2 v1,2,2,2 v1,2,3,2 T2,1,1 ifm T2,1,2 ifm T2,2,1 ifm T2,2,2 ifm T3,1,1 ofm T3,1,2 ofm T3,2,1 ofm T3,2,2 ofm T3,3,1 ofm T3,3,2 ofm v2,1,1,1 v2,1,2,1 v2,1,3,1 v2,1,1,2 v2,1,2,2 v2,1,3,2 v2,2,1,1 v2,2,2,1 v2,2,3,1 v2,2,1,2 v2,2,2,2 v2,2,3,2 v1,1,1,1 v1,1,2,1 v1,1,3,1 v1,1,1,2 v1,1,2,2 v1,1,3,2 v1,2,1,1 v1,2,2,1 v1,2,3,1 v1,2,1,2 v1,2,2,2 v1,2,3,2 v2,1,1,1 v2,1,2,1 v2,1,3,1 v2,1,1,2 v2,1,2,2 v2,1,3,2 v2,2,1,1 v2,2,2,1 v2,2,3,1 v2,2,1,2 v2,2,2,2 v2,2,3,2 PE1 PE2 start-time time OFM reuse IFM reuse (a) Ordered tile-based task graph (b) Schedule Figure 4: FNAS-Sched: (a) re-ordered graph from 3(e); (b) the generated schedule graph. data tiles and tasks, called tile-based task graph. To generate the graph, the generator \ufb01rst needs to de\ufb01ne each tile for a given design. Then, it can generate the tile-based task graph. According to FNAS-Design, there are two kinds of tiles: channel tile and row/col tile. For channel tile, letCHif m i = {1, 2, \u00b7 \u00b7 \u00b7 , \u2308CHi Tn \u2309} be a set of indices in the ith layer under tiling parameter Tn (considering ith layer\u2019s IFM); similarly CHof m i = {1, 2, \u00b7 \u00b7 \u00b7 , \u2308CHi Tm \u2309} is under tiling parameter Tm (considering ith layer\u2019s OFM). For row/col tile, let RCi = {1, 2, \u00b7 \u00b7 \u00b7 , \u2308Ri Tr \u2309\u00b7 \u2308Ci Tc \u2309} be a set of indices in layer i under tiling parameter Tr and Tc. Based on these parameters, we can de\ufb01ne the tiles as follows. We de\ufb01ne a tile in IFM as T if m i,j,m, indicating the tile is in the ith layer, the jth channel tile in CHif m i and mth row/col tile in RCi. Similarly, the tile in OFM is de\ufb01ned asT of m i,k,m, where k is the index of channel tile in CHof m i . Then, we can build the dependencies among data tiles and tasks. There are two kinds of dependencies. First, inter-layer dependencies describe the dependency between tasks and data tiles in two consecutive layers. We de\ufb01ne a task node to bevi,j,k,m, which process the tile T if m i,j,m and generate the tile T of m i+1,k,m. Next, the intra-layer dependencies describe dependency between two data tiles in one layer. If the tiling parameters Tm and Tn for a layer are di\ufb00erent, as shown in Layer 2 in Figure 3(d), the dependency would not be the simple one-to-one mapping Instead, it can be represented as follows. For the tile T if m i,j,m, its data is depending on tile T of m i,k,m if (j \u22121) \u00b7 Tn Tm + 1 \u2264k \u2264j \u00b7 Tn Tm , where k \u2208N. By following the above rules, the graph can be generated. For the tiles of three layers in Figure 3(d), its corresponding tile-based task graph is shown in Figure 3(e). 3.5 Scheduler design: \u2782FNAS-Sched FNAS-Sched is a scheduler to determine the sequence of tasks to be executed on multiple PEs, such that the schedule length (latency) can be minimized. FNAS-Sched tries to maximally exploit the parallelism among di\ufb00erent convolutional operations based on the tile-based task graph. The design follows three principles: \u2022 P1. Minimizing the start time of each PE to execute tasks as early as possible. for(row=0; row<R; row+=Tr) for(col=0;  col<C;   col+=Tc) for(to=0;    to<M;    to+Tm) for(ti=0;     ti<N;     ti+=Tn) // load OFM, WEI, IFM ID = PE1 SCH ={PE1:[v1,1,1,1, v1,2,1,1,  v1,1,2,1, ...], ...} for (v in SCH[ID]) // load IFM tile, OFM tile, WEI // extract trr,tcc,too,tii from task v Fixed scheduler on PS part The proposed scheduler in FNAS          for(i=0; i<K; i++)          for(j=0; j<K; j++)          for(trr=row;  trr<min(row+Tr, R);   trr++)          for(tcc=col;  tcc<min(col+Tc,C);    tcc++)          // Unrolling following MACs to process in parallel          for(too=to;   too<min(to+Tm,M);   too++)          for(tii=ti;      tii<min(ti+Tn,N);       tii++)                   OFM[too][trr][tcc] += WEI[too][tii][i][j]*        IFM[tii][trr+i][tcc+j] // store OFM Accelerator on PL part (PE1) replace (a) Fixed scheduler (c) Accelerator implementeation (b) The proposed scheduler Figure 5: FPGA-based CNN implementation. \u2022 P2. Maximizing the reuse of data on FPGA to reduce the communication bandwidth requirement. \u2022 P3. Minimizing the pipeline stall caused by the lack of input data in the execution on one PE. FNAS-Sched is carried out in three steps. Step 1: determine the sequence of IFM tiles in each layer P1. The order of executing IFM will a\ufb00ect the start time of the next layer. There are two possible strategies: i) increase the indices of channel tiles \ufb01rst; or ii) increase the indices of row/col tiles \ufb01rst. Because an OFM tile is related to all IFM channels, strategy i) is more favored than ii) to make the next layer start earlier. Step 2: determine the sequence of OFM tiles in each layer P1. The sequence of OFM tiles will determine the ready time of IFM tiles in the same layer. After Step 1, the sequence of IFM tiles has already been determined. Hence, we sequentially visit IFM tile in a layer and arrange the OFM tiles that are dependent on it. Step 3: determine the task sequence P2,P3. Task sequence will determine the data reuse rate. There are two data reuse strategies can be exploited: i) OFM reuse, and ii) IFM reuse. The OFM (or IFM) reuse indicates that the consecutively executed tasks have the same OFM (or IFM) tile and the same row/col tile. We observe the uniform reuse strategy for all layers will lead to the pipeline stall due to the lack of input data. In FNAS, we will alternatively apply the above two strategies for consecutive layers. Followed by the above three steps, we can obtain a schedule of tasks. For the tile-based task graph in Figure 3(e), Figure 4(a) gives the graph with the reordered IFM/OFM tiles. We also give the schedule for this graph in Figure 4(b). As shown in Figure 4(b), tasks in layer1 (PE1) can achieve OFM reuse, while IFM reuse can be achieved in layer2 (PE2). In addition, the start-time is only 4 time units, and there is no stall in the executions for both layers. In a FPGA implementation, the scheduler is usually implemented in the processing system (PS) end and the accelerator is implemented on the programming logics (PL) end. Figures 5(a),(c) illustrate the commonly used PS/PL designs proposed by [13]. In their design, di\ufb00erent data tiles are sent to PL part with a \ufb01xed order, called \u201c\ufb01xed scheduling\u201d. In order to implement the proposed scheduler, we modify the implementation of PS part, as shown in \fFigure 5(b), where we can specify the tasks to PEs and launch tasks in an order determined by our scheduler. 3.6 Latency analysis: \u2783FNAS-Analyzer FNAS-Analyzer aims to e\ufb03ciently and accurately compute the latency L of a neural architecture on target FPGAs with determined schedule. In the schedule, the latency of PE is the sum of three parts: (1) the processing time, (2) the start time, and (3) the stall time. We are going to analyze each of them in following sections. Processing Time. We \ufb01rst determine the execution time of tasks in the tile-based graph. Since all tasks in layer i utilize the same accelerator for execution, they have the same execution time, denoted as ETi which equals Kh,i \u00d7 Kw,i \u00d7 Tr,i \u00d7 Tc,i (see \u2780FNASDesign). The processing time PTi of a PE is the summation of execution time of all task nodes in the corresponding layer i, which can be directly calculated as follows. PTi = ETi \u00d7 |CHif m i | \u00d7 |CHof m i+1 | (2) where |CHif m i | \u00d7 |CHof m i+1 | is the task number in layer i (see \u2781). Start Time. The start time of a layer depends on the start time and the data reuse strategy of its previous layer. Let \u2206ti,of m be the di\ufb00erence between start time of layers i \u22121 and i. We de\ufb01ne \u2206tj,if m similarly. First, let us consider that layer i \u22121 applies OFM reuse, indicating one tile in OFM is ready after computing \u2308CHi\u22121 Tn,i\u22121 \u2309tasks. For one IFM in layer i, it requires \u2308Tn,i Tm,i \u2309OFM tiles. In consequence, \u2206ti,of m can be calculated as follows. \u2206ti,of m = \u2308CHi\u22121 Tn,i\u22121 \u2309\u00d7 \u2308Tn,i Tm,i \u2309\u00d7 ETi\u22121 (3) Next, consider layer i \u22121 applies IFM reuse to compute \u2206tj,if m. In this case, each IFM in layer i \u22121 will be reused to compute the partial sum of all related OFMs. After all IFM except the last one in the same row/col tile have completed, it will continuously generate OFMs in layer i. Thus, \u2206tj,if m can be calculated as follows. \u2206tj,if m = \u0014\u0012 \u2308CHi\u22121 Tn,i\u22121 \u2309\u22121 \u0013 \u00d7 \u2308CHj Tm,j \u2309+ \u2308Tn,i Tm,i \u2309 \u0015 \u00d7 ETj\u22121 (4) where the \ufb01rst term of multiplication indicates the number of tasks completed before the \ufb01rst OFM in layer i is produced. And \u2308Tn,i Tm,i \u2309 is the number of OFM required by one IFM in layer i. Stall Time. After a PE has been launched, it may be stalled because the data for the next task is not ready. However, there may exist another task that has already been ready to run. In our scheduler, we can maintain a ready-to-run queue. If a stall occurs, we will pick one task in the ready-to-run queue to avoid pipeline stall. Latency. We can then derive a tight lower bound on latency Latsys by summing up processing time and starting time. For a total of N processing elements (PE), assume the \ufb01rst and the last PEs apply OFM reuse, we can calculate Latsys as follows. Latsys = \u00d5 i=2,4, \u00b7\u00b7\u00b7,N \u22121 \u2206ti,of m + \u00d5 j=3,5, \u00b7\u00b7\u00b7,N \u2206tj,if m + PTN (5) After obtaining Latsys, we have the latency L = Latsys. Table 2: Data sets and parameter settings in FNAS. Data sets Training Para. Controller Parameters Para. Train Val. E L FS FN T [TS4,TS3,TS2,TS1] MNIST 60,000 10,000 25 4 [5,7,14] [9,18,36] 60 TS-High TS-Low [2,5,10,20] [1,4,10,20] CIFAR-10 45,000 5,000 25 10 [1,3,5,7] [24,36,48,64] 60 [1.5,2,2.5,10] ImageNet 4,500 500 25 15 [1,3,5,7] [16,32,64,128] 60 [2.5,5,7.5,10] \u2022 L: number of layers; FS: \ufb01lter size; FN: number of \ufb01lters; T: trails; E: epoch Summary.FNAS framework considers the performance of child networks on target FPGAs in the neural architecture search process. As shown in Formula 1, if the latency cannot satisfy the timing speci\ufb01cation, there is no need to train the generated child network. In addition, the controller will be guided to avoid searching architectures that have insu\ufb03cient performance. Consequently, the search process can be dramatically accelerated, and the performance of the resultant child network on target FPGAs can be guaranteed. The evaluations on the e\ufb03ciency of FNAS will be presented in the next section. 4 EXPERIMENTAL RESULTS This section will report the evaluation results of the proposedFNAS framework. Results on MNIST, CIFAR-10, and ImageNet data sets show that FNAS can achieve up to 11.13\u00d7, 10.89\u00d7 and 10.38\u00d7 speedup in the search process respectively. Moreover, for the case where NAS [16] generates architectures with latencies 9.85\u00d7 longer than the speci\ufb01cation, FNAS can meet the specs with less than 1% accuracy loss. 4.1 Experimental setup Datasets. FNAS will search convolutional neural network structures for three kinds of data sets, including MNIST, CIFAR-10, and ImageNet. Each data set is composed of training set and validation set, as shown in Table 2. For instance, there are 60,000 and 10,000 examples in training and validation sets, respectively. Kindly note that for ImageNet, we use a smaller data set to reduce the computation time. In training of the child networks, the number of epochs is set as 25, and the maximum validation accuracy in the last 5 epochs will be utilized to compute the reward for updating the controller. Controller. We implement the reinforcement learning based RNN controller based on [16] to generate child networks. For di\ufb00erent data sets, the controller has di\ufb00erent con\ufb01gurations as shown in Table 2. For instance, we explain the con\ufb01guration of MNIST: (1) its child network has 4 layers, (2) the possible \ufb01lter size (height and width) is 5, 7 or 14, (3) the possible channel number is 9, 18, or 36, and (4) it will \ufb01nd 60 child networks. FNAS Tool. We implement all the components described in Section 3 and integrate them into the controller to realize the FNAS framework. To compare FNAS with NAS, we employ both low-end and high-end FPGAs to implement the resultant architectures using MNIST data set. The low-end and high-end FPGAs selected are Xilinx 7A50T and 7Z020, respectively. To see how general our conclusions are, we have explored CIFAR-10 and ImageNet as additional data sets, where Xilinx ZU9ED is used. 4.2 E\ufb03ciency and accuracy evaluations of FNAS Figures 6(a),(b),(c) reports comparison results on search time, latencies of the resultant DNNs, and accuracies of resultant DNNs \f0 6 1.2 1.8 2.4 \u00b4102 0 10 20 30 min ms 0 25% 50% 75% 100% NAS FNAS-loose FNAS-med FNAS-tight 7Z020 7A50T 7Z020 7A50T 7Z020 7A50T (a) search time (b) latency (c) accuracy Figure 6: Comparison of search time, latency and accuracy between NAS and FNAS. 1.00% 0% 0.25% 0.50% 0.75% accuracy loss 12\u00b4 0 3\u00b4 6\u00b4 9\u00b4 search time reduction MNIST CIFAR-10 TS1 TS2 TS3 TS4 TS1 TS2 TS3 TS4 ImageNet 10.38 10.89 (a) (b) timing specification timing specification 11.13 Figure 7: (a) Accuracy loss and (b) search time reduction vs. timing speci\ufb01cations on three data sets. The architectures from NAS are used as the baseline cases. respectively on the MINST data set. In these \ufb01gures, the x-axis represents di\ufb00erent FPGAs. The FNAS-loose (TS2), FNAS-med (TS3), and FNAS-tight (TS4) correspond to three timing speci\ufb01cations (see Table 2). From Figure 6(a), we can see that FNAS can dramatically reduce the search time. For the target FPGA of 7Z020, for loose, medium and tight timing speci\ufb01cation (actual values in Table 2), the search times are reduced from 190 minutes to 74, 59, 17 minutes, achieving 2.56\u00d7, 3.22\u00d7, 11.13\u00d7 reductions respectively compared with NAS. There are two reasons for the improvement: (1) with earlystage pruning, we will not train the architectures whose inference latencies violate the speci\ufb01cation; (2) the structure of the valid DNNs are usually simpler than the ones with violations, and accordingly most complex architectures are naturally pruned without the need of training. From the \ufb01gure we can also see that for FNAS the search time decreases as the timing speci\ufb01cation gets tighter, which is also expected as tighter speci\ufb01cations prunes more potential architectures. Next, as shown in Figure 6(b), for di\ufb00erent timing speci\ufb01cations, FNAS can generate a speci\ufb01c architecture to satisfy the speci\ufb01cation, i.e., the latency of the architecture decreases as the speci\ufb01cation gets tighter. On the contrary, NAS can only generate a single architecture, which has a latency that is 2.54\u00d7, 4.19\u00d7, and 7.81\u00d7 longer than the speci\ufb01cation. The \ufb02exibility of FNAS provides more choices for designers and also helps to prune useless designs at the very early design stage. Figure 6(c) reports the accuracy of the generated architectures. We can observe that, even compared with the architectures generated by NAS that have timing violations, those generated by FNAS only have accuracy degradation of 0.08%, 0.24%, and 0.81% for loose, medium and tight timing timing speci\ufb01cation respectively. The above results clearly show that FNAS can achieve both high e\ufb03ciency and high accuracy in exploring neural architectures. Finally, we explore how the accuracy of the architectures generated by FNAS as well as the corresponding search time scales 12.20% 14.15% 14.51% 14.50% 13.27% 15.02% 14.53% 14.90% 13.42% 13.37% 13.23% 12.75% 12.76% 9.72% 8.59% 15.63% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 10 20 30 40 50 60 \u00b4105 Clock Cycles Explored Architectures FNAS-Sched Fixed sched Figure 8: Number of clock cycles on a set of architectures using FNAS-Sched and \ufb01xed scheduling [13] using four accelerators. with the timing speci\ufb01cations. The results are depicted in Figure 7 where three data sets MNIST, CIFAR-10, and ImageNet are used. For MNIST, the high-end FPGA is used. The x-axis represents different timing speci\ufb01cations, where TC1 is the loosest one while TC4 is the tightest one and the corresponding values are summarized in Table 2. When reporting both accuracy and search time, we use the architecture from NAS as reference and show the accuracy loss as well as the search time reduction. From Figure 7(a) we can observe that in general the architectures generated by FNAS without timing violations only have less than 1% accuracy loss compared with those generated by NAS that have violations. The accuracy loss gets higher as the timing constraint becomes tighter. Meanwhile, from Figure 7(b), we can see that the search time of FNAS can be signi\ufb01cantly reduced, achieving 11.18\u00d7 reduction for CIFAR-10 compared with NAS. 4.3 Improvements from the scheduler We also evaluate the e\ufb03ciency of the proposedscheduler against that uses the \ufb01xed scheduling technique [13] on four accelerators. We test a set of architectures with 4 convolution layers on PYNQ board [5]. For each convolution layer, the \ufb01lter size is 3\u00d73 and the number of \ufb01lters is 64 or 128. Figure 8 reports the number of clock cycles of each possible architecture. As shown in this \ufb01gure, our proposed scheduler can consistently achieve better performance. This convincingly demonstrates the e\ufb00ectiveness of the proposed scheduler for FPGA designs. 5"
+                }
+            ],
+            "Dawei Li": [
+                {
+                    "url": "http://arxiv.org/abs/2402.01729v3",
+                    "title": "Contextualization Distillation from Large Language Model for Knowledge Graph Completion",
+                    "abstract": "While textual information significantly enhances the performance of\npre-trained language models (PLMs) in knowledge graph completion (KGC), the\nstatic and noisy nature of existing corpora collected from Wikipedia articles\nor synsets definitions often limits the potential of PLM-based KGC models. To\nsurmount these challenges, we introduce the Contextualization Distillation\nstrategy, a versatile plug-in-and-play approach compatible with both\ndiscriminative and generative KGC frameworks. Our method begins by instructing\nlarge language models (LLMs) to transform compact, structural triplets into\ncontext-rich segments. Subsequently, we introduce two tailored auxiliary tasks,\nreconstruction and contextualization, allowing smaller KGC models to assimilate\ninsights from these enriched triplets. Comprehensive evaluations across diverse\ndatasets and KGC techniques highlight the efficacy and adaptability of our\napproach, revealing consistent performance enhancements irrespective of\nunderlying pipelines or architectures. Moreover, our analysis makes our method\nmore explainable and provides insight into generating path selection, as well\nas the choosing of suitable distillation tasks. All the code and data in this\nwork will be released at\nhttps://github.com/David-Li0406/Contextulization-Distillation",
+                    "authors": "Dawei Li, Zhen Tan, Tianlong Chen, Huan Liu",
+                    "published": "2024-01-28",
+                    "updated": "2024-02-24",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Knowledge graph completion (KGC) is a fundamental task in natural language processing (NLP), aiming at unveiling hidden insights within diverse knowledge graphs to explore novel knowledge patterns. Traditional KGC methods (Nickel et al., 2011; Bordes et al., 2013) typically predict the missing part of the triplets by learning the representation of each entity and relation based on their structural information. However, such embeddingbased methods tend to overlook the rich textual inMethods H@1 H@3 H@8/10 ChatGPT-1-shot 15.6 17.6 19.6 PaLM2-1-shot 15.7 20.8 25.4 KG-S2S (Chen et al., 2022a) 28.5 38.8 49.3 Table 1: ChatGPT and PaLM2\u2019s unsatisfactory performance on the test set of FB15k-237N compared to a smaller KGC model, KG-S2S (Chen et al., 2022a). formation of the knowledge graph. Therefore, pretrained language models (PLMs) have been introduced to KGC and achieved promising results (Kenton and Toutanova, 2019; Xie et al., 2022). J. G. Ballard Shanghai place_of_birth Problem: succinct Text: J.G. Ballard  was a novelist Problem: static Text: Shanghai is a 2010  American mystery/thriller  neonoir film directed by  Mikael H\u00e5fstr\u00f6m, ... Problem: noisy Text: In 1984, J.G. Ballard won broad,  ... British boy during the Japanese occup ation of Shanghai Figure 1: An example to illustrate the limitations of the current textual information for KGC. While it has been well-discovered that textual information can be beneficial for PLM-based KGC models (Yao et al., 2019; Wang et al., 2021b; Chen et al., 2022a; Li et al., 2022; Chen et al., 2023a), prior attempts to augment KGC models with textual data from Wikipedia article (Zhong et al., 2015) or synsets definitions (Yao et al., 2019) have encountered certain limitations: (i) Entity descriptions, often succinct and static, may inhibit the formation of a comprehensive understanding of entities within KGC models. (ii) The incorporation of triplet descriptions, albeit potentially enriching, can introduce substantial noise, particularly when derived through automatic entity alignment (Sun et al., 2020). Figure 1 demonstrates an example to illustrate the aforementioned limitations. The description for the head \u201cJ. G. Ballard\u201d is limited and for the tail \u201cShanghai\u201d, it mistakenly uses the definition of the movie also named \u201cShanghai\u201d. arXiv:2402.01729v3  [cs.CL]  24 Feb 2024 \fAlso, while the two entities show up in the triplet description, it falls short in conveying the semantic essence of the relation \u201cplace_of_birth\u201d. In light of these limitations, our attention shifts to Large Language Models (LLMs) (Brown et al., 2020; Zhang et al., 2022; Anil et al., 2023; Touvron et al., 2023), renowned for their capability in generating articulate and high-quality data (Dai et al., 2023; Shridhar et al., 2023; Zheng et al., 2023). Our exploration commences with a scrupulous evaluation of LLMs, such as ChatGPT and PaLM2, in KGC, benchmarking them across several esteemed KGC datasets (Dettmers et al., 2018; Garcia-Duran et al., 2018; Mahdisoltani et al., 2013). Utilizing 1-shot In-Context Learning (ICL), we deduce missing heads or tails in triplets and report evaluation metrics. It reveals a significant performance discrepancy of two LLMs in comparison to KGS2S (Chen et al., 2022a) despite its reliance on a smaller foundational model, T5-base (Raffel et al., 2020). This insight propels us toward the conclusion that direct utilization of LLMs for KGC tasks, while intuitive, is outperformed by the fine-tuning of more diminutive, specialized KGC models. This observation aligns with findings from Liang et al. (2022); Sun et al. (2023); Zhao et al. (2023); Wei et al. (2023), which highlighted the limitations of LLMs in knowledge-centric tasks. Experiment results and analysis on more KGC datasets can be found in Appendix A. To optimally harness LLMs for KGC, we draw inspiration from recent works (Xiang et al., 2022; Kim et al., 2022a) and introduce a novel approach, Contextualization Distillation. Contextualization Distillation first extracts descriptive contexts from LLMs with well-designed prompts, thereby securing dynamic, high-quality context for each entity and triplet. Subsequent to this, two auxiliary tasks are proposed to train smaller KGC models with these informative, descriptive contexts. The plug-in-and-play characteristic of our contextualization distillation enables us to apply and evaluate it on various KGC datasets and baseline models. Through extensive experiments, we affirm that Contextualization Distillation consistently enhances the performance of smaller KGC models, irrespective of architectural and pipeline disparities. Additionally, we provide an exhaustive analysis of each step of Contextualization Distillation, encouraging further insights and elucidations. The contributions of this work can be summarized into three main aspects: \u2022 We identify the constraints of the current corpus for PLMs-based KGC models and introduce a plug-in-and-play approach, Contextualization Distillation, to enhance smaller KGC models with extracted rationale from LLMs. \u2022 We conduct extensive experiments across several widely recognized KGC datasets and utilize various baseline models. Through these experiments, we validate the effectiveness of Contextualization Distillation in consistently improving smaller KGC models. \u2022 We delve into a comprehensive analysis of our proposed method and provide valuable insights and guidance on generating path selection for distillation, as well as the selection of suitable distillation tasks. 2 Related Work 2.1 Knowledge Graph Completion Traditional KGC methods (Nickel et al., 2011; Bordes et al., 2013) involve embedding entities and relations into a representation space. In pursuit of a more accurate depiction of entity-relation pairs, different representation spaces (Trouillon et al., 2016; Xiao et al., 2016) have been proposed considering various factors, e.g., differentiability and calculation possibility (Ji et al., 2021). During training, two primary objectives emerge to assign higher scores to true triplets than negative ones: 1) Translational distance methods gauge the plausibility of a fact by measuring the distance between the two entities under certain relations (Lin et al., 2015; Wang et al., 2014); 2) Semantic matching methods compute the latent semantics of entities and relations (Yang et al., 2015; Dettmers et al., 2018). To better utilize the rich textual information of knowledge graphs, PLMs have been introduced in KGC. Yao et al. (2019) first propose to use BERT (Kenton and Toutanova, 2019) to encode the entity and relation\u2019s name and adopt a binary classifier to predict the validity of given triplets. Following them, Wang et al. (2021a) leverage the Siamese network to encode the head-relation pair and tail in a triplet separately, aiming to reduce the time cost and make the inference scalable. Lv et al. (2022) convert each triple and its textual information into natural prompt sentences to fully inspire PLMs\u2019 potential in the KGC task. Chen et al. (2023a) design a conditional soft prompts framework to maintain a balance between structural information and textual \fGiven a triplet (Portishead | music , genre , parent genre | Ambient music),please gener ate a short paragraph to introduce \"Portishe ad \" and \"Ambient music\" and reflect their r elationship \"music , genre , parent genre\". LLM Contextualized Triplet Distillation Prompt Portishead <Sep> music , genre , parent ge nre <Sep> ? Discriminative  PLM Generative  PLM Portishead <Sep> music , genre , parent ge nre <Sep> Ambient music Ambient music Reconstruction KGC Contextualization ... Ambient music is a genre of music that e merged in the 1970s and is characterized by its atmospheric and often relaxing sound ... ... <Mask> music is a <Mask> of music th at <Mask> in the 1970s and is characterize d by its <Mask> and often relaxing sound ... ... Ambient music is a genre of music that e merged in the 1970s and is characterized by its atmospheric and often relaxing sound ... ... Ambient music is a genre of music that e merged in the 1970s and is characterized by its atmospheric and often relaxing sound ... Figure 2: An overview pipeline of our Contextualization Distillation. We first extract descriptive contexts from LLMs (Section 3.1). Then, two auxiliary tasks, reconstruction (Section 3.3.1) and contextualization (Section 3.3.2) are designed to train the smaller KGC models with the contextualized information. knowledge in KGC. Recently, there are also some works trying to leverage generative PLMs to perform KGC in a sequence-to-sequence manner and achieve promising results (Xie et al., 2022; Saxena et al., 2022; Chen et al., 2022a). 2.2 Distillation from LLMs Knowledge distillation has proven to be an effective approach for transferring expertise from larger, highly competent teacher models to smaller, affordable student models (Bucilu\u02c7 a et al., 2006; Hinton et al., 2015; Beyer et al., 2022). With the emergence of LLMs, a substantial body of research has concentrated on distilling valuable insights from these LLMs to enhance the capabilities of smaller PLMs. One of the most common methods is to prompt LLMs to explain their predictions and then use such rationales to distill their reasoning abilities into smaller models (Wang et al., 2022a; Ho et al., 2022; Magister et al., 2022; Hsieh et al., 2023; Shridhar et al., 2023). Distilling conversations from LLMs is another cost-effective method to build new dialogue datasets (Kim et al., 2022b; Chen et al., 2023b; Kim et al., 2022a) or augment existing ones (Chen et al., 2022b; Zhou et al., 2022; Zheng et al., 2023). There are also some attempts (Marjieh et al., 2023; Zhang et al., 2023) that focus on distilling domain-specific knowledge from LLMs for various downstream applications. Several recent studies have validated the contextualization capability of LLMs to convert structural data into raw text. Among them, Xiang et al. (2022) convert triplets in the data-to-text generation dataset into their corresponding descriptions to facilitate disambiguation. Kim et al. (2022a) design a pipeline for synthesizing a dialogue dataset by distilling conversations from LLMs, enhanced with a social commonsense knowledge graph. By contrast, we are the first to leverage descriptive context generated by LLMs as an informative auxiliary corpus to the KGC models. 3 Contextualization Distillation In this section, we first illustrate how we curate prompts to extract the descriptive context of each triplet from the LLM. Subsequently, we design a multi-task framework, together with two auxiliary tasks\u2014reconstruction and contextualization\u2014to train smaller KGC models with these high-quality context corpus. The overview pipeline of our method is illustrated in Figure 2. \fInput Output Portishead is a British trip hop band formed in Bristol in 1991. They are co nsidered one of the pioneers of the genre, along with Massive Attack and  Tricky. Ambient music is a genre of music that emerged in the 1970s and i s characterized by its atmospheric and often relaxing sound. Portishead's m usic is often described as ambient, due to its use of loops, drones, and othe r sound effects. Given a triplet (Portishead | music , genre , parent genre | Ambient music), please generate a short paragraph to introduce \"Portishead \" and \"Ambient  music\" and reflect their relationship \"music , genre , parent genre\". Figure 3: An example contains our instruction to LLMs and the generated descriptive context. We use green to highlight entity description prompt/ generation result and blue to highlight triplet description prompt/ generation result. 3.1 Extract Descriptive Context from LLMs Recent studies have highlighted the remarkable ability of LLMs to contextualize structural data and transform it into context-rich segments (Xiang et al., 2022; Kim et al., 2022a). Here we borrow their insights and extract descriptive context from LLMs to address the limitations of the existing KGC corpus we mentioned in Section 1. In particular, we focus on two commonly employed types of descriptions prevalent in prior methodologies: entity description (ED) (Yao et al., 2019; Chen et al., 2022a) and triplet description (TD) (Sun et al., 2020). Entity description refers to the definition and description of individual entities, while triplet description refers to a textual segment that reflects the specific relationship between two entities within a triplet. Given triplets of a knowledge graph ti \u2208T, we first curate prompt pi for the ith triplet by filling the pre-defined template: pi = Template(hi, ri, ti), (1) where hi, ri, ti are the head entity, relation, and tail entity of the ith triplet. Then, we use pi as the input to prompt the LLM to generate the descriptive context ci for each triplet: ci = LLM(pi), (2) 3.2 Generating Path Without loss of generalization, we consider different generating paths to instruct the LLMs to generate textual information and conduct an ablation study in Section 4.3. All the generating paths we adopt are as follows: T \u2212 \u2192(ED, T D) generates both entity description and triplet description at one time. As Figure 3 shows, this is the context generating path we use in the main experiment. T \u2212 \u2192ED curates prompt to instruct the LLM to generate the entity description only. T \u2212 \u2192T D curates prompt to instruct the LLM to generate the triplet description only. T \u2212 \u2192RA prompts the LLM to generate rationale rather than descriptive context. T \u2212 \u2192ED \u2212 \u2192T D produces entity description and triplet description in a two-step way. The final descriptive context is obtained by concatenating the two segments of text. We also give further details and examples of our prompt in Appendix F. 3.3 Multi-task Learning with Descriptive Context Different PLM-based KGC models adopt diverse loss functions and pipeline architectures (Yao et al., 2019; Chen et al., 2022a; Xie et al., 2022; Chen et al., 2023a). To ensure the compatibility of our Contextualization Distillation to be applied in various PLM-based KGC methods, we design a multi-task learning framework for these models to learn from both the KGC task and auxiliary descriptive context-based tasks. For the auxiliary tasks, we design reconstruction (Section 3.3.1) and contextualizatioin (Section 3.3.2) for discriminative and generative KGC models respectively. 3.3.1 Reconstruction The reconstruction task aims to train the model to restore the corrupted descriptive contexts. For the discriminative KGC models, we follow the implementation of Kenton and Toutanova (2019) and use masked language modeling (MLM). Previous studies have validated that such auxiliary selfsupervised tasks in the domain-specific corpus can benefit downstream applications (Han et al., 2021; Wang et al., 2021b). To be specific, MLM randomly identifies 15% of the tokens within the descriptive context. Among these tokens, 80% are tactically concealed with the special token \u201c< Mask >\u201d, 10% are seamlessly substituted with random tokens, while the remaining 10% keep unchanged. For each selected token, the objective of MLM is to restore the original content at that particular position, achieved through the cross-entropy loss. The aforementioned process can be formally expressed as follows: c \u2032 i = MLM(ci), (3) Lrec = 1 N N X i=1 \u2113(f(c \u2032 i), ci) (4) \fThe final loss of discriminative KGC models is the combination of the KGC loss1 and the proposed reconstruction loss: Ldis = Lkgc + \u03b1 \u00b7 Lrec, (5) where \u03b1 is a hyper-parameter to control the ratios between the two losses. 3.3.2 Contextualization The objective of contextualization is to instruct the model in generating the descriptive context ci when provided with the original triplet ti = h, r, t. Compared with reconstruction, contextualization demands a more nuanced and intricate ability from PLM. It necessitates the PLM to precisely grasp the meaning of both entities involved and the inherent relationship that binds them together, to generate fluent and accurate descriptions. Specifically, we concatenate head, relation and tail with a special token \u201c< Sep >\u201d as input: Ii = Con(hi, < Sep >, ri, < Sep >, ti) (6) Then, we input them into the generative PLM and train the model to generate descriptive context ci using the cross-entropy loss: Lcon = 1 N N X i=1 \u2113(f(Ii), ci) (7) The final loss of generative KGC models is the combination of the KGC loss2 and the proposed contextualization loss: Lgen = Lkgc + \u03b1 \u00b7 Lcon (8) For generative KGC models, it is also applicable to apply reconstruction as the auxiliary task. We have done an ablation study in Section 4.5 to examine the effectiveness of each auxiliary task on generative KGC models. 4 Experiment In this section, we apply our Contextualization Distillation across a range of PLM-based KGC baselines. We compare our enhanced model with our approach against the vanilla models using several KGC datasets. Additionally, we do further analysis of each component in our contextualized distillation and make our method more explainable by conducting case studies. 1We give the illustration of the discriminative KGC models we used in Appendix B.1 2We give the illustration of the generative KGC models we used in Appendix B.2 4.1 Experimental Settings Datasets We use WN18RR (Dettmers et al., 2018) and FB15k-237N (Lv et al., 2022) in our experiment. WN18RR serves as an enhanced version of its respective counterparts, WN18 (Bordes et al., 2013). The improvements involve the removal of all inverse relations to prevent potential data leakage. For FB15K-237N, it\u2019s a refine version of FB15k (Bordes et al., 2013), by eliminating concatenated relations stemming from Freebase mediator nodes (Akrami et al., 2020) to avoid Cartesian production relation issues. Baselines we adopt several PLM-based KGC models as baselines and apply the proposed Contextualization Distillation to them. KG-BERT (Yao et al., 2019) is the first to suggest utilizing PLMs for the KGC task. we also consider CSPromKG (Chen et al., 2023a), which combines PLMs with traditional Knowledge Graph Embedding (KGE) models, achieving a balance between efficiency and performance in KGC. In addition to these discriminative models, we also harness generative KGC models. GenKGC (Xie et al., 2022) is the first to accomplish KGC in a sequence-tosequence manner, with a fine-tuned BART (Lewis et al., 2020) as its backbone. Following them, KGS2S (Chen et al., 2022a) adopt soft prompt tuning and lead to a new SOTA performance among the generative KGC models. Implementation details All our experiments are conducted on a single GPU (RTX A6000), with CUDA version 11.1. We use PaLM2-540B(Anil et al., 2023) as the large language model to distill descriptive context. We tune the Contextualization Distillation hyper-parameter \u03b1 \u2208{0.1, 0.5, 1.0}. We follow the hyper-parameter settings in the original papers to reproduce each baseline\u2019s result. For all datasets, we follow the previous works (Chen et al., 2022a, 2023a) and report Mean Reciprocal Rank (MRR), Hits@1, Hits@3 and Hits@10. More details about our experiment implementation and dataset statistics are shown in Appendix C. 4.2 Main Result Table 2 displays the results of our experiments on WN18RR and FB15k-237N. We observe that our Contextualization Distillation consistently enhances the performance of all baseline methods, regardless of whether they are based on generative or discriminative models. This unwavering \fWN18RR FB15k-237N MRR H@1 H@3 H@10 MRR H@1 H@3 H@10 Traditional Methods TransE* (Bordes et al., 2013) 24.3 4.3 44.1 53.2 25.5 15.2 30.1 45.9 DisMult* (Yang et al., 2015) 44.4 41.2 47.0 50.4 20.9 14.3 23.4 33.0 ComplEx* (Trouillon et al., 2016) 44.9 40.9 46.9 53.0 24.9 18.0 27.6 38.0 ConvE* (Dettmers et al., 2018) 45.6 41.9 47.0 53.1 27.3 19.2 30.5 42.9 RotatE* (Sun et al., 2018) 47.6 42.8 49.2 57.1 27.9 17.7 32.0 48.1 CompGCN* (Vashishth et al., 2019) 47.9 44.3 49.4 54.6 31.6 23.1 34.9 48.0 PLMs-based Methods MTL-KGC* (Kim et al., 2020) 33.1 20.3 38.3 59.7 24.1 16.0 28.4 43.0 StAR* (Wang et al., 2021a) 40.1 24.3 49.1 70.9 PKGC* (Lv et al., 2022) 30.7 23.2 32.8 47.1 KGT5* (Saxena et al., 2022) 50.8 48.7 54.4 Our Implementation KG-BERT (Yao et al., 2019) 21.6 4.1 30.2 52.4 20.3 13.9 20.1 40.3 KG-BERT-CD 30.3 16.5 35.4 60.2 25.0 17.2 26.6 45.5 GenKGC (Xie et al., 2022) 28.6 44.4 52.4 18.7 27.3 33.7 GenKGC-CD 29.3 45.6 53.3 20.4 29.3 34.9 KG-S2S (Chen et al., 2022a) 57.0 52.5 59.7 65.4 35.4 28.5 38.8 49.3 KG-S2S-CD 57.6 52.6 60.7 67.2 35.9 28.9 39.4 50.2 CSProm-KG (Chen et al., 2023a) 55.2 50.0 57.2 65.7 36.0 28.1 39.5 51.1 CSProm-KG-CD 55.9 50.8 57.8 66.0 37.2 28.8 41.0 53.0 Table 2: Experiment results on WN18RR and FB15k-237. * denotes results we take from Chen et al. (2022a). Methods suffixed with \"-CD\" indicate the baseline models with our Contextualization Distillation applied. The best results of each metric are in bold. improvement demonstrates the robust generalization and compatibility of our approach across various PLMs-based KGC methods. Additionally, some baselines we choose to implement our Contextualization Distillation also utilize context information. For example, both KGBERT and CSProm-KG adopt entity descriptions to enhance entity embedding representation. Nevertheless, our approach manages to deliver additional improvements to these context-based baselines. Among them, it is worth noting that the application of our approach to KG-BERT achieves an overall 31.7% enhancement in MRR. All these findings lead us to the conclusion that Contextualization Distillation is not only compatible with context-based KGC models but also capable of further enhancing their performance. 4.3 Ablation Study on Generating Path We investigate the efficacy of different context types in the distillation process by employing various generative paths. As illustrated in Table 3, we initially explore the impact of entity description and triplet description when utilized separately as auxiliary corpora (T \u2212 \u2192ED and T \u2212 \u2192TD). The experimental findings underscore the critical roles played by both entity description and triplet description as distillation corpora, leading to noticeable enhancements in the performance of smaller KGC models. Furthermore, we ascertain that Paths FN15k-237N H@1 H@3 H@10 18.7 27.3 33.7 T \u2212 \u2192ED 20.0 28.9 34.5 T \u2212 \u2192TD 20.1 29.0 34.6 T \u2212 \u2192RA 19.4 28.2 34.2 T \u2212 \u2192ED \u2212 \u2192TD 19.8 28.6 34.5 T \u2212 \u2192(ED, TD) 20.4 29.3 34.9 Table 3: Ablation study results in GenKGC with different generating paths to distill corpus from LLMs. We conduct the experiment using FB15k-237N. We add the vallina GenKGC in the first row for comparison. our method\u2019s generating path T \u2212 \u2192(ED, TD), which utilizes these two corpora, achieves more improvements by endowing the models with a more comprehensive and richer source of information. To gain a comprehensive understanding of the effectiveness of our Contextualization Distillation, we also explored other alternative generative paths. While rationale distillation has demonstrated its potential in various NLP tasks (Hsieh et al., 2023; Shridhar et al., 2023), our investigation delves into the T \u2212 \u2192RA path, wherein we instruct the LLM to generate rationales for each training sample. Although the model utilizing rationale distillation exhibits improved performance compared to the vanilla one, it falls short when compared with our Contextualization Distillation incorporating entity \fdescriptions and triplet descriptions. One plausible explanation for this disparity lies in the intrinsic nature of rationales, which tend to be intricate and structurally complex. This complexity can pose a greater challenge for smaller models to fully comprehend, in contrast to the more straightforward descriptive text utilized in our approach. T \u2212 \u2192ED \u2212 \u2192TD borrows the insight from Chain-of-CoT (CoT) (Wei et al., 2022) that generates the content step by step. Interestingly, our findings indicate that this multi-step generative path also yields suboptimal performance when compared to the single-step generative path. This discrepancy can be attributed to the text incoherence resulting from the concatenation of three segments of descriptions. In light of the insights gained from these observations, we summarize our distillation guidance for KGC as follows: smaller models can benefit more from comprehensive, descriptive and coherent content generated by LLMs. 4.4 Ablation Study on Descriptive Context FN15k-237N H@1 H@3 H@10 GenKGC 18.7 27.3 33.7 GenKGC w/ Contextualization w/ Wikipedia 19.2 27.9 34.0 GenKGC-CD 20.4 29.3 34.9 Table 4: Ablation study results in GenKGC with descriptive context generated by our method and collected by Zhong et al. (2015). In this section, we replace the auxiliary corpus used in the auxiliary task with the Wikipedia corpus collected by (Zhong et al., 2015) to study the effectiveness of the distillation. As Table 4 shows, while the auxiliary task with Wikipedia corpus improves the model\u2019s performance, the overall enhancement is not as significant as that brought by our Contextualization Distillation. This further demonstrates the corpus generated by large language models effectively tackles the limitations of the preceding corpus for KGC, resulting in more pronounced improvements for the KGC model. 4.5 Ablation Study on Generative KGC Models In this section, we compare the effectiveness of reconstruction and contextualization in generative FN15k-237N MRR H@1 H@3 H@10 GenKGC 18.7 27.3 33.7 w/ Reconstruction 19.4 28.2 34.2 w/ Contextualization 20.4 29.3 34.9 KG-S2S 35.4 28.5 38.8 49.3 w/ Reconstruction 35.8 29.3 38.9 48.9 w/ Contextualization 35.9 28.9 39.4 50.2 Table 5: Ablation study results on GenKGC and KGS2S with reconstruction and contextualization as the auxiliary task respectively. We conduct the experiment using FB15k-237N. Figure 4: MRR scores on the validation set during the CSProm-KG training on FB15k-237N. We use thin bars to mark the epochs in which the models achieve the best performance in the validation set. KGC models. For GenKGC and KG-S2S, we employ the pre-trained tasks of their respective backbone models (BART for GenKGC and T5 for KGS2S) as the reconstruction objective. More details of our reconstruction implementation for generative KGC models can be found in Appendix D. Table 5 presents the ablation study results on FB15k-237N. We find reconstruction is also effective in improving the performance of generative KGC models, showing that KGC models can consistently benefit from the descriptive context with different auxiliary tasks. Comparing the two auxiliary tasks, models with contextualization outperform those with reconstruction on almost every metric, except for Hits@1 in KG-S2S. This implies that contextualization is a critical capability for generative KGC models to master for better KGC performance. Generative models have benefited more from the training of converting structural triplets into descriptive context than simply restoring the corrupted corpus. \ffrom Wikipedia (Zhong et al., 2015) Ours Head Ballard was a novelist. J.G. Ballard (1930-2009) was an English writer. He was born in Shanghai, China, and his early experiences there shaped his writing. His novels often explored themes of alienation, technology, and the future... Tail Shanghai is a 2010 American mystery/thriller neo-noir film directed by Mikael H\u00e5fstr\u00f6m, starring John Cusack and Gong Li... Shanghai is a city in China. It is one of the most populous cities in the world, and it is a major center of commerce and culture. Shanghai has a long history, and it has been home to many different cultures over the centuries... Triplet In 1984, J.G. Ballard won broad, critical recognition for the war novel Empire of the Sun, a semi-autobiographical story of the experiences of a British boy during the Japanese occupation of Shanghai. Ballard was born in Shanghai in 1930. He lived there until he was eight years old, when his family moved to England. Ballard\u2019s early experiences in Shanghai had a profound impact on his writing... Table 6: Descriptive context of the triplet (J.G. Ballard, place_of_birth, Shanghai). The text in green represents positive content and the text in red represents negative content. 4.6 Efficiency Analysis The additional training cost brought by the auxiliary distillation tasks may pose a potential constraint on our approach. However, we also notice baseline models with our method coverage faster on the validation set. Figure 4 presents the validation MRR vs epoch numbers during the CSPromKG training on FB15k-237N. It is obvious that CSProm-KG with Contextualization Distillation achieves a faster convergence and attains the best checkpoint earlier (at around 125 epochs) compared to the variant without our method (at around 220 epochs). This implies auxiliary distillation loss can also expedite model learning in KGC. This trade-off between batch processing time and training steps ultimately results in a training efficiency comparable to that of the vanilla models. 4.7 Case Study We conduct a comparative analysis between the description corpus collected from Wikipedia (Zhong et al., 2015) and those generated using our method to show the advantage of our Contextualization Distillation more straightforwardly. As presented in Table 6, entity descriptions generated by the LLM effectively address the limitations issue and static shortcomings, resulting in more informative and accurate content. Regarding the triplet description, although the \u201csemi-autobiographical\u201d used in Zhong et al. (2015) somewhat implies Query (The Devil\u2019s Double, genre, ?) Ground Truth Biographical film Baseline War film Ours Biographical film Our Context The Devil\u2019s Double is a biographical film that tells the story of Latif Yahia, a young Iraqi man who was forced to impersonate Saddam Hussein\u2019s son Uday Hussein... Table 7: Case study on FB15K-237N with KG-S2S. we also let the model generate a descriptive context for each test sample. The text in bold represents informative content in the generated descriptive context. J.G. Ballard\u2019s connection to Shanghai during his childhood, it still fails to express the semantics of \u201cplace_of_birth\u201d clearly. In contrast, the descriptive context generated by our method provides a more elaborate and coherent contextualization of the \u201cplace_of_birth\u201d between \u201cJ.G. Ballard\u201d and \u201cShanghai\u201d. These comparisons highlight the effectiveness of our method in addressing the previous corpus\u2019 limitation. Furthermore, We showcase how the auxiliary training with descriptive context enhances the baseline models. Table 7 presents the results of KGS2S performance in a test sample of FB15k-237N, both with and without our contextualization distil\flation. In this case, the vanilla KG-S2S wrongly predicts the genre of the film \u201cThe Devil\u2019s Double\u201d as \u201c\u2019War film\u2019, whereas the KG-S2S trained with our auxiliary task correctly labels it as \u201cBiographical film\u201d. Also, by making the model contextualize each triplet, we find the model with our method applied successfully captures many details about the movie, such as the genre and plot, and presents this information as fluent text. In summary, the model not only acquires valuable insights about the triplets but also gains the ability to adeptly contextualize this information through our Contextualization Distillation. Due to the space limitation, we put further analysis about LLMs\u2019 sizes in Appendix E. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2311.03319v1",
+                    "title": "DAIL: Data Augmentation for In-Context Learning via Self-Paraphrase",
+                    "abstract": "In-Context Learning (ICL) combined with pre-trained large language models has\nachieved promising results on various NLP tasks. However, ICL requires\nhigh-quality annotated demonstrations which might not be available in\nreal-world scenarios. To overcome this limitation, we propose \\textbf{D}ata\n\\textbf{A}ugmentation for \\textbf{I}n-Context \\textbf{L}earning\n(\\textbf{DAIL}). DAIL leverages the intuition that large language models are\nmore familiar with the content generated by themselves. It first utilizes the\nlanguage model to generate paraphrases of the test sample and employs majority\nvoting to determine the final result based on individual predictions. Our\nextensive empirical evaluation shows that DAIL outperforms the standard ICL\nmethod and other ensemble-based methods in the low-resource scenario.\nAdditionally, we explore the use of voting consistency as a confidence score of\nthe model when the logits of predictions are inaccessible. We believe our work\nwill stimulate further research on ICL in low-resource settings.",
+                    "authors": "Dawei Li, Yaxuan Li, Dheeraj Mekala, Shuyao Li, Yulin wang, Xueqi Wang, William Hogan, Jingbo Shang",
+                    "published": "2023-11-06",
+                    "updated": "2023-11-06",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Recently, the rapid development of large language models (LLMs) (Devlin et al., 2018; Radford et al., 2019) and their striking skill and knowledge have sparked significant interest in In-Context Learning (ICL) (Brown et al., 2020). Different from other training-based paradigms, under ICL, there is no parameter adjustment to LLMs. Instead, the model is given an instruction, which usually consists of a task description (or prompt), several in-context samples (or demonstration), and the test case that needs to be inferred. This tuning-free approach has gained prominence in various research domains within Natural Language Processing (NLP) due to its impressive performance in numerous downstream tasks (Bonifacio et al., 2022; Gao et al., 2023b,a). ICL requires demonstrations as part of the context. Many works focus on improving ICL by employing diverse methods to select suitable demonstrations, such as similarity-based retrieval (Liu et al., 2021; Rubin et al., 2021), score function learning (Li and Qiu, 2023), permutation search (Lu et al., 2021; Wu et al., 2022), and etc. However, in numerous real-world scenarios, the availability of annotated samples is severely constrained (Mekala and Shang, 2020; Su et al., 2022) for such demonstration selection. Moreover, in the fine-grained classification with a large target-label space, the context length of LLMs could not accommodate more than one demonstration per label. To address these problems, we leverage the insight of using the ensemble method and improve ICL for low-resource scenarios. We propose Data Augmentation for In-Context Learning (DAIL) that uses the large language model paraphrase the test sample multiple times. Subsequently, the paraphrased samples, along with the original test sample, are separately presented to the large language model for inference. Finally, a majority voting mechanism is employed to integrate all the single results and get the final label. In the experiments, we show the effectiveness and generalization of our proposed method by evaluating on coarse-grained and fine-grained classification tasks. Additionally, we demonstrate the potential to utilize voting consistency to serve as the confidence score of the model output when the logits are inaccessible, such as for closed LLMs (e.g. ChatGPT). Note that Wei et al. (2022); Wang et al. (2022) employ ensembling methods for reasoning. These studies employ various sampling methods to obtain multiple reasoning paths of the model and generate answers by ensembling them. However, in Section 3.3, we demonstrate that their effectiveness is heavily reliant on providing well-prepared reasoning samples to the model\u2019s input. arXiv:2311.03319v1  [cs.CL]  6 Nov 2023 \fIn-Context  Learning DAIL Language  Model Language  Model Language  Model NEG POS POS NEG It 's a charming and often affecting journey. Inference It is a lovely and frequently moving expedition. This is a delightful and frequently touching excursion. Original Sentence: Paraphrase1: Paraphrase2: Inference Confidence:  66.7% Voting Original Sentence: Paraphrase Prompt In-Context Sample \u6587\u672c It 's a charming and often affecting journey. Figure 1: An overview pipeline of our method, DAIL. The part above the dotted line represents the standard InContext Learning process, and the lower part represents our method. DAIL augments the test sample by generating multiple paraphrases, obtains individual predictions, and ensembles them to return the final answer. 2 DAIL: Data Augmentation for In-Context Learning The intuition behind our method is that LLM is more familiar with what it generates i.e. it would likely have better performance while inferencing with the text generated by itself. Here, we refer to this as self-paraphrase. We validate this empirically in Section 3.2 and observe it to be true. Figure 1 shows the overview pipeline of our DAIL. First, in line with the self-paraphrase assumption, we augment the data by paraphrasing the original text of the test sample using the LLM itself and getting multiple candidates for the next step. Then, each candidate is concatenated with a prompt corresponding to each task, along with m randomly selected demonstrations from the training set. To mimic the low-resource scenario, we consider no more than 1 demonstration per label. The final label is generated by considering the candidates from all the paraphrased texts in addition to the original text. This is achieved through a majority voting approach, where the final label is determined based on the majority consensus among the candidates. Finally, for LLMs whose logits are inaccessible, we consider the voting consistency as its confidence score. Mathematically, DAIL obtains n samples by paraphrasing the test sample s: Sp = LLMpara(s, n) (1) Then, DAIL uses the n paraphrased samples together with the original sample to perform inference independently and get n + 1 labels in total: ai = LLMinf(p; d; si), si \u2208Sp \u222a{s} (2) Here p and d represent the prompt and demonstrations respectively. Assume the total label space is L, then the final result through majority voting is: avote = argmax l\u2208L n+1 X i=1 1(ai = l) (3) Finally, we consider voting consistency as the confidence score using: c = Pn+1 i=1 1(ai = avote) n + 1 (4) 3 Experiments 3.1 Experimental Settings We use several classification tasks in our experiments, including SST2, SST5 (Socher et al., 2013), CR (Hu and Liu, 2004), Emotion (Saravia et al., 2018), TREC (Voorhees and Tice, 2000) and AGNews (Zhang et al., 2015). We also use two finegrained classification datasets, Empathetic Dialogues (Rashkin et al., 2018) and Yahoo Answer Topics (Sileo, 2023) to evaluate DAIL\u2019s performance with large label spaces. Table 2 shows the detailed statistics of the aforementioned datasets. For baseline models, we choose ChatGPT1 and 1https://platform.openai.com/docs/models/gpt-3-5 \fModel Method SST2 SST5 CR Emotion TREC AG News Emp Dialogue Yahoo ChatGPT Standard ICL 93.6 54.4 88.0 52.2 77.1 84.6 43.1 64.3 DAIL-1 93.8 52.2 89.4 52.4 79.6 84.7 43.0 69.6 DAIL-2 93.6 54.2 89.6 54.1 82.5 87.4 45.5 70.6 DAIL-4 93.6 55.4 89.9 54.6 84.6 87.6 45.7 70.3 DAIL-4 w/o SP 94.7 53.8 90.3 51.5 82.1 85.2 42.2 71.2 PaLM2-540B Standard ICL 93.2 48.7 91.7 50.5 70.1 86.2 51.4 67.5 DAIL-1 92.5 52.3 91.4 50.5 69.6 84.1 49.8 67.9 DAIL-2 93.4 51.8 91.1 52.2 73.9 87.2 51.9 68.4 DAIL-4 93.4 54.8 92.3 53.0 76.4 88.6 53.0 68.5 DAIL-4 w/o SP 93.2 52.9 92.3 52.1 73.3 85.5 53.6 67.7 Table 1: Comparison of DAIL and Standard ICL. We report the accuracy score for each dataset. The best results are in bold. Our DAIL-4 achieves the overall best results. Task Dataset Testset Size Sentiment Classification SST2 872 SST5 2210 CR 376 Topic Classification AGNews 7600 TREC 500 Yahoo 2000 Emotion Classification Emotion 2000 Emp Dialogue 10900 Table 2: Detailed statistics of the datasets we use in the experiment. PaLM2-540B (Anil et al., 2023). We use the official APIs of each model to conduct evaluations. For ChatGPT, we use a stable version GPT-3.5-turbo-0301 for our experiments. Due to the lack of a stable version of the PaLM2-540B model, we repeat the experiments three times and report the average results. To simulate a lowresource scenario, we establish the number of demonstrations for each label category in each dataset to be 1. We report the accuracy score for each dataset. We compare our method with standard ICL. For our method, we compare several numbers of paraphrases per sample ranging from 1 to 4 and DAIL-i indicates i paraphrases per sample. For DAIL-1, we simply replace the original test sample with a paraphrase generated by the LLM and perform inference with it. For DAIL-2 and DAIL-4, we follow the method we illustrate in Section to generate and ensemble paraphrases. More details about our paraphrase and prompt format can be found in Appendix A 3.2 Main Results Table 1 shows the comparison between DAIL and standard ICL method. To further validate our assumption underlying the self-paraphrase mechanism, we also conduct an ablation study on DAIL-4 by using paraphrases interchangeably, denoted by DAIL-4 w/o SP. We use the paraphrase generated by PaLM for ChatGPT and vice versa. Here DAIL-1 doesn\u2019t exhibit a significant improvement in all datasets, suggesting the necessity of adopting an ensemble-based approach in DAIL. When the number of paraphrase samples used in DAIL increases, the performance consistently improves. DAIL-4 attains the best overall results across various datasets for both models. For datasets such as Yahoo that have 10 target classes, the performance improvement is up to 6 points in the case of ChatGPT. It is also notable that the performance in both models drops when we replace the self-paraphrase mechanism in DAIL-4. This confirms the assumption we put forth that large language models are more accustomed to the content they generate. SST5 TREC Emotion Standard ICL 54.4 77.1 52.2 Self-Consistency (Wang et al., 2022) 54.8 78.2 55.6 Prompt-Ensemble (Gao et al., 2020) 54.2 81.9 55.5 DAIL 55.4 84.6 55.6 Table 3: Comparison of DAIL and other ensemble-based methods. The best results are in bold. We observe DAIL out-performing other ensemble-based methods for classification. 3.3 Comparison to Other Ensemble-based Methods We conduct a comparison between our method and two other ensemble-based approaches, namely Self-Consistency (Wang et al., 2022) and PromptEnsemble (Gao et al., 2020). Self-Consistency is a method that incorporates sampling different reasoning paths during decoding. The various sampled results are subsequently used for voting to determine the final result. Prompt\fText Prediction Ground Truth Org A well acted and well intentioned snoozer. Neutral Negative Para Although well-performed and well-meaning, it can be quite dull. Negative It\u2019s a yawn-inducing film, but the acting and themes are commendable. Neutral The movie is a bore, despite the admirable acting and good intentions. Negative While well-acted and with good intentions, the movie is a bit of a snooze fest. Negative Table 4: An example in SST5 dataset. The text in green represent positive content and the text in red represent negative content (a) Binary classification (b) Multi-class classification Figure 2: Model Accuracy vs Voting consistency score using ChatGPT. We observe a positive correlation between voting consistency score and accuracy, thus making it a reliable confidence metric. Ensemble (Gao et al., 2020), on the other hand, employs different prompts for multiple inferences and then uses the outputs of the model to determine the final prediction through voting. In our experiments, we manually write prompts for each dataset. More details about it can be found in Appendix C. Table 3 displays the comparison results. While the three methods demonstrate similar performance on the Emotion dataset, DAIL exhibits a significant advantage over the other two methods in terms of SST5 and TREC. Additionally, we observe the performance of Self-Consistency in the classification datasets we use is not as strong as reported in the original paper for several reasoning datasets. One possible explanation for this discrepancy is that the classification datasets we employ lack reasoning samples that are provided to the model. In contrast, DAIL does not rely on any external sources, making it a more comprehensive approach. 3.4 Voting Consistency as Confidence Score In this section, we assess the reliability of employing voting consistency as the confidence score for the model. We plot the accuracy of ChatGPT vs several voting consistency thresholds illustrating their relationship in Figure 2. For the binary classification datasets (SST2, CR), we utilize confidence values of 0.6, 0.8, and 1.0. For the multiple classification datasets (SST5, Emotion, TREC, AGNews), we use confidence values of 0.4, 0.6, 0.8, and 1.0. We put the results of PaLM in Appendix B. Although the confidence score doesn\u2019t completely correspond to the accuracy, we observe a significant positive correlation with each other. By considering the confidence level, we can roughly assess the reliability of the model\u2019s output. We believe this could serve as a confidence score for addressing the problems in LLMs such as hallucination (Ji et al., 2023) and confidence calibration (Wang et al., 2020; Mekala et al., 2022, 2023). 3.5 Case Study To explain our method, we provide a case study from ChatGPT\u2019s results on SST5. Consider a negative sentiment sentence: \u201cA well acted and well intentioned snoozer.\u201d. Here both positive expression (\u201cwell acted\u201d and \u201cwell intentioned\u201d) and negative expressions (\u201csnoozer\u201d) are present, making it a challenging one for predicting sentiment. For standard ICL, the model fails to capture this subtle sentiment and gives the wrong label \u2019Neutral\u2019. The paraphrases generated by DAIL are mentioned in Table 4. For DAIL, one of the paraphrases generated by ChatGPT is \u201cAlthough wellperformed and well-meaning, it can be quite dull\u201d. It emphasizes the switch in relationship with the word \u201calthough\u201d and gets the correct label \u2019Negative\u2019. By paraphrasing several times and using majority voting, DAIL predicts it correctly. 4"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2310.13231v1",
+                    "title": "Multi-level Contrastive Learning for Script-based Character Understanding",
+                    "abstract": "In this work, we tackle the scenario of understanding characters in scripts,\nwhich aims to learn the characters' personalities and identities from their\nutterances. We begin by analyzing several challenges in this scenario, and then\npropose a multi-level contrastive learning framework to capture characters'\nglobal information in a fine-grained manner. To validate the proposed\nframework, we conduct extensive experiments on three character understanding\nsub-tasks by comparing with strong pre-trained language models, including\nSpanBERT, Longformer, BigBird and ChatGPT-3.5. Experimental results demonstrate\nthat our method improves the performances by a considerable margin. Through\nfurther in-depth analysis, we show the effectiveness of our method in\naddressing the challenges and provide more hints on the scenario of character\nunderstanding. We will open-source our work on github at\nhttps://github.com/David-Li0406/Script-based-Character-Understanding.",
+                    "authors": "Dawei Li, Hengyuan Zhang, Yanran Li, Shiping Yang",
+                    "published": "2023-10-20",
+                    "updated": "2023-10-20",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL",
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction As one essential element in stories, character comprehension is a popular research topic in literary, psychological and educational research (McKee, 1997; Currie, 2009; Paris and Paris, 2003; Bower, 1978). To fully understand characters, individuals must empathize with characters based on personal experiences (Gernsbacher et al., 1998), construct profiles according to characters\u2019 identities, and inference about characters\u2019 future actions (Fiske et al., 1979; Mead, 1990). According to the data modality and format, character comprehension can be categorized into several classes (Sang et al., 2022a). In this work, we focus on character understanding in scripts (Chen and Choi, 2016; Sang et al., 2022b). Scripts are written text for plays, movies, or broadcasts (Onions et al., 1966). Typically, scripts are often structured with several text fields, including scene description, conversation, transition and summary (Saha, 2021). Character Sheldon Jennifer Story Title TBBT The Test Dataset TVSHOWGUESS ROCStories Text Length 528832 41 Character\u2019s Related Text Sheldon: \" ... we take on Koothrappaliand his dog. Really give ourselves a challenge.\" Jennifer has a big exam tomorrow. ... Jennifer felt bittersweet about it. Table 1: Comparison between a script from TVSHOWGUESS (Sang et al., 2022b) and a narrative from ROCStories (Mostafazadeh et al., 2016). Although pre-trained language models (PLMs) have demonstrated their effectiveness in language and vision research fields (Qiu et al., 2020; Min et al., 2021), script-based character understanding is yet a hard task, as shown in our experiments. Here we highlight two challenges. The first one is text type. As scripts mainly consist of conversations between different characters, at the core of script-based character understanding is conversation understanding. Especially, scripts often involve multi-party conversations where multiple characters talk and interact with each other in a single scene. Considering other common issues in conversation understanding, it is non-trivial for PLMs to comprehend characters based on finegrained conversation information (Li and Zhao, 2021; Ma et al., 2022; Li et al., 2022; Tu et al., 2022). The other challenge of applying PLMs to script-based character understanding is text length. Table 1 shows a comparison between a script from TVSHOWGUESS (Sang et al., 2022b) and a short story from ROCStories (Mostafazadeh et al., 2016). Typically, scripts are very long with even billion of words (Chen and Choi, 2016), and in turn character information are distributed globally throughout the entire script (Bai et al., 2021; Inoue et al., 2022). However, PLMs are ineffective in capturing such global information due to the sensitiveness of context modeling (Liu et al., 2019; Joshi et al., 2020) arXiv:2310.13231v1  [cs.CL]  20 Oct 2023 \fand the limitation of input length (Dai et al., 2019; Beltagy et al., 2020). To address the aforementioned challenges, we propose a multi-level contrastive learning framework and capture both fine-grained and global information using two devised contrastive losses. For fine-grained character information, we build a summary-conversation contrastive loss by comparing character representations from different sources. Specifically, we leverage two text fields in scripts, i.e., summary and conversation, and then extract character representations from the corresponding field. The representations of the same character are then treated as the postive pairs, while those of different characters are negative pairs. To model the global information, we also propose a novel cross-sample contrastive loss as inspired by (Bai et al., 2021; Inoue et al., 2022). By aligning the same character\u2019s representation in different samples, the model overcomes the input length limitation and learns the global information of each character. To validate the effectiveness of our framework, we benchmark the performances of several PLMs, including SpanBERT, Longformer, BigBird, and ChatGPT-3.5, on three widely-adopted character understanding tasks. In general, our contributions are as follows: \u2022 We identify two critical challenges for character understanding in scripts and propose a multilevel contrastive learning framework to address them. \u2022 Through extensive experiments, we demonstrate the effectiveness of our method across multiple datasets and downstream tasks. \u2022 With further analysis, we provide some insights into script-based character understanding. All codes will be open-sourced for future research. 2 Related Work 2.1 Character Understanding Character understanding has long been the subject of considerable interest and scrutiny. Some early works propose to extract keywords as characters\u2019 features from movies (Bamman et al., 2013) and novels (Flekova and Gurevych, 2015). Other works attempt to learn the relationship between characters in both supervised (Massey et al., 2015; Kim and Klinger, 2019) and unsupervised ways (Krishnan and Eisenstein, 2014; Iyyer et al., 2016). Recently, more challenging tasks in character understanding have emerged. Chen and Choi (2016) benchmark the character linking and coreference resolution tasks on TV show scripts. Brahman et al. (2021) collect dataset with storybooks and their summaries, and define the character description generation and character identification tasks. Sang et al. (2022b) extend the character guessing task into a multi-character scenario on TV show scripts. Additionally, some works attempt to combine traditional self-supervised learning methods (Mikolov et al., 2013) with language models (Liu et al., 2019) to learn contextual character embeddings and apply them in downstream tasks (Azab et al., 2019; Inoue et al., 2022). In this work, we focus on character understanding tasks in scripts. While some works benchmark summary-based tasks in narratives (Chen et al., 2021; Brahman et al., 2021), we are the first to leverage script summaries as auxiliary data and learn fine-grained and global character representations in a novel way. 2.2 Contrastive Learning In recent years, contrastive learning is widely used in various NLP tasks (Zhang et al., 2022b), including sentence representation (Gao et al., 2021; Kim et al., 2021), machine translation (Pan et al., 2021; Vamvas and Sennrich, 2021), text generation (Lee et al., 2020; Shu et al., 2021; Zhang et al., 2022a), and etc. Literatures in multimodal research field adopt contrastive learning for vision-language model training, constructing positive pairs with images and their corresponding captions (Li et al., 2020; Radford et al., 2021; Yang et al., 2022). In our work, we also regard characters in summaries and conversations as two different views of the same target and align them for a better representation. Moreover, some works aim to construct positive pairs in global manners. Both Qin et al. (2020) and Hogan et al. (2022) conduct document-level contrastive learning in the relation extraction task to align the representation of the same entity and relation. Pan et al. (2021) propose an aligned augmentation method that generates more positive sentence pairs in different languages to improve translation performances in non-English directions. Similarly, Qin et al. (2022) acquire multilingual views of the same utterance from bi-lingual dictionaries. Following this line of research, we propose the cross-sample contrastive learning in addition to the in-sample contrastive loss to learn character \f... ... Multi-level Contrastive Learning Summary Encoding Conversation Summary Character Embeddings Enhanced Character  Embeddings Cross-sample Maximize Sum-Conv Minimize Cross-sample Minimize Sum-Conv Maximize LSup LSum LCross + ... ... Inference with Character Embedding Sample 1 Sample 2 ... ... ... ... PLM Encoder Attention-Based Layer Conversation Encoding Figure 1: The overview pipeline of our method. Each color represents a character entity or embedding. The conversation and summary encoding parts correspond to Section 4.1 and 4.2 respectively. The multi-level contrastive learning part corresponds to Section 4.3. The inference with character embedding part corresponds to Section 4.4. representations globally. 3 Preliminaries Generally, character understanding tasks require the model to predict character\u2019s information given a segment of text. For script-based character understanding, the provided texts often consist of conversations within scripts. In this work, we also leverage script summaries as an additional source. We provide detailed examples in Appendix A. In practice, the model first generates character\u2019s embeddings e in the representation learning step. Subsequently, a feed-forward network FFN is often adopted as the classifier with the cross-entropy loss: p = Softmax(FFN(e)) (1) LSup = \u22121 N N X i=1 yi log(p) (2) 4 Method Our work presents a multi-level contrastive learning framework for character representation learning. Firstly, we follow a general encoding process to obtain character representations from conversations and summaries. Then, we describe two novel contrastive losses to capture fine-grained and global information at both in-sample and cross-sample levels. Finally, we propose a two-stage training paradigm that applies different losses in different learning stages. Figure 1 illustrates an overview pipeline of our method. 4.1 Character Representation in Conversation To obtain character representations from the conversation field in the scripts, we first concatenate each utterance (Joshi et al., 2020; Beltagy et al., 2020) and utilize a pre-trained language model PLM1 to produce the encoding of the whole text H: H = PLM(u1; u2; , ...; uT ) (3) Then, the character embeddings e1, e2, ...en are extracted from the contextual encoding H. After that, we follow previous works (Bai et al., 2021; Sang et al., 2022b) and use an attention-based layer to share the character-level information among each embedding2: e1, ...en = Extract(H) (4) e1, ...en = Attention(e1, ...en) (5) However, the conversations in the scripts are complex and thus the character embeddings solely based on the conversations are often insufficient for fine-grained character understanding. 1Without loss of generalization, we adopt several PLMs in experiments. 2We provide further details in Appendix B \f4.2 Character Representation in Summary To supply more information, we leverage scripts\u2019 summaries as auxiliary data and apply contrastive learning to capture the character intricacies. Similar with conversation encoding, given a summary S contains a group of character mentions {cms 1, cms 2, ..., cms n}, we also encode the whole summary and extract the character representations: Hs = PLM(S) (6) es i = tstarti + tendi, 1 <= i <= n (7) where tstarti and tendi are the first and last tokens of the ith character mention cms i in the summary. After that, we follow (Bai et al., 2021) and use a mention-level self-attention (MLSA) layer 3 to gather information for each character embedding: es 1, ..., es n = MLSA(es 1, ..., es n) (8) and the last layer\u2019s output es i is treated as the character\u2019s representation from the summary. 4.3 Multi-level Contrastive Learning To enhance the character representations learned from the conversation and the summary, we develop a novel multi-level contrastive learning to capture both fine-grained and global information. 4.3.1 Summary-conversation Contrastive Learning At the local in-sample level, we develop a summaryconversation contrastive loss to align representations of the same character. This gives the model an additional perspective on character representation and encourages it to find a general space where different representations of the same character are closer. Concretely, the loss function for the summary-conversation contrastive learning is: LSum = P X i=1 \u2212log expsim(eci,es ci)/\u03c4 PP j=1 expsim(eci,es cj )/\u03c4 (9) where ci denotes the ith character, and P here is the number of characters that appear in both scripts and summaries. Also, \u03c4 is a temperature hyperparameter, and sim(, ) stands for the similarity function4. Note that in samples where conversation and summary contain multiple representations of 3It is a transformer encoder layer with B repeated block. Please refer to Bai et al. (2021) for more details. 4Here we use Cosine similarity. character ci, we randomly select one as eci and es ci, respectively. By applying the summary-conversation contrastive loss, we are able to learn fine-grained character representations from both summary and conversation texts. 4.3.2 Cross-sample Contrastive Learning In addition to fine-grained information, global-level information is also crucial for character representation learning (Bai et al., 2021; Inoue et al., 2022). To this end, we also propose a cross-sample contrastive learning to align the same character representation in different samples within a batch: LCross = K X i=1 \u2212log expsim(e1 ci,e2 ci)/\u03c4 PK j=1 expsim(e1 ci,e2 cj )/\u03c4 (10) SI(e1 ci) \u0338= SI(e2 ci) (11) where SI(e) means the sample index of the character representation e5. When there are multiple representations of one given character in a batch, we randomly select two from them. For cross-sample learning, we impose a constraint that restricts e1 ci and e2 ci to originate from different samples. K is the number of characters appearing in at least two different samples within a batch. To this end, the cross-sample contrastive loss forces the model to utilize global information in a batch and thus obtain a comprehensive understanding of the characters. 4.4 Two-stage Training To fully train the model, we further propose a twostage training paradigm to apply different losses in different learning stages. Concretely, in the first stage, we combine the two contrastive losses with the supervised loss together, and post-train the pre-trained language model. The supervised loss serves as a guidance to facilitate the contrastive learning, and stabilize the training at the very beginning. The total loss of the first stage is: LTotal = \u03bb\u2217LSup +\u03b1\u2217LSum +\u03b2 \u2217LCross (12) where \u03bb, \u03b1, \u03b2 are hyper-parameters of task ratios, and we will analyze their effects in Section 6.3. After the first stage, only the supervised loss is 5e generally represents any character embedding. \fkept to train the model in the second stage. This makes the model concentrate on the downstream supervision signals. 5 Experiments Setup 5.1 Tasks and Datasets We evaluate the proposed method on three character understanding tasks, i.e., coreference resolution (Chen and Choi, 2016), character linking (Chen and Choi, 2016), and character guessing (Sang et al., 2022b). Coreference Resolution Given a conversation in scripts that contains multiple utterances and n character mention entity c1, c2, ..., cn within it, the objective of the coreference resolution task is to assemble all mention entities that refer to the same character in a cluster. Character Linking The input of the character linking task is the same as coreference resolution. Unlike coreference resolution, the goal of character linking is to accurately classify each mention entity to the character in a pre-defined character set Z = {z1, z2, ..., zm}. Character Guessing Distinct from previous tasks, the character guessing task focuses on identifying the speaker for each utterance in scripts. In this task, each utterance within a scene is segmented and fed into the model. The speaker\u2019s name preceding each utterance is masked and replaced with a special token. The same speaker within a scene is represented by the same special token. The objective of the character guessing task is to predict the identity of the speaker for each special token. Datasets We choose two TV show datasets to conduct experiments. For coreference resolution and character linking, we use the latest released version of the Character Identification dataset6. For character guessing, we adopt the TVSHOWGUESS dataset7 to conduct experiments. We follow all the training, development, and testing separation provided by the original datasets. The dataset statistics are given in Table 13 in Appendix. 5.2 Baseline Models Following previous works, we adopt several stateof-the-art (SOTA) models in character understanding as baselines and apply the proposed framework on them. For coreference resolution and 6https://github.com/emorynlp/ character-identification 7https://github.com/YisiSang/TVSHOWGUESS character linking, we choose SpanBERT (Joshi et al., 2020), a transformer-architecture pre-trained model with the contiguous random span mask strategy in the pre-training stage. We also adopt C2, which combines coreference resolution and character linking together and achieves the SOTA performance in both two tasks. For character guessing, we use BigBird (Zaheer et al., 2020) and Longformer (Beltagy et al., 2020), as they are specialized for long-form document input. We follow Sang et al. (2022b) and add a character-specific attentive pooling layer upon the the model encoders and denote them as BigBird-P and Longformer-P. Notably, we also design a zero-shot and one-shot instruction prompts and evaluate ChatGPT-3.5 (gpt3.5-turbo) via its official API8 as another strong large language model baseline. 5.3 Evaluation Metrics For coreference resolution, we follow the previous works (Zhou and Choi, 2018; Bai et al., 2021) and use B3, CEAF\u03d54, and BLANC as our evaluation metrics. These three metrics are first proposed by the CoNNL\u201912 shared task (Pradhan et al., 2012) to measure the clustering performance of the coreference resolution task. For character linking and character guessing, we use Macro and Micro F1 to evaluate the models\u2019 classification performances. 5.4 Implementation Details We employ both the base and large sizes of each model, and implement our proposed method on them. For summary-conversation contrastive loss, we use summary corpus collected by Chen et al. (2021). We follow the hyper-parameter settings in the original papers to reproduce each baseline\u2019s result. We repeat each experiment 3 times and report the average scores. For ChatGPT prompts and other implementation details, please refer to Appendix C and Appendix D. We will open-source all codes in this work. 6 Results and Analysis 6.1 Main Results Table 2 and Table 3 present the automatic evaluation results on the three tasks. Surprisingly, even with specialized instruction and one-shot demonstration, ChatGPT-3.5 performs the worst among all the baselines on each task. This implies that 8https://platform.openai.com/docs/ api-reference/completions/create \fMODEL Coreference Resolution Character Linking B3 CEAF\u03d54 BLANC MICRO MACRO PREC. REC. F1 PREC. REC. F1 PREC. REC. F1 ChatGPT-Zero-Shot 63.43 59.51 61.41 68.39 64.37 66.32 80.39 77.74 78.97 74.7 64.3 ChatGPT-One-Shot 66.43 62.54 64.43 68.47 64.44 66.40 82.19 79.40 80.70 76.2 63.6 SpanBERT-base 77.40 82.67* 79.94 74.69 67.93 71.15* 84.80* 89.96 87.20 85.0* 78.4 SpanBERT-base (Ours) 79.95* 84.71 82.26 76.67 70.38 73.39* 87.44 91.26 89.26 86.3 78.9* SpanBERT-large 81.92 85.56 83.69* 77.85 74.74 76.25* 88.61* 91.91 90.20 87.2* 82.8* SpanBERT-large (Ours) 83.55* 87.38* 85.42* 79.83 76.29 78.02* 89.18* 93.00 91.00 88.2* 83.7* C2-base 80.75 84.77* 82.71* 76.97 71.78 74.28 82.22* 91.52 89.80* 85.6 80.4* C2-base (Ours) 83.35 85.12* 84.23 76.88* 74.97 75.91 90.48 91.85* 91.15 86.4 81.1 C2-large 84.98 86.92* 85.94 79.63 78.16* 78.89 90.87* 93.05* 91.93 87.6* 82.5* C2-large (Ours) 86.42 86.44* 86.24* 78.82 80.42* 79.61 91.77* 93.13 92.45* 88.0* 83.2* Table 2: Automatic evaluation results on coreference resolution and character linking. The best results are in bold. We follow previous works to present the results of coreference resolution in a 2-digital decimal and the results of character linking in a 1-digital decimal. * denotes that p \u22640.01 in the statistical significance test. Model MICRO MACRO ChatGPT-Zero-Shot 48.58 42.17 ChatGPT-One-Shot 51.57 44.05 BigBird-P-base 71.01 70.32* BigBird-P-base (Ours) 72.61 73.00 BigBird-P-large 75.43* 75.24 BigBird-P-large (Ours) 77.68* 76.41 Longformer-P-base 71.80 73.75 Longformer-P-base (Ours) 73.65* 74.22 Longformer-P-large 77.58 75.92* Longformer-P-large (Ours) 78.92* 76.52* Table 3: Automatic evaluation results on character guessing. The best results are in bold. character understanding is still hard and complex to solve for large language models. Among the three tasks, models perform worse on character guessing than on coreference resolution and character linking tasks. In particular, ChatGPT achieves extremely low scores of 44.05 Macro-F1 in character guessing. Since character guessing requires a deeper understanding of each character and more varied narrative comprehension skills (Sang et al., 2022b), this suggests that the current pre-trained models, especially LLMs, have room for improvement in tasks that require global and indepth learning for a specific individual. Despite the discrepancies in model architecture and size, the proposed method brings significant improvements to each baseline model on almost every metric, except for B3 and CEAF\u03d54 in C2-large model. These results indicate the effectiveness and compatibility of our method. 6.2 Ablation Studies We also conduct an ablation study to examine the contributions of the two novel contrastive losses, i.e., the cross-sample loss and summaryconversation loss. To implement, we select SpanBERT-base and SpanBERT-large as backbone models and implement model variants by removing one of two contrastive losses in the training phases. Table 4 presents the results of our ablation study on the coreference resolution and character linking tasks. Compared with the vanilla SpanBERTbase and SpanBERT-large, adding one or two contrastive losses yield better performances. Additionally, we observe that when applied separately, models with the summary-conversation loss work better than models with the cross-sample loss only. More importantly, it is evident that the models trained with both contrastive losses together outperform the models with only one loss, indicating the necessity of our multi-level contrastive framework as well as its effectiveness in addressing the two challenges, i.e., text type and text length. We also conduct an ablation study on the twostage learning strategy. Table 5 shows the experiment results on C2-base using character linking and coreference resolution. While the one-stage multi-task training can also improve the baseline model\u2019s performance, we found it leads to a suboptimal result compared with that using our twostage learning strategy. This observation leads us to the conclusion that supervision-only fine-tuning is also very important in our method, consistently enhancing baseline models\u2019 performance. This aligns with the findings of prior research, which advocate for task-specific fine-tuning following multi-task \fMODEL Coreference Resolution Character Linking B3 CEAF\u03d54 BLANC MICRO MACRO PREC. REC. F1 PREC. REC. F1 PREC. REC. F1 SpanBERT-base 77.40 82.67 79.94 74.69 67.93 71.15 84.80 89.96 87.20 85.0 78.4 SpanBERT-base (Ours) 79.95 84.71 82.26 76.67 70.38 73.39 87.44 91.26 89.26 86.3 78.9 w/o cross-sample loss 79.48 83.06 81.23 74.72 71.07 72.85 87.68 90.59 89.08 86.1 80.0 w/o summary-conversation loss 79.00 83.36 81.11 74.68 70.33 72.44 85.45 90.61 87.85 85.6 78.8 SpanBERT-large 81.92 85.56 83.69 77.85 74.74 76.25 88.61 91.91 90.20 87.2 82.8 SpanBERT-large (Ours) 83.55 87.38 85.42 79.83 76.29 78.02 89.18 93.00 91.00 88.2 83.7 w/o cross-sample loss 83.85 86.68 85.24 79.44 76.37 77.88 90.68 92.47 90.65 87.4 83.6 w/o summary-conversation loss 85.29 83.96 84.62 74.65 79.20 76.69 91.45 91.15 90.80 87.9 82.8 Table 4: Ablation study results on two contrastive losses. The experiment is conducted using character resolution and character linking. Coreference Resolution Character Linking B3 CEAF\u03d54 BLANC MICRO MACRO C2-base 82.71 74.28 89.80 85.6 80.4 C2-base-OS 83.58 74.28 90.63 86.1 80.8 C2-base-TS 84.23 75.91 91.15 86.4 81.1 Table 5: Ablation study results on two-stage learning strategy. -OS and -TS represents the one-stage training and two-stage training respectively. For one-stage training, we remove the second supervised loss-only stage and adopt the multi-task training only. post-training (Guan et al., 2020; Han et al., 2021). \u03bb \u03b1 \u03b2 B3 CEAF\u03d54 BLANC MICRO MACRO \u2013 \u2013 \u2013 83.69 76.25 90.20 87.2 82.8 1.0 0.0 0.0 83.98 76.12 90.75 87.8 82.2 0.0 1.0 1.0 85.04 77.72 90.77 86.4 78.6 1.0 1.0 1.0 85.42 78.02 91.00 88.2 83.7 0.5 1.0 1.0 85.14 78.15 91.02 88.4 83.2 1.0 0.5 0.5 85.23 79.00 90.96 88.1 82.0 Table 6: Hyper-parameter analysis results on coreference resolution and character linking. For coreference resolution, we report the F1 scores of the B3, CEAF\u03d54 and BLANC metrics. 6.3 Analysis on Hyper-Parameters The task ratio setting is also an important component of our method. In this section, we investigate their impacts by testing various task ratios in the first training stage. We employ the SpanBERTlarge model and perform experiments on the coreference resolution and character linking tasks. The results of the hyper-parameter analysis are presented in Table 6. As defined in Equation 12, \u03bb, \u03b1, and \u03b2 represent the ratios of task-specific supervised loss, summary-conversation loss, and Figure 2: Evidence type analysis result. cross-sample loss, respectively. Accordingly, the first block (Row 1) presents the vanilla SpanBERTlarge performance w/o our framework, and the second block (Row 2 and Row 3) shows the model variants with only supervision loss or contrastive losses. Comparing the first and second block we can see, there is no obvious improvement when only keeping the supervised loss, a.k.a \u03bb = 1.0, \u03b1 = 0.0, \u03b2 = 0.0 in the first stage. Moreover, when \u03bb is set to 0.0, the model trained without supervised loss also exhibits inferior performances, e.g., there is a notable decrease in Macro F1 (from 82.8 to 78.6). This finding supports our hypothesis that the task-specific supervision signal plays a crucial role in guiding the two contrastive learning. When examining the last block (Row 4-6), we observe that the models w/ our framework under different task ratios consistently surpasses the others (except only one MARCO metric). This further demonstrates the robustness of our method on the task ratio hyper-parameter. \f6.4 Resource Availability Analysis The proposed summary-conversation contrastive learning relies on well-organized script datasets that include a summary of each scene. This prerequisite could potentially limit the applicability of our approach to datasets in other languages or domains. To address this constraint, we conduct an experiment in which we replaced the manually collected summary dataset with an automatically generated one, produced by ChatGPT. As depicted in Table 7, our results indicate that when using the auto-generated corpus in summary-conversation contrastive learning, a significant improvement is still observed when compared to the vanilla baseline. This discovery further validates the adaptability of our method, irrespective of whether golden or generated summaries are used. Coreference Resolution Character Linking B3 CEAF\u03d54 BLANC MICRO MACRO C2-base 82.71 74.28 89.80 85.6 80.4 C2-base-LLM 84.14 76.06 90.89 86.1 80.9 C2-base-G 84.23 75.91 91.15 86.4 81.1 Table 7: Experiment results with automatically generated summarization. -LLM and -G denote the model trained on summaries generated by ChatGPT and those trained using the dataset provided by (Chen et al., 2021). 6.5 Breakdown to Evidence Type To better understand when and how our method works on each sample, we conduct an evidence type analysis on the character guessing task based on the fine-grained annotation provided by Sang et al. (2022b). To remedy the scarcity issue in the original annotations, we merge the fine-grained annotation categories into two broader categories: Global & In-depth Evidence and Local & Textual Evidence. More details on evidence type merging is described in Appendix E. The results of evidence type analysis are presented in Figure 2. Note that our framework works better when Local & Textual evidence is required for character guessing than Global & In-depth evidence. This finding aligns with our intuition that Global & In-depth evidence is more challenging for the model to comprehend. It is also worth noting that our framework yields larger increases for samples requiring Global & In-depth evidences (2.4% and 2.7% for the base and large size models respectively), as compared to those requiring Local Figure 3: Character embedding visualization result. & Textual evidence (1.1% and 1.6% for the base and large models respectively). Based on these results, we safely conclude that our framework is effective in facilitating character information modeling, especially for global information. 6.6 Visualization The core of our method is to learn fine-grained and global character representations. To this end, we also visualize the learned character embeddings in the character guessing task. Specifically, we use character embeddings in the test set of the \u201cFRIENDS\u201d (a subset of TVSHOWGUESS dataset) and randomly choose 6 embeddings for each character from different samples. Figure 3 shows the visualization results using T-SNE (Van der Maaten and Hinton, 2008). We compare the character embeddings generated by Longformer-P-Large w/ and w/o our framework. One thing to note is that without our framework, some character embeddings of Ross overlap with those of Rachel. This is because that in the TV show \u201cFRIENDS\u201d, Ross and Rachel are partners and together appearing and engaging in many scenes. In contrast, this overlapping phenomenon is greatly mitigated. Overally speaking, our framework encourages the embeddings belonging to the same character exhibit a more compact clustering pattern. This finding provides a new perspective to understand the effectiveness of our proposed method in character comprehension tasks. 6.7 Case Study We also choose a challenging sample from \u201cThe Big Bang Theory\u201d subset of TVSHOWGUESS in the character guessing task, and analyze the predictions from Longformer-P-Large w/o and w/ our method, as well as that from ChatGPT. As shown in Table 8, all the predictions from ChatGPT are wrong, indicating ChatGPT lacks a fine-grained understanding of each character. Be\fsides, the only difference between the vanilla model w/ and w/o our framework is whether the speaker P1 is predicted correctly or not. In this case, predicting P1 is particularly challenging, as few utterances are spoken by this character. Hence, it is a must for the models to guess P1\u2019s identity using other details in the scene. By understanding the relationships between P1 and other characters, our method is able to correctly predict that P1 is Sheldon\u2019s partner, Amy. This demonstrates that our method benefits the fine-grained understanding on character relationships in script-based character understanding, e.g., character guessing tasks. P0 : Hey, sorry about that P1 : No, we\u2019re sorry. We never should have been comparing relationships in the first place. P2 : Why? We won. You know, I say, next, we take on Koothrappali and his dog. Really give ourselves a challenge. P3 : I just want to say one more thing about this. Just because Penny and I are very different people does not mean that we\u2019re a bad couple. P2 : The answer is one simple test away. Hmm? You know, it\u2019s like when I thought there was a possum in my closet. Did I sit around wondering? No, I sent Leonard in with a pointy stick and a bag. P3 : I killed his Chewbacca slippers. P0 : Let\u2019s just take the test. P3 : No, no, no, I don\u2019t want to. P0 : Oh, well, \u2019cause you know we\u2019re gonna do bad. P3 : Because it doesn\u2019t matter. I don\u2019t care if we\u2019re a ten or a two. P2 : Or a one. A one is possible. P3 : Marriage is scary. You\u2019re scared, I\u2019m scared. But it doesn\u2019t make me not want to do it. It, it just makes me want to hold your hand and do it with you. P0 : Leonard. P1 : It makes me so happy if you said things like that. P2 : We got an eight-point-two. Trust me, you\u2019re happy. ChatGPT:P0: Leonard, P1: Sheldon, P2: Penny, P3:Howard Vanilla: P0: Penny, P1: Howard, P2: Sheldon, P3:Leonard Ours: P0: Penny, P1: Amy, P2: Sheldon, P3:Leonard Golden: P0: Penny, P1: Amy, P2: Sheldon, P3:Leonard Table 8: An example chosen from \u201cThe Big Bang Theory\u201d in the character guessing task. We analyze the predictions made by ChatGPT (one-shot), LongformerP-Large (vanilla and with our framework). 7 Discussion about LLMs on Character Understanding In this section, we go deeper to discuss the unsatisfied performance when adopting the ICL of LLMs to perform character understanding tasks. One possible reason for this is the script-based character understanding we focus on requires the model to learn the character information globally. For example, in character guessing, anonymous speakers sometimes need to be identified with some global evidence, like linguistic style and the character\u2019s relationship with others. These subtle cues are usually not included in the current sample and thus require the model to learn them globally from other samples (Sang et al., 2022b). However, due to the fine-tuned unavailability of ICL, LLMs can only utilize local information from the current sample and limited demonstrations to make inferences. We believe this is the reason that LLMs don\u2019t perform well in our script-based character understanding scenario. Additionally, we notice ICL also falls short in some other tasks that involve learning a domain-specific entity or individual across multiple samples, like knowledge graph completion (Yao et al., 2023). In our work, we notice while the number of demonstrations increases, the performance of LLMs shows a corresponding improvement. It appears that augmenting the number of demonstrations in the prompt could be a potential strategy for enhancing the capabilities of LLMs in these global learning tasks. Nonetheless, it\u2019s essential to note that incorporating an excessive number of relevant samples as demonstrations faces practical challenges, primarily due to constraints related to input length and efficiency considerations. In the future, more efforts are needed to explore optimal ways of harnessing the ICL method of LLMs in such global learning scenarios. 8"
+                }
+            ],
+            "Cong Liu": [
+                {
+                    "url": "http://arxiv.org/abs/2402.10011v3",
+                    "title": "Clifford Group Equivariant Simplicial Message Passing Networks",
+                    "abstract": "We introduce Clifford Group Equivariant Simplicial Message Passing Networks,\na method for steerable E(n)-equivariant message passing on simplicial\ncomplexes. Our method integrates the expressivity of Clifford group-equivariant\nlayers with simplicial message passing, which is topologically more intricate\nthan regular graph message passing. Clifford algebras include higher-order\nobjects such as bivectors and trivectors, which express geometric features\n(e.g., areas, volumes) derived from vectors. Using this knowledge, we represent\nsimplex features through geometric products of their vertices. To achieve\nefficient simplicial message passing, we share the parameters of the message\nnetwork across different dimensions. Additionally, we restrict the final\nmessage to an aggregation of the incoming messages from different dimensions,\nleading to what we term shared simplicial message passing. Experimental results\nshow that our method is able to outperform both equivariant and simplicial\ngraph neural networks on a variety of geometric tasks.",
+                    "authors": "Cong Liu, David Ruhe, Floor Eijkelboom, Patrick Forr\u00e9",
+                    "published": "2024-02-15",
+                    "updated": "2024-03-12",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI"
+                    ],
+                    "main_content": "INTRODUCTION Graph Neural Networks (GNNs) have established themselves as effective tools for learning representations of relational data from a variety of domains, including social networks (Fan et al., 2019), bioinformatics (Zitnik & Leskovec, 2017), and physics (Battaglia et al., 2016). Recently, there has been a surge of interest in the theoretical foundations of GNNs, with a particular focus on their expressive power (Geerts & Reutter, 2022). Standard message passing leverages the sparsity of the underlying graph by exchanging \u2018messages\u2019 between nodes only when they are adjacent. As such, nodes with local structures that are alike will learn similar representations, restricting the expressivity of this framework. This limitation has been formalized by Xu et al. (2019); Morris et al. (2019), showing that Message Passing Neural Networks (MPNNs) are at most as expressive as the Weisfeiler-Lehman (1-WL) test at distinguishing unattributed graphs. As a consequence, such networks cannot identify graph structures such as triangles or their higher-dimensional equivalents (Chen et al., 2020). In response, Bodnar et al. (2021) developed simplicial message-passing networks operating on simplicial complexes, resulting in both theoretical and empirical improvements regarding expressive power. Further, their contributions illuminate mathematical intersections with algebraic and differential topology, as well as geometry (Ghrist, 2014). In the context of geometric graphs, the focus lies on graphs that are embedded in a geometry, such as a metric space or a manifold. These data points are often accompanied by geometric features, like positions or velocities. Such quantities transform predictably, though nontrivially, under rigid operations like rotations, reflections, or translations. E(n) Equivariant Graph Neural Networks (EGNNs) (Satorras et al., 2021) are designed to respect these symmetries by either exhibiting an equivariance or invariance characteristic tailored to the specific task. This field has witnessed continual advancements (Huang et al., 2022; Th\u00a8 olke & Fabritiis, 2022; Finzi et al., 2020; Brandstetter et al., 2022; Batzner et al., 2022), with one of the latest being the introduction of Clifford Group Equivariant Neural Networks (CGENNs): a neural network architecture operating on the Clifford (or geometric) algebra (Ruhe et al., 2023a; Brehmer et al., 2023). While these methods reap the benefits of geometric equivariance, they are still limited to the expressive power of traditional message-passing. \u2217Contributed equally. 1AMLab. 2AI4Science Lab. 3Anton Pannekoek Institute. 4UvA-Bosch Delta Lab. 1 arXiv:2402.10011v3  [cs.AI]  12 Mar 2024 \fPublished as a conference paper at ICLR 2024 Combining the merits of both simplicial message passing and geometric equivariance, Eijkelboom et al. (2023) developed a method based on EGNNs, called E(n) Equivariant Message Passing Simplicial Networks (EMPSNs). However, we can identify two limitations of this method. First, the higher-dimensional simplices are initialized using manually calculated geometric information. Second, EGNN is an architecture that falls under the umbrella of scalarization methods (Han et al., 2022), which operate mainly with invariant features and update vector features only through scaling. In this work, we go a step further and develop steerable Clifford algebra-based simplicial messagepassing networks: Clifford Group Equivariant Simplicial Message Passing Networks (CSMPNs). The Clifford algebra contains higher-order elements like bivectors and trivectors. These objects are computed through vector composition and can represent geometric quantities like areas and volumes. Analogously, simplices are also fully defined by their constituent vertices. The geometric product and the Clifford algebra are well-defined for any inner product space of any dimension. This versatility makes the geometric product a suitable option for computing simplex features and enables embedding simplices with geometric features across various spaces and dimensions. As such, we initialize the higher-order simplices using geometric products (as well as linear combinations) of their vertices. The network then refines these simplices by passing messages between simplices of different order. To do so efficiently, unlike EMPSN, which uses a separate model for each type of communication, we share the parameters of the message passing function across various simplex orders, conditioning it on the dimensionalities of the source and target simplices. Additionally, we restrict the final message to an aggregation of the incoming simplicial messages, enabling the use of modern parallelized graph reduction operations in the simplicial setting. We call the resulting algorithm shared simplicial message passing. Equivariance is achieved by restricting all message and update neural networks to be Clifford group equivariant networks (Ruhe et al., 2023a). Experimental results show that our method can outperform both equivariant and simplicial graph neural networks in geometric tasks spanning multiple domains and dimensionalities, including a convex hulls volume prediction task, human walking motion prediction, molecular motion prediction, and NBA player trajectory prediction. 2 BACKGROUND We provide a brief introduction to Clifford group equivariant neural networks and simplicial complexes. For a review on equivariant message passing networks, consider Appendix B. 2.1 CLIFFORD GROUP EQUIVARIANT NEURAL NETWORKS We start by introducing the Clifford algebra, also known as the geometric algebra, which is a powerful mathematical object with applications in various areas of science and engineering. For a full development, we refer the reader to Ruhe et al. (2023a). While the theory has been generally developed, we restrict our attention to the case where the underlying vector space is Rd. The Clifford algebra Cl(Rd, q) is the \u201cbiggest\u201d unitary, associative, non-commutative algebra generated by Rd such that for all v \u2208Rd, v2 = q(v), where q : Rd \u2192R is a quadratic form. In other words, vectors square to scalars. q has an associated bilinear form b : Rd \u00d7 Rd \u2192R. The algebra\u2019s elements, generally called multivectors, are linear combinations of non-commutative products of vectors while respecting the quadratic form: x \u2208Cl(Rd, q), x = P i\u2208I ci \u00b7 vi,1 . . . vi,k. Here, the index set I is finite, ci \u2208R are scalars, and vi,j \u2208Rd are vectors. Clifford multiplication is referred to as the geometric product. The Clifford algebra forms a graded vector space, with Cl(Rd, q) = d M k=0 Cl(k)(Rd, q), dim Cl(k)(Rd, q) = \u0012d k \u0013 , (1) where and dim Cl(Rd, q) = 2d. These subspaces are called grades, where elements of k = 0 are called scalars, elements of k = 1 are called vectors, elements of k = 2 are called bivectors, and so on. A multivector x \u2208Cl(Rd, q) can thus be written as a sum of its grades: x = Pd k=0 x(k), where x(k) \u2208Cl(k)(Rd, q). The operation (\u00b7)(k) : Cl(Rd, q) \u2192Cl(k)(Rd, q) is called grade projection, which selects the grade-k component of x. Beyond scalars and vectors, which only express magnitude and direction, respectively, bivectors express oriented area, trivectors express oriented volume, and so forth. Note that we really have the identifications: Cl(0)(Rd, q) = R and Cl(1)(Rd, q) = Rd. 2 \fPublished as a conference paper at ICLR 2024 \u03c1(w) Rotate \u03d5 \u03c1(w) Rotate \u03d5 Figure 1: Illustration of our proposed architecture. Top left: a set of vertices (and edges) is lifted to a simplicial complex. We highlight three simplex types: vertices (0-simplices, ), edges (1simplices, ), and triangles (2-simplices, ). In this case, the vertex feature is vector-valued and embedded as the grade 1 part of a Clifford algebra element: a multivector. In three dimensions, a multivector has scalar ( ), vector ( ), bivector ( ) and trivector ( ) components. Higher-order simplices are initialized using the geometric product of their constituent vertices. As such, edges in the top left visualization are bivector-valued, and triangles are trivector-valued. The simplicial message-passing framework, denoted by \u03d5, refines the multivector-valued simplices, as portrayed in the bottom-left, by passing messages between simplices of different order. Crucially, \u03d5 maintains equivariance to the Clifford group\u2019s orthogonal action \u03c1(w), representing a rotation here. In doing so, our method is ensured to respect the geometric symmetries of the input data. Ruhe et al. (2023a) then define the Clifford group \u0393(Rd, q). Its elements are invertible homogeneous1 multivectors of Cl(Rd, q) that preserve vectors under the group action \u03c1(w) : Cl(Rd, q) \u2192Cl(Rd, q), called the twisted conjugation. That is, if x \u2208Cl(1)(Rd, q) = Rd, then \u03c1(w)(x) \u2208Cl(1)(Rd, q) = Rd. It is further shown that all \u03c1(w) preserve the quadratic form q. Therefore, each \u03c1(w) defines an orthogonal automorphism. Let F \u2208R[T1, . . . , T\u2113] be a polynomial (non-commutative, where multiplication is performed by the geometric product) in \u2113variables, w \u2208\u0393(Rd, q), and x1, . . . , x\u2113\u2208Cl(Rd, q). We then have the following equivariance properties (Ruhe et al., 2023a). Theorem 2.1 (All polynomials are Clifford group equivariant). Let F \u2208R[T1, . . . , T\u2113] be a polynomial in \u2113variables with coefficients in R, w \u2208\u0393(Rd, q). Further, consider \u2113elements x1, . . . , x\u2113\u2208Cl(Rd, q). Then we have the following equivariance property: \u03c1(w) (F(x1, . . . , x\u2113)) = F(\u03c1(w)(x1), . . . , \u03c1(w)(x\u2113)). (2) Theorem 2.2 (All grade projections are Clifford group equivariant). For w \u2208\u0393(Rd, q), x \u2208 Cl(Rd, q) and k = 0, . . . , d we have the following equivariance property: \u03c1(w)(x(k)) = (\u03c1(w)(x))(k) . (3) In particular, for x \u2208Cl(k)(Rd, q) we also have \u03c1(w)(x) \u2208Cl(k)(Rd, q). That is, polynomials in multivectors, as well as their grade projections, are Clifford group equivariant, and therefore equivariant to the orthogonal group. It is worth mentioning that \u03c1(w) not only preserves vectors, but is generally grade-preserving. Using these two properties, Ruhe et al. (2023a) then introduce several linear, bilinear (through the geometric product), and nonlinear equivariant neural network layers. Composing these layers, we obtain a Clifford group equivariant neural network. 1We mean homogeneous with respect to parity. That is, multivectors that only have nonzero even grades (k = 0 mod 2) or nonzero odd grades. 3 \fPublished as a conference paper at ICLR 2024 Several other works that incorporate geometric algebra in deep learning are (Melnyk et al., 2021; Spellings, 2021; Brandstetter et al., 2023; Ruhe et al., 2023b). Finally, Brehmer et al. (2023) introduce the geometric algebra transformer, a method that uses the projective geometric algebra (Gunn, 2016) in combination with the transformer (Vaswani et al., 2017) architecture. 2.2 SIMPLICIAL COMPLEXES Definition 2.3 (Simplicial Complex). Let V be a finite set. An abstract simplicial complex K is a subset of the power set 2V that satisfies: 1. \u2200v \u2208V : {v} \u2208K; 2. \u2200\u03c3 \u2208K : \u2200\u03c4 \u2286\u03c3, \u03c4 \u0338= \u2205: \u03c4 \u2208K. In words, it is a space formed by a collection of subsets \u03c3 \u2286V , called simplices, which always including all the singletons, such that if \u03c3 \u2208K and \u03c4 \u2286\u03c3, then \u03c4 \u2208K as well. In other words, K is closed under taking subsets. For example, if a triangle (dimension 2 simplex) is in K, then all its edges and vertices (dimensions 1 and 0, respectively) are also in K. We also have dim \u03c3 := |\u03c3| \u22121 and dim K := max{dim \u03c3 | \u03c3 \u2208K}. Note that the 1-skeleton of a simplicial complex is a graph consisting of only vertices and edges, showing that simplicial complexes are a natural generalization of graphs. The act of creating a simplicial complex from a lower dimensional ones, like a vertex set, is called a lifting transformation. By constructing a rich adjacency structure on the simplicial complex, we can model interactions not only between vertices, but also between higher-dimensional simplices. E.g. a triangle in a simplicial complex can be thought of as a meta-vertex that represents the interaction between its three vertices. Bodnar et al. (2021) introduce message passing networks on simplicial complexes lifted from regular graphs. Exploring the connectivity within a simplicial complex, they identify the following adjacency types. Definition 2.4 (Simplicial Adjacencies). Let us define the boundary relation \u03c4 \u227a\u03c3 := \u03c4 \u2282\u03c3 \u2227 dim \u03c4 = dim \u03c3 \u22121. We define the following adjacency structures of a simplicial complex K: 1. Boundary adjacencies B(\u03c3) := {\u03c4 \u2208K | \u03c4 \u227a\u03c3}; 2. Coboundary adjacencies C(\u03c3) := {\u03c4 \u2208K | \u03c4 \u227b\u03c3}; 3. Lower adjacencies N\u2193(\u03c3) := {\u03c4 \u2208K | \u2203\u03b4 \u2208K : \u03b4 \u227a\u03c4, \u03b4 \u227a\u03c3}; 4. Upper adjacencies N\u2191(\u03c3) := {\u03c4 \u2208K | \u2203\u03b4 \u2208K : \u03b4 \u227b\u03c4, \u03b4 \u227b\u03c3}. For example, if \u03c3 is a triangle, then B(\u03c3) is the set of edges that make up the triangle. Lower adjacencies are simplices of the same dimensionality but with a common lower boundary, while upper adjacencies share a common upper boundary. Note that regular message passing uses only the upper adjacencies of nodes. Simplicial message passing networks satisfy the following theorems. Theorem 2.5 (Bodnar et al. (2021)). Simplicial Weisfeiler-Lehman (SWL) with a clique complex lifting is strictly more powerful than Weisfeiler-Lehman (WL), and is not less powerful than 3-WL. Theorem 2.6 (Bodnar et al. (2021)). Message Passing Simplicial Networks (MPSNs) with sufficient layers and injective neighborhood aggregators are as powerful as SWL. Moreover, MPSNs are strictly more powerful than WL. The Weisfeiler-Lehman algorithm (Leman & Weisfeiler, 1968) is a well-known graph isomorphism test. Here, a clique complex lift turns every clique in a graph into a simplex in a simplicial complex. We depict two isomorphic geometric graphs in Appendix C. 3 CLIFFORD GROUP EQUIVARIANT SIMPLICIAL MESSAGE PASSING NETWORKS 3.1 CREATING GEOMETRIC SIMPLICIAL COMPLEXES Starting with a finite set V , potentially part of a (geometric) graph G = (V, E), several ways exist to construct a simplicial complex K. For instance, we can form a simplicial complex by creating a 4 \fPublished as a conference paper at ICLR 2024 simplex for every possible subset of V . That is, we include all edges, triangles, tetrahedra, and so on. However, the resulting simplicial complex would have \u0000 |V | n+1 \u0001 number of n-simplices. One can easily see that this number can quickly become prohibitively big. As such, various methods exist to lift a set of points to a simplicial complex in a more tractable way. Examples are Vietoris-Rips, \u02c7 Cech, manual and algorithmic lifts. Furthermore, if we have access to a graph structure, we can use a clique lift. For a more detailed discussion of these, consider Appendix D. To create a Vietoris-Rips lift, we consider a node position xv \u2208X attached to V , together with a distance function d : X \u00d7 X \u2192R. Then, the Vietoris-Rips complex Vietoris-Rips(V, d, \u03f5) is the simplicial complex containing all simplices whose vertices are \u03f5-close for some \u03f5 > 0 (see Figure 5). The manual lift defines a simplicial complex by hand. For example, we can define the shape of a polyhedral object by its vertices, edges, and faces, making a simplicial complex also known as a mesh. Another alternative is to let expert knowledge guide us. Take, for example, the molecule of water H2O. The bond angle between the two hydrogen atoms, governed by triangular interaction involving all three atoms, is crucial for the molecule\u2019s properties. Finally, recent work has been done on algorithmically constructing simplicial complexes from data, for example, through the mapper procedure (Hajij et al., 2018). In this work, we use the Vietoris-Rips and manual lifts in our experiments. We typically cap the maximal simplex dimension to 2 for efficiency reasons. 3.2 EMBEDDING SIMPLICIAL DATA IN THE CLIFFORD ALGEBRA In the following, we elaborate on how we embed the usual scalar and vector features of each node v \u2208V in the Clifford algebra, as well as how we create simplex features. First, we have node features \u2200v \u2208V : hv \u2208Rk \u2295 \u0000Rd\u0001l. Here, let hv 1, . . . , hv k \u2208R denote scalar features, or invariants. Consider, for example, a particle\u2019s mass or charge in a physics setting. They transform trivially under the Clifford group action. On the other hand, let hv k+1, . . . , hv l \u2208Rd be vector features, such as position, velocity, and acceleration. They transform nontrivially under the Clifford group action. One of these is usually the position of the vertex, which (e.g., through distances) is used to construct a simplicial complex. Recall that the Clifford subspace Cl(0)(Rd, q) = R is the subspace of all scalars, and therefore we can embed our scalar features in this subspace. Cl(1)(Rd, q) = Rd is the vector subspace, and therefore we can embed our vector features here. Further, if we have access to higher-order features such as bivectors, we can embed them in the other subspaces of the Clifford algebra. As such, after embedding, we now denote \u2200v \u2208V : hv \u2208Cl(Rd, q)m, where m = k + l. We now consider a simplicial complex K lifted from V . Our goal is, analogously to the node features, to obtain a Clifford feature h\u03c3 for each simplex \u03c3 \u2208K. For the singletons {vi} \u2208K, we can directly put h{vi} := hvi. For the edges {vi, vj} \u2208K, we can put h{vi,vj} := hvihvj, denoting the geometric product of the two Clifford features. This process extends to triangles and higher-dimensional simplices, multiplying Clifford features of all vertices with geometric product. In Figure 1, we depict this embedding, illustrating how edge and triangle simplices relate to bivectorand trivector-valued features. Since there are multiple ways to embed simplices, we learn the embedding through Clifford group-equivariant layers, i.e., we take the Clifford simplicial features as input to learnable Clifford group-equivariant layers and use the outputs as the learnable Clifford simplicial features. These can be decomposed into parameterized linear combinations as well as parameterized geometric products, resulting in analogous embeddings to the ones described above, but including learnable parameters. To ensure permutation-invariant embeddings, one can aggregate permutations of geometric products. A more intricate approach involves passing messages between the vertices of a k-simplex. The aggregated readout is then used as the initialized feature for the k-simplex. Eijkelboom et al. (2023) take a similar approach as here, but they manually have to define the simplicial embeddings for all simplices. Specifically, distances, volumes, and angles between simplices are calculated. All such quantities can be expressed using linear combinations and geometric products (Doran & Lasenby, 2003), which are computed by Clifford group equivariant layers. Moreover, Eijkelboom et al. (2023) can only embed geometrically covariant information through relative positions, which are computed additively. In contrast, the use of geometric products and the fact that any polynomial in multivectors is Clifford group equivariant, we can also embed covariant information 5 \fPublished as a conference paper at ICLR 2024 0 0 0 1 1 1 2 0 0 0 1 1 1 2 Figure 2: Left: we show how a simple graph (three fully-connected nodes) is lifted to a simplicial complex. Using simplicial message passing, we allow communication between objects of different dimensions. That is, between vertices (0 \u21940) , nodes and edges (0 \u21921 and 1 \u21920) , edges (1 \u21941) , and between edges and triangles (1 \u21922 and 2 \u21921) . Right: same as left, but a top-down view. It illustrates the hypergraph associated with the complex with several meta-vertices representing the simplices of various dimensionality. Instead of running message passing separately for all different communication types, we share the parameters of a single neural network operating on the extended graph. By conditioning on the message type, it is still able to leverage the simplicial complex. extracted from three or more node positions multiplicatively. Since these networks generalize to inner-product spaces of any dimension, these notions also hold for Clifford algebras over higherdimensional spaces. 3.3 EQUIVARIANT SHARED SIMPLICIAL MESSAGE PASSING For all \u03c3 \u2208K, we now have a Clifford feature h\u03c3 \u2208Cl(Rd, q)m. We propose two techniques that enable efficient (equivariant) message passing on simplicial complexes. In doing so, we require access to a parameterized message function \u03d5m and an update function \u03d5h. First, the message m\u03c3 will be an equivariant aggregation of all information (processed by a neural network) from several adjacencies of different dimensions as defined in Definition 2.4 2. In contrast to, e.g., Eijkelboom et al. (2023); Bodnar et al. (2021), who iteratively run message passing for different adjacency types, we can leverage existing parallel implementations of classical message passing. In other words, we consider the adjacency matrix of the simplicial complex\u2019s corresponding hypergraph, where we have several meta-vertices that represent the different types of simplices. This corresponds to considering the 0-simplices of the barycentric subdivision of the simplicial complex, which is a common way for refining simplicial complexes (Ghrist, 2014). This idea is visualized in Figure 2. Algorithm 1 Shared Simplicial Message Passing Require: K, \u2200\u03c3 \u2208K : h\u03c3, \u03d5m, \u03d5h Repeat: m\u03c3 \u2190Agg \u03c4\u2208B(\u03c3) \u03c4\u2208C(\u03c3) \u03c4\u2208N\u2191(\u03c3) \u03c4\u2208N\u2193(\u03c3) \u03d5m(h\u03c3, h\u03c4, dim \u03c3, dim \u03c4) h\u03c3 \u2190\u03d5h (h\u03c3, m\u03c3, dim \u03c3) Secondly, instead of considering a different parameterization for each type of communication, we define a single message function \u03d5m that can handle all types of communication. However, by conditioning \u03d5m on the type of message, it can still leverage the simplicial complex. In doing so, we efficiently share parameters between different types of communication, which is in contrast with previous methods that defined a different neural network for each type of communication. 2Intriguingly, Bodnar et al. (2021) prove that only the boundary and upper adjacencies are required for full expressivity. However, we only use the boundary, coboundary, and upper adjacencies to keep consistent with Eijkelboom et al. (2023). 6 \fPublished as a conference paper at ICLR 2024 MSE (\u2193) MPNN (Gilmer et al., 2017) 0.212 GVP-GNN (Jing et al., 2021) 0.097 VN (Deng et al., 2021) 0.046 EGNN (Satorras et al., 2021) 0.011 CGENN (Ruhe et al., 2023a) 0.013 EMPSN (Eijkelboom et al., 2023) 0.007 CSMPN 0.002 Table 1: MSE (\u2193) of the tested models on the convex hulls experiment. Figure 3: In the convex hulls experiment, the task is to estimate the volume of the convex hull of eight five-dimensional random points. Here, we display a three-dimensional example, which is easier to visualize. To make the overall method equivariant, we utilize Clifford group equivariant neural networks from Ruhe et al. (2023a). Then, as long as the simplicial embedding, the aggregation operation (e.g., a summation), and \u03d5m and \u03d5h are equivariant, the overall method is Clifford group equivariant (see Appendix B). This then makes it equivariant to rotations, reflections, and other orthogonal transformations in any dimension. The algorithm is summarized in Algorithm 1. Note that it generalizes typical message passing, which only considers messages from the upper adjacencies between 0simplices, i.e. nodes. For a quick overview of how this compares to typical message passing and simplicial message passing, consider Appendix F. 4 EXPERIMENTS We selected a set of geometric experiments that involve different types of data from several domains and include both invariant and equivariant predictions. Currently, the QM9 (Ramakrishnan et al., 2014) and MD17 (Chmiela et al., 2017) datasets are highly popular benchmarks for equivariant graph neural networks. We found, however, that the state-of-the-art models incorporate a lot of domain knowledge, which is beyond the scope of the current work. Note that we ensure a fair comparison by maintaining a similar scale of parameters between CSMPN and the baseline models across all experiments. More experimental details than presented here can be found in Appendix E. 4.1 5D CONVEX HULLS We run this experiment based on the convex hull volumetric experiment of Ruhe et al. (2023a). We consider a five-dimensional space, where we sample eight points from a standard normal distribution. The task is to estimate the volume of the convex hull of these points. We give an example of the three-dimensional case in Figure 3. Note that this is an E(5)-invariant task. For the simplicial networks (EMPSN (Eijkelboom et al., 2023) and ours), we use the representation of the hull as a set of 4-simplices, and extract all their 1and 2-simplices. We then pass messages between these as well as the 0-simplices (the points). As shown in Table 1, CSMPN outperforms the other models, showing that in this setting the simplicial structure is a significantly improved way of presenting the data to the network. 4.2 CMU MOTION CAPTURE In this experiment, we evaluate our models on the CMU Human Motion Capture dataset (Gross & Shi, 2001). We demonstrate that CSMPN exhibits greater performance in motion prediction compared to equivariant architectures reliant on regular graphs. In agreement with, e.g., Huang et al. (2022), we use the 35th human subject of the dataset. Each graph in the dataset contains 31 equally connected nodes, with each representing a specific position on the human body during walking. The objective is to use the node positions of a random frame to predict the node positions after 30 timesteps. We manually lift the graph to a simplicial complex. Specifically, we connect elbow joint nodes with shoulder and palm nodes to create edges and triangles. Similarly, hip, knee, and heel nodes are linked, forming another set of edges and triangles. 7 \fPublished as a conference paper at ICLR 2024 Method MSE (\u2193) Radial Field (K\u00a8 ohler et al., 2020) 197.0 TFN (Thomas et al., 2018) 66.9 SE(3)-Tr (Fuchs et al., 2020) 60.9 GNN (Gilmer et al., 2017) 67.3 EGNN (200K) (Satorras et al., 2021) 31.7 GMN (200K) (Huang et al., 2022) 17.7 EMPSN (200K) 15.1 CGENN (200K) 9.41 CSMPN (200K) 7.55 8 6 4 2 0 2 10 8 6 4 5 10 15 20 25 30 T argets Predictions Table 2: Left: MSE (10\u22122) of the tested models on the CMU motion capture dataset. Right: Depiction (not cherry-picked) of an instance (the ground-truth target positions) vs. a CSMPN prediction. Aspirin Benzene Ethanol Malonaldehyde Radial Field (K\u00a8 ohler et al., 2020) 17.98 / 26.20 7.73 / 12.47 8.10 / 10.61 16.53 / 25.10 TFN (Thomas et al., 2018) 15.02 / 21.35 7.55 / 12.30 8.05 / 10.57 15.21 / 24.32 SE(3)-Tr (Fuchs et al., 2020) 15.70 / 22.39 7.62 / 12.50 8.05 / 10.86 15.44 / 24.47 EGNN (Satorras et al., 2021) 14.61 / 20.65 7.50 / 12.16 8.01 / 10.22 15.21 / 24.00 S-LSTM (Alahi et al., 2016) 13.12 / 18.14 3.06 / 3.52 7.23 / 9.85 11.93 / 18.43 NRI (Kipf et al., 2018) 12.60 / 18.50 1.89 / 2.58 6.69 / 8.78 12.79 / 19.86 NMMP (Hu et al., 2020) 10.41 / 14.67 2.21 / 3.33 6.17 / 7.86 9.50 / 14.89 GroupNet (Xu et al., 2022) 10.62 / 14.00 2.02 / 2.95 6.00 / 7.88 7.99 / 12.49 GMN-L (Huang et al., 2022) 9.76 / 48.12 / 4.83 / 13.11 / EqMotion (300K) (Xu et al., 2023) 5.95 / 8.38 1.18 / 1.73 5.05 / 7.02 5.85 / 9.02 EMPSN (300K) 9.53 / 12.63 1.03 / 1.12 8.80 / 9.76 7.83 / 10.85 CGENN (300K) 3.70 / 5.63 1.03 / 1.59 4.53 / 6.35 4.20 / 6.55 CSMPN (300K) 3.82 / 5.75 1.03 / 1.60 4.44 / 6.30 3.88 / 5.94 Table 3: ADE / FDE (10\u22122) (\u2193) of the tested models on the MD17 atomic motion dataset. We incorporated all baselines from Huang et al. (2022) and added EMPSN and CGENN ourselves. From Table 2, we note that EMPSN, the simplicial version of EGNN, already enjoys a significant boost from simplicial message passing. On top of this, Clifford layers further improve the performance, which CSMPN then again surpasses. We ensured that model sizes, in terms of the number of parameters, are comparable. 4.3 MD17 ATOMIC MOTION PREDICTION We turn to the molecular domain with the MD17 dataset Chmiela et al. (2017). However, we do not use the standard task of predicting the energy of a molecule, but instead we predict the motion of the atoms. In accordance with Han et al. (2022), we select four molecules: aspirin, benzene, ethanol, and malonaldehyde. The aim is to assess how well CSMPN can model molecular dynamics by directly predicting atom positions in future time steps. We take the starting positions of the heavy atoms from ten separate time frames for each molecule and predict their positions in the next ten frames. For aspirin, we use k-nearest neighbors with k = 3 to construct the regular graphs from the atom positions. The other molecules are fully connected. Since we have access to this graph structure, we use a clique complex lift to form the simplicial complex. The Average Displacement Error / Final Displacement Error (ADE / FDE) of the methods are displayed in Table 3. Here, ADE is the RMSE of the location predictions averaged over the number of 8 \fPublished as a conference paper at ICLR 2024 Attack Defense STGAT (Huang et al., 2019) 9.94 / 15.80 7.26 / 11.28 Social-Ways (Amirian et al., 2019) 9.91 / 15.19 7.31 / 10.21 Weak-Supervision (Zhan et al., 2019) 9.47 / 16.98 7.05 / 10.56 DAG-Net (200K) (Monti et al., 2020) 8.98 / 14.08 6.87 / 9.76 CGENN (200K) 9.17 / 14.51 6.64 / 9.42 CSMPN (200K) 8.88 / 14.06 6.44 / 9.22 Table 4: ADE / FDE (\u2193) of the tested models on the VUSport NBA player trajectory dataset. time-steps, and FDE the RMSE of the final time-step. We take all baselines from Xu et al. (2023) and included EMPSN and CGENN ourselves. Inspecting the results, we see that simplicial message passing yields a significant boost for EMPSN, the simplicial version of EGNN. Clifford layers then further improve the performance, the only exception being the case of benzene. We hypothesize that the restricted expressivity of EMPSN forms a good inductive bias here, since the molecule is rigid and planar. CSMPN achieves outstanding scores across the board, slightly surpassing clifford in the cases of ethanol and malonaldehyde. We ensured that the simplicial structure used in the comparison of EMPSN and CSMPN is equal, and that the number of parameters is comparable. Similarly, the architecture of the CGENN and CSMPN models is roughly identical. 4.4 NBA PLAYERS 2D TRAJECTORY PREDICTION Finally, we subject our CSMPN to testing on the STATS SportVU NBA Dataset (STATS Perform, 2023). This two-dimensional dataset contains the tracking positions of the NBA team players in regular seasons. Our preprocessing approach aligns with Monti et al. (2020), representing each player\u2019s position as a two-dimensional coordinate. Considering the distinct behaviors of players in offensive or defensive roles, we assess our model on these different contexts, utilizing ten observed time frames to predict the subsequent forty. Motion uncertainty and interactions between players make the task challenging. Each player\u2019s position is represented as a node in our graph, each node is connected to all the other nodes in the graph to make the regular graph fully-connected. We also include one fixed reference point in our graphs indicating the orientation of the basketball court. We create the simplicial complex using Vietoris-Rips with infinite \u03f5 with a maximum simplex dimension of 2. We compare our CSMPN with baselines provided by Monti et al. (2020) as well as CGENN and present the results in Table 4. 5"
+                }
+            ],
+            "Abiy Tasissa": [
+                {
+                    "url": "http://arxiv.org/abs/2104.13894v1",
+                    "title": "Weighed $\\ell_1$ on the simplex: Compressive sensing meets locality",
+                    "abstract": "Sparse manifold learning algorithms combine techniques in manifold learning\nand sparse optimization to learn features that could be utilized for downstream\ntasks. The standard setting of compressive sensing can not be immediately\napplied to this setup. Due to the intrinsic geometric structure of data,\ndictionary atoms might be redundant and do not satisfy the restricted isometry\nproperty or coherence condition. In addition, manifold learning emphasizes\nlearning local geometry which is not reflected in a standard $\\ell_1$\nminimization problem. We propose weighted $\\ell_0$ and weighted $\\ell_1$\nmetrics that encourage representation via neighborhood atoms suited for\ndictionary based manifold learning. Assuming that the data is generated from\nDelaunay triangulation, we show the equivalence of weighted $\\ell_1$ and\nweighted $\\ell_0$. We discuss an optimization program that learns the\ndictionaries and sparse coefficients and demonstrate the utility of our\nregularization on synthetic and real datasets.",
+                    "authors": "Abiy Tasissa, Pranay Tankala, Demba Ba",
+                    "published": "2021-04-28",
+                    "updated": "2021-04-28",
+                    "primary_cat": "eess.SP",
+                    "cats": [
+                        "eess.SP",
+                        "cs.IT",
+                        "cs.LG",
+                        "math.IT",
+                        "math.OC"
+                    ],
+                    "main_content": "Introduction The compressive sensing (CS) problem considers the recovery of a sparse vector x \u2208Rm given d undetermined measurements y = Ax. The optimization problem is given by min x\u2208Rm ||x||0 s.t. y = Ax, (1) where ||\u00b7||0 denotes the \u21130 norm. However, the \u21130 minimization is known to be intractable and a common approach is a convex relaxation based on \u21131 minimization. min x\u2208Rm ||x||1 s.t. y = Ax. (2) CS theory shows that the \u21131 relaxation exactly recovers the underlying sparse solution assuming certain conditions on A such as the restricted isometry property (RIP) or the coherence condition (Candes et al., 2006). RIP is known to hold with high probability for random matrices and the mutual coherence can be employed for deterministic matrices (Donoho and Elad, 2003; Gribonval and Nielsen, 2003). We highlight two limitations of the standard CS. The \ufb01rst is the assumption of a \ufb01xed measurement matrix which is constraining in cases where an optimal prede\ufb01ned measurement matrix is a priori unavailable. The more general sparse coding framework learns suitable dictionaries from data adapted to the task at hand (Engan et al., 2000; Aharon et al., 2006; Elad and Aharon, 2006; Jiang et al., 2013). Second, measurement matrices do not always satisfy RIP or coherence condition which are properties leveraged to guarantee the recovery of sparse solutions. In this paper, we illustrate these limitations in the context of the manifold learning problem. Sparse subspace clustering (Elhamifar and Vidal, 2013) is a method to cluster data that lie on the union of manifolds. It relies on the principle of self-representation, which expresses a given example as a sparse combination of its neighbors. A recent work (Tankala et al., 2020) utilizes dictionary learning and substitutes self representation with sparse convex combinations of dictionary columns. In this scalable approach, the size of the dictionary depends on intrinsic dimensions of data. In both frameworks, the dictionary atoms are in fact correlated and standard CS theory assuming RIP or coherence is not applicable. In addition, the \u21131 regularization to promote sparsity can be geometrically oblivious. For instance, a common step in manifold learning is estimate of local geometry by reconstructing a point from nearby pointsRoweis and Saul (2000). With that, a sparse reconstruction of a point that uses far away dictionary atoms is sub-optimal. Related work: Given the mentioned limitations of standard CS, the works in (Elhamifar and Vidal, 2011; Tankala et al., 2020) use a weighted \u21131 penalty which take into account the proximity of dictionary atoms. Our work is in the spirit of weighted \u21131 minimizations and structured compressive sensing (Khajehnejad et al., 2009; Vaswani and Lu, 2010; Friedlander et al., 2011; Pilanci et al., 2012). We note that the weighted \u21131 regularization employed in this paper resembles a Laplacian smoothness term (Dornaika and Weng, 2019; Cai et al., 2010). The proposed weighted \u21131 regularization in its exact form has also been used in (Zhong and Pun, 2020). Contributions: We propose weighted \u21130 and \u21131 metrics that account for locality and allow a representation of a point using neighborhood points. Under a certain generative model of the points, we show that the \u21130 and \u21131 problem are equivalent. To our knowledge, the analysis of the weighted \u21130 and weighted \u21131 metrics and applications to manifold learning is a novelty of this work. 2 Proposed Method We consider m landmark points a1, a2, ..., am with a unique Delaunay triangulation (Lee and Schachter, 1980). In this setting, each point in the set {yi}n i=1 \u2208Rd is generated from a convex combination of at most 1 arXiv:2104.13894v1  [eess.SP]  28 Apr 2021 \fd + 1 atoms. Figure 1 shows an example with d = 2 i.e. data points in R2. Compactly, we have yi = Axi where the d \u00d7 m matrix A is the dictionary de\ufb01ned as A = [a1, a2, ..., am] and xi \u2208Rm is the coe\ufb03cient vector xT i = \u0002xi1 xi2 . . . xim \u0003 such that xij \u22650 for all j and Pm j=1 xij = 1 i.e. the coe\ufb03cient vector xi is supported on the probability simplex in Rm. a1 a2 a3 a4 a5 a9 a10 a12 a11 a6 a7 a8 Figure 1: The red dots indicate the atoms which generate the data points. Each black dot, denoting a data point, is a convex combination of three atoms which are vertices of the triangle the point belongs to. We show that the optimal weighted \u21130 and \u21131 metric is based on a point representing itself using the vertices of the triangle it belongs to. Our aforementioned goal of representing by local dictionaries motivates the following de\ufb01nition of a weighted \u21130 metric. De\ufb01nition 1. Assume m landmark points a1, a2, ..., am in Rd with a unique Delaunay triangulation. Let the point z \u2208Rd have the following representation, z = Pm i=1 xiai with the coe\ufb03cient vector x supported on the probability simplex in Rm. The weighted \u21130 metric is de\ufb01ned as \u2113W,0(x) = m X i=1 1xi\u0338=0||z \u2212ai||2, (3) where 1xi\u0338=0 denotes the indicator function. 1xi\u0338=0 takes a value of 1 if xi > 0 and is 0 otherwise. Given the above de\ufb01nition of a weighted \u21130 metric and the fact that a given point z admits di\ufb00erent representations as a convex combination of the dictionary atoms, the natural question is the sense in which this metric is minimal i.e. among the di\ufb00erent representations, which ones admit minimal values in this metric? The following theorem shows that the local reconstruction is minimal in the weighted \u21130 metric. Theorem 1. Let y \u2208Rd lie in the convex hull of the points a1, . . . , am \u2208Rd with a unique a unique Delaunay triangulation. Consider the following \u2113W,0 minimization problem. min x\u2208Rm \u2113W,0(x) s.t. y = Ax. (4) The optimal solution x\u2217to the above program is such that x\u2217 j \u0338= 0 comprise the vertices of some face of this triangulation that contains y. Proof. Consider an arbitrary Delaunay cell containing y de\ufb01ned by the vertices {aj : j \u2208S, |S| = k} with S \u2286[m] denoting a set of at most d + 1 indices. Let x be a k-sparse feasible solution of the program with support S\u2032. It follows that X i\u2208S\u2032 1xi\u0338=0||y \u2212ai||2 > X i\u2208S 1xi\u0338=0||y \u2212ai||2. (5) Above, the inequality follows from the fact that atoms within S are the closest to the point y. Therefore, the k-sparse representation using the vertices in S is the optimal solution to the \u2113W,0 minimization problem. Motivated by the fact that the weighted \u21130 metric enforces local reconstructions and with the goal of obtaining a regularization amenable to optimization, we de\ufb01ne a convex relaxation of the weighted \u21130 problem. De\ufb01nition 2. Assume m landmark points a1, a2, ..., am in Rd with a unique Delaunay triangulation. Let the point z \u2208Rd have the following representation, z = Pm i=1 xiai with the coe\ufb03cient vector x supported on the probability simplex in Rm. The weighted \u21131 metric is de\ufb01ned as \u2113W,1(x) = m X i=1 xi||z \u2212ai||2, (6) Analogous to CS theory, the next question is the sense in which a weighted \u21131 minimization is equivalent to a weighted \u21130 minimization problem. This equivalency is summarized in the theorem below. Theorem 2. Let y \u2208Rd lie in the convex hull of the points a1, . . . , am \u2208Rd, and let x \u2208Rm be a solution to the following optimization problem. 2 \fminimize x\u2208Rm X j xj\u2225y \u2212aj\u22252 subject to y = X j xjaj X j xj = 1 xj \u22650, for all j. (7) If the points a1, . . . , am have a unique Delaunay triangulation, then the set of points aj such that xj \u0338= 0 comprise the vertices of some face of this triangulation that contains y. Proof. It su\ufb03ces to show that if a Delaunay cell of a1, . . . , am contains y, then each point aj such that xj \u0338= 0 is a vertex of this cell. To this end, consider an arbitrary Delaunay cell containing y de\ufb01ned by the vertices {aj : j \u2208S}. Here, S \u2286[m] denotes a set of d + 1 indices. We will prove that xj \u0338= 0 only if j \u2208S. Since y lies in the Delaunay cell, we may write it as a convex combination of only the vertices {aj : j \u2208S} using a coe\ufb03cient vector x\u2032 \u2208Rm, which may di\ufb00er from the solution x to the optimization problem. Observe that for any vector c \u2208Rd, the identities y = P j xjaj and P j xj = 1 imply X j xj\u2225y \u2212aj\u22252 = X j xj(\u2225aj \u2212c\u22252 \u22122\u27e8y \u2212c, aj \u2212c\u27e9+ \u2225y \u2212c\u22252) = X j xj\u2225aj \u2212c\u22252 \u2212\u2225y \u2212c\u22252. The above identity also holds if we replace xj with x\u2032 j. By the de\ufb01nition of a Delaunay cell, there is a sphere with center c and radius r such that \u2225aj \u2212c\u2225= r for each j \u2208S and \u2225aj \u2212c\u2225> r for each j / \u2208S (this is where we use the assumption that the triangulation is unique). Since the support of x\u2032 is contained in S and the support of x is not, we have X j x\u2032 j\u2225y \u2212aj\u22252 = X j x\u2032 j\u2225aj \u2212c\u22252 \u2212\u2225y \u2212c\u22252 < X j xj\u2225aj \u2212c\u22252 \u2212\u2225y \u2212c\u22252 = X j xj\u2225y \u2212aj\u22252, which contradicts the optimality of x. Thus, each point aj such that xj \u0338= 0 is a vertex of the cell. We conclude that {aj : xj \u0338= 0} are the vertices of some face of the triangulation containing y. 3 Dictionary Learning Algorithm The utility of the proposed model employing weighted \u21131 regularization discussed in the previous section depends on the generative dictionary. In practice, the dictionary A is not available and needs to be estimated from the input data points. Let Y = [y1, . . . , yn] \u2208Rd\u00d7n be a set of n data points in Rd. For a data point y \u2208Rd, we consider the following minimization problem. min A\u2208Rm\u00d7d,x\u2208Rm 1 2\u2225y \u2212Ax\u22252 + \u03bb m X j=1 xj\u2225y \u2212aj\u22252. (8) Let   L(A, y, x) = 1 2\u2225y \u2212Ax\u22252 + \u03bb Pm j=1 xj\u2225y \u2212aj\u22252 if x \u2208S and \u221eotherwise with S = {x \u2208Rm : x1, . . . , xm \u22650 and Pm j=1 xj = 1} denoting the probability simplex in Rm. We note that the parameter \u03bb balances the reconstruction loss with the weighted \u21131 regularization and determines the sparsity of x. If the dictionary is \ufb01xed, the resulting problem is a weighted \u21131 minimization problem and can be e\ufb03ciently solved (Salman Asif and Romberg, 2012). However, given a \ufb01xed coe\ufb03cient, learning the dictionary is a nontrivial task. We have proposed the K-Deep Simplex (KDS) algorithm for this purpose in an earlier work (Tankala et al., 2020). Given data, a classical method to learn the dictionary alternates between sparse approximation and dictionary update step (Agarwal et al., 2016). KDS utilizes this along with algorithm unrolling (Monga et al., 2019; Gregor and LeCun, 2010; Hershey et al., 2014), a framework to recast a structured optimization framework into a neural network, to solve the optimization problem in (8). We develop an autoencoder architecture to solve the optimization problem. We summarize the main steps of the KDS algorithm below and refer the interested reader to (Tankala et al., 2020) for details. Encoder: We employ the accelerated projected gradient descent (Bubeck, 2015) to compute the optimal coe\ufb03cient for x\u2217(A, y) \u2208argmin x   L(A, y, x). (9) Starting with initialization x(0) = \u02dc x(0) = 0, the algorithm updates can be written as follows x(t+1) = PS \u0010 \u02dc x(t) \u2212\u03b1\u2207x  L(A, y, \u02dc x(t)) \u0011 (10) \u02dc x(t+1) = x(t+1) + \u03b3(t)(x(t+1) \u2212x(t)), (11) 3 \fFigure 2: Circle and two moons. Autoencoder input (\ufb01rst and third) and output (second and fourth), with learned atoms marked in red. for 0 \u2264t \u2264T. The parameter \u03b1 is a step size, and the constants \u03b3(t) are given by the recurrence \u03b7(0) = 0, \u03b7(t+1) = 1 + p 1 + 4\u03b7(t) 2 , \u03b3(t) = \u03b7(t) \u22121 \u03b7(t+1) . (12) The gradient of the loss function   L is given by \u2207x  L(A, y, x) = A\u22a4(Ax \u2212y) + \u03bb m X j=1 \u2225y \u2212aj\u22252ej. (13) The operator PS projects onto S, the probability simplex and has the following form (Wang and CarreiraPerpin\u00b4 an, 2013) PS(x) = ReLU(x + b(x) \u00b7 1), (14) with b : Rm \u2192R denoting a piecewise-linear bias function. In summary, the approximate optimal code x(T )(A, y) \u2248x\u2217(A, y) is an output of a recurrent encoder with input y, weights A, and activation function PS. Decoder and Backward Pass : Given A and x, the decoder approximately reconstructs the input y by computing \u02c6 y = Ax(T ). The \ufb01nal step it to \ufb01nd the optimal network weights by solving A\u2217\u2208 argmin A 1 n n X i=1   L(A, yi, x\u2217(A, yi)). We minimize 1 n n X i=1   L(A, yi, x(T )(A, yi)) by backpropagation through the autoencoder. An advantage of this optimization framework is that the encoding and decoding steps are amenable to GPU parallelizations across the data points. 4 Numerical Experiments In this section, we show that the proposed regularization is useful for the manifold learning task. We \ufb01rst consider one-dimensional manifolds in R2. Figure 2 shows two such data sets. The \ufb01rst data is a unit circle in R2. The second data we consider is the classic two moon data set (Ng et al., 2001). The latter dataset consists of two disjoint semicircular arcs. To test the robustness of the proposed regularization and the KDS algorithm, a small Gaussian white noise is added to each data point. Figure 2 shows the results of training the autoencoder on these data sets. Using the weighted \u21131 metric and employing the KDS algorithm, each data point is accurately represented as sparse convex combinations of neighborhood atoms. In addition, the learned atoms are geometrically signi\ufb01cant and can be interpreted as the dictionary that generate the points up to additive noise. 4.1 Application: Clustering The learned sparse coe\ufb03cients can be used for downstream tasks. Here, we illustrate their use for the clustering task and compare our results to competitive clustering methods. The \ufb01rst is the k-means (KM) algorithm (Lloyd, 1982) which is a centroid based clustering algorithm. The second is the sparse manifold clustering and embedding (SMCE) algorithm proposed in (Elhamifar and Vidal, 2011). Therein, the authors also consider a proximity regularization. However, the dictionary is constituted from all data points. In contrast, our method learns few atoms from the input data. We evaluate the algorithms on two datasets. The \ufb01rst is a noisy two moon dataset. The second is the MNIST dataset (LeCun et al., 1998) which is a database of 10 di\ufb00erent digits each represented as a 28 \u00d7 28 grayscale image. We consider a subset of the data by considering 5 digits, 0, 3, 4, 6 and 7. For KDS and SMCE, we run spectral clustering (Ng et al., 2001) on a similarity graph derived from the coe\ufb03cients. Table 1 summarizes the results. We note that the proposed algorithm achieves the best accuracy over the baselines. As important as the accuracy is the fact that the clustering is performed on a sparse similarity graph. This ensures that KDS is fast and makes it an e\ufb03cient scalable algorithm for large datasets. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2101.03197v1",
+                    "title": "Deep Diffusion Processes for Active Learning of Hyperspectral Images",
+                    "abstract": "A method for active learning of hyperspectral images (HSI) is proposed, which\ncombines deep learning with diffusion processes on graphs. A deep variational\nautoencoder extracts smoothed, denoised features from a high-dimensional HSI,\nwhich are then used to make labeling queries based on graph diffusion\nprocesses. The proposed method combines the robust representations of deep\nlearning with the mathematical tractability of diffusion geometry, and leads to\nstrong performance on real HSI.",
+                    "authors": "Abiy Tasissa, Duc Nguyen, James Murphy",
+                    "published": "2021-01-08",
+                    "updated": "2021-01-08",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV",
+                        "cs.LG",
+                        "eess.IV"
+                    ],
+                    "main_content": "Introduction Machine learning has provided revolutionary new tools for remote sensing, but state-of-the-art methods often require huge labeled training sets. In particular, supervised deep learning methods can achieve near-perfect labeling accuracy on high-dimensional hyperspectral images (HSI), provided large libraries of labeled pixels are available [1]. This hinders the practicality of these methods, as in many settings, data is collected at a pace that far exceeds human ability to generate corresponding labeled training data. In order to account for this, methods that require only a very small number of labels are needed. The active learning regime is particularly attractive for HSI labeling problems. In active learning, an algorithm is provided with an unlabeled dataset, and the algorithm iteratively queries points for labels. By choosing query points intelligently, the active learning algorithm can yield the classi\ufb01cation performance of a much larger training set chosen uniformly at random. We propose an active learning method for HSI based on deep feature extraction and random walks on graphs. First, an unsupervised variational autoencoder is used to nonlinearly denoise and compress the highdimensional HSI. Then, the resulting features are considered as vertices of a graph, and a Markov di\ufb00usion process on the graph is used to determine label queries and label all data points. The proposed method combines the e\ufb03cient feature learning of deep autoencoders with the mathematical interpretability of graph di\ufb00usion processes, and leads to strong empirical performance on real HSI. 2 Background 2.1 Variational Autoencoders In an autoencoder architecture, input data is cascaded through nonlinear layers to obtain a latent representation. The latent representation is then cascaded through nonlinear layers to obtain output data. These two stages respectively de\ufb01ne the encoder and decoder. Typically, a loss function that enforces the reconstructed output to be similar to the input is minimized and the trained autoencoder learns a low-dimensional latent feature useful for downstream tasks. In contrast to the autoencoder, in the variational autoencoder (VAE) [2], the output of the encoder is not a deterministic map but parameters of a distribution. In particular, an encoding network maps x \u2208RN and obtains parameters of the latent variable distribution q(z|x). A latent feature z sampled from this distribution is an input to a decoder that outputs \u02c6 x \u223cp(x|z). A typical prior for the distribution of the latent variable is a Gaussian random variable p(z) \u223cN(0, I). Given this, the VAE optimization consists of two terms: (i) a reconstruction loss L1 = Ez\u223cq(z|x) log(p(x|z)) that enforces that the reconstructed output \u02c6 x is similar to the input x; and (ii) a Kullback-Leibler divergence loss L2 = KL(q(z|x), N(0, I)) that enforces that q(z|x) agrees with p(z). The encoder and decoder are jointly trained by maximizing the total loss L1 + L2. 2.2 Learning by Active Nonlinear Di\ufb00usion The active learning algorithm employed in this paper is based on the ideas in [3, 4, 5]. In [6], the authors propose a semisupervised algorithm, learning by active nonlinear di\ufb00usion (LAND), that obtains the most important data points to query for labels. Important features of LAND are (i) it is a principled algorithm with provable performance guarantees; (ii) it accounts for nonlinear clusters possibly in high dimensions; and (iii) it is robust to noise and outliers [6]. We represent an HSI as X = {xi}n i=1 \u2282RN where each pixel is a point in RN where N is the number of spectral bands. Let NNk(xi) denote the set of k-nearest neighbors of xi in X using the Euclidean distance metric. The n\u00d7n weight matrix W is de\ufb01ned as Wij = exp(\u2212\u2225xi \u2212xj\u22252 2/\u03c32), xj \u2208NNk(xi) with \u03c3 denoting \u2217This research is partially supported by the US National Science Foundation grants NSF-DMS 1912737, NSF-DMS 1924513, and NSF-CCF 1934553. 1 arXiv:2101.03197v1  [cs.CV]  8 Jan 2021 \fa scale parameter. With this, the notion of the degree of xi naturally follows as deg(xi) := P xj\u2208X Wij. To de\ufb01ne a random walk on X, we employ the n\u00d7n transition matrix Pij = Wij \u000e deg(xi). It can be easily veri\ufb01ed that P has a spectral decomposition {(\u03bb\u2113, \u03a8\u2113)}n \u2113=1. The di\ufb00usion distance at time t between xi, xj \u2208X is de\ufb01ned as Dt(xi, xj) = pPn \u2113=1 \u03bb2t \u2113(\u03a8\u2113(xi) \u2212\u03a8\u2113(xj))2. We note that t tells us how long the di\ufb00usion process runs. In this paper, we use t = 30 for experiments. The main part of the LAND algorithm is to identify points to query for labels. LAND uses a kernel density estimator (KDE) and di\ufb00usion geometry for this task. In particular, the KDE is de\ufb01ned as p(x) = P y\u2208NNk(x) exp(\u2212\u2225x \u2212y\u22252 2/\u03c32 0) with \u03c30 denoting a scale parameter. For x \u2208X, let \u03c1t(x) = \uf8f1 \uf8f2 \uf8f3 min p(y)\u2265p(x),x\u0338=y Dt(x, y), x \u0338= argmax z p(z), max y\u2208X Dt(x, y), x = argmax z p(z), (1) be the di\ufb00usion distance to the nearest neighbor of higher density. The maximizers of Dt(x) = p(x)\u03c1t(x) are queried for labels. These labels are propagated to other data points by proceeding from high to low density and assigning each unlabeled point the same label as its Dt-nearest neighbor of higher density that is labeled; see Algorithm 1 and [6] for details. 2.3 Related Work In recent years, deep generative methods, such as generative adversarial network (GANs) and variational autoencoder networks (VAEs), have been used for feature extraction in many machine learning tasks [7, 8]. In the context of clustering, a set of methods, known as deep clustering, propose learning features of the data and clustering simultaneously, showing strong empirical results [9, 10, 11]. For HSI images, several works have employed di\ufb00erent deep learning architectures to extract essential features for downstream tasks such as classi\ufb01cation [12, 13, 14, 15, 16]. Active learning is a learning paradigm where the user has the ability to select the training data [17, 18]. The underlying idea is that a few informative training samples could be su\ufb03cient for training an algorithm and obtaining accurate results. This framework has been used in remote sensing for HSI image classi\ufb01cation [19, 20, 21, 22]. The main idea in this paper is that the active learning process depends on the representation and geometry of the data. The closest work to ours is [23] where the authors combine active learning with VAEs. Therein, K-means clustering is \ufb01rst used to partition the space and then labels are acquired using uniform random sampling in each partition. Given the labels, a classi\ufb01er is then trained in the latent space for the prediction task. One of the highlights of the proposed method is that the clustering algorithm LAND handles a broader class of cluster geometries than K-means does. We note that in contrast to the similar work [24], our method is in the active learning framework and the feature extraction and di\ufb00usion process via LAND are decoupled. 3 Proposed Algorithm We propose an active learning algorithm, VAE-LAND (see Algorithm 1), which has two main stages. The \ufb01rst stage is feature extraction of an unlabeled high-dimensional dataset using a VAE. The second stage employs the LAND algorithm to infer the true labels. The proposed algorithm combines the power of VAEs to extract features with di\ufb00usion geometry on graphs to \ufb01nd impactful labels to query, which then propagate to other points. Algorithm 1: Variational Autoencoder Learning by Active Nonlinear Di\ufb00usion (VAE-LAND) Input: {xi}n i=1 (Unlabeled Data); t (Time Parameter); B (Budget); O (Labeling Oracle) Output: Y (Labels) 1: Run VAE on unlabeled data to obtain the latent representation {\u02c6 xi}n i=1. 2: Compute P and {(\u03bb\u2113, \u03c8\u2113)}M \u2113=1 using {\u02c6 xi}n i=1. 3: Compute kernel density estimate {p(\u02c6 xi)}n i=1 and {\u03c1t(\u02c6 xi)}n i=1 (1); 4: Compute Dt(\u02c6 xi) = p(\u02c6 xi)\u03c1t(\u02c6 xi). 5: Sort the data in decreasing Dt value to acquire the ordering {\u02c6 xmi}n i=1. 6: for i = 1 : B do 7: Query O for the label L(\u02c6 xmi) of \u02c6 xmi. 8: Set Y (\u02c6 xmi) = L(\u02c6 xmi). 9: end for 10: Sort X according to p in decreasing order as {\u02c6 x\u2113i}n i=1. 11: for i = 1 : n do 12: if Y (\u02c6 x\u2113i) = 0 then 13: Y (\u02c6 x\u2113i) = Y (zi), zi = argmin z {Dt(z, \u02c6 x\u2113i) | p(z) > p(\u02c6 x\u2113i) and Y (z) > 0}. 14: end if 15: end for 2 \f4 Experimental Results We demonstrate the accuracy of the proposed algorithm experimentally. The training of the VAE is done using Tensor\ufb02ow in Python. For doing the active learning via LAND, we use the publicly available MATLAB code at https://jmurphy.math.tufts.edu/Code/. Our code can be found at https://github.com/abiy-tasissa/ VAE-LAND. Our test HSI dataset is the Salinas A hyperspectral dataset. The Salinas scene was captured over Salinas Valley, California. The image has a spatial resolution of 3.7-meter pixels and contains 224 spectral bands. The ground truth consists of 16 classes. We consider the Salinas A dataset, which is a subset of the Salinas dataset, and contains 6 classes. The Salinas A dataset and the ground truth data are publicly available (http://www.ehu.eus/ccwintco/index.php/Hyperspectral Remote Sensing Scenes#Salinas-A scene). Figure 1 shows a visual of the high-dimensional data and the ground truth labels. The performance of the algorithm is assessed using overall accuracy, de\ufb01ned as the ratio of correctly estimated labels to total number of labels after optimally aligning with the ground truth. The Salinas A HSI dataset of size Figure 1: The 86 \u00d7 83 Salinas A HSI data consists of 6 classes. Left: the sum of all spectral bands. Right: the ground truth. Figure 2: A schematic of the VAE architecture. Input is a vector in R224. The encoder and decoder are fully connected neural networks. Both have three layers with 128 neurons in each layer. For all layers, the activation function is the recti\ufb01ed linear unit (ReLU). The input is cascaded through an encoder. The extracted latent feature in R40 is then cascaded through the decoder to obtain the output vector in R224. Unlike the standard autoencoder, the extraction of the latent feature is not deterministic (see Section 2.1 for discussion.) 83 \u00d7 86 \u00d7 224 is represented as a point cloud of size 7138 \u00d7 224. We use the unlabeled data to learn a latent space representation of Salinas A in R40 dimensions. A schematic of the VAE architecture is shown in Figure 2. We optimize the VAE loss function using the Adam algorithm with learning rate set to 10\u22124. After training the VAE, we input the optimal latent space representation of the Salinas A dataset to the LAND algorithm for the task of inferring the ground truth labels of the HSI data. We compare our result to the standard LAND algorithm that labels the Salinas A dataset in its original representation. Since LAND is an active learning framework, we consider varying number of labeled data points ranging from 10 to 2000. In addition, we compare the active learning methods to query the samples with randomly selected training data. Figure 1 compares the performance of LAND and performance of VAE-LAND. First, for both VAELAND and standard LAND, LAND queries lead to signi\ufb01cantly better accuracy than random queries. The proposed algorithm, VAE-LAND, attains an accuracy of 96.97% with just 10 labeled points. This is a 12.5% improvement to the accuracy of the standard LAND algorithm for the same number of labeled points. The standard LAND algorithm requires 400 labeled points to reach accuracy of 90% while for the same number of labeled points, VAE-LAND has an accuracy of 98.35%. Complexity and run time: The complexity of LAND is O(CNN + nKNN + n log(n)) where CNN is the cost of computing all KNN nearest neighbours [6]. The computational cost of VAE is di\ufb03cult to estimate as it depends on several factors (e.g architecture, activation function, choice of SGD algorithm). In our numerical experiments, the cost of VAE is the dominating cost. Since LAND runs on low-dimensional features extracted from VAE, it is e\ufb03cient. 5"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2006.09534v3",
+                    "title": "Towards improving discriminative reconstruction via simultaneous dense and sparse coding",
+                    "abstract": "Discriminative features extracted from the sparse coding model have been\nshown to perform well for classification. Recent deep learning architectures\nhave further improved reconstruction in inverse problems by considering new\ndense priors learned from data. We propose a novel dense and sparse coding\nmodel that integrates both representation capability and discriminative\nfeatures. The model studies the problem of recovering a dense vector\n$\\mathbf{x}$ and a sparse vector $\\mathbf{u}$ given measurements of the form\n$\\mathbf{y} = \\mathbf{A}\\mathbf{x}+\\mathbf{B}\\mathbf{u}$. Our first analysis\nproposes a geometric condition based on the minimal angle between spanning\nsubspaces corresponding to the matrices $\\mathbf{A}$ and $\\mathbf{B}$ that\nguarantees unique solution to the model. The second analysis shows that, under\nmild assumptions, a convex program recovers the dense and sparse components. We\nvalidate the effectiveness of the model on simulated data and propose a dense\nand sparse autoencoder (DenSaE) tailored to learning the dictionaries from the\ndense and sparse model. We demonstrate that (i) DenSaE denoises natural images\nbetter than architectures derived from the sparse coding model\n($\\mathbf{B}\\mathbf{u}$), (ii) in the presence of noise, training the biases in\nthe latter amounts to implicitly learning the $\\mathbf{A}\\mathbf{x} +\n\\mathbf{B}\\mathbf{u}$ model, (iii) $\\mathbf{A}$ and $\\mathbf{B}$ capture low-\nand high-frequency contents, respectively, and (iv) compared to the sparse\ncoding model, DenSaE offers a balance between discriminative power and\nrepresentation.",
+                    "authors": "Abiy Tasissa, Emmanouil Theodosis, Bahareh Tolooshams, Demba Ba",
+                    "published": "2020-06-16",
+                    "updated": "2022-12-14",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "cs.LG",
+                        "eess.SP",
+                        "math.IT"
+                    ],
+                    "main_content": "Introduction The problem of representing data as a linear mixture of components \ufb01nds applications in many areas of signal and image processing [7, 38]. Variational partial di\ufb00erential equations (PDE) approaches in [4, 62] develop an image decomposition model whereby a target image is decomposed into a piecewise smooth component (cartoon) and a periodic component (texture). Another approach is based on the idea that a desirable model would have multiple matrices each capturing di\ufb00erent structures of the image e.g. \u03a61 is a matrix modeling texture and \u03a62 is a matrix modeling cartoon. Along these methods is Morpohological Component Analysis (MCA) [25] which utilizes pre-speci\ufb01ed dictionaries and a a sparsity prior on the representation coe\ufb03cients. In the simplest setting, an image y is modeled in MCA as y \u2248P2 k=1 \u03a6kxk where \u03a61, \u03a62 are the pre-speci\ufb01ed structured dictionaries capturing di\ufb00erent morphologies of the image. The MCA model is then based on the idea that the sparsest representation of each component is attained in the pre-set dictionary component. In contrast to MCA which imposes sparsity for all mixture coe\ufb03cients, the proposed model in this paper adopts a combination of smooth and sparse regularization. Related motivating problems for representing data as a linear mixture are the background/foreground separation problem [67] and anomaly detection in images [16]. In the former problem, a widely used approach is the robust principal component analysis (RPCA) algorithm [15] which is based on decomposing a data matrix into low rank (background) and sparse (foreground) components. The latter problem, anomaly detection with images, arises in most manufacturing processes and methods based on decomposing the images into smooth and sparse components have been studied [64, 50]; however, no theoretical analysis has been conducted for that decomposition. In addition, in image anomaly detection the sparse and smooth components are localized, inhibiting the generality of the model. In this paper, we study a model which decomposes a signal or image in consideration into a smooth part, sparse part, and noise. The prototypical form of our model is y = Ax+Bu+e where y \u2208Rm is the measured 1 arXiv:2006.09534v3  [cs.IT]  14 Dec 2022 \fsignal, Ax is the smooth component, Bu is the sparse component and e is noise. The smooth component is generated from the dictionary A \u2208Rm\u00d7p using a smooth coe\ufb03cient x. In turn, the sparse component is generated from the dictionary B \u2208Rm\u00d7n using a sparse vector u. We use Tikhonov regularization [57] and \u21131 regularization to promote smoothness and sparsity, respectively. With that, the general non-noisy problem we study in this paper is min x\u2208Rp,u\u2208Rn ||Gx||2 2 + ||u||1 s.t. y = Ax + Bu, (1.1) where G is the Tikhonov regularization operator. Here on, we refer to the model in (1.1) as the dense and sparse coding problem. The word dense here refers to the smooth part and is used to emphasize that we do not require any sparsity. Both MCA and the dense and sparse coding problem naturally lend themselves to and share a common theme with sparse coding and dictionary learning. Sparsity regularized deep neural network architectures have been used for image denoising and discriminative tasks such as image classi\ufb01cation [51, 58, 49]. Recent work has highlighted some limitations of convolutional sparse coding (CSC) autoencoders and its multi-layer and deep generalizations [56, 65] for data reconstruction. The work in [51] argues that the sparsity levels that CSC allows can only accommodate very sparse vectors, making it unsuitable to capture all features of signals such as natural images, and propose to compute the minimum mean-squared error solution under the CSC model, which is a dense vector capturing a richer set of features. Moreover, the majority of CSC frameworks subtract low-frequency components of images prior to applying convolutionl dictionary learning (CDL) for representation purposes [29]; however, this is not desired for denoising tasks where noise corrupts low frequencies in addition to the high spectrum. The goals of the paper are twofold. First, it presents a theoretical analysis of the dense and sparse coding problem. Second, we argue the dense representation x in a dictionary A is useful for reconstruction and the sparse representation u has discriminative capability. 1.1 Organization of paper Section 2 discusses related work. In Section 3, we give some technical background to the main analysis. Section 4 presents the theoretical analysis of the paper. Phase transition, classi\ufb01cation, and denoising experiments appear in Section 5. We conclude in Section 6. We start by de\ufb01ning notations. 1.2 Notation Lowercase and uppercase boldface letters denote column vectors and matrices, respectively. Given a vector x \u2208Rn and a support set S \u2282{1, ..., n}, xS denotes the restriction of x to indices in S. supp(x) denotes the support of a vector i.e. supp(x) = {i : xi \u0338= 0}. For a matrix A \u2208Rm\u00d7p, AS is a submatrix of size m \u00d7 |S| with column indices in S. The column space of a matrix A is designated by Col(A), its null space by Ker(A). We denote the Euclidean, \u21130, \u21131, and \u2113\u221enorms of a vector x, respectively as ||x||2, ||x||0, ||x||1, and ||x||\u221e. The operator and in\ufb01nity norm of a matrix A are respectively denoted as ||A|| and ||A||\u221e. The sign function, applied componentwise to a vector x, is denoted by sgn(x). The indicator function is denoted by 1. The column vector ei denotes the vector of zeros except a 1 at the i-th location. The orthogonal complement of a subspace W denoted by W \u22a5. The operator PW denotes the orthogonal projection operator onto the subspace W . log(x) denotes the logarithm of x in base e. 2 Related work 2.1 Compressive sensing from union of dictionaries The dense and sparse coding problem is similar in \ufb02avor to sparse recovery in the union of dictionaries [22, 21, 24, 20, 33]. Most results in this literature take the form of an uncertainty principle that relates the sum of the sparsity of x and u to the mutual coherence between A and B, and which guarantees that the representation is unique and identi\ufb01able by \u21131 minimization. In [22], the authors study the problem of recovering x and u from y = Ax + Bu where A \u2208Rm\u00d7m is a discrete Fourier matrix (DFT) and B \u2208Rm\u00d7m is the identity matrix. Therein, under the assumption that the support of u is known, the result shows that x can be exactly 2 \frecovered if 2||x||0||u||0 < m. This result for the Fourier-identity pair was further generalized to the case where A and B are orthonormal bases [24] and when the measurements come from a union of non-orthogonal bases [20, 33]. We remark that all the aforementioned results assume sparsity on both x and u. 2.2 Error correction The problem of recovering a signal x given the measurement model y = Ax + u where u is a sparse error vector is the sparse error correction problem [10]. The work therein considers a tall measurement matrix A and assumes that the fraction of corrupted entries is suitably bounded. To obtain a sparse minimization program, the matrix A is eliminated by a matrix B, where BA = 0, resulting By = Bu. The resulting model is then solved via \u21131 minimization and exactness is shown under the restricted isometry condition on B. The works in [63, 43, 48, 55, 54] study the general error correction problem y = Ax + Bu under di\ufb00erent models of the signal x, the sparse interference vector u and under di\ufb00erent assumptions on the measurement matrices A and B. We note that all the aforementioned works consider a single sparsifying norm to recover the components x and u. In contrast, we use mixed norms that impose the Tikhonov and sparsity regularization. We also note that when A is fat, the setting we consider in this paper, our problem departs from the sparse error correction problem. 2.3 Weighted Lasso When there is some prior on the support of the underlying sparse vector y given measurements of the form y = Bu, the works in [36, 39, 61, 45, 28, 27] consider a weighted Lasso program. Assume that the estimate of the support of u is T \u2282{1, ..., n}. The standard weighted Lasso considers the following optimization program min u ||u||1,w subject to y = Bu, where ||u||1,w = Pn i=1 wiui and w \u2208Rn indicates the vector of weights. For instance, Mansour and Saab [39] set the weights as follows: wi \u2208[0, 1] if i \u2208T and 1 otherwise. The aforementioned works discuss the recovery of the underlying signal using weighted Lasso under some conditions on the weights, size and accuracy of the estimated support. One approach for the feasibility constraint in (1.1) is to reformulate it as a weighted Lasso problem in a combined dictionary [A B] where the support estimate is the support of x. This formulation has few limitations. First, the recovery guarantees in weighted Lasso require accuracy of the estimated support which would depend on the number of columns of A. Second, these guarantees impose uniform structure on the combined dictionary e.g. weighted null space conditions while our conditions allow deterministic dictionary A and random dictionary B. Finally, even in the regime where weighted Lasso might succeed in recovering the underlying x and u, it does not guarantee that Ax is smooth as that regularization is not re\ufb02ected in the Lasso optimization. 2.4 Morphological component analysis (MCA) Morphological Component Analysis (MCA) considers the decomposition of a signal or image into di\ufb00erent components with each component having a speci\ufb01c structure (morphology) [52, 53, 9, 25]. In MCA, the speci\ufb01c structure is imposed via sparsity with respect to pre-speci\ufb01ed dictionaries. Speci\ufb01cally, given K dictionaries {\u03a61, ..., \u03a6K}, we model the data y as a superposition of K linear components as follows: y = PK i=1 \u03a6iyi where yi is assumed to be sparse in \u03a6i but it is not as sparse (or not sparse at all) in other dictionaries. With this set up, the dictionaries discriminate between the di\ufb00erent components. In [52] a basis pursuit approach to solve the MCA is employed and its utility is illustrated in several examples e.g. separating texture from the smooth part of an image. The main di\ufb00erence of our model from MCA is a smooth-sparse decomposition as opposed to sparse-sparse decomposition with pre-set dictionaries that contain information about targeted morphologies. The theory and analysis of the two models is also markedly di\ufb00erent as we consider a smoothness regularizer. 2.5 Dictionary learning and unrolling Given a data set, learning a dictionary in which each example admits a sparse representation is useful in a number of tasks [3, 37]. This problem, known as sparse coding [44] or dictionary learning [2, 29], has been the 3 \fsubject of investigation in recent years in the signal processing community. A growing body of work, referred to as algorithm unrolling [42], has mapped the sparse coding problem into encoders for sparse recovery [30]. In this paper, we unroll the dense and sparse coding problem for a principled design of an autoencoder. The appeal of the autoencoder is that it learns dense representation x in a dictionary A useful for reconstruction and sparse representation u which has discriminative capability. 2.6 Contribution We focus on (1.1) and start by \ufb01rst providing theoretical guarantees for the feasibility problem y = Ax + Bu. Our \ufb01rst result is based on a geometric condition on the minimum principal angle between certain subspaces. Next, we prove that the convex program in (1.1) recovers the underlying components x and u under some mild assumptions on the measurement matrices and the Tikhonov regularizer. We empirically validate the e\ufb00ectiveness of our methods through phase transition curves and make a direct comparison to noisy compressive sensing, highlighting the latter\u2019s inability to recover signals adhering to our proposed model. We then apply our optimization algorithm to sense real data and validate the expected properties of the sparse and smooth components. We connect the proposed model to dictionary learning/algorithm unrolling by proposing a dense and sparse autoencoder (DenSaE). We demonstrate the superior discriminative and representation capabilities of DenSaE compared to sparse coding networks in the task of reconstruction and classifying MNIST dataset. In this task, we characterize the role of dense and sparse learned components; we argue that the dense representation x is useful for reconstruction and the sparse representation u has discriminative capability. Moreover, in an image denoising task, we show the outpefromance of DenSaE against networks strictly performing sparse coding. 3 Technical background In this section, we start by brie\ufb02y reviewing uniqueness results for compressive sensing with exact measurements. The typical assumption in these analysis is to assume the measurement matrices satisfy certain conditions such as the restricted isometry property (RIP) and coherence. In our setting, we employ RIPless recovery analysis [12, 32] which will be referred to and used in our the proof of Theorem 7. We note that existing anisotropic analysis is based on a single measurement matrix. For our model, the anisotropic analysis is applied to a certain matrix derived from A, B and the Tikhonov regularizer G. 3.1 Compressive sensing with exact measurements An underdetermined linear system y = Bu where y \u2208Rm and B \u2208Rm\u00d7n with n > m generally has in\ufb01nitely many solutions. One prior for the recovery of the solution is sparsity of the underlying vector u. This leads to the following optimization problem min u\u2208Rm ||u||0 s.t. y = Au. (3.1) While (3.1) is a natural program, it is known to be computationally intractable. Common alternatives are basis pursuit (BP) [59, 18] and iterative greedy methods such as orthogonal matching pursuit (OMP)[19, 46, 60]. In basis pursuit, (3.1) is modi\ufb01ed by considering a convex relaxation of the \u21130 norm leading to the \u21131 minimization problem. min u\u2208Rm ||u||1 s.t. y = Bu. (3.2) A fundamental question is under what conditions they identify the underlying solution for (3.1). A related part of this question is when the \u21130 minimization problem admits a unique solution. One condition is the restricted isometry property (RIP) [13]. The s-restricted isometry constant of an m \u00d7 n measurement matrix B is the smallest constant \u03b4s such that (1 \u2212\u03b4s)||u||2 \u2264||Bu\u22252 \u2264(1 + \u03b4s)||u||2, 4 \fholds for all s-sparse signals u (an s-sparse vector has at most s non-zero entries). Small RIP constants ensure unique solution of the \u21130 minimization problem and further provide the guarantee that the \u21131 relaxation in (3.2) is exact [11]. It is known that random measurement matrices have small RIP constants [14, 5]. Another condition for analysis is based on the coherence of a measurement matrix B de\ufb01ned as \u00b5 = max i\u0338=j |bT i bj|, where bi denotes the i-th column of B which is assumed to be unit-norm. To state the next result, consider y = Bu\u2217where u\u2217is the underlying sparse vector. If ||u\u2217||0 < 1 2 \u0010 1 + 1 \u00b5 \u0011 , the \u21130 problem admits a unique solution and both BP and OMP relaxations are exact [20, 31, 59]. 3.2 Compressive sensing with structured random matrices We review the technical results in [12, 32] which consider conditions for the exact recovery of the \u21131 minimization problem for the setting of structured measurement matrices. In this setting, we do not assume restricted isometry property (RIP) of the measurement matrices [13] and this broadens the sensing matrices that can be employed. We consider a sequence of i.i.d random vectors b1, ..., bm drawn from some distribution F on Rn and with measurements de\ufb01ned as yi = bT i u\u2217. Two properties are essential to the analysis. The \ufb01rst is the isotropy property which states that E[bbT ] = I for b \u223cF. The second property is the incoherence property. The incoherence parameter is the smallest number \u00b5(F) such that max 1\u2264i\u2264n |\u27e8b, ei\u27e9|2 \u2264\u00b5(F), (3.3) holds almost surely. Given these assumptions, the work in [12] provides the following theoretical result. Theorem 1 ([12]). Let B = 1 \u221am Pm i=1 bieT i be a measurement matrix and let u be a \ufb01xed but otherwise arbitrary s-sparse vector in Rn. Then with probability at least 1 \u22125 n \u2212e\u2212\u03b2 , u is the unique minimizer to min u ||u||1 subject to y = Bu, provided that m \u2265C\u03b2\u00b5(F)s log n. The constant C\u03b2 = C0(1 + \u03b2) where C0 is some constant. We note that the de\ufb01nition of B based on re-scaling the vectors b1, b2, ..., bm is done so that the columns of B are approximately unit-normed. The work in [32] extends the above result without assuming the isotropy property. Let \u03a3 = E[bbT ] 1 2 , where as before b is drawn from distribution F on Rn, denote the covariance matrix. The anisotropic analysis in [32] is based on two properties pertaining to the measurement matrix. The \ufb01rst is the incoherence property which is formally de\ufb01ned as follows. The incoherence parameter is the smallest number \u00b5(F) such that max 1\u2264i\u2264n |\u27e8b , ei\u27e9|2 \u2264\u00b5(F) and max 1\u2264i\u2264n |\u27e8b , E[bb\u2217]\u22121ei\u27e9|2 \u2264\u00b5(F) (3.4) hold almost surely. The second property is based on the conditioning of the covariance matrix. More precisely, it is based on an s-sparse condition number which we restate the de\ufb01nition in [32] below. De\ufb01nition 1. [32] The largest and smallest s-sparse eigenvalue of a matrix X are given by \u03bbmax(s, X) := max ||v||0\u2264s ||Xv||2 ||v||2 and \u03bbmin(s, X) := min ||v||0\u2264s ||Xv||2 ||v||2 . The s-sparse condition number of X is cond(s, X) = \u03bbmax(s, X) \u03bbmin(s, X) . Given these assumptions, the main result in [32] is stated below. Theorem 2. [32] Let B = 1 \u221am Pm i=1 bieT i be a measurement matrix and let u be a \ufb01xed but otherwise arbitrary s-sparse vector in Rn. De\ufb01ne \u03bas = max{cond(s, \u03a3), cond(s, \u03a3\u22121)} and let \u03b2 \u22651. If the number of measurements ful\ufb01lls m \u2265C\u03bas \u00b5(F) \u03b2 s log n, then the solution u of the convex program min u ||u||1 subject to y = Bu, is unique and equal to u\u2217with probability at least 1 \u2212e\u2212\u03b2. 5 \fThe proof of Theorem 1 and Theorem 2 is based on the dual certi\ufb01cate approach. The idea is to \ufb01rst propose a dual certi\ufb01cate vector v with su\ufb03cient conditions that ensure uniqueness of the minimization problem. It then remains to construct the dual certi\ufb01cate satisfying the conditions. For later reference, we note that the works in [12, 32] construct a dual certi\ufb01cate v satisfying the following conditions ||vS \u2212sgn(u\u2217 S)||2 \u22641 4 and ||vS\u22a5||\u221e\u22641 4. (3.5) The following condition follows from the assumptions in Theorem 1 and 2 ||\u2206S||2 \u22642||\u2206S\u22a5||2, for \u2206\u2208Ker(B). (3.6) 4 Theoretical Analysis The dense and sparse coding problem studies the solutions of the linear system y = Ax+Bu. Given matrices A \u2208Rm\u00d7p and B \u2208Rm\u00d7n and a vector y \u2208Rm, the goal is to provide conditions under which there is a unique solution (x\u2217, u\u2217), where u\u2217is s-sparse, and an algorithm for recovering it. 4.1 Feasibility problem In this subsection, we \ufb01rst study the uniqueness of solutions to the linear system accounting for the di\ufb00erent structures the measurement matrices A and B can have. A uniqueness result can be readily obtained by assuming orthogonality of Col(A) and Col(B). In what follows, we establish uniqueness results for the general setting. The main result of this subsection is Theorem 4 which, under a natural geometric condition based on the minimum principal angle between the column space of A and the span of s columns in B, establishes a uniqueness result for the dense and sparse coding problem. Since the vector u in the proposed model is sparse, we consider the classical setting of an overcomplete measurement matrix B with n \u226bm. The next theorem provides a uniqueness result assuming a certain direct sum representation of the space Rm. We \ufb01rst note that, for A \u2208Rm\u00d7p with p \u226bm, any z \u2208Ker(A) can be added to any solution x. With that, we consider uniqueness modulo the vectors in Ker(A) i.e. any feasible solution is assumed to be in Ker(A)\u22a5. Theorem 3. Assume that there exists at least one solution to y = Ax + Bu, namely the pair (x\u2217, u\u2217). Let S, with |S| = s, denote the support of u\u2217. If BS has full column rank and Rm = Col(A) \u2295Col(BS), the only unique solution to the linear system, with the condition that any feasible s-sparse vector u is supported on S and any feasible x is in Ker(A)\u22a5, is (x\u2217, u\u2217). Proof. Let (x, u), with u supported on S and x \u2208Ker(A)\u22a5, be another solution pair. It follows that A\u03b41 + BS(\u03b42)S = 0 where \u03b41 = x \u2212x\u2217and \u03b42 = u \u2212u\u2217. Let U \u2208Rm\u00d7r and V \u2208Rm\u00d7q be matrices whose columns are the orthonormal bases of Col(A) and Col(BS), respectively. The equation A\u03b41 + BS(\u03b42)S = 0 can equivalently be written as Pr i=1\u27e8A\u03b41 , Ui\u27e9Ui + Pq i=1\u27e8BS(\u03b42)S , Vi\u27e9Vi = 0 with Ui and Vi denoting the i-th columns of U and V , respectively. More compactly, we have \u0002 U V \u0003 \u0014 {\u27e8A\u03b41 , Ui\u27e9}r i=1 {\u27e8BS(\u03b42)S , Vi\u27e9}q i=1 \u0015 = 0. Noting that the matrix \u0002U V \u0003 has full column rank, the homogeneous problem admits the trivial solution implying that A\u03b41 = 0 and BS(\u03b42)S = 0. Since BS has full column rank and \u03b41 \u2208{Ker(A) \u2229Ker(A)\u22a5}, it folows that \u03b41 = \u03b42 = 0. Therefore, (x\u2217, u\u2217) is the unique solution. The uniqueness result in the above theorem hinges on the representation of the space Rm as the direct sum of the subspaces Col(A) and Col(BS). We use the de\ufb01nition of the minimal principal angle between two subspaces, and its formulation in terms of singular values [8], to derive an explicit geometric condition for the uniqueness analysis of the linear system in the general case. De\ufb01nition 2. Let U \u2208Rm\u00d7r and V \u2208Rm\u00d7q be matrices whose columns are the orthonormal basis of Col(A) and Col(B), respectively. The minimum principal angle between the subspaces Col(A) and Col(B) is de\ufb01ned as follows cos(\u00b5(U, V )) = max u\u2208Col(U),v\u2208Col(V ) uT v ||u||2||v||2 , (4.1) In addition, cos(\u00b5(U, V )) = \u03c31(U T V ) where \u03c31 denotes the largest singular value. 6 \fTheorem 4. Assume that there exists at least one solution to y = Ax + Bu, namely the pair (x\u2217, u\u2217). Let S, with |S| = s, denote the support of u\u2217. Assume that BS has full column rank . Let U \u2208Rm\u00d7r and V \u2208Rm\u00d7q be matrices whose columns are the orthonormal bases of Col(A) and Col(BS), respectively. If cos(\u00b5(U, V )) = \u03c31(U T V ) < 1, the only unique solution to the linear system, with the condition that any feasible s-sparse vector u is supported on S and any feasible x is in Ker(A)\u22a5, is (x\u2217, u\u2217). Proof. Consider any candidate solution pair (x\u2217+ \u03b41, u\u2217+ \u03b42) with \u03b42 supported in S. We will prove uniqueness by showing that A\u03b41 + BS(\u03b42)S = 0 if and only if \u03b41 = 0 and \u03b42 = 0. Using the orthonormal basis set U and V , A\u03b41 + BS(\u03b42)S can be represented as : A\u03b41 + BS(\u03b42)S = \u0002 U V \u0003 \u0014 U T A\u03b41 V T BS(\u03b42)S \u0015 . For simplicity of notation, let K denote the block matrix: K = \u0002U V \u0003 . If we can show that the columns of K are linearly independent, it follows that A\u03b41 + BS(\u03b42)S = 0 if and only if A\u03b41 = 0 and BS(\u03b42)S = 0. We now consider the matrix KT K which has the following representation KT K = \u0014 [I]r\u00d7r [U T V ]r\u00d7q [V T U]q\u00d7r [I]q\u00d7q \u0015 = \u0014[I]r\u00d7r [0]r\u00d7q [0]q\u00d7r [I]q\u00d7q \u0015 + \u0014 [0]r\u00d7r [U T V ]r\u00d7q [V T U]q\u00d7r [0]q\u00d7q \u0015 . With the singular value decomposition of U T V being U T V = Q\u03a3RT , the last matrix in the above representation has the following equivalent form \u0014 0 U T V V T U 0 \u0015 = \u0014Q 0 0 R \u0015 \u00140 \u03a3 \u03a3 0 \u0015 \u0014Q 0 0 R \u0015T . It now follows that \u0014 0 U T V V T U 0 \u0015 is similar to the matrix \u00140 \u03a3 \u03a3 0 \u0015 . Hence, the nonzero eigenvalues of KT K are 1 \u00b1 \u03c3i, 1 \u2264i \u2264min(p, q), with \u03c3i denoting the i-th largest singular value of U T V . Using the assumption \u03c31 < 1 results the bound \u03bbmin \u0000KT K \u0001 > 0. It follows that the columns of K are linearly independent, and hence A\u03b41 = 0 and BS(\u03b42)S = 0. Since BS is full column rank and \u03b41 \u2208{Ker(A) \u2229Ker(A)\u22a5}, it follows that \u03b41 = 0 and \u03b42 = 0. A restrictive assumption of the above theorem is that the support of the sought-after s-sparse solution u\u2217is known. We can remove this assumption by considering Col(A) and Col(BT ) where T is an arbitrary subset of {1, 2, ..., n} with |T| = s. More precisely, we state the following corollary whose proof is similar to the proof of Theorem 4. Corollary 5. Assume that there exists at least one solution to y = Ax + Bu, namely the pair (x\u2217, u\u2217). Let S, with |S| = s, denote the support of u\u2217and T be an arbitrary subset of {1, 2, ..., n} with |T| \u2264s. Assume that any 2s columns of B are linearly independent. Let U \u2208Rm\u00d7p and V \u2208Rm\u00d7q be matrices whose columns are the orthonormal bases of Col(A) and Col(BS\u222aT ), respectively. If \u00b5(U, V ) = \u03c31(U T V ) < 1, holds for all choices of T, the only unique solution to the linear system is (x\u2217, u\u2217) with the condition that any feasible u is s-sparse and any feasible x is in Ker(A)\u22a5. Of interest is the identi\ufb01cation of simple conditions such that \u03c31(U T V ) < 1. The following theorem proposes one such condition to establish uniqueness. Theorem 6. Assume that there exists at least one solution to y = Ax + Bu, namely the pair (x\u2217, u\u2217). Let S, with |S| = s, denote the support of u\u2217. Assume that BS has full column rank. Let U \u2208Rm\u00d7r and V \u2208Rm\u00d7q be matrices whose columns are the orthonormal bases of Col(A) and Col(BS), respectively. Let max i,j |uT i vj| = \u00b5. If s < 1 \u221ar\u00b5, the only unique solution to the linear system, with the condition that any feasible s-sparse vector u is supported on S and any feasible x is in Ker(A)\u22a5, is (x\u2217, u\u2217). Proof. It su\ufb03ces to show that \u03c31 < 1. Noting that \u03c31 = ||U T V ||2, we use the following matrix norm inequality ||U T V ||2 \u2264\u221ar||U T V ||\u221eas follows: \u03c31 \u2264\u221ar ||U T V ||\u221e\u2264\u221ar\u00b5s < 1. The constant \u00b5 is the coherence of the matrix U T V [23, 59]. The above result states that if the mutual coherence of U T V is small, we can accommodate increased sparsity of the underlying signal component u\u2217. 7 \fWe note that, up to a scaling factor, \u03c31(U T V ) is the block coherence of U and V [26]. However, unlike the condition in [26], we do not restrict the dictionaries A and B to have linearly independent columns. In the next subsection, we propose a convex program to recover the dense and sparse vectors. Theorem 7 establishes uniqueness results for the proposed optimization program. 4.2 Dense and sparse recovery via convex optimization We propose the following convex optimization program for the dense and sparse coding problem min x,u ||Gx||2 2 + ||u||1 s.t. y = Ax + Bu. (4.2) The Tikhonov matrix G is assumed to have full column rank. In what follows, we show that, under certain conditions, the above minimization problem admits a unique solution (x\u2217, u\u2217). We can eliminate the dense component in the linear constraint by projecting the vector y onto the orthogonal complement of Col(A) to obtain PCol(A)\u22a5(y) = PCol(A)\u22a5(Bu). For ease of notation, we de\ufb01ne \u02dc y = PCol(A)\u22a5(y) and \u02dc C = PCol(A)\u22a5(B). Before we discuss the main theoretical results, we discuss how the measurement matrices A and B are generated. Construction of measurement matrices First, note that Col(A)\u22a5is an r-dimensional subspace of Rm where r = dim Col(A)\u22a5. Given A \u2208Rm\u00d7p, either generated from a deterministic or random model, we \ufb01rst form the matrix H = AG\u2020. We sample r i.i.d random vectors, c1, c2, ..., cr from some distribution F on Rp i.e. each sampled vector is a random vector of size Rp. Let C = 1 \u221ar Pr i=1 cieT i be the scaled measurement matrix. We form a matrix \u02dc C of size m\u00d7p where its r rows are the sampled random vectors and the remaining m \u2212r rows are such that each column of \u02dc C lies in Col(H)\u22a5. It is important to note that the rows of \u02dc C are not i.i.d (except the r rows generated from the random model). We now form the matrix B \u2208Rm\u00d7n by constructing each of its columns as follows: bi = \u02dc ci + hi where bi is the i-th column of B, \u02dc ci is the i-th column of \u02dc C and hi is an arbitrary vector that is in column space of H. Our main result stated below assumes that the measurements c1, ..., cr satisfy the incoherence property and conditioning property of the covariance matrix (see (3.4) and De\ufb01nition 1). Theorem 7. Assume that there exists at least one solution to y = Ax + Bu, namely the pair (x\u2217, u\u2217) to the optimization program in (4.2). Let \u03b2 \u22651, r = dim Col(A)\u22a5and de\ufb01ne \u03bas = max{cond(s, \u03a3), cond(s, \u03a3\u22121)}. If the number of measurements ful\ufb01lls r \u2265C\u03bas \u00b5(F) \u03b22 s log n, then the solution of the convex program (4.2) is unique and equal to (x\u2217, u\u2217) with probability at least 1 \u2212e\u2212\u03b2. Proof. We make a change of variable z = Gx and rewrite the optimization in (4.2) as follows min z,u ||z||2 2 + ||u||1 s.t. y = AG\u2020z + Bu = Hz + Bu. (4.3) Given a feasible solution pair (\u02dc z, \u02dc u) and decompose \u02dc z as \u02dc z = z1 + z2 where z1 \u2208Ker(H)\u22a5and z2 \u2208Ker(H). We note that y = Hz1 + Bu = Hz + Bu and ||\u02dc z||2 = ||z1||2 2 + ||z2||2 2 \u2265||z1||2 2 with equality only attained if z2 = 0. With that, we will assume that, the optimal solution z\u2217lies in Ker(H)\u22a5. Further, for a \ufb01xed u, the least square solution \u02dc z to PCol(B)\u22a5(Hz) = PCol(B)\u22a5(y) lies in T \u2261Ker(PCol(B)\u22a5H)\u22a5. We now consider a generic feasible solution pair (z\u2217+ \u03b41, u\u2217+ \u03b42). Let the function f(z, u) de\ufb01ne the objective in the optimization program. The idea of the proof is to show that any feasible solution is not minimial in the objective value, f(z\u2217+ \u03b41, u\u2217+ \u03b42) \u2265f(z\u2217, u\u2217), with the inequality holding for all choices of \u03b41 and \u03b42 and equality obtained if and only if \u03b41 = \u03b42 = 0. Before we proceed, two remarks are in order. First, using the duality of the \u21131 norm and the \u2113\u221enorm, there exists a \u039b with supp(\u039b) \u2282S\u22a5, ||\u039b||\u221e= 1 and such that \u27e8\u039b , (\u03b42)S\u22a5\u27e9= ||(\u03b42)S\u22a5||1. Second, the subgradient of the \u21131 norm at u\u2217is characterized as follows: \u2202||u\u2217||1 = {sgn(u\u2217 S) + g | supp(g) \u2208S\u22a5, ||gS\u22a5||\u221e\u22641}. It follows that sgn(u\u2217) + \u039b is a subgradient of the \u21131 norm at u\u2217. It then follows that the inequality ||u||1 \u2265||u\u2217||1 + \u27e8sgn(u\u2217 S) + \u039b , u \u2212u\u2217\u27e9holds for any u. We lower bound f(z\u2217+ \u03b41, u\u2217+ \u03b42) as follows. ||z\u2217+ \u03b41||2 2 + ||u\u2217+ \u03b42||1 \u2265||z\u2217||2 2 + ||\u03b41||2 + 2\u27e8z\u2217, \u03b41\u27e9+ ||u\u2217||1 + \u27e8sgn(u\u2217 S) + \u039b , \u03b42\u27e9 =f(z\u2217, u\u2217) + ||\u03b41||2 + ||u\u2217||1 + \u27e8sgn(u\u2217 S) + \u039b , \u03b42\u27e9. (4.4) 8 \fAbove, the second equality uses the fact that z\u2217\u2208T and \u03b41 \u2208T \u22a5. We next use the feasibility condition that H\u03b41 + B\u03b42 = 0 and introduce the dual certi\ufb01cate v. To eliminate the component H\u03b41, we project it onto the orthogonal complement of the range of H and obtain \u02dc C\u03b42 = 0 where C = PCol(H)\u22a5(B). We further consider a reduced system by considering C\u03b42 = 0. We now assume that v \u2208Col(CT ). With this, we continue with lower bounding f(z\u2217+ \u03b41, u\u2217+ \u03b42). ||z\u2217+ \u03b41)||2 2 + ||u\u2217+ \u03b42||1 \u2265||z\u2217||2 2 + ||\u03b41||2 2 + ||u\u2217||1 + \u27e8sgn(u\u2217 S) + \u039b \u2212v, \u03b42\u27e9 =f(x\u2217, u\u2217) + \u27e8sgn(u\u2217 S) + \u039b \u2212v , \u03b42\u27e9 (4.5) It remains to show that \u27e8sgn(u\u2217 S) + \u039b \u2212v , \u03b42\u27e9> 0. By considering projections onto S and S\u22a5and using the fact that \u27e8\u039b , (\u03b42)S\u22a5\u27e9= ||(\u03b42)S\u22a5||1, we obtain \u27e8sgn(u\u2217 S) + \u039b \u2212v , \u03b42\u27e9=\u27e8sgn(u\u2217 S) \u2212vS , (\u03b42)S\u27e9\u2212\u27e8vS\u22a5, (\u03b42)S\u22a5\u27e9+ ||(\u03b42)S\u22a5||1 (4.6) \u2265\u2212|| sgn(u\u2217 S) \u2212vS||2||(\u03b42)S||2 \u2212||vS\u22a5||\u221e||(\u03b42)S\u22a5||1 + ||(\u03b42)S\u22a5||1 (4.7) \u2265\u22121 4 ||(\u03b42)S||2 \u22121 4||(\u03b42)S\u22a5||1 + ||(\u03b42)S\u22a5||1 (4.8) = \u22121 4 ||(\u03b42)S||2 + 3 4||(\u03b42)S\u22a5||1 \u22651 4||(\u03b42)S\u22a5||1. (4.9) The inequality in (4.8) follows from the conditions on the dual certi\ufb01cate ||vS \u2212sgn(u\u2217 S)||2 \u22641 4 and ||vS\u22a5||\u221e\u22641 4, (4.10) and the last inequality follows from the assumption on the deviation of \u03b42 as ||(\u03b42)S||2 \u22642||(\u03b42)S\u22a5||2. Combining (4.5) and the above bound with the \ufb01nal result noted in (4.9), we have f(z\u2217+ \u03b41, u\u2217+ \u03b42) \u2265f(z\u2217, u\u2217) + ||\u03b41||2 2 + 1 4||(\u03b42)S\u22a5||1. (4.11) We note that f(z\u2217+ \u03b41, u\u2217+ \u03b42) = f(z, u) if and only if ||(\u03b42)S\u22a5||1 = 0 and \u03b41 = 0. Since ||(\u03b42)S||2 \u2264 2||(\u03b42)S\u22a5||2, the equality ||(\u03b42)S\u22a5||1 = 0 implies that ||(\u03b42)S||2 = 0. With this, f(x\u2217+ \u03b41, u\u2217+ \u03b42) = f(x, u) if and only if \u03b42 = 0. Therefore, the solution (z\u2217, u\u2217) achieves the minimal value in the objective, and is a unique solution to (4.3). Finally, using the relation x\u2217= G\u2020z\u2217guarantees unique solution to (4.2). This concludes the proof. Remark 1. The existence of the dual certi\ufb01cate v satisfying the conditions (4.10) and the deviation inequality ||(\u03b42)S||2 \u22642||(\u03b42)S\u22a5||2 follow from the anisotropic compressive sensing analysis once it is assumed that (i) the covariance matrix de\ufb01ned obeys completeness, incoherence, and conditioning \u03bas denoting its s-sparse condition number, and (ii) the sample complexity is as noted in Theorem 7. 4.3 Special cases of analysis and discussion We note a few special cases of our main theorem in Theorem 7. First, when the Tikhonov matrix G is the identity matrix, the regularization on x is the standard \u21132 (minimum norm least squares) regularization. The main result continues to hold, with additional assumptions, when G = A. In this case, the optimization problem is min x\u2208Ker(A)\u22a5,u ||Ax||2 2 + ||u||1 s.t. y = Ax + Bu. (4.12) We note that the uniqueness result for the above problem is modulo restriction to the subspace Ker(A)\u22a5.We now state the following result. Theorem 8. Assume that there exists at least one solution to y = Ax + Bu, namely the pair (x\u2217, u\u2217) to the optimization program in (4.12). Let \u03b2 \u22651, r = dim Col(A)\u22a5and de\ufb01ne \u03bas = max{cond(s, \u03a3), cond(s, \u03a3\u22121)}. Assume the two conditions ||BT S A|| \u2264 1 32||x\u2217||2 , max i,j |(BT S\u22a5A)i,j| \u2264 1 32||x\u2217||\u221e . (4.13) If the number of measurements ful\ufb01lls r \u2265C\u03bas \u00b5(F) \u03b22 s log n, then the solution of the convex program (4.2) is unique and equal to (x\u2217, u\u2217) with probability at least 1 \u2212e\u2212\u03b2. 9 \fProof. The proof follows the proof of Theorem 7. The main di\ufb00erence is that we need to show that \u27e8sgn(u\u2217 S) + \u039b \u2212v \u22122BT Ax\u2217, \u03b42\u27e9> 0. By considering projections onto S and S\u22a5and using the fact that \u27e8\u039b , (\u03b42)S\u22a5\u27e9= ||(\u03b42)S\u22a5||1, we obtain \u27e8sgn(u\u2217 S) + \u039b \u2212v \u22122BT Ax\u2217, \u03b42\u27e9 =\u27e8sgn(u\u2217 S) \u2212vS , (\u03b42)S\u27e9\u2212\u27e8vS\u22a5, (\u03b42)S\u22a5\u27e9+ ||(\u03b42)S\u22a5||1 \u2212\u27e82[BT Ax\u2217]S , (\u03b42)S\u27e9 \u2212\u27e82[BT Ax\u2217]S\u22a5, (\u03b42)S\u22a5\u27e9 \u2265\u2212|| sgn(u\u2217 S) \u2212vS||2||(\u03b42)S||2 \u2212||vS\u22a5||\u221e||(\u03b42)S\u22a5||1 + ||(\u03b42)S\u22a5||1 (4.14) \u22122||BT S A|| ||x\u2217||2 ||(\u03b42)S||2 \u22122 max i,j |(BT S\u22a5A)i,j|||x\u2217||\u221e||(\u03b42)S\u22a5||1 \u2265\u22121 4 ||(\u03b42)S||2 \u22121 4||(\u03b42)S\u22a5||1 + ||(\u03b42)S\u22a5||1 \u22121 16 ||(\u03b42)S||2 \u22121 16 ||(\u03b42)S\u22a5||1 (4.15) = \u22125 16 ||(\u03b42)S||2 + 11 16||(\u03b42)S\u22a5||1 \u2265\u221210 16 ||(\u03b42)S\u22a5||1 + 11 16||(\u03b42)S\u22a5||1 = 1 16||(\u03b42)S\u22a5||1. (4.16) The rest of the proof is similar to the proof of Theorem 7. We now apply the proposed model to a set of data points y1, ..., yn. Let x1, ..., xn and u1, ....un denote the dense and sparse components of the data points respectively. We use Y , X and U to represent the data matrix, the dense coe\ufb03cient matrix and the sparse coe\ufb03cient matrix respectively. With this, the dense and sparse coding optimization problem is min X,U ||GX||2 F + ||U||1 s.t. Y = AX + BU, (4.17) where ||U||1 = Pn i=1 ||ui||1. When G = I, the works in [66, 47] establish that the minimization with respect to X leads to low rank solutions. The e\ufb00ect of the mixed regularizations and global structure promoted given a set of data points is left for future work. 5 Experiments 5.1 Phase transition curves We generate phase transition curves and present how the success rate of the recovery, using the proposed model, changes under di\ufb00erent scenarios. To generate the data, we \ufb01x the number of columns of B to be n = 100. Then, we vary the sampling ratio \u03c3 = m n+p \u2208[0.05, 0.95] and the sparsity ratio \u03c1 = s m in the same range. Note that the sensing matrix in our model is [A B]; therefore, our de\ufb01nition of \u03c3 takes into account both the size of A and B. In the case where we revert to the compressive sensing scenario (p = 0), the sampling ratios coincide. We generate random matrices A \u2208Rm\u00d7p and B \u2208Rm\u00d7n whose columns have expected unit norm. The vector u \u2208Rn has s randomly chosen indices, whose entries are drawn according to a standard normal distribution, and x \u2208Rp is generated as x = AT \u03b3 where \u03b3 \u2208Rm is a random vector. The construction ensures that x does not belong in the null space of A, and hence degenerate cases are avoided. We normalize both x and u to have unit norm, and generate the measurement vector y \u2208Rm as y = Ax + Bu. We solve the convex optimization problem of (4.12) to obtain the numerical solution pair (\u02c6 x, \u02c6 u) using CVXPY, and register a successful recovery if both \u2225\u02c6 x\u2212x\u22252 \u2225x\u22252 \u2264\u03f5 and \u2225\u02c6 u\u2212u\u22252 \u2225u\u22252 \u2264\u03f5, with \u03f5 = 10\u22123. For each choice of \u03c3 and \u03c1 we average 100 independent runs to estimate the success rate. Figure 1 shows the phase transition curves for p \u2208{0.1m, 0.5m} to highlight di\ufb00erent ratios between p and n. We observe that increasing p leads to a deterioration in performance. This is expected, as this creates a greater overlap on the spaces spanned by A and B. We can view our model as explicitly modeling the noise of the system. In such a case, the number of columns of A explicitly encodes the complexity of the noise model: as p increases, so does the span of the noise space. Extending the signal processing interpretation, 10 \fnote that we model the noise signal x as a dense vector, which can be seen as encoding smooth areas of the signal that correspond to low-frequency components. On the contrary, the signal u has, by construction, a sparse structure, containing high-frequency information, an interpretation that will be further validated on real data in Section 5.4. 0.1 0.26 0.42 0.58 0.74 0.9 \u03c3 = m n+p 0.95 0.77 0.59 0.41 0.23 0.05 \u03c1 = s m p = 0.1m 0.0 0.2 0.4 0.6 0.8 1.0 0.1 0.26 0.42 0.58 0.74 0.9 \u03c3 = m n+p 0.95 0.77 0.59 0.41 0.23 0.05 \u03c1 = s m p = 0.5m 0.0 0.2 0.4 0.6 0.8 1.0 Figure 1: Phase transition curves for p = 0.1m (left) and p = 0.5m (right). 5.2 Noisy compressive sensing If x can be interpreted as a noise vector, how does our model compare to noisy compressive sensing? Compressive sensing can be extended to the noisy case, which allows for the successful recovery of sparse signals under the presence of noise, assuming an upper bound on the noise level. In the rest of the section we examine how the proposed model, which incorporates both sparse and dense components, fares against this noisy variant. \u221240 \u221220 0 20 40 SNR (dB) 0.00 0.25 0.50 0.75 1.00 Normalized error \u2225\u02c6 u\u2212u\u2217\u22252 \u2225u\u2217\u22252 \u03c3 = 0.5, \u03c1 = 0.05, p = 0.1m Ours CS (a) \u221240 \u221220 0 20 40 SNR (dB) 0.4 0.6 0.8 1.0 Normalized error \u2225\u02c6 u\u2212u\u2217\u22252 \u2225u\u2217\u22252 \u03c3 = 0.5, \u03c1 = 0.5, p = 0.1m Ours CS (b) \u221240 \u221220 0 20 40 SNR (dB) 0.00 0.25 0.50 0.75 1.00 Normalized error \u2225\u02c6 u\u2212u\u2217\u22252 \u2225u\u2217\u22252 \u03c3 = 0.95, \u03c1 = 0.5, p = 0.1m Ours CS (c) \u221240 \u221220 0 20 40 SNR (dB) 0.00 0.25 0.50 0.75 1.00 Normalized error \u2225\u02c6 u\u2212u\u2217\u22252 \u2225u\u2217\u22252 \u03c3 = 0.5, \u03c1 = 0.05, p = 0.5m Ours CS (d) \u221240 \u221220 0 20 40 SNR (dB) 0.25 0.50 0.75 1.00 Normalized error \u2225\u02c6 u\u2212u\u2217\u22252 \u2225u\u2217\u22252 \u03c3 = 0.5, \u03c1 = 0.5, p = 0.5m Ours CS (e) \u221240 \u221220 0 20 40 SNR (dB) 0.00 0.25 0.50 0.75 1.00 Normalized error \u2225\u02c6 u\u2212u\u2217\u22252 \u2225u\u2217\u22252 \u03c3 = 0.95, \u03c1 = 0.5, p = 0.5m Ours CS (f) Figure 2: Normalized recovery error of u as the SNR varies (lower is better). We, again, \ufb01x the number of columns in B to be n = 100, and generate the matrices A \u2208Rm\u00d7p and B \u2208Rm\u00d7n, as well as the vectors x\u2217\u2208Rp and u\u2217\u2208Rn, as before. We de\ufb01ne the signal-to-noise ratio as SNR = 20 log10 ||u\u2217||2 ||x\u2217||2 , and iterate over the range [\u221240dB, 40dB]. To vary the SNR, we normalize both vectors and scale u\u2217by 10 SNR 20 . For our proposed method, we solve the optimization of (4.2), whereas for 11 \fnoisy compressive sensing we solve \u02c6 u = arg min u ||u||1 , s.t. ||y \u2212Bu||2 \u2264||Ax\u2217||2 , (5.1) and report the normalized error ||\u02c6 u\u2212u\u2217||2 ||u\u2217||2 for the two methods averaging 100 independent runs. We present the SNR curves in Figure 2. As an initial observation, note that noisy compressive sensing, in every case, recovers a vector when the energy of the noise is less than that of the signal (for a SNR> 0dB), and in most scenarios full recovery isn\u2019t achieved unless the ratio is very large (SNR> 25dB). That is expected, since the results for those settings assume an upper bound on the noise level. In contrast, our proposed model is able to recover both signals even when the norm of x is 100 times larger than that of u, at the cost of having access to the additional measurement matrix A. For low sparsity ratios (Figures 2(a) and (d)), we are able to recover both signals in every experiment using our model, whereas compressive sensing exhibits the behaviour we discussed above. Increasing the sparsity ratio while keeping the number of samples the same (Figures 2(b) and (e)) results in a signi\ufb01cant reduction in performance for both models (note the di\ufb00erent axis scaling for these \ufb01gures). This is in line with both the results for noisy compressive sensing and our results presented in Section 4 (as well as the phase transition curves of the previous subsection). When the overlap between A and B is greater (Figure 2(e)) our model su\ufb00ers a greater performance hit, as is expected from our analysis; note that noisy compressive sensing is una\ufb00ected by the relative size of A and B, since it only makes assumptions about the noise level and not its span. Finally, we observe that increasing the number of measurements (Figures 2(c) and (f)) restores performance, in line with our analysis. Remark: Note that the large sparsity ratios (\u03c1 = 0.5, corresponding to Figures 2(b), (c), (e), and (f)) violate our assumptions in Section 4 and recovery is not expected based on the transition curves of the previous subsection. However, we include them for illustration, as for smaller values of \u03c1 our model always recovered both vectors. As a key takeaway from this line of experimentation, we recommend the use of our model in every case where there is a structured interference with signi\ufb01cant norm. In cases where the norm of the signal of interest is dominating (SNR\u224840dB) and there is some form degeneracy (there is a large overlap between the signal spaces and u is barely sparse), noisy compressive sensing may be more appropriate. 5.3 Sensing with real data In this section, we present experiments on real data using random sensing matrices. The real data we consider is the MNIST databse of handwritten digits [34, 35]. Through these experiments we want to (i) showcase the performance of our model on actual, real data and (ii) show the e\ufb03cacy of our algorithm when using overcomplete sensing matrices. In order to generate overcomplete dictionaries, we proceeded as follows: as y Bu Ax u x Figure 3: Decomposition of an MNIST image to its sparse and dense components. MNIST images can be seen as vectors y \u2208R784, we \ufb01rst generated a random orthogonal matrix in R784\u00d7784. We used half of the columns of that matrix as a base for A \u2208R784\u00d71024 and the other half for B \u2208R784\u00d71024; 12 \fthe rest of the columns were generated as linear combinations of each base. To avoid arti\ufb01cial orderings that might result when using certain optimizers, we conclude the matrix generation with random column permutations. These matrices, while in combination they do span R784, were not the generating model for the MNIST dataset. As such, we slightly alter the optimization problem of (4.2) to relax the exact reconstruction, yielding: min x,u ||Ax + Bu \u2212y||2 2 + \u00b5 ||Ax||2 2 + \u03bb ||u||1 . (5.2) In our experiments, we use \u00b5 = 3 and \u03bb = 0.01. A visual decomposition of such an image is presented in Figure 3. Both u and x have their expected properties: u is visibly sparse, with few nonzero elements and x has a fairly dense structure. Moreover, we observe that the component of Ax seems slightly more fainted compared to that of Bu. In order to further validate this observation we computed the both the Euclidean norm and the total variation for each component, as a proxy for smoothness, and plot the distributions of total variation in Figure 4. The distributions were computed using 10000 training samples. The distribution of the Euclidean norm was very similar and can be found in the appendix, Figure 11. Observing the distributions of Ax and Bu we note that the distribution corresponding to the sparse component is skewed to higher values of total variation. This empirically validates the e\ufb00ectiveness of the smoothness regularization of (4.2), even when sensing using random matrices. As a \ufb01nal remark, we attempted to use B in a sparse coding framework 60 80 100 120 140 160 180 200 Total Variation 0.000 0.005 0.010 0.015 0.020 0.025 Density Bu Ax Figure 4: Total variation distribution for the components Ax and Bu of MNIST images. in order to recover a sparse vector. However, in every instance the solver failed to converge and produce a sprase vector; this is, to an extent, expected as B was generated using only half the columns of an orthogonal matrix and therefore is unable to fully span the image space, failing to adequately represent all the images in the data set. The main analysis in this paper is based on the measurement matrices and the Tikhonov matrix satisfying certain properties e.g. randomness is assumed in the matrix B. However, in practice, e\ufb03cient matrices are not always available, and random matrices might not optimal for every data set. Dictionary learning refers to the problem of learning A and B from data [1, 17, 29]. Based on unrolled neural architectures for the sparse coding model (B = 0) [58, 30], next section proposes an unrolled autoencoder to infer dense and sparse representations and learn dictionaries from data. 5.4 Dictionary learning based neural architecture for dense and sparse coding We mitigate the drawbacks of convolutional sparse coding model in capturing a wide range of features from natural images by proposing dense and sparse dictionary learning; the framework learns a diverse set of features (i.e., smooth features via A and high-frequency features appearing sparsely through B). We formulate the dense and sparse dictionary learning problem as minimizing the objective min A,B,X,U 1 2\u2225Y \u2212AX \u2212BU\u22252 F + 1 2\u03bbx \u2225AX\u22252 F + \u03bbu\u2225U\u22251, (5.3) 13 \fX(l), U (l) = arg min X,U 1 2\u2225Y \u2212A(l)X \u2212B(l)U\u22252 F + 1 2\u03bbx \u2225A(l)X\u22252 F + \u03bbu\u2225U\u22251. (5.4) A(l+1), B(l+1) = arg min A,B 1 2\u2225Y \u2212AX(l) \u2212BU (l)\u22252 F , (5.5) where as choice of design, we optimize only the reconstruction loss when updating the dictionaries. Based on the above alternating updates, we use deep unrolling to construct a unrolled neural network [58, 30], which we term the dense and sparse autoencoder (DenSaE), tailored to learning the dictionaries from the dense and sparse model. The encoder maps Y into a dense matrix XT and a sparse one UT using two sets of \ufb01lters of A and B through a recurrent network. A encodes the smooth part of the data (low frequencies), and B encodes the details of the signal (high frequencies). The encoding is achieved by unrolling T proximal gradient iterations shown below Xt = Xt\u22121 + \u03b1x(AT(Y \u2212(1 + 1 \u03bbx )AXt\u22121 \u2212BUt\u22121)), Ut = Sb \u0000Ut\u22121 + \u03b1uBT(Y \u2212AXt\u22121 \u2212BUt\u22121) \u0001 , (5.6) where \u03b1x and \u03b1u are step sizes, and Sb, with b = \u03b1u\u03bbu, is the activation function; for non-negative sparse coding, the activation is ReLUb(z) = (z \u2212b) \u00b7 1z\u2265b, and for general sparse coding, it is Shrinkageb(z) = ReLUb(z) \u2212ReLUb(\u2212z) [58]. Having a non-informative prior on Ax in DenSaE implies that \u03bbx \u2192\u221e. The parameters \u03b1x, \u03b1u, \u03bbu are tuned via grid search. The decoder reconstructs the image using the generative model (i.e., y = AxT +BuT ). For classi\ufb01cation, we use uT and xT as inputs to a linear classi\ufb01er C that maps them to the predicted class \u02c6 q. The dictionaries A and B are learned via backpropagation. We remark that b = \u03b1u\u03bbu. A larger value of b in the proximal mapping Sb enforces higher sparsity on u, and a smaller value of \u03bbx promotes smoothness on Ax. We learn the dictionaries A and B, as well as the classi\ufb01er C, by minimizing the weighted reconstruction (Rec.) and classi\ufb01cation (Logistic) loss (i.e., (1 \u2212\u03b2) Rec. + \u03b2 Logistic). Figure 5 shows the DenSaE architecture. We examined the following questions y \u03b1uBT Sb ut uT B B \u03b1xAT xt xT A A 1 + 1 \u03bbx \u02c6 y Decoder Encoder z C Smax \u02c6 q Classi\ufb01er Repeat T times Figure 5: DenSaE. The vector z comprises the normalized features stacked plus a 1 scalar for the classi\ufb01er bias, Sb and Smax are the soft-thresholding and soft-max operators, respectively. (i) How do the discriminative, reconstruction, and denoising capabilities change as we vary the number of \ufb01lters in A vs. B? (ii) What is the performance of DenSaE compared to sparse coding networks? (iii) What data characteristics does the model capture? As baselines, we trained two variants, CSCNethyp and CSCNetLS, of CSCNet [51], an architecture tailored to dictionary learning for the sparse coding problem y = Bu. In CSCNethyp, the bias is tuned as a shared hyper-parameter. In CSCNetLS, we learn a di\ufb00erent bias for each \ufb01lter by minimizing the reconstruction loss. When the dictionaries are non-convolutional, we call the network SCNet. 14 \fTable 1: DenSaE\u2019s performance on MNIST dataset from disjoint training. A A+B is de\ufb01ned as # columns of A # columns of A + # columns of B \u00d7 100. SCNetLS SCNethyp 5A 395B 25A 375B 200A 200B A A+B 1.25% 6.25% 50% Acc. 94.16 % 98.32% 98.18% 98.18% 96.98% Rec. 1.95 6.80 6.83 6.30 3.04 A contribution to important class features 0% 0% 0% A contribution to important rec. features 8% 28% 58% 5.4.1 DenSaE strikes a balance between discriminative capability and reconstruction We study the case when DenSaE is trained on the MNIST dataset for joint reconstruction and classi\ufb01cation. We show (i) how the explicit imposition of sparse and dense representations in DenSaE helps to balance discriminative and representation power, and (ii) that DenSaE outperforms SCNet. We warm start the training of the classi\ufb01er using dictionaries obtained by \ufb01rst training the autoencoder, i.e., with \u03b2 = 0. Characteristics of the representations xT and uT : To evaluate the discriminative power of the representations learned by only training the autoencoder, we \ufb01rst trained the classi\ufb01er given the representations (i.e., \ufb01rst train A and B with \u03b2 = 0, then train C with \u03b2 = 1). We call this disjoint training. Table 1 shows the classi\ufb01cation accuracy (Acc.), \u21132 reconstruction loss (Rec.), and the relative contributions, expressed as a percentage, of the dense or sparse representations to classi\ufb01cation and reconstruction for disjoint training. Each column of [A, B], and of C, corresponds to either a dense or a sparse feature. For reconstruction, we \ufb01nd the indices of the 50 most important columns and report the proportion of these that represent dense features. For each of the 10 classes (rows of C), we \ufb01nd the indices of the 5 most important columns (features) and compute the proportion of the total of 50 indices that represent dense features. The \ufb01rst row of Table 1 shows the proportion of rows of [A, B] that represent dense features. Comparing this row, respectively to the third and fourth row reveals the importance of x for reconstruction, and of u for classi\ufb01cation. Indeed, the Acc. and Rec. of Table 1 show that, as the proportion of dense features increases, DenSaE gains reconstruction capability but results in a lower classi\ufb01cation accuracy. Moreover, in DenSaE, the most important features in classi\ufb01cation are all from B, and the contribution of A in reconstruction is greater than its percentage in the model, which clearly demonstrates that dense and sparse coding autoencoders balance discriminative and representation power. Both Tables 1 and 2 show that DenSaE outperforms SCNetLS in classi\ufb01cation and SCNethyp in reconstruction. Speci\ufb01cally, for joint training, Table 2 (columns 2,3 and row J1) shows that DenSaE outperforms SCNethyp for reconstruction (32.61 \u226a47.70) while it is competitive for classi\ufb01cation (98.61 > 98.59). Moreover, Table 2 (cols 1, 3 and J0.75 and J1) highlights that DenSAE has signi\ufb01cantly better classi\ufb01cation (98.61 > 96.06) and reconstruction (32.61 < 71.20) performances than SCNetLS. Moreover, we observed that in the absence of noise, training SCNetLS results in dense features with small negative biases, hence, making its performance close to DenSaE with large number of atoms in A. We see that SCNetLS in absence of a supervised classi\ufb01cation loss fails to learn discriminative features useful for classi\ufb01cation. On the other hand, enforcing sparsity in SCNethyp suggests that sparse representations are useful for classi\ufb01cation. Table 2: DenSaE\u2019s performance on MNIST dataset from joint (J\u03b2) training. SCNetLS SCNethyp 5A 395B 25A 375B 200A 200B A A+B 1.25% 6.25% 50% J0.5 Acc. 96.61% 97.97% 97.07% 97.68% 96.46% Rec. 2.01 0.87 0.58 0.58 0.34 J0.75 Acc. 96.91% 98.18% 98.19% 98.23% 97.64% Rec. 2.17 1.24 0.75 1.11 0.51 J0.95 Acc. 97.23% 98.51% 98.23% 98.44% 97.81% Rec. 4.48 1.03 1.22 1.32 0.67 J1 Acc. 96.06% 98.59% 98.61% 98.56% 98.40% Rec. 71.20 47.70 32.61 30.20 25.57 How do reconstruction and classi\ufb01cation capabilities change as we vary \u03b2 in joint training?: In joint training of the autoencoder and the classi\ufb01er, it is natural to expect that the reconstruction loss should increase compared to disjoint training. This is indeed the case for SCNetLS; as we go from disjoint to joint training and as \u03b2 increases (Table 2), the reconstruction loss increases and classi\ufb01cation accuracy has an overall increase. However, for \u03b2 < 1, joint training of both 15 \fa) DenSaE 4A 60B Noisy Ax Bu Denoised A B b) CSCNetLS Original Implicit Ax Implicit Bu Denoised Implicit A Unused Implicit B Figure 6: Visualization of a test image for \u03c4 = 50. a) DenSaE (4A, 60B), b) CSCNetLS. Bu images are scaled and plotted with a di\ufb00erent colormap for visualization purposes. networks that enforce some sparsity on their representations, SCNethyp and DenSaE, improves reconstruction and classi\ufb01cation. For purely discriminative training (\u03b2 = 1), DenSaE outperforms both SCNetLS and SCNethyp in classi\ufb01cation accuracy and representation capability. We speculate that this likely results from the fact that, by construction, the encoder from DenSaE seeks to produce two sets of representations: namely a dense one, mostly important for reconstruction and a sparse one, useful for classi\ufb01cation. In some sense, the dense component acts as a prior that promotes good reconstruction. 5.4.2 Denoising We trained DenSaE for supervised image denoising when \u03b2 = 0 using BSD432 and tested it on BSD68 [40]. All the networks are trained for 250 epochs using the ADAM optimizer and the \ufb01lters are initialized using the random Gaussian distribution. The initial learning rate is set to 10\u22124 and then decayed by 0.8 every 50 epochs. We set \u03f5 of the optimizer to be 10\u22123 for stability. At every iteration, a random patch of size 128\u00d7128 is cropped from the training image and zero-mean Gaussian noise is added to it with the corresponding noise level. We varied the ratio of number of \ufb01lters in A and B as the overall number of \ufb01lters was kept constant. We evaluate the model in the presence of Gaussian noise with standard deviation of \u03c4 = {15, 25, 50, 75}. Table 3: DenSaE\u2019s denoising performance on BSD68 as the ratio of \ufb01lters changes. \u03c4 1A63B 4A60B 8A56B 16A48B 32A32B 15 30.21 30.18 30.18 30.14 29.89 25 27.70 27.70 27.65 27.56 27.26 50 24.81 24.81 24.43 24.44 23.68 75 23.31 23.33 23.09 22.09 20.09 Ratio of number of \ufb01lters in A and B: Unlike reconstruction, Table 3 shows that the smaller the number of \ufb01lters associated with A, the better DenSaE can denoise images. We hypothesize that this is a direct consequence of our \ufb01ndings from Section 4 that if the column spaces of A and B are suitably unaligned, the easier is the recovery x and u. Dense and sparse vs. sparse coding: Table 4 shows that DenSaE (best network from Table 3) denoises images better than CSCNethyp, suggesting that the dense and sparse coding model represents images better than sparse coding. Table 4: DenSaE vs. CSCNet on BSD68. \u03c4 DenSaE CSCNethyp CSCNetLS 15 30.21 30.12 30.34 25 27.70 27.51 27.75 50 24.81 24.54 24.81 75 23.33 22.83 23.32 Dictionary characteristics: Figure 6(a) shows the decomposition of a noisy test image (\u03c4 = 50) by DenSaE. The \ufb01gure demonstrates that Ax captures low-frequency content while Bu captures high-frequency details (edges). This is corroborated by the smoothness of the \ufb01lters associated with A, and the Gabor-like nature of those associated with B [41]. We observed similar performance when we tuned \u03bbx, and found that, as \u03bbx decreases, Ax captures a lower frequencies, and Bu a broader range. CSCNet deviates from sparse coding and implicitly learns Ax + Bu model in the presence of noise: By training the biases, CSCNetLS deviates from the sparse coding model; the neural network\u2019s bias is directly related 16 \fto the sparsity enforcing hyper-parameter \u03bbu. The larger this bias, the sparser the representations. We observed that CSCNetLS automatically segments \ufb01lters into three groups: one with small bias, one with intermediate ones, and a third with large values (see 10). We found that the feature maps associated with the large bias values are all zero. Moreover, the majority of features are associated with intermediate bias values, and are sparse, in contrast to the small number of feature maps with small bias values, which are dense. We call the dictionary atoms, corresponding such small biases, implicit A. Similarly, dictionary atoms with moderate biases can be seen as implicit B. These observations suggest that autoencoders implementing the sparse coding model (y = Bu), when learning the biases by minimizing reconstruction error, implicitly perform two functions. First, they select the optimal number of \ufb01lters. Second, they partition the \ufb01lters into two groups: one that yields a dense representation of the input, and another that yields a sparse one. In other words, the architectures trained in this manner implicitly learn the dense and sparse coding model. Figure 6(b) shows the \ufb01lters. The above-mentioned interpretation of CSCNet with learned biases o\ufb00ers to revisit the optimization-based model used to construct the autoencoder; if the network implicitly learns a bipartite representation, why not explicitly model that structure? Through our experiments we showed that doing so leads to increased performance, the recovered dictionaries and representations do not deviate from their expected behavior, and the resulting network is more interpretable, as we can directly visualize and analyze each component. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1812.05786v1",
+                    "title": "Low-rank Matrix Completion in a General Non-orthogonal Basis",
+                    "abstract": "This paper considers theoretical analysis of recovering a low rank matrix\ngiven a few expansion coefficients with respect to any basis. The current\napproach generalizes the existing analysis for the low-rank matrix completion\nproblem with sampling under entry sensing or with respect to a symmetric\northonormal basis. The analysis is based on dual certificates using a dual\nbasis approach and does not assume the restricted isometry property (RIP). We\nintroduce a condition on the basis called the correlation condition. This\ncondition can be computed in time $O(n^3)$ and holds for many cases of\ndeterministic basis where RIP might not hold or is NP hard to verify. If the\ncorrelation condition holds and the underlying low rank matrix obeys the\ncoherence condition with parameter $\\nu$, under additional mild assumptions,\nour main result shows that the true matrix can be recovered with very high\nprobability from $O(nr\\nu\\log^2n)$ uniformly random expansion coefficients.",
+                    "authors": "Abiy Tasissa, Rongjie Lai",
+                    "published": "2018-12-14",
+                    "updated": "2018-12-14",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT",
+                        "math.NA",
+                        "math.OC"
+                    ],
+                    "main_content": "Introduction Recovering low-rank matrices from given incomplete linear measurements plays an important role in many problems such as image and video processing [4], model reduction [14], phase retrieval [9], molecular conformation [17, 32, 13], localization in sensor networks [12, 3], dimensionality reduction [29], recommender systems [21] as well as solving PDEs on manifold-structured data represented as incomplete distance [23], just to name a few. A natural framework to the low-rank recovery problem is rank minimization under linear constraints. However, this problem is NP-hard [26] and thus motivates alternative solutions. A series of theoretical papers [7, 11, 18, 25, 26] showed that the NPhard rank minimization problem for matrix completion can be obtained by solving the following convex nuclear norm minimization problem: minimize X\u2208Rn\u00d7n \u2225X\u2225\u2217 subject to Xi, j = Mi, j (i, j) \u2208\u2126 (1) where \u2225X\u2225\u2217denotes the nuclear norm de\ufb01ned as the sum of the singular values of X, and \u2126\u2282{(i, j)|i, j = 1, ..., n}, |\u2126| = m, denotes a random set that consists of the sampled indices. The remarkable fact is that, under certain conditions, the underlying low-rank matrix can be reconstructed exactly with high probability from only O(nr log2(n)) uniformly sampled measurements. The idea to use the nuclear norm as an approximation of the rank function was \ufb01rst discussed in [14]. Loosely, minimizing the sum of singular values will likely lead to a solution with many zero singular values resulting a low rank matrix. One generalization of the matrix completion problem in [18] considers \u2217This work is supported in part by NSF DMS\u20131522645 and an NSF Career Award DMS\u20131752934. \u2020Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A. (tasisa@rpi.edu). \u2021Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A. (lair@rpi.edu). 1 \fmeasurements with respect to a symmetric orthonormal basis and gives comparable theoretical guarantees based on the elegant dual certi\ufb01cate analysis. In particular, it shows that the true low rank matrix can be recovered with high probability from O(nr log2(n)) uniformly sampled measurements. The starting point and inspiration for this work was our recent work in [28] which studies a matrix completion problem with respect to a speci\ufb01c non-orthogonal basis. In this paper, we consider the matrix completion problem with respect to any non-orthogonal basis. Given a general unit-norm basis {w\u03b1}L \u03b1=1 which spans an L dimensional subspace S of Rn\u00d7n, the nuclear norm minimization program for this general matrix completion problem is provided by minimize X\u2208S \u2225X\u2225\u2217 subject to \u27e8X , w\u03b1\u27e9= \u27e8M , w\u03b1\u27e9 \u03b1 \u2208\u2126 (2) where \u2126denotes a random set sampling the basis indices. We are interested in the following two problems. 1. Could we obtain comparable recovery guarantees for the general matrix completion problem? 2. If the answer to (1) is a\ufb03rmative, what conditions are needed on the basis {w\u03b1} and \u2126? The main goal of this paper is a theoretical analysis of these two problems. We start by discussing few examples which show how the problem naturally arises in several applications. Euclidean Distance Geometry Problem. Given partial information on pairwise distances, the Euclidean distance geometry problem is concerned with constructing the con\ufb01guration of points. The problem has applications in diverse areas [17, 32, 29, 12, 23]. Formally, consider a set of n points P = {p1, p2, ..., pn}T \u2208Rr. Let D = [d2 i, j] denote the Euclidean distance matrix. The inner product matrix, also known as the Gram matrix and de\ufb01ned as Xi, j = \u27e8pi , p j\u27e9, is a positive semide\ufb01nite matrix of rank r. A minor analysis reveals that D and X can be related in the following way: Di, j = Xi,i + X j, j \u22122Xi, j. For r \u226an, consider the following nuclear norm minimization program to recover X. minimize X\u2208Rn\u00d7n \u2225X\u2225\u2217 subject to Xi,i + X j, j \u22122Xi, j = Di, j (i, j) \u2208\u2126 (3) X \u00b7 1 = 0 ; X = XT ; X \u2ab00 The constraint X \u00b7 1 = 0 \ufb01xes the translation ambiguity. The above minimization problem can be equivalently interpreted as a general matrix completion problem with respect to some operator basis w\u03b1. minimize X\u2208S+ \u2225X\u2225\u2217 subject to \u27e8X , w\u03b1\u27e9= \u27e8M , w\u03b1\u27e9 \u2200\u03b1 \u2208\u2126 (4) where w\u03b1 = 1 2(e\u03b11, \u03b11 + e\u03b12, \u03b12 \u2212e\u03b11, \u03b12 \u2212e\u03b12, \u03b11) and S+ = {X \u2208Rn\u00d7n | X = XT & X \u00b7 1 = 0} \u2229{X \u2208Rn\u00d7n | X \u2ab00}. The constant 1 2 is a normalization constant and e\u03b11, \u03b12 is a matrix whose entries are all zero except a 1 at the (\u03b11, \u03b12)-th entry. It can be veri\ufb01ed that {w\u03b1}L \u03b1=1, L = n(n\u22121) 2 , is a non-orthogonal basis for the linear space {X \u2208Rn\u00d7n | X = XT & X \u00b7 1 = 0}. Theoretical analysis of this problem was recently conducted by the authors of this paper in [28] and in fact inspires this work. 2 \fSpectrally Sparse Signal Reconstruction. The problem of signal reconstruction has many important practical applications. When the underlying signal is assumed to be sparse, the theory of compressive sensing states that the signal can be recovered by solving the convex l1 minimization problem [8]. In [4], the authors consider the recovery of spectrally sparse signal x \u2208Rn of known order r where r \u226an. Let H : Cn \u2192Cn1\u00d7n2 be the linear operator that maps a vector z \u2208Cn to a Hankel matrix Hz \u2208Cn1\u00d7n2 with n + 1 = n1 + n2. Denote the orthonormal basis of n1 \u00d7 n2 Hankel matrices by {H\u03b1}n1\u00d7n2 \u03b1=1 . After some analysis, the reconstruction problem is formulated as low-rank Hankel matrix completion problem [4]. \ufb01nd Hz subject to rank(Hz) = r P\u2126(Hz) = P\u2126(Hx) P\u2126is the sampling operator de\ufb01ned as: P\u2126(Z) = P \u03b1\u2208\u2126\u27e8Z , H\u03b1\u27e9H\u03b1 where \u2126is the random set that consists of the sampled Hankel basis. For the case where r is not speci\ufb01ed, one can consider the following nuclear norm minimization problem. minimize X\u2208Rn1\u00d7n2 \u2225X\u2225\u2217 subject to \u27e8X , H\u03b1\u27e9= \u27e8M , H\u03b1\u27e9 \u2200\u03b1 \u2208\u2126 (5) where X = Hz and M = Hx. (5) is now in the form of a general matrix completion problem. Signal Recovery Under Quadratic Measurements. Given a signal x \u2208Rn, consider quadratic measurements of the form ai = \u27e8x , wi\u27e9with some vector wi \u2208Rn. Given that m random measurements are available, it is of interest to determine if the underlying signal can be recovered. Using the lifting idea in [5], the signal recovery problem can be written as the following nuclear norm minimization problem. minimize X\u2208Rn\u00d7n \u2225X\u2225\u2217 subject to \u27e8X , W\u03b1\u27e9= \u27e8M , W\u03b1\u27e9 \u2200\u03b1 \u2208\u2126 (6) X = XT ; X \u2ab00 Above W\u03b1 = w\u03b1w\u2217 \u03b1 and \u2126is the random set that consists of the indices of the sampled vectors. In the case that ai are sampled independently and uniformly at random on the unit sphere, the above minimization problem is the well known PhaseLift problem [9]. One can consider a general case where the assumption is simply that the wi\u2019s are structured and form a basis. The framework introduced in this paper allows, under certain conditions, to state results about the uniqueness of this general recovery problem. Weighted Nuclear Norm minimization. The usual assumption in matrix completion is uniform random measurements. In the case of general sampling models, it has been argued that the nuclear norm minimization is not a suitable approach [27]. An alternative which has been argued to promote more accurate low rank solutions [5, 15] is the weighted nuclear norm minimization [16, 27]. With D as a weight matrix, the weighted nuclear norm minimization 3 \fproblem is given by minimize \u2225DX\u2225\u2217 subject to \u27e8X , w\u03b1\u27e9= \u27e8M , w\u03b1\u27e9 \u2200\u03b1 \u2208\u2126 (7) For simplicity, let D be a diagonal matrix. In this case, \u2225DX\u2225can be interpreted as weighting certain rows of X more than others. Introducing the matrix Y = DX, the above minimization problem can equivalently be rewritten as follows. minimize \u2225Y \u2225\u2217 subject to \u27e8Y , D\u22121w\u03b1\u27e9= \u27e8M , D\u22121w\u03b1\u27e9 \u2200\u03b1 \u2208\u2126 (8) M is the weighted ground matrix M de\ufb01ned as M = DM. This is a general matrix completion problem with respect to the basis D\u22121w\u03b1. Since D is a diagonal matrix, the set {D\u22121w\u03b1} is a basis. Of interest is the following question: What kind of choices for D lead to successful recovery algorithms? Challenges The minimization problem in (2) has random linear constraints which can be expressed as L(X) where L is the appropriate linear operator. Using the work in [26], one approach to show uniqueness of the general matrix completion problem is to check if L obeys the restricted isometry property (RIP) condition. Our basis are structured and deterministic and so in general the RIP condition does not hold. As an example, consider the nuclear norm minimization program in (4) for the Euclidean distance geometry problem. We choose any (i, j) < \u2126and construct a matrix X with Xi, j = X j,i = Xi,i = X j, j = 1 and zero everywhere else. One can easily check that L(X) = 0 which shows that the RIP condition does not hold. The general matrix completion problem resembles the matrix completion problem with respect to a symmetric orthonormal basis \ufb01rst considered in [18]. However, the basis {w\u03b1}L \u03b1=1 in the general matrix completion problem are not necessarily orthogonal. This has the implication that the measurements \u27e8w\u03b1, X\u27e9are not compatible with the expansion coe\ufb03cients of M. One solution is to employ any of the basis orthogonalization algorithms and consider the minimization problem in the new orthonormal basis. This solution however is not useful since the measurements can not be treated as independent in the new basis. As such, the lack of orthogonality mandates an alternative analysis to show that the general matrix completion problem admits a unique solution. In this paper, the analysis is based on the dual certi\ufb01cate approach [7]. Motivated by the work of David Gross [18], where the author generalizes the matrix completion problem to any symmetric orthonormal basis, our recent work in [28] considered the Euclidean distance geometry problem (4). The main technical di\ufb00erence from the work of [18] is that the basis in the Euclidean distance geometry problem is non-orthogonal. More precisely, in terms of analysis, the di\ufb00erence is mainly due to the sampling operator, which is central in the analysis of the matrix completion problem. For the orthonormal basis case, the sampling operator is self-adjoint. For the Euclidean distance geometry problem, the sampling operator is not self-adjoint and requires alternative analysis. Much inspired by our previous work, we are interested in generalizing the result to any non-orthogonal basis. The current paper is a culmination of this e\ufb00ort. The work in [22] develops RIPless recovery analysis for the compressed sensing problem from anisotropic measurements. The current work could be interpreted as analogue of this work to the general matrix completion problem. In particular, the notion of anistropic measurements for compressive sensing corresponds to non-orthogonal matrix basis for matrix completion. There are however di\ufb00erences as a direct analogue of some of the technical estimates in [22] is not directly applicable to the general matrix completion problem. 4 \fContributions In this paper, under suitable sampling conditions, a dual basis approach is used to show that the general matrix completion problem admits a unique solution. Introducing a dual basis to w\u03b1, denoted by z\u03b1, ensures that the measurements \u27e8X , w\u03b1\u27e9in (2) are compatible with expansion coe\ufb03cients of M. Based on the framework of the dual basis approach, we show that the minimization problem recovers the underlying matrix under suitable conditions. Two main contributions of this paper are as follows. 1. A dual basis approach is used to prove a uniqueness result for the general matrix completion problem. The main result shows that if the number of random measurements m is of order O(nr log2 n), under certain assumptions, the nuclear norm minimization program recovers the underlying low-rank solution with very high probability. A key part of our proof uses the operator Cherno\ufb00bound. This part of the proof based on the Cherno\ufb00bound is simple and might \ufb01nd use in other problems. 2. An important condition, named the correlation condition, is introduced. This condition determines whether the nuclear norm minimization program succeeds for a given general matrix completion problem. The well-known RIP condition might not hold for the case of deterministic measurements. However, the correlation condition could hold for deterministic measurements and more importantly can be checked in polynomial time. Outline The outline of the paper is as follows. Section 2 introduces the correlation condition and discusses the dual basis approach. The proof of the main result is presented in section 3. The key components of the proof can be described as follows. The general matrix completion problem is a convex minimization problem for which a su\ufb03cient condition to optimality is the KKT condition. Namely, if one can show that there exists a dual certi\ufb01cate Y that satis\ufb01es certain conditions, it follows that the general matrix completion problem has a unique solution. The construction of Y follows the elegant gol\ufb01ng scheme proposed in [18]. The bulk of the theoretical work then focuses on using this scheme and proving that the conditions hold with very high probability. The implication of the proof is that there is a unique solution to the general matrix completion problem with very high probability. Section 4 concludes the work. Notation The notations used in the paper are summarized in Table 1. x Vector \u2225X\u2225F Frobenius norm X Matrix \u2225X\u2225\u221e sup\u2225v\u2225\u221e=1 \u2225Xv\u2225\u221e X Operator \u2225X\u2225 sup\u2225v\u22252=1 \u2225Xv\u22252 XT Transpose \u2225X\u2225\u2217 Nuclear norm Tr(X) Trace \u2225A\u2225 sup\u2225X\u2225F=1 \u2225AX\u2225F. \u27e8X , Y \u27e9 Trace(XTY ) \u03bbmax, \u03bbmin Maximum, Minimum eigenvalue 1 A vector or matrix of ones Sgn X Usign (\u03a3)V T; here [U, \u03a3, V ] = svd(X) 0 A vector or matrix of zeros \u2126, I Random sampled set, Universal set Table 1: Notations 2 Matrix Completion Problem under a Non-orthogonal Basis In this section, we introduce a new condition referred as a correlation condition for low-rank matrix completion in a non-orthogonal basis. We will also discuss a dual basis formulation which plays an important role in our main result. 5 \f2.1 Correlation Parameter We consider an L dimensional subspace S \u2282Rn\u00d7n as the feasible set of the general matrix completion. We allow L \u2264n2 such as the case where the feasible solutions naturally satisfy linear constraints. Given a complete set of unit-norm basis {w\u03b1}L \u03b1=1 of the subspace S, any X \u2208S can be determined if one speci\ufb01es all the measurements {\u27e8X , w\u03b1\u27e9}L \u03b1=1. The problem studied in this paper considers the case where we have random access to a few of these measurements. Leaving the precise notion of \u201ca few\u201d for later, we consider the following question: Are there non-orthogonal basis for which the matrix completion framework, learning from few measurements, still works? In this paper, we note that as long as a certain condition, named correlation condition, on the basis matrices is satis\ufb01ed, the sample complexity of low rank matrix completion with respect to non-orthogonalbasis is of the same order as sample complexity of low rank matrix completion with respect to an orthogonal basis. Intuitively, there is decoupling in orthogonality which means that every additional measurement is informative. However, with non-orthogonal basis, the basis might be correlated and every additional measurement might not necessarily be informative. In the speci\ufb01c case of the general matrix completion problem, the goal is to rigorously show that, under certain conditions, a low rank matrix can be recovered from few random non-orthogonal measurements. The intuitive arguments above motivate the following correlation condition which loosely informs how far the nonorthogonal basis is from an orthonormal basis. De\ufb01nition 1. The unit-norm basis {w\u03b1 \u2208Rn\u00d7n}L n=1 has correlation parameter \u00b5 if there is a constant \u00b5 \u22650 such that the following two equations hold \r \r \r \r \r \r \r 1 n X \u03b1 wT \u03b1w\u03b1 \u2212I \r \r \r \r \r \r \r \u2264\u00b5 & \r \r \r \r \r \r \r 1 n X \u03b1 w\u03b1wT \u03b1 \u2212I \r \r \r \r \r \r \r \u2264\u00b5 (9) Intuitively, the above de\ufb01nition describes that the operators 1 n P \u03b1 wT \u03b1w\u03b1 and 1 n P \u03b1 w\u03b1wT \u03b1 are nearly isometric to the identity operator. To understand the correlation condition, we \ufb01rst establish certain properties and follow by computing the correlation condition of certain basis matrices. Lemma 1. [Properties and examples of correlation condition] Given a unit-norm basis {w\u03b1 \u2208Rn\u00d7n}L n=1 with correlation parameter \u00b5, the following statements hold: a. The correlation parameter is bounded above by n, that is, \u00b5 \u2264n. b. If the correlation condition holds, it follows that \u03bbmax \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed L X \u03b1=1 wT \u03b1w\u03b1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2264(\u00b5 + 1)n & \u03bbmax \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed L X \u03b1=1 w\u03b1wT \u03b1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2264(\u00b5 + 1)n c. If {w\u03b1}L n=1 is an orthonormal basis L = n2, the correlation condition holds with \u00b5 = 0. d. For the basis matrices in the Euclidean distance geometry problem, the correlation condition holds with \u00b5 = 1. Proof. 1. For positive semide\ufb01nite matrices A and B, the norm inequality \u2225A \u2212B\u2225\u2264max(\u2225A\u2225, \u2225B\u2225) holds. Let A = 1 n P \u03b1 wT \u03b1w\u03b1 and let B = I. Applying the triangle inequality, it follows that \r \r \r 1 n P \u03b1 wT \u03b1w\u03b1 \r \r \r \u22641 n P \u03b1 ||wT \u03b1|| ||w\u03b1|| \u2264 n. Therefore, \r \r \r 1 n P \u03b1 wT \u03b1w\u03b1 \u2212I \r \r \r \u2264max(n, 1) = n. An analogous argument obtains \r \r \r 1 n P \u03b1 w\u03b1wT \u03b1 \u2212I \r \r \r \u2264n. 6 \fHence, \u00b5 \u2264n as desired. It should be remarked that, although \u00b5 is bounded above by n, the theoretical analysis requires \u00b5 = O(1). 2. Note that \u03bbmax \u0010PL \u03b1=1 wT \u03b1w\u03b1 \u0011 = \u2225PL \u03b1=1 wT \u03b1w\u03b1 \u2212nI + nI\u2225\u2264\u2225PL \u03b1=1 wT \u03b1w\u03b1 \u2212nI\u2225+ \u2225nI\u2225= (\u00b5 + 1)n. A similar argument results \u03bbmax \u0010PL \u03b1=1 w\u03b1wT \u03b1 \u0011 \u2264(\u00b5 + 1)n. 3. For an orthonormal basis in S = Rn\u00d7n, the completeness relation states that Pn2 \u03b1=1 w\u03b1wT \u03b1 = Pn2 \u03b1=1 wT \u03b1w\u03b1 = nI. The proof of this fact is standard. For ease of reference, a proof is included in Lemma A.1. Using the completeness relation, it follows that the correlation condition holds with \u00b5 = 0. 4. For the Euclidean distance geometry problem, the basis are symmetric so it su\ufb03ces to consider 1 n P \u03b1 w2 \u03b1. A short calculation results 1 n P \u03b1 w2 \u03b1 = 1 2n(nI \u221211T). The correlation condition bound amounts to \ufb01nding the operator norm of 1 2\u2225I + 1 n11T\u2225. Since the maximum eigenvalue of 1 n11T is 1, 1 2\u2225I + 1 n11T\u2225= 1. Hence, for the Euclidean distance geometry problem, the correlation condition holds with \u00b5 = 1. \u25a1 Comparison with RIP condition The restricted isometry property, RIP in short, was introduced \ufb01rst in [10] for the compressive sensing problem. If the measurement matrix in the compressive sensing problem, denoted by A, obeys the RIP condition, it implies that the underlying sparse vector can be recovered by solving a convex l1 minimization problem. The notion of the restricted isometry property can be extended to the matrix completion problem and its de\ufb01nition, which \ufb01rst appeared in [26], is restated below. De\ufb01nition 2. For every integer r with 1 \u2264r \u2264n, a linear map A : Rn\u00d7n \u2192Rm obeys the RIP condition with isometry constant \u03b4r if (1 \u2212\u03b4r)\u2225X\u22252 F \u2264\u2225AX\u22252 2 \u2264(1 + \u03b4r)\u2225X\u22252 F holds for all matrices X of rank at most r. In particular, the analysis in [26] shows that if the RIP condition holds for the measurement operator of the general matrix completion problem, the underlying low rank matrix can be recovered by solving the convex nuclear norm minimization problem. With this, one could only consider certifying that the measurement operator satis\ufb01es RIP. For example, the RIP condition is satis\ufb01ed with high probability for random measurement operators [26]. RIP condition could fail if one considers deterministic measurement operators. The intuition is that, for this case, one could in some fashion construct matrices which belong to the null space of A implying that RIP does not hold. As noted earlier, the RIP condition fails to hold for the EDG problem where the proof of this fact relied on constructing a counterexample, a sparse matrix X, which violates the RIP condition. On the other hand, Lemma 1 (4) shows that the correlation condition holds for the EDG problem with correlation parameter \u00b5 = 1. Additionally, one could construct counter examples that imply that the RIP condition does not hold for the standard matrix completion problem and the phase retrieval problem. Therefore, in certain deterministic settings where the RIP condition does not hold, an RIPless analysis based on correlation parameter can be employed. Another advantage of the correlation condition is its computational complexity. Given a measurement operator, checking whether RIP condition holds or not is hard. For example, for the compressive sensing problem, it has been shown that [2] checking whether RIP holds or not is NP-hard. To the best of our knowledge, there is no analogous work for the matrix RIP condition. However, using the relations of vector RIP to matrix RIP as detailed in [24], it can be anticipated that checking the matrix RIP is also 7 \fNP-hard. On the other hand, the computational complexity of checking the correlation condition is at most O(n3). As such, given a measurement operator, it can easily be checked whether it satis\ufb01es the correlation condition or not. If the result is positive, an RIPless analysis can be carried out as will be detailed in this paper. If the correlation condition does not hold, no conclusion can be made. The above comparison is meant to illustrate that, for the case of deterministic measurements, the correlation condition could be used and an RIPless analysis can be carried out for the general matrix completion problem. 2.2 Dual basis formulation For the general matrix completion problem in (2), the measurements are in the form \u27e8X , w\u03b1\u27e9. If w\u03b1 is orthonormal, X can be expanded as X = P \u03b1\u27e8X , w\u03b1\u27e9w\u03b1 where \u27e8X , w\u03b1\u27e9are the expansion coe\ufb03cients. However, for non-orthogonal basis, this expansion does not hold. Since the measurements are inherent to the problem, the ideal expansion will be of the form X = P \u03b1\u27e8X , w\u03b1\u27e9z\u03b1. The idea of the dual basis approach is to realize this form with the implication that the expansion coe\ufb03cients match the random measurements. This approach is brie\ufb02y summarized below and we refer the interested reader to [28] where the dual basis approach is \ufb01rst considered in the context of the Euclidean distance geometry problem. Given the basis {w\u03b1}L \u03b1=1 of S, we de\ufb01ne the matrix H as H\u03b1, \u03b2 = \u27e8w\u03b1 , w\u03b2\u27e9and write H\u22121 \u03b1, \u03b2 as H\u03b1, \u03b2. After minor analysis, it is straightforward to check z\u03b1 = P \u03b2 H\u03b1, \u03b2w\u03b2 is a dual basis satisfying \u27e8z\u03b1 , w\u03b2\u27e9= \u03b4\u03b1, \u03b2. We note the following relations which will be used in later analysis, H = W TW , H\u22121 = ZTZ and Z = W H\u22121 where W = [w1, w2, ..., wL] and Z = [z1, z2, ..., zL] denote the matrix of vectorized basis matrices and vectorized dual basis matrices respectively. The sampling operator, a central operator in the analysis of the general matrix completion problem, is de\ufb01ned as follows. R\u2126: X \u2208S \u2212\u2192L m X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9z\u03b1 (10) The sampling model, for the set \u2126, is uniform random with replacement from I = {1, \u00b7 \u00b7 \u00b7 , L}. The size of \u2126is denoted by m and the scaling factor L m is simply for convenience of analysis. The adjoint operator of the sampling operator also appears in the analysis and has the following form. R\u2217 \u2126: X \u2208S \u2212\u2192L m X \u03b1\u2208\u2126 \u27e8X , z\u03b1\u27e9w\u03b1 (11) Using the sampling operator R\u2126, we can write (2) as follows. minimize X\u2208S \u2225X\u2225\u2217 subject to R\u2126(X) = R\u2126(M) (12) Another operator which appears in the analysis is the restricted frame operator de\ufb01ned as follows. F\u2126: X \u2208S \u2212\u2192L m X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9w\u03b1 (13) It can be readily veri\ufb01ed that the restricted frame operator is self-adjoint and positive semide\ufb01nite. Coherence. One can not expect to have successful reconstruction for an arbitrary matrix M, in particular, when M has very few non-zero expansion coe\ufb03cients. Thus, the notion of coherence is introduced in [7] to guarantee successful 8 \fcompletion. Consider the singular value decomposition of M = Pr k=1 \u03bbkukvT k . Let\u2019s write U = span {u1, ..., ur}, U\u22a5= span{ur+1, ..., un} as the orthogonal complement of U, V = span {v1, ..., vr} and V \u22a5= span{vr+1, ..., vn} as the orthogonal complement of V . Projections onto these spaces appear frequently in the analysis and can be summarized as follows. Let PU and PV denote the orthogonal projections onto U and V respectively. PU \u22a5and PV \u22a5are de\ufb01ned analogously. De\ufb01ne T = {U\u0398T + \u0398V T : \u0398 \u2208Rn\u00d7r} to be the tangent space of the rank r matrix in Rn\u00d7n at M. The orthogonal projection onto T is given by PTX = PUX + XPV \u2212PUXPV (14) It then follows that PT\u22a5X = X \u2212PTX = PU \u22a5XPV \u22a5. We de\ufb01ne a coherence condition as follows. De\ufb01nition 3. The aforementioned rank r matrix M \u2208Rn\u00d7n has coherence \u03bd with respect to unit norm basis {w\u03b1}\u03b1\u2208I if the following estimates hold max \u03b1\u2208I X \u03b2\u2208I \u27e8PT w\u03b1 , w\u03b2\u27e92 \u2264\u03bd r n (15) max \u03b1\u2208I X \u03b2\u2208I \u27e8PT z\u03b1 , w\u03b2\u27e92 \u2264cv\u03bd r n (16) max \u03b1\u2208I \u27e8w\u03b1 , UV T\u27e92 \u2264\u03bd r n2 (17) where {z\u03b1}\u03b1\u2208I is the dual basis of {w\u03b1}\u03b1\u2208I and cv is a constant satisfying cv \u2265\u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e. Remark 1. Since the above coherence conditions are a central part of the analysis, we consider equivalent simpli\ufb01ed forms. We start with a bound on \u2225PT w\u03b1\u22252 F using (15) and Lemma A.3 which results \u03bbmin(H) \u2225PT w\u03b1\u22252 F \u2264max \u03b1\u2208I X \u03b2\u2208I \u27e8PT w\u03b1 , w\u03b2\u27e92 \u2264\u03bd r n =\u21d2 \u2225PT w\u03b1\u22252 F \u2264\u03bbmax(H\u22121) \u03bd r n Next, using the above inequality, the following bound for \u2225PT z\u03b1\u2225F follows. \u2225PT z\u03b1\u2225F \u2264 X \u03b2\u2208I \u2225H\u03b1, \u03b2 PT w\u03b2\u2225F = X \u03b2\u2208I |H\u03b1,\u03b2| \u2225PT w\u03b2\u2225F \u2264\u2225H\u22121\u2225\u221e p \u03bbmax(H\u22121) r\u03bdr n It should be noted that the analysis presented in this paper requires that \u2225H\u22121\u2225\u221eis at most O(1). Finally, we use the previous inequality and (17) to derive a bound for \u27e8z\u03b1 , UV T\u27e92. \u27e8z\u03b1 , UV T\u27e92 = \u27e8 X \u03b2\u2208I H\u03b1, \u03b2w\u03b2 , UV T\u27e92 \u2264max \u03b2\u2208I \u27e8w\u03b2 , UV T\u27e92 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X \u03b2\u2208I |H\u03b1,\u03b2| \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 2 \u2264\u2225H\u22121\u22252 \u221e \u03bdr n2 9 \fThe coherence conditions can now be summarized as follows. max \u03b1\u2208I \u2225PT w\u03b1\u22252 F \u2264\u03bbmax(H\u22121) \u03bd r n (18) max \u03b1\u2208I \u2225PT z\u03b1\u22252 F \u2264\u2225H\u22121\u22252 \u221e\u03bbmax(H\u22121) \u03bd r n (19) max \u03b1\u2208I \u27e8z\u03b1 , UV T\u27e92 \u2264\u2225H\u22121\u22252 \u221e \u03bdr n2 (20) The coherence parameter is indicative of concentration of information in the ground truth matrix. If the underlying matrix has low coherence, each measurement is equally informative as the other. On the other hand, if a matrix has high coherence, it means that the information is concentrated on few measurements. Sampling Model. For the general matrix completion problem, the basis matrices are sampled uniformly at random with replacement. The advantage of this model is that the sampling process is independent. This property is crucial since our analysis uses concentration inequalities for i.i.d matrix valued random variables. A disadvantage of this sampling process is that the same measurement could be repeated and the analysis needs to account for the number of duplicates. Remark 2. Although we choose uniform sampling with replacement model, the analysis in this paper also works for sampling with out replacement. The latter model has the advantage that there are no duplicate measurements but the choice also means that the sampling is no longer independent. This in turn has the implication that concentration inequalities for i.i.d matrix valued random variables can not be used freely. However, in the work of Hoe\ufb00ding [20], for Hoefdding inequality, it is argued that the results derived for the case of the sampling with replacement also hold true for the case of sampling without replacement. In [19], it is shown that matrix concentration inequalities resulting from the operator Cherno\ufb00bound technique [1] also hold true for uniform sampling without replacement. With this, the main analysis in this paper holds with or without replacement. The use of uniform sampling with out replacement model in the analysis leads to a gain in terms of the number of measurements m. However, this gain is rather minimal and for sake of streamlined presentation, the uniform sampling with replacement is adopted in this paper. 3 Main Result and proof The main result of this paper shows that the nuclear norm minimization program for the general matrix completion problem in (12) recovers the underlying matrix with very high probability. A precise statement is stated in the theorem below. With out loss of generality and for ease of analysis, the theorem considers square matrices. The proof for rectangular matrices follows with minor modi\ufb01cations. Theorem 1. Let M \u2208Rn\u00d7n be a matrix of rank r that obeys the coherence conditions (15), (16) and (17) with coherence \u03bd and satis\ufb01es the correlation condition (9) with correlation parameter \u00b5. De\ufb01ne C as follows: C = max \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u03bbmax(H\u22121)3, cv, (\u00b5 + 1)\u2225H\u22121\u2225\u221e min( (\u00b5 + 1)\u2225H\u22121\u2225\u221e, 1 4)2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8with parameter cv from (16). Assume m measurements, {\u27e8M , w\u03b1\u27e9}\u03b1\u2208\u2126, are sampled uniformly at random with replacement. For \u03b2 > 1, if m \u2265log2   4 \u221a 2L\u03bbmax(H) \u03bbmin(H) \u221ar ! nr   48\u0002C\u03bd + n Lr \u0003\u0002 \u03b2 log(n) + log   4 log2   4 \u221a 2L\u03bbmax(H) \u03bbmin(H) \u221ar !! \u0003! (21) the solution to (12) is unique and equal to M with probability at least 1 \u2212n\u2212\u03b2. 10 \fSince the general matrix completion problem in (12) is convex, the optimizer can be characterized using the KKT conditions. A compact and simple form of these conditions is derived in [7]. With this, a brief outline of the proof is as follows. The proof is divided into two main parts. In the former part, we show that if the aforementioned optimality conditions hold, then M is a unique solution to the minimization problem. The latter and main part of the proof is concerned with showing that, under certain assumptions, these conditions do hold with very high probability. The implication of this is that, for a suitable choice of m, M is a unique solution for the general matrix completion problem. Our proof adapts arguments from [18, 28]. Few remarks on the di\ufb00erence of our proof to the matrix completion proofs in [18, 25] and our previous work [28] are in order. 1. The operator R\u2126is not self-adjoint. The main implication of this is that the operator PTR\u2126PT, an important operator in matrix completion analysis, is no longer isometric to PT. It turns out the appropriate operator to consider is PTR\u2217 \u2126R\u2126PT where the goal is to show that this operator is nearly isometric to PT. However, this approach is not amenable to simple analysis. In this work, the main argument is based on showing that the minimum eigenvalue of the operator PTF\u2126PT is bounded away from zero with very high probability. To prove this fact, the operator Cherno\ufb00bound is employed. The interpretation of this bound is that, restricted to the space T, the operator PTF\u2126PT is full rank. If the measurement basis is orthogonal, F\u2126= R\u2126, the implication is that the operator PTR\u2126PT on T is invertible. With this, PTF\u2126PT can be understood as the operator analogue of PTR\u2126PT for non-orthogonal measurements. 2. The measurement basis is non-orthogonal. Since we use the dual basis approach, the spectrum of the matrices H and H\u22121 become important. However, since we do not work with a \ufb01xed basis, all the constants are unknown. This is particularly relevant and presents some challenge in the use of concentration inequalities and will be apparent in later analysis. For the matrix completion problem, with measurement basis ei j, the theoretical lower bound of O(nr\u03bd log n) was established in [11]. Note that, if C = O(1) and \u00b5 = O(1), Theorem 1 requires on the order of nr\u03bd log2 n measurements which is only log(n) factor away from the optimal lower bound. The order of theorem 1 is also the same order as those used in [18, 25]. These works consider the low rank recovery problem with any orthogonal basis and the matrix completion problem respectively. Before the proof of main result, we illustrate Theorem 1 on some of the examples discussed in the introduction. Euclidean Distance Geometry Problem: A matrix completion formulation and theoretical analysis of the Euclidean distance geometry problem appears in [28]. Using the existing analysis in [28], L = n(n\u22121) 2 , \u03bbmax(H\u22121) \u22644, \u03bbmin(H\u22121) = 1 8n and \u2225H\u22121\u2225\u221e= 8. The constant cv, which satis\ufb01es cv \u2265\u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e, is set to 32. Using Lemma 1(d), the correlation condition for the Euclidean Distance Geometry Problem holds with \u00b5 = 1. Using Theorem 1, it can be seen that the number of samples needed to recover the underlying low rank Gram matrix for the Euclidean distance geometry problem is O(nr\u03bd log2 n). Spectrally Sparse Signal Reconstruction: For simplicity, consider the case n1 = n2 = n. For this problem, the orthonormality of the Hankel basis implies that H = I. Therefore, \u03bbmax(H\u22121) = \u03bbmin(H\u22121) = \u2225H\u22121\u2225\u221e= 1 and L = n2. The correlation condition holds trivially with \u00b5 = 0. The number of samples needed to cover the underlying low rank matrix is O(nr\u03bd log2 n). 11 \fWeighted Nuclear Norm Minimization: As discussed earlier, the weight matrix D is diagonal. For simplicity, assume that Pn i=1 Di,i = 1. In this case, the size of a diagonal entry informs the proportion of weight assigned to the corresponding row of the true matrix. For the main result in Theorem 1 to hold, certain assumptions are necessary. First, the basis {D\u22121w\u03b1}L \u03b1=1 needs to satisfy the correlation condition. Second, the size of the maximum and minimum eigenvalues of the matrix \u00af H = (D\u22121W )T(D\u22121W ) = W T(D\u22121)2W is important. After minor analysis, with H = W TW , \u03bbmax( \u00af H) \u2264\u03bbmax(H) + \u03bbmax \u0010 W T[(D\u22121)2 \u2212I]W \u0011 . A similar analysis leads to \u03bbmin( \u00af H) \u2265\u03bbmin(H) + \u03bbmin \u0010 W T[(D\u22121)2 \u2212I]W \u0011 . Assume that, with no weighting, the original basis w\u03b1 satis\ufb01es the necessary conditions for Theorem 1 to hold. For the weighted nuclear norm minimization to hold, one choice of su\ufb03cient conditions is that \u03bbmax \u0010 W T[(D\u22121)2 \u2212I]W \u0011 = c1\u03bbmax(H) and \u03bbmin \u0010 W T[(D\u22121)2 \u2212I]W \u0011 = c2\u03bbmin(H) where c1, c2 are dimension-free constants. A given choice of D can be checked if it veri\ufb01es these criterion. If the result is positive, the complexity of O(nr\u03bd log2 n) from Theorem 1 can be attained. Note that the conditions above are su\ufb03cient but not necessary. Sharper and more explicit condition on D requires further analysis and is not within the scope of this paper. It can be surmised that, in practice, one is working with a \ufb01xed basis matrix W and controlling the spectrum of \u00af H in terms of D is more amenable to analysis. Remark 3. It should be remarked that the minimum number of samples noted in Theorem 1 can be lowered, the constants could be improved, if one is working with explicit basis. The analysis presented here is generic and does not assume speci\ufb01c structure of the basis. Where the latter is readily available, most inequalities appearing in the technical details can be tightened lowering the sample complexity. For instance, if the basis is orthonormal as in the problem of spectrally sparse signal construction, the analysis in [18] gives tight results. In general, for explicit basis with some structure, one can adopt the analysis in this paper and improve certain bounds. Now we return to the main proof. For ease, the proof is structured into several intermediate results. The starting result is Theorem 2 which shows that if certain conditions hold, M is a unique solution to (12). Theorem 2. Given X \u2208S, let \u2206= X \u2212M denote the deviation from the true low rank matrix M. \u2206T and \u2206T\u22a5 denote the orthogonal projection of \u2206to T and T\u22a5respectively. For any given \u2126with |\u2126| = m, the following two statements hold. (a). If \u2225\u2206T\u2225F \u2265 \u221a 2L\u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F and \u03bbmin (PT F\u2126PT) > 1 2\u03bbmin(H), then R\u2126\u2206, 0. (b). If \u2225\u2206T\u2225F < \u221a 2L\u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F for \u2206\u2208ker R\u2126, and there exists a Y \u2208range R\u2217 \u2126satisfying, \u2225PTY \u2212Sgn M\u2225F \u22641 4 r 1 2L \u03bbmin(H\u22121) \u03bbmax(H\u22121) and \u2225PT\u22a5Y \u2225\u22641 2 (22) then \u2225X\u2225\u2217= \u2225M + \u2206\u2225\u2217> \u2225M\u2225\u2217. Theorem 2(a) states that, for \u201clarge\u201d \u2206T, any deviation from M is not in the null space of the operator. Theorem 2(b) states that, for \u201csmall\u201d \u2206T, deviations from M increase the nuclear norm. The theorem at hand is deterministic and at this stage no assumptions are made on the construction of the set \u2126. As long as the assumptions of the theorem are satis\ufb01ed, the theorem will hold true. After proving the theorem, we proceed to argue that the conditions in the theorem hold with very high probability. This will require certain sampling conditions and a suitable choice of m = |\u2126|. 12 \f3.1 Proof of Theorem 2 Proof of Theorem 2(a). First, observe that \u2225R\u2126\u2206\u2225F = \u2225R\u2126\u2206T + R\u2126\u2206T\u22a5\u2225F \u2265\u2225R\u2126\u2206T\u2225F \u2212\u2225R\u2126\u2206T\u22a5\u2225F. Since we want to show that R\u2126\u2206, 0, the observation leads to considering a lower bound for \u2225R\u2126\u2206T\u2225F and an upper bound for \u2225R\u2126\u2206T\u22a5\u2225F. For any X, \u2225R\u2126X\u22252 F can be bounded as follows. \u2225R\u2126X\u22252 F = \u27e8X , R\u2217 \u2126R\u2126X\u27e9= L2 m2 X \u03b2\u2208\u2126 X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9\u27e8X , w\u03b2\u27e9\u27e8z\u03b1 , z\u03b2\u27e9= L2 m2 X \u03b2\u2208\u2126 X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9\u27e8X , w\u03b2\u27e9H\u03b1, \u03b2 where the last inequality uses the fact that \u27e8z\u03b1 , z\u03b2\u27e9= H\u03b1, \u03b2. The min-max theorem applied to the above equation results L2 m2 \u03bbmin(H\u22121) X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e92 \u2264\u2225R\u2126X\u22252 F \u2264L2 m2 \u03bbmax(H\u22121) X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e92 (23) Setting X = \u2206T\u22a5and using the right inequality above, we obtain \u2225R\u2126\u2206T\u22a5\u22252 F \u2264L2 m2 \u03bbmax(H\u22121) X \u03b1\u2208\u2126 \u27e8\u2206T\u22a5, w\u03b1\u27e92 \u2264m L2 m2 \u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u22252 F (24) where the last inequality uses the fact that P \u03b1\u2208I\u27e8X , w\u03b1\u27e92 \u2264\u03bbmax(H)\u2225X\u22252 F (Lemma A.3) and the constant m bounds the maximum number of repetitions for any given measurement. Analogously, setting X = \u2206T and using the left inequality in (23), we obtain \u2225R\u2126\u2206T\u22252 F \u2265L2 m2 \u03bbmin(H\u22121) X \u03b1\u2208\u2126 \u27e8\u2206T , w\u03b1\u27e92 = L m\u03bbmin(H\u22121)\u27e8\u2206T , F\u2126\u2206T\u27e9 (25) The next step considers the projection onto T of the restricted frame operator and applies the min-max theorem resulting the following inequality. \u2225R\u2126\u2206T\u22252 F \u2265L m\u03bbmin(H\u22121)\u27e8\u2206T , F\u2126\u2206T\u27e9= L m\u03bbmin(H\u22121)\u27e8\u2206T , PT F\u2126PT \u2206T\u27e9 \u2265L m\u03bbmin(H\u22121)\u03bbmin(PT F\u2126PT) \u2225\u2206T\u22252 F (26) Above, the \ufb01rst equality follows since PT is self adjoint and evidently \u2206T \u2208T. The inequality in (26) can be reduced further using the assumption \u03bbmin (PT F\u2126PT) > 1 2\u03bbmin(H) in the theorem resulting \u2225R\u2126\u2206T\u22252 F > L 2m\u03bbmin(H\u22121)\u03bbmin(H) \u2225\u2206T\u22252 F (27) Finally, use the inequalities in (24) and (27) and the assumption in the theorem to show that \u2225R\u2126\u2206\u2225F > 0 as follows. \u2225R\u2126\u2206\u2225F > r L 2m\u03bbmin(H\u22121)\u03bbmin(H) \u2225\u2206T\u2225F \u2212 L \u221am s \u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F \u2265 r L 2m\u03bbmin(H\u22121)\u03bbmin(H)  \u221a 2L\u03bbmax(H\u22121) \u03bbmin(H\u22121) ! \u2225\u2206T\u22a5\u2225F \u2212 L \u221am s \u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F = 0 This concludes proof of Theorem 2(a). 13 \f\u25a1 Remark 4. (a) The upper bound estimate for ||R\u2126\u2206T\u22a5||2 F is not optimal. This is so since the number of times a given measurement can be duplicated is set to m which is a worst case estimate. One approach to improve the estimate is to make use of standard concentration inequalities and argue that the expected number of duplicates for a given measurement is much smaller than m. A second option, as noted in the remark on the sampling model section, is to use a uniform sampling with out replacement model. With this, the estimate can be improved since the factor m is no longer necessary. While these alternatives lead to a better estimate of the upper bound, the \ufb01nal gain in terms of number of measurements is minor as the improved estimates are inside of a log2. For these reasons and ease of presentation, the current estimate is used in the forthcoming analysis. (b)Lower bounding the term P \u03b1\u2208\u2126\u27e8\u2206T , w\u03b1\u27e92 does not lend itself to simpler analysis. For example, the use of standard concentration inequalities results probability of failures that scale with n. Since \u2225R\u2126\u2206T\u22252 F = \u27e8\u2206T , PTR\u2217 \u2126R\u2126PT\u2206T \u2212PT\u2206T\u27e9+ \u2225PT\u2206T\u22252 F, equivalently, we can consider an upper bound of \u2225PTR\u2217 \u2126R\u2126PT\u2206T \u2212PT\u2225to lower bound \u2225R\u2126\u2206T\u22252 F. The upper bound calculations can be carried out but the calculations are involved and assume certain structure of the basis w\u03b1. The simple alternative approach shown above is general since it does not make restrictive assumptions on the measurement basis. Proof of Theorem 2(b). Let X = M + \u2206be a feasible solution to (12) with the condition that \u2225\u2206T\u2225F < \u221a 2L \u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F and \u2206\u2208ker R\u2126. The goal now is to show that, for any X that satis\ufb01es these assumptions, the nuclear norm minimization is violated meaning that \u2225X\u2225\u2217= \u2225M + \u2206\u2225\u2217> \u2225M\u2225\u2217. The proof of this fact makes use of the dual certi\ufb01cate approach in [18]. The idea is to endow a certain object, named a dual certi\ufb01cate Y , with certain conditions so as to ensure that any X satisfying the earlier made assumptions is not a solution to (12). It then becomes a task to construct the certi\ufb01cate which satis\ufb01es the preset conditions. For ease of later reference, we start with the former task reproducing a proof, with minor changes, in section 2E of [18]. First, using the duality of the spectral norm and the nuclear norm, note that there exists a \u039b \u2208T\u22a5with ||\u039b|| = 1 such that \u27e8\u039b , PT\u22a5(\u2206)\u27e9= ||PT\u22a5(\u2206)||\u2217. Second, using the characterization of the subgradient of the nuclear norm [33], \u2202\u2225M\u2225\u2217= {Sgn M + \u0393 | \u0393 \u2208T\u22a5& \u2225PT\u22a5\u0393\u2225\u22641}. With this, it can be readily veri\ufb01ed that Sgn M + \u039b is a subgradient of ||M||\u2217at M. Mathematically, we have \u2225M + \u2206\u2225\u2217\u2265||M||\u2217+ \u27e8Sgn M + \u039b , \u2206\u27e9. Using the condition Y \u2208range R\u2217 \u2126which implies \u27e8Y , \u2206\u27e9= 0 in the previous inequality, we obtain \u2225M + \u2206\u2225\u2217\u2265||M||\u2217+ \u27e8Sgn M + \u039b , \u2206\u27e9 = ||M||\u2217+ \u27e8Sgn M + \u039b \u2212Y , \u2206\u27e9 = ||M||\u2217+ \u27e8Sgn M \u2212PT Y , \u2206T\u27e9+ ||\u2206T\u22a5||\u2217\u2212\u27e8PT\u22a5Y , \u2206T\u22a5\u27e9 The third equality follows using the earlier choice of \u039b and the fact that Sgn M \u2208T (see Lemma A.2). Finally, we 14 \fapply the assumptions of the theorem to the last equation above to obtain \u2225M + \u2206\u2225\u2217\u2265||M||\u2217+ \u27e8Sgn M \u2212PT Y , \u2206T\u27e9+ ||\u2206T\u22a5||\u2217\u2212\u27e8PT\u22a5Y , \u2206T\u22a5\u27e9 \u2265||M||\u2217\u22121 4 r 1 2L \u03bbmin(H\u22121) \u03bbmax(H\u22121)\u2225\u2206T\u2225F + \u2225\u2206T\u22a5\u2225\u2217\u22121 2\u2225\u2206T\u22a5\u2225\u2217 \u2265||M||\u2217\u22121 4 r 1 2L \u03bbmin(H\u22121) \u03bbmax(H\u22121)\u2225\u2206T\u2225F + 1 2\u2225\u2206T\u22a5\u2225F > ||M||\u2217\u22121 4 r 1 2L \u03bbmin(H\u22121) \u03bbmax(H\u22121)  \u221a 2L\u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F ! + 1 2\u2225\u2206T\u22a5\u2225F = ||M||\u2217+ 1 4\u2225\u2206T\u22a5\u2225F It can be concluded that \u2225M + \u2206\u2225\u2217> \u2225M\u2225\u2217as desired. \u25a1 Next, we state and prove a corollary which shows that M is a unique solution to (12) if the deterministic assumptions of Theorem 2 hold. Corollary 1. If the conditions of Theorem 2 hold, M is a unique solution to (12). Proof. De\ufb01ne \u2206= X \u2212M for any X \u2208S. Using Theorem 2(a), R\u2126\u2206, 0 if \u2225\u2206T\u22252 F \u2265  \u221a 2L\u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F !2 . It then su\ufb03ces to consider the case \u2225\u2206T\u22252 F <  \u221a 2L\u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F !2 for \u2206\u2208ker R\u2126. For this case, using the proof of Theorem 2(b), \u2225X\u2225\u2217> \u2225M\u2225\u2217. Therefore, M is the unique solution to (12). \u25a1 3.2 Proof of Theorem 1 Using the corollary above, if the two conditions in Theorem 2 hold, it follows that M is a unique solution to (12). The \ufb01rst condition in Theorem 2(a) is the assumption that \u03bbmin (PT F\u2126PT) > 1 2\u03bbmin(H). This will ensure that the minimum eigenvalue of the operator PT F\u2126PT is bounded away from zero. Using the operator Cherno\ufb00bound in [30] restated below, Lemma 2 addresses the assumption. Theorem 3 (Cherno\ufb00bound in [30]). Consider a \ufb01nite sequence {Lk} of independent, random, self-adjoint operators, acting on matrices in Rn\u00d7n, that satisfy Lk \u2ab00 and \u2225Lk\u2225\u2264R almost surely Compute the minimum eigenvalue of the sum of the expectations, \u00b5min := \u03bbmin \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X k E[Lk] \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 Then, we have Pr \u0014 \u03bbmin \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X k Lk \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2264(1 \u2212\u03b4) \u00b5min \u0015 \u2264n \u0014 exp(\u2212\u03b4) (1 \u2212\u03b4)1\u2212\u03b4 \u0015 \u00b5min R for \u03b4 \u2208[0, 1] 15 \fFor \u03b4 \u2208[0, 1], using Taylor series of log(1 \u2212\u03b4), note that (1 \u2212\u03b4) log(1 \u2212\u03b4) \u2265\u2212\u03b4 + \u03b42 2 . This results the following simpli\ufb01ed estimate. Pr \u0014 \u03bbmin \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X k Lk \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2264(1 \u2212\u03b4) \u00b5min \u0015 \u2264n exp \u0012 \u2212\u03b42 \u00b5min 2R \u0013 for \u03b4 \u2208[0, 1] Lemma 2. Consider the operator PT F\u2126PT : T \u2192T. With \u03ba = mn Lr , the following estimate holds. Pr   \u03bbmin (PT F\u2126PT) \u22641 2\u03bbmin(H) ! \u2264n exp   \u2212\u03bbmin(H)2\u03ba 8\u03bd ! Proof. Recall the restricted frame operator F\u2126= P \u03b1\u2208\u2126 L m\u27e8X , w\u03b1\u27e9w\u03b1. For X \u2208T, PT F\u2126PT X can be equivalently represented as follows. PT F\u2126PT X = X \u03b1\u2208\u2126 L m\u27e8X , PT w\u03b1\u27e9PT w\u03b1 Let L\u03b1 = L m\u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1 denote the operator in the summand. Since L\u03b1 is positive semide\ufb01nite, the operator Cherno\ufb00bound can be used. The bound requires estimate of an upper bound R of the spectrum norm of L\u03b1 and \u00b5min = \u03bbmin(P \u03b2\u2208\u2126E[L\u03b2]). First, we estimate R as follows. \r \r \r \r \r L m\u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1 \r \r \r \r \r = L m\u2225PT w\u03b1\u22252 F \u2264L m\u03bbmax(H\u22121)\u03bdr n The last inequality follows from the coherence estimate in (18). With this, set R = L m\u03bbmax(H\u22121)\u03bdr n . Next, we consider the estimate of \u00b5min by \ufb01rst evaluating P \u03b2\u2208\u2126E[L\u03b2]. X \u03b2\u2208\u2126 E[L\u03b2] = X \u03b2\u2208\u2126 \u0014 X \u03b1\u2208I 1 m\u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1 \u0015 = X \u03b1\u2208I \u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1 For any X \u2208T, \u27e8X , X \u03b2\u2208\u2126 E[L\u03b2](X)\u27e9can be lower bounded as follows. \u27e8X , X \u03b2\u2208\u2126 E[L\u03b2](X)\u27e9= X \u03b2\u2208I \u27e8X, w\u03b2\u27e92 \u2265\u03bbmin(H)\u2225X\u22252 F with the last inequality following from Lemma A.3. The variational characterization of the minimum eigenvalue, along with the fact that P \u03b1\u2208\u2126E[L\u03b1] is a self-adjoint operator, implies that the minimum eigenvalue of P \u03b1\u2208\u2126E[L\u03b1] is at least \u03bbmin(H). With this, set \u00b5min = \u03bbmin(H). The \ufb01nal step is to apply the operator Cherno\ufb00bound with R = L m\u03bbmax(H\u22121) \u03bdr n and \u00b5min = \u03bbmin(H). Setting \u03b4 = 1 2, \u03bbmin (PT F\u2126PT) > 1 2\u03bbmin(H) with probability of failure at most p1 given by p1 = n exp   \u2212\u03bbmin(H)2\u03ba 8\u03bd ! This concludes the proof. \u25a1 Lemma 2 shows that \u03bbmin (PT F\u2126PT) > 1 2\u03bbmin(H) holds with probability at least 1 \u2212p1 where the probability of failure is at most p1 = n exp \u0012 \u2212\u03ba 8\u03bd \u0013 with \u03ba = mn Lr . In what follows, the conditions in Theorem 2(b) are analyzed. The statement there assumes the existence of a certain dual certi\ufb01cate Y that satis\ufb01es the conditions in (22). In [18], David Gross devised a novel scheme, the gol\ufb01ng 16 \fscheme, to construct the dual certi\ufb01cate Y . Before showing the scheme, some notations are in order: 1) The random set \u2126is partitioned into l batches. The i-th batch, denoted \u2126i, contains mi elements with l X i=1 mi = m. 2) For a given batch, the sampling operator can be de\ufb01ned as follows Ri = L mi X \u03b1\u2208\u2126i \u27e8X , w\u03b1\u27e9z\u03b1. The inductive scheme is shown below. Q0 = sgn M, Yi = i X j=1 R\u2217 jQ j\u22121, Qi = sgn M \u2212PTYi, i = 1, ..., l (28) The main idea of the remaining analysis is to employ the gol\ufb01ng scheme and certify that the conditions in (22) hold with very high probability. In the analysis of the gol\ufb01ng scheme, the initial task is to show that the \ufb01rst condition in (22) holds. This requires a probabilistic estimate of \u2225PTR\u2217 \u2126PTX \u2212PTX\u2225F \u2265t for a \ufb01xed matrix X \u2208S and will be addressed in Lemma 3 to follow shortly. The proof of the lemma relies on the vector Bernstein inequality in [18]. We use a slightly modi\ufb01ed version of this inequality which is stated below. Theorem 4 (Vector Bernstein inequality). Let x1, ..., xm be independent zero-mean vector valued random variables. Assume that max i \u2225xi\u22252 \u2264R and P i E[\u2225xi\u22252 2] \u2264\u03c32. For any t \u2264\u03c32 R , the following estimate holds. Pr \u0014 \r \r \r \r \r \r \r m X i=1 xi \r \r \r \r \r \r \r2 \u2265t \u0015 \u2264exp   \u2212t2 8\u03c32 + 1 4 ! , Lemma 3. Given an arbitrary \ufb01xed X \u2208S, for t \u22641 with \u03ba = mn Lr , the following estimate holds. Pr (\u2225PTR\u2217 \u2126PTX \u2212PTX\u2225F \u2265t\u2225X\u2225F) \u2264exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 t2\u03ba 8 \u0010 \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bd + n Lr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (29) Proof. In what follows, with out loss of generality, we assume \u2225X\u2225F = 1. Using the dual basis expansion, PTR\u2217 \u2126PTX\u2212 PTX can be represented as follows. PTR\u2217 \u2126PTX \u2212PTX = X \u03b1\u2208\u2126 \u0014 L m\u27e8PTX , z\u03b1\u27e9PT w\u03b1 \u22121 mPTX \u0015 (30) The summand, denoted Y\u03b1, can be written as Y\u03b1 = X\u03b1 \u2212E[X\u03b1]. Since E[Y\u03b1] = 0, it satis\ufb01es the condition for the vector Bernstein inequality and we proceed to consider appropriate bounds for \u2225Y\u03b1\u2225F and E[\u2225Y\u03b1\u22252 F]. First, we bound \u2225Y\u03b1\u2225F making use of the coherence conditions (18) and (19). \u2225Y\u03b1\u2225F = \r \r \r \r \r L m\u27e8PTX , z\u03b1\u27e9PT w\u03b1 \u22121 mPTX \r \r \r \r \rF \u2264 \r \r \r \r \r L m\u27e8PTX , z\u03b1\u27e9PT w\u03b1 \r \r \r \r \rF + \r \r \r \r \r 1 mPTX \r \r \r \r \rF \u2264L m max \u03b1\u2208I \u2225PT w\u03b1\u2225F max \u03b1\u2208I \u2225PT z\u03b1\u2225F + 1 m \u22641 m \u0012L n \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bdr + 1 \u0013 Therefore, set R = 1 m \u0010 L n \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bdr + 1 \u0011 . To upper bound E[\u2225Y\u03b1\u22252 F], we start with the de\ufb01nition of E[\u2225Y\u03b1\u22252 F] 17 \fand proceed as follows. E[\u2225Y\u03b1\u22252 F] = E \u0014 L2 m2 \u27e8PTX , z\u03b1\u27e92\u2225PT w\u03b1\u22252 F + 1 m2 \u2225PTX\u22252 F \u22122L m2 \u27e8PTX , z\u03b1\u27e9\u27e8PTX , w\u03b1\u27e9 \u0015 = E \" L2 m2 \u27e8PTX , z\u03b1\u27e92\u2225PT w\u03b1\u22252 F # + 1 m2 \u2225PTX\u22252 F \u22122 m2 \u27e8 X \u03b1\u2208I \u27e8PTX , w\u03b1\u27e9z\u03b1 , PTX\u27e9 = E \" L2 m2 \u27e8PTX , z\u03b1\u27e92\u2225PT w\u03b1\u22252 F # \u22121 m2 \u2225PTX\u22252 F \u2264L m2 max \u03b1\u2208I \u2225PT w\u03b1\u22252 F X \u03b1\u2208I \u27e8PTX , z\u03b1\u27e92 + 1 m2 \u2264L m2 \u03bbmax(H\u22121)2 \u03bdr n + 1 m2 \u2264L m2 \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e \u03bdr n + 1 m2 Above, the second inequality results from the coherence conditions (18) and (19) and application of Lemma A.3. With this, set \u03c32 = 1 m \u0010 L n \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bdr + 1 \u0011 . To conclude the proof, we apply the vector Bernstein inequality with the speci\ufb01ed R and \u03c3. For t \u2264\u03c32 R = 1, with \u03ba = mn Lr , the following estimate holds. Pr (\u2225PTR\u2217 \u2126PTX \u2212PTX\u2225F \u2265t) \u2264exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 t2\u03ba 8 \u0010 \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bd + n Lr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (31) \u25a1 Next, it will be argued that the gol\ufb01ng scheme (28) certi\ufb01es the conditions in (22) with very high probability. In particular, we have the following lemma. Lemma 4. Yl obtained from the gol\ufb01ng scheme (28) satis\ufb01es the conditions in (22) with failure probability which is at most p = l X i=1 p2(i) + p3(i) + p4(i) where p2(i) = exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 \u03bai 32 \u0010 \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bd + n Lr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8, p3(i) = 2n exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u22123 min \u0010 (\u00b5 + 1)\u2225H\u22121\u2225\u221e, 1 4 \u00112 \u03bai 8(\u00b5 + 1)\u2225H\u22121\u22252 \u221e\u03bd \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8and p4(i) = n2 exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 3\u03bai 32 \u0010 cv\u03bd + n Lr \u0011 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8with ki = min Lr . Proof. In what follows, we repeatedly make use of the fact that Qi \u2208T since sgn M \u2208T from Lemma A.2. The main idea for showing that the \ufb01rst condition in (22) holds relies on a recursive form of Qi which can be derived as follows. Qi = sgn M \u2212PT \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed i X j=1 R\u2217 jQ j\u22121 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8= sgn M \u2212PT \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed i\u22121 X j=1 R\u2217 jQ j\u22121 + R\u2217 i Qi\u22121 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = sgn M \u2212PT i\u22121 X j=1 R\u2217 jQ j\u22121 \u2212PTR\u2217 i Qi\u22121 = sgn M \u2212PTYi\u22121 \u2212PTR\u2217 i Qi\u22121 = Qi\u22121 \u2212PTR\u2217 i Qi\u22121 = (PT \u2212PTR\u2217 i PT)Qi\u22121 (32) The \ufb01rst condition in (22) is a bound on \u2225(PT \u2212PTR\u2217 l PT)Ql\u22121\u2225F = \u2225Ql\u2225F. Using Lemma 3 with t2,i = 1 2, \u2225(PT \u2212 PTR\u2217 i PT)Qi\u22121\u2225< t2,i\u2225Qi\u22121\u2225F holds with failure probability at most p2(i) = exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 \u03bai 32 \u0010 \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bd + n Lr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 18 \fwhere \u03ba = min Lr . Using the recursive formula in (32) repeatedly, \u2225Qi\u2225F can be upper bounded as follows. \u2225Qi\u2225F < \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed i Y k=1 t2,k \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2225Q0\u2225F = 2\u2212i \u221ar (33) Setting l = log2   4 \u221a 2L\u03bbmax(H) \u03bbmin(H) \u221ar ! , the \ufb01rst condition in (22) is now satis\ufb01ed. It can be concluded that, using a union bound on the failure probabilities p2(i), \u2225Ql\u2225F < \u221ar2\u2212l = 1 4 r 1 2L \u03bbmin(H\u22121) \u03bbmax(H\u22121) holds with failure probability that is at most Pl i=1 p2(i). This implies that the \ufb01rst condition in (22) also holds with the same failure probability. Next, we consider the second condition in (22) which requires a bound on \u2225PT\u22a5Yl\u2225. First, note that \u2225PT\u22a5Yl\u2225= \u2225PT\u22a5Pl j=1 R\u2217 jQ j\u22121\u2225= \u2225Pl j=1 PT\u22a5R\u2217 jQ j\u22121\u2225\u2264Pl j=1 \u2225PT\u22a5R\u2217 jQ j\u22121\u2225. As such, in what follows, the focus will be \ufb01nding a suitable bound for \u2225PT\u22a5R\u2217 jQ j\u22121\u2225. This will be analyzed in Lemma 5. A key element in the proof of Lemma 5 is an assumption on the size of max\u03b2 |\u27e8Qi , z\u03b2\u27e9|. For ease of notation in further analysis, let \u03b7(Qi) be de\ufb01ned as: \u03b7(Qi) = max\u03b2 |\u27e8Qi , z\u03b2\u27e9|. The assumption is that, at the i-th step of the gol\ufb01ng scheme, \u03b7(Qi)2 \u2264\u2225H\u22121\u22252 \u221e \u03bd n2 c2 i where c2 i is an upper bound for \u2225Qi\u22252 F, \u2225Qi\u22252 F \u2264c2 i . The idea is to argue that this assumption holds with very high probability. To show this, assume that \u03b7(Qi) \u2264t4,i with failure probability p4(i). Setting t4,i = 1 2\u03b7(Qi\u22121) and applying the inequality \u03b7(Qi) \u2264t4,i recursively results \u03b7(Qi)2 \u22642\u22122\u03b7(Qi\u22121)2 \u22642\u22122i\u03b7(sgn M)2 \u22642\u22122i\u2225H\u22121\u22252 \u221e \u03bdr n2 = \u2225H\u22121\u22252 \u221e \u03bd n2 (2\u22122ir) where the last inequality follows from the coherence estimate in (20). It can now be concluded that, noting (33), the inequality above ensures that \u03b7(Qi)2 \u2264\u2225H\u22121\u22252 \u221e \u03bd n2 c2 i with ci = 2\u2212i \u221ar. The failure probability p4(i) follows from Lemma A.4, noting that \u03b7(Qi) = \u03b7(Qi\u22121 \u2212PTR\u2217 i Qi\u22121), and is given by p4(i) = n2 exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 3\u03ba j 32 \u0010 cv\u03bd + n Lr \u0011 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \u2200i \u2208[1, l] Having justi\ufb01ed the assumption on the size of \u03b7(Qi), a key part of Lemma A.4, we now consider certifying the second condition in (22). Assume that \u2225PT\u22a5R\u2217 jQ j\u22121\u2225< t3, jc j\u22121, with \u2225Q j\u22121\u2225F \u2264c j\u22121 = 2\u2212(j\u22121), holds with failure probability p3(j). Fixing t3, j = min  (\u00b5 + 1)\u2225H\u22121\u2225\u221e \u221ar , 1 4 \u221ar ! , Lemma 5 gives p3(j) = 2n exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u22123 min \u0010 (\u00b5 + 1)\u2225H\u22121\u2225\u221e, 1 4 \u00112 \u03ba j 8(\u00b5 + 1)\u2225H\u22121\u22252 \u221e\u03bd \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 where \u03ba j = m jn Lr . \u2225PT\u22a5Yl\u2225can now be upper bounded as follows. \u2225PT\u22a5Yl\u2225\u2264 l X k=1 \u2225PT\u22a5R\u2217 kQk\u22121\u2225< l X k=1 t3,kck\u22121 \u2264 1 4 \u221ar l X k=1 ck\u22121 = 1 4 \u221ar l X k=1 \u221ar2\u2212(k\u22121) < 1 2 Applying the union bound over the failure probabilities, \u2225PT\u22a5Yl\u2225< 1 2 holds true with failure probability which is at most Pl j=1[p3(j) + p4(j)]. With the same failure probability, the second condition in (22) holds true. 19 \f\u25a1 Lemma 5 will be proved shortly using the Bernstein inequality in [31] which is restated below for convenience. Theorem 5 (Bernstein inequality). Consider a \ufb01nite sequence {Xi} of independent, random matrices with dimension n. Assume that E[Xi] = 0 and \u2225Xi\u2225\u2264R \u2200i Let the matrix variance statistic of the sum \u03c32 be de\ufb01ned as \u03c32 = max \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \r \r \r \r \r \r \r X i E[XT i Xi] \r \r \r \r \r \r \r , \r \r \r \r \r \r \r X i E[XiXT i ] \r \r \r \r \r \r \r \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 For all t \u22650, Pr \u0014 \r \r \r \r \r \r \r X i Xi \r \r \r \r \r \r \r > t \u0015 \u2264 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 2n exp   \u22123t2 8\u03c32 ! t \u2264\u03c32 R 2n exp   \u22123t 8R ! t \u2265\u03c32 R (34) Lemma 5. Consider a \ufb01xed matrix G \u2208T. Assume that max\u03b2 \u27e8G , z\u03b2\u27e92 \u2264\u2225H\u22121\u22252 \u221e \u03bd n2 c2 with c set as \u2225G\u22252 F \u2264c2. Then, with \u03ba j = m jn Lr , the following estimate holds for all t \u2264(\u00b5 + 1)\u2225H\u22121\u2225\u221e \u221ar . Pr (\u2225PT\u22a5R\u2217 jG\u2225\u2265t c) \u22642n exp   \u2212 3t2\u03ba jr 8(\u00b5 + 1)\u2225H\u22121\u22252 \u221e\u03bd ! Proof of Lemma 5. Using the dual basis representation, PT\u22a5R\u2217 jG = X \u03b1\u2208\u2126j L m j \u27e8G , z\u03b1\u27e9PT\u22a5w\u03b1. The summand, denoted X\u03b1, has zero expectation since G \u2208T. With this, the zero mean assumption for Bernstein inequality is satis\ufb01ed. Next, we consider suitable estimates for R and \u03c32. The latter necessitates an estimate for max(\u2225E[X\u03b1XT \u03b1 ]\u2225, \u2225E[XT \u03b1 X\u03b1]\u2225). First, we bound \u2225E[X\u03b1XT \u03b1 ]\u2225. Since X\u03b1XT \u03b1 = L2 m2 j \u27e8G , z\u03b1\u27e92(PT\u22a5w\u03b1)(PT\u22a5w\u03b1)T, using Lemma A.5 and the fact that w\u03b1wT \u03b1 is positive semide\ufb01nite, \u2225E[X\u03b1XT \u03b1 ]\u2225can be upper bounded as follows. \r \r \rE[X\u03b1XT \u03b1 ] \r \r \r \u2264L m2 j max \u2225\u03d5\u22252=1 X \u03b1\u2208I \u27e8G , z\u03b1\u27e92\u27e8\u03d5 , w\u03b1wT \u03b1\u03d5\u27e9\u2264L m2 j max \u03b1\u2208I \u27e8z\u03b1 , G\u27e92 max \u2225\u03d5\u22252=1 \u27e8\u03d5 , \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X \u03b1\u2208I w\u03b1wT \u03b1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u03d5\u27e9 Using the correlation condition, in particular Lemma 1(b), \u2225P \u03b1\u2208I w\u03b1wT \u03b1\u2225\u2264(\u00b5 + 1)n, we obtain \r \r \rE[X\u03b1XT \u03b1 ] \r \r \r \u2264L (\u00b5 + 1) n m2 j max \u03b1\u2208I \u27e8z\u03b1 , G\u27e92 \u2264L (\u00b5 + 1) n m2 j \u2225H\u22121\u22252 \u221e \u03bd n2 c2 An analogous calculation and reasoning as above yields \r \r \rE[XT \u03b1 X\u03b1] \r \r \r \u2264L (\u00b5+1)n m2 j \u2225H\u22121\u22252 \u221e \u03bd n2 c2. Therefore, using triangle inequality, set \u03c32 = L (\u00b5 + 1) m j \u2225H\u22121\u22252 \u221e \u03bd n c2 20 \fTo complete the proof, it remains to estimate R. \u2225X\u03b1\u2225\u2264L m j |max \u03b1\u2208I \u27e8z\u03b1 , G\u27e9| \u2225PT\u22a5w\u03b1\u2225\u2264L m j \u2225H\u22121\u2225\u221e \u221a\u03bd n c (35) Two pertinent cases have to be considered. If \u03bd \u22651 r , \u2225X\u03b1\u2225\u2264 L m j \u2225H\u22121\u2225\u221e \u03bd \u221ar n c = R1 and if \u03bd < 1 r , \u2225X\u03b1\u2225\u2264 L m j \u2225H\u22121\u2225\u221e \u221a\u03bd n c = R2. Note that \u03c32 R1 = (\u00b5 + 1)\u2225H\u22121\u2225\u221ec \u221ar and \u03c32 R2 = (\u00b5 + 1)\u2225H\u22121\u2225\u221ec \u221ar . To conclude the proof, we apply the Bernstein inequality. An application of the Bernstein inequality results the following estimate. Pr(\u2225PT\u22a5R\u2217 jG\u2225\u2265t) \u22642n exp   \u2212 3t2\u03ba jr 8(\u00b5 + 1)\u2225H\u22121\u22252 \u221e\u03bdc2 ! (36) for all t \u2264(\u00b5 + 1)\u2225H\u22121\u2225\u221ec \u221ar with \u03ba j = m jn Lr . This concludes the proof of Lemma 5. \u25a1 Next, it will be argued that M is a unique solution to (12) with very high probability. The argument considers two separate cases based on comparing \u2225\u2206T\u22252 F and \u2225\u2206T\u22a5\u22252 F. This motivates us to de\ufb01ne the following two sets: S1 = \u001a X \u2208S : \u2225\u2206T\u22252 F \u2265 \u0010 \u221a 2L \u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F \u00112 \u001b and S2 = \u001a X \u2208S : \u2225\u2206T\u22252 F < \u0010 \u221a 2L \u03bbmax(H\u22121) \u03bbmin(H\u22121) \u2225\u2206T\u22a5\u2225F \u00112 & R\u2126(\u2206) = 0 \u001b . With this, assuming that \u2126is sampled uniformly at random with replacement, the two cases are as follows. 1. For all X \u2208S1, set |\u2126| = m \u201csu\ufb03ciently large\u201d such that Pr \u0010n \u2126\u2282I \f \f \f |\u2126| = m, \u03bbmin (PT F\u2126PT) > \u03bbmin(H) 2 o\u0011 \u22651 \u2212p1 based on Lemma 2. Therefore, all X \u2208S1 are feasible solutions to (12) with probability at most p1 from Theorem 2(a). 2. For all X \u2208S2, we obtain Pr \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3\u2126\u2282I \f \f \f |\u2126| = m, Y \u2208range R\u2217 \u2126& \u2225PTY \u2212Sgn M\u2225F \u22641 4 r 1 2L \u03bbmin(H\u22121) \u03bbmax(H\u22121) & \u2225PT\u22a5Y \u2225\u22641 2 \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u22651 \u2212\u01eb with \u01eb = l X i=1 \u0002p2(i) + p3(i) + p4(i)\u0003 by setting |\u2126| = m \u201csu\ufb03ciently large\u201d based on Lemma 4. Then, the probability of all X \u2208S2 being solutions to (12) is at most \u01eb from Theorem 2(b). Using the above two cases and employing the union bound, any X \u2208S di\ufb00erent from M is a solution to the general matrix completion problem with probability at most p = p1 + Pl i=1[p2(i) + p3(i) + p4(i)]. In the arguments above, we have used the terms \u201csu\ufb03ciently large\u201d, \u201csmall probability\u201d and \u201cvery high probability\u201d with out being precise. The goal now is to set everything explicit. First, de\ufb01ne very high probability as a probability of at least 1 \u2212n\u2212\u03b2 for \u03b2 > 1. Analogously, de\ufb01ne small failure probability as a probability of at most n\u2212\u03b2 for some \u03b2 > 1. To recover the underlying matrix with high probability, the idea is to carefully set the remaining free parameters m, l and mi so that the p \u2264n\u2212\u03b2 for \u03b2 > 1. This necessitates revisiting all the failure probabilities in the analysis. p1, the \ufb01rst failure probability, is the probability that the condition in Theorem 2(a) does not hold. In the construction of the dual certi\ufb01cate Y via the gol\ufb01ng scheme, three failure probabilities, p2(i), p3(i) and p4(i), \u2200i \u2208[1, l], appear. With this, all 21 \fthe failure probabilities are noted below. p1 = n exp   \u2212\u03bbmin(H)2\u03ba 8\u03bd ! ; p2(i) = exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 \u03bai 32 \u0010 \u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e\u03bd + n Lr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 p3(i) = 2n exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u22123 min \u0010 (\u00b5 + 1)\u2225H\u22121\u2225\u221e, 1 4 \u00112 \u03bai 8(\u00b5 + 1)\u2225H\u22121\u22252 \u221e\u03bd \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ; p4(i) = n2 exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 3\u03bai 32 \u0010 cv\u03bd + n Lr \u0011 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 To suitably set mi, since ki = min Lr , we can set ki with the condition that all the failure probabilities are at most 1 4ln\u2212\u03b2 for \u03b2 > 1. A minor calculation results one suitable choice of ki, ki = 48\u0000C\u03bd + 1 nr \u0001\u0000\u03b2 log(n) + log(4l)\u0001, with C de\ufb01ned as C = max \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e, cv, (\u00b5 + 1)\u2225H\u22121\u2225\u221e min \u0010 (\u00b5 + 1)\u2225H\u22121\u2225\u221e, 1 4 \u00112 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (37) The total failure probability, applying the union bound, is bounded above by n\u2212\u03b2. The number of measurements, m = lrnki, is at least log2   4 \u221a 2L\u03bbmax(H) \u03bbmin(H) \u221ar ! nr   48\u0002C\u03bd + n Lr \u0003\u0002 \u03b2 log(n) + log   4 log2   4 \u221a 2L\u03bbmax(H) \u03bbmin(H) \u221ar !! \u0003! (38) This \ufb01nishes the proof of Theorem 1. It can be concluded that the minimization program in (12) recovers the underlying matrix with very high probability. 3.3 Noisy General Matrix Completion In practical applications, the measurements in the general matrix completion problem are prone to noise. This motivates the analysis of the robustness of the nuclear norm minimization program for the general matrix completion problem. In particular, consider the following noisy general matrix completion problem where noise is modeled as Gaussian noise with mean \u00b5 and variance \u03c3. minimize X\u2208S \u2225X\u2225\u2217 subject to \u2225R\u2126(X) \u2212R\u2126(M)\u2225\u2264\u03b4 (39) Above, \u03b4 characterizes the level of noise. In [6], under certain assumptions, the authors show the robustness of the nuclear norm minimization algorithm for the matrix completion problem. Following this existing analysis and the dual basis framework, we expect that a robustness result, such as the one below, can be attained. Theorem 6. Let M \u2208Rn\u00d7n be a matrix of rank r that obeys the coherence conditions (15), (16) and (17) with coherence \u03bd and satis\ufb01es the correlation condition (9) with correlation parameter \u00b5. De\ufb01ne C as follows: C = max \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u03bbmax(H\u22121)\u2225H\u22121\u2225\u221e, cv, (\u00b5 + 1)\u2225H\u22121\u2225\u221e min \u0010 (\u00b5 + 1)\u2225H\u22121\u2225\u221e, 1 4 \u00112 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8. Assume m measurements \u27e8M , w\u03b1\u27e9, sampled uniformly at random with replacement, are corrupted with Gaussian noise of mean \u00b5, variance \u03c3 and noise level \u03b4. For \u03b2 > 1, if m \u2265log2   4 \u221a 2L\u03bbmax(H) \u03bbmin(H) \u221ar ! nr   48\u0002C\u03bd + n Lr \u0003\u0002 \u03b2 log(n) + log   4 log2   4 \u221a 2L\u03bbmax(H) \u03bbmin(H) \u221ar !! \u0003! (40) 22 \fthen \u2225M \u2212\u00af M\u2225F \u2264f \u0012 n, m n2 , \u03b4 \u0013 where \u00af M is a solution to (39) with probability at least 1 \u2212n\u2212\u03b2. Remark: A proof of the robustness theorem above requires specifying the level of noise and making the function f explicit. We leave this as a future work. 4"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1804.04310v2",
+                    "title": "Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion",
+                    "abstract": "The Euclidean distance geometry problem arises in a wide variety of\napplications, from determining molecular conformations in computational\nchemistry to localization in sensor networks. When the distance information is\nincomplete, the problem can be formulated as a nuclear norm minimization\nproblem. In this paper, this minimization program is recast as a matrix\ncompletion problem of a low-rank $r$ Gram matrix with respect to a suitable\nbasis. The well known restricted isometry property can not be satisfied in this\nscenario. Instead, a dual basis approach is introduced to theoretically analyze\nthe reconstruction problem. If the Gram matrix satisfies certain coherence\nconditions with parameter $\\nu$, the main result shows that the underlying\nconfiguration of $n$ points can be recovered with very high probability from\n$O(nr\\nu\\log^{2}(n))$ uniformly random samples. Computationally, simple and\nfast algorithms are designed to solve the Euclidean distance geometry problem.\nNumerical tests on different three dimensional data and protein molecules\nvalidate effectiveness and efficiency of the proposed algorithms.",
+                    "authors": "Abiy Tasissa, Rongjie Lai",
+                    "published": "2018-04-12",
+                    "updated": "2018-10-28",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "cs.LG",
+                        "math.IT",
+                        "math.NA",
+                        "math.OC"
+                    ],
+                    "main_content": "Introduction The Euclidean distance geometry problem, EDG hereafter, aims at constructing the con\ufb01guration of points given partial information on pairwise inter-point distances. The problem has applications in diverse areas, such as molecular conformation in computational chemistry [1], [2], [3], dimensionality reduction in machine learning [4] and localization in sensor networks [5], [6]. For instance, in the molecular conformation problem, the goal is to determine the structure of protein given partial interatomic distances obtained from nuclear magnetic resonance (NMR) spectroscopy experiments. Since the structure of protein determines its physical and chemical properties, the molecular conformation problem is crucial for biological applications such as drug design. In recent works, novel applications of EDG like solving partial di\ufb00erential equations (PDEs) on manifolds represented as incomplete inter-point distance information have been explored in [7]. For the mathematical setup of the problem, consider a set of n points P = {p1, p2, ..., pn}T \u2208Rn\u00d7r. The squared distance between any two points pi and p j is given by d2 i, j = ||pi \u2212p j||2 2 = pT i pi +pT j p j \u22122pT i p j. The Euclidean distance matrix D = [d2 i,j] can be written compactly as follows D = 1 diag(P P T)T + diag(P P T) 1T \u22122P P T (1) where 1 is a column vector of ones and diag(\u00b7) is a column vector of diagonal entries of the matrix in consideration. X = P P T is the inner product matrix well known as the Gram matrix. If the complete distance information is available, the coordinates can be recovered from the eigendecomposition of the Gram matrix using classical multidimensional scaling (MDS) [8]. It should be noted that this solution is unique up to rigid transformations and translations. In practice, the distance matrix is incomplete and \ufb01nding the underlying point coordinates with partial information is in general not possible. One approach proposed in [7] is based on a matrix completion applied to the Gram matrix which we brie\ufb02y summarize here. Assume that the entries of D are sampled uniformly at random. By construction, X is symmetric and positive semide\ufb01nite. The solution for X is unique up to translations. This ambiguity is \ufb01xed by considering the constraint that the centroid of the points is located at the origin, Pn k=1 pk = 0, which leads to X \u00b7 1 = 0. Assuming that X is low-rank, r \u226an, the work in [7] considers the following nuclear norm minimization program to recover X. minimize ||X||\u2217 subject to Xi ,i + X j ,j \u22122Xi , j = Di ,j (i, j) \u2208\u2126 (2) X \u00b7 1 = 0 ; X = XT ; X \u2ab00 A. Tasissa is with the Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY 12180, (tasisa@rpi.edu). R. Lai is with the Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A. (lair@rpi.edu). This work is partially supported by NSF DMS\u20131522645 and an NSF CAREER award, DMS\u20131752934. arXiv:1804.04310v2  [cs.IT]  28 Oct 2018 \f2 Here ||X||\u2217denotes the nuclear norm and \u2126\u2282{(i, j)|i, j = 1, ..., n, i < j}, |\u2126| = m, denotes the random set that consists of all the sampled indices. One characterization of the Euclidean distance matrix, due to Gower [9], states that D is an Euclidean distance matrix if and only if D = Xi ,i+X j , j\u22122Xi ,j = Di ,j \u2200i, j for some positive semide\ufb01nite matrix X satisfying X \u00b71 = 0. As such, the above nuclear norm minimization can be interpreted as a regularization of the rank with a prior assumption that the true Gram matrix is low rank. An alternative approach based on the matrix completion method is direct completion of the distance matrix [10], [11], [12]. Compared to this approach, an advantage of the above minimization program can be seen by comparing the rank of the Gram matrix X and the distance matrix D. Using (1), the rank of D is at most r +2 while the rank of X is simply r. Using matrix completion theory, it can be surmised that relatively less number of samples are required for the Gram matrix completion. Numerical experiments in [7] con\ufb01rm this observation. In [13], the authors consider a theoretical analysis of a speci\ufb01c instance of localization problem and propose an algorithm similar to (2). The paper considers a random geometric model and derives interesting results of bound of errors in reconstructing point coordinates. While the localization problem and EDG problem share a common theme, we remark that the EDG problem is di\ufb00erent and our analysis adopts the matrix completion framework. The main task of this paper is a theoretical analysis of the above minimization problem. In particular, under appropriate conditions, we will show that the above nuclear norm minimization recovers the underlying inner product matrix. a) Related Work: The EDG problem has been well studied in various contexts. Early works establish mathematical properties of Euclidean distance matrices (EDM) [9] and prove conditions for a symmetric hollow matrix to be EDM [14], [15]. Particularly, the important result due to Schoenberg states that a symmetric hollow matrix D is EDM if and only if the matrix \u22121 2JDJ is positive semide\ufb01nite. J is the geometric centering matrix de\ufb01ned as J = I \u22121 n11T [14]. In follow-up theoretical papers, the EDM completion problem was considered. Given a symmetric hollow matrix with some entries speci\ufb01ed, the EDM completion problem asks if the partial matrix can be completed to an EDM. A work of Hendrickson relates the EDM completion problem to a graph realization problem and provides necessary graph theoretic conditions on the partial matrix that ensures a unique solution [16]. Bakonyi and Johnson establish a link between EDM completion and the graph of partial distance matrix. Speci\ufb01cally, they show that there is a completion to a distance matrix if the graph is chordal [17]. The connection between the EDM completion problem and the positive semide\ufb01nite matrix completion was established in [18] and [19]. In [20], Alfakih establishes necessary and su\ufb03cient conditions for the solution of EDM completion problem to be unique. The emphasis in the above theoretical works is analytic characterization of EDMs and the EDM completion problem mostly employing graph theoretic methods. In the numerical side, a variety of algorithms using di\ufb00erent optimization techniques have been developed to solve the EDM completion problem [3], [21], [22], [23], [24], [25], [26]. The above review is by no means exhaustive and we recommend the interested reader to refer to [27] and [28]. This paper adopts the low-rank matrix recovery framework. While the well-known matrix completion theory can be applied to reconstruct D [10], [12], it can not be directly used to analyze (2). In particular, the measurements in (2) are with respect to the Gram matrix while the measurements in the work of [10], [12] are entry wise sampling of the distance matrix. The emphasis in this paper is on theoretical understanding of the nuclear norm minimization formulation for the EDG problem as stated in (2). In particular, the goal is to provide rigorous probabilistic guarantees that give precise estimates for the minimum number of measurements needed for a certain success probability of the recovery algorithm. b) Main Challenges: The random linear constraints in (2) can be equivalently written as a linear map L acting on X. A \ufb01rst approach to showing uniqueness for the problem (2) is to check if L satis\ufb01es the restricted isometry property (RIP) [12]. However, the RIP does not hold for the completion problem (2). This can be simply veri\ufb01ed by choosing any (i, j) < \u2126 and construct a matrix X with Xi, j = X j,i = Xi,i = X j,j = 1 and zero everywhere else. Then, it is clear that L(X) = 0 implying that the RIP condition does not hold. In general, RIP based analysis is not suitable for deterministic structured measurements. Adopting the framework introduced in [29], the nuclear norm minimization problem in (2) can be written as a matrix completion problem with respect to a basis {w\u03b1}. It, however, turns out that w\u03b1 is not orthogonal. With non-orthogonal basis, the measurements \u27e8w\u03b1, X\u27e9are not compatible with the expansion coe\ufb03cients of the true Gram matrix. A possible remedy is orthogonalization of the basis, say using the Gram-Schmidt orthogonalization. Unfortunately, the orthogonalization process does not preserve the structure of the basis w\u03b1. This has the implication that the modi\ufb01ed minimization problem (2) no longer corresponds to the original problem. As such, the lack of orthogonality is critical in this problem. In addition, it is necessary that the solution of (2) is symmetric positive semide\ufb01nite satisfying the constraint X \u00b7 1 = 0. On the basis of the above considerations, an alternative approach is considered to show that (2) admits a unique solution. The analysis presented in this paper is not based on RIP but on the dual certi\ufb01cate approach introduced in [10]. Our proof was inspired by the work of David Gross [29] where the author generalizes the matrix completion problem to any orthonormal basis. In the case of the EDG problem, one main challenge is that the sampling operator, an important operator in matrix completion, is no longer self-adjoint. This necessitates modi\ufb01cations and alternative proofs to some of the technical statements that appear in [29]. c) Major Contributions: In this paper, a dual basis approach is introduced to show that (2) has a unique solution under appropriate sampling conditions. First, the minimization problem in (2) is written as matrix completion problem with respect to a basis w\u03b1. Second, by introducing a dual basis {v\u03b1} to {w\u03b1}, one can ensure that the measurements \u27e8X , w\u03b1\u27e9in (2) are compatible with expansion coe\ufb03cients of the true Gram matrix M. The two main contributions of this paper are as follows. 1) A dual basis approach is introduced to address the EDG problem. Under certain assumptions, we show that the nuclear \f3 norm minimization problem succeeds in recovering the underlying low rank solution. Precisely, the main result states that if |\u2126| = m \u2265O(nr log2 n), the nuclear norm minimization program (2) recovers the underlying inner product matrix with very high probability, 1 \u2212n\u2212\u03b2 for \u03b2 > 1(see more details in Theorem 1). The minimization problem has a positive semide\ufb01nite constraint. The proof describes how this constraint is handled. In addition to the Bernstein bound which appears in matrix completion analysis, a crucial part in the proof uses the operator Cherno\ufb00bound. We would like to emphasize that this part of the proof and the estimate using Cherno\ufb00bound is simple and could be useful in a broader setting. 2) We develop simple and fast algorithms to solve the EDG problem under two instances. The \ufb01rst instance considers the case of exact partial information. The second instance considers the more realistic setup of a noisy partial information. Numerical tests on various data show that the algorithms are accurate and e\ufb03cient. The outline of the paper is as follows. In section II, we introduce a dual basis approach and formulate a well-de\ufb01ned matrix completion problem for the EDG problem. We conclude the section by proposing coherence conditions for the EDG problem and explaining the sampling scheme. In section III, the proof of exact reconstruction is presented. In brief terms, the main parts are summarized as follows. From convex optimization theory, showing uniqueness of the EDG problem is equivalent to showing that there exists a dual certi\ufb01cate, denoted by Y , satisfying certain conditions. Y is constructed using the gol\ufb01ng scheme proposed in [29]. Next, we show that these conditions hold with very high probability by employing concentration inequalities. This implies that there is a unique solution with very high probability. In section IV, fast numerical algorithms for the EDG matrix completion problem are proposed. Section V validates the e\ufb03ciency and e\ufb00ectiveness of the proposed numerical algorithms. Finally, we conclude the work in the last section. d) Notation: To make our notation consistent in the paper, we summarize notations used in this paper in Table I. TABLE I Notations x Vector ||x||2 l2 norm X Matrix ||X||F Frobenius norm X Operator ||X|| Operator norm XT Transpose ||X||\u2217 Nuclear norm Tr(X) Trace ||A|| sup||X||F=1 ||AX||F. \u27e8X , Y \u27e9 Trace(XT Y ) \u03bbmax, \u03bbmin Maximum, Minimum eigenvalue 1 A vector or matrix of ones Sgn X Usign (\u03a3)V T ; here [U, \u03a3, V ] = svd(X) 0 A vector or matrix of zeros \u2126, I Random sampled set, Universal set II. Dual basis formulation The aim of this section is to show that the EDG problem (2) can be equivalently stated as a matrix completion problem with respect to a special designed basis. The matrix completion problem is an instance of the low rank matrix recovery problem where the goal is to recover a low rank matrix given random partial linear observations. The natural minimization problem for low rank recovery problem is rank minimization with linear constraints. Unfortunately, this problem is NP hard [12] motivating other solutions. In a series of remarkable theoretical papers [10], [12], [29], [30], [31], it was shown that, under certain conditions, the solutions of the NP hard rank minimization problem can be obtained by solving the convex nuclear norm minimization problem which is computationally tractable [32], [33]. Our theoretical analysis is inspired by the work [29] where the author extends the nuclear norm minimization formulation to recovering a low rank matrix given that a few of its coe\ufb03cients in a \ufb01xed orthonormal basis are known. a) Dual basis: To write the EDG problem (2) as a matrix completion problem with respect to an appropriate operator basis, let us introduce few notations. We write I = {\u03b1 = (\u03b11, \u03b12) | 1 \u2264\u03b11 < \u03b12 \u2264n} and de\ufb01ne w\u03b1 = e\u03b11, \u03b11 + e\u03b12, \u03b12 \u2212e\u03b11, \u03b12 \u2212e\u03b12, \u03b11 (3) where e\u03b11, \u03b12 is a matrix whose entries are all zero except a 1 in the (\u03b11, \u03b12)-th entry. It is clear that {w\u03b1} forms a basis of the linear space S = {X \u2208Rn\u00d7n | X = XT & X \u00b7 1 = 0} and the number of basis is L = n(n\u22121) 2 . For conciseness and ease in later analysis, we further de\ufb01ne S+ = S \u2229{X \u2208Rn\u00d7n | X \u2ab00}. Therefore, for any given subsample \u2126of I, the EDG problem (2) can be written as the following nuclear norm minimization problem minimize X\u2208S+ ||X||\u2217 subject to \u27e8w\u03b1, X\u27e9= \u27e8w\u03b1, M\u27e9 \u2200\u03b1 \u2208\u2126 (4) where M is the true underlying rank r Gram matrix satisfying Mi, i + Mj, j \u22122Mi, j = Di, j \u2200(i, j). The EDG problem can now be equivalently interpreted as a matrix completion problem with respect to the basis w\u03b1. \f4 By construction, w\u03b1 is symmetric and satis\ufb01es w\u03b1 \u00b71 = 0. The latter condition naturally enforces the constraint X \u00b71 = 0. It is clear that any X \u2208S can be expanded in the basis {w\u03b1} as X = P \u03b2\u2208I c\u03b2w\u03b2. After minor algebra, c\u03b2 = P \u03b1\u2208I\u27e8X , w\u03b1\u27e9H\u03b1, \u03b2 where we de\ufb01ne H\u03b1, \u03b2 = \u27e8w\u03b1 , w\u03b2\u27e9and H\u03b1, \u03b2 = H\u22121 \u03b1, \u03b2. Note that, since {w\u03b1} is a basis, the inverse H\u22121 is well de\ufb01ned. This results in the following representation of X. X = X \u03b2\u2208I X \u03b1\u2208I H\u03b1, \u03b2\u27e8w\u03b1 , X\u27e9w\u03b2 (5) The crux of the dual basis approach is to simply consider (5) and rewrite it as follows X = X \u03b1\u2208I \u27e8X , w\u03b1\u27e9v\u03b1 (6) where v\u03b1 = P \u03b2 H\u03b1, \u03b2w\u03b2. It can be easily veri\ufb01ed that {v\u03b1} is a dual basis of {w\u03b1} satisfying \u27e8v\u03b1 , w\u03b2\u27e9= \u03b4\u03b1, \u03b2. Let W = [w1, w2, ..., wL] and V = [v1, v2, ..., vL] denote the matrix of vectorized basis matrices and vectorized dual basis matrices respectively. The following basic relations are useful in later analysis, H = W TW , H\u22121 = V TV and V = W H\u22121. In the context of the EDG completion problem, the dual basis approach ensures that the expansion coe\ufb03cients match the measurements while preserving the condition that the matrix in consideration is symmetric with zero row sums. With this, (4) turns into a well formulated matrix completion problem with respect to the basis w\u03b1. An alternative way to rewrite (4) makes use of the sampling operator de\ufb01ned as follows. R\u2126: X \u2208S \u2212\u2192L m X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9v\u03b1 (7) where we assume that \u2126are sampled uniformly at random from I with replacement and m = |\u2126| is the size of \u2126. The scaling factor L m is for ease in later analysis. It can be easily veri\ufb01ed that m2 L2 R2 \u2126= m L R\u2126. The adjoint operator of R\u2126, R\u2217 \u2126, can be simply derived and is given by R\u2217 \u2126: X \u2208S \u2212\u2192L m X \u03b1\u2208\u2126 \u27e8X , v\u03b1\u27e9w\u03b1 (8) Using the sampling operator R\u2126, we can write (4) as follows minimize X\u2208S+ \u2225X\u2225\u2217 subject to R\u2126(X) = R\u2126(M) (9) In addition to the sampling operator, the restricted frame operator appears in the analysis and is de\ufb01ned as follows F\u2126: X \u2208S \u2212\u2192L m X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9w\u03b1 (10) Note that the restricted frame operator is self-adjoint and positive semide\ufb01nite. b) Coherence: Pathological situations can arise in (9) when M has very few non-zero coe\ufb03cients. In the context of EDG, in extreme cases, this happens if only the diagonal elements of D are sampled and/or there are many overlapping points. This motivates the notion of coherence \ufb01rst introduced in [10]. Let M = Pr k=1 \u03bbkukuT k , where the eigenvectors have been chosen to be orthonormal and \u03bbk \u22650 as M \u2208S+. We write U = span {u1, ..., ur} as the column space of M, U \u22a5= span{ur+1, ..., un} as the orthogonal complement of U and further denote PU and PU \u22a5as the orthogonal projections onto U and U \u22a5, respectively. Let T = {UZT + ZU T : Z \u2208Rn\u00d7r} be the tangent space of the rank r matrix in S+ at M. The orthogonal projection onto T is given by PT X = PUX + XPU \u2212PUXPU (11) It can be readily veri\ufb01ed that PT\u22a5X = X \u2212PTX = PU \u22a5XPU \u22a5. The coherence conditions can now be de\ufb01ned as follows. De\ufb01nition 1. The aforementioned rank r matrix M \u2208S+ has coherence \u03bd with respect to basis {w\u03b1}\u03b1\u2208I if the following estimates hold max \u03b1\u2208I X \u03b2\u2208I \u27e8PT w\u03b1 , w\u03b2\u27e92 \u22642\u03bd r n (12) max \u03b1\u2208I X \u03b2\u2208I \u27e8PT v\u03b1 , w\u03b2\u27e92 \u22644\u03bd r n (13) max \u03b1\u2208I \u27e8w\u03b1 , UU T\u27e92 \u22641 4\u03bd r n2 (14) where {v\u03b1}\u03b1\u2208I is the dual basis of {w\u03b1}\u03b1\u2208I. \f5 Remark 1. If {w\u03b1} is an orthonormal basis, it follows trivially that the dual basis {v\u03b1} is also {w\u03b1}. For this speci\ufb01c case, the above coherence conditions are equivalent, up to constants, to the coherence conditions in [29]. However, in the most general setting, the coherence conditions (12) and (13) depart from their orthonormal counterparts. This is because these conditions depend on the spectrum of the correlation matrix H. Since the analysis in the sequel makes repeated use of the coherence conditions, the above equations are further simpli\ufb01ed to convenient forms as allows. First, using (12), consider a bound on ||PT w\u03b1||2 F. Using Lemma A.4 and Lemma A.6, one obtains ||PT w\u03b1||2 F \u2264max \u03b1\u2208I X \u03b2\u2208I \u27e8PT w\u03b1 , w\u03b2\u27e92 \u22642\u03bd r n \u2212\u2192||PT w\u03b1||2 F \u22642\u03bd r n Using the above inequality and Lemma A.4, one obtains the following bound for ||PT v\u03b1||F. ||PT v\u03b1||F \u2264 X \u03b2\u2208I ||H\u03b1, \u03b2 PT w\u03b2||F = X \u03b2\u2208I |H\u03b1, \u03b2| ||PT w\u03b2||F \u22642 r 2\u03bdr n Next, consider a bound on \u27e8v\u03b1 , UU T\u27e92. Using (14) and Lemma A.4, one arrives at the following inequality. \u27e8v\u03b1 , UU T\u27e92 = \u27e8 X \u03b2\u2208I H\u03b1, \u03b2w\u03b2 , UU T\u27e92 \u2264max \u03b2\u2208I \u27e8w\u03b2 , UU T\u27e92 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X \u03b2\u2208I |H\u03b1, \u03b2| \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 2 \u2264\u03bdr n2 All in all, the simpli\ufb01ed forms of the coherence conditions are summarized as follows. max \u03b1\u2208I ||PT w\u03b1||2 F \u22642\u03bd r n (15) max \u03b1\u2208I ||PT v\u03b1||2 F \u22648\u03bd r n (16) max \u03b1\u2208I \u27e8v\u03b1 , UU T\u27e92 \u2264\u03bdr n2 (17) The simpli\ufb01ed coherence conditions presented above are the same as the standard coherence assumptions up to constants (see [29], [31]). If the matrix M has coherence \u03bd with respect to the standard basis, comparable bounds could be derived for the above coherence conditions. This is true since the basis {w\u03b1} is not \u201cfar\u201d from the standard basis. For a precise statement of this fact, we refer the interested reader to Lemma A.5. The implication of this fact is that an incoherent M with respect to the standard basis is also incoherent with respect to the EDG basis. Intuitively, the coherence parameter is fundamentally about the concentration of information in the underlying matrix. For a matrix with low coherence(incoherent), each measurement is equally as informative as the other. In contrast, the information is concentrated on few measurements for a coherent matrix. Remark 2 (Coherence and Geometry). In the context of the EDG problem, an interesting question is the relationship, if any, between coherence and the geometry of the underlying points. One analytical task is to relate the coherence, using the compressive sensing de\ufb01nition, of the Gram matrix to the coherence conditions (12)-(14). Computationally, numerical tests indicate that points arising from random distributions or points which are a random sample of structured points have comparable coherence values irrespective of the underlying distribution. However, since estimating coherence accurately and e\ufb03ciently for large n is computationally expensive, the numerical tests were limited to relatively small values of n. The question of the geometry of points and how it relates to coherence is important to us and future work will explore this problem through analysis and extensive numerical tests. c) Sampling Model: The sampling scheme is an important element of the matrix completion problem. For the EDG completion problem, it is assumed that the basis vectors are sampled uniformly at random with replacement. Previous works have considered uniform sampling with replacement [29], [31] and Bernoulli sampling [10]. The implication of our choice is that the sampling process is independent. This property is essential as we make use of concentration inequalities for i.i.d matrix valued random variables. Remark 3. For the EDG completion problem, we have considered uniform sampling with out replacement. In this case, the sampling process is no longer independent. The implication is that one could not simply use concentration inequalities for i.i.d matrix valued random variables. However, as noted in the classic work of Hoe\ufb00ding [34, section 6], the results derived for the case of the sampling with replacement also hold true for the case of sampling without replacement. An analogous argument for matrix concentration inequalities, resulting from the operator Cherno\ufb00bound technique [35], under uniform sampling with out replacement is shown in [36]. The analysis based on this choice leads to a \u201cslightly lower\u201d number of measurements m. Since the gain is minimal, which will be made apparent in later analysis, we use the uniform sampling with replacement model. \f6 III. Proof of Main Result The main goal of this section is to show that the nuclear norm minimization problem (9) admits a unique solution. Thus, the optimization problem provides the desired reconstruction. Under certain conditions, our main result guarantees a unique solution for the minimization problem (9). More precisely, we have the following theorem. Theorem 1. Let M \u2208Rn\u00d7n be a matrix of rank r that obeys the coherence conditions (12), (13) and (14) with coherence \u03bd. Assume m measurements, {\u27e8M , w\u03b1\u27e9}\u03b1\u2208\u2126, are sampled uniformly at random with replacement. For \u03b2 > 1, if m \u2265nr log2 \u0010 4 \u221a 8Lrn \u0011 \" 96   \u03bd + 1 nr ! \u0010 \u03b2 log(n) + log \u0010 4 log2(4L \u221ar) \u0011\u0011# the solution to (9), equivalently (2) and (4), is unique and equal to M with probability at least 1 \u2212n\u2212\u03b2. The optimization problem in (9) is a convex minimization problem for which the optimality of a feasible solution X is equivalent to the su\ufb03cient KKT conditions. For details, we refer the interested reader to [10] where the the authors derive a compact form of these optimality conditions. The over all structure of the proof is as follows. First, we show that if certain deterministic optimality and uniqueness conditions hold, then it certi\ufb01es that M is a unique solution to the minimization problem. The remaining part of the proof will focus on showing that these conditions hold with very high probability under certain conditions. This in turn will imply that M is a unique solution with very high probability for an appropriate choice of m. The proof of Theorem 1 closely follows the approach in [29]. For ease of presentation, the proof is divided into several intermediate results. Readers familiar with matrix completion proof in [29], [31] will recall that one crucial part of the proof is bounding the spectral norm of ||PTR\u2126PT\u2212PT||. The interpretation of this bound is that the operator PTR\u2126PT is almost isometric to PT. In the case of our problem, R\u2126is no longer self-adjoint and the equivalent statement is a bound on ||PTR\u2217 \u2126R\u2126PT \u2212PT||. Unfortunately, the term ||PTR\u2217 \u2126R\u2126PT \u2212PT|| is not amenable to simpler analysis as standard concentration inequalities result sub-optimal success rates. The idea is instead to consider the operator PTF\u2126PT and show that the minimum eigenvalue is bounded away from 0 with high probability using the operator Cherno\ufb00bound. The implication is that the operator PTF\u2126PT is full rank on the space T. For non orthogonal measurements, PTF\u2126PT can be interpreted as the analogue of the operator PTR\u2126PT. In [30], for the standard matrix completion problem with measurement basis ei j, it was argued theoretically that a lower bound for m is O(nr\u03bd log n). Theorem 1 requires on the order of nr\u03bd log2 n measurements which is log(n) multiplicative factor away from this sharp lower bound. We remark that, despite the aforementioned technical challenges, our result is of the same order as the result in [29], [31] which consider the low rank recovery problem with any orthogonal basis and the matrix completion problem respectively. We start the proof by stating and proving Theorem 2 which rigorously shows that if appropriate conditions as discussed earlier hold, then M is a unique solution to (9). Theorem 2. Given X \u2208S+, we write \u2206= X \u2212M as the deviation from the underlying low rank matrix M. Let \u2206T and \u2206T\u22a5be the orthogonal projection of \u2206to T and T\u22a5, respectively. For any given \u2126with |\u2126| = m, the following two statements hold. (a). If ||\u2206T||2 F \u2265 \u0010 n \u221a 8L||\u2206T\u22a5||F \u00112 and \u03bbmin (PT F\u2126PT) > 1 2, then R\u2126\u2206, 0. (b). If ||\u2206T||2 F < \u0010 n \u221a 8L||\u2206T\u22a5||F \u00112 for \u2206\u2208ker R\u2126, and there exists a Y \u2208range R\u2217 \u2126satisfying, ||PTY \u2212Sgn M||F \u2264 1 4n \u221a 8L and ||PT\u22a5Y || \u22641 2 (18) then \u2225X\u2225\u2217= \u2225M + \u2206\u2225\u2217> \u2225M\u2225\u2217. To interpret the above theorem, note that Theorem 2(a) states that any deviation from M fails to be in the null space of the sampling operator for \u201clarge\u201d \u2206T. For \u201csmall\u201d \u2206T, Theorem 2(b) claims that any deviation from M increases the nuclear norm thereby violating the minimization condition. We would like to emphasize that this is a deterministic theorem which does not depend on the random sampling of \u2126as long as the conditions in the statement are satis\ufb01ed. In fact, we will later show that these conditions will hold with very high probability under certain sampling conditions and an appropriate choice of m = |\u2126|. A. Proof of Theorem 2 Proof of Theorem 2(a). Suppose \u2225\u2206T\u22252 F \u2265 \u0010 n \u221a 8L\u2225\u2206T\u22a5\u2225F \u00112. To prove R\u2126\u2206, 0, it su\ufb03ces to show \u2225R\u2126\u2206\u2225F > 0. Note that \u2225R\u2126\u2206\u2225F = \u2225R\u2126\u2206T + R\u2126\u2206T\u22a5\u2225F \u2265\u2225R\u2126\u2206T\u2225F \u2212\u2225R\u2126\u2206T\u22a5\u2225F. This motivates \ufb01nding a lower bound for ||R\u2126\u2206T||F and an upper bound for ||R\u2126\u2206T\u22a5||F. For any X, ||R\u2126X||2 F can be bounded as follows ||R\u2126X||2 F = \u27e8R\u2126X , R\u2126X\u27e9= \u27e8X , R\u2217 \u2126R\u2126X\u27e9= L2 m2 X \u03b2\u2208\u2126 X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9\u27e8v\u03b1 , v\u03b2\u27e9\u27e8X , w\u03b2\u27e9= L2 m2 X \u03b2\u2208\u2126 X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e9H\u03b1, \u03b2\u27e8X , w\u03b2\u27e9 \f7 noting that \u27e8v\u03b1 , v\u03b2\u27e9= H\u03b1, \u03b2. Using the min-max theorem in the above equation, one obtains L2 m2 \u03bbmin(H\u22121) X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e92 \u2264||R\u2126X||2 F \u2264L2 m2 \u03bbmax(H\u22121) X \u03b1\u2208\u2126 \u27e8X , w\u03b1\u27e92 \u2264L2 m2 \u03bbmax(H\u22121) X \u03b1\u2208I \u27e8X , w\u03b1\u27e92 (19) Using the fact \u03bbmax(H\u22121) = 1 (Lemma A.4) and P \u03b1\u2208I\u27e8X , w\u03b1\u27e92 \u22642n\u2225X\u22252 F (Lemma A.6) and setting X = \u2206T\u22a5in the right of the above inequality results ||R\u2126\u2206T\u22a5||2 F \u22642L2 m2 nm||\u2206T\u22a5||2 F (20) In the above inequality, the constant m bounds the maximum number of repetitions for any given measurement. Next, the lower bound for ||R\u2126\u2206T||F is considered using the left inequality in (19). ||R\u2126\u2206T||2 F \u2265 L2 2m2n X \u03b1\u2208\u2126 \u27e8\u2206T , w\u03b1\u27e92 = L 2mn\u27e8\u2206T , L m X \u03b1\u2208\u2126 \u27e8\u2206T , w\u03b1\u27e9w\u03b1\u27e9= L 2mn\u27e8\u2206T , F\u2126\u2206T\u27e9 (21) with the fact that \u03bbmin(H\u22121) = 1 2n (Lemma A.4). Using the min-max theorem on the projection onto T of the restricted frame operator, we have ||R\u2126\u2206T||2 F \u2265 L 2mn\u27e8\u2206T , F\u2126\u2206T\u27e9= L 2mn\u27e8\u2206T , PT F\u2126PT \u2206T\u27e9\u2265 L 2mn\u03bbmin(PT F\u2126PT) ||\u2206T||2 F (22) Above, the \ufb01rst equality holds since PT \u2206T = \u2206T and PT is self-adjoint. The last inequality simply applies the min-max theorem on the self-adjoint operator PT F\u2126PT. Using the assumption that \u03bbmin (PT F\u2126PT) > 1 2, the inequality in (22) reduces to ||R\u2126\u2206T||2 F > L 4mn||\u2206T||2 F (23) Combining (20) and (23), we have \u2225R\u2126\u2206\u2225F > r L 4mn||\u2206T||F \u2212L m \u221a 2n||\u2206T\u22a5||F \u2265 r L 4mn \u0010 n \u221a 8L \u0011 ||\u2206T\u22a5||F \u2212L m \u221a 2nm||\u2206T\u22a5||F = 0 where the last inequality follows from the theorem\u2019s assumption. This concludes proof of Theorem 2(a). \u25a1 Remark 4. (1) The estimate of the upper bound of ||R\u2126\u2206T\u22a5||2 F is not sharp. Speci\ufb01cally, the constant m can be much lowered. This can be achieved, for example, by employing standard concentration inequalities and arguing that the expected number of duplicate measurements is substantially smaller than m. Alternatively, as noted earlier in the sampling model section, uniform sampling with out replacement can be adopted which has the implication that the constant m is not necessary. Both analysis lead to a better estimate of the upper bound. However, the gain in terms of measurements, m, is minimal, resulting lower constants inside of a log2. We use the current estimate, albeit not sharp, for sake of more compact analysis. (2) Lower bounding the term P \u03b1\u2208\u2126\u27e8\u2206T , w\u03b1\u27e92 is not amenable to simpler analysis. In fact, a mere application of the standard concentration inequalities result an undesirable probability of failure that increases with n. Note that ||R\u2126\u2206T||2 F = \u27e8\u2206T , PTR\u2217 \u2126R\u2126PT\u2206T \u2212PT\u2206T\u27e9+ ||PT\u2206T||2 F. A lower bound for ||R\u2126\u2206T||2 F motivates an upper bound of ||PTR\u2217 \u2126R\u2126PT\u2206T \u2212PT||. This bound can be achieved, albeit long calculations, but it relies on the special structure of w\u03b1. To make the technical analysis simple and more general, we use an alternative approach shown in the above proof. This is a di\ufb00erent way to handle the lower bound estimation of ||R\u2126\u2206T||2 F. Proof of Theorem 2(b). Consider a feasible solution X = M + \u2206\u2208S+ satisfying ||\u2206T||F < n \u221a 8L\u2225\u2206T\u22a5\u2225F and \u2206\u2208ker R\u2126. We need to show that \u2225M + \u2206\u2225\u2217> \u2225M\u2225\u2217which implies that nuclear minimization is violated. The proof of this requires the construction of a dual certi\ufb01cate Y satisfying certain conditions. The proof is similar to the proof in section 2E of [29] but it is shown below for completeness and ease of reference in later analysis. Using the pinching inequality [37], \u2225M +\u2206\u2225\u2217\u2265||PU(M +\u2206)PU||\u2217+||PU \u22a5(M +\u2206)PU \u22a5||\u2217. Noting that PUMPU = M, PU \u22a5MPU \u22a5= 0 and PU \u22a5\u2206PU \u22a5= \u2206T\u22a5, the above inequality reduces to \u2225M + \u2206\u2225\u2217\u2265||M + PU\u2206PU||\u2217+ ||\u2206T\u22a5||\u2217 (24) Note that ||\u2206T\u22a5||\u2217= \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9. The \ufb01rst term on right hand side can be lower bounded using the fact that the nuclear norm and the spectral norm are dual to one another. Stated precisely, ||X2||\u2217= sup ||X1||\u22641 \u27e8X1 , X2\u27e9for any X1. Using this inequality with X1 = Sgn M and X2 = M + PU\u2206PU in (24) results ||M + PU\u2206PU||\u2217+ ||\u2206T\u22a5||\u2217\u2265\u27e8Sgn M , M + PU\u2206PU\u27e9+ \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9 = ||M||\u2217+ \u27e8Sgn M , PU\u2206PU\u27e9+ \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9 \f8 Note that Sgn M \u2208T (see Lemma A.1). Using this fact, it can be veri\ufb01ed that \u27e8Sgn M , PU\u2206T\u22a5PU\u27e9= 0 and \u27e8Sgn M , PU\u2206T PU\u27e9= \u27e8Sgn M , \u2206T\u27e9. The above inequality can now be written as ||M + PU\u2206PU||\u2217+ ||\u2206T\u22a5||\u2217\u2265||M||\u2217+ \u27e8Sgn M , \u2206T\u27e9+ \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9 (25) Using (24) and (25), it can be concluded that ||M +\u2206||\u2217> ||M||\u2217if it can be shown that \u27e8Sgn M , \u2206T\u27e9+\u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9> 0. Since \u27e8Y , \u2206\u27e9= 0 for any Y \u2208range R\u2217 \u2126, we have \u27e8Sgn M , \u2206T\u27e9+ \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9= \u27e8Sgn M , \u2206T\u27e9+ \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9\u2212\u27e8Y , \u2206\u27e9 = \u27e8Sgn M \u2212PTY , \u2206T\u27e9+ \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9\u2212\u27e8PT\u22a5Y , \u2206T\u22a5\u27e9 Assuming the conditions in the statement of the theorem and considering the last equation, we obtain \u27e8Sgn M , \u2206T\u27e9+ \u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9\u2265\u2212 1 4n \u221a 8L ||\u2206T||F + ||\u2206T\u22a5||\u2217\u22121 2||\u2206T\u22a5||\u2217 \u2265\u2212 1 4n \u221a 8L ||\u2206T||F + 1 2||\u2206T\u22a5||F > \u2212 1 4n \u221a 8L \u0010 n \u221a 8L\u2225\u2206T\u22a5\u2225F \u0011 + 1 2||\u2206T\u22a5||F = 1 4||\u2206T\u22a5||F Above, the \ufb01rst inequality follows from the duality of the spectral norm and the nuclear norm. It has been shown that \u27e8Sgn M , \u2206T\u27e9+\u27e8Sgn \u2206T\u22a5, \u2206T\u22a5\u27e9> 0 under the assumptions of Theorem 2(b). Using (24) and (25), it follows that \u2225M +\u2206\u2225\u2217> \u2225M\u2225\u2217concluding the proof of Theorem 2(b). \u25a1 Consequently, if the deterministic conditions in Theorem 2 hold, M is a unique solution to (9). We formally write it as a corollary which can help us to prove the probabilistic statement in Theorem 1. Corollary 1. If the conditions of Theorem 2 hold, M is a unique solution to (9). Proof. For any X \u2208S+, de\ufb01ne \u2206= M \u2212X. Using Theorem 2(a), R\u2126\u2206, 0 if ||\u2206T||2 F \u2265 \u0010 n \u221a 8L||\u2206T\u22a5||F \u00112. It then su\ufb03ces to consider the case ||\u2206T||2 F < \u0010 n \u221a 8L||\u2206T\u22a5||F \u00112 for \u2206\u2208ker R\u2126. For this case, the proof of Theorem 2(b) shows that \u2225X\u2225\u2217> \u2225M\u2225\u2217. It can then be concluded that M is the unique solution to (9). \u25a1 B. Proof of Theorem 1 It follows that certifying that the two conditions in Theorem 2 hold implies a unique solution to (9). Hence, the main task in the proof is to show that, under the assumptions of Theorem 1, these two conditions hold with very high probability. If this can be achieved, it implies that the conclusion of Theorem 1 holds true with the same high probability. The \ufb01rst condition in Theorem 2 is that \u03bbmin (PT F\u2126PT) > 1 2. The goal is to show that the minimum eigenvalue of the operator PT F\u2126PT is bounded away from zero. This will be proven in Lemma 1 to follow shortly. The lemma makes use of the matrix Cherno\ufb00bound in [38] and is restated below for convenience. Theorem 3. Let {Lk} be a \ufb01nite sequence of independent, random, self-adjoint operators satisfying Lk \u2ab00 and ||Lk|| \u2264R almost surely with the minimum and maximum eigenvalues of the sum of the expectations, \u00b5min := \u03bbmin \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X k E[Lk] \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 and \u00b5max := \u03bbmax \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X k E[Lk] \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 Then, we have Pr \u0014 \u03bbmin \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X k Lk \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2264(1 \u2212\u03b4) \u00b5min \u0015 \u2264n \u0014 exp(\u2212\u03b4) (1 \u2212\u03b4)1\u2212\u03b4 \u0015 \u00b5min R for \u03b4 \u2208[0, 1] For \u03b4 \u2208[0, 1], using Taylor series of log(1 \u2212\u03b4), note that (1 \u2212\u03b4) log(1 \u2212\u03b4) \u2265\u2212\u03b4 + \u03b42 2 . This results the following simpli\ufb01ed estimate Pr \u0014 \u03bbmin \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X k Lk \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2264(1 \u2212\u03b4) \u00b5min \u0015 \u2264n exp \u0012 \u2212\u03b42 \u00b5min 2R \u0013 for \u03b4 \u2208[0, 1] Lemma 1. Given the operator PT F\u2126PT : T \u2192T with \u03ba = m nr, the following estimate holds. Pr   \u03bbmin (PT F\u2126PT) \u22641 2 ! \u2264n exp \u0012 \u2212\u03ba 8\u03bd \u0013 Proof. Using the de\ufb01nition of F\u2126(X) = L m P \u03b1\u2208\u2126\u27e8X , w\u03b1\u27e9w\u03b1, for X \u2208T, PT F\u2126PT X can be written as follows PT F\u2126PT X = X \u03b1\u2208\u2126 L m\u27e8X , PT w\u03b1\u27e9PT w\u03b1 \f9 Let L\u03b1 denote the operator in the summand, L\u03b1 = L m\u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1. It can be easily veri\ufb01ed that L\u03b1 is positive semide\ufb01nite. The operator Cherno\ufb00bound requires estimate of R and \u00b5min. R can be estimated as follows. \f \f \f \f \f \f \f \f \f \f L m\u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1 \f \f \f \f \f \f \f \f \f \f = L m||PT w\u03b1||2 F \u2264n\u03bdr m The last inequality follows from the coherence estimate in (15). Therefore, R = n\u03bdr m . Next, we consider \u00b5min. The \ufb01rst step is to compute P \u03b1\u2208\u2126E[L\u03b1]. X \u03b1\u2208\u2126 E[L\u03b1] = X \u03b1\u2208\u2126 \u0014 X \u03b1 1 m\u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1 \u0015 = X \u03b1 \u27e8\u00b7 , PT w\u03b1\u27e9PT w\u03b1 For any X \u2208T, lower bound \u27e8X , X \u03b1\u2208\u2126 E[L\u03b1](X)\u27e9as follows. \u27e8X , X \u03b1\u2208\u2126 E[L\u03b1](X)\u27e9= X \u03b1 \u27e8X, w\u03b1\u27e92 \u2265||X||2 F The inequality results from Lemma A.4 and Lemma A.6. Noting that P \u03b1\u2208\u2126E[L\u03b1] is a self-adjoint operator, and using the variational characterization of the minimum eigenvalue, it can be concluded that the minimum eigenvalue of P \u03b1\u2208\u2126E[L\u03b1] is at least 1. With this, set \u00b5min = 1. Finally, apply the matrix Cherno\ufb00bound with R = n\u03bdr m and \u00b5min = 1. Setting \u03b4 = 1 2, \u03bbmin (PT F\u2126PT) > 1 2 with probability of failure at most p1 given by p1 = n exp \u0012 \u2212\u03ba 8\u03bd \u0013 This concludes the proof. \u25a1 Under the assumptions of Theorem 1 and using Lemma 1, \u03bbmin (PT F\u2126PT) > 1 2 holds with probability 1 \u2212p1 where the probability of failure is p1 = n exp \u0012 \u2212\u03ba 8\u03bd \u0013 with \u03ba = m nr. The proof of the second condition in Theorem 2 is more technically involved and requires the construction of Y satisfying the conditions in (18). The construction follows the ingenious gol\ufb01ng scheme introduced in [29]. Partition the basis elements in \u2126into l batches with the i-th batch \u2126i containing mi elements and l X i=1 mi = m. The restriction operator for a batch \u2126i is de\ufb01ned as Ri = L mi X \u03b1\u2208\u2126i \u27e8X , w\u03b1\u27e9v\u03b1. Then Y = Yl is constructed using the following recursive scheme Q0 = sgn M, Yi = i X j=1 R\u2217 jQ j\u22121, Qi = sgn M \u2212PTYi, i = 1, \u00b7 \u00b7 \u00b7 , l (26) Using the gol\ufb01ng scheme, it will be shown that the conditions in (18) hold with very high probability. The technical analysis of the scheme, particularly certifying the \ufb01rst condition in (18), requires a probabilistic estimate of ||PTR\u2217 \u2126PTX \u2212PTX||F \u2265t for a \ufb01xed matrix X \u2208S. Using Lemma 2 to follow, a suitable estimate can be achieved. The lemma uses the vector Bernstein inequality in [29]. An easier but slightly weaker version is restated below for convenience of reference. Theorem 4 (Vector Bernstein inequality). Let x1, ..., xm be independent zero-mean vector valued random variables. Assuming that max i ||xi||2 \u2264R and let P i E[||xi||2 2] \u2264\u03c32, then for any t \u2264\u03c32 R , the following inequality holds Pr \u0014\f \f \f \f \f \f \f \f \f \f m X i=1 xi \f \f \f \f \f \f \f \f \f \f2 \u2265t \u0015 \u2264exp   \u2212t2 8\u03c32 + 1 4 ! , Lemma 2. For a \ufb01xed matrix X \u2208S, the following estimate holds Pr (||PTR\u2217 \u2126PTX \u2212PTX||F \u2265t||X||F) \u2264exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 t2\u03ba 16 \u0010 \u03bd + 1 nr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (27) for all t \u22641 with \u03ba = m nr. Proof. It is clear that PTX \u2208S from the de\ufb01nition of PT. Thus, we can expand PTR\u2217 \u2126PTX as follows. PTR\u2217 \u2126PTX = L m X \u03b1\u2208\u2126 \u27e8PTX , v\u03b1\u27e9PT w\u03b1 (28) \f10 With this, PTR\u2217 \u2126PTX \u2212PTX can be written as follows PTR\u2217 \u2126PTX \u2212PTX = X \u03b1\u2208\u2126 \u0014 L m\u27e8PTX , v\u03b1\u27e9PT w\u03b1 \u22121 mPTX \u0015 (29) Let X\u03b1 = L m\u27e8PTX , v\u03b1\u27e9PT w\u03b1. Noting that E[ L m\u27e8PTX , v\u03b1\u27e9PT w\u03b1] = 1 mPTX, let Y\u03b1 = X\u03b1 \u2212E[X\u03b1] denote the summand. It is clear that E[Y\u03b1] = 0 by construction. The vector Bernstein inequality requires a suitable bound for ||Y\u03b1||F and E[||Y\u03b1||2 F]. Without loss of generality, assume ||X||F = 1. The \ufb01rst step is to bound ||Y\u03b1||F. ||Y\u03b1||F = \f \f \f \f \f \f \f \f \f \f L m\u27e8PTX , v\u03b1\u27e9PT w\u03b1 \u22121 mPTX \f \f \f \f \f \f \f \f \f \fF \u2264 \f \f \f \f \f \f \f \f \f \f L m\u27e8PTX , v\u03b1\u27e9PT w\u03b1 \f \f \f \f \f \f \f \f \f \fF + \f \f \f \f \f \f \f \f \f \f 1 mPTX \f \f \f \f \f \f \f \f \f \fF (30) < n2 2m max \u03b1\u2208I ||PT w\u03b1 ||F max \u03b1\u2208I ||PT v\u03b1||F + 1 m \u22641 m(2n\u03bdr + 1) (31) Above, the last inequality follows from the coherence estimates (15) and (16). Using the above estimate, set R in the vector Bernstein inequality as R = 1 m(2n\u03bdr + 1). Using the parallelogram identity and monotonicity of expectation, E[||Y\u03b1||2 F] \u2264 2E[||X\u03b1||2 F] + 2||E[X\u03b1]||2 F. With this, E[||Y\u03b1||2 F] can be upper bounded as follows E[||Y\u03b1||2 F] \u22642L2 m2 E[\u27e8PTX , v\u03b1\u27e92||PT w\u03b1||2 F] + 2 m2 ||PTX||2 F \u22642L m2 max \u03b1\u2208I ||PTw\u03b1||2 F X \u03b1\u2208I \u27e8PTX , v\u03b1\u27e92 + 2 m2 \u2264n2 m2 2\u03bdr n + 2 m2 \u22642 m2 (1 + n\u03bdr) Above, the third inequality follows from the coherence estimates, (15) and (16), and Lemma A.6. Using the above estimate, set \u03c32 in vector Bernstein inequality as \u03c32 = 2 m (1 + n\u03bdr). Finally, apply the vector Bernstein inequality with R = 1 m(2n\u03bdr + 1) and \u03c32 = 2 m (1 + n\u03bdr). For t \u2264\u03c32 R > 1, Pr (||PTR\u2217 \u2126PTX \u2212PTX||F \u2265t) \u2264exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed\u2212 t2\u03ba 16 \u0010 \u03bd + 1 nr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (32) where \u03ba = m nr. This concludes the proof of Lemma 2. \u25a1 Having proved Lemma 2, the next step is to show that the gol\ufb01ng scheme (26) certi\ufb01es the conditions in (18) with very high probability. The precise statement is stated in Lemma 3 below. Lemma 3. Yl obtained from the gol\ufb01ng scheme (26) satis\ufb01es the conditions in (18) with failure probability at most p = l X i=1 (p2(i) + p3(i) + p4(i)) where p2(i) = exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u2212\u03bai 64 \u0010 \u03bd + 1 nr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8, p3(i) = n exp  \u22123ki 128\u03bd ! and p4(i) = n2 exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u22123\u03bai 64 \u0010 \u03bd + 1 nr \u0011 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8with ki = mi nr . Proof. Note that sgn M is symmetric. It is easy to verify that Qi is symmetric as Yi and PTYi are symmetric. In addition, using Lemma A.1, Qi is in T since sgn M \u2208T. To show that the \ufb01rst condition in (18) holds, we derive a recursive formula for Qi as follows. Qi = sgn M \u2212PT \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed i X j=1 R\u2217 jQ j\u22121 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8= sgn M \u2212PT \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed i\u22121 X j=1 R\u2217 jQ j\u22121 + R\u2217 i Qi\u22121 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = sgn M \u2212PT i\u22121 X j=1 R\u2217 jQ j\u22121 \u2212PTR\u2217 i Qi\u22121 = sgn M \u2212PTYi\u22121 \u2212PTR\u2217 i Qi\u22121 = Qi\u22121 \u2212PTR\u2217 i Qi\u22121 = (PT \u2212PTR\u2217 i PT)Qi\u22121 (33) Observe that the \ufb01rst condition in (18) is equivalent to a bound on \u2225Ql\u2225F. Using the bound \u2225(PT\u2212PTR\u2217 i PT)Qi\u22121\u2225F < t2,i\u2225Qi\u22121\u2225F with failure probability p2(i) obtained from Lemma 2 and setting t2,i = 1/2, p2(i) = exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u2212\u03bai 64 \u0010 \u03bd + 1 nr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8with \u03bai = mi nr , we have the following bound of \u2225Qi\u2225F using the above recursive formula. \u2225Qi\u2225F < \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed i Y k=1 t2,k \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u2225Q0\u2225F = 2\u2212i \u221ar (34) \f11 To satisfy the \ufb01rst condition in (18), set l = log2 \u0010 4n \u221a 8Lr \u0011 . Using the union bound on the failure probabilities p2(i), we have ||PTY \u2212Sgn M||F = ||Ql||F \u2264\u221ar2\u2212l = 1 4n \u221a 8L holds true with failure probability at most Pl i=1 p2(i). To complete the proof, it remains to show that Y satis\ufb01es the second condition in (18). The condition is equivalent to controlling the operator norm of PT\u22a5Yl. First, it is clear that \u2225PT\u22a5Yl\u2225= \u2225Pl j=1 PT\u22a5R\u2217 jQ j\u22121\u2225\u2264Pl j=1 \u2225PT\u22a5R\u2217 jQ j\u22121\u2225. This motivates bounding the operator norm of \u2225PT\u22a5R\u2217 jQ j\u22121\u2225which will be the focus of Lemma 4 below. Before proving the lemma, we start by addressing an assumption the lemma entails on the size of \u03b7(Qi) = max\u03b2 |\u27e8Qi , v\u03b2\u27e9|. Speci\ufb01cally, at the i-th stage of the gol\ufb01ng scheme, we need to show that the assumption \u03b7(Qi)2 \u2264\u03bd n2 c2 i with ||Qi||2 F \u2264c2 i holds with very high probability. To enforce this, let \u03b7(Qi) < t4,i with probability 1 \u2212p4(i). We set t4,i = 1 2\u03b7(Qi\u22121) to obtain \u03b7(Qi)2 < 2\u22122\u03b7(Qi\u22121)2 < 2\u22122i\u03b7(sgn M)2 \u2264\u03bdr n2 (2\u22122i) = \u03bd n2 (2\u22122ir) Above, the second inequality is obtained by applying the inequality \u03b7(Qi) < 1 2\u03b7(Qi\u22121) recursively and the last inequality follows from the coherence condition in (17). Using (34), at the i-th stage of the gol\ufb01ng scheme, the above inequality precisely enforces the condition that \u03b7(Qi)2 \u2264\u03bd n2 c2 i with ci = 2\u2212i \u221ar. Using Lemma A.8, noting that \u03b7(Qi) = \u03b7(Qi\u22121 \u2212PTR\u2217 i Qi\u22121), the failure probability p4(i) is given by p4(i) = n2 exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u22123\u03bai 64 \u0010 \u03bd + 1 nr \u0011 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \u2200i \u2208[1, l] This concludes the analysis on the assumption of the size of \u03b7(Qi) which will be used freely in Lemma 4. Using the bound ||PT\u22a5R\u2217 jQ j\u22121|| \u2264t3,j cj\u22121, where once again cj\u22121 satis\ufb01es ||Q j\u22121||F \u2264cj\u22121 = 2\u2212(j\u22121), with failure probability p3( j) obtained from Lemma 4 and setting t3,j = 1 4 \u221ar, p3(j) = n exp  \u22123\u03ba j 128\u03bd ! with \u03ba = mj nr , it follows that ||PT\u22a5Yl|| \u2264 l X k=1 ||PT\u22a5R\u2217 kQk\u22121|| \u2264 l X k=1 t3, k ck\u22121 = 1 4 \u221ar l X k=1 ck\u22121 = 1 4 \u221ar l X k=1 \u221ar2\u2212(k\u22121) < 1 2 where the second to the last inequality uses (34) to bound ||Qk\u22121||F. Using the union bound of failure probabilities, ||PT\u22a5Yl|| < 1 2 holds true with failure probability which is at most Pl j=1[p3( j) + p4( j)]. The second condition in (18) now holds with this failure probability. This completes the proof of Lemma 3. \u25a1 The proof of Lemma 4 uses the Bernstein inequality. A simpli\ufb01ed version of the inequality was derived in [38]. Our analysis uses this simpler version and is restated below for ease of reference. Theorem 5 (Bernstein inequality). Consider a \ufb01nite sequence {Xi} of independent, random, self-adjoint matrices with dimension n. Assuming that E[Xi] = 0 and \u03bbmax(Xi) \u2264R almost surely. with the norm of the total variance \u03c32 = \f \f \f \f \f \f \f \f \f \f P i E(X2 i ) \f \f \f \f \f \f \f \f \f \f, then, for all t \u22650, the following estimate holds: Pr \u0014\f \f \f \f \f \f X i Xi \f \f \f \f \f \f > t \u0015 \u2264 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 n exp   \u22123t2 8\u03c32 ! t \u2264\u03c32 R n exp   \u22123t 8R ! t \u2265\u03c32 R (35) Now we are ready to estimate ||PT\u22a5R\u2217 jQ j\u22121|| formally described in the following lemma. Lemma 4. Consider a \ufb01xed matrix G \u2208T. Assume that max \u03b2 \u27e8G , v\u03b2\u27e92 \u2264\u03bd n2 c2 with c satisfying ||G||2 F \u2264c2. Then, the following estimate holds for all t \u22641 \u221ar with \u03ba j = mj nr . Pr(||PT\u22a5R\u2217 jG|| > t c) \u2264n exp   \u22123t2\u03ba jr 8\u03bd ! Proof of Lemma 4. We expand PT\u22a5R\u2217 jG in the dual basis as PT\u22a5R\u2217 jG = X \u03b1\u2208\u2126j L mj \u27e8G , v\u03b1\u27e9PT\u22a5w\u03b1. Let X\u03b1 denote the summand. The proof makes use of Bernstein inequality (35) which mandates appropriate bound on \u2225X\u03b1\u2225and \u2225E[X2 \u03b1]\u2225. First consider the \f12 bound on the latter term. Noting that X2 \u03b1 = L2 m2 j \u27e8G , v\u03b1\u27e92(PT\u22a5w\u03b1)2, we have the following bound for E[X\u03b1]2 using Lemma A.7 and the fact that w2 \u03b1 is positive semide\ufb01nite. \f \f \f \f \f \fE[X2 \u03b1] \f \f \f \f \f \f \u2264n2 2m2 j max \u2225\u03d5\u22252=1 X \u03b1\u2208I \u27e8G , v\u03b1\u27e92\u27e8\u03d5 , w2 \u03b1\u03d5\u27e9\u2264n2 2m2 j max \u03b1\u2208I \u27e8v\u03b1 , G\u27e92 max \u2225\u03d5\u22252=1 \u27e8\u03d5 , \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X \u03b1\u2208I w2 \u03b1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8\u03d5\u27e9 (36) Due to the special structure of w\u03b1 in the EDG problem, it is straightforward to verify that X \u03b1\u2208I w2 \u03b1 = 2nI \u2212211T It now follows that \u03bbmax(P \u03b1\u2208I w2 \u03b1) = 2n which implies that \f \f \f \f \f \fE[X2 \u03b1] \f \f \f \f \f \f \u2264n3 m2 j max \u03b1\u2208I \u27e8v\u03b1 , G\u27e92 \u2264n3 m2 j \u03bd n2 c2 = n\u03bd m2 j c2 Next, consider a bound on \u2225X\u03b1\u2225which results \u2225X\u03b1\u2225\u2264n2 2mj |max \u03b1\u2208I \u27e8v\u03b1 , G\u27e9| \u2225PT\u22a5w\u03b1\u2225\u2264n2 2mj \u221a\u03bd n \u2225w\u03b1\u2225c (37) We consider two cases. If \u03bd \u22651 r , \u2225X\u03b1\u2225\u2264n\u03bd \u221ar mj c = R1 and if \u03bd < 1 r , \u2225X\u03b1\u2225< n \u221a\u03bd mj c = R2. Finally, we apply the Bernstein inequality with R1, R2 and \u03c32 = n\u03bd mj c2. It can be easily veri\ufb01ed that \u03c32 R1 = 1 \u221arc and \u03c32 R2 < 1 \u221arc. Therefore, for all t \u2264 1 \u221arc, the following estimate holds. Pr(\u2225PT\u22a5R\u2217 jG\u2225\u2265t) \u2264n exp   \u22123t2\u03ba jr 8\u03bdc2 ! (38) with \u03ba j = mj nr . This concludes the proof of Lemma 4. \u25a1 In what follows, it will be argued that M is a unique solution to (9) with \u201cvery high\u201d probability. It is su\ufb03ces to show all X \u2208S+ \u2212{M} are solutions with a very small probability by choosing \u2126\u201csu\ufb03ciently large\u201d. For X \u2208S+, write the deviation \u2206= X \u2212M. We de\ufb01ne S1 = \bX \u2208S+ : ||\u2206T||2 F \u2265 \u0012 n q 8 L m||\u2206T\u22a5||F \u00132 \t and S2 = \bX \u2208S+ : ||\u2206T||2 F < \u0012 n q 8 L m||\u2206T\u22a5||F \u00132 & R\u2126(\u2206) = 0} be two spaces de\ufb01ned based on deterministic comparisons of ||\u2206T||2 F and ||\u2206T\u22a5||2 F. Assuming that \u2126is sampled uniformly at random with replacement, we consider the following two cases. 1) Choose |\u2126| = m \u201csu\ufb03ciently large\u201d such that Pr \u0010n \u2126\u2282I \f \f \f |\u2126| = m, \u03bbmin (PT F\u2126PT) > 1/2 o\u0011 \u22651 \u2212p1 based on Lemma 1. Using Theorem 2(a), all X \u2208S1 are feasible solutions to (9) with probability p1. 2) Recall that using the gol\ufb01ng scheme Pr \u0010n \u2126\u2282I \f \f \f |\u2126| = m, Y \u2208range R\u2217 \u2126& ||PTY \u2212Sgn M||F \u2264 1 4L & ||PT\u22a5Y || \u22641 2 o\u0011 \u22651\u2212\u03f5 with \u03f5 = l X i=1 \u0002p2(i) + p3(i) + p4(i)\u0003 by choosing |\u2126| = m \u201csu\ufb03ciently large\u201d. Using Theorem 2(b), all X \u2208S2 are not solutions to (9) with probability \u03f5. Using the union bound, X , M is a solution to the EDG nuclear minimization problem with probability which is at most p = p1 + Pl i=1[p2(i) + p3(i) + p4(i)]. In what follows, the goal is to ensure that p is \u201csmall\u201d with m su\ufb03ciently large implying the uniqueness of M to (9) with very high probability. This is attained by a suitable choice of the parameters m, l and mi which are detailed below. The \ufb01rst failure probability is the failure that condition in Theorem 2(a) does not hold and is given by p1 = n exp \u0012\u2212\u03ba 8\u03bd \u0013 In the proof of Lemma 3, the failure probabilities p2(i), p3(i) and p4(i) are given by p2(i) = exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u2212\u03bai 64 \u0010 \u03bd + 1 nr \u0011 + 1 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ; p3(i) = n exp  \u22123\u03bai 128\u03bd ! ; p4(i) = n2 exp \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \u22123\u03bai 64 \u0010 \u03bd + 1 nr \u0011 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \u2200i \u2208[1, l] To prove Theorem 1, it remains to specify ki and show that the total probability of failure is very \u201csmall\u201d. In precise terms, this means that the probability of failure is bounded above by n\u2212\u03b2 for some \u03b2 > 1. ki is chosen in such a way that all the failure probabilities, p1, p2(i), p3(i) and p4(i), are at most 1 4ln\u2212\u03b2. With this, one appropriate choice for ki is ki = 96\u0000\u03bd + 1 nr \u0001\u0000\u03b2 log(n) + \f13 log(4l)\u0001. Using the union bound, it can be veri\ufb01ed that the total failure is bounded above by n\u2212\u03b2. The number of measurement m = lnrki must be at least log2 \u0010 4n \u221a 8Lr \u0011 nr   96\u0000\u03bd + 1 nr \u0001\u0002 \u03b2 log(n) + log \u0010 4 log2(4L \u221ar) \u0011 \u0003! (39) This \ufb01nishes the proof of Theorem 1 and concludes that the minimization program in (9) recovers the true inner product matrix with very high probability. C. Noisy EDG Completion In a practical settings, the available partial information is not exact but noisy. For simplicity, consider an additive Gaussian noise Z with mean \u00b5 and variance \u03c3. The modi\ufb01ed nuclear norm minimization for the EDG problem can now be written as minimize X\u2208S+ \u2225X\u2225\u2217 subject to ||R\u2126(X) \u2212R\u2126(M)||F \u2264\u03b4 (40) where \u03b4 characterizes the level of noise. Unlike the noisy matrix completion problem, the noise parameters for the EDG problem can not be chosen arbitrarily. The reason is that the perturbed distance matrix needs to be non-negative. In the context of numerically solving the noisy EDG problem, details of how to set these parameters will be discussed in the next section. Under the assumption that \u00b5 and \u03c3 are chosen appropriately, we posit that the theoretical guarantees for the exact case extend to noisy EDG completion. Following the analysis in [39] and using the dual basis framework, it can be surmised that Theorem 1 holds true with failure probability proportional to noise level \u03b4. A sketch of such a theorem is stated below. Theorem 6. Let M \u2208Rn\u00d7n matrix of rank r that obeys the coherence conditions (12), (13) and (14) with coherence \u03bd. Assume m measurements \u27e8M , w\u03b1\u27e9, sampled uniformly at random with replacement are corrupted with Gaussian noise of mean \u00b5, variance \u03c3 and noise level \u03b4. For any \u03b2 > 1, if m \u2265log2 \u0010 4 \u221a 8Lrn \u0011 nr   96[\u03bd + 1 nr][\u03b2 log(n) + log \u0010 4 log2(4L \u221ar) \u0011 ] ! then ||M \u2212\u00af M||F \u2264f \u0012 n, m n2 , \u03b4 \u0013 where \u00af M is a solution to (40) with probability at least 1 \u2212n\u2212\u03b2. The interpretation of the above theorem is that the EDG problem is stable to noise. Numerical experiments con\ufb01rm this consideration. However, a precise analysis mandates characterization of the level of noise and an exact speci\ufb01cation of f in the above theorem. This is left for future work. IV. Numerical Algorithms The theoretical analysis so far shows that the nuclear minimization approach yields a unique solution for the EDG problem. In this section, we aim at developing a practical algorithm to recover the inner product matrix X given partial pairwise distance information supported on some random set \u2126. Since the available partial information might be noisy in applications, we also extend the algorithm to this case. A. Exact partial information An algorithm similar to ours appears in [7] where the authors design an algorithm employing the alternative directional minimization (ADM) method to recover the Gram matrix. A crucial part of their algorithm uses the hard thresholding operator which computes eigendecompositions at every iteration and is computationally intensive. Comparatively, an advantage of our algorithm is that it does not require an eigendecomposition making it fast and suitable for tests on large data. Since the nuclear norm of a positive semide\ufb01nite matrix equates to its trace, we consider the following minimization problem. min \u00af X\u2208Rn\u00d7n Trace ( \u00af X) subject to R\u2126( \u00af X) = R\u2126(M) , \u00af X \u00b7 1 = 0 and \u00af X \u2ab00 Consider a matrix C \u2208Rn\u00d7(n\u22121) satisfying CTC = I and CT1 = 0, we rewrite the above minimization problem as follows by a change of variable \u00af X = CXCT. min X\u2208R(n\u22121)\u00d7(n\u22121) Trace (CXCT) subject to R\u2126(CXCT) = R\u2126(M) , X \u2ab00 Note that the sum to one constraint drops out since CT1 = 0 and CXCT \u2ab00 if X \u2ab00. To enforce that X is positive semide\ufb01nite, let X = \u00af P \u00af P T where \u00af P \u2208R(n\u22121)\u00d7q with q unknown a priori. Since \u00af P \u00af P T has at most rank q, it entails a good \f14 estimate for q which ideally should be a reasonable estimate of the rank. Due to the trace regularization, our algorithm only needs a rough guess of q. The above minimization problem now reduces to min \u00af P \u2208R(n\u22121)\u00d7q Trace (C \u00af P \u00af P TCT) subject to R\u2126(C \u00af P \u00af P TCT) = R\u2126(M) (41) With P = C \u00af P , consider the simpli\ufb01ed minimization problem. min P \u2208Rn\u00d7q Trace (P P T) subject to R\u2126(P P T) = R\u2126(M) (42) The above technique, the change of variables employing C, has been previously used [21], [27]. In [27], the authors remark that the reparamterization leads to numerically stable interior point algorithms. Note that C is simply an orthonormal basis for the space {x \u2208Rn | xt1 = 0}. For the EDG problem, the goal is to \ufb01nd the Gram matrix \u00af X = CXCT = CCTP P TCCT. CCT is simply the orthogonal projection onto the aforementioned space given by CCT = I \u22121 n11T. Given \u00af X, classical MDS can then be used to \ufb01nd the point coordinates. Therefore, our focus is on solving for P in (42). We employ the method of augmented Lagrangian to solve the minimization problem. The constraint R\u2126(P P T) = R\u2126(M) can be written compactly using the linear operator A de\ufb01ned as A : Rn\u00d7n \u2212\u2192R|\u2126| with A(X) = f \u2208R|\u2126|\u00d71, fi = \u27e8X , w\u03b1i\u27e9for \u03b1i \u2208\u2126. For latter use, the adjoint of A , A\u2217, can be derived as follows. For y \u2208R|\u2126|\u00d71, \u27e8AX , y\u27e9= P i\u27e8X , w\u03b1i\u27e9yi = \u27e8X , P i yiw\u03b1i\u27e9. It follows that A\u2217y = P i yiw\u03b1i. Thus, we write (42) as min P Trace (P P T) subject to A(P P T) = b (43) The augmented Lagrangian is given by L(P ; \u039b) = Trace (P P T) + r 2||A(P P T) \u2212b + \u039b||2 2 where \u039b \u2208R|\u2126|\u00d71 denotes the Lagrangian multiplier and r is the penalty term. The augmented Lagrangian step is simply P k = argmin L(P ; \u039bk\u22121) followed by updating of the multiplier \u039bk. To solve the \ufb01rst problem with respect to P , the BarzilaiBorwein steepest descent method [40] is employed with objective function Trace (P P T)+ r 2||A(P P T)\u2212b+\u039bk\u22121||2 2 and gradient 2P + 2rA\u2217\u0010 A(P P T) \u2212b + \u039bk\u22121\u0011 P . The iterative scheme is summarized in Algorithm 1. Algorithm 1 Augmented Lagrangian based scheme to solve (43) 1: Initialization: Set \u039b0 = 0 , q = 10, P 0 = rand(n, q), E0 = E0 Total = 0. Set maxiterations , bbiterations , r , Tol 2: for k = 1:maxiterations do 3: Barzilai-Borwein(BB) descent for P k = argmin L(P ; \u039bk\u22121). 4: \u039bk = \u039bk\u22121 + A(P k(P k)T) \u2212b 5: Ek = r 2||A(P k(P k)T) \u2212b||2 2 6: Ek Total = Trace (P k(P k)T) + r 2||A(P k(P k)T) \u2212b + \u039bk||2 2 7: if Ek < Tol & Ek Total < Tol then 8: break B. Partial information with Gaussian noise Assume that the available partial information is noisy. Formally, R\u2126(X) = R\u2126(M) + R\u2126(Z) where R\u2126(Z) is an additive Gaussian noise. Proceeding analogously to the case of exact partial information, the following minimization problem is obtained. min P Trace (P P T) + \u03bb 2||R\u2126(P P T) \u2212R\u2126(M)||2 F (44) Using the operator A introduced earlier, (44) can be rewritten as min P Trace (P P T) + \u03bb 2||A(P P T) \u2212b||2 F (45) where b = A(M). The augmented Lagrangian is given by L(P ; \u039b) = Trace (P P T) + \u03bb 2||A(P P T) \u2212b||2 2 where \u039b \u2208Rn\u00d71 denotes the Lagrangian multiplier and r is a penalty term. The augmented Lagrangian step is simply P k = argmin L(P ; \u039bk\u22121). As before, P is computed using the Barzilai-Borwein steepest descent method with objective function Trace (P P T)+ \u03bb 2||A(P P T)\u2212b||2 2 and gradient 2P +2\u03bbA\u2217\u0010 A(P P T) \u2212b \u0011 P . The iterative scheme is summarized in Algorithm 2. \f15 Algorithm 2 Augmented Lagrangian based scheme to solve (45) 1: Initialization: Set q = 10 , P 0 = rand(n,q) , E0 Total = 0. Set maxiterations , bbiterations , r , \u03bb , Tol 2: for k = 1:maxiterations do 3: Barzilai-Borwein(BB) descent for Pk. 4: Ek Total = Trace (P k(P k)T) + \u03bb 2||A(P k(P k)T) \u2212b||2 2 5: if Ek Total < Tol then 6: break V. Numerical Results In this section, we demonstrate the e\ufb03ciency and accuracy of the proposed algorithms. All the numerical experiments are ran in MATLAB on a laptop with an Intel Core I7 2.7 GHz processor and a 16GB RAM. A. Experiments on synthetic and real data We \ufb01rst test the Euclidean distance geometry problem on di\ufb00erent three-dimensional objects. These objects include a sphere, a cow, and a map of a subset of US cities. Given n points from these objects, the full n\u00d7n distance matrix D has n(n \u22121) 2 = L degrees of freedom. The objective is to recover the full point coordinates given \u03b3L entries of D, \u03b3 \u2208[0, 1], chosen uniformly at random. Algorithm 1 and Algorithm 2 output P from which X = P P T is constructed. Classical MDS is then used to \ufb01nd the global coordinates. In all of the numerical experiments, a rank estimation q = 10 is used. This choice shows that the algorithms recover the ground truth despite a rough estimate of the true rank. The stopping criterion is a tolerance on the relative total energy de\ufb01ned as (Ek Total \u2212Ek\u22121 Total)/Ek Total. For algorithm 1, an additional stopping criterion is a tolerance on Ek = r 2||A( \u00af Pk) \u2212b||2 2. In all of the numerical experiments, the tolerance is set to 10\u22125. The maximum iteration is set to 100. Accuracy of the reconstruction is measured using the relative error of the inner product matrix ||X \u2212M||F ||M||F where X is the numerically computed inner product matrix and M is the ground truth. For di\ufb00erent sampling rates, Figure 1 displays the reconstructed three-dimensional objects assuming that exact partial information is provided. For all experiments, the penalty term r is set to 1.0. Table II shows the relative error of the inner product matrix for all the di\ufb00erent cases. The algorithm provides very good reconstruction except the 1% sphere. For this speci\ufb01c case, the distance matrix for the sphere is comparatively small. This means that 1% provides very little information and more samples are needed for reasonable reconstruction. TABLE II Relative error of the inner product matrix for different three-dimensional objects under different sampling rates. The partial information is exact. Results are averages over 50 runs. \u03b3 1% 2% 3% 5% Sphere 3.27e \u221201 2.10e \u221203 7.98e \u221205 3.21e \u221205 Cow 1.53e \u221204 7.08e \u221205 5.03e \u221205 3.65e \u221205 US Cities 1.94e \u221204 1.13e \u221204 7.53e \u221205 5.82e \u221205 For the case of partial information with the additive Gaussian noise N(\u00b5, \u03c3), the noisy distance matrix can be written as \u00af D = D + N(\u00b5, \u03c3). The standard deviation \u03c3 is a critical parameter determining the extent to which the underlying information is noisy. In the EDG problem, the perturbed distance matrix \u00af D must have non-negative entries. Thus \u00b5 and \u03c3 need to be chosen carefully to satisfy this condition. In our numerical experiments, \u03c3 is set to be the minimum non-zero value of the partial distance matrix and \u00b5 is 3\u03c3. It can be easily veri\ufb01ed that, with very high probability, these choices ensure a non-negative noisy distance matrix. The choice of \u03c3 corresponds to the case where the error of the measurement is in the order of the minimum distance. While this choice might not re\ufb02ect practical measurements, the setting allows us to test the extent to which the algorithm handles a noisy data. The parameter \u03bb is a penalty term which needs to be chosen carefully. It can be surmised that \u03bb needs to increase with increasing sampling rate. For our numerical experiments, a simple heuristic is to set \u03bb = 100\u03b3. While a more sophisticated analysis might result an optimal choice of \u03bb, the simple choice is found to be su\ufb03cient and e\ufb00ective in the numerical tests. For di\ufb00erent sampling rates, Figure 2 displays the reconstructed three-dimensional objects assuming that a noisy partial information is provided. Table III shows the relative error of the inner product matrix for all the di\ufb00erent cases. The algorithm results good reconstruction except the 1% sphere. As discussed earlier, this speci\ufb01c case requires relatively more samples for reasonable reconstruction. We further apply the proposed algorithm to the molecular conformation problem [1]. In this problem, the aim is to determine the three-dimensional structure of a molecule given partial information on pairwise distances between the atoms. An instance of \f16 (a) 1% (b) 2% (c) 3% (d) 5% (e) 1% (f) 2% (g) 3% (h) 5% (i) 1% (j) 2% (k) 3% (l) 5% Fig. 1. Reconstruction results of di\ufb00erent three-dimensional objects from exact partial information with sampling rates 1% (1st column), 2% (2nd column), 3% (3rd column) and 5% (4th column), respectively. TABLE III Relative error of the inner product matrix for different three-dimensional objects under different sampling rates. The relative error is an average of 50 runs and the partial information is noisy. 1% 2% 3% 5% Sphere 5.21e \u221201 5.55e \u221202 1.86e \u221202 8.80e \u221203 Cow 2.84e \u221202 6.60e \u221203 3.80e \u221203 2.00e \u221203 US Cities 2.80e \u221202 9.00e \u221203 5.50e \u221203 3.10e \u221203 this problem is determining the structure of proteins using nuclear magnetic resonance (NMR) spectroscopy or X-ray di\ufb00raction experiments. The problem is challenging since the partial distance matrix obtained from experiments is sparse, non-uniform, noisy and prone to outliers [1], [41]. We test our method on a simple version of the problem to illustrate that the algorithm can also work on real data. Our numerical experiment considers two protein molecules identi\ufb01ed as 1AX8 and 1RGS obtained from the Protein Data Bank [42]. We use a pre-processed version of the data taken from [41]. Given the full distance matrix, the partial data is a random uniform sample with 3% sampling rate. The setup of the numerical experiments is the same as before. Figure 3 displays the reconstructed three-dimensional structure of 1AX8 and 1RGS under exact partial information and noisy partial information. The results demonstrate that the algorithm provides good reconstruction of the underlying three-dimensional structures. B. Computational comparisons of Algorithms To evaluate the e\ufb03ciency of the proposed algorithms, numerical tests are carried out using Algorithm1 and Algorithm 2. For the test, the input is a random uniform sample of the exact/noisy distance matrix of one of the three-dimensional objects discussed earlier(a sphere, a cow and a map of a subset of US cities). The sampling rate is set to 5% and the algorithms are run until a stopping criterion discussed before is met. In what follows, the reported results are averages of 50 runs and the \f17 (a) 1% (b) 2% (c) 3% (d) 5% (e) 1% (f) 2% (g) 3% (h) 5% (i) 1% (j) 2% (k) 3% (l) 5% Fig. 2. Reconstruction results of di\ufb00erent three-dimensional objects from noisy partial information with di\ufb00erent sampling rates 1% (1st column), 2% (2nd column), 3% (3rd column) and 5% (4th column), respectively. (a) 1AX8 (b) 1AX8 (c) 1RGS (d) 1RGS Fig. 3. Reconstructed three-dimensional structure of proteins identi\ufb01ed as 1AX8 and 1RGS in the Protein Data Bank. Exact partial information (3% sample rate) for the \ufb01rst and third, noisy partial information for the second and fourth. number of iterations is the ceiling of the average number of iterations. Tables IV summarizes the results of the computational experiments for the three-dimensional objects. It can be concluded that the algorithm is fast and converges in few iterations to the desired solution. We also conduct comparisons with the algorithm for the EDG problem proposed in [7] that also employs the augmented Lagrangian method. As remarked earlier, the main di\ufb00erence between the two algorithms is on the way the positive semide\ufb01nite condition is imposed on the inner product matrix X. In [7], the positive semide\ufb01nite condition is imposed on X directly while algorithm 1 uses the factorization P P T to enforce this condition. To compare the two algorithms, we use the same input of data which is a random uniform sample of the distance matrix of one of the three-dimensional objects. The sampling rate is set to 5%. For both algorithms, the stopping criterion is the relative error of the total energy and the tolerance is set 10\u22125. Table V summarizes the comparison of these two algorithms. The reported results are averages of 10 runs. We see that our algorithm converges to the desired solution faster and with signi\ufb01cantly less number of iterations. Finally, we consider the molecular conformation problem and compare our algorithm to the DISCO algorithm proposed in [41]. The DISCO algorithm is an SDP based divide and conquer algorithm. In [41], the authors demonstrate that the algorithm \f18 is e\ufb00ective on sparse and highly noisy molecular conformation problems. For the numerical tests, the input for both algorithms is a protein molecule. The sampling rate is set to 5% and it is assumed that the underlying partial information is exact. We downloaded the DISCO code, a MATLAB mex code version 1.4, from http://www.math.nus.edu.sg/\u223cmattohkc/disco.html which provides an input \ufb01le and executables. For our algorithm, the stopping criterion is maximum number of iterations set to 200. We emphasize that the rank estimate using our method for all experiments is 5. This means our method has 5/3 times variables to the method used in DISCO which compute molecular coordinates directly (i.e. rank number is 3). The DISCO algorithm has a radius parameter which implies that the input distance matrix consists pairwise distances less than or equal to the radius. For consistent comparison with the EDG problem and our algorithm, the radius is set large. In lack of a source code for DISCO, under the above setups, we run the algorithm as it is. Table VI summarizes the comparison of these two algorithms. The reported results are averages of 20 runs. We see that our algorithm attains a relative error of the same order as DISCO but is faster on all the tests. Some caveats about the comparison are the assumption on the radius and a very sparse partial exact information. In [41], the radius is set to 6 \u00c5 since NMR measurements have a limited range of validity estimated to be 6 \u00c5. With this choice, the problem departs from the EDG problem since there is localization. For this localization problem with a noisy input data and relatively sparse input (20% of distance within the radius), we note that DISCO results in excellent reconstruction of the protein molecules. The above comparison is meant to illustrate that, for a simplistic setup, our algorithm is very fast and has the potential to handle tests on large protein molecules TABLE IV Computational summary of Algorithm 1 from 50 runs. The data are the different three-dimensional objects. The sampling rate is 5%. 3D Object Number of Points Computational Time(Sec) Number of iterations Exact Sphere 1002 3.27 9 Cow 2601 68.08 26 US Cities 2920 101.92 34 Noisy Sphere 1002 10.56 19 Cow 2601 62.63 17 US Cities 2920 62.33 18 TABLE V Comparison of the proposed Algorithm 1 and the algorithm in [7] with 10 runs. The data are the different three-dimensional objects. It is assumed that the partial information is exact. The sampling rate is 5%. 3D Object Number of Points Computational Time(Sec) Number of iterations Alg. 1 Alg. [7] Alg. 1 Alg.[7] Sphere 1002 3.06 101.85 9 204 Cow 2601 51.40 1387.80 20 257 US Cities 2920 72.51 2417.30 23 250 TABLE VI Comparison of the proposed Algorithm 1 and the algorithm in [41] with 20 runs. The data are the different protein molecules. It is assumed that the partial information is exact. The sampling rate is 5%. Protein Molecule Number of Points Computational Time(Sec) Relative error of the inner product matrix Alg. 1 Alg. [41] Alg. 1 Alg. [41] 1PTQ 402 3.38 11.97 8.03e \u221207 1.63e \u221207 1AX8 1003 9.44 45.94 1.02e \u221208 8.01e \u221207 1RGS 2015 52.62 214.30 5.41e \u221209 1.62e \u221207 1KDH 2923 104.60 438.04 2.86e \u221209 8.16e \u221209 1BPM 3672 223.14 392.29 3.46e \u221209 3.27e \u221208 C. Phase transition of Algorithm 1 Last but not least, we numerically investigate the optimality of the proposed algorithm by plotting the phase transition. Given the number of points and the underlying rank, the theory provides the sampling rate which leads to successful recovery with very high probability (see Theorem 1). To investigate the optimality of Algorithm 1, the following numerical experiment was \f19 carried out. Consider sampling rates ranging from 1% to 100% and rank ranging from 1 to 40. For each pair, Algorithm 1 is run 50 times. Successful recovery refers to the case where the relative error of the inner product matrix is within tolerance. As remarked earlier, the tolerance is set to 10\u22125. Out of the 50 runs, the number of times the algorithm succeeds provides us with a probability of success. This procedure is repeated for all combination of sampling rate and rank. Figure 4 shows the optimality result of Algorithm 1. Namely, for a large portion in the sampling rate-rank domain, the proposed algorithm can provide successful reconstruction. Fig. 4. Success probability of Algorithm 1 given sampling rate and rank. For each sampling rate, Algorithm 1 is run for ranks ranging from 1 to 40. The average success probability, of 50 runs, is shown above. White indicates perfect recovery and black indicates failure of the algorithm. VI."
+                }
+            ],
+            "Shuchin Aeron": [
+                {
+                    "url": "http://arxiv.org/abs/0804.3439v5",
+                    "title": "Information theoretic bounds for Compressed Sensing",
+                    "abstract": "In this paper we derive information theoretic performance bounds to sensing\nand reconstruction of sparse phenomena from noisy projections. We consider two\nsettings: output noise models where the noise enters after the projection and\ninput noise models where the noise enters before the projection. We consider\ntwo types of distortion for reconstruction: support errors and mean-squared\nerrors. Our goal is to relate the number of measurements, $m$, and $\\snr$, to\nsignal sparsity, $k$, distortion level, $d$, and signal dimension, $n$. We\nconsider support errors in a worst-case setting. We employ different variations\nof Fano's inequality to derive necessary conditions on the number of\nmeasurements and $\\snr$ required for exact reconstruction. To derive sufficient\nconditions we develop new insights on max-likelihood analysis based on a novel\nsuperposition property. In particular this property implies that small support\nerrors are the dominant error events. Consequently, our ML analysis does not\nsuffer the conservatism of the union bound and leads to a tighter analysis of\nmax-likelihood. These results provide order-wise tight bounds. For output noise\nmodels we show that asymptotically an $\\snr$ of $\\Theta(\\log(n))$ together with\n$\\Theta(k \\log(n/k))$ measurements is necessary and sufficient for exact\nsupport recovery. Furthermore, if a small fraction of support errors can be\ntolerated, a constant $\\snr$ turns out to be sufficient in the linear sparsity\nregime. In contrast for input noise models we show that support recovery fails\nif the number of measurements scales as $o(n\\log(n)/SNR)$ implying poor\ncompression performance for such cases. We also consider Bayesian set-up and\ncharacterize tradeoffs between mean-squared distortion and the number of\nmeasurements using rate-distortion theory.",
+                    "authors": "Shuchin Aeron, Venkatesh Saligrama, Manqi Zhao",
+                    "published": "2008-04-22",
+                    "updated": "2010-04-28",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "Introduction In this paper we derive information theoretic bounds on the performance of the Compressed Sensing problem, [1],[2],[3], Y = GX + N \u221a SNR (1) where the measurements Y \u2208Rm\u00d71, the desired signal X \u2208Rn, and the compression (sensing) matrix G \u2208Rm\u00d7n. The noise N d \u223cN(0, Im), where Im is an identity matrix of size m, is assumed to be a Gaussian random vector with independent identically distributed (IID) components. We characterize results for both deterministic and stochastic compression matrices G = [gij]. For deterministic, G, the columns, gj, are normalized to have unit \u21132 norm. For the stochastic setting we consider matrices drawn from IID (independent identically distributed) Gaussian ensembles. Each component here is assumed to be distributed as gij d \u223cN(0, 1/m), i = 1, . . . , m, j = 1, 2, . . . , n. Note that under this normalized sensing matrix scenario, \u2217The authors are with the department of Electrical and Computer Engineering at Boston University, MA -02215. They can be reached at {shuchin, srv, mqzhao}@bu.edu. This research was supported by the Presidential Early Career Award (PECASE) N00014-02-100362, NSF CAREER award ECS 0449194. Venkatesh Saligrama was also supported by the MIT-Portugal Program while he was visiting MIT. 1 arXiv:0804.3439v5  [cs.IT]  28 Apr 2010 \fthe term SNR also denotes the inverse of the noise variance. We refer to the signal model of Equation (1) as the output noise model. In parallel we also consider the input noise model given by, Y = G \u0012 X + N \u221a SNR \u0013 (2) where N d \u223cN(0, In) is a Gaussian random vector with IID components. Evidently the noise here enters before the \u201ccompression\u201d operator, G, is applied. This model is motivated by fusion problems that arise in sensor networks [4], where noisy observations are compressed. The support of the signal X is denoted by Supp(X) = {j | Xj \u0338= 0}. We assume that the cardinality of the signal support, |Supp(X)| \u2264k < n. It is often convenient to state and interpret results in terms of the sparsity ratio \u03b1n = k n. The regime when \u03b1n n\u2192\u221e \u2212 \u2192\u03b1 > 0 is referred to as the linear regime and the regime when \u03b1n n\u2192\u221e \u2212 \u21920 is referred to as the sub-linear regime. We consider two types of distortions in signal reconstruction, namely, (a) Support distortion, i.e., d( \u02c6 X, X)) = 1 k Pn j=1 |I{Xi\u0338=0} \u2212I{ \u02c6 Xi\u0338=0}|, where, I{\u00b7} is the indicator function. (b) Mean squared distortion, d( \u02c6 X, X)) = 1 n\u2225\u02c6 X \u2212X\u22252 = 1 n Pn j=1 | \u02c6 Xj \u2212Xj|2. These two distortion metrics address two di\ufb00erent issues in signal recovery. The \ufb01rst metric penalizes solely the support detection part while the second metric penalizes both support detection and amplitude estimation. We will now highlight the main contributions and results of the paper. 1.1 Bounds for Exact and Approximate Support Recovery In this part we further restrict the signal X to be bounded away from zero by a constant \u03b2 > 0 on its support. This is a standard assumption employed by other researchers(see [5, 6, 7, 8]) since it is impossible to identify the support of a signal X from noisy measurements with arbitrarily small non-zero components. We derive necessary and su\ufb03cient conditions for exact and approximate support recovery for this case under both output and input noise models. A central contribution of our work in this setting is that we explicitly quantify the required SNR and the number of measurements, m for exact support recovery. For the output noise model we show that the minimum SNR required for support recovery is \u2126(log(n)) regardless of m. In addition for this minimum SNR level, the number of measurements, m, must scale as \u2126(k log(n/k)) to guarantee exact support recovery. Furthermore, we derive su\ufb03cient conditions and show that with SNR = \u2126(log(n)) and m = \u2126(k log(n/k)) the maximum-likelihood decoder can exactly identify the signal support with high probability. These results are depicted in Table 1. While not depicted in this table it is interesting to consider what happens as SNR increases. The bounds derived in this paper show that we cannot get signi\ufb01cant improvement in m unless SNR is scaled substantially (as a fractional power of n). We also derive conditions for support recovery for input noise models. Here our necessary conditions say that if m = o \u0010 n log(n) SNR \u0011 then recovery would be impossible. Evidently, either the SNR or the number of measurements must scale linearly with n to ensure support recovery. Thus either we must operate in an essentially noiseless regime or forsake all compression. We also extend our results to approximate support recovery. Here a tradeo\ufb00between the number of measurements, SNR and support errors for di\ufb00erent sparsity ratios. These tradeo\ufb00s are summarized in Column 2 of Table 2. An interesting aspect of these results is that a constant SNR is su\ufb03cient if we could tolerate a constant fraction of errors in the support recovery. To establish the necessary conditions we use Fano\u2019s inequality and its variations [9]. For deriving su\ufb03cient conditions we analyze the performance of the Maximum-Likelihood (ML) estimator based on a novel insight that every large support error event is essentially contained in the union of single support error events. This leads to a sharp bound that is order-wise optimal. Our necessary and su\ufb03cient conditions for di\ufb00erent sparsity levels require similar scaling of SNR, and the number of measurements(see Table 1). Related LiteratureThe necessary condition that SNR = \u2126(log(n)) irrespective of the number of measurements was \ufb01rst reported by the authors in [10]. This paper extends these results to include necessary conditions on the number of measurements. Similar conditions have also been reported by Fletcher et. al. [11] but due to the constraints imposed on the signal space\u2014the signal is limited to have small amplitude variations on its support elements\u2014their conditions are conservative (see discussion in [5]) for our setup. Necessary conditions have also been derived by Wainwright [6]. When the bounds of [6] (see Theorem 2 2 \fEXACT SUPPORT RECOVERY (Output Noise Model) Linear Sparsity Sub-Linear Sparsity 0 < \u03b1 = \u03b1n = k n \u03b1n = k n = n\u2212\u03b3, \u03b3 > 0 Necessity (this paper) SNR = \u2126 \u0010 log(n) \u03b22 \u0011 SNR = \u2126 \u0010 log(n) \u03b22 \u0011 m = \u2126(n) m = \u2126(k log n k ) Su\ufb03ciency (this paper) SNR = 32 log(2n) \u03b22 SNR = 32 log(2n) \u03b22 m = 6nH2(2\u03b1), \u03b1 \u22640.04 m = 6k log n 2k Table 1: Summary of fundamental bounds for exact support recovery in the worst-case setting described in Equation (1). \u03b2 is the minimum absolute value of the signal X on its support set; H2(\u00b7) denotes the binary entropy function; k is the maximum allowable cardinality (sparsity) of the support of X; \u03b1 is the maximum sparsity ratio and; 1/SNR is the noise variance in each noise dimension. The necessary conditions are stated for arbitrary (not necessarily IID) matrices, G, such that the marginal distribution of each component has zero mean and variance 1/m. The su\ufb03cient conditions are stated for the case when each element of G is drawn IID \u223cN(0, 1 m). in [6]) are applied to our setup, it implies that the number of measurements scale as \u2126(log(n)), which is conservative. In addition [6], primarily imposes conditions on the number of measurements but does not impose separate bounds on SNR. In contrast we show that unless SNR scales as \u2126(log(n)) support recovery is impossible regardless of m. Furthermore, for SNR = O(log(n)) we show that m must scale as \u2126(k log(n/k)). Su\ufb03cient conditions for support recovery for output noise models has been described in [6, 12, 11, 7, 8] as well. Nevertheless, these upper bounds are also signi\ufb01cantly weaker than that appearing here. Both Wainwright [6] and Akcakaya et. al. [12] use union bounding to derive error bounds for exact recovery. Union bounds are generally conservative and results in requiring signi\ufb01cantly high SNR, i.e. signi\ufb01cantly low admissible noise variance (see for instance, Theorem 1 in [6]). The su\ufb03cient conditions of Fletcher et. al. [11] is based on Greedy Basis Pursuit algorithm. However, their analysis, as described earlier, constrains the signals, X, to have small amplitude variations on its support elements and when applied to our output noise setup is conservative (see again discussion in [5]). While [13, 12] derive some results for approximate support recovery, the achievable region in terms of number of measurements and SNR as a function of achievable distortion is implicitly stated and is therefore not comparable to the results presented here. 1.2 Rate distortion bounds In the second part of the paper, we consider sparse Bayesian signal models for X to fully exploit the power of information theoretic methods. This naturally leads us to characterizing necessary and su\ufb03cient conditions in terms of the rate distortion function. We \ufb01rst consider arbitrary pointwise distortion metrics, i.e., 1 nd( \u02c6 X, X) = 1 n P j d( \u02c6 Xj, Xj), j = 1, 2, . . . , n, where Xj, \u02c6 Xj are the j-th components of X, \u02c6 X respectively. For deriving necessary conditions we develop a new modi\ufb01ed Fano\u2019s inequality that provides us with a worst case lower bound to the probability of error in reconstruction to within a distortion 1 nd( \u02c6 X, X) \u2264d0 in terms of the scalar rate distortion function RX(d0) and mutual information I(X, Y), between X and Y. This bound is of independent interest since it can be applied to non-sparsifying distributions as well. In particular we show that, P \u0012 1 nd( \u02c6 X, X) \u2265d0 \u0013 \u2265RX(d0) \u2212c0 \u22121 nI(X; Y) RX(d0) for some small constant c0 < RX(d0). For deriving su\ufb03cient conditions we compute upper bounds to the probability of error subject to a tolerable distortion based on the so called covering property of rate distortion theory. In particular we formalize a minimum distance decoder (distance measured in terms of given distortion metric) over the set of rate distortion quantization points. We then specialize our bounds to the mean squared distortion metric. The results are summarized in the second column of Table 2. Our necessary and su\ufb03cient conditions for the number of measurements and SNR match within a constant factor for the linear sparsity regime. 3 \fAPPROXIMATE SUPPORT RECOVERY Su\ufb03cient conditions (Linear Sparsity Regime) Support Error Distortion 1 k Pn j=1 |I{Xi\u0338=0} \u2212I{ \u02c6 Xi\u0338=0}| \u2264d0 Mean Squared Distortion 1 n\u2225\u02c6 X \u2212X\u22252 \u2264d0 SNR = \u2126 \u0010 H2(2\u03b1nd0) \u03b22 \u0011 and m = \u2126(nH2(2\u03b1n)) m = \u2126(nRX(d0/2)), for SNR = \u2126( RX(d0/2) d0 ). Table 2: The \ufb01rst column describes the achievable rate regions for approximate support recovery. Support error distortion d0 is the fraction of the true support in error. The second column describes the results for the Bayesian set-up in terms of the scalar rate distortion function for varying mean squared distortion. Here the distortion d0 is the desired mean-squared-distortion. Related LiteratureRate distortion analysis has been reported in [14, 15] for mean squared error and for a Gaussian source. In contrast our expressions apply to general distortion measures and to any source for which a rate distortion function is de\ufb01ned. These results appeared in our preliminary work [16]. In addition the results in [14] for the case when G is random are only proven for k = 1. In contrast in this paper we prove results for general k = \u03b1n. For a \ufb01xed problem size (n, k, m) the results in [15] are stated in terms of a critical SNR threshold. This makes the expressions implicit in the number of measurements required as a function of signal sparsity and therefore the scaling laws are unclear. The rest of the paper is organized as follows. In Section 2 we present our problem set-up. Here the notion of Sensing Capacity is introduced to study the asymptotic behavior of both the output noise and input noise models. Section 3 presents necessary and su\ufb03cient conditions for support recovery. In Section 4 we consider the Bayesian setup and derive bounds for signal recovery under arbitrary distortion measures. This requires us to generalize the traditional Fano\u2019s inequality to general (average) distortion measures and continuous signal spaces. We also provide extensions of Fano\u2019s inequality for discrete signal spaces with Hamming distortion in reconstruction. Section 4.2 presents a novel ML upper bound for signal recovery to within a given squared distortion level. Using these results, in Section 5 we evaluate bounds for SNR and number of measurements required to reconstruct X to di\ufb00erent levels of distortion level for output and input noise models. We then comment on the di\ufb00erences between worst-case and Bayesian setups. 2 Problem Set-up We consider output and input noise models described in Equations (1) and (2). The sparsity of X is modeled both deterministically and stochastically as is the compression matrix G. We use bold-face to denote vectors and matrices, while regular font is used to denote scalar components of the vector and matrices. The jth component of a vector X is denoted Xj, the jth column of a matrix G is denoted gj and its ij-th component is denoted as gij. The cardinality of a set S is denoted by |S|. Given a set S \u2282{1, 2 . . . , n}, XS denotes the signal, X, restricted to the set of components indexed by S. Similarly, we denote by GS the matrix formed from columns indexed by S. We use Pr(\u00b7) and P(\u00b7) interchangeably to denote the probability of an event. Non-Random Sparsity Signal Model: We say that \u039e{k} \u2282Rn is a family of k-sparse sequences if for every X \u2208\u039e{k}, the support of X is smaller than or equal to k. Formally, let Supp(X) = {j | Xj \u0338= 0} Then \u039e{k} is a family of k-sparse sequences if, \u039e{k} = {X : |Supp(X)| \u2264k} (3) We will refer to the ratio, \u03b1n = k/n as the sparsity ratio. We will often work with subsets of \u039e{k} \u03b2 \u2282\u039e{k}. These are sequences whose minimum absolute value is bounded away from zero by a constant \u03b2 \u22650: \u039e{k} \u03b2 = {X \u2208Rn : |Supp(X)| \u2264k, |Xj| \u2265\u03b2, \u2200j \u2208Supp(X)} (4) We will see when we derive necessary conditions that \u03b2 > 0 is necessary for support recovery. This is mainly because it is impossible to determine the support of a signal with arbitrarily small components under noisy measurements. This condition is also assumed by other authors [17, 7]. 4 \fWe denote by \u039ek \u2282\u039e{k} the set consisting of exactly k-sparse sequences. \u039ek = {X : |Supp(X)| = k} (5) This distinction is important and the reader should keep this in mind. The subset \u039ek \u03b2 \u2282\u039ek is analogously de\ufb01ned. Bayesian signal model: We say that a prior distribution on X is an asymptotically sparsifying distribution if for su\ufb03ciently large k, n the distribution concentrates all the measure on a subset of \u039e{k}. In this paper we will provide general results for arbitrary sparsifying priors and explicit bounds for the following Gaussian mixture model, namely, each component of the signal is distributed as: Xi d \u223cPX = \u03b1N(\u00b51, \u03c32 1) + (1 \u2212\u03b1)N(\u00b50, \u03c32 0) The corresponding n dimensional distribution of X is realized as a product measure on Rn. As an example note that for \u00b51 = 1, \u00b50 = 0 and \u03c31 = \u03c30 \u21920 this mixture model asymptotically models binary sparse sequences with sparsity highly concentrated around k = \u03b1n. The main reason for using a Bayesian signal model is that it lends itself to information theoretic tools and allows us to study the tradeo\ufb00s between the number of measurements at di\ufb00erent distortion levels for a given SNR. 2.1 Sensing Capacity The nature of the results developed in the paper are asymptotic, namely, we let the signal dimension n and the sparsity k each approach in\ufb01nity at di\ufb00erent rates and derive bounds on the number of measurements, m, and SNR, for exact/approximate reconstruction of X. In this context we also derive bounds for m and SNR for reconstruction of functions Z = f(X) of X. For instance, we consider functions f(\u00b7) that indicate the support or sign function of X. We denote \u02c6 X(Y) (resp. \u02c6 Z(Y)) as an estimate of X (resp Z) based on the observation Y. The distortion between the estimate Z and the estimate \u02c6 Z is denoted by 1 nd(\u02c6 Z, Z) = 1 n P j d( \u02c6 Zj, Zj) for some scalar distortion metric d(\u00b7, \u00b7). The sensing capacity involves determining the largest ratio nH2(\u03b1n) m = nH2( k n ) m , required for reconstruction to within a desired distortion. To build motivation on this ratio, consider again the maximum sparsity ratio \u03b1n = k n. The cardinality of the support set is 2log(Pk j=0 ( n j))) = O(2nH2(k/n)), where H2(\u00b7) denotes the binary entropy function [18]. The term nH2(k/n) is a measure of the entropy of the support set, i.e., the average number of bits required to uniquely encode the support set. The sensing capacity measures the number of source bits/measurement required for accurate decoding to a desired distortion level from compressed measurements. If sensing capacity is a constant, it implies that the number of measurements required is proportional to the source entropy. On the other hand if the sensing capacity approaches zero, it means that the number of measurements must increase signi\ufb01cantly faster than the source entropy. This also implies that the compression operator G o\ufb00ers poor compression. We next de\ufb01ne the \u03f5-sensing capacity for a signal X of dimension n and with maximum sparsity k. We use \u039e to denote a suitable subset of admissible signals, X. This could be any subset such as those described in Equations (4) and (5). C1 n,\u03f5(SNR, \u03b1n, d0) \u2206 = C1 n,\u03f5(SNR, k, d0) = sup m \u001anH(k/n) m : EG sup X\u2208\u039e P \u0012 1 nd(Z, \u02c6 Z) \u2264d0|G, X \u0013 \u22651 \u2212\u03f5 \u001b (6) where the probability is over N. Note that one may choose a less conservative notion by interchanging the order of maxX\u2208\u039e{k} and EG: C2 n,\u03f5(SNR, \u03b1n, d0) \u2206 = C2 n,\u03f5(SNR, k, d0) = sup m \u001anH(k/n) m : sup X\u2208\u039e EGP \u0012 1 nd(Z, \u02c6 Z) \u2264d0|G, X \u0013 \u22651 \u2212\u03f5 \u001b (7) For the Bayesian set-up the sensing capacity is de\ufb01ned as, C3 n,\u03f5(SNR, \u03b1n, d0) \u2206 = C3 n,\u03f5(SNR, k, d0) = sup m \u001anH(k/n) m : EG,XP \u0012 1 nd(Z, \u02c6 Z) \u2264d0|G, X \u0013 \u22651 \u2212\u03f5 \u001b (8) 5 \fwhere the probability is again over N. Since EG sup X\u2208\u039e{k} P \u0012 1 nd(Z, \u02c6 Z) \u2265d0|G, X \u0013 \u2265 sup X\u2208\u039e{k} EGP \u0012 1 nd(Z, \u02c6 Z) \u2265d0|G, X \u0013 \u2265EG,XP \u0012 1 nd(Z, \u02c6 Z) \u2265d0|G, X \u0013 This implies that C1 n,\u03f5(SNR, k, d0) \u2264C2 n,\u03f5(SNR, k, d0) \u2264C3 n,\u03f5(SNR, k, d0) (9) This chain of inequalities implies that an upper bound for the Bayesian sensing capacity is an upper bound for the other notions as well. A lower bound for the worst-case sensing capacity (Equation (6)) is a lower bound for the other notions as well. To derive the lower bound to sensing capacity we derive an upper bound on the probability of error using Maximum Likelihood (ML) analysis that uniformly holds for all X \u2208\u039e{k}. For this reason we primarily focus on the notion of Equation (6) and Equation (8). To avoid cumbersome notation we drop the superscript denoting the di\ufb00erent notions, namely, we employ Cn,\u03f5(\u00b7) \u2206 = Ci n,\u03f5(\u00b7), since it is usually clear from the context. We propose an asymptotic de\ufb01nition for sensing capacity by letting n \u2212 \u2192\u221eas follows. De\ufb01nition 2.1. Let {\u03b1n}, be any sequence of sparsity ratios where k is either \ufb01xed or approaching in\ufb01nity linearly or sub-linearly with n. Sensing capacity is the supremum over all the sensing rates such that as the signal dimension, n, the number of measurements, m, and the dimension of the (possibly) random sensing matrix, G \u2208Rm\u00d7n, approaches in\ufb01nity, there exists a sequence of estimators \u02c6 Z such that the probability that the distortion, 1 nd(Z, \u02c6 Z) is below d0 approaches one. Formally, C(SNR, {\u03b1n}, d0) = lim \u03b5\u21920 lim sup m,n Cn,\u03f5(SNR, \u03b1n, d0) where we explicitly denote the dependence of capacity on SNR, sparsity sequence \u03b1n, and distortion level d0. In the following we begin by considering the case of exact support recovery for the family of k-sparse sequences. 3 Support Recovery: Worst-Case Setting In this section we consider the problem of exact support recovery under the models of Equations (1) and (2) for the non-random parameter set, \u039e{k} \u03b2 given by Equation (4). Suppose, \u02c6 X is the estimate for X based on measurements Y. Recall that by exact support recovery we mean that, Pe = EG sup X\u2208\u039e{k} \u03b2 P{Supp( \u02c6 X) \u0338= Supp(X) | X, G} \u2212 \u21920 where the probability is over N. In this context one may also talk about sign pattern recovery, Pe = EG sup X\u2208\u039e{k} \u03b2 P{Sgn( \u02c6 X) \u0338= Sgn(X) | X, G} \u2212 \u21920 Here the Sgn function is described by Sgn(X) = \uf8f1 \uf8f2 \uf8f3 1, if X > 0 \u22121, if X < 0 0, if X = 0 It is easy to see that the results derived below also hold for sign pattern recovery with appropriate adaptation of the proof methodology and the subsequent results only di\ufb00er by constant factors and in particular does not change the resulting scaling laws. Therefore we will focus on the problem of support recovery. For this set-up following are our main results for the output and input noise models. Theorem 3.1 (Output Noise Model:Necessity). Consider the output noise model of Equation (1) with the signal set de\ufb01ned by Equation (4). Let G be any matrix such that the marginal distribution for each component has zero mean with variance 1 m. Then there exists no estimator that can recover the support if SNR = o(log(n)). Furthermore, for SNR = O(log(n)) support recovery is impossible if m = o(k log(n/k)). 6 \fFigure 1: Figure illustrating the intuition behind our ML analysis for support recovery using binary X as an example. In the Figure X0 is the true signal that is taken to be the origin. Support error events with support errors more than 1 are contained in union of events with support error of 1 before the sensing/compression operator G is applied. This property is essentially preserved under the transformation by G if the minimum singular value of matrix G is well behaved. The proof can found in Section 3.1.2. Note that we do not have to assume that the components of the sensing matrix are distributed IID. The proof of the theorem also shows that the number of measurements can not be decreased signi\ufb01cantly unless SNR scales as n\u03b3 for some \u03b3 > 0. It is interesting to point out that in contrast to the noiseless case where 2k + 1 are required for signal reconstruction, the presence of even small noise (namely, variance scaling as 1/ log(n)) signi\ufb01cantly alters this fundamental bound. The following result characterizes a partial converse of Theorem 3.1. Theorem 3.2 (Output Noise Model:Su\ufb03ciency). Suppose the sensing matrix, G, in Equation (1) is drawn from an IID Gaussian ensemble with each component gij d \u223cN(0, 1 m) and the signal set is given by Equation (4). If m = \u2126(nH2( k n)) = \u2126(k log(n/k)) and SNR = \u2126(log(n)) then the ML algorithm can exactly recover the support with high probability for all k n = \u03b1n \u2264.04. Alternatively, for any sensing matrix G with m \u22652k + 1 and SNR = \u2126(log(n)) the ML algorithm can recover the support with high probability, if the minimum singular value, \u03c3G,min = minX\u2208\u039e{2k} \u2225GX\u22252 2 \u2225X\u22252 2 is bounded way from zero. Remark 3.1. Note that Theorem 3.2 for the deterministic case requires \u03c3G,min to be bounded away from zero. One may question whether this requirement is fundamental. We argue that this is so here. Note that the optimal decoder must compare di\ufb00erent signals with supports smaller than k and pick the most likely. If \u03c3G,min is arbitrarily small, it implies that there are k columns which are badly conditioned. In the presence of noise a worst-case signal emanating from these k sparse columns will go virtually undetected relative to noise. The proof for the deterministic and stochastic sensing matrices appear in Sections 3.2 and 3.3. A geometric intuition of the proof for deriving the su\ufb03cient condition is shown in Figure (1) for binary X. The proof is based on the fact that for Gaussian noise N, before the compression operator G is applied, the support errors larger than one are contained in the union of events with support error equal to one. We show that this is largely true when the compression is applied as well. The proof for the random Gaussian matrix G is based on the deterministic case. It turns out that the sparsity ratio \u03b1n < 0.04 controls the singular values of the sub-matrix, namely, we can ensure \u03c3G,min > 0 with high probability for sparsity ratios below this number. 7 \fNote that we can also state these results in terms of sensing capacity. Formally, given any \u03b5 > 0, there is an n(\u03b5) such that for all n \u2265n(\u03b5) and any monotonic sequence \u03b1n < 0.04, there are positive constants, c1, c2, so 0 < c1 \u2264Cn,\u03b5(log(n), \u03b1n, 0) \u2264c2 In contrast to the optimistic results for output noise models, we have the following pessimistic result for the input noise model whose proof can be found in Section 3.1.1. Theorem 3.3 (Input Noise Model:Necessity). Consider the input noise model of Equation (2) with the signal set de\ufb01ned by Equation (4) and G drawn from an IID Gaussian ensemble with each component gij d \u223cN(0, 1/m). Let \u03b1n be any positive monotonic sequence of sparsity ratios. Then recovery fails if m = o \u0010 n max(log(n),log( 1 \u03b1n )) \u03b22SNR \u0011 . Alternatively, the sensing capacity is zero. This says that for the input noise model one cannot expect meaningful compression in a noisy regime. To ensure support recovery either the SNR has to scale linearly with n, which implies essentially a noiseless regime, or the number of measurements must scale linearly with n with any meaningful level of noise. This calls into question the sensor network motivated compression schemes such as those presented in [4] where the raw noisy measurements are randomly projected and transmitted to a fusion center. 3.0.1 Achievable Distortion Regions for Support Recovery In this section we will describe results for approximate support recovery, namely, we allow some distortion in support recovery. An important implication of our result is that in the constant sparsity regime it is su\ufb03cient for SNR to be a constant independent of n if we accommodate a constant fraction of support errors. We account for the support distortion as d( \u02c6 X, X)) = 1 k n X j=1 |I{Xi\u0338=0} \u2212I{ \u02c6 Xi\u0338=0}| where, I{\u00b7} is the indicator function. Theorem 3.4. Consider the observation model of Equation (1) with G drawn from a Gaussian ensemble. Let X \u2208\u039e{k} \u03b2 and let d0 be as described above. It follows that if SNR \u2265 64H2( 2kd0 n ) \u03b22 and m \u22656nH2 \u00002 k n \u0001 the probability of support error greater than distortion d0 goes to zero. Consequently, it follows that for support recovery with constant distortion, d0, in the linear sparsity regime, i.e, \u03b1n = k/n \u2265\u03b1 > 0, it is su\ufb03cient for the SNR to be a constant independent of the signal dimension n. Proof. The proof is based on the proof of Theorem 3.2 and we refer the reader to the appendix. Note that Theorem 3.4 only trades o\ufb00SNR with the distortion. However one would expect that with allowable distortion in support recovery it is possible to tradeo\ufb00number of measurements with distortion. In the following sections we will develop this tradeo\ufb00of number of measurements with the rate-distortion function by considering a Bayesian set-up. The main reason why this tradeo\ufb00is possible in a Bayesian set-up is due to the fact that before we analyzed a worst case set-up while in Bayesian case we analyze an average case scenario and it turns out that on an average the number of measurements can indeed be traded o\ufb00with distortion. 3.1 Proof of Theorems 3.3 and 3.1: Necessary Conditions We derive necessary conditions based on lower bounds to probability of error. As we pointed out in Equation (9) putting a suitable measure on the signal X can provide necessary conditions for the worst-case setup. This motivates employing di\ufb00erent versions of Fano\u2019s Lemma to establish the results. The standard version of the lemma appears in [18] and we repeat it here for the sake of completion: 8 \fLemma 3.1. Suppose X is a \ufb01nite discrete set and X \u2208X is distributed uniformly over this \ufb01nite set. Let the observation Y be distributed according to the conditional distribution P(Y|X), with X \u2208X. let \u02c6 X(Y) denote the estimate of X given Y. Then the probability of error in estimating X from Y is lower bounded by, P( \u02c6 X(Y) \u0338= X) \u22651 \u2212I(X; Y) + log 2 log(|X| \u22121) where I(X; Y) denotes the mutual information between X and Y. An alternate version of Fano\u2019s lemma stated in [19] provides a lower bound for N-ary hypothesis testing. Lemma 3.2. Let (Y, B) be a \u03c3\u2212\ufb01eld and let P1, . . . , PN be probability measures on B thought of as induced by N hypotheses {1, 2, . . . , N}. Denote by \u03b8(y) the estimator of the measures de\ufb01ned on Y. Then max 1\u2264i\u2264n Pi(\u03b8(y) \u0338= Pi) \u22651 N N X i=1 Pi(\u03b8(y) \u0338= Pi) \u22651 \u2212 1 N 2 P i,j D(Pi\u2225Pj) + log 2 log(N \u22121) where Pi means the distribution conditioned on the hypothesis i and D(Pi\u2225Pj) is the Kullback-Liebler (KL) distance between the distributions Pi and Pj. Note that the use of these Lemmas requires a \ufb01nite number of hypothesis or discrete alphabets. Therefore, in order to use these Lemmas for general k-sparse sequences X \u2208\u039e{k} \u03b2 we \ufb01rst show that the worst case probability of error in support recovery is lower bounded by the probability of error in support recovery for X belonging to k-sparse sequences in {0, \u03b2}n. To this end we have the following Lemma. Lemma 3.3. Let \u039e{k} \u03b2 be the family of k sparse non-random sequences as de\ufb01ned in Equation (4). Denote the conditional distribution of Y given X as P(Y | X). Let \u039e{k} {0,\u03b2} = {X \u2208\u039e{k} \u03b2 | Xj = \u03b2, j \u2208Supp(X)} be a subset of \u039e{k} \u03b2 consisting of binary valued sequences. Let \u02c6 X denote an estimator for X based on observation Y. Then, Pe|G = min \u02c6 X\u2208\u039e{k} \u03b2 max X\u2208\u039e{k} \u03b2 P{Supp( \u02c6 X) \u0338= Supp(X)|G, X} \u2265 min \u02c6 X\u2208\u039e{k} \u03b2 max X\u2208\u039e{k} {0,\u03b2} P( \u02c6 X \u0338= X, \u02c6 X \u2208\u039e{k} {0,\u03b2}|G, X) \u2265 min \u02c6 X\u2208\u039e{k} {0,\u03b2} max X\u2208\u039e{k} {0,\u03b2} P( \u02c6 X \u0338= X, \u02c6 X \u2208\u039e{k} {0,\u03b2}|G, X) (10) Proof. See Appendix. The main idea behind the proofs of the results that follow below is to \ufb01rst lower bound the error probability by using Lemma 3.3 and restrict attention to binary sequences. Next we further restrict the signal class to a smaller subset of \u039e{k} {0,\u03b2} of cardinality n. Then, \ufb01nally using Lemma 3.2 we derive the lower bounds for the set of binary sequences. The lower bound thus obtained yields the necessary conditions. 3.1.1 Input Noise Model(Proof of Theorem 3.3) From Lemma 3.3 it is su\ufb03cient to focus on the case when X belongs to the set of k-sparse sequences in {0, \u03b2}n and any subset of these sequences. We will establish the \ufb01rst part of the Theorem as follows:Let \u039e{\u03b7} {0,\u03b2} be the subset of \u03b7 < k sparse binary valued sequences. Let X0 \u2208\u039e{\u03b7} {0,\u03b2}, be an arbitrary element with support Supp(X0) = \u03b7 \u22121. Next choose n elements Xj, j = 1, 2, . . . , n with support equal to \u03b7 and at a unit Hamming distance from X0. Denote by the probability kernel Pj, 0 \u2264j \u2264n the induced observed distributions. Under the AWGN noise model, for a given G, and a \ufb01xed set of elements, Xj, the probability kernels are Gaussian distributed, i.e., Hj : Y d \u223cPj \u2261N \u0012 GXj, \u03a3 SNR \u0013 , j = 0, 1, . . . , n 9 \fwhere \u03a3 = GGT . Furthermore we have n + 1 hypotheses. Consider now the support recovery problem. It is clear that the error probability can be mapped into a corresponding hypothesis testing problem. For this we consider \u03b8(Y) as estimate of one of the n + 1 distributions above and we have the following set of inequalities. Pe|G = max X\u2208\u0393\u03b7 PX( \u02c6 X \u0338= X | G) = max j Pj(\u03b8(Y) \u0338= Pj | G) \u2265 1 n + 1 n X j=0 Pj(\u03b8(Y) \u0338= Pj | G) where we write Pe|G to point out that the probability of error is conditioned on G. Applying Lemma 3.2 it follows that the probability of error in exact support recovery is lower bounded by, Pe|G \u2265 log(n) \u2212 1 (n+1)2 P i,j,i\u0338=j D(Pi\u2225Pj) \u2212log 2 log(n) We observe that under AWGN noise N that, D(Pi\u2225Pj) = SNR(Xi \u2212Xj)T GT \u03a3\u22121G(Xi \u2212Xj) = SNR(Xi \u2212Xj)T V \u0014 I 0 0 0 \u0015 V\u2217(Xi \u2212Xj) (11) where \u03a3 = GGT , G = U[\u039b, 0]V\u2217is the SVD of G with V = [v1, v2, v3, . . . , vn] = [vrs]. The last equality in Equation (11) follows from straightforward algebraic manipulations. Now by noting that (Xi \u2212Xj) is at most a 2-sparse vector with its non-zero entries equal to \u03b2 at some locations q and p, we can further reduce the last expression to D(Pi\u2225Pj) = SNR\u03b22 Pm l=1(vpl \u2212vql)2. Now using the standard rotational invariance properties of IID Gaussian matrices [20], that its singular vectors are uniformly distributed over a sphere, it follows by taking expectations and using symmetry that, Pe = EGPe|G \u2265 log(n) \u2212 n n+1 2\u03b22SNR m n \u2212log 2 log(n) (12) Now, the error probability is bounded away from zero by \u03f5 if the number of measurements scales as follows: m = o \u0012(n + 1) log(n) \u03b22SNR \u0013 To establish the second upper bound we consider the family, \u039ek {0,\u03b2} of exact k-sparse binary valued sequences which form a subset of \u039e{k} {0,\u03b2}. Following similar logic as in the proof of the \ufb01rst part, for the set of exactly k-sparse sequences, we form the corresponding \u0000n k \u0001 hypotheses. Then, Pe = EGPe|G \u2265 log( \u0000n k \u0001 \u22121) \u2212 1 ( n k) 2 P i,j,i\u0338=j D(Pi\u2225Pj) \u2212log 2 log( \u0000n k \u0001 \u22121) (13) We compute the average pairwise KL distance, 1 \u0000n k \u00012 X i,j,i\u0338=j D(Pi\u2225Pj) = 1 \u0000n k \u0001 k X j=1 SNR(X \u2212X\u2032)T GT \u03a3\u22121G(X \u2212X\u2032).\u266f(sequences X\u2032 at hamming distance 2j from X) The equality above follows from symmetry. Again using the standard rotational invariance properties of IID Gaussian matrices [20], the above equation implies that , 1 \u0000n k \u00012 X i,j,i\u0338=j D(Pi\u2225Pj) = m n 1 \u0000n k \u0001 k X j=1 SNR\u03b22 \u0012n \u2212k j \u0013\u0012k j \u0013 (2j) = m n 2\u03b22SNR\u03b1nn(1 \u2212\u03b1n) where the last equality follows from standard combinatorial identity. The proof then follows by noting that for large enough value of n, log( \u0000n k \u0001 \u22121) \u2265\u03b1nn log 1 \u03b1n . 10 \f3.1.2 Output Noise Model (Proof of Theorem 3.1) We will now establish Theorem 3.1 namely, that if SNR = o(log(n)) support recovery is impossible. Furthermore, if SNR = O(log(n)) support recovery will be impossible if the number of measurements scales as o(k log(n/k)). The \ufb01rst part follows from the following Proposition. Proposition 3.1 (Output noise model SNR Bound). For the observation model of Equation (1) with the signal set of Equation (4) the SNR must scale with log(n) 2\u03b22 for perfect support recovery irrespective of which sensing matrix is used. Proof. The proof follows along the same lines as the proof of Theorem 3.3 with \u03a3 = I up to Equation (11). In the Kullback Leibler distance calculation we are now left with the term GT G. Since G is normalized its expected value is identity. Therefore, we no longer get the factor n/m in Equation 12. Consequently, following the rest of the steps we have that, 2\u03b22SNR \u2265log(n) for exact support recovery. Next we establish what happens for SNR = O(log(n)) to prove the second part of Theorem 3.1. First, note that if the sparsity, k, grows linearly with the signal dimension, n, there is nothing to prove, since it is well-known [1] that the number of measurements must scale at least as 2k + 1 = \u2126(n) even when there is no noise to guarantee support recovery. For this reason we focus on the sub-linear case namely, k = n\u2212\u03b3, \u03b3 < 1. We consider the subset \u039ek {0,\u03b2} consisting of strictly k-sparse sequences taking values in {0, \u03b2}n. From Lemma 3.3 we see that it is su\ufb03cient to focus on this set. Applying Lemma 3.1 with a uniform prior on the support set we get max X\u2208X P( \u02c6 X \u0338= X|X, G) \u2265P( \u02c6 X \u0338= X|G) \u22651 \u2212I(X; Y|G) + log 2 log(|X| \u22121) (14) where X = \u039ek {0,\u03b2} \u2282{0, \u03b2}n is the discrete alphabet in which values of X are realized. The \ufb01rst inequality follows because the worst-case probability of error is larger than the Bayesian error. Note that strictly speaking since we are interested in the support errors, the probability of error events and the mutual information term must contain the support of X as the variable but since we are restricting ourselves to binary valued sequences X \u2208\u039ek {0,\u03b2}, knowing the support implies that we know X. Now log |X| = log \u0000n k \u0001 since there are \u0000n k \u0001 such hypothesis consisting of all the possible support locations with cardinality k. We will now upper bound the mutual information term. It follows that, I(X; Y|G) = h(Y|G) \u2212h(Y|X, G) \u2264h(Y) \u2212h(N) (a) \u2264 m X i=1 h(Yi) \u2212m 2 log \u0012 2\u03c0e 1 SNR \u0013 (b) \u2264m 2 log \u0012 2\u03c0e \u0012k\u03b22 m + 1 SNR \u0013\u0013 \u2212m 2 log(2\u03c0e 1 SNR) = m 2 log(1 + k\u03b22SNR m ) where h(\u00b7) is the di\ufb00erential entropy; (a) follows from the fact that the noise is Gaussian and the chain rule together with the fact that conditioning reduces entropy; (b) follows from the fact that Gaussian distributions maximizes di\ufb00erential entropy. Now from Equation (14) it follows that the number of measurements must satisfy, m \u2265 log \u0000\u0000n k \u0001 \u22121 \u0001 log(1 + k\u03b22SNR m ) + log 2 m (15) Next unless SNR = \u2126(log(n)) we know from Proposition 3.1 that support recovery is impossible. Hence we set SNR = log(n), which is the minimum possible. We next establish the theorem by contradiction. To this end let the number of measurements scale as m = \u03c1n log( \u0000n k \u0001 ), with \u03c1n \u21920, then, by rearranging the terms in Equation (15) we get log   1 + k\u03b22SNR \u03c1n log( \u0000n k \u0001 ) ! + log 2 \u03c1n log( \u0000n k \u0001 ) \u2265log \u0000\u0000n k \u0001 \u22121 \u0001 \u03c1n log( \u0000n k \u0001 ) (16) 11 \fNext note that the expression on the left can be simpli\ufb01ed by noting that k\u03b22SNR \u03c1n log( \u0000n k \u0001 ) = \u0398 \uf8eb \uf8ed 1 \u03c1n(1 \u2212log(k) log(n)) \uf8f6 \uf8f8 while the expression on the right has the scaling \u0398(\u03c1\u22121 n ). Consequently, if maximum admissible sparsity, k, grows sub-linearly with n then log(1+ k\u03b22SNR \u03c1n log(( n k))) = \u0398(log(1+\u03c1\u22121 n )) and Equation (16) can never be satis\ufb01ed since \u03c1n \u21920. This shows that for sub-linear cases recovery is impossible if m = o(log( \u0000n k \u0001 )) = o(nH2(\u03b1n)). Remark 3.2. Note that unless SNR scales as n\u03b4 for some \u03b4 > 0 we will still need the measurements to scale as \u2126(k log(n/k)) to guarantee support recovery. 3.2 Proof of Theorem 3.2: Deterministic Case In this section we derive su\ufb03cient conditions for support recovery for the output noise model for any given arbitrary deterministic matrix G and for general noise covariance \u03a3. For the output noise model of Equation (1), we assume that each column of the deterministic G is normalized. Subsequently we specialize these results to the case when G is chosen from the Gaussian ensemble and with \u03a3 = I. To simplify the exposition we introduce several new variables. We associate each admissible signal, X \u2208\u039e{k} by its support, S. We denote by XS the signal, X, restricted to the set of components indexed by S. Similarly, we denote by GS the matrix formed from columns indexed by S. Since the maximum sparsity level is k the number of di\ufb00erent support sets is equal to Pk j=1 \u0000n j \u0001 \u22121. We index the di\ufb00erent support sets as S\u03c9 with \u03c9 \u2208I = n 0, 1, 2, . . . , Pk j=1 \u0000n j \u0001 \u22121 o . Also we denote by Xmin S\u03c9 the minimum absolute value of the components of the signal X on the support set S\u03c9, i.e., Xmin S\u03c9 = min{|Xj| : j \u2208S\u03c9}. Without loss of generality we assume that the true signal is X0, the support set of the true signal to be S0 corresponding to \u03c9 = 0. We denote by X0,j the jth component of the true signal. For any \u03c9 \u0338= 0, we denote the overlapping support by, S0,\u03c9, false detection by, S0c,\u03c9 and missed detection by, S0,\u03c9c, namely, Overlap\u2212S0,\u03c9 = S0 \u2229S\u03c9 False Alarms\u2212S0c,\u03c9 = Sc 0 \u2229S\u03c9 Misses\u2212S0,\u03c9c = S0 \u2229Sc \u03c9 For a given noise covariance \u03a3 the ML estimator is given by, \u02c6 X = min X\u2208\u039e{k} \u03b2 (Y \u2212GX)T \u03a3\u22121(Y \u2212GX) The above ML estimator is hard to analyze. In order to simplify the analysis we will consider a sub-optimal ML estimator. To this end consider the set, \u039e{k} \u03b2/2. Clearly, \u039e{k} \u03b2 \u2282\u039e{k} \u03b2/2. We propose the following suboptimal ML estimator, \u02c6 X = arg min X\u2208\u039e{k} \u03b2/2 \u2225Y \u2212GX\u22252 (17) and report Supp( \u02c6 X) as the \ufb01nal solution. Note that this estimator is sub-optimal since it is prone to more errors. To see this note that we consider a larger signal set and we ignore possible noise correlation \u03a3 in our estimator. Consequently, the error probability in detecting the correct support can only be larger than the optimal ML estimator. The performance of the relaxed estimator provides an upper bound for the performance of ML estimator. Hence, we can write, PML e|G \u2264Pe|G = P   N : min \u03c9\u0338=0, Xmin S\u03c9 \u2265\u03b2/2 \u2225Y \u2212GS\u03c9XS\u03c9\u22252 \u2264 min Xmin S0 \u2265\u03b2/2 \u2225Y \u2212GS0XS0\u22252 ! (18) Note that in the above expression XS0 is not the true signal, X0, but any other signal whose support is identical to that of the true signal. We then have the following result. 12 \fLemma 3.4. PML e|G \u2264Pe|G \u2264P(E1) + P(E2) where E1 = {N : min \u03c9\u0338=0, Xmin S\u03c9 \u2265\u03b2/2 \u2225Y \u2212GS\u03c9XS\u03c9\u22252 \u2264min \u02dc X \u2225Y \u2212GS0 \u02dc X\u22252} E2 = \b N : \u2225(GT S0GS0)\u22121GT S0N\u2225\u221e\u2265\u03b2/2 \t Proof. First note the following qualitative points. In the event E1 we have replaced the constrained minimization on the R.H.S. of the inequality in the error event with an unconstrained one. This will simplify the subsequent analysis as closed form expressions can be obtained. The event E2 captures the probability that the unconstrained minimization in E1 is very far from the constrained one. Here we use the fact that the minimum component on the support of the true signal X0 is greater than \u03b2. We also relax our ML estimator so that we \ufb01nd a best \ufb01t with any signal sharing the same support set, S0, as X0 but with Xmin S0 \u2265\u03b2/2. Now, denote A \u2206 = ( N : min \u03c9\u0338=0, Xmin S\u03c9 \u2265\u03b2/2 \u2225Y \u2212GS\u03c9XS\u03c9\u22252 \u2264 min Xmin S0 \u2265\u03b2/2 \u2225Y \u2212GS0XS0\u22252 ) B \u2206 = ( N : min Xmin S0 \u2265\u03b2/2 \u2225Y \u2212GS0XS0\u22252 = min \u02dc X \u2225Y \u2212GS0 \u02dc X\u22252 ) Then we have Pe = P(A) = P(A \u2229B) + P(A \u2229\u00af B) \u2264P(A \u2229B) + P( \u00af B) The Lemma then follows by noting that, A \u2229B = ( N : min Xmin S\u03c9 \u2265\u03b2/2,\u03c9\u0338=0 \u2225Y \u2212GS\u03c9XS\u03c9\u22252 \u2264 min Xmin S0 \u2265\u03b2/2 \u2225Y \u2212GS0XS0\u22252 ) \u2229B = ( N : min Xmin S\u03c9 \u2265\u03b2/2,\u03c9\u0338=0 \u2225Y \u2212GS\u03c9XS\u03c9\u22252 \u2264min \u02dc X \u2225Y \u2212GS0 \u02dc X\u22252 ) = E1 and, \u00af B = ( N : min Xmin S0 \u2265\u03b2/2 \u2225Y \u2212GS0XS0\u22252 \u0338= min \u02dc X \u2225Y \u2212GS0 \u02dc X\u22252 ) \u2282E2 From the above Lemma, it is su\ufb03cient to focus on events E1 and E2 separately. The following lemma provides a result that considerably simpli\ufb01es the error event E1. It turns out that the event E1 is a subset of the union of atomic events, namely, Lemma 3.5. For m \u22652k + 1, E1 \u2286\u02dc E1 = [ X\u2208{\u03b2/2,\u2212\u03b2/2} n [ j=1 \b N : 2NT gjX \u2265\u03c3G,min|X|2\t where, gj is the j \u2212th column of the matrix G and \u03c3G,min = min |S|\u22642k \u03c3min(GT SGS) (19) where \u03c3min(GT SGS) denotes the minimum singular values of GT SGS. 13 \fProof. See Appendix. We now have the following Lemma. Lemma 3.6. Consider the output noise model for a deterministic matrix G with m \u22652k + 1 and N distributed as N(0, \u03a3). The probability of the error event E1 is upper bounded by, P(E1) \u2264exp \u001a \u2212\u03c32 G,min \u03bbmin(\u03a3\u22121)\u03b22SNR 32 \u001b exp{log 2n} (20) where \u03bbmin(\u03a3\u22121) is the minimum eigenvalue value of the matrix \u03a3\u22121. Proof. See Appendix. We now have the following Lemma for the error event E2. Again note that the result applies to any matrix G (not necessarily Gaussian). Lemma 3.7. For the setup of Lemma 3.6, we have, P(E2) \u2264exp \u001a \u2212\u03c32 G,min \u03bbmin(\u03a3\u22121)\u03b22SNR 8 + log 2n \u001b Proof. See Appendix. By combining Lemmas 3.6 and 3.7 we can prove the deterministic case of Theorem 3.2. We state it as a proposition since we will refer to it later. Proposition 3.2. Consider the setup of Lemma 3.6. Then for exact support recovery it is su\ufb03cient that m \u22652k + 1 and SNR = \u2126   1 \u03c32 G,min 64 log 2n \u03b22\u03bbmin(\u03a3\u22121) ! . Proof. From Lemmas 3.6 and 3.7 it follows that for m \u22652k + 1, Pe|G \u2264exp \u001a \u2212\u03c32 G,min \u03bbmin(\u03a3\u22121)\u03b22SNR 32 + log 2n \u001b + exp \u001a \u2212\u03c32 G,min \u03bbmin(\u03a3\u22121)\u03b22SNR 8 + log 2n \u001b \u22642 exp \u001a \u2212\u03c32 G,min \u03bbmin(\u03a3\u22121)\u03b22SNR 32 + log 2n \u001b Therefore for SNR = 2 \u00b7 \u03c3\u22122 G,min 32 log 2n \u03b22\u03bbmin(\u03a3\u22121) the probability of error Pe|G \u22642e\u2212log 2n. Thus with n \u2192\u221e, Pe|G goes to zero as 1 n. This implies that SNR scaling of \u2126 \u0012 \u03c3\u22122 G,min 64 log 2n \u03b22\u03bbmin(\u03a3\u22121) \u0013 is su\ufb03cient. 3.3 Proof of Theorem 3.2: Gaussian Case We will now focus on sensing matrices, G, drawn from an IID Gaussian ensemble. As in the deterministic case we need to bound the probabilities of events, E1 and E2. We will \ufb01rst focus our attention on event E1. We point out that the proof for the deterministic case cannot be directly applied. First, note that \u03c3G,min of Equation (19) is now a random variable. Therefore, we need to average over this random variable in computing an upperbound to the probability of events E1, E2. A second problem is that in the deterministic case we assumed that the \u21132 norm of each column, gj is deterministically normalized to unity. In the Gaussian case only the expected power is normalized to unity. Note also that for the output noise model considered in this paper \u03a3 = I. Therefore \u03bbmin(\u03a3\u22121) = 1. Following along the lines of the proof of Lemma 3.6 we see that, P(E1 | G) \u2264exp ( \u2212 \u03c32 G,min\u03b22SNR 32 maxj \u2225gj\u22252 ) exp{log 2n} 14 \fWe need to now characterize a lower bound for \u03c32 G,min maxj \u2225gj\u22252 . To this end we observe that, Pr   \u03c32 G,min maxj \u2225gj\u22252 \u2265(1 \u2212\u03b7)2 1 + \u03f5 ! \u2265Pr(\u03c3G,min \u2265(1 \u2212\u03b7)2, max j \u2225gj\u22252 \u22641 + \u03f5) (21) \u22651 \u2212(Pr(\u03c3G,min \u2264(1 \u2212\u03b7)2) + Pr(max j \u2225gj\u22252 \u22651 + \u03f5)) This implies that we should characterize \u03c3G,min and maxj \u2225gj\u22252 separately. We appeal to the following lemma in [2], to characterize \u03c3G,min. Lemma 3.8. Suppose the sparsity is \u03b1n = k/n and we consider a function f(q) := p n/m \u0010\u221aq + p 2H2(q) \u0011 , where H2(q) := \u2212q log q \u2212(1 \u2212q) log(1 \u2212q). Let G be an m \u00d7 n matrix drawn from a Gaussian ensemble with gij d \u223cN(0, 1/m). Then it follows that \u03c3G,min described in Equation 19 has the following concentration property, P (\u03c3G,min \u22641 \u2212\u03b7) \u22642 exp \u0012 \u2212n\u03f5H2(\u03b1n) 2 \u0013 \u2206 = \u03b41(n, \u03b1n, \u03f5) (22) where, \u03b7 = 2(1 + \u03b5)f(2\u03b1) + (1 + \u03b5)2f 2(2\u03b1). We consider the following concentration result to characterize maximum power of the columns of G. Lemma 3.9. Let G be drawn from an IID Gaussian ensemble with gij d \u223cN(0, 1/m). Let gj, j = 1, 2 . . . , n be the columns of G. Then, for any \u03f5 > 0, it follows that, P(max j \u2225gj\u22252 2 \u22651 + \u03f5) \u2264exp \u0010 \u2212m 2 (log(1 + \u03f5) + \u03f5) + log n \u0011 \u2206 = \u03b42(m, n, \u03f5) Proof. Clearly X := m\u2225g\u22252 2 is \u03c72 distributed with degree m and its moment generating function is E(etX) = (1 \u22122t)\u2212m/2. From Cherno\ufb00bound, Pr(X \u2265a) \u2264E(etX) eta = (1 \u22122t)\u2212m/2 eta Choosing a = m(1 + \u03f5) and t = 1 2(1 \u2212m/a) = \u03f5 2(1+\u03f5), we have Pr(\u2225g\u22252 2 \u22651 + \u03f5) \u2264exp \u0010 \u2212m 2 (log(1 + \u03f5) + \u03f5) \u0011 The proof then follows by employing the union bound. Putting Lemmas 3.8 and 3.9 together with Equation (21) and taking the expectation with respect to G we get, P(E1) =EG (P(E1 | G)I\u0393 + P(E1 | G)I\u0393c) \u2264exp \u001a \u2212(1 \u2212\u03b7)2\u03b22SNR 32(1 + \u03b5) \u001b exp{log 2n}(1 \u2212\u03b4) + \u03b4 where \u0393 = {G : \u03c3G,min maxj \u2225gj\u22252 \u2264(1\u2212\u03b7)2 (1+\u03b5) } and \u03b4 = \u03b41(n, \u03b1n, \u03f5) + \u03b42(m, n, \u03f5). Note that P(\u0393c) \u2264\u03b4 and \u03b4 can be made arbitrarily small for m = \u2126(log(n)) and k su\ufb03ciently large. We are now left to ensure that the \ufb01rst term in the RHS of the above equation can be made small as well. For this purpose we need (1 \u2212\u03b7)2\u03b22SNR (1 + \u03f5)32 = (1 + \u03b3) log 2n (23) 15 \ffor some arbitrary \u03b3 > 0. Let \u03b71 = \u0010 32(1+\u03b3)(1+\u03f5) log 2n \u03b22SNR \u00111/2 . This implies that it is su\ufb03cient that, 1 \u2212\u03b7 \u2265\u03b71 = \u21d2\u03b7 \u22641 \u2212\u03b71 (24) = \u21d2(1 + \u03b5)f(2\u03b1)(2 + (1 + \u03b5)f(2\u03b1)) + 1 \u22641 + (1 \u2212\u03b71) (25) = \u21d2(1 + (1 + \u03b5)f(2\u03b1))2 \u22642 \u2212\u03b71 = \u21d2(1 + \u03b5)f(2\u03b1) \u2264 p 2 \u2212\u03b71 \u22121 (26) For this inequality to be satis\ufb01ed we need \u03b71 \u22641. A su\ufb03cient condition for support recovery can be obtained by substituting for \u03b7 and we get \u03b71 = 32(1 + \u03b3)(1 + \u03f5) log(2n) \u03b22SNR < 1, n m \u2264 1 (1 + \u03b5)2( \u221a 2\u03b1 + p 2H2(2\u03b1))2 ( p 2 \u2212\u03b71 \u22121)2 Since ( \u221a 2\u03b1 + p 2H2(2\u03b1))2 \u22646H2(2\u03b1) and \u03b3, \u03b5 can be made arbitrarily small, the result now follows for event E1. We are now left to bound the probability of event E2. This case is simple since the normalizing factor maxj \u2225gj\u22252 is no longer relevant as seen from the proof of Lemma 3.7. It su\ufb03ces to ensure that \u03c3G,min needs to be bounded away from zero. However, note that we already have this from bounding the probability of event E1. The result now follows. 4 Recovery for Arbitrary Distortions: Bayesian signal model In this section we switch to a Bayesian signal model from the worst-case setting considered in the previous section. There are a number of reasons for considering such a model: (A) For both the input and output noise models we need the SNR to scale as \u2126(log(n)) for exact support recovery regardless of the number of measurements. (B) For exact support recovery in the worst-case setup we require that the minimum singular values of all sub-matrices of G as described in Equation (19) be uniformly bounded away from zero (Theorem 3.2). This arises because a worst-case signal, X, matched to the smallest singular value can be chosen. However, this problem may not arise in the average case setting. (C) The situation is worse for the input noise model. Even with SNR of \u2126(log(n)) the number of measurements required is linearly proportional to signal dimension. (D) Theorem 3.4 points out that even with distortion we can only hope to reduce the SNR but not the number of measurements. Consequently, it is worth exploring whether these results can be improved in the average Bayesian case. Fundamentally, the idea is that if we remove a su\ufb03ciently small set of signals then it is conceivable that the results could be more promising. In the following we \ufb01rst develop novel lower and upper bounds to probability of error subject to a distortion in reconstruction. The main ingredient in realizing these bounds is the use of the minimal covering property of the rate distortion function. We begin with a minimal cover as a functional mapping of the source to the set of rate distortion quantization points. Then for the lower bound to the probability of error we follow the steps of the proof Fano\u2019s inequality, [18] which we appropriately modify to address detection of the correct quantization point corresponding to the true X. Similarly for the upper bound to the probability of error we propose a minimum distance decoder (ML decoder for AWGN noise) over the set of rate distortion quantization points and derive a closed form result for the particular case of \u21132 distortion. 4.1 Lower boundmodi\ufb01ed Fano\u2019s inequality In the following we will use X and Xn interchangeably. The main reason for introducing this notation is that we will deal with n-dimensional probability distributions over X induced by the product measure PXn = PX \u00d7 ... \u00d7 PX(n times). Lemma 4.1. Given observation(s) Y for the sequence Xn \u225c{X1, . . . , Xn} of random variables drawn IID with Xi d \u223cPX. Let \u02c6 Xn(Y) be the reconstruction of Xn from Y. Let the distortion measure be given by 16 \fd(Xn, \u02c6 Xn(Y)) = Pn i=1 d(Xi, \u02c6 Xi(Y)). Then given \u03f5 > 0 for su\ufb03ciently large n we have P \u0012 1 nd( \u02c6 Xn(Y), Xn) \u2265d0 \u0013 \u2265RX(d0) \u2212K(d0, n) \u22121 nI(Xn; Y) RX(d0) + \u03f5 + \u03f5 where K(d0, n) is the logarithm of the number of neighbors of a quantization point in the n-dimensional rate-distortion mapping) and RX(d0) is the corresponding (scalar) rate distortion function for X. We have the following result for the special case of \ufb01nite alphabets with Hamming distortion. Lemma 4.2. Given observation(s) Y for the sequence Xn \u225c{X1, ..., Xn} of random variables drawn i.i.d. according to PX and Xi \u2208X, |X| < \u221e. Let \u02c6 Xn(Y) be the reconstruction of Xn from Y. For hamming distortion dH(\u00b7, \u00b7) and for distortion levels, d0 \u2264min \u001a 1/2, (|X| \u22121) min x\u2208X PX(x) \u001b we have P \u0012 1 ndH(Xn, \u02c6 Xn(Y)) \u2265d0 \u0013 \u2265 nRX(d0) \u2212I(Xn; Y) \u22121 \u2212log nd0 n log(|X|) \u2212n \u0010 H2(d0) + d0 log(|X| \u22121) + log nd0 n \u0011 4.2 Constructive upper bound to probability of error for \u21132 distortion In this section we will provide a constructive upper bound to the probability of error in reconstruction subject to an average squared distortion level for the output noise model. To this end assume that we are given a minimal d0 cover as described in Theorem 8.1 of [21]. Speci\ufb01cally, we have a set of balls, Bi \u2282Rn, i = 1, 2 . . . , 2n(RX(d0)+\u03f5), of diameter 2\u221and0 such that, for any \u03f5 > 0 we have for su\ufb03ciently large n that, Pr{ N\u03f5(n,d0) [ i=1 Bi} \u22651 \u2212\u03f5 where RX(d0) is the (scalar) rate distortion function for X d \u223cPX and N\u03f5(n, d0) = 2n(RX(d0)+\u03f5). Each ball Bi is represented by a quantization points Zi . = Zn i . Thus with high probability for any X there exists a point, Zi to which it can be mapped to such that the distortion is less than d0. We consider a modi\ufb01ed maximum likelihood estimator to establish an achievable upper bound. Given G and the rate distortion points Zi, we enumerate the set of points, GZi \u2208Rm. Then given the observation Y we map it to the nearest point GZi \u2208Rm\u00d71. Our estimator \u02c6 X(Y) then outputs Zi. We refer to Figure 2 for an illustration. Lemma 4.3. Given observation Y = GX + N SNR for the sequence X . = Xn \u225c{X1, . . . , Xn} of random variables drawn IID with Xi d \u223cPX. Let \u02c6 Xn(Y) be the reconstruction of Xn from Y. Then for any \u03f5 > 0 we have for su\ufb03ciently large n, P(\u2225\u02c6 X(Y) \u2212X\u22252 \u22652nd0) \u2264(1 \u2212\u03f5) exp \u001a \u2212SNR\u2225G(Zi \u2212Zj)\u22252 32 \u001b 2nRX(d0) + \u03f5 (27) where Zi and Zj are any two quantization points such that \u2225Zi \u2212Zj\u2225= 4\u221and0. Proof. To compute the probability of error we \ufb01rst consider a pairwise error probability, namely, Pe(i, j) = P {N : X \u2208Bi \u2192Zj | d(Bi, Bj) \u22652nd0, G} (28) where, d(Bi, Bj) is the minimum squared distance between any two points, Xi \u2208Bi and Xj \u2208Bj. Under the minimum distance estimator we have, Pe(i, j) = P \u001a N : \u2225GX + N \u221a SNR \u2212GZi\u22252 \u2265\u2225GX + N \u221a SNR \u2212GZj\u22252 \u001b (29) 17 \fFigure 2: Figure showing the rate distortion cover by balls B of radius \u221and0. The ML decoding over the set of rate distortion quantization points (identi\ufb01ed as centers of the distortion balls) consists of mapping Y to the correct distortion ball for X using a minimum distance decoder. Shown in the \ufb01gure is a pair-wise error event for mapping X \u2208Bi to quantization point Zj \u2208Bj that is at a set distance of 2nd0 from Bi to which X belongs. where we have omitted the conditioning variables and equations for brevity. Simplifying the expression inside the probability of error we get that, Pe(i, j) = P \u001a 2 NT \u221a SNR G(Zj \u2212Zi) \u2225G(Zj \u2212Zi)\u2225\u2265\u2225G(X \u2212Zj)\u22252 \u2212\u2225G(X \u2212Zi)\u22252 \u2225G(Zj \u2212Zi)\u2225 \u001b (30) In other words we are asking for the pairwise probability of error in mapping a signal that belongs to the distortion ball Bi to the quantization point Zj of the distortion ball Bj under the noisy mapping GX + N such that the set (squared) distance between the distortion balls is \u22652nd0, see Figure 2. Under the assumption that the noise N is an AWGN noise with unit power in each dimension, its projection N onto the unit vector G(Zj\u2212Zi) \u2225G(Zj\u2212Zi)\u2225is also AWGN with unit power. Thus we have Pe(i, j) = P \u001a N \u221a SNR \u2265\u2225G(X \u2212Zj)\u22252 \u2212\u2225G(X \u2212Zi)\u22252 2\u2225G(Zj \u2212Zi)\u2225 \u001b \u2264P \u001a N \u221a SNR \u2265min X\u2208Bi \u2225G(X \u2212Zj)\u22252 \u2212\u2225G(X \u2212Zi)\u22252 2\u2225G(Zj \u2212Zi)\u2225 \u001b where we have further upper bounded the probability of the pairwise error via choosing the worst case X that minimizes the distance between the ball Bi and the quantization point Zj and maximizes the distance from the quantization point Zi within the distortion ball Bi. For the case of squared distortion and covering via spheres of average radius d0, it turns out that the worst case X is given by X = 3Zi+Zj 4 and \u2225Zi \u2212Zj\u2225= 4\u221and0. Plugging this value in the expression we have for the worst case pairwise probability of error that Pe(i, j) \u2264P ( N \u2265 \u221a SNR\u2225G(Zi \u2212Zj)\u2225 4 ) \u2264exp \u001a \u2212SNR\u2225G(Zi \u2212Zj)\u22252 32 \u001b where the second inequality follows by the standard upper bound to the error function. Now we apply the union bound over the set of rate distortion quantization points Zj minus the set of points that are the 18 \fneighbors of Zi (see \ufb01gure 2). The maximum number of such points is given by N\u03f5(n, d0) = 2n(RX(d0)+\u03f5), where RX(d0) is the scalar rate distortion function, [18]. Hence we have, P(\u2225\u02c6 X \u2212X\u22252 \u22652nd0 | X \u2208 [ i Bi) \u2264exp \u001a \u2212SNR\u2225G(Zi \u2212Zj)\u22252 32 \u001b 2n(RX(d0)+\u03f5) with \u2225Zi \u2212Zj\u2225= 4\u221and0. To \ufb01nish the proof we note that with probability (1 \u2212\u03f5), the signal X belongs to one of the balls Bj. Thus taking expectations with respect to X the result follows. 5 Approximate Recovery: Bayesian Bounds In this paper we will consider the following mixture model for explicit evaluation of the bounds. Xi d \u223cPX = \u03b1N(\u00b51, \u03c32 1) + (1 \u2212\u03b1)N(\u00b50, \u03c32 0) (31) i.e., each component Xi of X is IID PX de\ufb01ned above. It is easy to see that for \u00b51 = 1, \u00b50 = 0 for \u03c30 = 0 this mixture model for large enough n results in an approximately k = \u03b1n sparse sequence. We use \u03c31 = 0 to model a binary discrete case and \u03c31 = 1 to model a continuous valued case. It is worth pointing out that this model has been used previously in several papers, e.g. see [22, 14] to probabilistically model sparse signals. 5.1 Discrete X: Support recovery It is easy to see that using a binary signal model for X one can address the support recovery problem in the Bayesian setting. Under this case X is drawn IID according to, PX = \u03b1\u03b4(X \u2212\u03b2) + (1 \u2212\u03b1)\u03b4(X) , : \u03b1 \u22640.5 (32) where, \u03b4(\u00b7) is the usual singular measure. Note that it follows from Asymptotic Equipartition Property (AEP), see [18], that asymptotically the n-dimensional probability distribution uniformly concentrates on the set of exactly k-sparse sequences \u039e\u03b1n {0,\u03b2}, i.e. given \u03f5 > 0, \u2203n such that PXn \u0010 \u039e\u03b1n {0,\u03b2} \u0011 \u22651 \u2212\u03f5. Thus these bounds can be compared to the worst-case setup of Section 3 when X \u2208\u039ek {0,\u03b2} , k = \u03b1n. For this discrete case we have the following main results stated in terms of the scalar rate distortion function RX(d0) with Hamming distance as the distortion measure. Note that for this case RX(d0) = H2(\u03b1) \u2212H2(d0) : d0 \u2264\u03b1. Theorem 5.1. Consider the input noise model of Equation (2) and the binary model for X as described above. Then, a. Necessity: Asymptotically as n \u2192\u221eif m \u2264 nRX(d0) 0.5 log(1 + \u03b1\u03b22SNR) there does not exist any algorithm that recovers the signal to within an average Hamming distortion of d0. b. Su\ufb03ciency: Asymptotically as n \u2192\u221e, it is su\ufb03cient that m \u2265 nRX(d0/2) 0.5 log(1 + d0\u03b22SNR 2 ) for the constructive ML estimator of section 4.2 to reliably recover the signal to within Hamming distortion of d0. Proof. To prove part (a) note that from Lemma 4.2 for the probability of error to approach zero implies that the numerator in the lower bound approach zero. This implies that we need, n m \u2264 1 mI(X; Y|G) RX(d0) \u22121 n \u2212log nd0 n (33) To this end recall that Y = G(X+ 1 \u221a SNRN). Consider the SVD of G = USV\u2217, where U, V are orthonormal matrices and S = [D 0], with D a positive diagonal matrix. From [20] it follows that U, S, V are independent 19 \frandom matrices. Furthermore U and V are isotropically random. By linearly transforming Y by premultiplying by D\u22121U\u2217we get an equivalent system of equations with \u02dc Y = V\u2217 1X + 1 \u221a SNR V\u2217 1N (34) where V\u2217 1 is the matrix formed from the \ufb01rst m rows of V\u2217. Now note that since the rows of V\u2217 1 are orthogonal and normalized \u02dc N = 1 \u221a SNRV\u2217 1N is IID Gaussian with each component having zero mean and variance 1/SNR. This transformation implies that I(X; Y|G) = I( \u02dc Y; \u02dc X|V1) since V is independent of U and S. Now by direct computation it follows that, EVI( \u02dc Y; \u02dc X | V1) \u2264h( \u02dc Y | V1) \u2212h( \u02dc Y | V, \u02dc X) \u2264m 2 log(1 + SNR\u03b1\u03b22) where to get the last inequality we have used the fact that h( \u02dc Y | V, \u02dc X) is the entropy of noise \u02dc N and for the \ufb01rst term, h( \u02dc Y | V1), we have used the fact that a Gaussian distribution maximizes the entropy over all other random variables with zero mean and identical variance [18]. Finally, for su\ufb03ciently large n the term 1 n + log nd0 n can be made arbitrarily small and the result follows. We will now prove part (b). In order to simplify the derivation we again focus on Equation (34). Following the proof of Lemma 4.3 the pairwise error can now be computed as follows Pe(i, j) \u2264P ( N \u2265 \u221a SNR\u2225V\u2217 1(Zi \u2212Zj)\u2225 4 | V1 ) \u2264exp \u001a \u2212SNR\u2225V\u2217 1(Zi \u2212Zj)\u22252 32 \u001b (35) To compute the error probability we will need to take the expectation over V1 and apply the union bound to bound the error probability over all error patterns. To simplify the expectation over V1 we let, \u03c6(D, V1) = exp \u001a \u2212SNR\u2225DV\u2217 1(Zi \u2212Zj)\u22252 32 \u001b (36) where, D is a positive diagonal random matrix independent of V\u2217 1. Note that our problem reduces to bounding expectation of \u03c6(Im, V1) over V1. Note that when \u03c3max(D) \u22641 we have \u03c6(Im, V1) \u2264\u03c6(D, V1). Next, note that trivially we have, \u03c6(Im, V1))I{\u03c3max(D)\u22641} \u2264\u03c6(D, V1)I{\u03c3max(D)\u22641} + \u03c6(D, V1)I{\u03c3max(D)\u22651} (37) where I{\u00b7} denotes the indicator function. Consequently, we can take expectations over the two independent matrices D and V1 to obtain, EV1(\u03c6(Im, V1)))Prob(\u03c3max(D) \u22641) \u2264ED,V1 exp \u001a \u2212SNR\u2225DV\u2217 1(Zi \u2212Zj)\u22252 32 \u001b (38) Note that we can introduce a isotropically random unitary matrix U, namely, exp \b \u2212SNR 32 \u2225DV\u2217 1(Zi \u2212Zj)\u22252\t = exp \b \u2212SNR 32 \u2225UDV\u2217 1(Zi \u2212Zj)\u22252\t without modifying the result. Now the matrix H = UDV\u2217 1 can be identi\ufb01ed by a suitable IID Gaussian matrix when U, D, V are chosen independently and U and V are chosen uniformly from set of all unitary matrices; the positive diagonal matrix D is distributed according to the distribution of singular values of a Gaussian matrix. To ensure a tight approximation we need to choose a Gaussian matrix such that P(\u03c3max(D) \u22641) approaches one. This can be accomplished by choosing H as an IID Gaussian ensemble with each component hij d \u223cN(0, 1 (1+\u221a m/n)n). Then following similar steps as in the proof of Lemma 4.3 we arrive at a similar upper bound, P(\u2225\u02c6 X \u2212X\u22252 \u22652nd0) \u2264(1 \u2212\u03f5) exp \u001a \u2212SNR\u2225H(Zi \u2212Zj)\u22252 32 \u001b 2n(RX(d0)+\u03f5) + \u03f5 where, \u2225Zi \u2212Zj\u2225= 4\u221and0. Since \u03f5 is arbitrary, the result then follows by taking expectation with respect to H and using the moment generating function of the \u03c72 random variable, [23]. 20 \fTheorem 5.2. Consider the output noise model of Equation (1) and the binary model for X as described above. Then, a. Necessity: Asymptotically as n \u2192\u221eif m \u2264 nRX(d0) 0.5 log(1 + n m\u03b1\u03b22SNR) there does not exist any algorithm that recovers the signal to within an average Hamming distortion of d0. b. Su\ufb03ciency: Asymptotically as n \u2192\u221eit is su\ufb03cient that m \u2265 nRX(d0/2) 0.5 log(1 + n m d0\u03b22SNR 2 ) for the constructive ML estimator of section 4.2 to reliably recover the signal to within Hamming distortion of d0. Proof. The proof of part (a) follows along the same lines as that of 5.1 with the following modi\ufb01cation to the upper bound of the mutual information expression, EGI(X; Y|G) \u2264m 2 log(1 + n\u03b1\u03b22SNR m ) (39) The proof of part (b) follows from the upper bound to the probability of error in Lemma 4.3 by taking expectation with respect to G and using the moment generating function of the \u03c72 random variable, see [23]. We will now reduce the implicit expression in the above Lemma to derive some explicit conditions on the number of measurements m. To this end we have the following corollary. Corollary 5.1. Consider the output noise model of Equation (1) and the binary model for X as described above. Then, (a) Asymptotically as n \u2192\u221eif SNR \u22642RX(d0) \u03b1\u03b22 and m \u22642nRX(d0) there exists no algorithm that can recover X to within an average Hamming distortion of d0; (b) On the other hand asymptotically as n \u2192\u221eit is su\ufb03cient that SNR \u2265200RX(d0/2) d0\u03b22 with m \u22652.08nRX(d0/2) for the constructive ML estimator of section 4.2 to recover X to within an average Hamming distortion of d0. Proof. To begin with we will focus on the su\ufb03cient conditions. Denote by c = nRX(d0/2) m . Also let \u03b7 = d0\u03b22SNR 2RX(d0/2). Then from part (b) of Theorem 5.2 we have as a su\ufb03cient condition that, f(c) = 0.5 log(1 + c\u03b7) \u2212c \u22650 (40) In particular we want to \ufb01nd max{c|f(c) \u22650}. To this end note that f(c) = 0 at c = 0. Also for there to exist any positive c such that f(c) > 0 it is required that \u03b7 \u22652. In particular \u03b7 \u22652 is the condition for a positive derivative near zero. This implies that d0\u03b22SNR 2RX(d0/2) \u22652 or SNR \u22654RX(d0/2) d0\u03b22 . Given that this condition is satis\ufb01ed, c = 1\u22122/\u03b7 2 lies in the feasible region. Therefore m = 2 1\u22122/\u03b7nRx(d0/2) \u22482nRX(d0/2) is a su\ufb03cient condition for reliable recovery for some su\ufb03ciently large \u03b7 > 2, i.e. for SNR \u22654RX(d0/2) d0\u03b22 . In particular if we choose \u03b7 = 100 then SNR \u2265200RX(d0/2) d0\u03b22 and m \u22652.08nRX(d0/2) is su\ufb03cient for reliable recovery. Analyzing part (a) of the Theorem 5.2 in a similar manner, one can show that if SNR \u22642RX(d0) \u03b1\u03b22 and m \u22642nRX(d0) there exits no algorithm that can reliably recover X to within the desired distortion level. Remark 5.1. One immediate observation from the above analysis is that unlike the worst case set-up one can indeed tradeo\ufb00the number of measurements with distortion in the Bayesian set-up. 5.2 Continuous X: \u21132 recovery Under this case X is drawn IID according to, PX = \u03b1N(0, \u03b22) + (1 \u2212\u03b1)\u03b4(X) (41) 21 \fFor this case we have the following main results. The results are stated in terms of the scalar rate distortion function RX(d0) given by RX(d0) = H2(\u03b1) + \u03b1 2 log \u03b1 d0 : d0 < \u03b1, (see section 8.4 for the derivation of this result). Notice in the following that in contrast to the discrete case where d0 \u2264\u03b1 here we impose d0 \u2264\u03b1/2 and for reasonable reconstruction one typically desires d0 = \u03f5\u03b1 for some small \u03f5 > 0. The reason that we require d0 \u2264\u03b1 2 is due to the additional term of K(n, d0) in the modi\ufb01ed Fano\u2019s inequality 4.1 which appears in the continuous setting. Theorem 5.3. Consider the input noise model of Equation (2) and the mixture model for X as described above. Then, a. Necessity: Asymptotically as n \u2192\u221eif m \u2264n(RX(d0) \u2212\u03b1 2 log 2) 0.5 log(1 + \u03b1\u03b22SNR) there does not exist any algorithm that recovers the signal to within an average \u21132 distortion of d0. b. Su\ufb03ciency: Asymptotically as n \u2192\u221eit is su\ufb03cient that m \u2265 nRX(d0/2) 0.5 log(1 + d0\u03b22SNR 2 ) for the constructive ML estimator of section 4.2 to reliably recover the signal to within an average \u21132 distortion of d0. Proof. For part (a) \ufb01rst note that from Theorem 5.1 we have EGI(X; Y|G) \u2264m 2 log(1 + \u03b22\u03b1SNR). From Lemma 4.1 it follows that for feasibility of recovery to with distortion d0 (asymptotically) it is required that, n m \u2264 1 mEGI(X; Y|G) RX(d0) \u2212K(d0, n) (42) The result then follows by noting that |K(d0, n) \u22120.5\u03b1 log 2| < \u03f5 with \u03f5 arbitrarily small for large enough n, see e.g. [24]. Note that for the case at hand in order for the expression RX(d0)\u2212K(d0, n) to remain positive and hence meaningful, d0 \u2264\u03b1/2. The proof of part (b) follows exactly along the same lines as the proof of part (b) in Theorem 5.1. Note that unlike the case of support recovery where the number of measurements had to grow with signal dimension even with SNR of log(n) here we see that the number of measurements does scale with the distortion for moderate signal to noise ratios. This maybe acceptable in cases where either a probability model for the signal set is available. Theorem 5.4. Consider the output noise model of Equation (1) and the mixture model for X as described above. Then, a. Necessity: Asymptotically as n \u2192\u221eif m \u2264 n(RX(d0) \u2212\u03b1 2 log 2) 0.5 log(1 + n m\u03b1\u03b22SNR) there does not exist any algorithm that recovers the signal to within an average \u21132 distortion of d0. b. Su\ufb03ciency: Asymptotically as n \u2192\u221eit is su\ufb03cient that m \u2265 nRX(d0/2) 0.5 log(1 + n m d0\u03b22SNR 2 ) for the constructive ML estimator of section 4.2 to reliably recover the signal to within an average \u21132 distortion of d0. Proof. The proof is similar to the proof of Theorem 5.3. It is easy to see that Corollary 5.1 holds true for this case too with appropriate modi\ufb01cations to the necessary conditions in terms of RX(d0) \u2212\u03b1 2 log 2 instead of RX(d0). 5.3 Comparison between Worst-Case and Bayesian Setups Based on the worst-Case and Bayesian results we can comment on the main di\ufb00erences. The situation is slightly complicated since we considered two di\ufb00erent types of distortions in these cases. We recall the items (A)\u2014(D) listed in the beginning of Section 4 as a means for comparison. Note that by adopting a Bayesian setup we no longer need that the minimum singular value of sub-matrices of G be uniformly 22 \fbounded away from zero. This can be attributed to the fact that we are taking expectation with respect to G in Equation (27). However, note that the number of quantization points N\u03f5(n, d0) in Theorem 8.1 will go to in\ufb01nity if we insist on nearly exact support recovery. Second, note that the measurements do scale with the distortion-level, larger the admissible distortion, smaller the number of measurements. This is even more surprising for input noise models since in the worst-case setup we required the number of measurements to scale with signal dimension. Finally, for signal reconstruction to within a distortion level d0 we only need a constant SNR in contrast to the worst-case setup. However, this issue can be attributed to the fact that our mean-squared distortion metric is less stringent in comparison to support errors. 6 Appendix 6.1 Proof of Lemma 3.3 Consider any arbitrary G and N. Let for each X \u2208\u039e{k} \u03b2 denote by PX the observed distribution of Y given X as induced by the relation Y = GX + N. We next consider the equivalence class of all sequences with the same support and lump the corresponding class of observation probabilities into a single composite hypothesis, i.e., [X] = {X\u2032 \u2208\u039e{k} \u03b2 | Supp(X\u2032) = Supp(X)} (43) Each equivalence class bears a one-to-one correspondence with binary valued k-sparse sequences, \u039e{k} {0,\u03b2} = {X \u2208\u039e{k} \u03b2 | Xi = \u03b2, i \u2208Supp(X)} (44) Our task is to lower bound the worst-case error probability Pe|G = min \u02c6 X max X\u2208\u039e{k} \u03b2 PX([ \u02c6 X] \u0338= [X]|G) (45) Now note that, max X\u2208\u039e{k} \u03b2 PX([ \u02c6 X] \u0338= [X]|G) \u2265 max X\u2208\u039e{k} {0,\u03b2} PX([ \u02c6 X] \u0338= [X]|G) = max X\u2208\u039e{k} 0,\u03b2 PX( \u02c6 X \u0338= X, \u02c6 X \u2208\u039e{0,\u03b2}|G) (46) This implies that Pe|G = min \u02c6 X\u2208\u039e{k} \u03b2 max X\u2208\u039e{k} \u03b2 PX{Supp( \u02c6 X) \u0338= Supp(X)|G} (47) \u2265 min \u02c6 X\u2208\u039e{k} \u03b2 max X\u2208\u039e{k} {0,\u03b2} PX( \u02c6 X \u0338= X, \u02c6 X \u2208\u039e{k} {0,\u03b2}|G) (48) = min \u02c6 X\u2208\u039e{k} {0,\u03b2} max X\u2208\u039e{k} {0,\u03b2} PX( \u02c6 X \u0338= X, \u02c6 X \u2208\u039e{k} {0,\u03b2}|G) (49) \u2265 min \u02c6 X\u2208\u039e{k} {0,\u03b2} max X\u2208\u039e\u2032 {0,\u03b2} PX( \u02c6 X \u0338= X, \u02c6 X \u2208\u039e{k} {0,\u03b2}|G) (50) 6.2 Proof of Lemma 3.5 Denote by E1\u03c9 the error event when a signal from the \u03c9th support set is more likely, i.e., E1\u03c9 = ( N : min \u03c9\u2208IXmin S\u03c9 \u2265\u03b2/2 \u2225Y \u2212GS\u03c9XS\u03c9\u22252 \u2264min \u02dc X \u2225Y \u2212GS0 \u02dc X\u22252, \u03c9 \u0338= 0, ) (51) In the following we will drop stating the obvious fact that \u03c9 \u2208I. Now note that, E1 = [ \u03c9\u0338=0 E1\u03c9 (52) 23 \fWe \ufb01rst upperbound E1\u03c9 by a more manageable event, namely, F\u03c9 = ( N : min Xmin S0c,\u03c9 \u2265\u03b2/2 min XS0,\u03c9 \u2225Y \u2212GS0,\u03c9XS0,\u03c9 \u2212GS0c,\u03c9XS0c,\u03c9\u22252 \u2264min \u02dc X \u2225Y \u2212GS0 \u02dc X\u22252, \u03c9 \u0338= 0 ) (53) It is clear that, E1\u03c9 \u2282F\u03c9 (54) This is because the signal on the common support S0,\u03c9 is relaxed to take on any value and not necessarily those that are bounded away from zero by \u03b2/2. We will now simplify the events in F\u03c9 by analytically carrying out the unconstrained minimizations. Recall that Y = GX0 + N. Let X0 S0 denote the true signal X0 restricted to its support. Then Y = GS0X0 S0. Note that X0 S0 is composed of X0 S0,\u03c9 corresponding to the overlap and X0 0,\u03c9c corresponding to the misses. We have the following Lemma. Lemma 6.1. For m \u22652k + 1 F\u03c9 \u2282\u02dc F\u03c9 = [ Xmin S0c,\u03c9 \u2265\u03b2/2,X0,min S0,\u03c9c \u2265\u03b2 \b N : 2NT \u03a01G\u2032X\u2032 \u2265\u2225\u03a01G\u2032X\u2032\u22252, \t (55) where \u03a01 = (I \u2212GS0,\u03c9(GT S0,\u03c9GS0,\u03c9)\u22121GT S0,\u03c9) (56) is a projection operator and G\u2032 = [GS0c,\u03c9GS0,\u03c9c ], X\u2032 = \u0014\u2212XS0c,\u03c9 X0 S0,\u03c9c \u0015 , Xmin S0c,\u03c9 \u2265\u03b2/2, X0,min S0,\u03c9c \u2265\u03b2 (57) Proof. Consider the error region, F\u03c9 = ( N : min Xmin S0c,\u03c9 \u2265\u03b2/2 min XS0,\u03c9 \u2225Y \u2212GS0,\u03c9XS0,\u03c9 \u2212GS0c,\u03c9XS0c,\u03c9\u22252 \u2264min \u02dc X \u2225Y \u2212GS0 \u02dc X\u22252, \u03c9 \u0338= 0 ) (58) Fixing XS0c,\u03c9 we perform the inner minimization \ufb01rst on the L.H.S in the above equation. It can be shown that the inner minimum is achieved at, X0 S0,\u03c9 \u2212XS0,\u03c9 = \u2212(GT S0,\u03c9GS0,\u03c9)\u22121GT S0,\u03c9(N + GS0,\u03c9c X0 S0,\u03c9c \u2212GS0c,\u03c9XS0c,\u03c9) (59) Also the unconstrained minimum on the R.H.S. is given by, min \u02dc X ||(Y \u2212GS0 \u02dc X)||2 = NT \u03a00N (60) where \u03a00 = (I \u2212GS0(GT S0GS0)\u22121GT S0) (61) is a projection operator. Substituting these results in the expression for F\u03c9 we obtain, F\u03c9 = ( N : min Xmin S0c,\u03c9 \u2265\u03b2/2(G\u2032X\u2032)T \u03a01G\u2032X\u2032 \u22122NT \u03a01G\u2032X\u2032 + NT (\u03a00 \u2212\u03a01)N \u22640 ) (62) A simple application of the matrix lemma shows that (\u03a00 \u2212\u03a01) is a positive semi-de\ufb01nite matrix. This implies NT (\u03a00 \u2212\u03a01)N \u22650, \u2200N. Ignoring this non-negative term can only increase the probability of error. 24 \fTherefore ignoring this term we obtain, F\u03c9 = ( N : min Xmin S0,\u03c9c \u2265\u03b2/2(G\u2032X\u2032)T \u03a01G\u2032X\u2032 \u22122NT \u03a01G\u2032X\u2032 \u22640 ) (63) \u2286 [ Xmin S0c,\u03c9 \u2265\u03b2/2,X0,min S0,\u03c9c \u2265\u03b2 \b N : (G\u2032X\u2032)T \u03a01G\u2032X\u2032 \u22122NT \u03a01G\u2032X\u2032 \u22640 \t (64) = [ Xmin S0c,\u03c9 \u2265\u03b2/2,X0,min S0,\u03c9c \u2265\u03b2 \b N : 2NT \u03a01G\u2032X\u2032 \u2265\u2225\u03a01G\u2032X\u2032\u22252\t = \u02dc F\u03c9 (65) where the last equality follows from the fact that \u03a01 is a projection. Now note that if any column of G\u2032 falls into the null space of GS0,\u03c9 then probability of the event F\u03c9 is 1 and therefore the probability of error is 1 in the worst case. This will not happen as long as G is full rank and m \u22652k + 1. We now have the following Lemma. Lemma 6.2. \u02dc F\u03c9 \u2286L\u03c9 = L [ j=1 n N : 2NT g \u2032 jX\u2032 \u2265\u03c3min((G\u2032)T G\u2032)X\u20322, |X\u2032| = \u03b2/2 o (66) where L = |S0c,\u03c9 \u222aS0,\u03c9c| is the total number of location errors and, \u03c3min((G\u2032)T G\u2032) is the minimum singular value of the matrix (G\u2032)T G\u2032. Proof. Let \u02dc G = \u03a01G\u2032. Then note that for any X\u2032, n \u02dc N : 2NT \u02dc GX\u2032 \u2265\u2225\u02dc G X\u2032\u22252o \u2286 n N : 2NT \u02dc GX\u2032 \u2265\u03c3min( \u02dc GT \u02dc G)\u2225X\u2032\u22252o (67) Now note that \u02dc GX\u2032 = L X j=1 \u02dc gjX\u2032 j where \u02dc gj is the j-th column of the matrix \u02dc G and X\u2032 = [X\u2032 1, . . . , X\u2032 j, . . . , X\u2032 L]T . Note also that \u2225X\u2032\u22252 = X j |X\u2032 j|2 By a simple superposition of events this implies that \u02dc F\u03c9 \u2286 [ Xmin S0c,\u03c9 \u2265\u03b2/2,X0,min S0,\u03c9c \u2265\u03b2 L [ j=1 n N : 2NT \u02dc gjX\u2032 j \u2265\u03c3min( \u02dc GT \u02dc G)|X\u2032 j|2o \u2286 [ Xmin S0c,\u03c9 \u2265\u03b2/2,X0,min S0,\u03c9c \u2265\u03b2 L [ j=1 n N : 2NT \u02dc gjX\u2032 j \u2265\u03c3min( \u02dc GT \u02dc G)|X\u2032 j|2o (68) \u2286 L [ j=1 n N : 2NT \u02dc gjX\u2032 \u2265\u03c3min( \u02dc GT \u02dc G)|X\u2032|2 : |X\u2032| = \u03b2/2 o (69) where the last inequality follows from the fact all the events with X\u2032 \u2265\u03b2/2 are contained in the event X\u2032 = \u03b2/2. Now note that since \u03a01 is a projection and m \u22652k + 1 and L \u22642k it implies \u03c3min( \u02dc GT \u02dc G) = \u03c3min((G\u2032)T G\u2032). This implies that, \u02dc F\u03c9 \u2286 L [ j=1 n N : 2NT \u02dc gjX\u2032 \u2265\u03c3min( \u02dc GT \u02dc G)|X\u2032|2 : |X\u2032| = \u03b2/2 o (70) = [ X\u2032=\u00b1\u03b2/2 L [ j=1 \b N : 2NT g\u2032 jX\u2032 \u2265\u03c3G,min((G\u2032)T G\u2032)|X\u2032|2\t (71) 25 \fSince L \u22642k \u2264n and \b g\u2032 1, .., g\u2032 j, ..., g\u2032 L \t = {gi : i \u2208S0c,\u03c9 \u222aS0,\u03c9c} \u2286{g1, ..., gn}, F\u03c9 \u2286 [ X\u2032=\u00b1\u03b2/2 n [ j=1 \b N : 2NT gjX\u2032 \u2265\u03c3G,min|X\u2032|2\t (72) = L\u03c9 (73) where \u03c3G,min = minS:|S|\u22642k \u03c3min(GT SGS). The result then follows by noting that, E1 = [ \u03c9 E1\u03c9 \u2286 [ \u03c9 F\u03c9 \u2286 [ \u03c9 L\u03c9 (74) and replacing the notation X\u2032 by X. 7 Proof of Lemma 3.6 From Lemma 3.5 we have , P(E1) \u2264 [ X=\u00b1\u03b2/2 n [ j=1 n N : 2NT gjX \u2265\u03c3G,min \u221a SNR|X|2o = [ X=\u00b1\u03b2/2 n [ j=1 n W : 2WT \u03a31/2gjX \u2265\u03c3G,min \u221a SNR|X|2o = [ X=\u00b1\u03b2/2 n [ j=1 \uf8f1 \uf8f2 \uf8f3w : 2wX \u2265 \u221a SNR\u03c3G,min X2 q g \u2032 j\u03a3gj \uf8fc \uf8fd \uf8fe Note that W is IID normally distributed Gaussian vector and we let w = WT \u03a31/2gj \u221a gT j \u03a3gj . Next noting that \u2225gj\u2225= 1 \u2200j we have, \uf8f1 \uf8f2 \uf8f3w : 2wX \u2265 \u221a SNR\u03c3G,min X2 q \u2225gT j \u03a3gj\u2225 \uf8fc \uf8fd \uf8fe\u2286 ( w : 2wX \u2265 \u221a SNR\u03c3G,min s 1 \u03bbmax(\u03a3)X2 ) (75) = n w : 2wX \u2265 \u221a SNR\u03c3G,min p \u03bbmin(\u03a3\u22121)X2o (76) We now apply the union bound over all the possible 2n error events corresponding to each j \u2208{1, 2, ..., n} and X = \u00b1\u03b2/2 and obtain, P(E1) \u2264P ( w : w \u2265 p \u03bbmin(\u03a3\u22121)\u03c3G,min \u221a SNR\u03b2 4 ) exp(log 2n) (77) = 1 \u221a 2\u03c0 Z \u221e \u221a \u03bbmin(\u03a3\u22121)\u03c3G,min \u221a SNR\u03b2 4 exp(\u2212y2/2)dy \u00b7 exp(log 2n) (78) \u2264exp \u001a \u2212\u03bbmin(\u03a3\u22121)\u03c32 G,min \u03b22SNR 32 \u001b exp {log 2n} (79) Note that the probability is only taken over the noise W (N) as G is given and is \ufb01xed. Here we have used the approximation Q(x) \u2264exp(\u2212x2/2) for the standard error function de\ufb01ned as Q(x) = 1 \u221a 2\u03c0 R \u221e x exp(\u2212x2/2)dx. 8 Proof of Lemma 3.7 For any X0 supported on the submatrix GS0 the probability of the error event E2 is given by, P(E2) = P n N : \u2225(GT S0GS0)\u22121GT S0N\u2225\u221e\u2265 \u221a SNR\u03b2/2 o (80) 26 \fTo this end let GS0 = U\u03a3S0V\u2217, U \u2208Cm\u00d7k, V\u2217\u2208Ck\u00d7m. Then (GT S0GS0)\u22121GT S0 = U\u03a3\u22121 S0 V\u2217. Then let \u02dc N = V\u2217N. Then since V is orthonormal matrix \u02dc N has the same distribution as that of N. Now note that if N \u223cN(0, \u03a3), then P n \u2225U\u03a3\u22121 S0 \u02dc N\u2225\u221e\u2265SNR\u03b2/2 o \u2264 m X i=1 P \u001a \u2225(U\u03a3\u22121 S0 \u03a31/2W)i\u2225\u2265SNR\u03b2 2 \u001b (81) (a) \u22642m exp \u001a \u2212SNR\u03b22 8 \u03bbmin(\u03a3\u22121)\u03c32 GS0,min \u001b (82) (b) \u2264exp ( \u2212 SNR\u03b22\u03bbmin(\u03a3\u22121)\u03c32 G,min 8 + log 2n ) (83) where (a) follows from the following facts applied in succession(1) Maximum variance among the noise components (U\u03a3\u22121 S0 \u03a31/2W)i is given by \u03c3\u22121 GS0,min\u03bbmax(\u03a31/2) and \u03bbmax(\u03a31/2) = \u03bbmin(\u03a3\u22121/2); (2) Q(x) \u2264 e\u2212x2/2 for the standard error function de\ufb01ned as Q(x) = 1 \u221a 2\u03c0 R \u221e x exp(\u2212x2/2)dx. (b) follows from the fact that m \u2264n and \u03c3G,min \u2264\u03c3GS0,min. 8.1 Proof of Theorem 3.4 We follow along the lines of the proof for the deterministic case presented in Section 3.2. Basically we modify Lemma 3.5. We follow the same steps till Lemma 6.2. Then following similar algebraic steps as used in Lemma 6.2 it turns out that the support error events with Hamming distortion \u22652kd0 + 1 are almost contained in the union of support error events with Hamming distortion kd0 \u2264dH \u22642kd0. Then in this case the upper bound in Proposition 3.2 is modi\ufb01ed to, Pe|G \u22642 exp \u001a \u2212\u03a3\u22121)\u03c32 G,min \u03b22kd0SNR 32 \u001b e2kd0H2(2kd0/n) (84) The result for Gaussian G is then identical to the development in Section 3.3. 8.2 Proof of lemma 4.1 Let Xn = {X1, . . . , Xn} be an IID sequence where each variable Xi is distributed according to a distribution PX de\ufb01ned on the alphabet X. Denote PXn \u225c(PX)n the n-dimensional distribution induced by PX. Let the space X n be equipped with a distance measure d(., .) with the distance in n dimensions given by d(Xn, Zn) = Pn k=1 d(Xk, Zk) for Xn, Zn \u2208X n. For this setting we have the following Theorem taken from [21]. Theorem 8.1. Given \u03f5 > 0, there exist a set of points n Zn 1 , ..., Zn N\u03f5(n,d0) o \u2282X n such that, PXn \uf8eb \uf8ed N\u03f5(n,d0) [ i=1 Bi \uf8f6 \uf8f8\u22651 \u2212\u03f5 (85) where Bi \u225c \b Xn : 1 nd(Xn, Zn i ) \u2264d0 \t with the property that 1 n log N\u03f5(n, d0) \u2264RX(d0) + \u03f5. This implies that for all Xn, \u2203a mapping f(Xn) : Xn \u2192Zn i s.t. P \u0000 1 nd(Xn, Zn i ) \u2264d0 \u0001 \u22651 \u2212\u03f5 Now we are given that there is an algorithm \u02c6 Xn(Y) that produces an estimate of Xn given the observation Y. To this end de\ufb01ne an error event on the algorithm as follows, En = \u001a 1 if 1 nd(Xn, \u02c6 Xn(Y)) \u2265d0 0 otherwise Now, consider the following expansion, H(f(Xn), En, |Y) = H(f(Xn)|Y) + H(En, An|f(Xn), Y) = H(En|Y) + H(f(Xn)|En, Y) 27 \fThis implies that H(f(Xn)|Y) \u2264H(En) + H(f(Xn)|En, Y) Note that since H(En) \u22641 H(f(Xn)|Y) \u22641 + PeH(f(Xn)|Y, En = 1) + (1 \u2212Pe)H(f(Xn)|Y, En = 0) (86) Note that by construction H(f(Xn)|Y, En = 1) \u2264log N\u03f5(n, d0) and (1 \u2212Pe)H(f(Xn)|Y, En = 0) \u2264 (1 \u2212Pn e ) log (|S|) where S is the set given by, S = \b i : dset \u0000Bf(Xn), Bi \u0001 \u2264nd0 \t where dset(S1, S2) = mins\u2208S1,s\u2032\u2208S2 dn(s, s\u2032) is the set distance between two sets. Now note that H(f(Xn)|Y) = H(f(Xn)) \u2212I(f(Xn); Y) \u2265H(f(Xn)) \u2212I(Xn; Y) where the second inequality follows from data processing inequality over the Markov chain f(Xn) \u2194Xn \u2194Y. Thus we have, Pe \u2265H(f(Xn)) \u2212log |S| \u2212I(Xn; Y) \u22121 log N\u03f5(n, d0) \u2212log |S| (87) \u2265I(f(Xn); Xn) \u2212log |S| \u2212I(Xn; Y) \u22121 nRX(d0) + \u03f5 (88) The proof then follows by noting that by de\ufb01nition of the rate distortion function I(f(Xn); Xn) \u2265nRX(d0) (see [18]) and by identifying K(n, d0) = 1 n log |S|. 8.3 Proof of lemma 4.2 Proof. De\ufb01ne the error event, E = \u001a 1 if 1 ndH(Xn, \u02c6 Xn(Y)) \u2265d0 0 otherwise Expanding H(Xn, E|Y) in two di\ufb00erent ways we get that, H(Xn|Y) \u22641 + nPe log(|X|) + (1 \u2212Pe)H(Xn|E = 0, Y) Now the term (1 \u2212Pe)H(Xn|E = 0, Y) \u2264(1 \u2212Pe) log nd0\u22121 X j=0 \u0012 n d0n \u2212j \u0013 (|X| \u22121)nd0\u2212j (89) \u2264(1 \u2212Pe) log nd0 \u0012 n d0n \u22121 \u0013 (|X| \u22121)nd0 (90) \u2264n(1 \u2212Pe) \u0012 H2(d0) + d0 log(|X| \u22121) + log nd0 n \u0013 (91) where the second inequality follows from the fact that d0 \u22641/2 and \u0000n d0n\u2212j \u0001 (|X| \u22121)nd0\u2212j is a decreasing function in j for d0 \u22641/2. Then we have for the lower bound on the probability of error that, Pe \u2265 H(Xn|Y) \u2212n \u0010 H2(d0) + d0 log(|X| \u22121) + log nd0 n \u0011 \u22121 n log(|X|) \u2212n \u0010 H2(d0) + d0 log(|X| \u22121) + log nd0 n \u0011 Since H(Xn|Y) = H(Xn) \u2212I(Xn; Y) we have Pe \u2265 n \u0010 H(X) \u2212H2(d0) \u2212d0 log(|X| \u22121) \u2212log nd0 n \u0011 \u2212I(Xn; Y) \u22121 n log(|X|) \u2212n \u0010 H2(d0) + d0 log(|X| \u22121) + log nd0 n \u0011 28 \fIt is known that RX(d0) \u2265H(X) \u2212H2(d0) \u2212d0 log(|X| \u22121), with equality i\ufb00 d0 \u2264(|X| \u22121) min x\u2208X PX(x) see e.g., [25]. Thus for values of distortion d0, d0 \u2264min \u001a 1/2, (|X| \u22121) min x\u2208X PX(x) \u001b (92) we have for all n, Pe \u2265 nRX(d0) \u2212I(Xn; Y) \u22121 \u2212log nd0 n log(|X|) \u2212n \u0010 H2(d0) + d0 log(|X| \u22121) + log nd0 n \u0011 8.4 Rate distortion function for the mixture Gaussian source under squared distortion measure It has been shown in [26] that the rate distortion function for a mixture of two Gaussian sources with variances given by \u03c31 with mixture ratio \u03b1 and \u03c30 with mixture ratio 1 \u2212\u03b1, is given by Rmix(D) = ( H2(\u03b1) + (1\u2212\u03b1) 2 log( \u03c32 0 D ) + \u03b1 2 log( \u03c32 1 D ) if D < \u03c32 0 H2(\u03b1) + \u03b1 2 log( \u03b1\u03c32 1 D\u2212(1\u2212\u03b1)\u03c32 0 ) if \u03c32 0 < D \u2264(1 \u2212\u03b1)\u03c32 0 + \u03b1\u03c32 1 For a strict sparsity model we have \u03c32 0 \u21920 we have Rmix(D) = H2(\u03b1) + \u03b1 2 log( \u03b1\u03c32 1 D ) if 0 < D \u2264\u03b1\u03c32 1"
+                },
+                {
+                    "url": "http://arxiv.org/abs/0704.3434v3",
+                    "title": "On sensing capacity of sensor networks for the class of linear observation, fixed SNR models",
+                    "abstract": "In this paper we address the problem of finding the sensing capacity of\nsensor networks for a class of linear observation models and a fixed SNR\nregime. Sensing capacity is defined as the maximum number of signal dimensions\nreliably identified per sensor observation. In this context sparsity of the\nphenomena is a key feature that determines sensing capacity. Precluding the SNR\nof the environment the effect of sparsity on the number of measurements\nrequired for accurate reconstruction of a sparse phenomena has been widely\ndealt with under compressed sensing. Nevertheless the development there was\nmotivated from an algorithmic perspective. In this paper our aim is to derive\nthese bounds in an information theoretic set-up and thus provide algorithm\nindependent conditions for reliable reconstruction of sparse signals. In this\ndirection we first generalize the Fano's inequality and provide lower bounds to\nthe probability of error in reconstruction subject to an arbitrary distortion\ncriteria. Using these lower bounds to the probability of error, we derive upper\nbounds to sensing capacity and show that for fixed SNR regime sensing capacity\ngoes down to zero as sparsity goes down to zero. This means that\ndisproportionately more sensors are required to monitor very sparse events. Our\nnext main contribution is that we show the effect of sensing diversity on\nsensing capacity, an effect that has not been considered before. Sensing\ndiversity is related to the effective \\emph{coverage} of a sensor with respect\nto the field. In this direction we show the following results (a) Sensing\ncapacity goes down as sensing diversity per sensor goes down; (b) Random\nsampling (coverage) of the field by sensors is better than contiguous location\nsampling (coverage).",
+                    "authors": "Shuchin Aeron, Manqi Zhao, Venkatesh Saligrama",
+                    "published": "2007-04-25",
+                    "updated": "2007-06-04",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION In this paper we study fundamental limits to the performance of sensor networks for a class of linear sensing models under a \ufb01xed SNR regime. Fixed SNR is an important and necessary ingredient for sensor network applications where the observations are inevitably corrupted by external noise and clutter. In addition we are motivated by sensor network applications where the underlying phenomena exhibits sparsity. Sparsity is manifested in many applications for which sensor networks are deployed, e.g. localization of few targets in a large region, search for targets from among a large number of sites e.g. land mine detection, estimation of temperature variation for which few spline coef\ufb01cients may suf\ufb01ce to represent the \ufb01eld , i.e. phenomena is sparse under a suitable transformation. More recent applications such as that considered in [1] also involve imaging a sparse scattering medium. The motivation for considering linear sensing models comes from the fact that in most cases the observation at a sensor is a superposition of signals that emanate from different sources, locations etc. For e.g., in seismic and underground borehole sonic applications, each sensor receives signals that is a superposition of signals arriving from various point/extended sources located at different places. In radar applications [1], [2], under a far \ufb01eld assumption the observation system is linear and can be expressed as a matrix of steering vectors. In this case the directions becomes the variable space and one looks for strategies to optimally search using many such radars. Statistical modulation of gain factors in different directions is feasible in these scenarios and is usually done to control the statistics of backscattered data. In other scenarios the scattering medium itself induces random gain factors in different directions. In relation to signal sparsity compressive sampling, [3], [4] has shown to be very promising in terms of acquiring minimal information, which is expressed as minimal number of random projections, that suf\ufb01ces for adequate reconstruction of sparse signals. Thus in this case too, the observation model is linear. In [5] this set-up was used in a sensor network application for realizing ef\ufb01cient sensing and information distribution system by combining with ideas from linear network coding. Also it was used in [6] to build a wireless sensor network architecture using a distributed source-channel matched communication scheme. For applications related to wireless sensor networks where power limited sensors are deployed, it becomes necessary to compress the data at each sensor. For e.g. consider a parking surveillance system where a network of wireless low resolution cameras are deployed, [7]. With each camera taking several October 27, 2018 DRAFT \f3 Fig. 1. A schematic of I-Park: a parking lot monitoring system. snapshots in space and transmitting all of them to a base station will overwhelm the wireless link to the base station. Instead transmission overhead is signi\ufb01cantly reduced by sending a weighted sum of the observations. An illustration is shown in \ufb01gure 1. A similar set-up was also considered in [8] for a robotic exploration scenario. Motivated by the scenarios considered above we start with sensing (observation) models where at a sensor the information about the signal is acquired as a projection of the signal onto a weight vector. Under this class of observation model, the sensing model is linear and is essentially a matrix, G \u2208Rm\u00d7n chosen from some appropriate class particular to the application. In this work we consider a \ufb01xed SNR model (see also [9]) where the observations at m sensors for the signal X \u2208X n are given by, Y = \u221a SNR GX + N (1) where each row of the matrix G is restricted to have a unit \u21132 norm and where N is the noise vector with unit noise power in each dimension. It is important to consider \ufb01xed SNR scenario particularly for applications related to sensor networks. Practically each sensor is power limited. In an active sensing scenario the sensors distribute this power to sense different modalities, or to look (beamform) in various directions. Thus we restrict the \u21132 norm of each row of G to be unity and then scale the system model October 27, 2018 DRAFT \f4 appropriately by SNR. For a networked setting we assume that the observations made at the sensors are available for processing at a centralized location or node. In case when this is infeasible or costly, information can be exchanged or aggregated at each sensor using distributed consensus type algorithms, such as that studied in [10]. In order utilize the information theoretic ideas and tools, we adopt a Bayesian perspective and assume a prior distribution on X. Another motivation for considering a Bayesian set-up is that one can potentially model classi\ufb01cation/detection scenarios where prior information is usually available and is useful. Note that under some technical conditions it can be shown that a lower bound to the Bayesian error is also lower bound to worst case probability of error for the parametric set-up. Therefore the lower bounds presented in this paper also provide lower bounds to the parameter estimation problem. In this paper we capture the system performance via evaluating asymptotic upper and lower bounds to the ratio C(d0) = n m such that reconstruction to within a distortion level d0 is feasible. We call the ratio C(d0) as sensing capacity : the number of signal dimensions reliably identi\ufb01ed per projection (sensor). This term was coined in [11] in the context of sensor networks for discrete applications. Alternatively, bounds to C(d0) can be interpreted as providing scaling laws for the minimal number of sensors/projections required for reliable monitoring/signal reconstruction. For a signal sparsity level of k, a different ratio of k m also seems to be a reasonable choice, but in most cases k is unknown and needs to be determined, e.g., target density, or sparsest signal reconstruction. Here it is important to penalize false alarms, misclassi\ufb01cation costs. Furthermore, n and m are known and part of the problem speci\ufb01cation, while signal complexity is governed by k, and one of our goals is to understand performance as a function of signal complexity. In this paper we show that sensing capacity C(d0) is also a function of signal sparsity apart from SNR. The upper bounds to C(d0) are derived via \ufb01nding lower bounds to the probability of error in reconstruction subject to a distortion criteria, that apply to any algorithm used for reconstruction. The achievable (lower) bounds to C(d0) are derived via upper bounding the probability of error in a max-likelihood detection set-up over the set of rate distortion quantization points. Since most of the development for these classes of problems has been algorithmic, [3], [9], our motivation for the above development is driven by the need to \ufb01nd fundamental algorithm independent bounds for these classes of problems. In particular, under an i.i.d model on the components of X that models a priori information, e.g. sparsity of X, and letting \u02c6 X(Y) denote the reconstruction of X from Y, then we show that, October 27, 2018 DRAFT \f5 Pr \u0012 1 nd( \u02c6 X(Y), X) \u2265d0 \u0013 \u2265RX(d0) \u2212K(d0, n) \u22121 nI(X; Y|G) RX(d0) \u2212o(1) (2) for some appropriate distortion measure d(., .) and where RX(d0) is the corresponding scalar rate distortion function; K(n, d0) is bounded by a constant and it depends on the number of neighbors of a quantization point in an optimal n\u2212dimensional rate distortion mapping. Next, we consider the effect of structure of G on the performance. Using the result on the lower bound on the probability of error given by equation (2), a necessary condition is immediately identi\ufb01ed in order that the reconstruction to within an average distortion level d0 is feasible, which is, RX(d0)\u2212K(n, d0) \u2264 1 nI(X; Y|G). For a \ufb01xed prior on X the performance is then determined by the mutual information term that in turn depends on G. This motivates us to consider the effect of the structure of G on the performance and via evaluation of I(X; Y|G) for various ensembles of G we quantify the performance of many different scenarios that restrict the choice of G for sensing. Under the case when G is chosen independently of X and randomly from an ensemble of matrices (to be speci\ufb01ed later in the problem set-up), we have I(X; Y, G) = I(X; G) | {z } =0 + I(X; Y|G) (3) = I(X; Y) + I(X; G|Y) (4) \u21d2I(X; Y|G) = I(X; Y) + I(X; G|Y) (5) This way of expanding allow us to isolate the effect of structure of the sensing matrix G on the performance which in principle in\ufb02uences bounds on C(d0) through the change in mutual information as captured via the equations 3-5 and as applied to satisfy the necessary conditions prescribed by the lower bound in equation (2). Using the above idea, in this paper we will show the effect of sensing diversity on the performance, a concept which is explained next. Under the sensing model as prescribed above, at each sensor one can relate each component of the corresponding projection vector as contributing towards diversity in sensing. The total number of non-zero components in the projection vector is called sensing diversity. This terminology is analogous to that used in MIMO systems in the context of communications. As will be shown later on that loss in sensing capacity is not very signi\ufb01cant at reasonable levels of sensing diversity (with randomization in sampling per sensor). In fact there is a saturation effect that comes into play, which implies that most of the gains can be obtained at diversity factor close to 0.5. Now if one October 27, 2018 DRAFT \f6 considers the noiseless case, i.e. Y = GX, then it was shown in [3] that for some m and for some sparsity k as a function of n and the coherence of the sensing matrix, an \u21131 optimization problem : min ||X||1 subject to : Y = GX, X \u22650 yields exact solution. To this end note that if G is sparse then solving the above system is computationally faster as is shown in [12]. There are other types of modalities that arise in the context of resource constrained sensor networks. As an example consider the application in [7] where each camera may be physically restricted to sample contiguous locations in space or under limited memory it is restricted to sample few locations, possibly at random. This motivates us to consider other structures on G under such modalities of operation. In this paper we will contrast random sampling and contiguous sampling and show that random sampling is better than contiguous sampling. In such scenarios it becomes important to address a coverage question and in some cases may lead to a poor performance. In highly resource constrained scenarios randomization in elements of G is not feasible. In this direction we also consider an ensemble of {0, 1} matrices, with and without randomization in the locations of non-zero entries in each row. To facilitate the reading of the paper we itemize the organization as follows. 1. We present the problem set-up in section II where we make precise the signal models and the ensembles of sensing matrices that will be considered in relation to different sensor networking scenarios. 2. In section III we will present the lower bounds to the probability of error in reconstruction subject to an average distortion criteria. The development is fairly general and is self-contained. 3. In section IV we will present a constructive upper bound to the probability of error in reconstruction subject to an average \u21132 distortion criteria. The development there is particular to the \ufb01xed SNR linear sensing model that is the subject of the present paper, though the ideas are in general applicable to other sensing models and to other classes of distortion measures. 4. Once we establish the upper and lower bounds, we will use the results to obtain upper and lower bounds to sensing capacity for the \ufb01xed SNR linear sensing models, in sections V and VI. In these sections we will consider the full diversity Gaussian ensemble for sensing matrix. The motivation to consider this model is that the mutual information and moment generating functions are easier to evaluate for the Gaussian ensemble. This is thus useful to gain initial insights into the tradeoffs of signal sparsity and SNR. October 27, 2018 DRAFT \f7 5. Since the bounds to sensing capacity can be interpreted as providing bounds for number of projections/sensors for reliable monitoring, in section VII we will compare the scaling implied by bounds to sensing capacity to that obtained in [9] in the context of complexity penalized regularization framework. 6. In section VIII we consider the effect of the structure of the sensing matrix G on sensing capacity. The section is divided into several subsections. We begin by considering the effect of sensing diversity on sensing capacity. Following that we consider the effect of correlation in the columns of G on achievable sensing capacity. Then we consider a very general case of a deterministic sensing matrix and via upper bounding the mutual information we comment on the performance of various types of sensing architectures of interest. 7. In section IX we consider the {0, 1} ensemble for sensing matrices and provide upper bounds to sensing capacity for various modalities in sensing. 8. In section X we give an example of how our methods can be extended to handle cases when one is interested in reconstruction of functions of X rather than X itself. In this direction we will consider the case of recovery of sign patterns of X. II. PROBLEM SET-UP Assume that the underlying signal X lies in an n-dimensional space X n, where X can be discrete or continuous. Discrete X models scenarios of detection or classi\ufb01cation and continuous X models scenarios of estimation. a) Fixed SNR model: : The observation model for the sensors is a linear observation model and is given by, Y = \u221a SNR GX + N (6) which is the \ufb01xed SNR model as described in the introduction. The matrix G \u2208Rm\u00d7n is a random matrix selected from an ensemble which we will state subsequently. For all m, n each row of G is restricted to have a unit \u21132 norm. The noise vector N is i.i.d. Gaussian unit variance in each dimension. A. Discussion about \ufb01xed SNR model At this point it is important to bring out an important distinction of the assumption and subsequently analysis of a \ufb01xed SNR model in contrast to similar scenarios considered but in albeit high SNR setting. October 27, 2018 DRAFT \f8 The observation model of equation 1 studied in this paper is related to a class of problems that have been central in statistics. In particular it is related to the problem of regression for model order selection. In this context the subsets of columns of the sensing matrix G form a model for signal representation which needs to be estimated from the given set of observations. The nature selects this subset in a weighted/non-weighted way as modeled by X. The task is then to estimate this model order and thus X. In other words estimate of X in most cases is also linked to the estimate of the model order under some mild assumptions on G. Several representative papers in this direction are [13], [14], [15] that consider the performance of several (signal) complexity penalized estimators in both parametric and nonparametric framework. One of the key differences to note here is that the analysis of these algorithms is done for the case when SNR \u2192\u221e, i.e. in the limit of high SNR which is re\ufb02ected by taking the additive noise variance to go to zero or not considering the noise at all. However SNR is an important and necessary ingredient for applications related to sensor networks and therefore we will not pursue a high SNR development here. Nevertheless the results obtained are directly applicable to such scenarios. In the next section we will \ufb01rst outline prior distribution(s) on X, that re\ufb02ect the sparsity of the signal X and the model for realizing sensing diversity in the sensing matrix G. Then we will outline the choices of ensembles for the sensing matrix G. In the following N(m, \u03c32) denotes the Gaussian distribution with mean m and variance \u03c32. B. Generative models of signal sparsity and sensing diversity b) Signal sparsity: In a Bayesian set-up we model the sparsity of the phenomena by assuming a mixture distribution on the signals X. In particular the n dimensional vector X = X1, ..., Xn is a sequence drawn i.i.d from a mixture distribution PX = \u03b1N(m1, \u03c32 1) + (1 \u2212\u03b1)N(m0, \u03c32 0) where \u03b1 \u22641 2. In this paper we consider two cases. 1) Discrete Case: m1 = 1 and m0 = 0 and \u03c31 = \u03c30 = 0. This means that X is a Bernoulli(\u03b1) sequence. This models the discrete case for addressing problems of target localization, search, etc. 2) Continuous Case: m1 = m2 = 0 but \u03c32 1 = 1 and \u03c32 0 = 0. This models the continuous case. In this context we call \u03b1 the sparsity ratio which is held \ufb01xed for all values of n. Under the above model, on an average the signal will be k sparse where k = \u03b1n. Note that k \u2192\u221eas n \u2192\u221e. October 27, 2018 DRAFT \f9 c) Sensing diversity and ensemble for G: In connection to the model for diversity, the sensing matrix G is random matrix such that for each row i, Gij, j = 1, 2, .., n are distributed i.i.d according to a mixture distribution, (1 \u2212\u03b2)N(m0, \u03c32 0) + \u03b2N(m1, \u03c32 1). We consider three cases: 1) Gaussian ensemble: m1 = m0 = 0 and \u03c31 = 1; \u03c30 = 0 2) Deterministic G: The matrix G is deterministic. 3) {0, 1}m\u00d7n ensemble: m1 = 1; m0 = 0 and \u03c31 = \u03c30 = 0. The matrix is then normalized so that each row has a unit \u21132 norm. In this context we call \u03b2 as the (sensing) diversity ratio. Under the above model, on an average each sensor will have a diversity of l = \u03b2n. Note that l \u2192\u221eas n \u2192\u221e. Given the set-up as described above the problem is to \ufb01nd upper and lower bounds to C(d0) = lim sup \u001a n m : Pr \u0012 1 nd( \u02c6 X(Y), X) > d0 \u0013 \u21920 \u001b where \u02c6 X(Y) is the reconstruction of X from observation Y and where d(X, \u02c6 X(Y) = Pn i=1 d(Xi, \u02c6 Xi(Y)) for some distortion measure d(., .) de\ufb01ned on X \u00d7X. In this paper we will consider Hamming distortion measure for discrete X and squared distortion measure for the continuous X. Under this set-up we exhibit the following main results: 1) Sensing capacity C(d0) is also a function of SNR, signal sparsity and sensing diversity. 2) For a \ufb01xed SNR sensing capacity goes to zero as sparsity goes to zero. 3) Low diversity implies low sensing capacity. 4) Correlations across the columns and across the rows of G leads to decrease in sensing capacity. 5) For the {0, 1} ensemble for sensing matrices, sensing capacity for random sampling is higher than for contiguous sampling. In the next section we will provide asymptotic lower bounds on the probability of error in reconstruction subject to a distortion criteria. Following that we will provide a constructive upper bound to the probability of error. We will then use these results to evaluate upper and lower bounds to sensing capacity. In the following we will use X and Xn interchangeably. III. BOUNDS TO THE PERFORMANCE OF ESTIMATION ALGORITHMS: LOWER BOUNDS Lemma 3.1: Given observation(s) Y for the sequence Xn \u225c{X1, ..., Xn} of random variables drawn i.i.d. according to PX. Let \u02c6 Xn(Y) be the reconstruction of Xn from Y. Also is given a distortion measure d(Xn, \u02c6 Xn(Y)) = Pn i=1 d(Xi, \u02c6 Xi(Y)) then, October 27, 2018 DRAFT \f10 Pr \u0010 1 nd( \u02c6 Xn(Y), Xn) \u2265d0 \u0011 \u2265RX(d0) \u2212K(d0, n) \u22121 nI(Xn; Y) RX(d0) \u2212o(1) where K(d0, n) is bounded by a constant and where RX(d0) is the corresponding (scalar) rate distortion function for X. Proof: See Appendix. Essentially, K(n, d0) = 1 n \u00d7 log(\u266fneighbors of a quantization point in an optimal n-dimensional ratedistortion mapping). NOTE: The assumption of a scalar valued process in lemma 3.1 is taken for the sake of simplicity. The results are easily generalizable and can be extended to the case of vector valued processes. For the simpler case of discrete parameter space, the lower bound to the minimax error in a parameter estimation framework is related to the Bayesian error as follows, min \u02c6 X(Y) max X\u2208\u0398 Pr \u0012 1 nd(X, \u02c6 X(Y )) \u2265d0 \u0013 = min \u02c6 X(Y) max P\u0398\u2208P\u03b8 X X\u2208\u0398 P(X)Pr \u0012 1 nd(X, \u02c6 X(Y )) \u2265d0 \u0013 (7) \u2265 min \u02c6 X(Y) X X\u2208\u0398 \u03c0(X)Pr \u0012 1 nd(X, \u02c6 X(Y )) \u2265d0 \u0013 (8) where \u0398 is the parameter space and P\u0398 is the class of probability measures over \u0398 and \u03c0 \u2208P is any particular distribution. The above result holds true for the case of continuous parameter space under some mild technical conditions. Thus a lower bound to the probability of error as derived in this paper also puts a lower bound on the probability of error for the parametric set-up. In our set-up we will choose \u03c0 as a probability distribution that appropriately models the a priori information on X, e.g. signal sparsity. For modeling simple priors such as sparsity on X one can choose distributions that asymptotically put most of the mass uniformly over the relevant subset of \u0398 and is a key ingredient in realization of the lower bound on probability of error derived in this paper. We have the following corollary that follows from lemma 3.1. Corollary 3.1: Let Xn = X1, .., Xn be an i.i.d. sequence where each Xi is drawn according to some distribution PX(x) and Xn \u2208X n, where |X| is \ufb01nite. Given observation Y about Xn we have, Pr(Xn \u0338= \u02c6 Xn(Y)) \u2265H(X) \u22121 nI(Xn; Y) \u22121/n H(X) + o(1) \u2212o(1) October 27, 2018 DRAFT \f11 A. Tighter bounds for discrete X under hamming distortion The results in the previous section can be stated for any \ufb01nite n without resorting to the use of AEP for the case of discrete alphabets, with hamming distortion as the distortion measure and for certain values of the average distortion constraint d0. We have the following lemma. Lemma 3.2: Given observation(s) Y for the sequence Xn \u225c{X1, ..., Xn} of random variables drawn i.i.d. according to PX. Then for hamming under distortion measure dH(., .), for Xi \u2208X, |X| < \u221e and for distortion levels, d0 \u2264(|X| \u22121) minX\u2208X PX, Pr( 1 ndH(Xn, \u02c6 Xn(Y) \u2265d0)) \u2265 nRX(d0) \u2212I(Xn; Y) \u22121 n log(|X|) \u2212n (h(d0) + d0 log(|X| \u22121)) Proof: See Appendix. B. Comment on the proof technique The proof of lemma 3.1 closely follows the proof of Fano\u2019s inequality [16], where we start with a distortion error event based on 1 nd( \u02c6 X(Y), X) \u2265d0 and then evaluate conditional entropy of a ratedistortion mapping conditioned on the error event and the observation Y. To bound K(n, d0), we use results in [17] for the case of squared distortion measure. In relation to the lower bounds presented in this paper for the probability of reconstruction subject to an average distortion level one such development was considered in [18] in the context of a non-parametric regression type problem. Let \u03b8 be an element of the metric space (d, \u0398). Then given {Yi, Gi}m i=1 for some random or non-random vectors Gi \u2208Rn and Yi being the responses to these vectors under \u03b8. Also is given the set of conditional pdfs given by p\u03b8(Gi)(Yi) where the notation means that that the pdfs are parametrized by \u03b8(Gi). The task is to \ufb01nd a lower bound on the minimax reconstruction distortion under measure d, in reconstruction of \u03b8 given Y and G. In our case one can identify X \u225c\u03b8 and \u0398 \u225cX n with squared metric d. For such a set-up lower bounds on the asymptotic minimax expected distortion in reconstruction (not the probability of such an event) was derived in [18] using a variation of Fano\u2019s bound (see [19]) under a suitable choice of worst case quantization for the parameter space \u0398 = {space of q-smooth functions in [0, 1]n} meterized with \u2113r, 1 \u2264r \u2264\u221edistance. Our derivation has a \ufb02avor of this method in terms of identifying the right quantization, namely the rate distortion quantization for a given level of average distortion in a Bayesian setting. Although we evaluate the lower bounds to the probability of error and not the expected distortion itself, the lower bound on the expected distortion in reconstruction follows immediately. Moreover our method works for October 27, 2018 DRAFT \f12 any distortion metric d, though in this paper we will restrict ourselves to cases of interest particular to sensor networks applications. IV. CONSTRUCTIVE UPPER BOUND TO THE PROBABILITY OF ERROR In this section we will provide a constructive upper bound to the probability of error in reconstruction subject to an average squared distortion level. Unlike the lower bounds in this section we will provide upper bounds for the particular observation model of equation (6). This could potentially be generalized but we will keep our focus on the problem at hand. To this end, given \u01eb > 0 and n, assume that we are given the functional mapping f(Xn) (or f(X)) that corresponds to the minimal cover at average distortion level d0 as given by lemma 11.2. Upon receiving the observation Y the aim is to map it to the index corresponding index f(X), i.e. we want to detect which distortion ball the true signal belongs to. Clearly if X is not typical there is an error. From lemma 11.1, the probability of this event can be bounded by an arbitrary \u03b4 > 0 for a large enough n. So we will not worry about this a-typical event in the following. Since all the sequences in the typical set are equiprobable, we covert the problem to a max-likelihood detection set-up over the set of rate-distortion quantization points given by the minimal cover as follows. Given G we and the rate distortion points corresponding to the functional mapping f(Xn), we enumerate the set of points, GZn i \u2208Rm. Then given the observation Y we map Y to the nearest point (in Rm) GZn i . Then we ask the following probability, Pr \u0010\u221a SNRGf(X) \u2192 \u221a SNRGf(X\u2032)|G, X \u2208Bi, X\u2032 \u2208Bj : 1 ndset(Bi, Bj) \u22652d0 \u0011 that is, we are asking what is the probability that the in typical max-likelihood detection set-up we will map signals from distortion ball Bi to signals in distortion ball Bj that is at an average set distance \u22652d0 from Bi, where dset(Bi, Bj) = minX\u2208Bi,X\u2032\u2208Bj d(X, X\u2032). For sake of brevity we denote the above probability via Pe(pair) to re\ufb02ect it as a pairwise error probability. Since the noise is additive Gaussian noise we have Pe(pair) = Pr \u0012 NT G(X \u2212X\u2032) \u22651 2 \u221a SNR||G(X \u2212X\u2032)||2 : X \u2208Bi, X\u2032 \u2208Bj \u0013 Pe(pair) = Pr   NT G(X \u2212X\u2032) ||G(X \u2212X\u2032)|| \u2265 \u221a SNR 2||G(X1 \u2212X2)||||G(X \u2212X\u2032)||2 : X \u2208Bi, X\u2032 \u2208Bj ! October 27, 2018 DRAFT \f13 Since noise N is AWGN noise with unit variance in each dimension, its projection onto the unit vector G(X\u2212X\u2032) ||G(X\u2212X\u2032)|| is also Gaussian with unit variance. Thus we have Pe(pair) = Pr   N \u2265 \u221a SNR 2 ||G(X \u2212X\u2032)|| : X \u2208Bi, X\u2032 \u2208Bj ! By a standard approximation to the Q(.) (error) function, we have that, Pe \u0012 f(X) \u2192f(X\u2032)|X \u2208Bi, X\u2032 \u2208Bj, G : 1 ndset(Bi, Bj) \u22652d0 \u0013 \u2264exp \u001a \u2212SNR||G(X \u2212X\u2032)||2 4 \u001b In the worst case we have the following bound, Pe \u0012 f(X) \u2192f(X\u2032)|X \u2208Bi, X\u2032 \u2208Bj, G : 1 ndset(Bi, Bj) \u22652d0 \u0013 \u2264exp \u001a \u2212 min X\u2208Bi,X\u2032\u2208Bj SNR||G(X \u2212X\u2032)||2 4 \u001b Now note that from above construction it implies that the average distortion in reconstruction of X is bounded by 2d0 if the distortion metric obeys triangle inequality. To evaluate the total probability of error we use the union bound to get, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u22652d0 \u0013 \u2264exp \u001a \u2212 min X\u2208Bi,X\u2032\u2208Bj SNR||G(X \u2212X\u2032)||2 4 \u001b 2n(RX(d0)\u2212K(n,d0)) We will use this general form and apply it to particular cases of ensembles of the sensing matrix G. In the following sections we begin by providing upper and lower bounds to the sensing capacity for the Gaussian ensemble for full diversity. V. SENSING CAPACITY: UPPER BOUNDS, GAUSSIAN ENSEMBLE A. Discrete X, full diversity, Gaussian ensemble For this case we have the following main lemma. Lemma 5.1: Given X \u2208{0, 1}n drawn Bernoulli (\u03b1, 1 \u2212\u03b1) and G chosen from the Gaussian ensemble. Then, with the distortion measure as the hamming distortion, for a diversity ratio of \u03b2 = 1 and for d0 \u2264\u03b1, the sensing capacity C is upper bounded by C(d0) \u2264 1 2 log(1 + \u03b1SNR) RX(d0) Proof: From lemma 3.2 the probability of error is lower bounded by zero if the numerator in the lower bound is negative, this implies for any m, n that October 27, 2018 DRAFT \f14 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 \u03b1 Sensing Capacity SNR increasing Fig. 2. The plot of sparsity versus upper bounds to the sensing capacity for various SNRs for the binary case (X = {0, 1}) for zero Hamming distortion. Cm,n(d0, G) \u2264 1 mI(X; Y|G) RX(d0) Since G is random we take expectation over G. It can be shown that the mutual information EGI(Xn; Y|G) \u2264 maxPX:P 1 nEX2 i \u2264\u03b1 1 2EG log det(Im\u00d7m + GXXT GT ) = E\u03bb1,..,\u03bbm Pm i=1 1 2 log(1 + \u03bbi\u03b1SNR) where \u03bbi are singular values of GGT . Since rows of G have a unit norm \u21d2\u03bbi \u22641 \u2200i. Hence EGI(Xn; Y|G) \u2264m 2 log(1 + \u03b1SNR). Thus the result follows. B. Continuous X, full diversity, Gaussian ensemble Lemma 5.2: Given X \u2208Rn drawn i.i.d. according to PX = \u03b1N(0, 1) + (1 \u2212\u03b1)N(0, 0) and G chosen from the Gaussian ensemble. Then, for squared distortion measure, for diversity ratio \u03b2 = 1 and for d0 \u2264\u03b1 2 , the sensing capacity C(d0) obeys, C(d0) \u2264 1 2 log(1 + \u03b1SNR) H(\u03b1) + \u03b1 2 log \u03b1 2d0 Proof: From lemma 5.1 we have that EGI(X; Y|G) \u2264 m 2 log(1 + \u03b1SNR). In order that the probability of error be lower bounded by zero, from lemma 3.1 it follows that asymptotically n m \u2264 EGI(X; Y|G) RX(d0) \u2212K(d0, n) October 27, 2018 DRAFT \f15 It can be shown that |K(d0, n) \u2212log 2| < \u01eb with \u01eb very small for large enough n, see e.g. [17]. The lemma then follows by plugging in the results from section XI-C. It can be easily seen that as \u03b1 \u21930 the sensing capacity goes to zero. We illustrate this by plotting the upper bounds in \ufb01gure 2 for the discrete case. We will revisit this phenomena in section VII in relation to the bounds derived in [5] in the context of compressed sensing. VI. SENSING CAPACITY: LOWER BOUNDS, GAUSSIAN ENSEMBLE A. Discrete alphabet, full diversity The discrete X with hamming distortion is a special case where we can provide tighter upper bounds. The proof follows from the development in section IV and identifying that for the discrete case one can choose the discrete set of points instead of the distortion balls. We have the following lemma. Lemma 6.1: Given X \u2208X n with |X| < \u221e, for \u03b2 = 1 and G chosen from a Gaussian ensemble. Then for d0 \u2264minx\u2208X PX(x), a sensing capacity of C(d0) = 1 2 log(1 + SNRd0 2 ) H(X) \u2212d0 log |X \u22121| \u2212d0 log 1 d0 is achievable in that the probability of error goes down to zero exponentially for choices of C = n m = C(d0) \u2212\u03b7 for any \u03b7 > 0. Proof: We have Pr \u0012 1 nd(X, \u02c6 X(Y)) \u2265d0|G \u0013 \u2264exp \u001a \u2212SNR||G(X \u2212X\u2032)||2 4 \u001b 2nH(X)\u2212nd0 log |X\u22121|\u2212log( n nd0) where we have applied the union bound to all the typical sequences that are outside the hamming distortion ball of radius d0. Taking the expectation with respect to G we get, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u2265d0 \u0013 \u2264EG exp \u001a \u2212SNR||G(X \u2212X\u2032)||2 4 \u001b 2nH(X)\u2212nd0 log |X\u22121|\u2212log( n nd0) Now note that since G is a Gaussian random matrix where each row has a unit \u21132 norm, ||G(X \u2212 X\u2032)||2 = Pm i=1 | Pn j=1 Gij(Xi \u2212X\u2032 j)|2 is a sum of m independent \u03c72 random variables with mean ||X \u2212X\u2032||2. Thus from the moment generating function of the \u03c72 random variable we get that, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u2265d0 \u0013 \u2264   1 1 + SNR||X\u2212X\u2032||2 2 n !m/2 2nH(X)\u2212nd0 log |X\u22121|\u2212log( n nd0) This implies, October 27, 2018 DRAFT \f16 Pr \u0012 1 nd(X, \u02c6 X(Y)) \u2265d0 \u0013 \u22642\u2212m 2 log(1+ SNRd0 2 )2nH(X)\u2212nd0 log |X\u22121|\u2212log( n nd0) Now note that for d0 \u2264\u03b1, log \u0000 n nd0 \u0001 \u2265nd0 log 1 d0 . Then from above one can see that the probability of error goes down to zero if, n m < 1 2 log(1 + SNRd0 2 ) H(X) \u2212d0 log |X \u22121| \u2212d0 log 1 d0 Thus a sensing capacity of C(d0) = 1 2 log(1 + SNRd0 2 ) H(X) \u2212d0 log |X \u22121| \u2212d0 log 1 d0 is achievable in that the probability of error goes down to zero exponentially for choices of C = n m = C(d0) \u2212\u03b7 for any \u03b7 > 0. B. Continuous X, full diversity Lemma 6.2: [Weak Achievability] For X \u2208Rn and drawn i.i.d. according to Px(X), G chosen from the Gaussian ensemble and \u03b2 = 1, a sensing capacity of C(2d0) = 1 2 log(1 + d0SNR) RX(d0) \u2212K(n, d0) is achievable in that the probability of error goes down to zero exponentially with n for C = n m \u2264 C(2d0) \u2212\u01eb for some arbitrary \u01eb > 0. Proof: For this case we invoke the construction as outlined in section IV. From the results in that section we get that, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u22652d0 \u0013 \u2264exp \u001a \u2212 min X\u2208Bi,X\u2032\u2208Bj SNR||G(X \u2212X\u2032)||2 4 \u001b 2n(RX(d0)\u2212K(n,d0)) Note that the result is little weaker in that guarantees are only provided to reconstruction within d0, but one can appropriately modify the rate distortion codebook to get the desired average distortion level. Proceeding as in the case of discrete X and , by taking the expectation over G and noting that minX\u2208Bi,X\u2032\u2208Bj ||X \u2212X\u2032||2 \u22652nd0, we get that, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u22652d0 \u0013 \u2264 \u0012 1 1 + SNRd0 \u0013m/2 2n(RX(d0)\u2212K(n,d0)) October 27, 2018 DRAFT \f17 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 distortion Sensing Capacity \u03b1 = 0.3 SNR = 10 upper bound lower bound 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 distortion Sensing Capacity \u03b1 = 0.3 SNR = 10 upper bound lower bound (a) (b) Fig. 3. (a) Plots of upper and lower bounds to sensing capacity for the Gaussian mixture model. (b) Plots of upper and lower bounds for sensing capacity for the Bernoulli model. The distortion on the x-axis is mean squared distortion for the Gaussian case and hamming distortion for the Bernoulli case. Note that zero distortion achievable sensing capacity is zero and there is an SNR gap in the upper and lower bounds. This implies, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u22652d0 \u0013 \u2264 \u0012 1 1 + SNRd0 \u0013m/2 2n(RX(d0)\u2212K(n,d0)) Pr \u0012 1 nd(X, \u02c6 X(Y)) \u22652d0 \u0013 \u22642\u2212m 2 log(1+SNRd0)2n(RX(d0)\u2212K(n,d0)) This implies that for n m < 1 2 log(1 + d0SNR) RX(d0) \u2212K(n, d0) the probability of error goes to zero exponentially. This means that a sensing capacity of C(2d0) = 1 2 log(1 + d0SNR) RX(d0) \u2212K(n, d0) is achievable in that the probability of error goes down to zero exponentially with n for C = n m \u2264 C(d0) \u2212\u03b7 for some arbitrary \u03b7 > 0. A plot of upper and lower bounds are shown in \ufb01gure 3. October 27, 2018 DRAFT \f18 VII. COMPARISON WITH EXISTING BOUNDS Note that the results in this paper are stated for d0 \u2264\u03b1 for the discrete case and for d0 \u2264\u03b1 2 for the continuous case. This is because one must consider stricter average distortion measures as the phenomena becomes sparser. To bring out this point concretely and for purposes of comparison with existing bounds, we consider the result obtained in [5] based on optimal complexity regularized estimation framework. They show that the expected mean squared error in reconstruction is upper bounded by, E \u0014 1 n||X \u2212\u02c6 X||2 \u0015 \u2264C1C2 k log n m (9) where C1 \u223c1 and C2 \u223c50(P + \u03c3)2 {(1 + p) log 2 + 4}, under normalization of the signal and the noise power and p is the number of quantization levels, [9]. To this end consider an extremely sparse case, i.e., k = 1. Then the average distortion metric in equation 9, does not adequately capture the performance, as one can always declare all zeros to be the estimated vector and the distortion then is upper bounded by O( 1 n). Consider the case when X is extremely sparse, i.e. \u03b1 \u21930 as 1 n. Then a right comparison is to evaluate the average distortion per number of non-zero elements, E h 1 \u03b1n||X \u2212\u02c6 X||2i . Using this as the performance metric we have from equation 9, E \u0014 1 \u03b1n||X \u2212\u02c6 X||2 \u0015 \u2264C1C2 n log n m (10) When \u03b1 is small then the average number of projections required such that the per non-zero element distortion is bounded by a constant, scales as O(n log n). This is indeed consistent with our results, in that the Sensing Capacity goes down to zero as 1 log n. X is sparse, i.e. \u03b1 < 1 but not very small. From results on achievable sensing capacity we have that Pr \u0010 1 n||X \u2212\u02c6 X||2 \u2265d0 \u0011 \u2264\u2212m 2 log(1 + d0SNR/2) + n(RX(d0) \u2212K(n, d0)) In order to compare the results we \ufb01x, performance guarantee of Pr(d(X, \u02c6 X) \u2265d0) \u2264\u01eb for a given \u01eb > 0, we have for the minimal number of projections required that, m \u22652 (log(1/\u01eb) + n(RX(d0) \u2212K(n, d0))) log(1 + d0SNR/2) from our results. From results in [9] it follows that, October 27, 2018 DRAFT \f19 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 \u03b1 number of projections (log 10 ) scale n = 10000 Maximum Likelihood Decoding Complexity Regularized Estimator   Fig. 4. The difference in scaling of the number of projections with the sparsity rate from bounds derived from Sensing Capacity and from bounds obtained in [9]. Our bounds are sharper. m \u2265C1C2 \u03b1n log n d0\u01eb For the special case of binary alphabet we have the following scaling orders for the number of projections in both cases, from achievable sensing capacity we have m1 \u2265O(nH2(\u03b1)) and from results in [9] we have m2 \u2265O(\u03b1n log n). A plot of these orders as a function of \u03b1 for a \ufb01xed n is shown in \ufb01gure, 4. VIII. EFFECT OF STRUCTURE OF G In this section we will show that effect of structure of G on sensing capacity. This section is divided into several subsections and the discussion is self-contained. In section VIII-A we will show that for the Gaussian ensemble, the sensing capacity reduces for when diversity is low. Following that in section VIII-B we will show the effect of correlation across columns in the sensing matrix for the Gaussian ensemble on achievable sensing capacity. In section VIII-C we will present a general result for a generic sensing matrix G which will subsequently be used to highlight the effect of structures such as that induced via random \ufb01ltering using a FIR \ufb01lter with/without downsampling as considered in [20]. A. Effect of sensing diversity, Gaussian ensemble In order to show the effect of sensing diversity we evaluate the mutual information EGI(X; Y|G) using the intuition described in the introduction. To this end we have the following lemma. October 27, 2018 DRAFT \f20 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 \u03b1 Sensing Capacity SNR  = 10 0 0.2 0.4 0.6 0.8 1 0.7 0.8 0.9 1 1.1 1.2 1.3 Diversity ratio \u03b2 \u03b1 = 0.3, SNR = 10 \u03b2 =1 \u03b2 = 1/n Fig. 5. The gap between upper bounds to sensing capacity in very low diversity and full diversity for the binary alphabet case. Shown also is the Sensing Capacity as a function of diversity for \ufb01xed sparsity. Note the saturation effect with diversity ratio. Lemma 8.1: For a diversity ratio of \u03b2, with l = \u03b2n as the average diversity per sensor and an average sparsity level of k = \u03b1n , we have EGI(X; Y|G) \u2264m 2 Ej \u0014 log \u0012SNR l j + 1 \u0013\u0015 , (11) where the expectation is evaluated over the distribution Pr(j) = \u0000k j \u0001\u0000n\u2212k l\u2212j \u0001 \u0000n l \u0001 Proof: See Appendix. In the above lemma j plays the role of number of overlaps between the projection vector and the sparse signal. As the diversity reduces this overlap reduces and the mutual information decreases. We will illustrate this by considering the extreme case when \u03b2 \u2193with n as 1 n. For this case we have, I(X; Y|G) \u2264m 2 Ej h log \u0010 j SNR l + 1 \u0011i = m 2 [(1 \u2212\u03b1) log(SNR \u00b7 0 + 1) + \u03b1 log(SNR + 1)] = m\u03b1 2 log(1 + SNR) The effect is illustrated in \ufb01gure 5. Thus low sensing diversity implies low sensing capacity. October 27, 2018 DRAFT \f21 B. Effect of correlation in G on achievable sensing capacity In this section we will show that correlation in sensing matrix G reduces achievable capacity. Correlation in G can arise due to many physical reasons such as correlated scattering, correlation of gains across modalities in sensing which may arise due to the physical construction of the sensor. Naturally there can be direct relations between various phenomena that can lead to such correlation. This is captured by assuming that there is correlation across the columns of G. Consider the upper bound to the probability of error as derived in section IV, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u22652d0 \u0013 \u2264exp \u001a \u2212 min X\u2208Bi,X\u2032\u2208Bj SNR||G(X \u2212X\u2032)||2 4 \u001b 2n(RX(d0)\u2212K(n,d0)) In the above expression, the term SNR||G(X \u2212X\u2032)||2 = SNR n X i=1 | n X j=1 Gij(Xi \u2212X\u2032 j)|2 where Pn j=1 Gij(Xi \u2212X\u2032 j) for each i are independent Gaussian random variables with zero mean and variance given by\u2206T \u03a3Gi\u2206where \u2206is the vector \u2206= X \u2212X\u2032 and \u03a3Gi is the covariance matrix (symmetric and positive semi-de\ufb01nite) of the i-th row of G. By construction, we know that 1 n\u2206T \u2206\u22652d0 and note that in the worst case, min \u2206T \u02dc \u03a3Gi\u2206= \u03bbmin\u2206T \u2206 where \u03bbmin is the minimum eigenvalue of the normalized covariance matrix \u02dc \u03a3Gi. Proceeding in a manner similar to that in the proof of lemma 6.2 we have that, Pr \u0012 1 nd(X, \u02c6 X(Y)) \u22652d0 \u0013 \u2264 \u0012 1 1 + d0SNR\u03bbmin \u0013m/2 2n(RX(d0)\u2212K(n,d0)) From the above expression one can see that achievable sensing capacity falls in general, since \u03bbmin \u22641 as compared to the case when the elements of G are uncorrelated in which case \u03bbmin = 1 = \u03bbmax. C. Deterministic G In this section we will consider deterministic matrices G and provide upper bounds to sensing capacity for the general case. To this end denote the rows of G as Gi, i = 1, 2, . . . , m. Let the cross-correlations of these rows be denoted as: ri = GT i Gi+1 GT i Gi October 27, 2018 DRAFT \f22 As before to ensure the SNR, to be \ufb01xed we impose GT i Gi = 1 for all i. Then we have the following result: Lemma 8.2: For the generative models for the signal X as outlined in the problem set-up, an upper bound for the sensing capacity for a deterministic sensing matrix G \u2208Rm\u00d7n is given by: C(d0) \u2264 m\u22121 X i=1 log \u0010 1 + SNR\u03b1(1 \u2212ri) + ri\u03b1SNR \u03b1SNR+1(1 + \u03b1SNR(1 \u2212ri)) \u0011 RX(d0) \u2212K(n, d0) (12) Proof: We will evaluate I(X; Y|G) via the straightforward method, I(X; Y|G) = h(Y|G) \u2212h(Y|G, X) Note that h(Y|G, X) = h(N). Note that h(Y|G) \u2264h(Y) \u2264h(Y\u2217) where Y\u2217is a Gaussian random vector obtained via GX\u2217where X\u2217is now a Gaussian random vector with i.i.d components and with the same covariance as X under the generative model(s). We will now upper bound the entropy of Y via, h(Y) \u2264h(Y\u2217) \u2264h(Y \u2217 1 ) + m\u22121 X i=1 h(Y \u2217 i+1 | Y \u2217 i ) \u2264h(Y \u2217 1 ) + h(Y \u2217 i+1 \u2212\u03b7iY \u2217 i ) where \u03b7iY \u2217 i is the best MMSE estimate for Y \u2217 i+1. The MMSE estimate of Y \u2217 i+1 from Y \u2217 i is given by, \u02c6 Y \u2217 i+1 = \u03a3Y \u2217 i Y \u2217 i+1 \u03a3Y \u2217 i Y \u2217 i \u03a3Y \u2217 i Y \u2217 i+1 = ri\u03b1SNR and \u03a3Y \u2217 i = \u03b1SNR + 1. The result then follows by evaluating the MMSE error given by, E(Y \u2217 i+1 \u2212\u02c6 Y \u2217 i+1)2 = E \u0012 Y \u2217 i+1 \u2212 ri\u03b1SNR \u03b1SNR + 1Y \u2217 i \u00132 E \u0010 Y \u2217 i+1 \u2212ri\u03b1SNR \u03b1SNR+1Y \u2217 i \u00112 = \u03b1SNR + 1 + (ri\u03b1SNR)2 \u03b1SNR+1 \u22122(ri\u03b1SNR)2 \u03b1SNR+1 = 1 + \u03b1SNR(1 \u2212ri) + ri\u03b1SNR \u03b1SNR+1 (1 + (1 \u2212ri)\u03b1SNR) Plugging in the quantities the result follows. Let us see the implications of the above result for one particular type of sensing matrix architecture induced via a random \ufb01ltering and downsampling, considered in [20]. The output of the \ufb01lter of length L < n can be modeled via multiplication of X via a Toeplitz matrix (with a banded structure). The overlap between successive rows of the matrix G is L\u22121 in this case implying a large cross correlation ri. From lemma 12 it follows that larger cross correlation in rows implies poor sensing capacity. Also note that October 27, 2018 DRAFT \f23 Fig. 6. Illustration of random sampling Vs contiguous sampling in a sensor network. This leads to different structures on the sensing matrix and that leads to different performance. for a \ufb01ltering architecture one has to address a coverage issue wherein it is required that m > n \u2212L + 1. This implies that L > n \u2212m + 1. Thus the \ufb01lter length has to be suf\ufb01ciently large which implies that cross-correlation is also large. Indeed randomizing each row will lead to low cross-correlation (in an expected sense) but the coverage issue still needs to be addressed. On the other hand one can subsample the output signal of length n\u2212L+1 by some factor so as to reduce the cross correlation yet ensuring coverage. In this case the matrix almost becomes like a upper triangular matrix and there is a signi\ufb01cant loss of sensing diversity. A loose tradeoff between the \ufb01lter-length L and the sampling factor d (say) immediately follows from lemma 12 where the cross correlation changes according to ri = L(1 \u2212d) n IX. UPPER BOUNDS ON SENSING CAPACITY FOR {0, 1} ENSEMBLE The main motivation for considering this ensemble comes from scenarios where randomization in the elements of G is not feasible, e.g. \ufb01eld estimation from smoothed data. In this case each sensor measures a superposition of the signals that are in the sensing range of the sensor. This leads us to consider other October 27, 2018 DRAFT \f24 types of modalities, e.g. contiguous sampling of X by each sensor Vs random sampling for \u03b2 < 1. An illustration of the two types of sampling is shown in \ufb01gure 6. We reveal the following contrast for the two cases for same \u03b2 < 1 Lemma 9.1: Random Sampling: For the {0, 1} ensemble for sensing matrices consider the case when each row of G has \u03b2n ones randomly placed in n positions. Then for discrete X \u2208{0, 1}n drawn Bernoulli(\u03b1) and for d0 < \u03b1, Crand(d0) \u2264 H(J) h2(\u03b1) \u2212h2(d0) where H(.) is the discrete entropy function and where J is a random variable with distribution given by Pr(J = j) = \u0000\u03b1n j \u0001\u0000n(1\u2212\u03b1) \u03b2n\u2212j \u0001 \u0000 n \u03b2n \u0001 Proof: See Appendix. Lemma 9.2: Contiguous Sampling: For the {0, 1} ensemble for sensing matrices consider the case where each row of G has \u03b2n consecutive ones randomly placed with wrap around. Then for discrete X \u2208{0, 1}n drawn Bernoulli(\u03b1) and d0 < \u03b1, Ccontg.(d0) \u2264 h2(\u03b1 + \u03b2) h2(\u03b1) \u2212h2(d0) Proof: See Appendix. As seen the upper bound, Crand(d0) \u2265Ccontg.(d0). Thus randomization in G performs better. The difference is shown in \ufb01gure 7 for a low sparsity scenario. The proofs of the lemmas 9.1 and 9.2 follow from the upper bounds to the mutual information terms as provided in section XII and then applying the necessary conditions for the lower bound on the probability of error to be lower bounded by zero. X. ESTIMATION OF FUNCTIONS OF X The analysis of lower bounds to the probability of error presented in this paper extend in a straightforward way to estimation of functions of X. In this section we will consider one such scenario that has received attention in relation to problems arising in physics. The discussion below will reveal the power of the method presented in this work and it is easily capable of handling more complicated cases and scenarios, though the computation of the terms involved in the analysis may become hard. October 27, 2018 DRAFT \f25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 10 diversity ratio Upper bound to sensing capacity contiguous Vs randomized sampling, \u03b1 = 1/12     randomized sampling contiguous sampling Fig. 7. A comparison of the upper bounds to sensing capacity for the randomized sampling Vs contiguous sampling case. X is the Bernoulli model and the ensemble for G is the {0, 1} ensemble. We have selected the case of low sparsity in this case. Note that due to loose overbounding of mutual information (we basically got rid of noise) the upper bounds are greater than in the case of Gaussian ensemble. A. Detecting the sign pattern of X Of particular interest is to estimate the sign pattern of the underlying signal X. To this end de\ufb01ne a new random variable U, via Ui = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 if Xi > 0 \u22121 if Xi < 0 0 if Xi = 0 The corresponding n dimensional extension and probability distribution on U is induced directly via PX. In such a case note that U \u2192X \u2192Y \u2192\u02c6 U(Y) forms a Markov chain. To this end consider an error event de\ufb01ned via, E = \uf8f1 \uf8f2 \uf8f3 1 if U \u0338= \u02c6 U(Y) 0 otherwise Then we have, October 27, 2018 DRAFT \f26 H(U, E|Y) = H(E|Y) | {z } \u22641 + H(U|E, Y) = H(U|Y) + H(E|U, Y) | {z } =0 Thus we have H(U|Y) \u22641 + PeH(U|E = 1, Y) | {z } \u2264n log 3 + (1 \u2212Pe)H(U|E = 0, Y) | {z } =0 This implies, Pe \u2265H(U) \u2212I(U; Y|G) \u22121 n log 3 In order to evaluate the I(U; Y|G) we note that I(U, X; Y|G) = I(X; Y|G). This follows from , I(U, X; Y|G) = H(U, X) \u2212H(X, U|Y, G) = H(X) \u2212H(X|G, Y) \u2212H(U|G, Y, X) = I(X; Y|G). Thus I(U; Y|G) = I(X; Y|G)\u2212I(X; Y|G, U) and both these terms can be adequately bounded/evaluated. XI. APPENDIX A. Proof of lemma 3.1 Let Xn = {X1, ..., Xn} be an i.i.d. sequence where each variable Xi is distributed according to a distribution PX de\ufb01ned on the alphabet X. Denote PXn \u225c(PX)n the n-dimensional distribution induced by PX. Let the space X n be equipped with a distance measure d(., .) with the distance in n dimensions given by dn(Xn, Zn) = Pn k=1 d(Xk, Zk) for Xn, Zn \u2208X n. Given \u01eb > 0, there exist a set of points \b Zn 1 , ..., ZN\u01eb(n,d0) \t \u2282X n such that, PXn \uf8eb \uf8ed N\u01eb(n,d0) [ i=1 Bi \uf8f6 \uf8f8\u22651 \u2212\u01eb (13) where Bi \u225c \b Xn : 1 ndn(Xn, Zn i ) \u2264d0 \t , i.e., the d0 balls around the set of points cover the space X n in probability exceeding 1 \u2212\u01eb. Given such set of points there exists a function f(Xn) : Xn \u2192Zn i s.t. Pr \u0000 1 ndn(Xn, Zn i ) \u2264d0 \u0001 \u2265 1 \u2212\u01eb. To this end, let TPXn denote the set of \u03b4 typical sequences in X n that are typical PXn, i.e. TPXn = \u001a Xn : | \u22121 n log \u02c6 P(Xn) \u2212H(X)| \u2264\u03b4 \u001b October 27, 2018 DRAFT \f27 where \u02c6 P(Xn) is the empirical distribution induced by the sequence Xn. We have the following lemma from [21]. Lemma 11.1: For any \u03b7 > 0 there exists an n0 such that for all n \u2265n0, such that Pr \u0012 Xn : | \u22121 n log \u02c6 P(Xn) \u2212H(X)| < \u03b4 \u0013 > 1 \u2212\u03b7 In the following we choose \u03b7 = \u03b4. Given that there is an algorithm \u02c6 Xn(Y) that produces an estimate of Xn given the observation Y. To this end de\ufb01ne an error event on the algorithm as follows, En = \uf8f1 \uf8f2 \uf8f3 1 if 1 ndn(Xn, \u02c6 Xn(Y)) \u2265d0 0 otherwise De\ufb01ne another event An as follows An = \uf8f1 \uf8f2 \uf8f3 1 if Xn \u2208TPXn 0 otherwise Note that since Xn is drawn according to PXn and given \u03b4 > 0 we choose n0 such that conditions of lemma 11.1 are satis\ufb01ed. In the following we choose n \u2265n0(\u03b4). Then a priori, Pr(An = 1) \u2265(1 \u2212\u03b4). Now, consider the following expansion, H(f(Xn), En, An|Y) = H(f(Xn)|Y) + H(En, An|f(Xn), Y) = H(En, An|Y) + H(f(Xn)|En, An, Y) This implies that H(f(Xn)|Y) = H(En, An|Y) \u2212H(En, An|f(Xn), Y) + H(f(Xn)|En, An, Y) = I(En, An; f(Xn)|Y) + H(f(Xn)|En, An, Y) \u2264H(En, An) + H(f(Xn)|En, An, Y) \u2264H(En) + H(An) + H(f(Xn)|En, An, Y) Note that H(En) \u22641 and H(An) = \u03b4 log 1 \u03b4 + (1 \u2212\u03b4) log 1 1\u2212\u03b4 \u223c\u03b4. Thus we have H(f(Xn)|Y) \u22641 + \u03b4 + P n e H(f(Xn)|Y, En = 1, An) +(1 \u2212P n e )H(f(Xn)|Y, En = 0, An) October 27, 2018 DRAFT \f28 Now the term P n e H(f(Xn)|Y, En = 1, An) \u2264P n e log N\u01eb(n, d0). Note that the second term does not go to zero. For the second term we have that, (1 \u2212P n e )H(f(Xn)|Y, En = 0, An) = P(An = 1)(1 \u2212P n e )H(f(Xn)|Y, En = 0, An = 1) +P(An = 0)(1 \u2212P n e )H(f(Xn)|Y, En = 0, An = 0) \u2264(1 \u2212P n e )H(f(Xn)|Y, En = 0, An = 1) +\u03b4(1 \u2212P n e ) log (N\u01eb(n, d0)) The \ufb01rst term on R.H.S in the above inequality is bounded via, (1 \u2212P n e )H(f(Xn)|Y, En = 0, An = 1) \u2264(1 \u2212P n e ) log (|S|) where S is the set given by, S = n i : dset \u0010 Bf(Xn), Bi \u0011 \u2264d0 o where dset(S1, S2) = mins\u2208S1,s\u2032\u2208S2 dn(s, s\u2032) is the set distance between two sets. Now note that I(f(Xn); Xn) = H(f(Xn)) and H(f(Xn)|Y) = H(f(Xn))\u2212I(f(Xn); Xn) \u2265H(f(Xn))\u2212I(Xn; Y) where the second inequality follows from data processing inequality over the Markov chain f(Xn) \u2194 Xn \u2194Y. Thus we have, P n e \u2265 I(f(Xn); Xn) \u2212log |S| \u2212I(Xn; Y) \u22121 (1 \u2212\u03b4) log N\u01eb(n, d0) \u2212log |S| \u2212 \u03b4(1 + log N\u01eb(n, d0)) (1 \u2212\u03b4) log N\u01eb(n, d0) \u2212log |S| The above inequality is true for all the mappings f satisfying the distortion criteria for mapping Xn and for all choices of the set satisfying the covering condition given by 11.2. We now state the following lemma for a minimal covering, taken from [16]. Lemma 11.2: Given \u01eb > 0 and the distortion measure dn(., .), let N\u01eb(n, d0) be the minimal number of points Zn 1 , ..., Zn N\u01eb(n,d0) \u2282X n satisfying the covering condition, PXn \uf8eb \uf8ed N\u01eb(n,d0) [ i=1 Bi \uf8f6 \uf8f8\u22651 \u2212\u01eb Let N\u01eb(n, d0) be the minimal such number. Then, October 27, 2018 DRAFT \f29 lim sup n 1 nN\u01eb(n, d0) = RX(\u01eb, d0) where RX(\u01eb, d0) is the in\ufb01mum of the \u01ebachievable rates at distortion level d0. Note that lim\u01eb\u21930 RX(\u01eb, d0) = RX(d0) where RX(d0) = minp( \u02c6 X|X) I( \u02c6 X; X) subject to 1 nE(d(Xn, \u02c6 Xn)) \u2264 d0. In order to lower bound P n e we choose the mapping f(Xn) to correspond to the minimal cover. Also w.l.o.g we choose \u03b4 = \u01eb. We note the following. 1) From lemma 11.1, given \u01eb > 0, \u2203n0(\u01eb) such that for all n \u2265n0(\u01eb), we have Pr(TPXn) \u22651 \u2212\u01eb. 2) Given \u01eb > 0 and for all \u03b2 > 0, for the minimal cover we have from lemma 11.2 that \u2203n1(\u03b2) such that for all n \u2265n1(\u03b2), N\u01eb(n, d0) \u2264n(RX(\u01eb, d0) + \u03b2). 3) From the de\ufb01nition of the rate distortion function we have for the choice of the functions f(Xn) that satis\ufb01es the distortion criteria, I(f(Xn); Xn) \u2265nRX(\u01eb, d0). Therefore we have for n \u2265max(n0, n1), P n e \u2265 nRX(\u01eb, d0) \u2212log |S| \u2212I(Xn; Y) \u22121 (1 \u2212\u01eb)(n(RX(\u01eb, d0) + \u03b2) \u2212log |S| \u2212 \u01eb(1 + n(RX(\u01eb, d0) + \u03b2) (1 \u2212\u01eb)n(RX(\u01eb, d0) + \u03b2) \u2212log |S| Clearly, log |S| \u2264n 2 RX(\u01eb, d0). d) Limiting case: Since the choice of \u01eb, \u03b2 is arbitrary we can choose them to be arbitrary small. In fact we can choose \u01eb, \u03b2 \u21930. Also note that for every \u01eb > 0 and \u03b2 > 0 there exists n2(\u03b2) such that RX(d0) + \u03b2 \u2265RX(\u01eb, d0) \u2265RX(d0) \u2212\u03b2. Therefore for all n \u2265max(n0, n1, n2) in the limiting case when \u01eb, \u03b2 \u21930, we have Pe \u2265RX(d0) \u22121 n log |S| \u22121 nI(Xn; Y) RX(d0) \u22121 n log |S| \u2212o(1) This implies that Pe \u2265RX(d0) \u22121 n log |S| \u22121 nI(Xn; Y) RX(d0) \u2212o(1) The proof then follows by identifying K(n, d0) = 1 n log |S|, and is bounded above by a constant. October 27, 2018 DRAFT \f30 B. Proof of lemma 3.2 Proof: Given an observation Y about the event Xn. De\ufb01ne an error event, E = \uf8f1 \uf8f2 \uf8f3 1 if 1 ndH(Xn, \u02c6 Xn(Y)) \u2265d0 0 otherwise Expanding H(Xn, E|Y) in two different ways we get that, H(Xn|Y) \u22641 + nPe log(|X|) + (1 \u2212Pe)H(Xn|E = 0, Y) Now the term (1 \u2212Pe)H(Xn|E = 0, Y) \u2264(1 \u2212Pe) \u0000 n d0n \u0001 (|X| \u22121)nd0 \u2264n(1 \u2212Pe) (h(d0) + d0 log(|X| \u22121)) Then we have for the lower bound on the probability of error that, Pe \u2265H(Xn|Y) \u2212n (h(d0) + d0 log(|X| \u22121))) \u22121 n log(|X|) \u2212n (h(d0) + d0 log(|X| \u22121)) Since H(Xn|Y) = H(Xn) \u2212I(Xn; Y) we have Pe \u2265n (H(X) \u2212h(d0) \u2212d0 log(|X| \u22121)) \u2212I(Xn; Y) \u22121 n log(|X|) \u2212n (h(d0) + d0 log(|X| \u22121)) It is known that RX(d0) \u2265H(X) \u2212h(d0) \u2212d0 log(|X| \u22121), with equality iff d0 \u2264(|X| \u22121) min X\u2208X PX see e.g., [16]. Thus for those values of distortion we have for all n, Pe \u2265 nRX(d0) \u2212I(Xn; Y) \u22121 n log(|X|) \u2212n (h(d0) + d0 log(|X| \u22121)) C. Rate distortion function for the mixture Gaussian source under squared distortion measure It has been shown in [22] that the rate distortion function for a mixture of two Gaussian sources with variances given by \u03c31 with mixture ratio \u03b1 and \u03c30 with mixture ratio 1 \u2212\u03b1, is given by October 27, 2018 DRAFT \f31 Rmix(D) = \uf8f1 \uf8f2 \uf8f3 H(\u03b1) + (1\u2212\u03b1) 2 log(\u03c32 0 D ) + \u03b1 2 log(\u03c32 1 D ) if D < \u03c32 0 H(\u03b1) + \u03b1 2 log( \u03b1\u03c32 1 D\u2212(1\u2212\u03b1)\u03c32 0 ) if \u03c32 0 < D \u2264(1 \u2212\u03b1)\u03c32 0 + \u03b1\u03c32 1 For a strict sparsity model we have \u03c32 0 \u21920 we have that, Rmix(D) = H(\u03b1) + \u03b1 2 log(\u03b1\u03c32 1 D ) if 0 < D \u2264\u03b1\u03c32 1 D. Bounds on Mutual information In this section we will evaluate bounds on mutual information that will be useful in characterization of the Sensing Capacity. Given that the matrix G is chosen independently of X we expand the mutual information between X and Y, G in two different ways as follows \u2013 I(X; Y, G) = I(X; G) | {z } =0 + I(X; Y|G) = I(X; Y) + I(X; G|Y) This way of expanding gives us handle onto evaluating the mutual information with respect to the structure of the resulting sensing matrix G. From above we get that, I(X; Y|G) = I(X; Y) + I(X; G|Y) = h(Y) \u2212h(Y|X) + h(G|Y) \u2212h(G|X, Y) To this end we have the following lemma. Lemma 11.3: For a sparsity level of \u03b1 and diversity factor of \u03b2 = 1, I(X; Y|G) \u2264m 2 log(1 + \u03b1P N0 ) Proof: First note that, h(Y) \u2264m 2 log 2\u03c0e(N0 + \u03b1P) Since conditioned on X, Y is distributed with a Gaussian density we have, h(Y|X) = m 2 log 2\u03c0e   N0 + Pk i=1 X2 i P n ! h(G|Y) \u2264h(G) = mn 2 log \u0012 2\u03c0eP n \u0013 October 27, 2018 DRAFT \f32 Note also that conditioned on X and Y the G has a Gaussian distribution. Now note that, h(G|Y, X). First note that, rows of G are independent of each other given X and Y. So we can write, h(G|Y, X) = mh(g1|Y, X) where g1 is the \ufb01rst row of the matrix G. Since g is Gaussian one can \ufb01nd the residual entropy in terms of the residual MMSE error in estimation of g given X and Y. This error is given by \u2013 MMSEg1|Y,X = \u03a3g1|X \u2212\u03a3g1Y|X\u03a3\u22121 Y|X\u03a3T g1Y|X = \u03a3g1 \u2212\u03a3g1Y1|X\u03a3\u22121 Y1|X\u03a3T g1Y1|X The second equation follows from the fact that G is independent of X and given X the row g1 is independent of other observations, Y2, ..., Ym. First note that given X we also know which positions of X are zeros. So without lossof generality we can assume that the \ufb01rst k elements of X are non-zeros and the rest are zeros. Now note the following, \u03a3g1 = P n In \u03a3g1Y1|X = P n \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X1 . . . Xk 0n\u2212k \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 where 0n\u2212k is a column vector of n \u2212k zeros. \u03a3Y1|X = P n k X i=1 X2 i + N0 Therefore we have, h(g1|Y1, X) = 1 2 log(2\u03c0e)k det \u0010 P n Ik \u2212P n X1:k\u03a3\u22121 Y1|X P n XT 1:k \u0011 + n\u2212k 2 log 2\u03c0eP n Note that the second term on the R.H.S in the above equation corresponds to the entropy of those elements of the row g1 that have no correlation with Y, i.e. nothing can be inferred about these elements since they overlap with zero elements of X. Now, using the equation det(I + AB) = det(I + BA), we have that October 27, 2018 DRAFT \f33 h(g1|Y1, X) = 1 2 log(2\u03c0eP n )k det \u0010 1 \u2212XT 1:k\u03a3\u22121 Y1|X P n X1:k \u0011 = 1 2 log \u0010 (2\u03c0eP n )k N0 P n Pk i=1 X2 i +N0 \u0011 Plugging in all the expressions we get a lower bound on the mutual information I(X; Y|G) I(X; Y|G) \u2264m 2 log(1 + \u03b1P N0 ) In contrast to the upper bound derived in the proof of lemmas 5.1 and 5.2, this alternate derivation provides a handle to understand the effect of the structure of G on the mutual information when one is not allowed to pick a maximizing input distribution on X. Moreover the above derivation can potentially handle scenarios of correlated G. Below we will use the above result in order to prove lemma 8.1. E. Proof of lemma 8.1 To this end let l = \u03b2n and is \ufb01xed, i.e. there are only l non-zero terms in each row of matrix G. We have h(G) = ml 2 log 2\u03c0eP l + mh2(\u03b2) Now we will \ufb01rst evaluate h(G|Y, X). Proceeding as in derivation of lemma 11.3, we have that, h(G|X, Y) = mh(g1|Y1, X) + mh2(\u03b2) where one can see that if the matrix G is chosen from a Gaussian ensemble then given X and Y it tells nothing about the positions of the non-zeros in each row. Hence the additive term h2(\u03b2) appears in both terms and is thus canceled in the overall calculations. So we will omit this term in the subsequent calculations. To this end, let j denote the number of overlaps of the vector g1 and the k-sparse vector X. Given Y1 and X one can only infer something about those elements of G that contribute to Y1. Given the number of overlaps j we then have h(g1|X, Y1, j) = l\u2212j 2 log 2\u03c0eP l + 1 2 log \u0010 (2\u03c0eP l )j N0 P l Pj i=1 X2 j+N0 \u0011 where we have assumed without loss of generality that the \ufb01rst j elements of X are non-zero and overlap with elements of the \ufb01rst row. Now note that, h(Y|j) \u2264m 2 log 2\u03c0e(Pj l + N0) October 27, 2018 DRAFT \f34 h(Y|X, j) = m 2 log 2\u03c0e   P l j X i=1 X2 i + N0 ! From above we have that, I(X; Y|G, j) = m 2 log(1 + jP lN0 ) Taking the expectation with respect to the variable j we have, I(X; Y|G) = m 2 Ej log(1 + jP lN0 ) Note that j \u2264min {k, l} and has a distribution given by, Pr(j) = \u0000k j \u0001\u0000n\u2212k l\u2212j \u0001 \u0000n l \u0001 XII. UPPER BOUNDS TO MUTUAL INFORMATION FOR {0, 1} ENSEMBLE In this section we will derive upper bounds to the mutual information I(X; Y|G) for the case when the matrix is chosen from a {0, 1} ensemble. First it is easily seen that for this ensemble a full diversity leads to loss of rank and thus the mutual information is close to zero. So we will only consider the case \u03b2 < 1. A. Random locations of 1\u2019s in G In this section we will provide simple upper bounds to the mutual information I(X; Y|G) for the case of {0, 1} ensemble of sensing matrices. Note that, I(X; Y|G) \u2264I(X; GX|G) Let \u02dc Y = GX. Then we have, I(X; \u02dc Y|G) = I(X; \u02dc Y) + I(X; G| \u02dc Y) Now note that 1 nI(X; \u02dc Y) = o(1). Then we need to evaluate I(G; X| \u02dc Y) \u2264H(G)\u2212H(G| \u02dc Y, X). Now note that since each row of G is an independent Bernoulli\u223c\u03b2 sequence we can split the entropy into sum of entropies each individual rows. To this end focus on the \ufb01rst row. Then conditioned on there being l 1\u2019s in the row we have, H(G1|l) \u2264 \u0000n l \u0001 . Given that X is k-sparse we have, October 27, 2018 DRAFT \f35 H(G1|X, \u02dc Y, l, k) = min(k,l) X j=0 \u0000k j \u0001\u0000n\u2212k l\u2212j \u0001 \u0000n l \u0001 log \u0012k j \u0013\u0012n \u2212k l \u2212j \u0013 Thus we have I(X; G| \u02dc Y, k, l) \u2264 \u0012n l \u0013 \u2212 min(k,l) X j=0 \u0000k j \u0001\u0000n\u2212k l\u2212j \u0001 \u0000n l \u0001 log \u0012k j \u0013\u0012n \u2212k l \u2212j \u0013 = H(J|k, l) where J is a random variable with distribution given by, Pr(J = j) = \u0000k j \u0001\u0000n\u2212k l\u2212j \u0001 \u0000n l \u0001 For large enough n, k = \u03b1n and l = \u03b2n w.h.p. Thus I(X; G|Y) \u2264H( \u02dc J), where \u02dc J has a limiting distribution given by, Pr( \u02dc J = j) = \u0000\u03b1n j \u0001\u0000n(1\u2212\u03b1) \u03b2n\u2212j \u0001 \u0000 n \u03b2n \u0001 In other words given \u01eb > 0 there exists an n0 such that for all n \u2265n0, sup j |PJ(j)\u2212P \u02dc J(j)| \u2264\u01eb and by continuity of the entropy function, [[16], pp. 33, Lemma 2.7], it follows that |H(J) \u2212H( \u02dc J)| \u2264\u2212\u01eb log \u01eb n B. Contiguous sampling In this case for each row we have H(G1) = log n. To evaluate H(G1|X, \u02dc Y), \ufb01x the number of ones in G1 to be equal to l and the number of non-zero elements in X to be equal to k. Now note that if \u02dc Y1 = 0 then there is no overlap in G1 and X. This means that the row of G can have contiguous ones in n \u2212k \u2212l positions equally likely. The probability of no overlap is n\u2212k\u2212l n . On the other hand if \u02dc Y1 > 0, then uncertainty in locations of ones in G1 reduces to log(k + l). The probability that Y > 0 is k+l n . Thus we have, I(G1; X| \u02dc Y) \u2264mH(O) where O is a binary random variable with distribution (1 \u2212k+l n ), k+l n . For large enough n this comes close to 1 \u2212(\u03b1 + \u03b2), \u03b1 + \u03b2. Thus we have, I(G; X|Y) \u2264mH(\u03b1 + \u03b2) October 27, 2018 DRAFT \f36"
+                }
+            ]
+        },
+        "edge_feat": {}
+    }
+}
\ No newline at end of file