diff --git "a/abs_29K_G/test_abstract_long_2405.03958v1.json" "b/abs_29K_G/test_abstract_long_2405.03958v1.json" new file mode 100644--- /dev/null +++ "b/abs_29K_G/test_abstract_long_2405.03958v1.json" @@ -0,0 +1,697 @@ +{ + "url": "http://arxiv.org/abs/2405.03958v1", + "title": "Simple Drop-in LoRA Conditioning on Attention Layers Will Improve Your Diffusion Model", + "abstract": "Current state-of-the-art diffusion models employ U-Net architectures\ncontaining convolutional and (qkv) self-attention layers. The U-Net processes\nimages while being conditioned on the time embedding input for each sampling\nstep and the class or caption embedding input corresponding to the desired\nconditional generation. Such conditioning involves scale-and-shift operations\nto the convolutional layers but does not directly affect the attention layers.\nWhile these standard architectural choices are certainly effective, not\nconditioning the attention layers feels arbitrary and potentially suboptimal.\nIn this work, we show that simply adding LoRA conditioning to the attention\nlayers without changing or tuning the other parts of the U-Net architecture\nimproves the image generation quality. For example, a drop-in addition of LoRA\nconditioning to EDM diffusion model yields FID scores of 1.91/1.75 for\nunconditional and class-conditional CIFAR-10 generation, improving upon the\nbaseline of 1.97/1.79.", + "authors": "Joo Young Choi, Jaesung R. Park, Inkyu Park, Jaewoong Cho, Albert No, Ernest K. Ryu", + "published": "2024-05-07", + "updated": "2024-05-07", + "primary_cat": "cs.CV", + "cats": [ + "cs.CV", + "cs.AI", + "cs.LG" + ], + "label": "Original Paper", + "paper_cat": "Diffusion AND Model", + "gt": "Current state-of-the-art diffusion models employ U-Net architectures\ncontaining convolutional and (qkv) self-attention layers. The U-Net processes\nimages while being conditioned on the time embedding input for each sampling\nstep and the class or caption embedding input corresponding to the desired\nconditional generation. Such conditioning involves scale-and-shift operations\nto the convolutional layers but does not directly affect the attention layers.\nWhile these standard architectural choices are certainly effective, not\nconditioning the attention layers feels arbitrary and potentially suboptimal.\nIn this work, we show that simply adding LoRA conditioning to the attention\nlayers without changing or tuning the other parts of the U-Net architecture\nimproves the image generation quality. For example, a drop-in addition of LoRA\nconditioning to EDM diffusion model yields FID scores of 1.91/1.75 for\nunconditional and class-conditional CIFAR-10 generation, improving upon the\nbaseline of 1.97/1.79.", + "main_content": "Introduction In recent years, diffusion models have led to phenomenal advancements in image generation. Many cuttingedge diffusion models leverage U-Net architectures as their backbone, consisting of convolutional and (qkv) self-attention layers Dhariwal & Nichol (2021); Kim et al. (2023); Saharia et al. (2022); Rombach et al. (2022); Podell et al. (2024). In these models, the U-Net architecture-based score network is conditioned on the time, and/or, class, text embedding Ho & Salimans (2021) using scale-and-shift operations applied to the convolutional layers in the so-called residual blocks. Notably, however, the attention layers are not directly affected by the conditioning, and the rationale behind not extending conditioning to attention layers remains unclear. This gap suggests a need for in-depth studies searching for effective conditioning methods for attention layers and assessing their impact on performance. Meanwhile, low-rank adaptation (LoRA) has become the standard approach for parameter-efficient fine-tuning of large language models (LLM) Hu et al. (2022). With LoRA, one trains low-rank updates that are added to frozen pre-trained dense weights in the attention layers of LLMs. The consistent effectiveness of LoRA for LLMs suggests that LoRA may be generally compatible with attention layers used in different architectures and for different tasks Chen et al. (2022); Pan et al. (2022); Lin et al. (2023); Gong et al. (2024). In this work, we introduce a novel method for effectively conditioning the attention layers in the U-Net architectures of diffusion models by jointly training multiple LoRA adapters along with the base model. We call these LoRA adapters TimeLoRA and ClassLoRA for discrete-time settings, and Unified Compositional LoRA (UC-LoRA) for continuous signal-to-ratio (SNR) settings. Simply adding these LoRA adapters in a drop-in fashion without modifying or tuning the original model brings consistent enhancement in FID scores across several popular models applied to CIFAR-10, FFHQ 64x64, and ImageNet datasets. In particular, adding LoRA-conditioning to the EDM model Karras et al. (2022) yields improved FID scores of 1.75, 1.91, 2.31 for class-conditional CIFAR-10, unconditional CIFAR-10, and FFHQ 64x64 datasets, respectively, outperforming the baseline scores of 1.79, 1.97, 2.39. Moreover, we find that LoRA conditioning by itself is 2 \fScale-Shift Group Norm SiLU Convolution Group Norm SiLU Convolution Input Conditioning Linear Input QKV Group Norm \u03c9-scale LoRA LoRA Dot Product Projection \u03c9-scale LoRA LoRA MLP MLP Conditioning A1 Am B1 Bm \u03c91(t) \u03c9m(t) W A\u2032 c B\u2032 c \u00b7 \u00b7 \u00b7 A1 Am B1 Bm \u03c91(cond) \u03c9m(cond) W \u00b7 \u00b7 \u00b7 cond. MLP Unified compositional LoRA TimeLoRA and ClassLoRA Attn. Block LoRA U-Net block LoRA conditioning of attention block Figure 2: Conditioning of U-Net Block: (left) scale-and-shift conditioning on the convolutional block (middle) LoRA conditioning on the attention block (right) top: TimeLoRA and ClassLoRA for the discrete-time setting, bottom: unified composition LoRA for the continuous-SNR setting. powerful enough to perform effectively. Our experiments show that only conditioning the attention layers using LoRA adapters (without the conditioning convolutional layers with scale-and-shift) achieves comparable FID scores compared to the baseline scale-and-shift conditioning (without LoRA). Contribution. Our experiments show that using LoRA to condition time and class information on attention layers is effective across various models and datasets, including nano diffusion Lelarge et al. (2024), IDDPM Nichol & Dhariwal (2021), and EDM Karras et al. (2022) architectures using the MNIST Deng (2012), CIFAR-10 Krizhevsky et al. (2009), and FFHQ Karras et al. (2019) datasets. Our main contributions are as follows. (i) We show that simple drop-in LoRA conditioning on the attention layers improves the image generation quality, as measured by lower FID scores, while incurring minimal (\u223c10%) added memory and compute costs. (ii) We identify the problem of whether to and how to condition attention layers in diffusion models and provide the positive answer that attention layers should be conditioned and LoRA is an effective approach that outperforms the prior approaches of no conditioning or conditioning with adaLN Peebles & Xie (2023). Our results advocate for incorporating LoRA conditioning into the larger state-of-the-art U-Net-based diffusion models and the newer experimental architectures. 2 Prior work and preliminaries 2.1 Diffusion models Diffusion models Sohl-Dickstein et al. (2015); Song & Ermon (2019); Ho et al. (2020); Song et al. (2021b) generate images by iteratively removing noise from a noisy image. This denoising process is defined by the reverse process of the forward diffusion process: given data x0 \u223cq0, progressively inject noise to x0 by q(xt | xt\u22121) = N \u0010p 1 \u2212\u03b2txt\u22121, \u03b2tI \u0011 for t = 1, . . . , T and 0 < \u03b2t < 1. If \u03b2t is sufficiently small, we can approximate the reverse process as q(xt\u22121 | xt) \u2248N (\u00b5t(xt), \u03b2tI) 3 \fwhere \u00b5t(xt) = 1 \u221a1 \u2212\u03b2t (xt + \u03b2t\u2207log pt(xt)). A diffusion model is trained to approximate the score function \u2207log pt(xt) with a score network s\u03b8, which is often modeled with a U-Net architecture Ronneberger et al. (2015); Song & Ermon (2019). With s\u03b8 \u2248\u2207log pt(xt), the diffusion model approximates the reverse process as p\u03b8(xt\u22121|xt) = N \u0012 1 \u221a1 \u2212\u03b2t (xt + \u03b2ts\u03b8(xt, t)), \u03b2tI \u0013 \u2248q(xt\u22121 | xt). To sample from a trained diffusion model, one starts with Gaussian noise xT \u223cN (0, (1 \u2212\u00af \u03b1T )I), where \u00af \u03b1t = Qt s=1(1\u2212\u03b2s), and progressively denoise the image by sampling from p\u03b8(xt\u22121|xt) with t = T, T \u22121, . . . , 2, 1 sequentially to obtain a clean image x0. The above discrete-time description of diffusion models has a continuous-time counterpart based on the theory of stochastic differential equation (SDE) for the forward-corruption process and reversing it based on Anderson\u2019s reverse-time SDE Anderson (1982) or a reverse-time ordinary differential equation (ODE) with equivalent marginal probabilities Song et al. (2021a). Higher-order integrators have been used to reduce the discretization errors in solving the differential equations Karras et al. (2022). Architecture for diffusion models. The initial work of Song & Ermon (2019) first utilized the CNN-based U-Net architecture Ronneberger et al. (2015) as the architecture for the score network. Several improvements have been made by later works Ho et al. (2020); Nichol & Dhariwal (2021); Dhariwal & Nichol (2021); Hoogeboom et al. (2023) incorporating multi-head self-attention Vaswani et al. (2017), group normalization Wu & He (2018), and adaptive layer normalization (adaLN) Perez et al. (2018). Recently, several alternative architectures have been proposed. Jabri et al. (2023) proposed Recurrent Interface Network (RIN), which decouples the core computation and the dimension of the data for more scalable image generation. Peebles & Xie (2023); Bao et al. (2023); Gao et al. (2023); Hatamizadeh et al. (2023) investigated the effectiveness of transformer-based architectures Dosovitskiy et al. (2021) for diffusion models. Yan et al. (2023) utilized state space models Gu et al. (2022) in DiffuSSM to present an attention-free diffusion model architecture. In this work, we propose a conditioning method for attention layers and test it on several CNN-based U-Net architectures. Note that our proposed method is applicable to all diffusion models utilizing attention layers. 2.2 Low-rank adaptation Using trainable adapters for specific tasks has been an effective approach for fine-tuning models in the realm of natural language processing (NLP) Houlsby et al. (2019); Pfeiffer et al. (2020). Low-rank adpatation (LoRA, Hu et al. (2022)) is a parameter-efficient fine-tuning method that updates a low-rank adapter: to fine-tune a pre-trained dense weight matrix W \u2208Rdout\u00d7din, LoRA parameterizes the fine-tuning update \u2206W with a low-rank factorization W + \u2206W = W + BA, where B \u2208Rdout\u00d7r, A \u2208Rr\u00d7din, and r \u226amin{din, dout}. LoRA and diffusion. Although initially proposed for fine-tuning LLMs, LoRA is generally applicable to a wide range of other deep-learning modalities. Recent works used LoRA with diffusion models for various tasks including image generation Ryu (2023); Gu et al. (2023); Go et al. (2023), image editing Shi et al. (2023), continual learning Smith et al. (2023), and distillation Golnari (2023); Wang et al. (2023b). While all these works demonstrate the flexibility and efficacy of the LoRA architecture used for fine-tuning diffusion models, to the best of our knowledge, our work is the first attempt to use LoRA as part of the core U-Net for diffusion models for full training, not fine-tuning. 4 \f2.3 Conditioning the score network For diffusion models to work properly, it is crucial that the score network s\u03b8 is conditioned on appropriate side information. In the base formulation, the score function \u2207xpt(x), which the score network s\u03b8 learns, depends on the time t, so this t-dependence must be incorporated into the model via time conditioning. When class-labeled training data is available, class-conditional sampling requires class conditioning of the score network Ho & Salimans (2021). To take advantage of data augmentation and thereby avoid overfitting, EDM Karras et al. (2022) utilizes augmentation conditioning Jun et al. (2020), where the model is conditioned on the data augmentation information such as the degree of image rotation or blurring. Similarly, SDXL Podell et al. (2024) uses micro-conditioning, where the network is conditioned on image resolution or cropping information. Finally, text-to-image diffusion models Saharia et al. (2022); Ramesh et al. (2022); Rombach et al. (2022); Podell et al. (2024) use text conditioning, which conditions the score network with caption embeddings so that the model generates images aligned with the text description. Conditioning attention layers. Prior diffusion models using CNN-based U-Net architectures condition only convolutional layers in the residual blocks by applying scale-and-shift or adaLN (see (left) of Figure 2). In particular, attention blocks are not directly conditioned in such models. This includes the stateof-the-art diffusion models such as Imagen Saharia et al. (2022), DALL\u00b7E 2 Ramesh et al. (2022), Stable Diffusion Rombach et al. (2022), and SDXL Podell et al. (2024). To clarify, Latent Diffusion Model Rombach et al. (2022) based models use cross-attention method for class and text conditioning, but they still utilize scale-and-shift for time conditioning. There is a line of research proposing transformer-based architectures (without convolutions) for diffusion models, and these work do propose methods for conditioning attention layers. For instance, DiT Peebles & Xie (2023) conditioned attention layers using adaLN and DiffiT Hatamizadeh et al. (2023) introduced time-dependent multi-head self-attention (TMSA), which can be viewed as scale-and-shift conditioning applied to attention layers. Although such transformer-based architectures have shown to be effective, whether conditioning the attention layers with adaLN or scale-and-shift is optimal was not investigated. In Section 5.5 of this work, we compare our proposed LoRA conditioning on attention layers with the prior adaLN conditioning on attention layers, and show that LoRA is the more effective mechanism for conditioning attention layers. Diffusion models as multi-task learners. Multi-task learning Caruana (1997) is a framework where a single model is trained on multiple related tasks simultaneously, leveraging shared representations between the tasks. If one views the denoising tasks for different timesteps (or SNR) of diffusion models as related but different tasks, the training of diffusion models can be interpreted as an instance of the multi-task learning. Following the use of trainable lightweight adapters for Mixture-of-Expert (MoE) Jacobs et al. (1991); Ma et al. (2018), several works have utilized LoRA as the expert adapter for the multi-task learning Caccia et al. (2023); Wang et al. (2023a; 2024); Zadouri et al. (2024). Similarly, MORRIS Audibert et al. (2023) and LoRAHub Huang et al. (2023) proposed using the weighted sum of multiple LoRA adapters to effectively tackle general tasks. In this work, we took inspiration from theses works by using a composition of LoRA adapters to condition diffusion models. 3 Discrete-time LoRA conditioning Diffusion models such as DDPM Ho et al. (2020) and IDDPM Nichol & Dhariwal (2021) have a predetermined number of discrete timesteps t = 1, 2, . . . , T used for both training and sampling. We refer to this setting as the discrete-time setting. We first propose a method to condition the attention layers with LoRA in the discrete-time setting. In particular, we implement LoRA conditioning on IDDPM by conditioning the score network with (discrete) time and (discrete) class information. 5 \f3.1 TimeLoRA TimeLoRA conditions the score network for the discrete time steps t = 1, . . . , T. In prior architectures, time information is typically injected into only the residual blocks containing convolutional layers. TimeLoRA instead conditions the attention blocks. See (right) of Figure 2. Non-compositional LoRA. Non-compositional LoRA instantiates T independent rank-r LoRA weights A1, A2, . . . , AT , B1, B2, . . . , BT . The dense layer at time t becomes Wt = W + \u2206W(t) = W + BtAt for t = 1, . . . , T. To clarify, the trainable parameters for each linear layer are W, A1, A2, . . . , AT , and B1, B2, . . . , BT . In particular, W is trained concurrently with A1, A2, . . . , AT , and B1, B2, . . . , BT . However, this approach has two drawbacks. First, since T is typically large (up to 4000), instantiating T independent LoRAs can occupy significant memory. Second, since each LoRA (At, Bt) is trained independently, it disregards the fact that LoRAs of nearby time steps should likely be correlated/similar. It would be preferable for the architecture to incorporate the inductive bias that the behavior at nearby timesteps are similar. Compositional LoRA. Compositional LoRA composes m LoRA bases, A1, . . . , Am and B1, . . . , Bm, where m \u226aT. Each LoRA basis (Ai, Bi) corresponds to time ti for 1 \u2264t1 < \u00b7 \u00b7 \u00b7 < tm \u2264T. The dense layer at time t becomes Wt = W + \u2206W(t) = W + m X i=1 (\u03c9t)i BiAi, where \u03c9t = ((\u03c9t)1 , . . . , (\u03c9t)m) is the time-dependent trainable weights composing the LoRA bases. To clarify, the trainable parameters for each linear layer are W, A1, A1, . . . , Am, B1, B1, . . . , Bm, and \u03c9t. Since the score network is a continuous function of t, we expect \u03c9t \u2248\u03c9t\u2032 if t \u2248t\u2032. Therefore, to exploit the task similarity between nearby timesteps, we initialize (\u03c9t)i with a linear interpolation scheme: for tj \u2264t < tj+1, (\u03c9t)i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 tj+1 \u2212t tj+1 \u2212tj i = j t \u2212tj tj+1 \u2212tj i = j + 1 0 otherwise. In short, at initialization, \u2206W(t) uses a linear combination of the two closest LoRA bases. During training, \u03c9t can learn to utilize more than two LoRA bases, i.e., \u03c9t can learn to have more than two non-zeros through training. Specifically, (\u03c91, . . . , \u03c9T ) \u2208Rm\u00d7T is represented as an m \u00d7 T trainable table implemented as nn.Embedding in Pytorch. 3.2 ClassLoRA Consider a conditional diffusion model with C classes. ClassLoRA conditions the attention layers in the score network with the class label. Again, this contrasts with the typical approach of injecting class information only into the residual blocks containing convolutional layers. See (right) of Figure 2. Since C is small for CIFAR-10 (C = 10) and the correlations between different classes are likely not strong, we only use the non-compositional ClassLoRA: Wc = W + \u2206W(c) = W + B\u2032 cA\u2032 c for c = 1, . . . , C. In other words, each LoRA (A\u2032 c, B\u2032 c) handles a single class c. When C is large, such as in the case of ImageNet1k, one may consider using a compositional version of ClassLoRA. 6 \f4 Continuous-SNR LoRA conditioning Motivated by (Kingma et al., 2021), some recent models such as EDM Karras et al. (2022) consider parameterizing the score function as a function of noise or signal-to-noise ratio (SNR) level instead of time. In particular, EDM Karras et al. (2022) considers the probability flow ODE Xt = \u2212\u02d9 \u03c3(t)\u03c3(t)s\u03b8(x; \u03c3(t)) dt, where s\u03b8(x; \u03c3) is the score network conditioned on the SNR level \u03c3. We refer to this setting as the continuousSNR setting. The main distinction between Sections 3 and 4 is in the discrete vs. continuous parameterization, since continuous-time and continuous-SNR parameterizations of score functions are equivalent. We choose to consider continuous-SNR (instead of continuous-time) parameterizations for the sake of consistency with the EDM model Karras et al. (2022). Two additional issues arise in the present setup compared to the setting of Section 3. First, by considering a continuum of SNR levels, there is no intuitive way to assign a single basis LoRA to a specific noise level. Second, to accommodate additional conditioning elements such as augmentations or even captions, allocating independent LoRA for each conditioning element could lead to memory inefficiency. 4.1 Unified compositional LoRA (UC-LoRA) Consider the general setting where the diffusion model is conditioned with N attributes cond1, . . . , condN, which can be a mixture of continuous and discrete information. In our EDM experiments, we condition the score network with N = 3 attributes: SNR level (time), class, and augmentation information. Unified compositional LoRA (UC-LoRA) composes m LoRA bases A1, . . . , Am and B1, . . . , Bm to simultaneously condition the information of cond1, . . . condN into the attention layer. The compositional weight \u03c9 = (\u03c91, . . . , \u03c9m) of the UC-LoRA is obtained by passing cond1, . . . condN through an MLP. Prior diffusion models typically process cond1, . . . , condN with an MLP to obtain a condition embedding v, which is then shared by all residual blocks for conditioning. For the j-th residual block, v is further processed by an MLP to get scale and shift parameters \u03b3j and \u03b2j: v = SharedMLP(cond1, . . . , condN) (\u03b3j, \u03b2j) = MLPj(v). The (\u03b3j, \u03b2j) is then used for the scale-and-shift conditioning of the j-th residual block in the prior architectures. In our UC-LoRA, we similarly use the shared embedding v and an individual MLP for the j-th attention block to obtain the composition weight \u03c9j(v): v = SharedMLP(cond1, \u00b7 \u00b7 \u00b7 , condN) \u03c9j(v) = MLPj(v). Then, the j-th dense layer of the attention block becomes W(cond1, . . . , condN) = W + \u2206W(cond1, . . . , condN) = W + m X i=1 \u03c9j,i(v)BiAi. To clarify, the trainable parameters for the j-th dense layer are W, A1, A2, . . . , Am, B1, B2, . . . , Bm, and the weights in MLPj. Shared across the entire architecture, the weights in SharedMLP are also trainable parameters. 7 \f5 Experiments In this section, we present our experimental findings. Section 5.1 describes the experimental setup. Section 5.2 first presents a toy, proof-of-concept experiment to validate the proposed LoRA conditioning. Section 5.3 evaluates the effectiveness of LoRA conditioning on attention layers with a quantitative comparison between diffusion models with (baseline) conventional scale-and-shift conditioning on convolutional layers; (only LoRA) LoRA conditioning on attention layers without conditioning convolutional layers; and (with LoRA) conditioning both convolutional layers and attention layers with scale-and-shift and LoRA conditioning, respectively. Section 5.4 investigates the effect of tuning the LoRA rank and the number of LoRA bases. Section 5.5 compares our proposed LoRA conditioning with the adaLN conditioning on attention layers. Section 5.6 explores the robustness of ClassLoRA conditioning compared to conventional scale-and-shift conditioning in extrapolating conditioning information. 5.1 Experimental Setup Diffusion models. We implement LoRA conditioning on three different diffusion models: nano diffusion Lelarge et al. (2024), IDDPM Nichol & Dhariwal (2021), and EDM-vp Karras et al. (2022). With nano diffusion, we conduct a proof-of-concept experiment. With IDDPM, we test TimeLoRA and ClassLoRA for the discrete-time setting, and with EDM, we test UC-LoRA for the continuous-SNR setting. Datasets. For nano diffusion, we use MNIST. For IDDPM, we use CIFAR-10 for both unconditional and class-conditional sampling, and ImageNet64, a downsampled version of the ImageNet1k, for unconditional sampling. For EDM-vp, we also use CIFAR-10 for both unconditional and class-conditional sampling and FFHQ64 for unconditional sampling. Configurations. We follow the training and architecture configurations proposed by the baseline works and only tune the LoRA adapters. For IDDPM, we train the model for 500K iterations for CIFAR-10 with batch size of 128 and learning rate of 1 \u00d7 10\u22124, and 1.5M iterations for ImageNet64 with batch size of 128 and learning rate of 1 \u00d7 10\u22124. For EDM, we train the model with batch size of 512 and learning rate of 1 \u00d7 10\u22123 for CIFAR-10, and with batch size of 256 and learning rate of 2 \u00d7 10\u22124 for FFHQ64. For sampling, in IDDPM, we use 4000 and 4001 timesteps for the baseline and LoRA conditioning respectively, and in EDM, we use the proposed Heun\u2019s method and sample images with 18 timesteps (35 NFE) for CIFAR-10 and 40 timesteps (79 NFE) for FFHQ64. Here, NFE is the number of forward evaluation of the score network and it differs from the number of timesteps by a factor of 2 because Heun\u2019s method is a 2-stage Runge\u2013Kutta method. Appendix A provides further details of the experiment configurations. Note that the baseline works heavily optimized the hyperparameters such as learning rate, dropout probability, and augmentations. Although we do not modify any configurations of the baseline and simply add LoRA conditioning in a drop-in fashion, we expect further improvements from further optimizing the configuration for the entire architecture and training procedure. LoRA. We use the standard LoRA initialization as in the original LoRA paper Hu et al. (2022): for the LoRA matrices (A, B) with rank r, A is initialized as Aij \u223cN(0, 1/r) and B as the zero matrix. Following Ryu (2023), we set the rank of each basis LoRA to 4. For TimeLoRA and ClassLoRA, we use 11 and 10 LoRA bases, and for UC-LoRA we use 18 and 20 LoRA bases for CIFAR-10 and FFHQ. Due to our constrained computational budget, we were not able to conduct a full investigation on the optimal LoRA rank or the number LoRA bases. However, we experiment with the effect of rank and number of LoRA bases to limited extent and report the result in Section 5.4. 5.2 Proof-of-concept experiments We conduct toy experiments with nano diffusion for both discrete-time and continuous-SNR settings. Nano diffusion is a small diffusion model with a CNN-based U-Net architecture with no skip connections with about 500, 000 trainable parameters. We train nano diffusion on unconditional MNIST generation with 8 \f3 different conditioning methods: conventional scale-and-shift, TimeLoRA, and UC-LoRA. As shown in Figure 3, conditioning with TimeLoRA or UC-LoRA yields competitive result compared to the conventional scale-and-shift conditioning. Figure 3: MNIST samples generated by nano diffusion trained with (1st row) conventional scale-and-shift conditioning; (2nd row) TimeLoRA with linear interpolation initialization; (3rd row) UC-LoRA; and (4th row) TimeLoRA with random initialization. Initialization of \u03c9i(t) for TimeLoRA. As shown in Figure 3 the choice of initialization of \u03c9i(t) for TimeLoRA impacts performance. With randomly initialized \u03c9i(t), nano diffusion did not converge after 100 epochs, whereas with \u03c9i(t) initialized with the linear interpolation scheme, it did converge. Moreover, Figure 4 shows that even in UC-LoRA, \u03c9(t) shows higher similarity between nearby timesteps than between distant timesteps after training. This is consistent with our expectation that \u03c9i(t) \u2248\u03c9i(t\u2032) if t \u2248t\u2032. 250 500 750 1000 t1 200 400 600 800 1000 t2 250 500 750 1000 t1 1.0 0.5 0.0 0.5 1.0 Figure 4: Cosine similarity between \u03c9(t1) and \u03c9(t2) for UC-LoRA applied to nano diffusion (left) at initialization and (right) after training. At initialization, the cosine similarity between \u03c9(t1) and \u03c9(t2) has no discernible pattern. After training, however, the cosine similarity between \u03c9(t1) and \u03c9(t2) for t1 \u2248t2 is close to 1, implying their high similarity. 5.3 Main quantitative results Simply adding LoRA conditioning yields improvements. To evaluate the effectiveness of the drop-in addition of LoRA conditioning to the attention layers, we implement TimeLoRA and ClassLoRA to IDDPM and UC-LoRA to EDM, both with the conventional scale-and-shift conditioning on the convolutional layers unchanged. We train IDDPM with CIFAR-10, ImageNet64 and EDM with CIFAR-10, FFHQ64. As reported in Table 1, the addition of LoRA conditioning to the attention layers consistently improves the image generation quality as measured by FID scores Heusel et al. (2017) across different diffusion models and datasets with only (\u223c10%) addition of the parameter counts. Note these improvements are achieved without tuning any hyperparameters of the base model components. 9 \fInitializing the base model with pre-trained weights. We further test UC-LoRA on pre-trained EDM base models for unconditional CIFAR-10 and FFHQ64 generations. As reported in Table 1, using pre-trained weights showed additional gain on FID score with fewer number of interations (\u223c50%). To clarify, although we initialize the base model with pre-trained weights, we fully train both base model and LoRA modules rather than finetuning. LoRA can even replace scale-and-shift. We further evaluate the effectiveness of LoRA conditioning by replacing the scale-and-shift conditioning for the convolutional layers in residual blocks with LoRA conditioning for the attention blocks. The results of Table 1 suggest that solely using LoRA conditioning on attention layers achieves competitive FID scores while being more efficient in memory compared to the baseline score network trained with scale-and-shift conditioning on convolutional layers. For IDDPM, using LoRA in place of the conventional scale-and-shift conditioning consistently produces better results. Significant improvement is observed especially for class-conditional generation of CIFAR-10. For EDM, replacing the scale-and-shift conditioning did not yield an improvement, but nevertheless performed comparably. We note that in all cases, LoRA conditioning is more parameter-efficient (\u223c10%) than the conventional scale-and-shift conditioning. 5.4 Effect of LoRA rank and number of LoRA bases We investigate the effect of tuning the LoRA rank and the number of LoRA bases on the EDM model for unconditional CIFAR-10 generation and report the results in Table 2. Our findings indicate that using more LoRA bases consistently improves the quality of image generations. On the other hand, increasing LoRA rank does not guarantee better performance. These findings suggest an avenue of further optimizing and improving our main quantitative results of Section 5.3 and Table 1, which we have not yet been able to pursue due to our constrained computational budget. # basis rank FID # Params Varying # basis 9 4 1.99 57185519 18 4 1.96 57745499 36 4 1.95 58865459 Varying rank 18 2 1.93 57192539 18 4 1.96 57745499 18 8 1.96 58851419 Table 2: Effect of the number of LoRA bases and the LoRA rank on unconditional CIFAR-10 sampling of EDM with LoRA 5.5 Comparison with adaLN We compare the effectiveness of our proposed LoRA conditioning with adaLN conditioning applied to attention layers. Specifically, we conduct an experiment on EDM with scale-and-shift conditioning on convolutional layers removed and with (i) adaLN conditioning attention layers or (ii) LoRA conditioning attention layers. We compare the sample quality of unconditional and class-conditional CIFAR-10 generation and report the results in Table 3. We find that LoRA conditioning significantly outperforms adaLN conditioning for both unconditional and conditional CIFAR-10 generation. This indicates that our proposed LoRA conditioning is the more effective mechanism for conditioning attention layers in the U-Net architectures for diffusion models. Type uncond. cond. adaLN conditioning 2.16 2.0 LoRA conditioning 1.99 1.82 Table 3: Comparison of adaLN conditioning and LoRA conditioning on attention layers on EDM (without conditioning convolutional layers). We consider both unconditional and conditional CIFAR-10 generation. 10 \f5.6 Extrapolating conditioning information We conduct an experiment comparing two class-conditional EDM models each conditioned by scale-and-shift and ClassLoRA, for the CIFAR-10 dataset. During training, both models receive size-10 one-hot vectors (ci)j = \u03b4ij representing the class information. First, we input the linear interpolation \u03b1ci +(1\u2212\u03b1)cj (0 \u2264\u03b1 \u22641) of two class inputs ci and cj (corresponding to \u2018airplane\u2019 and \u2018horse\u2019, respectively) to observe the continuous transition between classes. As shown in the top of Figure 5, both the scale-and-shift EDM and ClassLoRA EDM models effectively interpolate semantic information across different classes. However, when a scaled input \u03b2ci is received, with \u03b2 ranging from -1 to 1, scale-and-shift EDM generates unrecognizable images when \u03b2 < 0, while ClassLoRA EDM generates plausible images throughout the whole range, as shown in the bottom of Figure 5. This toy experiment shows that LoRA-based conditioning may be more robust to extrapolating conditioning information beyond the range encountered during training. Appendix C provides further details. Figure 5: Results of (Top) interpolation of class labels in class-conditional EDM with (row1) ClassLoRA; (row2) scale-and-shift; (bottom) extrapolation of class labels in class-conditional EDM with (row1) ClassLoRA; (row2) scale-and-shift 6", + "additional_graph_info": { + "graph": [ + [ + "Joo Young Choi", + "Jaewoong Cho" + ], + [ + "Joo Young Choi", + "Albert No" + ], + [ + "Joo Young Choi", + "Ernest K. Ryu" + ], + [ + "Jaewoong Cho", + "Changho Suh" + ], + [ + "Jaewoong Cho", + "Kartik Sreenivasan" + ], + [ + "Jaewoong Cho", + "Keon Lee" + ], + [ + "Jaewoong Cho", + "Soheun Yi" + ], + [ + "Albert No", + "Tsachy Weissman" + ], + [ + "Albert No", + "Taeho Yoon" + ], + [ + "Albert No", + "Sehyun Kwon" + ], + [ + "Albert No", + "Ernest K. Ryu" + ], + [ + "Albert No", + "Amir Ingber" + ], + [ + "Ernest K. Ryu", + "Bang Cong Vu" + ], + [ + "Ernest K. Ryu", + "Kun Yuan" + ], + [ + "Ernest K. Ryu", + "Jialin Liu" + ] + ], + "node_feat": { + "Joo Young Choi": [ + { + "url": "http://arxiv.org/abs/2405.03958v1", + "title": "Simple Drop-in LoRA Conditioning on Attention Layers Will Improve Your Diffusion Model", + "abstract": "Current state-of-the-art diffusion models employ U-Net architectures\ncontaining convolutional and (qkv) self-attention layers. The U-Net processes\nimages while being conditioned on the time embedding input for each sampling\nstep and the class or caption embedding input corresponding to the desired\nconditional generation. Such conditioning involves scale-and-shift operations\nto the convolutional layers but does not directly affect the attention layers.\nWhile these standard architectural choices are certainly effective, not\nconditioning the attention layers feels arbitrary and potentially suboptimal.\nIn this work, we show that simply adding LoRA conditioning to the attention\nlayers without changing or tuning the other parts of the U-Net architecture\nimproves the image generation quality. For example, a drop-in addition of LoRA\nconditioning to EDM diffusion model yields FID scores of 1.91/1.75 for\nunconditional and class-conditional CIFAR-10 generation, improving upon the\nbaseline of 1.97/1.79.", + "authors": "Joo Young Choi, Jaesung R. Park, Inkyu Park, Jaewoong Cho, Albert No, Ernest K. Ryu", + "published": "2024-05-07", + "updated": "2024-05-07", + "primary_cat": "cs.CV", + "cats": [ + "cs.CV", + "cs.AI", + "cs.LG" + ], + "main_content": "Introduction In recent years, diffusion models have led to phenomenal advancements in image generation. Many cuttingedge diffusion models leverage U-Net architectures as their backbone, consisting of convolutional and (qkv) self-attention layers Dhariwal & Nichol (2021); Kim et al. (2023); Saharia et al. (2022); Rombach et al. (2022); Podell et al. (2024). In these models, the U-Net architecture-based score network is conditioned on the time, and/or, class, text embedding Ho & Salimans (2021) using scale-and-shift operations applied to the convolutional layers in the so-called residual blocks. Notably, however, the attention layers are not directly affected by the conditioning, and the rationale behind not extending conditioning to attention layers remains unclear. This gap suggests a need for in-depth studies searching for effective conditioning methods for attention layers and assessing their impact on performance. Meanwhile, low-rank adaptation (LoRA) has become the standard approach for parameter-efficient fine-tuning of large language models (LLM) Hu et al. (2022). With LoRA, one trains low-rank updates that are added to frozen pre-trained dense weights in the attention layers of LLMs. The consistent effectiveness of LoRA for LLMs suggests that LoRA may be generally compatible with attention layers used in different architectures and for different tasks Chen et al. (2022); Pan et al. (2022); Lin et al. (2023); Gong et al. (2024). In this work, we introduce a novel method for effectively conditioning the attention layers in the U-Net architectures of diffusion models by jointly training multiple LoRA adapters along with the base model. We call these LoRA adapters TimeLoRA and ClassLoRA for discrete-time settings, and Unified Compositional LoRA (UC-LoRA) for continuous signal-to-ratio (SNR) settings. Simply adding these LoRA adapters in a drop-in fashion without modifying or tuning the original model brings consistent enhancement in FID scores across several popular models applied to CIFAR-10, FFHQ 64x64, and ImageNet datasets. In particular, adding LoRA-conditioning to the EDM model Karras et al. (2022) yields improved FID scores of 1.75, 1.91, 2.31 for class-conditional CIFAR-10, unconditional CIFAR-10, and FFHQ 64x64 datasets, respectively, outperforming the baseline scores of 1.79, 1.97, 2.39. Moreover, we find that LoRA conditioning by itself is 2 \fScale-Shift Group Norm SiLU Convolution Group Norm SiLU Convolution Input Conditioning Linear Input QKV Group Norm \u03c9-scale LoRA LoRA Dot Product Projection \u03c9-scale LoRA LoRA MLP MLP Conditioning A1 Am B1 Bm \u03c91(t) \u03c9m(t) W A\u2032 c B\u2032 c \u00b7 \u00b7 \u00b7 A1 Am B1 Bm \u03c91(cond) \u03c9m(cond) W \u00b7 \u00b7 \u00b7 cond. MLP Unified compositional LoRA TimeLoRA and ClassLoRA Attn. Block LoRA U-Net block LoRA conditioning of attention block Figure 2: Conditioning of U-Net Block: (left) scale-and-shift conditioning on the convolutional block (middle) LoRA conditioning on the attention block (right) top: TimeLoRA and ClassLoRA for the discrete-time setting, bottom: unified composition LoRA for the continuous-SNR setting. powerful enough to perform effectively. Our experiments show that only conditioning the attention layers using LoRA adapters (without the conditioning convolutional layers with scale-and-shift) achieves comparable FID scores compared to the baseline scale-and-shift conditioning (without LoRA). Contribution. Our experiments show that using LoRA to condition time and class information on attention layers is effective across various models and datasets, including nano diffusion Lelarge et al. (2024), IDDPM Nichol & Dhariwal (2021), and EDM Karras et al. (2022) architectures using the MNIST Deng (2012), CIFAR-10 Krizhevsky et al. (2009), and FFHQ Karras et al. (2019) datasets. Our main contributions are as follows. (i) We show that simple drop-in LoRA conditioning on the attention layers improves the image generation quality, as measured by lower FID scores, while incurring minimal (\u223c10%) added memory and compute costs. (ii) We identify the problem of whether to and how to condition attention layers in diffusion models and provide the positive answer that attention layers should be conditioned and LoRA is an effective approach that outperforms the prior approaches of no conditioning or conditioning with adaLN Peebles & Xie (2023). Our results advocate for incorporating LoRA conditioning into the larger state-of-the-art U-Net-based diffusion models and the newer experimental architectures. 2 Prior work and preliminaries 2.1 Diffusion models Diffusion models Sohl-Dickstein et al. (2015); Song & Ermon (2019); Ho et al. (2020); Song et al. (2021b) generate images by iteratively removing noise from a noisy image. This denoising process is defined by the reverse process of the forward diffusion process: given data x0 \u223cq0, progressively inject noise to x0 by q(xt | xt\u22121) = N \u0010p 1 \u2212\u03b2txt\u22121, \u03b2tI \u0011 for t = 1, . . . , T and 0 < \u03b2t < 1. If \u03b2t is sufficiently small, we can approximate the reverse process as q(xt\u22121 | xt) \u2248N (\u00b5t(xt), \u03b2tI) 3 \fwhere \u00b5t(xt) = 1 \u221a1 \u2212\u03b2t (xt + \u03b2t\u2207log pt(xt)). A diffusion model is trained to approximate the score function \u2207log pt(xt) with a score network s\u03b8, which is often modeled with a U-Net architecture Ronneberger et al. (2015); Song & Ermon (2019). With s\u03b8 \u2248\u2207log pt(xt), the diffusion model approximates the reverse process as p\u03b8(xt\u22121|xt) = N \u0012 1 \u221a1 \u2212\u03b2t (xt + \u03b2ts\u03b8(xt, t)), \u03b2tI \u0013 \u2248q(xt\u22121 | xt). To sample from a trained diffusion model, one starts with Gaussian noise xT \u223cN (0, (1 \u2212\u00af \u03b1T )I), where \u00af \u03b1t = Qt s=1(1\u2212\u03b2s), and progressively denoise the image by sampling from p\u03b8(xt\u22121|xt) with t = T, T \u22121, . . . , 2, 1 sequentially to obtain a clean image x0. The above discrete-time description of diffusion models has a continuous-time counterpart based on the theory of stochastic differential equation (SDE) for the forward-corruption process and reversing it based on Anderson\u2019s reverse-time SDE Anderson (1982) or a reverse-time ordinary differential equation (ODE) with equivalent marginal probabilities Song et al. (2021a). Higher-order integrators have been used to reduce the discretization errors in solving the differential equations Karras et al. (2022). Architecture for diffusion models. The initial work of Song & Ermon (2019) first utilized the CNN-based U-Net architecture Ronneberger et al. (2015) as the architecture for the score network. Several improvements have been made by later works Ho et al. (2020); Nichol & Dhariwal (2021); Dhariwal & Nichol (2021); Hoogeboom et al. (2023) incorporating multi-head self-attention Vaswani et al. (2017), group normalization Wu & He (2018), and adaptive layer normalization (adaLN) Perez et al. (2018). Recently, several alternative architectures have been proposed. Jabri et al. (2023) proposed Recurrent Interface Network (RIN), which decouples the core computation and the dimension of the data for more scalable image generation. Peebles & Xie (2023); Bao et al. (2023); Gao et al. (2023); Hatamizadeh et al. (2023) investigated the effectiveness of transformer-based architectures Dosovitskiy et al. (2021) for diffusion models. Yan et al. (2023) utilized state space models Gu et al. (2022) in DiffuSSM to present an attention-free diffusion model architecture. In this work, we propose a conditioning method for attention layers and test it on several CNN-based U-Net architectures. Note that our proposed method is applicable to all diffusion models utilizing attention layers. 2.2 Low-rank adaptation Using trainable adapters for specific tasks has been an effective approach for fine-tuning models in the realm of natural language processing (NLP) Houlsby et al. (2019); Pfeiffer et al. (2020). Low-rank adpatation (LoRA, Hu et al. (2022)) is a parameter-efficient fine-tuning method that updates a low-rank adapter: to fine-tune a pre-trained dense weight matrix W \u2208Rdout\u00d7din, LoRA parameterizes the fine-tuning update \u2206W with a low-rank factorization W + \u2206W = W + BA, where B \u2208Rdout\u00d7r, A \u2208Rr\u00d7din, and r \u226amin{din, dout}. LoRA and diffusion. Although initially proposed for fine-tuning LLMs, LoRA is generally applicable to a wide range of other deep-learning modalities. Recent works used LoRA with diffusion models for various tasks including image generation Ryu (2023); Gu et al. (2023); Go et al. (2023), image editing Shi et al. (2023), continual learning Smith et al. (2023), and distillation Golnari (2023); Wang et al. (2023b). While all these works demonstrate the flexibility and efficacy of the LoRA architecture used for fine-tuning diffusion models, to the best of our knowledge, our work is the first attempt to use LoRA as part of the core U-Net for diffusion models for full training, not fine-tuning. 4 \f2.3 Conditioning the score network For diffusion models to work properly, it is crucial that the score network s\u03b8 is conditioned on appropriate side information. In the base formulation, the score function \u2207xpt(x), which the score network s\u03b8 learns, depends on the time t, so this t-dependence must be incorporated into the model via time conditioning. When class-labeled training data is available, class-conditional sampling requires class conditioning of the score network Ho & Salimans (2021). To take advantage of data augmentation and thereby avoid overfitting, EDM Karras et al. (2022) utilizes augmentation conditioning Jun et al. (2020), where the model is conditioned on the data augmentation information such as the degree of image rotation or blurring. Similarly, SDXL Podell et al. (2024) uses micro-conditioning, where the network is conditioned on image resolution or cropping information. Finally, text-to-image diffusion models Saharia et al. (2022); Ramesh et al. (2022); Rombach et al. (2022); Podell et al. (2024) use text conditioning, which conditions the score network with caption embeddings so that the model generates images aligned with the text description. Conditioning attention layers. Prior diffusion models using CNN-based U-Net architectures condition only convolutional layers in the residual blocks by applying scale-and-shift or adaLN (see (left) of Figure 2). In particular, attention blocks are not directly conditioned in such models. This includes the stateof-the-art diffusion models such as Imagen Saharia et al. (2022), DALL\u00b7E 2 Ramesh et al. (2022), Stable Diffusion Rombach et al. (2022), and SDXL Podell et al. (2024). To clarify, Latent Diffusion Model Rombach et al. (2022) based models use cross-attention method for class and text conditioning, but they still utilize scale-and-shift for time conditioning. There is a line of research proposing transformer-based architectures (without convolutions) for diffusion models, and these work do propose methods for conditioning attention layers. For instance, DiT Peebles & Xie (2023) conditioned attention layers using adaLN and DiffiT Hatamizadeh et al. (2023) introduced time-dependent multi-head self-attention (TMSA), which can be viewed as scale-and-shift conditioning applied to attention layers. Although such transformer-based architectures have shown to be effective, whether conditioning the attention layers with adaLN or scale-and-shift is optimal was not investigated. In Section 5.5 of this work, we compare our proposed LoRA conditioning on attention layers with the prior adaLN conditioning on attention layers, and show that LoRA is the more effective mechanism for conditioning attention layers. Diffusion models as multi-task learners. Multi-task learning Caruana (1997) is a framework where a single model is trained on multiple related tasks simultaneously, leveraging shared representations between the tasks. If one views the denoising tasks for different timesteps (or SNR) of diffusion models as related but different tasks, the training of diffusion models can be interpreted as an instance of the multi-task learning. Following the use of trainable lightweight adapters for Mixture-of-Expert (MoE) Jacobs et al. (1991); Ma et al. (2018), several works have utilized LoRA as the expert adapter for the multi-task learning Caccia et al. (2023); Wang et al. (2023a; 2024); Zadouri et al. (2024). Similarly, MORRIS Audibert et al. (2023) and LoRAHub Huang et al. (2023) proposed using the weighted sum of multiple LoRA adapters to effectively tackle general tasks. In this work, we took inspiration from theses works by using a composition of LoRA adapters to condition diffusion models. 3 Discrete-time LoRA conditioning Diffusion models such as DDPM Ho et al. (2020) and IDDPM Nichol & Dhariwal (2021) have a predetermined number of discrete timesteps t = 1, 2, . . . , T used for both training and sampling. We refer to this setting as the discrete-time setting. We first propose a method to condition the attention layers with LoRA in the discrete-time setting. In particular, we implement LoRA conditioning on IDDPM by conditioning the score network with (discrete) time and (discrete) class information. 5 \f3.1 TimeLoRA TimeLoRA conditions the score network for the discrete time steps t = 1, . . . , T. In prior architectures, time information is typically injected into only the residual blocks containing convolutional layers. TimeLoRA instead conditions the attention blocks. See (right) of Figure 2. Non-compositional LoRA. Non-compositional LoRA instantiates T independent rank-r LoRA weights A1, A2, . . . , AT , B1, B2, . . . , BT . The dense layer at time t becomes Wt = W + \u2206W(t) = W + BtAt for t = 1, . . . , T. To clarify, the trainable parameters for each linear layer are W, A1, A2, . . . , AT , and B1, B2, . . . , BT . In particular, W is trained concurrently with A1, A2, . . . , AT , and B1, B2, . . . , BT . However, this approach has two drawbacks. First, since T is typically large (up to 4000), instantiating T independent LoRAs can occupy significant memory. Second, since each LoRA (At, Bt) is trained independently, it disregards the fact that LoRAs of nearby time steps should likely be correlated/similar. It would be preferable for the architecture to incorporate the inductive bias that the behavior at nearby timesteps are similar. Compositional LoRA. Compositional LoRA composes m LoRA bases, A1, . . . , Am and B1, . . . , Bm, where m \u226aT. Each LoRA basis (Ai, Bi) corresponds to time ti for 1 \u2264t1 < \u00b7 \u00b7 \u00b7 < tm \u2264T. The dense layer at time t becomes Wt = W + \u2206W(t) = W + m X i=1 (\u03c9t)i BiAi, where \u03c9t = ((\u03c9t)1 , . . . , (\u03c9t)m) is the time-dependent trainable weights composing the LoRA bases. To clarify, the trainable parameters for each linear layer are W, A1, A1, . . . , Am, B1, B1, . . . , Bm, and \u03c9t. Since the score network is a continuous function of t, we expect \u03c9t \u2248\u03c9t\u2032 if t \u2248t\u2032. Therefore, to exploit the task similarity between nearby timesteps, we initialize (\u03c9t)i with a linear interpolation scheme: for tj \u2264t < tj+1, (\u03c9t)i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 tj+1 \u2212t tj+1 \u2212tj i = j t \u2212tj tj+1 \u2212tj i = j + 1 0 otherwise. In short, at initialization, \u2206W(t) uses a linear combination of the two closest LoRA bases. During training, \u03c9t can learn to utilize more than two LoRA bases, i.e., \u03c9t can learn to have more than two non-zeros through training. Specifically, (\u03c91, . . . , \u03c9T ) \u2208Rm\u00d7T is represented as an m \u00d7 T trainable table implemented as nn.Embedding in Pytorch. 3.2 ClassLoRA Consider a conditional diffusion model with C classes. ClassLoRA conditions the attention layers in the score network with the class label. Again, this contrasts with the typical approach of injecting class information only into the residual blocks containing convolutional layers. See (right) of Figure 2. Since C is small for CIFAR-10 (C = 10) and the correlations between different classes are likely not strong, we only use the non-compositional ClassLoRA: Wc = W + \u2206W(c) = W + B\u2032 cA\u2032 c for c = 1, . . . , C. In other words, each LoRA (A\u2032 c, B\u2032 c) handles a single class c. When C is large, such as in the case of ImageNet1k, one may consider using a compositional version of ClassLoRA. 6 \f4 Continuous-SNR LoRA conditioning Motivated by (Kingma et al., 2021), some recent models such as EDM Karras et al. (2022) consider parameterizing the score function as a function of noise or signal-to-noise ratio (SNR) level instead of time. In particular, EDM Karras et al. (2022) considers the probability flow ODE Xt = \u2212\u02d9 \u03c3(t)\u03c3(t)s\u03b8(x; \u03c3(t)) dt, where s\u03b8(x; \u03c3) is the score network conditioned on the SNR level \u03c3. We refer to this setting as the continuousSNR setting. The main distinction between Sections 3 and 4 is in the discrete vs. continuous parameterization, since continuous-time and continuous-SNR parameterizations of score functions are equivalent. We choose to consider continuous-SNR (instead of continuous-time) parameterizations for the sake of consistency with the EDM model Karras et al. (2022). Two additional issues arise in the present setup compared to the setting of Section 3. First, by considering a continuum of SNR levels, there is no intuitive way to assign a single basis LoRA to a specific noise level. Second, to accommodate additional conditioning elements such as augmentations or even captions, allocating independent LoRA for each conditioning element could lead to memory inefficiency. 4.1 Unified compositional LoRA (UC-LoRA) Consider the general setting where the diffusion model is conditioned with N attributes cond1, . . . , condN, which can be a mixture of continuous and discrete information. In our EDM experiments, we condition the score network with N = 3 attributes: SNR level (time), class, and augmentation information. Unified compositional LoRA (UC-LoRA) composes m LoRA bases A1, . . . , Am and B1, . . . , Bm to simultaneously condition the information of cond1, . . . condN into the attention layer. The compositional weight \u03c9 = (\u03c91, . . . , \u03c9m) of the UC-LoRA is obtained by passing cond1, . . . condN through an MLP. Prior diffusion models typically process cond1, . . . , condN with an MLP to obtain a condition embedding v, which is then shared by all residual blocks for conditioning. For the j-th residual block, v is further processed by an MLP to get scale and shift parameters \u03b3j and \u03b2j: v = SharedMLP(cond1, . . . , condN) (\u03b3j, \u03b2j) = MLPj(v). The (\u03b3j, \u03b2j) is then used for the scale-and-shift conditioning of the j-th residual block in the prior architectures. In our UC-LoRA, we similarly use the shared embedding v and an individual MLP for the j-th attention block to obtain the composition weight \u03c9j(v): v = SharedMLP(cond1, \u00b7 \u00b7 \u00b7 , condN) \u03c9j(v) = MLPj(v). Then, the j-th dense layer of the attention block becomes W(cond1, . . . , condN) = W + \u2206W(cond1, . . . , condN) = W + m X i=1 \u03c9j,i(v)BiAi. To clarify, the trainable parameters for the j-th dense layer are W, A1, A2, . . . , Am, B1, B2, . . . , Bm, and the weights in MLPj. Shared across the entire architecture, the weights in SharedMLP are also trainable parameters. 7 \f5 Experiments In this section, we present our experimental findings. Section 5.1 describes the experimental setup. Section 5.2 first presents a toy, proof-of-concept experiment to validate the proposed LoRA conditioning. Section 5.3 evaluates the effectiveness of LoRA conditioning on attention layers with a quantitative comparison between diffusion models with (baseline) conventional scale-and-shift conditioning on convolutional layers; (only LoRA) LoRA conditioning on attention layers without conditioning convolutional layers; and (with LoRA) conditioning both convolutional layers and attention layers with scale-and-shift and LoRA conditioning, respectively. Section 5.4 investigates the effect of tuning the LoRA rank and the number of LoRA bases. Section 5.5 compares our proposed LoRA conditioning with the adaLN conditioning on attention layers. Section 5.6 explores the robustness of ClassLoRA conditioning compared to conventional scale-and-shift conditioning in extrapolating conditioning information. 5.1 Experimental Setup Diffusion models. We implement LoRA conditioning on three different diffusion models: nano diffusion Lelarge et al. (2024), IDDPM Nichol & Dhariwal (2021), and EDM-vp Karras et al. (2022). With nano diffusion, we conduct a proof-of-concept experiment. With IDDPM, we test TimeLoRA and ClassLoRA for the discrete-time setting, and with EDM, we test UC-LoRA for the continuous-SNR setting. Datasets. For nano diffusion, we use MNIST. For IDDPM, we use CIFAR-10 for both unconditional and class-conditional sampling, and ImageNet64, a downsampled version of the ImageNet1k, for unconditional sampling. For EDM-vp, we also use CIFAR-10 for both unconditional and class-conditional sampling and FFHQ64 for unconditional sampling. Configurations. We follow the training and architecture configurations proposed by the baseline works and only tune the LoRA adapters. For IDDPM, we train the model for 500K iterations for CIFAR-10 with batch size of 128 and learning rate of 1 \u00d7 10\u22124, and 1.5M iterations for ImageNet64 with batch size of 128 and learning rate of 1 \u00d7 10\u22124. For EDM, we train the model with batch size of 512 and learning rate of 1 \u00d7 10\u22123 for CIFAR-10, and with batch size of 256 and learning rate of 2 \u00d7 10\u22124 for FFHQ64. For sampling, in IDDPM, we use 4000 and 4001 timesteps for the baseline and LoRA conditioning respectively, and in EDM, we use the proposed Heun\u2019s method and sample images with 18 timesteps (35 NFE) for CIFAR-10 and 40 timesteps (79 NFE) for FFHQ64. Here, NFE is the number of forward evaluation of the score network and it differs from the number of timesteps by a factor of 2 because Heun\u2019s method is a 2-stage Runge\u2013Kutta method. Appendix A provides further details of the experiment configurations. Note that the baseline works heavily optimized the hyperparameters such as learning rate, dropout probability, and augmentations. Although we do not modify any configurations of the baseline and simply add LoRA conditioning in a drop-in fashion, we expect further improvements from further optimizing the configuration for the entire architecture and training procedure. LoRA. We use the standard LoRA initialization as in the original LoRA paper Hu et al. (2022): for the LoRA matrices (A, B) with rank r, A is initialized as Aij \u223cN(0, 1/r) and B as the zero matrix. Following Ryu (2023), we set the rank of each basis LoRA to 4. For TimeLoRA and ClassLoRA, we use 11 and 10 LoRA bases, and for UC-LoRA we use 18 and 20 LoRA bases for CIFAR-10 and FFHQ. Due to our constrained computational budget, we were not able to conduct a full investigation on the optimal LoRA rank or the number LoRA bases. However, we experiment with the effect of rank and number of LoRA bases to limited extent and report the result in Section 5.4. 5.2 Proof-of-concept experiments We conduct toy experiments with nano diffusion for both discrete-time and continuous-SNR settings. Nano diffusion is a small diffusion model with a CNN-based U-Net architecture with no skip connections with about 500, 000 trainable parameters. We train nano diffusion on unconditional MNIST generation with 8 \f3 different conditioning methods: conventional scale-and-shift, TimeLoRA, and UC-LoRA. As shown in Figure 3, conditioning with TimeLoRA or UC-LoRA yields competitive result compared to the conventional scale-and-shift conditioning. Figure 3: MNIST samples generated by nano diffusion trained with (1st row) conventional scale-and-shift conditioning; (2nd row) TimeLoRA with linear interpolation initialization; (3rd row) UC-LoRA; and (4th row) TimeLoRA with random initialization. Initialization of \u03c9i(t) for TimeLoRA. As shown in Figure 3 the choice of initialization of \u03c9i(t) for TimeLoRA impacts performance. With randomly initialized \u03c9i(t), nano diffusion did not converge after 100 epochs, whereas with \u03c9i(t) initialized with the linear interpolation scheme, it did converge. Moreover, Figure 4 shows that even in UC-LoRA, \u03c9(t) shows higher similarity between nearby timesteps than between distant timesteps after training. This is consistent with our expectation that \u03c9i(t) \u2248\u03c9i(t\u2032) if t \u2248t\u2032. 250 500 750 1000 t1 200 400 600 800 1000 t2 250 500 750 1000 t1 1.0 0.5 0.0 0.5 1.0 Figure 4: Cosine similarity between \u03c9(t1) and \u03c9(t2) for UC-LoRA applied to nano diffusion (left) at initialization and (right) after training. At initialization, the cosine similarity between \u03c9(t1) and \u03c9(t2) has no discernible pattern. After training, however, the cosine similarity between \u03c9(t1) and \u03c9(t2) for t1 \u2248t2 is close to 1, implying their high similarity. 5.3 Main quantitative results Simply adding LoRA conditioning yields improvements. To evaluate the effectiveness of the drop-in addition of LoRA conditioning to the attention layers, we implement TimeLoRA and ClassLoRA to IDDPM and UC-LoRA to EDM, both with the conventional scale-and-shift conditioning on the convolutional layers unchanged. We train IDDPM with CIFAR-10, ImageNet64 and EDM with CIFAR-10, FFHQ64. As reported in Table 1, the addition of LoRA conditioning to the attention layers consistently improves the image generation quality as measured by FID scores Heusel et al. (2017) across different diffusion models and datasets with only (\u223c10%) addition of the parameter counts. Note these improvements are achieved without tuning any hyperparameters of the base model components. 9 \fInitializing the base model with pre-trained weights. We further test UC-LoRA on pre-trained EDM base models for unconditional CIFAR-10 and FFHQ64 generations. As reported in Table 1, using pre-trained weights showed additional gain on FID score with fewer number of interations (\u223c50%). To clarify, although we initialize the base model with pre-trained weights, we fully train both base model and LoRA modules rather than finetuning. LoRA can even replace scale-and-shift. We further evaluate the effectiveness of LoRA conditioning by replacing the scale-and-shift conditioning for the convolutional layers in residual blocks with LoRA conditioning for the attention blocks. The results of Table 1 suggest that solely using LoRA conditioning on attention layers achieves competitive FID scores while being more efficient in memory compared to the baseline score network trained with scale-and-shift conditioning on convolutional layers. For IDDPM, using LoRA in place of the conventional scale-and-shift conditioning consistently produces better results. Significant improvement is observed especially for class-conditional generation of CIFAR-10. For EDM, replacing the scale-and-shift conditioning did not yield an improvement, but nevertheless performed comparably. We note that in all cases, LoRA conditioning is more parameter-efficient (\u223c10%) than the conventional scale-and-shift conditioning. 5.4 Effect of LoRA rank and number of LoRA bases We investigate the effect of tuning the LoRA rank and the number of LoRA bases on the EDM model for unconditional CIFAR-10 generation and report the results in Table 2. Our findings indicate that using more LoRA bases consistently improves the quality of image generations. On the other hand, increasing LoRA rank does not guarantee better performance. These findings suggest an avenue of further optimizing and improving our main quantitative results of Section 5.3 and Table 1, which we have not yet been able to pursue due to our constrained computational budget. # basis rank FID # Params Varying # basis 9 4 1.99 57185519 18 4 1.96 57745499 36 4 1.95 58865459 Varying rank 18 2 1.93 57192539 18 4 1.96 57745499 18 8 1.96 58851419 Table 2: Effect of the number of LoRA bases and the LoRA rank on unconditional CIFAR-10 sampling of EDM with LoRA 5.5 Comparison with adaLN We compare the effectiveness of our proposed LoRA conditioning with adaLN conditioning applied to attention layers. Specifically, we conduct an experiment on EDM with scale-and-shift conditioning on convolutional layers removed and with (i) adaLN conditioning attention layers or (ii) LoRA conditioning attention layers. We compare the sample quality of unconditional and class-conditional CIFAR-10 generation and report the results in Table 3. We find that LoRA conditioning significantly outperforms adaLN conditioning for both unconditional and conditional CIFAR-10 generation. This indicates that our proposed LoRA conditioning is the more effective mechanism for conditioning attention layers in the U-Net architectures for diffusion models. Type uncond. cond. adaLN conditioning 2.16 2.0 LoRA conditioning 1.99 1.82 Table 3: Comparison of adaLN conditioning and LoRA conditioning on attention layers on EDM (without conditioning convolutional layers). We consider both unconditional and conditional CIFAR-10 generation. 10 \f5.6 Extrapolating conditioning information We conduct an experiment comparing two class-conditional EDM models each conditioned by scale-and-shift and ClassLoRA, for the CIFAR-10 dataset. During training, both models receive size-10 one-hot vectors (ci)j = \u03b4ij representing the class information. First, we input the linear interpolation \u03b1ci +(1\u2212\u03b1)cj (0 \u2264\u03b1 \u22641) of two class inputs ci and cj (corresponding to \u2018airplane\u2019 and \u2018horse\u2019, respectively) to observe the continuous transition between classes. As shown in the top of Figure 5, both the scale-and-shift EDM and ClassLoRA EDM models effectively interpolate semantic information across different classes. However, when a scaled input \u03b2ci is received, with \u03b2 ranging from -1 to 1, scale-and-shift EDM generates unrecognizable images when \u03b2 < 0, while ClassLoRA EDM generates plausible images throughout the whole range, as shown in the bottom of Figure 5. This toy experiment shows that LoRA-based conditioning may be more robust to extrapolating conditioning information beyond the range encountered during training. Appendix C provides further details. Figure 5: Results of (Top) interpolation of class labels in class-conditional EDM with (row1) ClassLoRA; (row2) scale-and-shift; (bottom) extrapolation of class labels in class-conditional EDM with (row1) ClassLoRA; (row2) scale-and-shift 6" + } + ], + "Jaewoong Cho": [ + { + "url": "http://arxiv.org/abs/2307.05906v1", + "title": "Mini-Batch Optimization of Contrastive Loss", + "abstract": "Contrastive learning has gained significant attention as a method for\nself-supervised learning. The contrastive loss function ensures that embeddings\nof positive sample pairs (e.g., different samples from the same class or\ndifferent views of the same object) are similar, while embeddings of negative\npairs are dissimilar. Practical constraints such as large memory requirements\nmake it challenging to consider all possible positive and negative pairs,\nleading to the use of mini-batch optimization. In this paper, we investigate\nthe theoretical aspects of mini-batch optimization in contrastive learning. We\nshow that mini-batch optimization is equivalent to full-batch optimization if\nand only if all $\\binom{N}{B}$ mini-batches are selected, while sub-optimality\nmay arise when examining only a subset. We then demonstrate that utilizing\nhigh-loss mini-batches can speed up SGD convergence and propose a spectral\nclustering-based approach for identifying these high-loss mini-batches. Our\nexperimental results validate our theoretical findings and demonstrate that our\nproposed algorithm outperforms vanilla SGD in practically relevant settings,\nproviding a better understanding of mini-batch optimization in contrastive\nlearning.", + "authors": "Jaewoong Cho, Kartik Sreenivasan, Keon Lee, Kyunghoo Mun, Soheun Yi, Jeong-Gwan Lee, Anna Lee, Jy-yong Sohn, Dimitris Papailiopoulos, Kangwook Lee", + "published": "2023-07-12", + "updated": "2023-07-12", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG" + ], + "main_content": "Introduction Contrastive learning has been widely employed in various domains as a prominent method for selfsupervised learning [25]. The contrastive loss function is designed to ensure that the embeddings of two samples are similar if they are considered a \u201cpositive\u201d pair, in cases such as coming from the same class [30], being an augmented version of one another [9], or being two different modalities of the same data [48]. Conversely, if two samples do not form a positive pair, they are considered a \u201cnegative\u201d pair, and the contrastive loss encourages their embeddings to be dissimilar. In practice, it is not feasible to consider all possible positive and negative pairs when implementing a contrastive learning algorithm due to the quadratic memory requirement O(N 2) when working with N samples. To mitigate this issue of full-batch training, practitioners typically choose a set of N/B mini-batches, each of size B = O(1), and consider the loss computed for positive and negative pairs within each of the N/B batches [7, 9, 24, 69, 8, 72, 17]. For instance, Gadre et al. [17] train a model on a dataset where N = 1.28 \u00d7 107 and B = 4096. This approach results in a memory \u2217Equal Contributions. Emails: . \u2020Correspondence to: Kangwook Lee . Preprint. Under review. arXiv:2307.05906v1 [cs.LG] 12 Jul 2023 \frequirement of O(B2) = O(1) for each mini-batch, and a total computational complexity linear in the number of chosen mini-batches. Despite the widespread practical use of mini-batch optimization in contrastive learning, there remains a lack of theoretical understanding as to whether this approach is truly reflective of the original goal of minimizing full-batch contrastive loss. This paper examines the theoretical aspects of optimizing mini-batches loaded for the contrastive learning. Main Contributions. The primary contributions of this paper are twofold. First, we show that under certain parameter settings, mini-batch optimization is equivalent to full-batch optimization if and only if all \u0000N B \u0001 mini-batches are selected. These results are based on an interesting connection between contrastive learning and the neural collapse phenomenon [36]. From a computational complexity perspective, the identified equivalence condition may be seen as somewhat prohibitive, as it implies that all \u0000N B \u0001 = O(N B) mini-batches must be considered. Our second contribution is to show that Ordered SGD (OSGD) [29] can be effective in finding mini-batches that contain the most informative pairs and thereby speeding up convergence. OSGD, proposed in a work by Kawaguchi & Lu [29], is a variant of SGD that modifies the model parameter updates. Instead of using the gradient of the average loss of all samples in a mini-batch, it uses the gradient of the average loss over the top-q samples in terms of individual loss values. We show that the convergence result from Kawaguchi & Lu [29] can be applied directly to contrastive learning. We also show that OSGD can improve the convergence rate of SGD by a constant factor in certain scenarios. Furthermore, in a novel approach to address the challenge of applying OSGD to the \u0000N B \u0001 mini-batch optimization (which involves examining O(N B) batches to select high-loss ones), we reinterpret the batch selection as a min-cut problem in graph theory [13]. This novel interpretation allows us to select high-loss batches efficiently via a spectral clustering algorithm [43]. The following informal theorems summarize our main findings. Theorem 1 (informal). Under certain parameter settings, the mini-batch optimization of contrastive loss is equivalent to full-batch optimization of contrastive loss if and only if all \u0000N B \u0001 mini-batches are selected. Although \u0000N B \u0001 mini-batch contrastive loss and full-batch loss are neither identical nor differ by a constant factor, the optimal solutions for both mini-batch and full-batch are identical (see Sec. 4). Theorem 2 (informal). In a demonstrative toy example, OSGD operating on the principle of selecting high-loss batches, can potentially converge to the optimal solution of mini-batch contrastive loss optimization faster by a constant factor compared to SGD (see Sec. 5.1). We validate our theoretical findings and the efficacy of the proposed spectral clustering-based batch selection method by conducting experiments on both synthetic and real data. On synthetic data, we show that our proposed batch-selection algorithms do indeed converge to the optimal solution of full-batch optimization significantly faster than the baselines. We also apply our proposed method to ResNet pre-training with CIFAR-100 [31] and Tiny ImageNet [33]. We evaluate the performance on downstream retrieval tasks, demonstrating that our batch selection method outperforms vanilla SGD in practically relevant settings. 2 Related Work Contrastive losses. Contrastive learning has been used for several decades to learn a similarity metric to be used later for applications such as object detection and recognition [41, 1]. Chopra et al. [12] proposed one of the early versions of contrastive loss which has been updated and improved over the years [60, 61, 59, 30, 45]. More recently, contrastive learning has been shown to rival and even surpass traditional supervised learning methods, particularly on image classification tasks [10, 3]. Further, its multi-modal adaptation leverages vast unstructured data, extending its effectiveness beyond image and text modalities [48, 27, 47, 39, 56, 16, 18, 34, 51, 52]. Unfortunately, these methods require extremely large batch sizes in order to perform effectively. Follow-up works showed that using momentum or carefully modifying the augmentation schemes can alleviate this issue to some extent [22, 10, 19, 64]. Effect of batch size. While most successful applications of contrastive learning use large batch sizes (e.g., 32,768 for CLIP and 8,192 for SimCLR), recent efforts have focused on reducing batch 2 \fsizes and improving convergence rates [65, 7]. Yuan et al. [68] carefully study the effect of the requirements on the convergence rate when a model is trained for minimizing SimCLR loss, and prove that the gradient of the solution is bounded by O( 1 \u221a B ). They also propose SogCLR, an algorithm with a modified gradient update where the correction term allows for an improved convergence rate with better dependence on B. It is shown that the performance for small batch size can be improved with the technique called hard negative mining [55, 28, 70]. Neural collapse. Neural collapse is a phenomenon observed in [46] where the final classification layer of deep neural nets collapses to the simplex Equiangular Tight Frame (ETF) when trained well past the point of zero training error [26, 71]. Lu & Steinerberger [36] prove that this occurs when minimizing cross-entropy loss over the unit ball. We extend their proof techniques and show that the optimal solution for minimizing contrastive loss under certain conditions is also the simplex ETF. Optimal permutations for SGD. The performance of SGD without replacement under different permutations of samples has been well studied in the literature [5, 53, 54, 42, 66, 2, 49, 40, 58, 57, 20, 44, 37, 50, 63, 38, 6, 11]. One can view batch selection in contrastive learning as a method to choose a specific permutation among the possible \u0000N B \u0001 mini-batches of size B. However, it is important to note that these bounds do not indicate an improved convergence rate for general non-convex functions and thus would not apply to the contrastive loss, particularly in the setting where the embeddings come from a shared embedding network. We show that in the case of OSGD [29], we can indeed prove that contrastive loss satisfies the necessary conditions in order to guarantee convergence. 3 Problem Setting Suppose we are given a dataset {(xi, yi)}N i=1 of N positive pairs (data sample pairs that are conceptually similar or related), where xi and yi are two different views of the same object. Note that this setup includes both the multi-modal setting (e.g., CLIP [48]) and the uni-modal setting (e.g., SimCLR [9]) as follows. For the multi-modal case, one can view (xi, yi) as two different modalities of the same data, e.g., xi is the image of a scene while yi is the text description of the scene. For the uni-modal case, one can consider xi and yi as different augmented images from the same image. We consider the contrastive learning problem where the goal is to find embedding vectors for {xi}N i=1 and {yi}N i=1, such that the embedding vectors of positive pairs (xi, yi) are similar, while ensuring that the embedding vectors of other (negative) pairs are well separated. Let ui \u2208Rd be the embedding vector of xi, and vi \u2208Rd be the embedding vector of yi. In practical settings, one typically considers parameterized encoders so that ui = f\u03b8(xi) and vi = g\u03d5(yi). We define embedding matrices U := [u1, u2, . . . uN] and V := [v1, v2, . . . , vN] which are the collections of embedding vectors. Now, we focus on the simpler setting of directly optimizing the embedding vectors instead of model parameters \u03b8 and \u03d5 in order to gain theoretical insights into the learning embeddings. This approach enables us to develop a deeper understanding of the underlying principles and mechanisms. Consider the problem of directly optimizing the embedding vectors for N pairs which is given by min U,V Lcon(U, V ) s.t. \u2225ui\u2225= 1, \u2225vi\u2225= 1 \u2200i \u2208[N], (1) where \u2225\u00b7\u2225denotes the \u21132 norm, the set [N] denotes the set of integers from 1 to N, and the contrastive loss (the standard InfoNCE loss [45]) is defined as Lcon(U, V ) := \u22121 N N X i=1 log eu\u22ba i vi PN j=1 eu\u22ba i vj ! \u22121 N N X i=1 log ev\u22ba i ui PN j=1 ev\u22ba i uj ! . (2) Note that Lcon(U, V ) is the full-batch version of the loss which contrasts all embeddings with each other. However, due to the large computational complexity and memory requirements during optimization, practitioners often consider the following mini-batch version instead. Note that there exist \u0000N B \u0001 different mini-batches, each of which having B samples. For k \u2208 h\u0000N B \u0001i , let Bk be the k-th mini-batch satisfying Bk \u2282[N] and |Bk| = B. Let UBk := {ui}i\u2208Bk and VBk := {vi}i\u2208Bk. Then, the contrastive loss for the k-th mini-batch is Lcon(UBk, VBk). 3 \f4 Relationship Between the Optimization for Full-Batch and Mini-Batch Recall that we focus on finding the optimal embedding matrices (U, V ) that minimize the contrastive loss. In this section, we investigate the relationship between the problem of optimizing the full-batch loss Lcon(U, V ) and the problem of optimizing the mini-batch loss Lcon(UBk, VBk). Towards this goal, we prove three main results, the proof of which are in Appendix B.1. \u2022 We derive the optimal solution that minimizes the full-batch loss (Lem. 1, Thm. 3). \u2022 We show that the solution that minimizes the average of \u0000N B \u0001 mini-batch losses is identical to the one that minimizes the full-batch loss (Prop. 1, Thm. 4). \u2022 We show that minimizing the mini-batch loss summed over only a strict subset of \u0000N B \u0001 mini-batches can lead to a sub-optimal solution that does not minimize the full-batch loss (Thm. 5). 4.1 Full-batch Contrastive Loss Optimzation In this section, we characterize the optimal solution for the full-batch loss minimization in Eq. (1). We start by providing the definition of the simplex equiangular tight frame (ETF) which turns out to be the optimal solution in certain cases. The original definition of ETF [62] is for N vectors in a d-dimensional space where N \u2265d + 1 3. Papyan et al. [46] defines the ETF for the case where N \u2264d+1 to characterize the phenomenon of neural collapse. In our work, we use the latter definition of simplex ETFs which is stated below. Definition 1 (Simplex ETF). We call a set of N vectors {ui}N i=1 form a simplex Equiangular Tight Frame (ETF) if \u2225ui\u2225= 1, \u2200i \u2208[N] and u\u22ba i uj = \u22121/(N \u22121), \u2200i \u0338= j. In the following Lemma, we first prove that the optimal solution of full-batch contrastive learning is the simplex ETF for N \u2264d + 1 which follows almost directly from Lu & Steinerberger [36]. Lemma 1 (Optimal solution when N \u2264d + 1). Suppose N \u2264d + 1. Then, the optimal solution (U \u22c6, V \u22c6) of the full-batch contrastive learning problem in Eq. (1) satisfies two properties: (i) U \u22c6= V \u22c6, and (ii) the columns of U \u22c6form a simplex ETF. Actually, many practical scenarios satisfy N > d + 1. However, the approach used in Lu & Steinerberger [36] cannot be directly applied for N > d + 1, leaving it as an open problem. While solving the open problem for the general case seems difficult, we characterize the optimal solution for the specific case of N = 2d, subject to the conditions stated below. Definition 2 (Symmetric and Antipodal). Embedding matrices U and V are called symmetric and antipodal if (U, V ) satisfies two properties: (i) Symmetric i.e., U = V ; (ii) Antipodal i.e., for each i \u2208[N], there exists j(i) such that uj(i) = \u2212ui. We conjecture that the optimal solutions for N = 2d are symmetric and antipodal. Note that the symmetric property holds for N \u2264d+1 case, and the antipodality is frequently assumed in geometric problems such as the sphere covering problem in [4]. Thm. 3 shows that when N = 2d, the optimal solution for the full-batch loss minimization, under a symmetric and antipodal configuration, form a cross-polytope which is defined as the following. Definition 3 (Simplex cross-polytope). We call a set of N vectors {u}N i=1 form a simplex crosspolytope if, for all i, the following three conditions hold: \u2225ui\u2225= 1; there exists a unique j such that u\u22ba i uj = \u22121; and u\u22ba i uk = 0 for all k / \u2208{i, j}. Theorem 3 (Optimal solution when N = 2d). Let (U \u22c6, V \u22c6) := arg min (U,V )\u2208A Lcon(U, V ) s.t. \u2225ui\u2225= 1, \u2225vi\u2225= 1 \u2200i \u2208[N], (3) where A := {(U, V ) : U, V are symmetric and antipodal}. Then, the columns of U \u22c6form a simplex cross-polytope for N = 2d. Proof Outline. By the antipodality assumption, we can apply Jensen\u2019s inequality to N \u22122 indices without itself ui and antipodal point \u2212ui for a given i \u2208[N]. Then we show that the simplex crosspolytope also minimizes this lower bound while satisfying the conditions that make the applications of Jensen\u2019s inequality tight. 3See Def. 4 in Appendix A for the full definition 4 \f\u2212\u03c0 \u2212\u03c0/2 0 \u03c0/2 \u03c0 \u03b8 = arctan(u1,2/u1,1) [radian] 1.5 1.6 1.7 1.8 1.9 loss full batch mini batch (a) ACxHicjVHLSsNAFD2Nr1pfVZdugkVwVRIp6kYoCuKyBdsKtUiSTmtoXsxMhFL0B9zqt4l/oH/hnXEKahGdkOTMufecmXuvn0Wh kI7zWrDm5hcWl4rLpZXVtfWN8uZW6Q5D1grSKOUX/meYFGYsJYMZcSuMs682I9Yx+dqXjnjnERpsmlHGesF3vDJByEgSeJap7cl CtO1dHLngWuARWY1UjL7hGHykC5IjBkEASjuB0NOFCwcZcT1MiOEQh1nuEeJtDlMcrwiB3Rd0i7rmET2itPodUBnRLRy0lpY4 80KeVxwuo0W8dz7azY37wn2lPdbUx/3jFxErcEvuXbpr5X52qRWKAY1DSDVlmlHVBcYl1RN7e/VCXJISNO4T7FOeFAK6d9trV G6NpVbz0df9OZilX7wOTmeFe3pAG7P8c5C9oHVfewWmvWKvVTM+oidrCLfZrnEeq4QAMt7f2IJzxb51ZkCSv/TLUKRrONb8t6+AD h249L= 2 1 3 1 4 2 3 4 2 1 2 3 1 3 4 1 4 3 2 3 4 2 4 4 3 4 Full Batch (B=4) (B=2) (B=2) All Combinations of Mini-Batches Some Mini-Batches Train Data Config. Solution u1 u3 u2 u4 u1 u3 u2 u4 u1 u2 u3 u4 2 1 3 1 4 2 3 4 1 2 1 1 2 3 ADLnicjVFNb9QwEH1NgZblo0s59hKxQuJQrZLtCrgVeXCsRVsW6muVo7X3Vp1PuQ4lVZR/hP/hBuIA+JafgBXYGy8ElAhcJTk+c 28Z89MVmlV2yT5uBKt3rh5a239du/O3Xv3N/oPNg/rsjFCTkSpS3Oc8VpqVciJVbL48pInmdaHmUXL1386FKaWpXFG7uo5GnO54U6U4JboqZ9xXJuzwX7etuhe/iHsk3NVtNwYvuharbsea1mbo9Ytx07sLMEYwKMUXi05EaeI7DjACOpLGbBatofJMPEr/g6SAMYIKz9sv8BDOUEGiQ6KAJazBUdNzghQJKuJO0RJnCkfl+jQI21DWZIyOLEX9J3T7iSwBe2dZ+3Vgk7R9BpSxnhMmpLyDGF3WuzjXd27 N+8W+/p7ragfxa8cmItzon9l26Z+b86V4vFGZ7GhTVHnGVSeCS+O74m4e/1KVJYeKOIdnFDeEhVcu+x7Te1rd73lPn7lMx3r9iLkNvjibkDTv8c53VwOBqmT4fjg/Fgdy+Meh1beIQnNM9n2MUr7GNC3u/wFd/wPXobvY8+RZ9/pkYrQfMQv63o6gfVd7ZS SB = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}} AC53icjVHLSsRAEKyN7/W16tFLcBE8LEui3oRC8eV3RVMLJMxlEH8yKZCBL27s2bePUHvOqfiH+gf2HPGMEHohOS1FR31U x3+0kgM+U4zxWr39gcGh4pDo6Nj4xWZua3sviPOWiw+MgTg98lolARqKjpArEQZIKFvqB2PfPN3V8/0KkmYyjXWZiKOQnUbyRHKmiOrW5ryQqTPOgmKn192w1+yqV3iF21hsLDVaXs/2etVure40HbPsn8AtQR3lase1J3g4RgyOHCEIijCARgyeg7hwkFC3BEK4lJC0sQFeqiSNqcsQRmM2HP6ntLusGQj2mvPzKg5nRLQm5LSxjxpYspLCevTbBPjbNmf/MujKe+2yX9/dIrJFbhjNi/dB+Z/9XpW hROsGpqkFRTYhdHS9dctMVfXP7U1WKHBLiND6meEqYG+VHn2jyUzturfMxF9Mpmb1npe5OV71LWnA7vdx/gR7i013udnabtXN8pRD2MWc1igea5gHVto0PeV7jHAx4taV1bN9bte6pVKTUz+LKsuzd6JsKSB = {{1, 2, 3, 4}} AC63icjVHLSsRAECzja1fqx69BfFw7IkuqgXYdGLR0VXBSPLZBw1bF4kE0FC/sCbN/HqD3jV/xD/QP/CnjGCD0QnJKmp7q qZ7nZj30ulZT3Gf0Dg0PDlZHq6Nj4xGRtano/jbKEiw6P/Cg5dFkqfC8UHelJXxzGiWCB64sDt7ep4gcXIkm9KNyTl7E4DthZ6J16nEmiurUFJ2DynDM/3y26G+a6WXVyJ7cbS07RcPLlRspTKeodmt1q2npZf4EdgnqKNd2VHuCgxNE4MgQCEJOyDIaXnCDYsxMQdIycuIeTpuECBKmkzyhKUwYjt0feMdkclG9JeaZazekUn96ElCbmSRNRXkJYnWbqeKadFfubd6491d0u6e+WXgGxEufE/qX7y PyvTtUicYo1XYNHNcWaUdXx0iXTXVE3Nz9VJckhJk7hE4onhLlWfvTZ1JpU1656y3T8RWcqVu15mZvhVd2SBmx/H+dPsL/UtFearZ1Wvb1RjrqCWcxhkea5ija2sI0OeV/hHg94NALj2rgxbt9Tjb5SM4Mvy7h7A0dinOI=SB = {{1, 2}, {3, 4}} ACx3icjVHLSsNAFD2Nr1pfVZdugkVwVRIp6rLoRncV7APaIk6bUPzcjIpluLCH3Crfyb+gf6Fd8YpqEV0Q pIz595zZu69bhL4qbCs15yxsLi0vJfLaytb2xuFbd3GmcY/VvTiIect1Uhb4EasLXwSslXDmhG7Amu7oXMabY8ZT P46uxSRh3dAZRH7f9xwhqU7Ebm+KJatsqWXOA1uDEvSqxcUXdNBDA8ZQjBEIQDOEjpacOGhYS4LqbEcUK+ijPco0D ajLIYZTjEjug7oF1bsxHtpWeq1B6dEtDLSWnigDQx5XHC8jRTxTPlLNnfvKfKU95tQn9Xe4XECgyJ/Us3y/yvTtYi0M epqsGnmhLFyOo87ZKprsibm1+qEuSQECdxj+KcsKeUsz6bSpOq2mVvHRV/U5mSlXtP52Z4l7ekAds/xzkPGkdl+7hcu aqUqmd61HnsYR+HNM8TVHGBGurkPcQjnvBsXBqxMTbuPlONnNbs4tsyHj4AoNKQzA=6= (b) Figure 1: (a) Comparing mini-batch loss and full-batch loss when N = 10, B = 2, and d = 2. We illustrate this by manipulating a single embedding vector u1 while maintaining all other embeddings (v1 and {ui, vi}10 i=2) at their optimal solutions. Specifically, u1 = [u1,1, u1,2] is varied as [cos(\u03b8), sin(\u03b8)] for \u03b8 \u2208[\u2212\u03c0, \u03c0]. While the two loss functions are not identical, corroborating Prop.1, their minimizers align, providing empirical support for Thm. 4; (b) The relationship between full-batch and mini-batch optimization in contrastive learning. Consider optimizing N = 4 pairs of d = 3 dimensional embedding vectors {(ui, vi)}N i=1 where ui and vi are shown as colored square and circle, respectively. The index i is written in the square/circle. The black rounded box represents a batch. We compare three batch selection options: (i) full batch, i.e., B = 4, (ii) all \u0000N B \u0001 = 6 mini-batches with size B = 2, and (iii) some mini-batches. Here, SB is the set of mini-batches where each mini-batch is represented by the set of constituent samples\u2019 indices. Our theoretical/empirical findings are: the optimal embedding that minimizes full-batch loss and the one that minimizes the sum of \u0000N B \u0001 mini-batch losses are identical, while the one that minimizes the mini-batch losses summed over only a strict subset of \u0000N B \u0001 batches does not guarantee the negative correlation between ui and uj for i \u0338= j. This illustration is supported by our mathematical results in Thms. 4 and 5. For the general case of N > d + 1, excluding N = 2d, we still leave it as an open problem. 4.2 Mini-batch Contrastive Loss Optimization Here we consider the mini-batch contrastive loss optimization problem, where we first choose multiple mini-batches of size B and then find U, V that minimize the sum of contrastive losses computed for the chosen mini-batches. Note that this is the loss that is typically considered in the contrastive learning since computing the full-batch loss is intractable in practice. Let us consider a subset of all possible \u0000N B \u0001 mini-batches and denote their indices by SB \u2286 h\u0000N B \u0001i . For a fixed SB, the mini-batch loss optimization problem is formulated as: min U,V Lcon mini(U, V ; SB) s.t. \u2225ui\u2225= 1, \u2225vi\u2225= 1 \u2200i \u2208[N], (4) where the loss of given mini-batches is Lcon mini(U, V ; SB) := 1 |SB| P i\u2208SB Lcon(UBi, VBi). To analyze the relationship between the full-batch loss minimization in Eq. (1) and the mini-batch loss minimization in Eq. (4), we first compare the objective functions of two problems as below. Proposition 1. The mini-batch loss and full-batch loss are not identical, nor is one a simple scaling of the other by a constant factor. In other words, when SB = h\u0000N B \u0001i , for all B \u22652, there exists no constant c such that Lcon mini(U, V ; SB) = c \u00b7 Lcon(U, V ) for all U, V . We illustrate this proposition by visualizing the two loss functions in Fig. 1(a) when N = 10, B = 2, and d = 2. We visualize it along a single embedding vector u1 by freezing all other embeddings (v1 and {ui, vi}10 i=2) at the optimal solution and varying u1 = [u1,1, u1,2] as [cos(\u03b8), sin(\u03b8)] for \u03b8 \u2208[\u2212\u03c0, \u03c0]. One can confirm that two losses are not identical (even up to scaling). Interestingly, the following result shows that the optimal solutions of both problems are identical. Theorem 4 (Optimization with all possible \u0000N B \u0001 mini-batches). Suppose B \u22652. The set of minimizers of the \u0000N B \u0001 mini-batch problem in Eq. (4) is the same as that of the full-batch problem in Eq. (1) for 5 \ftwo cases: (i) N \u2264d + 1, and (ii) N = 2d and the pairs (U, V ) are restricted to those satisfying the conditions stated in Def. 2. In such cases, the solutions (U, V ) for the \u0000N B \u0001 mini-batch optimization problem satisfies the following: Case (i) {ui}N i=1 forms a simplex ETF and ui = vi for all i \u2208[N]; Case (ii): {ui}N i=1 forms a simplex cross-polytope. Proof Outline. Similar to the proof of Lem. 1, we bound the objective function from below using Jensen\u2019s inequality. Then, we show that this lower bound is equivalent to a scaling of the bound from the proof of Lem. 1, by using careful counting arguments. Then, we can simply repeat the rest of the proof to show that the simplex ETF also minimizes this lower bound while satisfying the conditions that make the applications of Jensen\u2019s inequality tight. Now, we present mathematical results specifying the cases when the solutions of mini-batch optimization and full-batch optimization differ. First, we show that when B = 2, minimizing the mini-batch loss over any strict subset of \u0000N B \u0001 batches, is not equivalent to minimizing the full-batch loss. Theorem 5 (Optimization with fewer than \u0000N B \u0001 mini-batches). Suppose B = 2 and N \u2264d + 1. Then, the minimizer of Eq. (4) for SB \u228a h\u0000N B \u0001i is not the minimizer of the full-batch optimization in Eq. (1). Proof Outline. We show that there exist embedding vectors that are not the simplex ETF, and have a strictly lower objective value. This implies that the optimal solution of any set of mini-batches that does not contain all \u0000N 2 \u0001 mini-batches is not the same as that of the full-batch problem. The result of Thm. 5 is extended to the general case of B \u22652, under some mild assumption; please check Prop. 2 and 3 in Appendix B.1. Fig. 1(b) summarizes the main findings in this section. 5 Ordered Stochastic Gradient Descent for Mini-Batch Contrastive Learning Recall that the optimal embeddings for the full-batch optimization problem in Eq. (1) can be obtained by minimizing the sum of \u0000N B \u0001 mini-batch losses, according to Thm. 4. An easy way of approximating the optimal embeddings is using gradient descent (GD) on the sum of losses for \u0000N B \u0001 mini-batches, or to use a stochastic approach which applies GD on the loss for a randomly chosen mini-batch. Recent works found that applying GD on selective batches outperforms SGD in some cases [29, 37, 35]. A natural question arises: does this hold for mini-batch contrastive learning? Specifically, (i) Is SGD enough to guarantee good convergence on mini-batch contrastive learning?, and (ii) Can we come up with a batch selection method that outperforms vanilla SGD? To answer this question: \u2022 We show that Ordered SGD (OSGD) [29] can potentially accelerate convergence compared to vanilla SGD in a demonstrative toy example (Sec. 5.1). We also show that the convergence results from Kawaguchi & Lu [29] can be extended to mini-batch contrastive loss optimization (Sec. 5.2). \u2022 We reformulate the batch selection problem into a min-cut problem in graph theory [13], by considering a graph with N nodes where each node is each positive pair and each edge represents a proxy to the contrastive loss between two nodes. This allows us to devise an efficient batch selection algorithm by leveraging spectral clustering [43] (Sec. 5.3). 5.1 Convergence Comparison in a Toy Example: OSGD vs. SGD This section investigates the convergence of two gradient-descent-based methods, OSGD and SGD. The below lemma shows that the contrastive loss is geodesic non-quasi-convex, which implies the hardness of proving the convergence of gradient-based methods for contrastive learning in Eq. (1). Lemma 2. Contrastive loss Lcon(U, V ) is a geodesic non-quasi-convex function of U, V on T = {(U, V ) : \u2225ui\u2225= \u2225vi\u2225= 1, \u2200i \u2208[N]}. We provide the proof in Appendix B.2. In order to compare the convergence of OSGD and SGD, we focus on a toy example where convergence to the optimal solution is achievable with appropriate initialization. Consider a scenario where we have N = 4 embedding vectors {ui}N i=1 with ui \u2208R2. Each embedding vector is defined as u1 = (cos \u03b81, sin \u03b81); u2 = (cos \u03b82, \u2212sin \u03b82); u3 = (\u2212cos \u03b83, \u2212sin \u03b83); u4 = (\u2212cos \u03b84, sin \u03b84) for parameters {\u03b8i}n i=1. Over time step t, we consider updating the parameters \u03b8(t) := [\u03b8(t) 1 , \u03b8(t) 2 , \u03b8(t) 3 , \u03b8(t) 4 ] 6 \f(a) 0 25 50 75 100 125 150 175 200 Update steps 1.3 1.4 1.5 1.6 Contrastive loss OSGD SGD N choose B (full) (b) Figure 2: (a) Toy example considered in Sec. 5.1; (b) The training loss curves of three algorithms (OSGD, SGD, and \u0000N B \u0001 full-batch gradient descent) applied on the toy example when N = 4 and B = 2. The x-axis represents the number of update steps, while the y-axis displays the loss in Eq. (2). OSGD converges the fastest among the three methods. using gradient descent based methods. For all i, the initial parameters are set as \u03b8(0) i = \u03f5 > 0, and the other embedding vectors are initialized as v(0) i = u(0) i . This setting is illustrated in Fig. 2(a). At each time step t, each learning algorithm begins by selecting a mini-batch B(t) \u2282{1, 2, 3, 4} with batch size |B(t)| = 2. SGD randomly selects a mini-batch, while OSGD selects a mini-batch as follows: B(t) = arg max B\u2208S Lcon(UB(\u03b8(t)), VB(\u03b8(t))). Then, the algorithms update \u03b8(t) using gradient descent on Lcon(UB, VB) with a learning rate \u03b7: \u03b8(t+1) = \u03b8(t) \u2212\u03b7\u2207\u03b8Lcon(UB(t), VB(t)). For a sufficiently small margin \u03c1 > 0, let TOSGD, TSGD be the minimal time required for the algorithms to reach the condition E[\u03b8(T )] \u2208(\u03c0/4 \u2212\u03c1, \u03c0/4)N. Under this setting, the following theorem compares OSGD and SGD, in terms of the lower bound on the time required for the convergence to the optimal solution. Theorem 6. Consider the described setting where the parameters \u03b8(t) of embedding vectors are updated, as shown in Fig. 2(b). Suppose there exist \u02dc \u03f5, T such that for all t satisfying B(t) = {1, 3} or {2, 4}, \u2225\u2207\u03b8(t)Lcon(UB(t), VB(t))\u2225\u2264\u02dc \u03f5, and TOSGD, TSGD < T. Then, we have the following inequalities: TOSGD \u2265\u03c0/4 \u2212\u03c1 \u2212\u03f5 + O(\u03b72\u03f5 + \u03b7\u03f53) \u03b7\u03f5 , TSGD \u22653(e2 + 1) e2 \u22121 \u03c0/4 \u2212\u03c1 \u2212\u03f5 + O(\u03b72\u03f5 + \u03b72\u02dc \u03f5) \u03b7\u03f5 + O(\u03b7\u03f53 + \u03b7\u02dc \u03f5) . Corollary 1. Suppose lower bounds of TOSGD, TSGD in Thm. 6 are tight, and the learning rate \u03b7 is small enough. Then, TOSGD/TSGD = (e2 \u22121)/3(e2 + 1) \u22481/4. In Fig. 2(b), we present training loss curves of the full-batch contrastive loss in Eq. (2) for various algorithms implemented on the toy example. One can observe that the losses of all algorithms eventually converge to 1.253, the optimal loss achievable when the solution satisfies ui = vi and {ui}N i=1 form simplex cross-polytope. As shown in the figure, OSGD converges faster than SGD to the optimal loss. This empirical evidence corroborates our theoretical findings in Corollary 1. 5.2 Convergence of OSGD in Mini-batch Contrastive Learning Setting Recall that it is challenging to prove the convergence of gradient-descent-based methods for contrastive learning problem in Eq. (1) due to the non-quasi-convexity of the contrastive loss Lcon. Instead of focusing on the contrastive loss, we consider a proxy, the weighted contrastive loss defined as e Lcon(U, V ) := 1 q P( N B) j=1 \u03b3jLcon(UB(j), VB(j)) with \u03b3j = Pq\u22121 l=0 \u0000j\u22121 l \u0001\u0000( N B)\u2212j k\u2212l\u22121 \u0001 / \u0000( N B) k \u0001 for two arbitrary natural numbers k, q \u2264 \u0000N B \u0001 where B(j) is a mini-batch with j-th largest loss among batches of size B. Indeed, this is a natural objective obtained by applying OSGD to our problem, and we show the convergence of such an algorithm by extending the results in Kawaguchi & Lu [29]. OSGD updates the embedding vectors using the gradient averaged over q batches that have the largest losses among randomly chosen k batches (see Algo. 2 in Appendix B.2). Let U (t), V (t) be the updated embedding matrices when applying OSGD for t steps starting from U (0), V (0), using the learning rate \u03b7t. Then the following theorem, proven in Appendix B.2, holds. 7 \f2 4 6 contrastive loss 0 100 200 # of batches epoch = 0 2 4 0 100 200 epoch = 1 0.0 0.5 1.0 0 100 200 epoch = 51 0.0 0.5 1.0 0 100 200 epoch = 101 random spectral clustering Figure 3: Histograms of batch counts for N/B batches, for the contrastive loss measured from ResNet-18 models trained on CIFAR-100 using SGD, where N=50,000 and B=20. Each plot is derived from a distinct training epoch. Here we compare two batch selection methods: (i) randomly shuffling N samples and partition them into N/B batches of size B, (ii) our SC method given in Algo. 1. The histograms show that batches generated through the proposed spectral clustering method tend to contain a higher proportion of large loss values when compared to random batch selection. Similar results are observed in different settings, details of which are given in Appendix D.1. Theorem 7 (Convergence results). Consider sampling t\u22c6from [T \u22121] with probability proportional to {\u03b7t}T \u22121 t=0 , that is, P(t\u22c6= t) = \u03b7t/(PT \u22121 i=0 \u03b7i). Then \u2200\u03c1 > \u03c10 = 2 p 2/B + 4e2/B, we have E \u0014\r \r \r\u2207e Lcon(U (t\u22c6), V (t\u22c6)) \r \r \r 2\u0015 \u2264(\u03c1 + \u03c10)2 \u03c1(\u03c1 \u2212\u03c10) \u0010 e Lcon(U (0), V (0)) \u2212e Lcon \u22c6\u0011 + 8\u03c1 PT \u22121 t=0 \u03b72 t PT \u22121 t=0 \u03b7t , where e Lcon \u22c6denotes the minimized value of e Lcon. Given sufficiently small learning rate \u03b7t \u223cO(t\u22121/2), E\u2225\u2207e Lcon\u22252 decays at the rate of e O(T \u22121/2). Therefore, this theorem guarantees the convergence of OSGD for mini-batch contrastive learning. 5.3 Suggestion: Spectral Clustering-based Approach Algorithm 1: Spectral Clustering Method Input: the number of positive pairs N, batch size B, embedding matrices: U, V Output: selected batches {Bj}N/B j=1 1 Construct the affinity matrix A: Aij = 1{i \u0338= j} \u00d7 w(i, j) 2 Construct the degree matrix D from A: Dij = 1{i = j} \u00d7 (PN j=1 Aij) 3 L \u2190D \u2212A; k \u2190N/B 4 {Bj}N/B j=1 \u2190Apply the even-sized spectral clustering algorithm with L and k 5 return {Bj}N/B j=1 Applying OSGD to mini-batch contrastive learning has a potential benefit as shown in Sec. 5.1, but it also has some challenges. Choosing the best q batches with high loss in OSGD is only doable after we evaluate losses of all \u0000N B \u0001 combinations, which is computationally infeasible for large N. A naive solution to tackle this challenge is to first randomly choose k batches and then select q highloss batches among k batches. However, this naive random batch selection method does not guarantee that the chosen q batches are having the highest loss among all \u0000N B \u0001 candidates. Motivated by these issues of OSGD, we suggest an alternative batch selection method inspired by graph theory. Note that the contrastive loss Lcon(UB, VB) for a given batch B is lower bounded as follows: 1 B(B \u22121) \uf8f1 \uf8f2 \uf8f3 X i\u2208B X j\u2208B\\{i} log \u0010 1 + (B \u22121)eu\u22ba i (vj\u2212vi)\u0011 + log \u0010 1 + (B \u22121)ev\u22ba i (uj\u2212ui)\u0011 \uf8fc \uf8fd \uf8fe. (5) This lower bound is derived using Jensen\u2019s inequality. Detailed derivation is provided in Appendix C.1. A nice property of this lower bound is that it can be expressed as a summation of terms over a pair (i, j) of samples within batch B. Consider a graph G with N nodes, where the weight between node k and l is defined as w(k, l) := P (i,j)\u2208{(k,l),(l,k)} log \u0010 1 + (B \u22121)eu\u22ba i (vj\u2212vi)\u0011 + log \u0010 1 + (B \u22121)ev\u22ba i (uj\u2212ui)\u0011 . Recall that our goal is to choose q batches having the highest contrastive loss among \u0000N B \u0001 batches. We relax this problem by reducing our search space such that the q = N/B chosen batches B1, \u00b7 \u00b7 \u00b7 , Bq form a partition of N samples, i.e., Bi \u2229Bj = \u2205and 8 \fd = 2N -1.0 0.0 1.0 0 100 200 300 400 500 0.0 0.5 1.0 1.5 2.0 2.5 \u2225U\u2217TV \u2217\u2212UTV \u2225F OSGD Spectral Clustering Method SGD d = N/2 (a) optimal (b) full-batch (c) \u0000N B \u0001 -all -1.0 0.0 1.0 (d) \u0000N B \u0001 -sub 0 100 200 300 400 500 Update steps 0 1 2 3 4 \u2225U\u2217TV \u2217\u2212UTV \u2225F (e) norm difference Figure 4: The behavior of embedding matrices U, V optimized by different batch selection methods for N = 8 and B = 2 (Top: d = 2N, Bottom: d = N/2). (a)-(d): Heatmap of N \u00d7 N matrix visualizing the pairwise inner products u\u22ba i vj, where (a): ground-truth solution (ETF for d = 2N, cross-polytope for d = N/2), (b): optimized the full-batch loss with GD, (c): optimized the sum of \u0000N B \u0001 mini-batch losses with GD, (d): optimized a partial sum of \u0000N B \u0001 mini-batch losses with GD. Note that both (b) and (c) reach the ground-truth solution in (a), while (d) does not, supporting our theoretical results in Sec. 4.2. Further, (e) compares the convergence of three mini-batch selection algorithms: 1) SGD, 2) OSGD, and 3) our spectral clustering method, when updating embeddings for 500 steps. OSGD and our method nearly converge to the optimal solution, while SGD does not. Here, y-axis represents the Frobenius norm of the difference between the heatmaps of the optimal solution and the updated embeddings, denoted by \u2225U \u22c6\u22baV \u22c6\u2212U \u22baV \u2225F . \u222ai\u2208[q]Bi = [N]. In such scenario, our target problem is equivalent to the problem of clustering N nodes in graph G into q clusters with equal size, where the objective is to minimize the sum of weights of inter-cluster edges. This problem is nothing but the min-cut problem [13], and we can employ even-sized spectral clustering algorithm which solves it efficiently. The pseudo-code of our batch selection method4 is provided in Algo. 1, and further details of the algorithm are provided in Appendix C. Fig. 3 shows the histogram of contrastive loss for N/B batches chosen by the random batch selection method and the proposed spectral clustering (SC) method. One can observe that the SC method favors batches with larger loss values. 6 Experiments We validate our theoretical findings and the effectiveness of our proposed batch selection method by providing experimental results on synthetic and real datasets. We first show that our experimental results on synthetic dataset coincide with two main theoretical results: (i) the relationship between the full-batch contrastive loss and the mini-batch contrastive loss given in Sec. 4, (ii) the analysis on the convergence of OSGD and the proposed SC method given in Sec. 5. To demonstrate the practicality of our batch selection method, we provide experimental results on CIFAR-100 [31] and Tiny ImageNet [33]. Details of the experimental setting can be found in Appendix D, and our code is available at https://github.com/krafton-ai/mini-batch-cl. 6.1 Synthetic Dataset Consider the problem of optimizing the embedding matrices U, V using GD, where each column of U, V is initialized as a multivariate normal vector and then normalized as \u2225ui\u2225= \u2225vi\u2225= 1, \u2200i. We use learning rate \u03b7 = 0.5, and apply the normalization step at every iteration. First, we compare the minimizers of three optimization problems: (i) full-batch optimization in Eq.(1); (ii) mini-batch optimization in Eq. (4) with SB = h\u0000N B \u0001i ; (iii) mini-batch optimization with SB \u228a h\u0000N B \u0001i . We apply GD algorithm to each problem for N = 8 and B = 2, obtain the updated embedding matrices, and then show the heatmap plot of N \u00d7 N gram matrix containing all the 4Our algorithm finds N/B good clusters at once, instead of only finding a single best cluster. Compared with such alternative approach, our method is (i) more efficient when we update models for multiple iterations, and (ii) guaranteed to load all samples with N/B batches, thus expected to have better convergence [5, 21, 20]. 9 \fTable 1: Top-1 retrieval accuracy on CIFAR-100-C (or Tiny ImageNet-C) [23], when each algorithm uses CIFAR-100 (or Tiny ImageNet) to pretrain ResNet-18 with SimCLR and SogCLR objective. SC algorithm proposed in Sec. 5.3 outperforms all baselines. CIFAR-100 Tiny ImageNet SimCLR SogCLR SimCLR SogCLR OSGD 31.4 \u00b1 0.03 23.8 \u00b1 0.02 33.6 \u00b1 0.04 29.7 \u00b1 0.04 SGD 31.3 \u00b1 0.02 23.6 \u00b1 0.05 33.2 \u00b1 0.03 28.6 \u00b1 0.03 SC 32.5 \u00b1 0.05 30.0 \u00b1 0.04 33.8 \u00b1 0.04 33.3 \u00b1 0.03 pairwise inner products u\u22ba i vj in Fig. 4(b)-(d). Here, we plot for two regimes: d = 2N for the top row, and d = N/2 for the bottom row. In Fig. 4(a), we plot the gram matrix for the optimal solution obtained in Sec. 4.2. One can observe that when either full-batch or all \u0000N B \u0001 mini-batches are used for training, the trained embedding vectors reach a simplex ETF and simplex cross-polytope solutions for d = 2N and d = N/2, respectively, as proved in Thm 4. In contrast, when a strict subset of \u0000N B \u0001 mini-batches are used for training, these solutions are not achieved. Second, we compare the convergence speed of three algorithms in mini-batch optimization: (i) OSGD; (ii) the proposed SC method; and (iii) SGD (see details of the algorithms in Appendix C). Fig. 4(e) shows the \u2225U \u22c6\u22baV \u22c6\u2212U (t)\u22baV (t)\u2225F which is the Frobenius norm of the difference between heatmaps of the ground-truth solution (U \u22c6, V \u22c6) and the embeddings at each step t. We restrict the number of updates for all algorithms, specifically 500 steps. We observe that both OSGD and the proposed method nearly converge to the ground-truth solutions proved in Thm. 4 within 500 steps, while SGD does not. We obtain similar results for other values of N and d, given in Appendix D.2. 6.2 Real Datasets Here we show that the proposed SC method is effective in more practical settings where the embedding is learned by a parameterized encoder, and can be easily applied to existing uni-modal frameworks, such as SimCLR [9] and SogCLR [68]. We conduct mini-batch contrastive learning on CIFAR-100 and Tiny ImageNet datasets and report the performances in the image retrieval downstream task on corrupted datasets, the results of which are in Table 1. Due to the page limit, we provide detailed experimental information in the Appendix D.3. 7" + }, + { + "url": "http://arxiv.org/abs/2210.05936v1", + "title": "Equal Experience in Recommender Systems", + "abstract": "We explore the fairness issue that arises in recommender systems. Biased data\ndue to inherent stereotypes of particular groups (e.g., male students' average\nrating on mathematics is often higher than that on humanities, and vice versa\nfor females) may yield a limited scope of suggested items to a certain group of\nusers. Our main contribution lies in the introduction of a novel fairness\nnotion (that we call equal experience), which can serve to regulate such\nunfairness in the presence of biased data. The notion captures the degree of\nthe equal experience of item recommendations across distinct groups. We propose\nan optimization framework that incorporates the fairness notion as a\nregularization term, as well as introduce computationally-efficient algorithms\nthat solve the optimization. Experiments on synthetic and benchmark real\ndatasets demonstrate that the proposed framework can indeed mitigate such\nunfairness while exhibiting a minor degradation of recommendation accuracy.", + "authors": "Jaewoong Cho, Moonseok Choi, Changho Suh", + "published": "2022-10-12", + "updated": "2022-10-12", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG", + "cs.IR", + "stat.ML" + ], + "main_content": "Introduction Recommender systems are everywhere, playing a crucial role to support decision making and to decide what we experience in our daily life. One recent challenge concerning fairness arises when the systems are built upon biased historical data. Biased data due to polarized preferences of particular groups for certain items may often yield limited recommendation service. For instance, if female students exhibit high ratings on literature subjects and less interest in math and science relative to males, the subject recommender system trained based on such data may provide a narrow scope of recommended subjects to the female group, thereby yielding unequal experience. This unequal experience across groups may result in amplifying the gender gap issue in science, technology, engineering, and mathematics 1KRAFTON Inc., Seoul, Korea 2KAIST, Daejeon, Korea. Correspondence to: Changho Suh . *This work was done when Jaewoong Cho was with KAIST as a PhD candidate. (STEM) \ufb01elds. Among various works for fair recommender systems (Yao & Huang, 2017; Li et al., 2021; Kamishima & Akaho, 2017; Xiao et al., 2017; Beutel et al., 2019; Burke, 2017), one recent and most relevant work is (Yao & Huang, 2017). They focus on a scenario in which unfairness occurs mainly due to distinct recommendation accuracies across different groups. They propose novel fairness measures that quantify the degree of such unfairness via the difference between recommendation accuracies, and also develop an optimization framework that well trades the fairness measures against the average accuracy. However, it comes with a challenge in ensuring fairness w.r.t. the unequal experience. This is because similar accuracy performances between different groups do not guarantee a variety of recommendations to an underrepresented group with historical data bearing low preferences and/or scarce ratings for certain items. For instance, in the subject recommendation, the fairness notion may not serve properly, as long as female students exhibit low ratings (and/or lack of ratings) on math and science subjects due to societal/cultural in\ufb02uences (and/or sampling biases). Furthermore, if the recommended items are selected only according to the overall preference, the biased preference for a speci\ufb01c item group will further increase, and the exposure to the unpreferred item group will gradually decrease. Contribution: In an effort to address the challenge, we introduce a new fairness notion that we call equal experience. At a high level, the notion represents how equally various items are suggested even for an underrepresented group preserving such biased historical data. Inspired by an information-theoretic notion \u201cmutual information\u201d (Cover, 1999) and its key property \u201cchain rule\u201d, we quantify our notion so as to control the level of independence between preference predictions and items for any group of users. Speci\ufb01cally, the notion encourages prediction e Y (e.g., 1 if a user prefers an item; 0 otherwise) to be independent of the following two: (i) user group Zuser (e.g., 0 for male; and 1 for female); and (ii) item group Zitem (e.g., 0 for mathematics; and 1 for literature). In other words, it promotes e Y \u22a5(Zuser, Zitem); which in turns ensures all of the following four types of independence that one can think of: e Y \u22a5Zitem, e Y \u22a5Zuser, e Y \u22a5Zitem|Zuser, and e Y \u22a5Zuser|Zitem. This is inspired by the fact that mutual arXiv:2210.05936v1 [cs.LG] 12 Oct 2022 \fEqual Experience in Recommender Systems information being zero is equivalent to the independence between associated random variables, as well as the chain rule: I(e Y ; Zuser, Zitem) = I(e Y ; Zitem) + I(e Y ; Zuser|Zitem) = I(e Y ; Zuser) + I(e Y ; Zitem|Zuser). (1) See Section 3.1 for details. The higher independence, the more diverse recommendation services are offered for every group. We also develop an optimization framework that incorporates the quanti\ufb01ed notion as a regularization term into a conventional optimization in recommender systems (e.g., the one based on matrix completion (Koren, 2008; Koren et al., 2009)). Here one noticeable feature of our framework is that the fairness performances w.r.t. the above four types of independence conditions can be gracefully controlled via a single uni\ufb01ed regularization term. This is in stark contrast to prior works (Yao & Huang, 2017; Li et al., 2021; Kamishima & Akaho, 2017; Mehrotra et al., 2018), each of which promotes only one independence condition or two via two separate regularization terms. See below Related works for details. In order to enable an ef\ufb01cient implementation of the fairness constraint, we employ recent methodologies developed in the context of fair classi\ufb01ers, such as the ones building upon kernel density estimation (Cho et al., 2020a), mutual information (Zhang et al., 2018; Kamishima et al., 2012; Cho et al., 2020b), or covariance (Zafar et al., 2017a;b). We also conduct extensive experiments both on synthetic and two benchmark real datasets: MovieLens 1M (Harper & Konstan, 2015) and Last FM 360K (Celma, 2010). As a result, we \ufb01rst identify two primary sources of biases that incur unequal experience: population imbalance and observation bias (Yao & Huang, 2017). In addition, we demonstrate that our fairness notion can help improve the fairness measure w.r.t. equal experience (to be de\ufb01ned in Section 3.1; see De\ufb01nition 3.2) while exhibiting a small degradation of recommendation accuracy. Furthermore, we provide an extension of our fairness notion to the context of top-K recommendation from an end-ranked list. We also demonstrate the effectiveness of the proposed framework in top-K recommendation setting. Related works: In addition to (Yao & Huang, 2017), numerous fairness notions and algorithms have been proposed for fair recommender systems (Xiao et al., 2017; Beutel et al., 2019; Singh & Joachims, 2018; Zehlike et al., 2017; Narasimhan et al., 2020; Biega et al., 2018; Li et al., 2021; Kamishima & Akaho, 2017; Mehrotra et al., 2018; Schnabel et al., 2016). (Xiao et al., 2017) develop fairness notions that encourage similar recommendations for users within the same group. (Beutel et al., 2019) consider similar metrics as that in (Yao & Huang, 2017) yet in the context of pairwise recommender systems wherein pairewise preferences are given as training data. (Li et al., 2021) propose a fairness measure that quanti\ufb01es the irrelevancy of preference predictions to user groups, like demographic parity in the fairness literature (Feldman et al., 2015; Zafar et al., 2017a;b). Speci\ufb01cally, they consider the independence condition between prediction e Y and user group Zuser: e Y \u22a5Zuser. Actually this was also considered as another fairness measure in (Yao & Huang, 2017). Similarly, other works with a different direction consider the similar notion concerning the independence w.r.t. item group Zitem: e Y \u22a5Zitem (Kamishima & Akaho, 2017; Singh & Joachims, 2018; Biega et al., 2018). (Mehrotra et al., 2018) incorporate both measures to formulate a multi-objective optimization. In Section 2.2, we will elaborate on why the above prior fairness notions cannot fully address the challenge w.r.t. unequal experience. There has been a proliferation of fairness notions in the context of fair classi\ufb01ers: (i) group fairness (Feldman et al., 2015; Zafar et al., 2017b; Hardt et al., 2016; Woodworth et al., 2017); (ii) individual fairness (Dwork et al., 2012; Garg et al., 2018); (iii) causality-based fairness (Kusner et al., 2017; Nabi & Shpitser, 2018; Russell et al., 2017; Wu et al., 2019; Zhang & Bareinboim, 2018b;a). Among various prominent group fairness notions, demographic parity and equalized odds give an inspiration to our work in the process of applying the chain rule, re\ufb02ected in (1). Concurrently, a multitude of fairness algorithms have been developed with the use of covariance (Zafar et al., 2017a;b), mutual information (Zhang et al., 2018; Kamishima et al., 2012; Cho et al., 2020b), kernel density estimation (Cho et al., 2020a) or R\u00b4 enyi correlation (Mary et al., 2019) to name a few. In this work, we also demonstrate that our proposed framework (to be presented in Section 3) embraces many of these approaches; See Remark 3.4 for details. 2. Problem Formulation As a key technique for operating recommender systems, we consider collaborative \ufb01ltering which estimates user ratings on items. We \ufb01rst formulate an optimization problem building upon one prominent approach, matrix completion. We then introduce a couple of fairness measures proposed by recent prior works (Yao & Huang, 2017; Li et al., 2021; Kamishima & Akaho, 2017), and present an extended optimization framework that incorporates the fairness measures as regularization terms. 2.1. Optimization based on matrix completion As a well-known approach for operating recommender systems, we consider matrix completion (Fazel, 2002; Koren et al., 2009; Cand` es & Recht, 2009). Let M \u2208Rn\u00d7m be the ground-truth rating matrix where n and m denote the number of users and items respectively. Each entry, denoted by Mij, can be of any type. It could be binary, \ufb01ve-star \fEqual Experience in Recommender Systems rating, or any real number. Denote by \u2126the set of observed entries of M. For simplicity, we assume noiseless observation. Denote by c M \u2208Rn\u00d7m an estimate of the rating matrix. Matrix completion can be done via the rank minimization that exploits the low-rank structure of the rating matrix. However, since the problem is NP-hard (Fazel, 2002), we consider a well-known relaxation approach that intends to minimize instead the squared error between M and c M in the observed entries: min c M X (i,j)\u2208\u2126 (Mij \u2212c Mij)2. (2) There are two well-known approaches for solving the optimization in (2): (i) matrix factorization (Abadir & Magnus, 2005; Koren et al., 2009); and (ii) neural-net-based parameterization (Salakhutdinov et al., 2007; Sedhain et al., 2015; He et al., 2017). Matrix factorization assumes a certain structure on the rating matrix: M = LR where L \u2208Rn\u00d7r and R \u2208Rr\u00d7m. One natural way to search for optimal L\u2217and R\u2217is to apply gradient descent (Robbins & Monro, 1951) w.r.t. all of the Lij\u2019s and Rij\u2019s, although it does not ensure the convergence of the optimal point due to non-convexity. The second approach is to parameterize c M via neural networks such as restricted Boltzmann machine (Salakhutdinov et al., 2007) and autoencoder (Sedhain et al., 2015; Lee et al., 2018). For instance, one may employ an autoencoder-type neural network which outputs a completed matrix c M fed by the partially-observed version of M. For a user-based autoencoder (Sedhain et al., 2015), an observed row vector of M is fed into the autoencoder, while an observed column vector serves as an input for an item-based autoencoder (Sedhain et al., 2015). In this work, we consider the two approaches in our experiments: matrix factorization with gradient descent; and autoencoder-based parameterization. One common way to promote a fair recommender system is to incorporate a fairness measure, say Lfair (which we will relate to an estimated matrix c M), as a regularization term into the above base optimization in (2): min c M (1 \u2212\u03bb) X (i,j)\u2208\u2126 (Mij \u2212c Mij)2 + \u03bb \u00b7 Lfair (3) where \u03bb \u2208[0, 1] denotes a normalized regularization factor that balances prediction accuracy against the fairness constraint. For the fairness-regularization term Lfair, several fairness measures have been introduced. 2.2. Fairness measures in prior works (Yao & Huang, 2017; Kamishima & Akaho, 2017; Li et al., 2021) We list three of them, which are mostly relevant to our framework to be presented in Section 3. For illustrative purpose, we will explain them in a simple setting where there are two groups of users, say the male group M and the female group F. The \ufb01rst is value unfairness proposed by (Yao & Huang, 2017). It quanti\ufb01es the difference between prediction errors across the two groups of users over the entire items: VAL := 1 m m X j=1 \f \f \f \f 1 |M\u2126| X (i,j)\u2208\u2126:i\u2208M (Mij \u2212c Mij) | {z } prediction error w.r.t. M \u2212 1 |F\u2126| X (i,j)\u2208\u2126:i\u2208F (Mij \u2212c Mij) | {z } prediction error w.r.t. F \f \f \f \f (4) where M\u2126and F\u2126denote the male and female group w.r.t. observed entries, respectively. While the measure promotes fairness w.r.t. prediction accuracy across distinct groups, it may not ensure fairness w.r.t. the diversity of recommended items to users. To see this clearly, consider an extreme scenario in which the ground truth rating is very small Mij\u2217\u22480 for a certain item j\u2217(say science subject) for all i \u2208F. In this case, minimizing VAL may encourage c Mij\u2217\u22480 for all i \u2208F. This then incurs almost no recommendation of the science subject to the females, thus giving no opportunity to experience the subject. This motivates us to propose a new fairness measure (to be presented in Section 3.1) that helps mitigate such unfairness. On the other hand, (Kamishima & Akaho, 2017) introduce another fairness measure, which bears a similar spirit to demographic parity in the fairness literature (Feldman et al., 2015; Zafar et al., 2017a;b). The measure, named Calders and Verwer\u2019s discrimination score (CVS), quanti\ufb01es the level of irrelevancy between preference predictions and item groups. To describe it in detail, let us introduce some notations. Let Zitem be a sensitive attribute w.r.t. item groups, e.g., Zitem = 0 (literature) and Zitem = 1 (science). Let b Y be a generic random variable w.r.t. estimated ratings c Mij\u2019s. To capture the preference prediction, let us consider a simple binary preference setting in which e Y := 1{b Y \u2265\u03c4} where \u03c4 indicates a certain threshold. Specializing the measure into the one like demographic parity, it can be quanti\ufb01ed as: CVS := |P(e Y = 1|Zitem = 1) \u2212P(e Y = 1|Zitem = 0)|. (5) Minimizing the measure encourages the independence between e Y and Zitem, thereby promoting the same rating statistics across different groups. However, it does not necessarily ensure the same statistics when we focus on a certain group of users. It guarantees the independence only in the average sense. To see this clearly, consider a simple scenario in which there are two groups of users, say female and male. Let Zuser be another sensitive attribute w.r.t. user groups, \fEqual Experience in Recommender Systems 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 Figure 1. An example in which e Y \u22a5Zitem but e Y \u0338\u22a5Zitem|Zuser. Here e Y := 1{b Y \u2265\u03c4}; b Y is a generic random variable w.r.t. estimated ratings c Mij\u2019s; and \u03c4 indicates a certain threshold. The (i, j) entry of an estimated preference matrix with 6 users (row) and 6 items (column) indicates 1{c Mij \u2265\u03c4}. e.g., Zuser = 0 (female) and Zuser = 1 (male). Fig. 1 illustrates a concrete example where e Y is independent of Zitem. Notice that the number of 1\u2019s w.r.t. Zitem = 0 (over the entire user groups) is the same as that w.r.t. Zitem = 1. However, focusing on a certain user group, say Zuser = 0, e Y is highly correlated with Zitem. Observe in the case Zuser = 0 that the number of 1\u2019s is 9 for Zitem = 0, while it reads 0 for Zitem = 1. (Li et al., 2021) consider a similar measure, named useroriented group fairness (UGF), yet which targets the independence w.r.t. user groups. Similar to CVS, we can de\ufb01ne it by replacing Zitem with Zuser in (5): UGF := |P(e Y = 1|Zuser = 1) \u2212P(e Y = 1|Zuser = 0)|. (6) However, by symmetry, the high correlation issue discussed via Fig. 1 still arises. 3. Proposed Framework We \ufb01rst propose new fairness measures that can regulate fairness w.r.t. the opportunity to experience inherently-low preference items, as well as address the high correlation issue discussed as above. We then develop an integrated optimization framework that uni\ufb01es the fairness measures as a single regularization term. Finally we introduce concrete methodologies that can implement the proposed optimization. 3.1. New fairness measures The common limitation of the prior fairness measures (Yao & Huang, 2017; Kamishima & Akaho, 2017; Li et al., 2021) is that the independence between preference predictions and item groups may not be guaranteed for a certain group of users. This motivates us to consider the conditional independence as a new fairness notion, formally de\ufb01ned as below. De\ufb01nition 3.1 (Equalized Recommendation). A recommender system is said to respect \u201cequalized recommendation\u201d if its prediction e Y is independent of item\u2019s sensitive attribute Zitem given user\u2019s sensitive attribute Zuser: e Y \u22a5Zitem|Zuser. Inspired by the quanti\ufb01cation methods w.r.t. equalized odds in the fairness literature (Jiang et al., 2019; Donini et al., 2018; Hardt et al., 2016; Woodworth et al., 2017), we quantify the new notion via: DER := X z1\u2208Zuser X z2\u2208Zitem \f \f \fP(e Y = 1|Zuser = z1) \u2212P(e Y = 1|Zitem = z2, Zuser = z1) \f \f \f, (7) for arbitrary alphabet sizes |Zuser| and |Zitem|. Here DER stands for the difference w.r.t. two interested probabilities that arise in equalized recommendation, and this naming is similar to those in prior fairness metrics (Donini et al., 2018; Jiang et al., 2019). It captures the degree of violating equalized recommendation via the difference between the conditional probability and its marginal given Zuser. Notice that the minimum DER = 0 is achieved under \u201cequalized recommendation\u201d. One may consider another measure which takes \u201cmax\u201d operation instead of \u201cP\u201d in (7) or a different measure based on the ratio of the two associated probabilities. We focus on DER in (7) for tractability of an associated optimization problem that we will explain in Section 3.2. The constraint of \u201cequalized recommendation\u201d encourages the same prediction statistics of items for every user group, thereby promoting the equal chances of experiencing a variety of items for all individuals. However, the notion comes with a limitation. The limitation comes from the fact that conditional independence does not necessarily imply independence (Cover, 1999): e Y \u22a5Zitem|Zuser \u0338 = \u21d2 e Y \u22a5Zitem. (8) Actually, the ultimate goal of a fair recommender system is to ensure all of the following four types of independence: e Y \u22a5Zitem, e Y \u22a5Zuser|Zitem, e Y \u22a5Zuser, e Y \u22a5Zitem|Zuser. (9) \fEqual Experience in Recommender Systems One natural question that arises is then: What is a proper fairness notion which allows us to respect all of the above four conditions preferably in one shot? In an attempt to succinctly represent all of the four conditions, we invoke an information-theoretic notion, mutual information (Cover, 1999). One key property of mutual information, called the chain rule, gives an insight: I(e Y ; Zuser, Zitem) = I(e Y ; Zitem) + I(e Y ; Zuser|Zitem) = I(e Y ; Zuser) + I(e Y ; Zitem|Zuser). (10) From this, we can readily see that I(e Y ; Zuser, Zitem) = 0 = \u21d2 I(e Y ; Zitem) = 0, I(e Y ; Zuser|Zitem) = 0, I(e Y ; Zuser) = 0, I(e Y ; Zitem|Zuser) = 0. (11) This is due to the non-negativity property of mutual information. The key observation in (11) motivates us to propose a new fairness notion that we call equal experience. De\ufb01nition 3.2 (Equal Experience). A recommender system is said to respect \u201cequal experience\u201d if its prediction e Y is independent of both Zitem and Zuser: e Y \u22a5(Zitem, Zuser). Similar to DER, we also quantify the notion as the difference between the conditional probability and its marginal: DEE := X z1\u2208Zuser X z2\u2208Zitem \f \f \fP(e Y = 1) \u2212P(e Y = 1|Zitem = z2, Zuser = z1) \f \f \f, (12) for arbitrary alphabet sizes |Zuser| and |Zitem|. We also coin the similar naming: DEE (difference w.r.t. two interested probabilities that arise in equal experience). 3.2. Fairness-regularized optimization Taking DEE as the fairness-regularization term Lfair in the focused framework ((3)), we get: min c M (1 \u2212\u03bb) X (i,j)\u2208\u2126 (Mij \u2212c Mij)2 + \u03bb \u00b7 DEE (13) where \u03bb \u2208[0, 1] denotes a normalized regularization factor. Here one challenge that arises in (13) is that expressing DEE in terms of an optimization variable c M is not that straightforward. To overcome the challenge, we take the kernel density estimation (KDE) technique (Cho et al., 2020a) which enables faithful quanti\ufb01cation of fairness-regularization terms. One key bene\ufb01t of the KDE approach is that the computed measures based on KDE is differentiable w.r.t. model parameters, thus enjoying a family of gradient-based optimizers (G\u00b4 eron, 2019; Kingma & Ba, 2014b). Since the problem context where the KDE technique (Cho et al., 2020a) was introduced is different from ours, we describe below details on the technique, tailoring it to our framework. Implementation of DEE via the KDE technique (Cho et al., 2020a): We \ufb01rst parameterize prediction output c M via matrix factorization or a neural network. Let w be a collection of parameters w.r.t. c M. It could be a collection of matrix entries of L and R when employing matrix factorization c M = LR. Or it could be a collection of neural network parameters in the latter case. The key idea of the KDE technique is to approximate the interested probability distributions via kernel density estimator de\ufb01ned below: De\ufb01nition 3.3 (Kernel Density Estimator (KDE) (Davis et al., 2011)). Let (\u02c6 y(1), . . . , \u02c6 y(s)) be i.i.d. examples drawn from a distribution with an unknown density f. Its KDE is de\ufb01ned as: b f(\u02c6 y) := 1 sh Ps i=1 fk \u0010 \u02c6 y\u2212\u02c6 y(i) h \u0011 where fk is a kernel function (e.g., Gaussian kernel function (Davis et al., 2011)) and h > 0 is a smoothing parameter called bandwidth. For DEE, the interested probability distributions are P(e Y = 1) and P(e Y = 1|Zitem = z2, Zuser = z1). Let us \ufb01rst consider P(e Y = 1). Remember e Y := 1{b Y \u2265\u03c4}, so b Y should be taken into consideration initially. Using the KDE, we can estimate the probability density function of b Y , say fb Y (\u02c6 y): b fb Y (\u02c6 y) = 1 nmh n X i=1 m X j=1 fk \u02c6 y \u2212c Mij h ! . (14) This together with e Y := 1{b Y \u2265\u03c4} gives: b P(e Y = 1) = Z \u221e \u03c4 b fb Y (\u02c6 y)d\u02c6 y = 1 nm n X i=1 m X j=1 Fk \u03c4 \u2212c Mij h ! (15) where Fk(\u02c6 y) := R \u221e \u02c6 y fk(t)dt. Since the approach relies upon a family of gradient-based optimizers, the gradients of b P(e Y = 1) and b P(e Y = 1|Zitem = z2, Zuser = z1) need to be computed explicitly. Using the technique in (Cho et al., 2020a) (Proposition 1 therein), we can readily approximate \u2207wDEE. See Appendix A.2 for details. Remark 3.4 (Other choices for a measure of \u201cequal experience\u201d). Instead of DEE, one can resort to other measures based on prominent tools employed in the fairness literature: covariance (Zafar et al., 2017a;b), mutual information (Zhang et al., 2018; Kamishima et al., 2012; Cho et al., \fEqual Experience in Recommender Systems 2020b), Wasserstein distance (Jiang et al., 2020) and R\u00b4 enyi correlation (Mary et al., 2019). For instance, the covariancebased approach allows us to take Lfair as: Lfair = E h (b Y \u2212E[b Y ])(Z \u2212E[Z]) i , (16) where Z := (Zitem, Zuser). Here we use b Y instead of e Y , as e Y incurs non-differentiability, hindering implementation. In the case of mutual information, one can take Lfair as: Lfair = I(b Y ; Zitem, Zuser) \u2265I(e Y ; Zitem, Zuser). (17) Again, for ease of implementation, we employ b Y . This choice is also relevant because it serves as an upper bound of I(e Y ; Zitem, Zuser). Notice that e Y is a function of b Y . Reducing I(b Y ; Zitem, Zuser) yields the minimization of the interested quantity I(e Y ; Zitem, Zuser). For faithful implementation of I(b Y ; Zitem, Zuser), we may employ the variational optimization technique in (Zhang et al., 2018; Cho et al., 2020b) to translate it into a function optimization which can also be parameterized. Other choices can also be dealt with properly relying upon the associated techniques in (Jiang et al., 2020; Mary et al., 2019). \u25a0 4. Experiments We conduct experiments both on synthetic and two benchmark real datasets: MovieLens 1M (Harper & Konstan, 2015) and Last FM 360K (Celma, 2010). We generate synthetic data so as to pose fairness issues. Algorithms are implemented in PyTorch (Paszke et al., 2019), and experiments are performed on a server with Titan RTX GPUs. All the simulation results (to be reported) are the ones averaged over \ufb01ve trials with distinct random seeds. In Appendix A.5, we also present the running times of our algorithm and baselines on the synthetic and real datasets. 4.1. Synthetic dataset Here we highlight two types of bias: population imbalance and observation bias (Yao & Huang, 2017). For illustrative purpose, let us explain them in a simple subjectrecommendation example where there are two user groups (Zuser = 0 for male and Zuser = 1 for female) and two item groups (Zitem = 0 for science and Zitem = 1 for literature). Population imbalance refers to the difference in the groundtruth preferences between two user groups, e.g., for the science subject, male students exhibit higher ratings relative to females. Observation bias is the one that occurs due to the stereotype formed by societal and cultural in\ufb02uences. To understand what it means, let us consider a scenario where a male student equally likes science and literature subjects. But due to the stereotype that male students prefer science to literature in general, there may be very sparse ratings from male students for literature. The system trained based on such data might incorrectly interpret as if male students dislike literature. Such data is said to have observation bias. We generate synthetic data that bear the two biases in the context of binary ratings, i.e., Mij \u2208 {+1 (like), \u22121 (dislike)}. We divide n users into male and female groups each of n 2 , say M and F. Items are also divided into two groups of m 2 , say male-preferred group M\u2032 and female-preferred group F\u2032. To account for population imbalance, we \ufb01rst generate the ground-truth rating matrix M \u2208Rn\u00d7m using the following four probabilities: {pMM\u2032, pMF\u2032, pFM\u2032, pFF\u2032} where pMM\u2032 indicates the probability that a male student likes a male-preferred subject (science). More precisely, for i \u2208M and j \u2208M\u2032, Mij = \u001a +1, w.p. pMM\u2032; \u22121, w.p. 1 \u2212pMM\u2032. (18) Similarly the other probabilities are de\ufb01ned. To ensure the low-rank structure, say rank r, of the rating matrix, we generate r 2 basis rating vectors for male group as per the above preference probabilities, and similarly another set of r 2 basis rating vectors is generated for female group. Every male student picks one of the r 2 basis vectors w.r.t. the male group uniformly at random, and similarly for every female student. This then yields rank(M) = r. To control observation bias, we introduce another probability set: {qMM\u2032, qMF\u2032, qFM\u2032, qFF\u2032} where qMM\u2032 denotes the probability that a male student\u2019s rating is observed for a male-preferred subject. More precisely, for i \u2208M and j \u2208M\u2032, (i, j) \u2208 \u001a \u2126, w.p. qMM\u2032; \u2126c, w.p. 1 \u2212qMM\u2032. (19) Similarly the other probabilities are de\ufb01ned. For simplicity, throughout all the synthetic data simulations, we assume a symmetric setting in which pMM\u2032 = pFF\u2032(= p0), pMF\u2032 = pFM\u2032(= p1), qMM\u2032 = qFF\u2032(= q0), and qMF\u2032 = qFM\u2032(= q1). We consider a setting in which (r, n, m) = (20, 600, 400). We leave a more detailed explanation of experiments on the synthetic dataset in Appendix A.3. First, we check whether the bias actually incurs unfair recommendations. For ease of understanding, we consider two scenarios: (i) population imbalance varies without observation bias; and (ii) observation bias varies without population imbalance. Fig. 2 (Left) presents the 1st scenario, demonstrating that the fairness performance measured in DEE decreases with an increase in population imbalance, controlled by |p0 \u2212p1|. Fig. 2 (Right) considers the 2nd scenario. We see the same trend yet now w.r.t. the variation of observation bias. Table 1 presents the prediction error (RMSE) and fairness performances on the synthetic dataset having observation \fEqual Experience in Recommender Systems 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Figure 2. (Left) DEE as a function of (p0, p1) which controls the degree of population imbalance while \ufb01xing q0 = q1 = 0.2; (Right) DEE as a function of (q0, q1) w.r.t. observation bias while \ufb01xing p0 = p1 = 0.4. Here p0 = pMM\u2032 = pFF\u2032, p1 = pMF\u2032 = pFM\u2032, q0 = qMM\u2032 = qFF\u2032 and q1 = qMF\u2032 = qFM\u2032. bias (q0, q1) = (0.2, 0.01) yet without population imbalance (p0, p1) = (0.4, 0.4). We consider four fairness measures: (i) DEE in (12); (ii) VAL in (4); (iii) UGF in (6); (iv) CVS in (5). We also compare our algorithm with four baselines: (i) unfair (no fairness constraint); (ii) VAL-based algorithm (Yao & Huang, 2017); (iii) UGF-based algorithm (Li et al., 2021); (iv) CVS-based algorithm (Kamishima & Akaho, 2017). Each baseline, say VAL-based algorithm, achieves the best fairness performance only for VAL, while it does not work well under the other fairness measures. On the other hand, our algorithm offers great fairness performances for all the measures, except for VAL, which our framework does not target. 4.2. Real datasets We consider two benchmark datasets: MovieLens 1M (Harper & Konstan, 2015), and Last FM 360K (Celma, 2010): \u2022 MovieLens 1M: The associated task is to predict the movie rating on a 5-star scale. This dataset contains 6,040 users, 3,900 movies, and 1,000,209 ratings, i.e., rating matrix is 4.26% full.1 We divide user and item groups based on gender and genre, respectively. Action, crime, \ufb01lme-noir, war are selected as malepreferred genre, whereas children, fantasy, musical, romance are selected as female-preferred genre. \u2022 Last FM 360K: The associated task is to predict whether the user likes the artist or not. This dataset contains 359,347 users, 294,015 artists, and 17,559,530 play counts, i.e., rating matrix is 0.02% full.2 The data for play counts is converted to binary rating: +1 if counts > average, otherwise \u22121. We divide user and 1http://www.movielens.org/ 2http://ocelma.net/MusicRecommendationDataset/lastfm360K.html item groups based on gender and genre, respectively. Since this dataset only contains gender information, we use Last.fm API3 to collect the genre of corresponding artist\u2019s music; the tag was associated with 5,706 artists. We run experiments employing both matrix factorization (MF) based and autoencoder (AE) based techniques. In Appendix A.4, we leave a more detailed explanation of experiments on the real datasets and the results of AE-based technique. The results of real data experiments are listed in Tables 2 and 3. As in the synthetic data setting, we can make two relevant observations. All baseline algorithms fail to respect our metric (DEE) while meeting their own. We also see that our algorithm exhibits signi\ufb01cant performances for all the fairness measures except for VAL which does not have close relationship with the equal experience that we aim at. 5. Extension In this section, we discuss the extension of our work: (i) introducing a new notion that bears a similar spirit to equalized odds, and (ii) applying our notion to the fair ranking context (Zehlike et al., 2017). Our fairness notion equal experience aims at recommending a variety of items for all user groups. Demographic parity is similar to our notion in the sense of considering the irrelevancy of predictions to groups. On the other hand, equalized odds is the fairness notion that encourages equal error rates (e.g., true/negative positive rate) across user groups by employing the ground-truth label Y . Similar to equalized odds, our measure in recommender systems can readily be extended to a setting in which the ground-truth label is available to exploit. The key idea is to promote e Y \u22a5(Zuser, Zitem)|Y . We can employ the proposed op3http://www.last.fm/api \fEqual Experience in Recommender Systems Table 1. Prediction error (RMSE) and fairness performances on the synthetic dataset preserving observation bias (q0, q1) = (0.2, 0.01) while exhibiting no population imbalance (p0, p1) = (0.4, 0.4). The boldface indicates the best result and the underline denotes the second best. Measure RMSE DEE VAL UGF CVS Unfair 0.8889 \u00b1 0.0111 0.1201 \u00b1 0.0405 0.4646 \u00b1 0.0108 0.0120 \u00b1 0.0050 0.0118 \u00b1 0.0047 Ours (DEE) 0.9020 \u00b1 0.0081 0.0025 \u00b1 0.0006 0.4609 \u00b1 0.0076 0.0006 \u00b1 0.0004 0.0002 \u00b1 0.0001 Ours (DER) 0.8887 \u00b1 0.0042 0.0401 \u00b1 0.0056 0.4540 \u00b1 0.0110 0.0201 \u00b1 0.0028 0.0002 \u00b1 0.0002 VAL-based 0.8837 \u00b1 0.0045 0.1099 \u00b1 0.0045 0.0003 \u00b1 6.48e-6 0.0090 \u00b1 0.0004 0.0088 \u00b1 0.0015 UGF-based 0.8961 \u00b1 0.0067 0.0144 \u00b1 0.0144 0.4709 \u00b1 0.0182 0.0004 \u00b1 0.0003 0.0217 \u00b1 0.0027 CVS-based 0.9003 \u00b1 0.0061 0.1390 \u00b1 0.0413 0.4722 \u00b1 0.0055 0.0206 \u00b1 0.0022 0.0002 \u00b1 0.0001 Table 2. Prediction error (RMSE) and fairness performances of the matrix factorization based algorithm on MovieLens 1M dataset. The boldface indicates the best result and the underline denotes the second best. Our algorithm offers great fairness performances for all the measures, except for VAL, which our framework does not target. Measure RMSE DEE VAL UGF CVS Unfair 0.8541 \u00b1 0.0033 0.2447 \u00b1 0.0134 0.3227 \u00b1 0.0031 0.0058 \u00b1 0.0042 0.1291 \u00b1 0.0079 Ours (DEE) 0.8641 \u00b1 0.0047 0.0014 \u00b1 0.0008 0.2941 \u00b1 0.0024 0.0018 \u00b1 0.0016 0.0007 \u00b1 0.0004 Ours (DER) 0.8526 \u00b1 0.0029 0.0114 \u00b1 0.0041 0.3332 \u00b1 0.0050 0.0055 \u00b1 0.0022 0.0014 \u00b1 0.0001 VAL-based 0.8529 \u00b1 0.0011 0.3659 \u00b1 0.0033 0.0942 \u00b1 0.0016 0.0261 \u00b1 0.0020 0.1388 \u00b1 0.0030 UGF-based 0.8550 \u00b1 0.0015 0.2492 \u00b1 0.0100 0.3285 \u00b1 0.0051 0.0001 \u00b1 0.0001 0.1355 \u00b1 0.0038 CVS-based 0.8549 \u00b1 0.0018 0.0721 \u00b1 0.0069 0.3319 \u00b1 0.0046 0.0065 \u00b1 0.0042 0.0002 \u00b1 3.45e-5 Table 3. Prediction error (RMSE) and fairness performances of the matrix factorization based algorithm on Last FM 360K dataset. Measure RMSE DEE VAL UGF CVS Unfair 0.6720 \u00b1 0.0024 0.0840 \u00b1 0.0110 0.2297 \u00b1 0.0020 0.0404 \u00b1 0.0033 0.0204 \u00b1 0.0104 Ours (DEE) 0.6892 \u00b1 0.0040 0.0082 \u00b1 0.0023 0.2777 \u00b1 0.0048 0.0161 \u00b1 0.0009 0.0040 \u00b1 0.0011 Ours (DER) 0.6830 \u00b1 0.0033 0.0711 \u00b1 0.0140 0.2588 \u00b1 0.0024 0.0356 \u00b1 0.0072 0.0047 \u00b1 0.0007 VAL-based 0.6802 \u00b1 0.0006 0.1461 \u00b1 0.0216 0.0016 \u00b1 2.03e-5 0.0234 \u00b1 0.0020 0.0324 \u00b1 0.0202 UGF-based 0.6705 \u00b1 0.0030 0.0644 \u00b1 0.0313 0.2366 \u00b1 0.0016 0.0011 \u00b1 4.03e-5 0.3221 \u00b1 0.0157 CVS-based 0.6758 \u00b1 0.0037 0.0791 \u00b1 0.0136 0.2448 \u00b1 0.0034 0.0373 \u00b1 0.0063 0.0012 \u00b1 0.0003 timization framework via a new measure to promote the notion. See Appendix A.6 for details. Many end-to-end recommender systems offer a recommendation list via two processes: (i) candidate generation and (ii) ranking. In this work, we focus on the \ufb01rst candidate generation for which we built collaborative \ufb01ltering. But our proposed notion can also be applicable in generating an end ranked list. The idea behind the end-ranked list generation is to de\ufb01ne an indicator function, say R, which returns 1 when the item of interest belongs to, say top-K item set (0 otherwise). In this case, a similar notion of the same structure R \u22a5(Zuser, Zitem) serves a proper role. We also provide experimental results which demonstrate the effectiveness of the proposed method in the top-K recommendation setting (See Appendix A.7). 6." + } + ], + "Albert No": [ + { + "url": "http://arxiv.org/abs/2102.07541v2", + "title": "WGAN with an Infinitely Wide Generator Has No Spurious Stationary Points", + "abstract": "Generative adversarial networks (GAN) are a widely used class of deep\ngenerative models, but their minimax training dynamics are not understood very\nwell. In this work, we show that GANs with a 2-layer infinite-width generator\nand a 2-layer finite-width discriminator trained with stochastic gradient\nascent-descent have no spurious stationary points. We then show that when the\nwidth of the generator is finite but wide, there are no spurious stationary\npoints within a ball whose radius becomes arbitrarily large (to cover the\nentire parameter space) as the width goes to infinity.", + "authors": "Albert No, TaeHo Yoon, Sehyun Kwon, Ernest K. Ryu", + "published": "2021-02-15", + "updated": "2021-06-09", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG", + "math.OC" + ], + "main_content": "Introduction Generative adversarial networks (GAN) (Goodfellow et al., 2014), which learn a generative model mimicking the data distribution, have found a broad range of applications in machine learning. While supervised learning setups solve minimization problems in training, GANs solve minimax optimization problems. However, the minimax training dynamics of GANs are poorly understood. Empirically, training is tricky to tune, as reported in (Mescheder et al., 2018, Section 1) and (Goodfellow, 2016, Section 5.1). Theoretically, prior analyses of minimax training have established few guarantees. In the supervised learning setup, the limit in which the deep neural networks\u2019 width is in\ufb01nite has been utilized to analyze the training dynamics. Another line of work establishes guarantees showing that no spurious local minima exist, and such results suggest (but do not formally guarantee) that training converges to the global minimum despite the non-convexity. In this work, we study a Wasserstein GAN (WGAN) (Arjovsky et al., 2017; Gulrajani et al., 2017) with an in1Department of Electronic and Electrical Engineering, Hongik University, Seoul, Korea 2Department of Mathematical Sciences, Seoul National University, Seoul, Korea. Correspondence to: Ernest K. Ryu . Proceedings of the 38 th International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s). \ufb01nitely wide generator trained with stochastic gradient ascent-descent. Speci\ufb01cally, we show that a WGAN with a 2-layer generator and a 2-layer discriminator both with random features and sigmoidal1 activation functions and with the width of the generator (but not the discriminator) being large or in\ufb01nite has no spurious stationary points2 when trained with stochastic gradient ascent-descent. The theoretical analysis utilizes ideas from universal approximation theory and random feature learning. 1.1. Prior work The classical universal approximation theorem establishes that a 2-layer neural network with a sigmoidal activation function can approximate any continuous function when the hidden layer is suf\ufb01ciently wide (Cybenko, 1989). This universality result was extended to broader classes of activation functions (Hornik, 1991; Leshno et al., 1993), and quantitative bounds on the width of such approximations were established (Pisier, 1980-1981; Barron, 1993; Jones, 1992). Random feature learning (Rahimi & Recht, 2007; 2008a;b) combines these ingredients into the following implementable algorithm: generate the hidden layer weights randomly and optimize the weights of the output layers while keeping the hidden layer weights \ufb01xed. In recent years, there has been intense interest in the analysis of in\ufb01nitely wide neural networks, primarily in the realm of supervised learning. In the \u201clazy training regime\u201d, in\ufb01nitely wide neural networks behave as Gaussian processes at initialization (Neal, 1996; Lee et al., 2018) and are essentially linear in the parameters, but not the inputs, during training. The limiting linear network can be characterized with the neural tangent kernel (NTK) (Jacot et al., 2018; Du et al., 2019; Li & Liang, 2018). In a different \u201cmean-\ufb01eld regime\u201d, the training dynamics of in\ufb01nitely wide 2-layer neural networks are characterized with a Wasserstein gradient \ufb02ow. This idea was concurrently developed by several groups (Chizat & Bach, 2018; Mei et al., 2018; Rotskoff & Vanden-Eijnden, 2018; Rotskoff et al., 2019; Sirignano & Spiliopoulos, 2020a;b). Specif1We say an activation function is sigmoidal if it satis\ufb01es assumptions (AG) and (AD), which we later state. The standard sigmoid and tanh activations functions are sigmoidal. 2A stationary point is spurious if it is not a global minimum. arXiv:2102.07541v2 [cs.LG] 9 Jun 2021 \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points ically relevant to GANs, this mean-\ufb01eld machinery was applied to study the dynamics of \ufb01nding mixed Nash equilibria of zero-sum games (Domingo-Enrich et al., 2020). Finally, Geiger et al. (2020) provides a uni\ufb01cation of the NTK and mean-\ufb01eld limits. Another line of analysis in supervised learning establishes that no spurious local minima, non-global local minima, exist. The \ufb01rst results of this type were established for the matrix and tensor decomposition setups (Ge et al., 2016; 2017; Wu et al., 2018; Sanjabi et al., 2019). Later, these analyses were extended to neural networks through the notion of no spurious \u201cbasins\u201d (Nguyen et al., 2019; Liang et al., 2018b;a; Li et al., 2018a; Sun, 2020; Sun et al., 2020b) and \u201cmode connectivity\u201d (Garipov et al., 2018; Kuditipudi et al., 2019; Shevchenko & Mondelli, 2020). Prior works have established convergence guarantees for GANs. The work of (Lei et al., 2020; Hsieh et al., 2019; Domingo-Enrich et al., 2020; Feizi et al., 2020; Sun et al., 2020a) establish global convergence as described in Section 1.2. Cho & Suh (2019) establish that the solution to the Wasserstein GAN is equivalent to PCA in the setup of learning a Gaussian distribution but do not make explicit guarantees on the training dynamics. Sanjabi et al. (2018) use a maximization oracle on a regularized Wassertstein distance to obtain an algorithm converging to stationary points, but did not provide any results relating to global optimality. Although we do not make the connection formal, there is a large body of work establishing convergence for non-convex optimization problems with no spurious local minima solved with gradient descent (Lee et al., 2016; 2019) and stochastic gradient descent (Ge et al., 2015; Jin et al., 2017). The implication of having no spurious stationary points is that stochastic gradient descent \ufb01nds a global minimum. 1.2. Contribution The key technical challenge of this work is the nonconvexity of the loss function in the generator parameters, caused by the fact that the discriminator is nonlinear and non-convex in the input. Prior work avoided this dif\ufb01culty by using a linear discriminator (Lei et al., 2020) or by lifting the generator into the space of probability measures (Hsieh et al., 2019; Sun et al., 2020a; Domingo-Enrich et al., 2020), also described as \ufb01nding mixed Nash equilibria, but these are modi\ufb01cations not commonly used in the empirical training of GANs. Feizi et al. (2020) seems to be the only exception, as they establish convergence guarantees for a WGAN with a linear generator and quadratic discriminator, but their setup is restricted to learning Gaussian distributions. In contrast, we use a nonlinear discriminator and directly optimize the parameters without lifting to \ufb01nd mixed Nash equilibria (we \ufb01nd pure Nash equilibria), while using standard stochastic gradient ascent-descent. To the best of our knowledge, our work is the \ufb01rst to use in\ufb01nite-width analysis to establish theoretical guarantees for GANs with a nonlinear discriminator trained with stochastic gradient-type methods. Our proof technique, distinct from the NTK or mean-\ufb01eld techniques, utilizes universal approximation theory and random feature learning to establish that there are no spurious stationary points. The only other prior work to use in\ufb01nite-width analysis to study GANs was presented in (Domingo-Enrich et al., 2020), where the mean-\ufb01eld limit was used to establish guarantees on \ufb01nding mixed Nash equilibria. We point out that considering the NTK or mean-\ufb01eld limits of the generator and/or the discriminator networks does not resolve the non-convexity of the loss in the generator parameters. We adopt the random feature learning setup, where the hidden layer features are \ufb01xed, and optimize only the output layers for both the generator and the discriminator. Doing so allows us to focus on the key challenge of establishing guarantees on the optimization landscape despite the non-convexity. 2. Problem setup We consider a WGAN whose generator and the discriminator are two-layer networks as illustrated in Figure 1. Let X \u2208Rn be a random vector with a true (target) distribution PX. Let Z \u2208Rk be a continuous random vector from the latent space satisfying the following assumption. (AL) The latent vector Z \u2208Rk has a Lipschitz continuous probability density function qZ(z) satisfying qZ(z) > 0 for all z \u2208Rk. The standard Gaussian is a possible choice satisfying (AL). 2.1. Generator Class Let G = {\u03c6(\u00b7; \u03ba) | \u03ba \u2208Rp}, where \u03c6(\u00b7; \u03ba): Rk \u2192Rn, be a collection of generator feature functions satisfying the following assumption. (AG) All generator feature functions \u03c6 \u2208G are of form \u03c6(z; \u03ba) = \u03c3g(\u03bawz + \u03bab), where \u03ba = (\u03baw, \u03bab) \u2208 Rn\u00d7k \u00d7 Rn, and \u03c3g : R \u2192R is a bounded continuous activation function satisfying limr\u2192\u2212\u221e\u03c3g(r) < limr\u2192\u221e\u03c3g(r). (So p = nk + n.) De\ufb01nition 1 (Generator class, \ufb01nite width). Consider the generator feature functions \u03c61, . . . , \u03c6Ng \u2208G, where 1 \u2264 Ng < \u221e. For \u03b8 \u2208RNg, let g\u03b8(z) = Ng X i=1 \u03b8i\u03c6i(z). Write span({\u03c6i}Ng i=1) = \b g\u03b8 | \u03b8 \u2208RNg\t \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points Generator Discriminator Rk Z Ng \u2248\u221e \u03c61 \u03c62 \u03c6Ng . . . . . . \u02dc X \u03b81 \u03b82 \u03b8Ng Rn Rn . . . \u03c81 \u03c82 \u03c8Nd . . . R1 Nd < \u221e \u03b71 \u03b72 \u03b7Nd . . . . . . True distribution X \u223cPX Figure 1. Illustration of the generator and discriminator architectures. Z represents the latent variable, \u02dc X the generated samples, and X the true samples. Respectively, {\u03c6i} Ng i=1 and {\u03c8j}Nd j=1 are the generator and discriminator (post-activation) feature functions. Respectively, {\u03b8i} Ng i=1 and {\u03b7j}Nd j=1 are the trainable parameters of the generator and discriminator networks. for the class of generators constructed from the feature functions in {\u03c6i}Ng i=1. Note that there exists \u03bai \u2208Rp such that \u03c6i(z) = \u03c6(z; \u03bai) for 1 \u2264i \u2264Ng. We can view the generator g\u03b8 as a twolayer network, where \u03c6(z; \u03ba) = \u03c3g(\u03bawz + \u03bab) represents the post-activation values of the hidden layer. De\ufb01nition 2 (Generator class, in\ufb01nite width). For \u03b8 \u2208 M(Rp), where M(Rp) is the set of measures on Rp with \ufb01nite total mass, let g\u03b8(z) = Z \u03c6(z; \u03ba) d\u03b8(\u03ba). Write span(G) = {g\u03b8(z) | \u03b8 \u2208M(Rp)} for the class of in\ufb01nite-width generators constructed from the feature functions in G. We assume the class of generator feature functions G satis\ufb01es the following universality property. (Universal approximation property) For any function f : Rk \u2192Rn such that EZ [\u2225f(Z)\u22252] < \u221eand \u03b5 > 0, there exists \u03b8\u03b5 \u2208M(Rp) such that EZ [\u2225g\u03b8\u03b5(Z) \u2212f(Z)\u22252] < \u03b5. This assumption holds quite generally. In particular, the following lemma holds as a consequence of (Hornik, 1991). Lemma 1. (AG) implies G satis\ufb01es the (Universal approximation property). In functional analytical terms, (Universal approximation property) states that span(G) is dense in L1(qZ(z) dz; Rn). Later in the proof of Theorem 4, we instead use the following dual characterization of denseness. Lemma 2. Assume (AL) and (Universal approximation property). If a bounded continuous function h: Rk \u2192Rn satis\ufb01es EZ [\u03c6\u22ba(Z)h(Z)] = 0 \u2200\u03c6 \u2208G, then h(z) = 0 for all z \u2208Rk. 2.2. Discriminator Class Let D = {\u03c81, . . . , \u03c8Nd} be a class of discriminator feature functions \u03c8j : Rn \u2192R for 1 \u2264j \u2264Nd satisfying the following assumption. (AD) For all 1 \u2264j \u2264Nd, the discriminator feature functions are of form \u03c8j(x) = \u03c3(a\u22ba j x + bj) for some aj \u2208Rn and bj \u2208R. The twice differentiable activation function \u03c3 satis\ufb01es \u03c3\u2032(x) > 0 for all x \u2208R and supx\u2208R |\u03c3(x)| + |\u03c3\u2032(x)| + |\u03c3\u2032\u2032(x)| < \u221e. The weights a1, . . . , aNd and biases b1, . . . , bNd are sampled (IID) from a distribution with a probability density function. The sigmoid or tanh activation functions for \u03c3 and the standard Gaussian for the distribution of a1, . . . , aNd and b1, . . . , bNd are possible choices satisfying (AD). To clarify with measure-theoretic terms, we are assuming that a1, . . . , aNd and b1, . . . , bNd are sampled from a probability distribution that is absolutely continuous with respect to the Lebesgue measure. De\ufb01nition 3 (Discriminator Class). For \u03b7 \u2208RNd, let \u03a8(x) = (\u03c81(x), . . . , \u03c8Nd(x)) \u2208RNd and f\u03b7(x) = Nd X j=1 \u03b7j\u03c8j(x) = \u03b7\u22ba\u03a8(x). \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points Write span(D) = \b f\u03b7 | \u03b7 \u2208RNd\t for the class of discriminators constructed from the feature functions in D. In contrast with the generators, we only consider \ufb01nitewidth discriminators. 2.3. Adversarial training with stochastic gradients Consider the loss L(\u03b8, \u03b7) = EX [f\u03b7(X)] \u2212EZ [f\u03b7(g\u03b8(Z))] \u22121 2\u2225\u03b7\u22252 2 = EX [\u03b7\u22ba\u03a8(X)] \u2212EZ [\u03b7\u22ba\u03a8(g\u03b8(Z))] \u22121 2\u2225\u03b7\u22252 2. This is a variant of the WGAN loss with the Lipschitz constraint on the discriminator replaced with an explicit regularizer. This loss and regularizer were also considered in (Lei et al., 2020). We train the two networks adversarially by solving the minimax problem minimize \u03b8 maximize \u03b7 L(\u03b8, \u03b7) using stochastic gradient ascent-descent3 \u03b3t \u03b7 = \u03a8(Xt) \u2212\u03a8(g\u03b8(Zt 1)) \u2212\u03b7t \u03b7t+1 = \u03b7t + \u03b3t \u03b7 \u03b3t \u03b8 = (D\u03b8\u03a8(g\u03b8(Zt 2)))\u22ba\u03b7t+1 \u03b8t+1 = \u03b8t \u2212\u03b1\u03b3t \u03b8 for t = 0, 1, . . . , where Xt \u223cPX, Zt 1 \u223cPZ, and Zt 2 \u223cPZ are independent. We \ufb01x the maximization stepsize to 1 while letting the minimization stepsize be \u03b1 > 0. Note that \u03b3t \u03b7 and \u03b3t \u03b8 are stochastic gradients in the sense that E \u0002 \u03b3t \u03b7 \u0003 = \u2207\u03b7L(\u03b8t, \u03b7t) and E [\u03b3t \u03b8] = \u2207\u03b8L(\u03b8t, \u03b7t+1). We can also form \u03b3t \u03b7 and \u03b3t \u03b8 with batches. To clarify, D\u03b8\u03a8(g\u03b8(Z)) = \uf8ee \uf8ef \uf8f0 (\u2207\u03b8 (\u03c81(g\u03b8(Z))))\u22ba . . . (\u2207\u03b8 (\u03c8Nd(g\u03b8(Z))))\u22ba \uf8f9 \uf8fa \uf8fb = \uf8ee \uf8ef \uf8f0 (\u2207x\u03c81(g\u03b8(Z)))\u22ba . . . (\u2207x\u03c8Nd(g\u03b8(Z)))\u22ba \uf8f9 \uf8fa \uf8fb \u0002\u03c61(Z) \u00b7 \u00b7 \u00b7 \u03c6Ng(Z)\u0003 . 3In the in\ufb01nite-width case, where \u03b8 \u2208M(Rp), the stochastic gradient method we describe is not well de\ufb01ned as \u2207\u03b8, if formally de\ufb01ned, is not an element of M(Rp). For a rigorous treatment of analogs of gradient descent in M(Rp), see (Mei et al., 2019; Chizat, 2021) and reference therein. In this work, we apply stochastic gradient ascent-descent only in the \ufb01nite-width setup, but we analyze stationary points for both the \ufb01nite and in\ufb01nite setups. The minimax problem is equivalent to the minimization problem inf \u03b8 sup \u03b7 L(\u03b8, \u03b7) = inf \u03b8 J(\u03b8), where J(\u03b8) \u2206 = sup \u03b7 L(\u03b8, \u03b7) = 1 2 \u2225EX [\u03a8(X)] \u2212EZ [\u03a8(g\u03b8(Z))]\u22252 2 . Interestingly, stochastic gradient ascent-descent applied to L(\u03b8, \u03b7) is equivalent to stochastic gradient descent applied to J(\u03b8): eliminate the \u03b7-variable in the iteration to get \u03b8t+1 = \u03b8t \u2212\u03b1(D\u03b8\u03a8(g\u03b8(Zt 2)))\u22ba(\u03a8(Xt) \u2212\u03a8(g\u03b8(Zt 1))) and note E [D\u03b8\u03a8(g\u03b8(Zt 2))\u22ba(\u03a8(Xt) \u2212\u03a8(g\u03b8(Zt 1)))] = EZ [(D\u03b8\u03a8(g\u03b8(Z))\u22ba](EX [\u03a8(X)] \u2212EZ [\u03a8(g\u03b8(Z))]) = \u2207\u03b8J(\u03b8). In the following sections, we show that J(\u03b8) has no spurious stationary points under suitable conditions. Finally, we introduce the notation r(\u03b8) = EX [\u03a8(X)] \u2212EZ [\u03a8(g\u03b8(Z))], i.e., rj(\u03b8) = EX [\u03c8j(X)] \u2212EZ [\u03c8j(g\u03b8(Z))] for 1 \u2264j \u2264 Nd. This allows us to write J(\u03b8) = 1 2\u2225r(\u03b8)\u22252 2. 3. In\ufb01nite-width generator Consider a GAN with a two-layer in\ufb01nite-width generator g\u03b8 \u2208span(G) and a two-layer \ufb01nite-width discriminator f\u03b7(x) \u2208span(D). In this section, we show that under suitable conditions, J(\u03b8) has no spurious stationary points, i.e., a stationary point of J(\u03b8) is necessarily a global minimum. We say \u03b8s is a stationary point of J if J(\u03b8s + \u03bb\u00b5), as a function of \u03bb \u2208R, is differentiable and has zero gradient at \u03bb = 0 for any \u00b5 \u2208M(Rp). 3.1. Small discriminator (Nd \u2264n) We \ufb01rst consider the case where the discriminator has width Nd \u2264n. Consider the following condition. (Jacobian kernel point condition) The Jacobian D\u03a8(x)\u22ba= \u0002\u2207\u03c81(x) \u2207\u03c82(x) \u00b7 \u00b7 \u00b7 \u2207\u03c8Nd(x)\u0003 satis\ufb01es ker(D\u03a8(x)\u22ba) = {0} for all x \u2208Rn. We can interpret (Jacobian kernel point condition) to imply that there is no redundancy in the discriminator feature functions \u03c81, . . . , \u03c8Nd. This condition holds quite generally when sigmoidal activation functions are used, as characterized by the following lemma. \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points (AG) (UAP) denseness of G no spurious local min Ng = \u221e, Nd \u2264n (AD) (JKP) (AL) (JKB) \u03c3 = sigmoid tanh no spurious local min Ng = \u221e, Nd > n Lem 1 Lem 2 Thm 4 Lem 3 Lem 5 Thm 6 Figure 2. The diagrammatic summary of proofs in Section 3. We denote (Universal approximation property) by (UAP), (Jacobian kernel point condition) by (JKP), and (Jacobian kernel ball condition) by (JKB). Lemma 3. (AD) implies (Jacobian kernel point condition) with probability 1. Proof. Since \u2207\u03c8j(x) = aj\u03c3\u2032(a\u22ba j x + bj), we have D\u03a8(x)\u22ba= \u0002\u2207\u03c81(x) \u00b7 \u00b7 \u00b7 \u2207\u03c8Nd(x)\u0003 = \u0002a1 \u00b7 \u00b7 \u00b7 aNd \u0003 diag(\u03c3\u2032(a\u22ba 1x + b1), . . . , \u03c3\u2032(a\u22ba Ndx + bNd)). By (AD), \u0002a1 \u00b7 \u00b7 \u00b7 aNd \u0003 has full rank with probability 1, and therefore D\u03a8(x)\u22bahas full rank. We are now ready to state and prove the \ufb01rst main result of this work: our GAN with an in\ufb01nite-width generator has no spurious stationary points. Theorem 4. Assume (AL), (AG), and (AD). Then the following statement holds4 with probability 1: any stationary point \u03b8s satis\ufb01es J(\u03b8s) = 0. Proof. Let \u03b8s be a stationary point of J. Then, for any \u00b5 \u2208M(Rp), \u2202 \u2202\u03bbJ(\u03b8s + \u03bb\u00b5) \f \f \f \f \u03bb=0 = 0. 4The randomness comes from the random generation of \u03c8j\u2019s described in (AD) and is unrelated to randomness of SGD. Once \u03c8j\u2019s have been generated and the (Jacobian kernel point condition) holds by Lemma 3, the conclusion of Theorem 4 holds without further probabilistic quanti\ufb01ers. Since g\u03b8s+\u03bb\u00b5 = g\u03b8s + \u03bbg\u00b5, \u2202 \u2202\u03bbJ(\u03b8s + \u03bb\u00b5) = \u2202 \u2202\u03bb 1 2\u2225r(\u03b8s + \u03bb\u00b5)\u22252 2 = \u2212r(\u03b8s + \u03bb\u00b5)\u22baEZ \u0014 \u2202 \u2202\u03bb\u03a8(g\u03b8s+\u03bb\u00b5(Z)) \u0015 = \u2212r(\u03b8s + \u03bb\u00b5)\u22baEZ [D\u03a8(g\u03b8s+\u03bb\u00b5(Z))\u22bag\u00b5(Z)] = \u2212EZ \uf8ee \uf8f0 Nd X j=1 rj(\u03b8s + \u03bb\u00b5)\u2207\u03c8j(g\u03b8s+\u03bb\u00b5(Z))\u22bag\u00b5(Z) \uf8f9 \uf8fb. Thus, for all \u00b5 \u2208M(Rp), EZ \uf8ee \uf8f0 Nd X j=1 rj(\u03b8s)\u2207\u03c8j(g\u03b8s(Z))\u22bag\u00b5(Z) \uf8f9 \uf8fb= 0. By Lemmas 1 and 2, Nd X j=1 rj(\u03b8s)\u2207\u03c8j(g\u03b8s(z)) = 0 (1) for all z \u2208Rk. Thus, \u0002\u2207\u03c81(g\u03b8s(z)) \u00b7 \u00b7 \u00b7 \u2207\u03c8Nd(g\u03b8s(z))\u0003 \uf8ee \uf8ef \uf8f0 r1(\u03b8s) . . . rNd(\u03b8s) \uf8f9 \uf8fa \uf8fb= 0. By Lemma 3, the (Jacobian kernel point condition) holds with probability 1. Therefore, r(\u03b8s) = 0 and we conclude J(\u03b8s) = 0. 3.2. Large discriminator (n < Nd < \u221e) Next, consider the case where the discriminator has width Nd > n. In the small discriminator case, we used the (Jacobian kernel point condition), which states rank(D\u03a8(x)\u22ba) = \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points (AG) (AL) (AD) (fUAP) with high probability no spurious local min in large ball Ng < \u221e, Nd \u2264n with high probability (JKP) Lem 3 Lem 7 Thm 8 Figure 3. The diagrammatic summary of the proof in Section 4. We denote (Finite universal approximation property) by (fUAP), (Jacobian kernel point condition) by (JKP), and (Jacobian kernel ball condition) by (JKB). Nd. However, this is not possible in the large discriminator case as rank(D\u03a8(x)\u22ba) \u2264n < Nd. Therefore, we consider the following weaker condition. (Jacobian kernel ball condition) For any open ball B \u2282Rn, \\ x\u2208B ker(D\u03a8(x)\u22ba) = {0}. Since \u2207x(\u03b7\u22ba\u03a8(x)) = D\u03a8(x)\u22ba\u03b7, the (Jacobian kernel ball condition) implies that \u03b7\u22ba\u03a8(x) with \u03b7 \u0338= 0 is not a constant function within any open ball B, and we can interpret the condition to imply that there is no redundancy in the discriminator feature functions \u03c81, . . . , \u03c8Nd. The condition holds generically under mild conditions, as characterized by the following lemma. Lemma 5. Assume \u03c3: R \u2192R is the sigmoid or the tanh function. Then (AD) implies (Jacobian kernel ball condition) with probability 1. Proof outline of Lemma 5. The random generation of (AD) implies that with probability 1, all nonzero linear combinations of \u03c81, . . . , \u03c8Nd are nonconstant, i.e., f\u03b7(x) = \u03b7\u22ba\u03a8(x) with \u03b7 \u0338= 0 is not globally constant (Sussmann, 1992, Lemma 1). Since \u03c3 is an analytic function, this implies f\u03b7 with \u03b7 \u0338= 0 is not constant within any open ball B. So \u2207xf\u03b7(x) = D\u03a8(x)\u22ba\u03b7 is not identically zero in B and we conclude \u03b7 / \u2208T x\u2208B ker(D\u03a8(x)\u22ba) for any \u03b7 \u0338= 0. We are now ready to state and prove the second main result of this work. Theorem 6. Assume (AL), (AG), and (AD). Then the following statement holds5 with probability 1: for any stationary point \u03b8s, if the range of g\u03b8s(Z) contains an open-ball in Rn, then J(\u03b8s) = 0. 5Once \u03c8j\u2019s have been generated and the (Jacobian kernel ball condition) holds by Lemma 5, the conclusion of Theorem 6 holds without further probabilistic quanti\ufb01ers. Proof. Following the same steps as in the proof of Theorem 4, we arrive at (1), which we rewrite as D\u03a8(g\u03b8(z))\u22bar(\u03b8) = 0. Thus, for all z \u2208Rk, r(\u03b8) \u2208ker(D\u03a8(g\u03b8(z))\u22ba). Since the range of X\u03b8 = g\u03b8(Z) contains an open ball, the (Jacobian kernel ball condition), which holds with probability 1 by Lemma 5, implies r(\u03b8) = 0. Theorem 6 implies that a stationary point may be a spurious stationary point only when the generator\u2019s output is degenerate. One can argue that when PX, the target distribution of X, has full-dimensional support, the generator should not converge to a distribution with degenerate support. Indeed, this is what we observe in our experiments of Section 5. 4. Finite-width generator Consider a GAN with a two-layer \ufb01nite-width generator g\u03b8 \u2208span({\u03c6i}Ng i=1) and a two-layer \ufb01nite-width discriminator f\u03b7(x) \u2208span(D). In this section, we show that J(\u03b8) has no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the generator\u2019s width Ng goes to in\ufb01nity. The \ufb01nite-width analysis relies on a \ufb01nite version of the (Universal approximation property) that implies we can approximate a given function as a linear combination of {\u03c6i}Ng i=1. Let \u03b4(l) : Rk \u2192Rn have the delta function on the l-th component and zero functions for all other components, i.e., h \u03b4(l)(z) i i = ( 0 if i \u0338= l \u03b4(z) if i = l for 1 \u2264l \u2264n. (Finite universal approximation property) For a given \u03b5 > 0, there exists a large enough Ng \u2208N and \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points \u03c61, . . . , \u03c6Ng \u2208G such that there exists {\u03b8(\u03b5,l) i \u2208R | 1 \u2264 i \u2264Ng, 1 \u2264l \u2264n} satisfying \f \f \f \f \f \f EZ \uf8ee \uf8f0 \uf8eb \uf8ed Ng X i=1 \u03b8(\u03b5,l) i \u03c6i(Z) \u2212\u03b4(l)(Z) \uf8f6 \uf8f8 \u22ba f(Z) \uf8f9 \uf8fb \f \f \f \f \f \f < \u03b5 sup z\u2208Rk{\u2225f(z)\u22252 + \u2225Df(z)\u2225} for all coordinates l = 1, . . . , n, and for any continuously differentiable f : Rk \u2192 Rn such that supz\u2208Rk{\u2225f(z)\u22252 + \u2225Df(z)\u2225} < \u221e. This (Finite universal approximation property) holds with high probability when the width Ng is suf\ufb01ciently large and the weights and biases of the generator feature functions \u03c61, . . . \u03c6Ng are randomly generated. Lemma 7. Assume (AL) and (AG). Assume the \ufb01rst n parameters {\u03bai}n i=1 are chosen so that {\u03c6i}n i=1 are constant functions spanning the sample space Rn. Assume the remaining parameters {\u03bai}Ng i=n+1 are sampled (IID) from a probability distribution that has a continuous and strictly positive density function. Then for any \u03b5 > 0 and \u03b6 > 0, there exists large enough 6 Ng such that (Finite universal approximation property) with \u03b5 holds with probability at least 1 \u2212\u03b6. Remember that the parameters de\ufb01ne the generator feature functions through \u03c6i(x) = \u03c6i(x; \u03bai) for 1 \u2264i \u2264Ng. By choosing the \ufb01rst n parameters in this way, we are effectively providing a trainable bias term in the output layer of the generator. Note that most universal approximation results consider the approximation of functions, while (Finite universal approximation property) requires the approximation of the delta function, which is not truly a function. Proof outline of Lemma 7. Here, we illustrate the proof in the case of n = 1. The general n \u22651 case requires similar reasoning but more complicated notation. First, we de\ufb01ne the smooth approximation of \u03b4 by \u02dc \u03b4\u03b5(z) = C \u03b5k e\u2212\u2225z/\u03b5\u22252 2, where C is a constant (depending on k but not \u03b5) such that Z Rk \u02dc \u03b4\u03b5(z) dz = 1. We argue that \u02dc \u03b4\u03b5(z) \u2248\u03b4 in the sense made precise in Lemma 11 of the appendix. 6A quantitative bound on Ng can be established with careful bookkeeping. Speci\ufb01cally, using (10) of Appendix A.4, we can quantify Ng as a function of CK. This CK, which serves a similar role as the C of (Rahimi & Recht, 2007, Theorem 1), can also be quanti\ufb01ed as a function of \u03f5. However, the resulting bound is complicated and loose. Next, we approximate \u02dc \u03b4\u03b5 with the random feature functions. Using the arguments of (Barron, 1993, Theorem 2) and (Telgarsky, 2020, Section 4.2), we show that there exists a bounded density m(\u03ba) and \u03ba1 \u2208Rk+1 such that \u03c61 = \u03c6(z; \u03ba1) is a nonzero constant function and \u02dc \u03b4\u03b5(z) \u2248\u03b8\u03b5 1\u03c61(z) + Z \u03c6(z; \u03ba) m(\u03ba) d\u03ba for some \u03b8\u03b5 1 \u2208R. For large K > 0, Z \u03c6(z; \u03ba) m(\u03ba)d\u03ba \u2248 Z \u03c6(z; \u03ba) m(\u03ba)1{\u2225\u03ba\u2225\u2264K}(\u03ba) d\u03ba, where 1{\u2225\u03ba\u2225\u2264K} is the 0-1 indicator function. Write p(\u03ba) for the continuous and strictly positive density function of the distribution generating \u03ba. Then sup\u03ba{m(\u03ba)1{\u2225\u03ba\u2225\u2264K}(\u03ba)/p(\u03ba)} < \u221e, and this allows us to use random feature learning arguments of (Rahimi & Recht, 2008b). By (Rahimi & Recht, 2008b, Lemma 1), there exists a large enough Ng such that there exist weights {\u03b8\u03b5 i }Ng i=2 such that Ng X i=2 \u03b8\u03b5 i \u03c6(z; \u03bai) \u2248 Z \u03c6(z; \u03ba) m(\u03ba)1{\u2225\u03ba\u2225\u2264K}(\u03ba) d\u03ba with probability 1 \u2212\u03b6. Finally, we complete the proof by chaining the \u2248steps. We are now ready to state and prove the third main result of this work: our GAN with an \ufb01nite-width generator and small discriminator has no spurious stationary points within a large ball around the origin. Theorem 8. Let Nd \u2264n. Assume (AL), (AG), and (AD). Assume the generator feature functions are generated randomly as in Lemma 7. For any C > 0 and \u03b6 > 0, there exists a large enough Ng \u2208N such that the following statement holds with probability at least 1 \u2212\u03b6: any stationary point \u03b8s \u2208RNg satisfying \u2225\u03b8s\u22251 \u2264C is a global minimum.7 Proof of Theorem 8. Since \u03c6 is bounded by (AG) and \u2225\u03b8s\u22251 is bounded, the output of g\u03b8s is also bounded, and sup \u2225\u03b8s\u22251\u2264C \u2225g\u03b8s(0)\u22252 < \u221e. The (Jacobian kernel point condition), which holds with probability 1 by Lemma 3, implies C1 \u2206 = inf \u2225x\u22252\u2264 sup \u2225\u03b8s\u22251\u2264C \u2225g\u03b8s(0)\u22252 ! \u03c4min(D\u03a8(x)\u22ba) > 0, 7The randomness comes from the random generation of \u03c8j\u2019s and \u03c6i\u2019s described in (AD) and Lemma 7. Once the (Jacobian kernel point condition) and (Finite universal approximation property) holds (an event with probability at least 1 \u2212\u03b6) the conclusion of Theorem 8 holds without further probabilistic quanti\ufb01ers. Again, the size of Ng can be quanti\ufb01ed through careful bookkeeping. \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points (a) Samples from true distribution PX (b) Samples from generator g\u03b8(Z) (c) Convergence of the loss functions J and L Figure 4. Samples and loss functions with a mixture of 8 Gaussians, X \u2208R2, Z \u2208R2, Ng = 5, 000, and Nd = 1, 000. The generator accurately learns the sampling distribution, and the loss functions converge to 0. The code is available at https://github.com/ sehyunkwon/Infinite-WGAN. where \u03c4min denotes the Nd-th singular value. We use the fact that \u03c4min(D\u03a8(x)\u22ba) is a continuous function of x and the in\ufb01mum over a compact set of a continuous positive function is positive. (We use \u03c4min to denote the minimum singular value, rather than the standard \u03c3min to avoid confusion with the \u03c3 denoting the activation function.) By (AD), C2 \u2206 = max j=1,...,Nd sup x\u2208Rn{\u2225\u2207\u03c8j(x)\u2225+ \u2225\u22072\u03c8j(x)\u2225} \u2208(0, \u221e) By Lemma 7, there exists a large enough Ng such that (Finite universal approximation property) with \u03b5 = C1qZ(0) 2C2Ndn holds with probability 1 \u2212\u03b6. Let \u03b8s be a stationary point satisfying \u2225\u03b8s\u22251 \u2264C. However, assume for contradiction that J(\u03b8s) \u0338= 0, i.e., r(\u03b8s) \u0338= 0. Then \u2202 \u2202\u03b8i J(\u03b8s) = EZ \uf8ee \uf8f0 Nd X j=1 rj(\u03b8s)\u2207\u03c8j(g\u03b8s(Z))\u22ba\u03c6i(Z) \uf8f9 \uf8fb= 0 for all 1 \u2264i \u2264Ng. De\ufb01ne the normalized residual vector \u02c6 r = (1/\u2225r\u22252)r, and write EZ \uf8ee \uf8f0 Nd X j=1 \u02c6 rj(\u03b8)\u2207\u03c8j(g\u03b8(Z))\u22ba\u03c6i(Z) \uf8f9 \uf8fb= 0 (2) for all 1 \u2264i \u2264Ng. Now consider \f \f \f \f \f \f Nd X j=1 \u02c6 rj(\u03b8) \u2202 \u2202xl \u03c8j(g\u03b8(0))qZ(0) \f \f \f \f \f \f = \f \f \f \f \fEZ \" Nd X j=1 \u02c6 rj(\u03b8)\u2207\u03c8j(g\u03b8(Z))\u22ba | {z } \u2206 =f(Z)\u22ba \u03b4(l)(Z) #\f \f \f \f \f = \f \f \f \f \f \f EZ \uf8ee \uf8f0f(Z)\u22ba \uf8eb \uf8ed\u03b4(l)(Z) \u2212 Ng X i=1 \u03b8(l,\u03b5) i \u03c6i(z) \uf8f6 \uf8f8 \uf8f9 \uf8fb \f \f \f \f \f \f < \u03b5 sup z\u2208Rk{\u2225f(z)\u2225+ \u2225Df(z)\u2225} \u2264\u03b5 Nd X j=1 |\u02c6 rj(\u03b8)| sup x\u2208Rn{\u2225\u2207\u03c8j(x)\u2225+ \u2225\u22072\u03c8j(x)\u2225} \u2264\u03b5C2Nd, where the \ufb01rst equality follows from the de\ufb01nition of \u03b4(l), the second equality follows from (2), the \ufb01rst inequality follows from the the (Finite universal approximation property), the second inequality follows from the triangle inequality of the norm, and the third inequality follows from the de\ufb01nition of C2 and the fact that the normalized residual satis\ufb01es |\u02c6 rj(\u03b8)| \u22641 for all j. By summing this result over 1 \u2264l \u2264n and using the bound \u2225\u00b7 \u22252 \u2264\u2225\u00b7 \u22251, we get \u2225D\u03a8(g\u03b8(0))\u22ba\u02c6 r(\u03b8)\u22252 < \u03b5C2Ndn qZ(0) . Finally, we arrive at C1 = C1\u2225\u02c6 r(\u03b8)\u22252 \u2264\u2225D\u03a8(g\u03b8(0))\u22ba\u02c6 r(\u03b8)\u22252 < \u03b5C2Ndn qZ(0) = C1 2 , which is a contradiction. \fWGAN with an In\ufb01nitely Wide Generator Has No Spurious Stationary Points (a) Loss landscape with Ng = 2 (b) Loss landscape with Ng = 10 Figure 5. The landscape of J(\u03b8) for a mixture of two Gaussians with generator widths Ng = 2 and Ng = 10. The \ufb01rst Ng = 2 example has multiple non-global local minima. The second Ng = 10 example has no spurious stationary points despite the landscape being clearly non-convex. We provide corresponding contour plots in the appendix. 5. Experiments Figure 6 presents an experiment with a mixture of 8 Gaussians and Ng = 5, 000. The experiments demonstrate the suf\ufb01ciency of two-layer networks with random features and that the training does not encounter local minima when Ng is large. Figure 5 visualizes the loss landscape with generator widths Ng = 2 and Ng = 10. For the Ng = 10 case, the parameter space was projected down to a 2D space de\ufb01ned by random directions, as recommended by Li et al. (2018b). We observe the landscape becomes more favorable with larger width. 6." + }, + { + "url": "http://arxiv.org/abs/1709.00134v1", + "title": "Universality of Logarithmic Loss in Lossy Compression", + "abstract": "We establish two strong senses of universality of logarithmic loss as a\ndistortion criterion in lossy compression: For any fixed length lossy\ncompression problem under an arbitrary distortion criterion, we show that there\nis an equivalent lossy compression problem under logarithmic loss. In the\nsuccessive refinement problem, if the first decoder operates under logarithmic\nloss, we show that any discrete memoryless source is successively refinable\nunder an arbitrary distortion criterion for the second decoder.", + "authors": "Albert No, Tsachy Weissman", + "published": "2017-09-01", + "updated": "2017-09-01", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION In the lossy compression problem, logarithmic loss distortion is a criterion allowing a \u201csoft\u201d reconstruction of the source, a departure from the classical setting of a deterministic reconstruction. Although logarithmic loss plays a crucial role in the theory of learning and prediction, relatively little work has been done in the context of lossy compression, notwithstanding the two-encoder multiterminal source coding problem under logarithmic loss [1], [2], or recent work on the single-shot approach to lossy courec coding under logarithmic loss [3]. Note that lossy compression under logarithmic loss is closely related to the information bottleneck method [4]\u2013[6]. In this paper, we focus on universal properties of logarithmic loss in two lossy compression problems. First, we consider the \ufb01xed-length lossy compression problem. We show that there is a one-to-one correspondence between any \ufb01xed-length lossy compression problem under an arbitrary distortion measure and that under logarithmic loss. The correspondence is in the following strong sense: \u2022 Optimum schemes for the two problems are the same. \u2022 A good scheme for one problem is also a good scheme for the other. We will be more precise about \u201coptimum\u201d and \u201cgoodness\u201d of the scheme in later sections. This \ufb01nding essentially implies that it is enough to consider the lossy compression problem under logarithmic loss. We point out that our The material in this paper has been presented in part at the 2015 International Symposium on Information Theory. This work was supported by the Hongik University new faculty research support fund. A. No is with the Department of Electronic and Electrical Engineering, Hongik University, Seoul, Korea (email: albertno@hongik.ac.kr) T. Weissman is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA (email: tsachy@stanford.edu) \f2 result is different from that of [7] which justi\ufb01es logarithmic loss by showing it is the only loss function that satis\ufb01es a natural data processing requirement. We also consider the successive re\ufb01nement problem under logarithmic loss. Successive re\ufb01nement is a network lossy compression problem where one encoder wishes to describe the source to two decoders [8], [9]. Instead of having two separate coding schemes, the successive re\ufb01nement encoder designs a code for the decoder with a weaker link, and sends extra information to the second decoder on top of the message of the \ufb01rst decoder. In general, successive re\ufb01nement coding cannot do as well as two separate encoding schemes optimized for the respective decoders. However, if we can achieve the point-to-point optimum rates using successive re\ufb01nement coding, we say the source is successively re\ufb01nable. We show that any discrete memoryless source is successively re\ufb01nable as long as the weaker link employs logarithmic loss, regardless of the distortion criterion used for the stronger link. This idea can be useful to construct practical point-to-point lossy compression since it allows a smaller codebook and lower encoding complexity [10]\u2013[12]. The remainder of the paper is organized as follows. In Section II, we revisit some of the known results pertaining to logarithmic loss and \ufb01xed-length lossy compression. Section III is dedicated to the equivalence between lossy compression under arbitrary distortion measures and that under logarithmic loss. Section IV is dedicated to successive re\ufb01nement under logarithmic loss in the weaker link. Notation: Xn denotes an n-dimensional random vector (X1, X2, . . . , Xn) while xn denotes a speci\ufb01c possible realization of the random vector Xn. Similarly, Q denotes a random probability mass function while q denotes a speci\ufb01c probability mass function. We use natural logarithm and nats instead of bits. II. PRELIMINARIES A. Logarithmic Loss Let X be a set of discrete source symbols (|X| < \u221e), and M(X) be the set of probability measures on X. Logarithmic loss \u2113: X \u00d7 M(X)\u2192[0, \u221e] is de\ufb01ned by \u2113(x, q) = log 1 q(x) for x \u2208X and q \u2208M(X). Logarithmic loss between n-tuples is de\ufb01ned by \u2113n(xn, qn) = 1 n n X i=1 log 1 qi(xi), i.e., the symbol-by-symbol extension of the single letter loss. Let Xn be the discrete memoryless source with distribution PX. Consider the lossy compression problem under logarithmic loss where the reconstruction alphabet is M(X). The rate-distortion function is given by R(D) = inf PQ|X:E[\u2113(X,Q)]\u2264D I(X; Q) =H(X) \u2212D. The following lemma provides a property of the rate-distortion function achieving conditional distribution. \f3 Lemma 1. The rate-distortion function achieving conditional distribution PQ\u22c6|X satis\ufb01es PX|Q\u22c6(\u00b7|q) =q (1) H(X|Q\u22c6) =D (2) for PQ\u22c6almost every q \u2208M(X). Conversely, if PQ|X satis\ufb01es (1) and (2), then it is a rate-distortion function achieving conditional distribution, i.e., I(X; Q) =R(D) = H(X) \u2212D E [\u2113(X, Q)] =D. The key idea is that we can replace Q by PX|Q(\u00b7|Q), and have lower rate and distortion, i.e., I(X; Q) \u2265I(X; PX|Q(\u00b7|Q)) E [\u2113(X, Q)] \u2265E \u0002 \u2113(X, PX|Q(\u00b7|Q) \u0003 , which directly implies (1). Interestingly, since the rate-distortion function in this case is a straight line, a simple time sharing scheme achieves the optimal rate-distortion tradeoff. If the encoder losslessly compresses only the \ufb01rst H(X)\u2212D H(X) fraction of the source sequence components, while the decoder perfectly recovers those components and uses PX as its reconstruction for the remaining components then the resulting scheme obviously achieves distortion D with rate H(X) \u2212D. Moreover, this simple scheme directly implies successive re\ufb01nability of the source. For D1 > D2, suppose the encoder losslessly compresses the \ufb01rst H(X)\u2212D2 H(X) fraction of the source. Then the \ufb01rst decoder can perfectly reconstruct H(X)\u2212D1 H(X) fraction of the source with the message of rate H(X)\u2212D1 and distortion D1 while the second decoder can achieve distortion D2 with rate H(X) \u2212D2. Since both decoders can achieve the best rate-distortion pair, it follows that any discrete memoryless source under logarithmic loss is successively re\ufb01nable. B. Fixed-Length Lossy Compression In this section, we brie\ufb02y introduce the basic settings of the \ufb01xed-length lossy compression problem (see [13] and references therein for more details). In the \ufb01xed-length lossy compression setting, we have a source X with \ufb01nite alphabet X = {1, . . . , r} and source distribution PX. An encoder f : X \u2192{1, . . . , M} maps the source symbol to one of M messages. On the other side, a decoder g : {1, . . . , M} \u2192\u02c6 X maps the message to the actual reconstruction \u02c6 X where the reconstruction alphabet is also \ufb01nite \u02c6 X = {1, . . . , s}. Let d : X \u00d7 \u02c6 X \u2192[0, \u221e) be a distortion measure between the source and the reconstruction. First, we can think of the code that the expected distortion is lower than a given distortion level. De\ufb01nition 1 (Average distortion criterion). An (M, D) code is a pair of an encoder f with |f| \u2264M and a decoder g such that E [d(X, g(f(X)))] \u2264D. \f4 The minimum number of codewords required to achieve average distortion not exceeding D is de\ufb01ned by M \u22c6(D) = min{M : \u2203(M, D) code}. Similarly, we can de\ufb01ne the minimum achievable average distortion given the number of codewords M. D\u22c6(M) = min{D : \u2203(M, D) code}. One may consider a stronger criterion that restricts the probability of exceeding a given distortion level. De\ufb01nition 2 (Excess distortion criterion). An (M, D, \u01eb) code is a pair of an encoder f with |f| \u2264M and a decoder g such that Pr [d(X, g(f(X))) > D] \u2264\u01eb. The minimum number of codewords required to achieve excess distortion probability \u01eb and distortion D is de\ufb01ned by M \u22c6(D, \u01eb) = min{M : \u2203(M, D, \u01eb) code}. Similarly, we can de\ufb01ne the minimum achievable excess distortion probability given the target distortion D and the number of codewords M. \u01eb\u22c6(M, \u01eb) = min{\u01eb : \u2203(M, D, \u01eb) code}. Given the target distortion D and PX, the informational rate-distortion function is de\ufb01ned by R(D) = inf P \u02c6 X|X:E[d(X, \u02c6 X)]\u2264D I(X; \u02c6 X) We make the following benign assumptions: \u2022 There exists a unique rate-distortion function achieving conditional distribution P \u22c6 \u02c6 X|X. \u2022 We assume that P \u02c6 X\u22c6(\u02c6 x) > 0 for all \u02c6 x \u2208\u02c6 X since we can always discard the reconstruction symbol with zero probability. \u2022 If d(x, \u02c6 x1) = d(x, \u02c6 x2) for all x \u2208X, then \u02c6 x1 = \u02c6 x2. De\ufb01ne the information density of the joint distribution PX, \u02c6 X by \u0131X; \u02c6 X(x; \u02c6 x) = log PX, \u02c6 X(x, \u02c6 x) PX(x)P \u02c6 X(\u02c6 x). Then, we are ready to de\ufb01ne the D-tilted information which plays a key role in \ufb01xed-length lossy compression. De\ufb01nition 3. [13, De\ufb01nition 6] The D-tilted information in x \u2208X is de\ufb01ned as \uf6beX(x, D) = log 1 E h exp \u0010 \u03bb\u22c6D \u2212\u03bb\u22c6d(x, \u02c6 X\u22c6) \u0011i where the expectation is with respect to the marginal distribution of \u02c6 X\u22c6and \u03bb\u22c6= \u2212R\u2032(D). \f5 Theorem 2. [14, Lemma 1.4] For all \u02c6 x \u2208\u02c6 X , \uf6beX(x, D) = \u0131X; \u02c6 X\u22c6(x; \u02c6 x) + \u03bb\u22c6d(x, \u02c6 x) \u2212\u03bb\u22c6D, (3) and therefore we have R(D) = E [\uf6be(X, D)]. Let P \u22c6 X| \u02c6 X be the induced conditional probability from P \u22c6 \u02c6 X|X. Then (3) can equivalently be expressed as log 1 P \u22c6 X| \u02c6 X(x|\u02c6 x) = log 1 PX(x) \u2212\uf6beX(x, D) + \u03bb\u22c6d(x, \u02c6 x) \u2212\u03bb\u22c6D. (4) The following lemma shows that P \u22c6 X| \u02c6 X(\u00b7|\u02c6 x) are all distinct. Lemma 3. For all \u02c6 x1 \u0338= \u02c6 x2, there exists x \u2208X such that P \u22c6 X| \u02c6 X(x|\u02c6 x1) \u0338= P \u22c6 X| \u02c6 X(x|\u02c6 x2). Proof. Suppose by contradiction that P \u22c6 X| \u02c6 X(x|\u02c6 x1) = P \u22c6 X| \u02c6 X(x|\u02c6 x2) for all x \u2208X. Then, (4) implies log 1 PX(x) \u2212\uf6beX(x, D) + \u03bb\u22c6d(x, \u02c6 x1) \u2212\u03bb\u22c6D = log 1 PX(x) \u2212\uf6beX(x, D) + \u03bb\u22c6d(x, \u02c6 x2) \u2212\u03bb\u22c6D for all x \u2208X, i.e., d(x, \u02c6 x1) = d(x, \u02c6 x2) for all x \u2208X, which violates our assumption on the distortion measure. C. Successive Re\ufb01nability In this section, we review the successive re\ufb01nement problem with two decoders. Let the source Xn be i.i.d. random vector with distribution PX. The encoder wants to describe Xn to two decoders by sending a pair of messages (m1, m2) where 1 \u2264mi \u2264Mi for i \u2208{1, 2}. The \ufb01rst decoder reconstructs \u02c6 Xn 1 (m1) \u2208\u02c6 X n 1 based only on the \ufb01rst message m1. The second decoder reconstructs \u02c6 Xn 2 (m1, m2) \u2208\u02c6 X n 2 based on both m1 and m2. The setting is described in Figure 1. Xn Enc Dec 1 Dec 2 \u02c6 Xn 1 \u02c6 Xn 2 m1 m2 Fig. 1: Successive Re\ufb01nement Let di(\u00b7, \u00b7) : X \u00d7 \u02c6 Xi \u2192[0, \u221e) be a distortion measure for i-th decoder. The rates of code (R1, R2) are simply de\ufb01ned as R1 = 1 n log M1 \f6 R2 = 1 n log M1M2. An (n, R1, R2, D1, D2, \u01eb)-successive re\ufb01nement code is a coding scheme with block length n and excess distortion probability \u01eb where rates are (R1, R2) and target distortions are (D1, D2). Since we have two decoders, the excess distortion probability is de\ufb01ned by Pr h di(Xn, \u02c6 Xn i ) > Di for some i i . De\ufb01nition 4. A rate-distortion tuple (R1, R2, D1, D2) is said to be achievable, if there is a family of (n, R(n) 1 , R(n) 2 , D1, D2, \u01eb(n))-successive re\ufb01nement code where lim n\u2192\u221eR(n) i = Ri for all i, lim n\u2192\u221e\u01eb(n) = 0. For some special cases, both decoders can achieve the point-to-point optimum rates simultaneously. De\ufb01nition 5. Let Ri(Di) denote the rate-distortion function of the i-th decoder for i \u2208{1, 2}. If the rate-distortion tuple (R1(D1), R2(D2), D1, D2) is achievable, then we say the source is successively re\ufb01nable at (D1, D2). If the source is successively re\ufb01nable at (D1, D2) for all D1, D2, then we say the source is successively re\ufb01nable. The following theorem provides a necessary and suf\ufb01cient condition of successive re\ufb01nable sources. Theorem 4 ([8], [9]). A source is successively re\ufb01nable at (D1, D2) if and only if there exists a conditional distribution P \u02c6 X1, \u02c6 X2|X such that X \u2212\u02c6 X2 \u2212\u02c6 X1 forms a Markov chain and Ri(Di) = I(X; \u02c6 Xi) E h di(X, \u02c6 Xi) i \u2264Di for i \u2208{1, 2}. The condition in the theorem holds for all discrete memoryless sources under logarithmic loss. Note that the above results of successive re\ufb01nability can easily be generalized to the case of k decoders. III. ONE TO ONE CORRESPONDENCE BETWEEN GENERAL DISTORTION AND LOGARITHMIC LOSS A. Main Results Consider \ufb01xed-length lossy compression under an arbitrary distortion d(\u00b7, \u00b7) as described in Section II-B. We have a source X with the \ufb01nite alphabet X = {1, . . . , r}, source distribution PX, and the \ufb01nite reconstruction alphabet \u02c6 X = {1, . . . , s}. For the \ufb01xed number of messages M, let f \u22c6and g\u22c6be the encoder and decoder that achieve the optimum average distortion D\u22c6(M). I.e., E [d(X, g\u22c6(f \u22c6(X)))] = D\u22c6(M). Let P \u22c6 \u02c6 X|X denote the rate-distortion function achieving conditional distribution at distortion D = D\u22c6(M). I.e., PX \u00d7 P \u22c6 \u02c6 X|X achieves the in\ufb01mum in R(D\u22c6(M)) = inf P \u02c6 X|X:E[d(X, \u02c6 X)]\u2264D\u22c6(M) I(X; \u02c6 X). (5) \f7 Note that R(D\u22c6(M)) may be strictly smaller than log M in general since R(\u00b7) is an informational rate-distortion function which does not characterize the best achievable performance for the \u201cone-shot\u201d setting in which D\u22c6(M) is de\ufb01ned. Now, we de\ufb01ne the corresponding \ufb01xed-length lossy compresson problem under logarithmic loss. In the corresponding problem, the source alphabet X = {1, . . . , r}, source distribution PX, and the number of messages M remain the same. However, we have different reconstruction alphabet Y = {P \u22c6 X| \u02c6 X(\u00b7|\u02c6 x) : \u02c6 x \u2208\u02c6 X} \u2282M(X) where P \u22c6pertains to the achiever of the in\ufb01mum in (5) associated with the original loss function. Let the distortion now be logarithmic loss. We can also connect the encoding and decoding schemes between the two problems. Suppose f : X \u2192 {1, . . . , M} and g : {1, . . ., M} \u2192 \u02c6 X are an encoder and decoder pair in the original problem. Let f\u2113= f and g\u2113: {1, . . ., M} \u2192Y where g\u2113(m) = P \u22c6 X| \u02c6 X(\u00b7|g(m)). Then, f\u2113and g\u2113are an valid encoder and decoder pair for the corresponding \ufb01xed-length lossy compression problem under logarithmic loss. Conversely, given f\u2113and g\u2113, we can \ufb01nd corresponding f and g because Lemma 3 guarantees that PX| \u02c6 X(\u00b7|\u02c6 x) are distinct. The following result shows the relation between the corresponding schemes. Theorem 5. For any encoder-decoder pair (f\u2113, g\u2113) for the corresponding \ufb01xed-length lossy compression problem under logarithmic loss, we have E [\u2113(X, g\u2113(f\u2113(X)))] = H(X| \u02c6 X\u22c6) + \u03bb\u22c6(E [d(X, g(f(X)))] \u2212D\u22c6(M)) \u2265H(X| \u02c6 X\u22c6) where (f, g) is the corresponding encoder-decoder pair for the original lossy compression problem. Note that H(X| \u02c6 X\u22c6) and the expectations are with respect to the distribution PX \u00d7 P \u22c6 \u02c6 X|X. Moreover, equality holds if and only if f\u2113= f \u22c6and g\u2113(m) = P \u22c6 X| \u02c6 X(\u00b7|g\u22c6(m)). Proof. We have E [\u2113(X, g\u2113(f\u2113(X)))] = E h \u2113 \u0010 X, P \u22c6 X| \u02c6 X(\u00b7|g(f(X))) \u0011i = E \" log 1 P \u22c6 X| \u02c6 X(x|g(f(x))) # . Then, (4) implies that E [\u2113(X, g\u2113(f\u2113(X)))] = E \u0014 log 1 PX(X) \u2212\uf6beX(X, D\u22c6(M)) \u0015 \f8 + E [\u03bb\u22c6d(X, g(f(X))) \u2212\u03bb\u22c6D\u22c6(M)] = H(X| \u02c6 X\u22c6) + \u03bb\u22c6(E [d(X, g(f(X)))] \u2212D\u22c6(M)) (6) \u2265H(X| \u02c6 X\u22c6) (7) where (6) is because E [\uf6beX(X, D\u22c6(M))] = R(D\u22c6(M)) = I(X; \u02c6 X\u22c6) with respect to the distribution PX \u00d7 P \u22c6 \u02c6 X|X. The last inequality (7) is because D\u22c6(M) is the minimum achievable average distortion with M codewords. Equality holds if and only if E [d(X, g(f(X)))] = D\u22c6(M) which can be achieved by the optimum scheme for the original lossy compression problem. In other words, equality holds if f \u22c6 \u2113=f \u22c6 g\u22c6 \u2113(m) =P \u22c6 X| \u02c6 X(\u00b7|g\u22c6(m)). Remark 1. In the corresponding \ufb01xed-length lossy compression problem under logarithmic loss, the minimum achievable average distortion given the number of codewords M is D\u22c6 \u2113(M) = H(X| \u02c6 X\u22c6) where the conditional entropy is with respect to the distribution PX \u00d7 P \u22c6 \u02c6 X|X. B. Discussion 1) One-to-One Correspondence: Theorem 5 implies that for any \ufb01xed-length lossy compression problem, we can \ufb01nd an equivalent problem under the logarithmic loss where optimum encoding schemes are the same. Thus, without loss of generality, we can restrict our attention to the problem under logarithmic loss with reconstruction alphabet Y = {q(1), . . . , q(s)} \u2282M(X). 2) Suboptimality of the Scheme: Suppose f and g are sub-optimal encoder and decoder for the original \ufb01xedlength lossy compression problem, then the theorem implies E [\u2113(X, g\u2113(X))] \u2212H(X| \u02c6 X\u22c6) = \u03bb\u22c6(E [d(X, g(f(X)))] \u2212D\u22c6(M)) . (8) The left hand side of (8) is the cost of sub-optimality for the corresponding lossy compression problem. On the other hand, the right hand side is proportional to the cost of sub-optimality for the original problem. In Section III-B1, we discussed that the optimum schemes of the two problems coincide. Equation (8) shows stronger equivalence that the cost of sub-optimalities are linearly related. This simply implies a good code for one problem is also good for another. 3) Operations on \u02c6 X: In general, the reconstruction alphabet \u02c6 X does not have any algebraic structure. However, in the corresponding rate-distortion problem, the reconstruction alphabet is the set of probability measures where we have very natural operations such as convex combinations of elements, projection to a convex hull, etc. \f9 C. Exact Performance of Optimum Scheme In the previous section, we showed that there is a corresponding lossy compression problem under logarithmic loss which shares the same optimum coding scheme. In this section, we investigate the exact performance of the optimum scheme for the \ufb01xed-length lossy compression problem under logarithmic loss, where the reconstruction alphabet is the set of all measures on X, i.e., M(X). Although the optimum scheme associated with M(X) may differ from the optimum scheme with the restricted reconstruction alphabets Y, it may provide some insights. Note that we are not allowing randomization, but restrict attention to deterministic schemes. 1) Average Distortion Criterion: In this section, we characterize the minimum average distortion D\u22c6(M) when we have \ufb01xed number of messages M. Let an encoder and a decoder be f : X \u2192{1, . . . , M} and g : {1, . . . , M} \u2192 M(X) where g(m) = q(m) \u2208M(X). Then, we have E [\u2113(X, g(f(X)))] = X x\u2208X PX(x) log 1 q(f(x))(x) = H(X) + M X m=1 X x\u2208f \u22121(m) PX(x) log PX(x) q(m)(x) = H(X) + M X m=1 um log um + M X m=1 um X x\u2208f \u22121(m) PX(x) um log PX(x)/um q(m)(x) where um = P x\u2208f \u22121(m) PX(x). Since PX|f(X)(x|m) = PX(x) um for all x \u2208f \u22121(m), we have E [\u2113(X, g(f(X)))] = H(X) \u2212H(f(X)) + M X m=1 umD \u0010 PX|f(X)(\u00b7|m) \r \r \r q(m)\u0011 \u2265H(X) \u2212H(f(X)). Equality can be achieved by choosing q(m) = PX|f(X)(\u00b7|m) which can be done no matter what f is. Thus, we have D\u22c6(M) = H(X) \u2212 max f:|f|\u2264M H(f(X)). Note that one trivial lower bound is D\u22c6(M) \u2265H(X) \u2212log M. 2) Excess Distortion Criterion: In this section, we characterize the minimum number of codewords M \u22c6(D, \u01eb) that can achieve the distortion D and the excess distortion probability \u01eb. Let an encoder and a decoder be f : X \u2192 \f10 {1, . . . , M} and g : {1, . . . , M} \u2192M(X) where g(m) = q(m) \u2208M(X). Since \u2113(x, q) \u2264D is equivalent to q(x) \u2265e\u2212D, we have 1 \u2212Pe = X x\u2208X PX(x)1 \u0010 q(f(x))(x) \u2265e\u2212D\u0011 = M X m=1 X x\u2208f \u22121(m) PX(x)1 \u0010 q(m)(x) \u2265e\u2212D\u0011 . However, at most \u230aeD\u230bof the q(m)(x) can be larger than e\u2212D where \u230ax\u230bis the largest integer that is smaller than or equal to x. Thus, we can cover at most M \u00b7 \u230aeD\u230bof the source symbols with M codewords. Suppose PX(1) \u2265PX(2) \u2265\u00b7 \u00b7 \u00b7 \u2265PX(r), then the optimum scheme is f(x) = \u0018 x \u230aeD\u230b \u0019 q(m)(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1/\u230aeD\u230b if f(x) = m 0 otherwise where q(m) = g(m). The idea is that each reconstruction symbol q(m) covers \u230aeD\u230bnumber of source symbols by assigning probability mass 1/\u230aeD\u230bto each of them. The above optimum scheme satis\ufb01es 1 \u2212Pe = M\u00b7\u230aeD\u230b X x=1 PX(x) =FX \u0000M \u00b7 \u230aeD\u230b \u0001 where FX(\u00b7) is the cumulative distribution function of X. This implies that the minimum error probability is \u01eb\u22c6(M, D) = 1 \u2212FX \u0000M \u00b7 \u230aeD\u230b \u0001 . On the other hand, if we \ufb01x the target error probability \u01eb, the minimum number of codewords is M \u22c6(D, \u01eb) = \u0018F \u22121 X (1 \u2212\u01eb) \u230aeD\u230b \u0019 where F \u22121 X (y) = argmin 1\u2264x\u2264r {x : FX(x) \u2265y} and \u2308x\u2309is the smallest integer that is larger than or equal to x. Note that if we allow variable length coding without pre\ufb01x condition, the optimum coding scheme is similar to the optimum nonasymptotic lossless coding introduced in [15]. IV. SUCCESSIVE REFINABILITY We considered the \ufb01xed-length lossy compression problem so far. In this section, we provide another universal property of logarithmic loss where the source is discrete memoryless. \f11 A. Main Results Consider the successive re\ufb01nement problem with a discrete memoryless source as described in Section II-C. Speci\ufb01cally, we are instrested in the case where the \ufb01rst decoder is under logarithmic loss and the second decoder is under some arbitrary distortion measure d. Using the result from previous section, there is an equivalent ratedistortion problem under logarithmic loss for the second decoder. Since any discrete memoryless source under logarithmic loss is successively re\ufb01nable, one may argue that the source is successively re\ufb01nable under this setting. However, this can be misleading since we cannot directly apply our result to discrete memoryless source. This is mainly because the decoder only considers product measures when the source is discrete memoryless. Moreover, the equivalent rate-distortion problem under logarithmic loss has restricted reconstruction set with only \ufb01nitely many measures, while successive re\ufb01nability of logarithmic loss is guaranteed when the reconstruction alphabet is the set of all measures on X. Despite of these misconceptions, we show that our initial guess was correct, i.e., it is successively re\ufb01nable. This provides an additional universal property of logarithmic loss in the context of the successive re\ufb01nement problem. Theorem 6. Let the source be arbitrary discrete memoryless. Suppose the distortion criterion of the \ufb01rst decoder is logarithmic loss while that of the second decoder is an arbitrary distortion criterion d : X \u00d7 \u02c6 X\u2192[0, \u221e]. Then the source is successively re\ufb01nable. Proof. The source is successively re\ufb01nable at (D1, D2) if and only if there exists a X \u2212\u02c6 X \u2212Q such that I(X; Q) =R1(D1), E [\u2113(X, Q)] \u2264D1 I(X; \u02c6 X) =R2(D2), E h d(X, \u02c6 X) i \u2264D2. Let P \u02c6 X\u22c6|X be the informational rate-distortion function achieving conditional distribution for the second decoder. I.e., I(X; \u02c6 X\u22c6) = R2(D2), E h d(X, \u02c6 X\u22c6) i = D2. Consider a random variable Z \u2208Z such that the joint distribution of X, \u02c6 X\u22c6, Z is given by PX, \u02c6 X,Z(x, \u02c6 x, z) = PX, \u02c6 X(x, \u02c6 x)PZ| \u02c6 X(z|\u02c6 x) and H(X|Z) = D1. We assume that all the PX|Z(\u00b7|z) are distinct for all z \u2208Z. If we let Q = PX|Z(\u00b7|Z) and q(z) = PX|Z(\u00b7|z) for all z \u2208Z, then X \u2212\u02c6 X\u22c6\u2212Q forms a Markov chain and PX|Q(x|q(z)) = q(z)(x). Since Q = PX|Z(\u00b7|Z) is a one-to-one mapping, we have I(X; Q) =I(X; Z) = H(X) \u2212D1 = R1(D1). Also, we have E [\u2113(X, Q)] =E h log 1 PX|Z(X|Z) i = H(X|Z) = D1. \f12 We have no constraints on the set Z and the only requirements for the random variable Z is H(X|Z) = D1. Therefore, we can always \ufb01nd such random variable Z, and we can say that the source and respective distortion measures are successively re\ufb01nable. The key idea of the theorem is that (1) is the only loose required condition for the rate-distortion function achieving conditional distribution. Thus, for any distortion criterion in the second stage, we are able to choose an appropriate distribution PX, \u02c6 X,Q that satis\ufb01es both (1) and the condition for successive re\ufb01nability. Remark 2. We would like to point out that the way of constructing the joint distribution PX, \u02c6 X,Q in the proof using random variable Z is the only possible way. More precisely, consider a Markov chain X \u2212\u02c6 X \u2212Q that satis\ufb01es the condition for successive re\ufb01nability, then there exists a random variable Z such that Q = PX|Z(\u00b7|Z) where X\u2212\u02c6 X\u2212Z forms a Markov chain. This is because we can have Z = Q, in which case PX|Z(\u00b7|Z) = PX|Q(\u00b7|Q) = Q. Theorem 6 can be generalized to successive re\ufb01nement problem with K intermediate decoders. Consider random variables Zk \u2208Zk for 1 \u2264k \u2264K such that X \u2212\u02c6 X \u2212ZK \u2212\u00b7 \u00b7 \u00b7\u2212Z1 forms a Markov chain and the joint distribution of X, \u02c6 X\u22c6, Z1, . . . , ZK is given by PX, \u02c6 X,Z1,...,ZK(x, \u02c6 x, z1, . . . , zK) = PX, \u02c6 X(x, \u02c6 x)PZ1| \u02c6 X(z1|\u02c6 x) K\u22121 Y k=1 PZk+1|Zk(zk+1|zk) where H(X|Zk) = Dk and all the PX|Zk(\u00b7|zk) are distinct for all zk \u2208Zk. Similar to the proof of Theorem 6, we can show that Qk = PX|Zk(\u00b7|Zk) for all 1 \u2264k \u2264K satisfy the condition for successive re\ufb01nability. Thus, we can conclude that any discrete memoryless source with K intermediate decoders is successively re\ufb01nable as long as all the intermediate decoders are under logarithmic loss. V." + }, + { + "url": "http://arxiv.org/abs/1506.03407v2", + "title": "Strong Successive Refinability and Rate-Distortion-Complexity Tradeoff", + "abstract": "We investigate the second order asymptotics (source dispersion) of the\nsuccessive refinement problem. Similarly to the classical definition of a\nsuccessively refinable source, we say that a source is strongly successively\nrefinable if successive refinement coding can achieve the second order optimum\nrate (including the dispersion terms) at both decoders. We establish a\nsufficient condition for strong successive refinability. We show that any\ndiscrete source under Hamming distortion and the Gaussian source under\nquadratic distortion are strongly successively refinable.\n We also demonstrate how successive refinement ideas can be used in\npoint-to-point lossy compression problems in order to reduce complexity. We\ngive two examples, the binary-Hamming and Gaussian-quadratic cases, in which a\nlayered code construction results in a low complexity scheme that attains\noptimal performance. For example, when the number of layers grows with the\nblock length $n$, we show how to design an $O(n^{\\log(n)})$ algorithm that\nasymptotically achieves the rate-distortion bound.", + "authors": "Albert No, Amir Ingber, Tsachy Weissman", + "published": "2015-06-10", + "updated": "2016-03-15", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION In the successive re\ufb01nement problem, an encoder wishes to send a source to two decoders with different target distortions. Instead of designing separate coding schemes, the successive re\ufb01nement encoder uses a code for the \ufb01rst decoder which has a weaker link and sends extra information to the second decoder on top of the message of the \ufb01rst decoder. In general, the performance of a successive re\ufb01nement coding scheme is worse than separate coding for each decoder. However, for some cases, we can simultaneously achieve the optimum rates for both The material in this paper has been presented in part at the 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton), and at the 2014 International Symposium on Information Theory. This work was supported by the NSF Center for Science of Information under Grant Agreement CCF-0939370. \f2 decoders as if the optimum codes were used separately. In this case, we say the source is successively re\ufb01nable. Necessary and suf\ufb01cient conditions for successive re\ufb01nement were independently proposed by Kosh\u00b4 elev [1], [2] and Equitz and Cover [3]. Rimoldi [4] found the full rate-distortion region of the successive re\ufb01nement problem including non-successively re\ufb01nable sources. Kanlis and Narayan [5] extended the result to the error exponent that quanti\ufb01es \u201chow fast the excess distortion probability decays\u201d. Tuncel [6] characterized the entire region of ratedistortion-exponents with separate handling of the two error events. Both lines of work considered error exponents in the spirit of Marton [7], which characterized the error exponent for the point-to-point case. For the point-to-point source coding problem, Ingber and Kochman [8] and Kostina and Verd` u [9] independently proposed an asymptotic analysis that complements the error exponent analysis. In this setting, the \ufb01gure of merit is the minimum achievable rate when the excess distortion probability \u01eb and the block length n are \ufb01xed. This can be quanti\ufb01ed by the source dispersion. For an i.i.d. source with law P, the minimum rate can be approximated by R(P, D) + p V (P, D)/nQ\u22121(\u01eb), where R(P, D) and V (P, D) are, respectively, the rate-distortion function and dispersion of a source P at distortion level D. We can consider this rate as a \u201csecond order\u201d optimum rate (where the classical rate-distortion function is the \ufb01rst order result). With this stronger notion of optimality, it is natural to ask whether successive re\ufb01nement schemes can achieve the second order optimum rates at both decoders simultaneously. An obvious necessary condition for the existence of such schemes is that the source be successively re\ufb01nable, so we refer to such a source as \u201cstrongly successively re\ufb01nable\u201d (formal de\ufb01nitions follow in the sequel). In this paper, we present a second order achievability result for the successive re\ufb01nement problem. As a corollary, we derive a suf\ufb01cient condition for strong successive re\ufb01nability and show that a source P is strongly successively re\ufb01nable if all sources \u02dc P in the neighborhood of P are successively re\ufb01nable. In the second part of the paper, we show that successive re\ufb01nement codes can be useful in the point-to-point source coding problem when we want to achieve lower encoding complexity. The idea is that \ufb01nding the best representing codeword in a successive manner is often easier than \ufb01nding a codeword from the set of all codewords, which normally has exponential complexity. Moreover, storing exponentially many codewords is often prohibitive, while successive re\ufb01nement encoding can reduce the size of codebooks. Our \ufb01ndings here contribute to the recent line of work on reducing the complexity of rate-distortion codes, cf. [10]\u2013[12] and references therein. We aim to study the general approach of using successive encoding to reduce complexity. We denote this approach by \u201clayered coding\u201d, a family that includes all coding schemes that can be implemented in a successive manner. Basically, the layered coding scheme is searching for an appropriate codeword over a tree structure where the number of decoders corresponds to the level of the tree. The larger the tree, the faster the codeword can be found, and therefore the lower decoding complexity. In order to reduce the encoding complexity signi\ufb01cantly, we generalize the result to the case where the number of decoders is increasing with block length n. This is different from the classical successive re\ufb01nability where only a \ufb01xed number of decoders are considered. On the other hand, the larger tree structure restricts the class of coding schemes, and therefore too many decoders may cause a rate loss. Our result for this setting characterizes an achievable trade-off between encoding complexity (how fast can we \ufb01nd the \f3 codeword) and performance (how much do we end up compressing). Note that SPARC [12] and CROM [13] are manifestations of the layered coding approach that attain good performance. The rest of the paper is organized as follows. In Section II, we revisit the known results about successive re\ufb01nement and source dispersion. Section III provides the problem setting. We present our main results in Section IV, where proof details are given in Section V. Section VI is dedicated to a layered coding scheme, and we conclude in Section VII. Notation: Xn and X denotes an n-dimensional random vector (X1, X2, . . . , Xn) while xn and x denotes a speci\ufb01c realization of it. When we have two random vectors, we use the notation such as \u02c6 Xn 1 = ( \u02c6 X1,1, \u02c6 X1,2, . . . , \u02c6 X1,n) and \u02c6 Xn 2 = ( \u02c6 X2,1, \u02c6 X2,2, . . . , \u02c6 X2,n). II. PRELIMINARIES A. Source Dispersion Consider an i.i.d. source Xn with law P where the source alphabet is X and the reconstruction alphabet is \u02c6 X. Let d : X \u00d7 \u02c6 X \u2192[0, \u221e) be a distortion measure where d(xn, \u02c6 xn) = (1/n) Pn i=1 d(xi, \u02c6 xi). It is well known that the rate-distortion function R(P, D) is the optimal asymptotic compression rate for which distortion D can be achieved. However, this \ufb01rst order optimum rate can be achieved only when the block length n goes to in\ufb01nity. Beyond the \ufb01rst order rate, we can consider two1 asymptotic behaviors which are excess distortion exponent [7] and the source dispersion [9], [16]. The former considers how fast the excess distortion probability Pr h d(Xn, \u02c6 Xn) > D i is decaying, while the latter considers how fast the minimum number of codewords converges to R(P, D) when excess distortion probability \u01eb and block length n are given. It was shown that the difference between the minimum rate for \ufb01xed n and R(P, D) is inversely proportional to square root of n. More formally, let RP,D,\u01eb(n) be the minimum compression rate for which the excess distortion probability is smaller than \u01eb. The result is given by: Theorem 1 ([16]): Suppose R(P, D) is twice differentiable2 with respect to D and the elements of P in some neighborhood of (P, D). Then RP,D,\u01eb(n) = R(P, D) + r V (P, D) n Q\u22121(\u01eb) + O \u0012log n n \u0013 (1) where V (P, D) is the source dispersion, given by V (P, D) \u225cVAR [R\u2032(X, D)] (2) = X x\u2208X P(x)(R\u2032(x, D))2 \u2212 \"X x\u2208X P(x)R\u2032(x, D) #2 (3) 1These asymptotic approaches analyze the excess distortion probability. Other approaches exist which analyze the average achievable distortion [14], [15]. 2We say R(P, D) is differentiable at P if there is an extension \u02dc R(\u00b7, D) : Rm \u2192R which is differentiable. Under this de\ufb01nition, R\u2032(x, D) and V (P, D) are well and uniquely de\ufb01ned. Details are given in Appendix A. \f4 and R\u2032(x, D) denotes the derivative of R(P, D) with respect to the probability P(x): R\u2032(x, D) \u225c \u0014\u2202R(Q, D) \u2202Q(x) \u0015 Q=P . (4) We have a similar result for the Gaussian source under quadratic distortion: Theorem 2 ([8]): Consider an i.i.d. Gaussian source Xn distributed according to N(0, \u03c32), and quadratic distortion, i.e., d(xn, \u02c6 xn) = (1/n) Pn i=1(xi \u2212\u02c6 xi)2. Then RP,D,\u01eb(n) = 1 2 log \u03c32 D + r 1 2nQ\u22121(\u01eb) + O \u0012log n n \u0013 . (5) Note that the dispersion of the Gaussian source is V (P, D) = 1/2 nats2/source symbol for all D \u2264\u03c32. B. Successive Re\ufb01nement The successive re\ufb01nement problem with two decoders can be formulated as follows. Again, let Xn be i.i.d. with law P. The encoder sends a pair of messages (m1, m2) where 1 \u2264mi \u2264Mi for i \u2208{1, 2}. The \ufb01rst decoder takes m1 and reconstructs \u02c6 Xn 1 (m1) \u2208\u02c6 X n 1 where the second decoder takes (m1, m2) and reconstructs \u02c6 Xn 2 (m1, m2) \u2208\u02c6 X n 2 . Note that \u02c6 X1 and \u02c6 X2 denote the respective reconstruction alphabets of the decoders. The i-th decoder employs the distortion measure di(\u00b7, \u00b7) : X \u00d7 \u02c6 Xi \u2192[0, \u221e) and wants to recover the source xn with distortion Di, i.e., di(xn, \u02c6 Xn i ) \u2264Di for i \u2208{1, 2}. (6) The rates of the code are de\ufb01ned as R1 = 1 n log M1 (7) R2 = 1 n log M1M2. (8) An (n, R1, R2, D1, D2, \u01eb)-successive re\ufb01nement code is a coding scheme with block length n and excess distortion probability \u01eb where rates are (R1, R2) and target distortions are (D1, D2). Since we have two decoders, the excess distortion probability is de\ufb01ned by Pr h di(Xn, \u02c6 Xn i ) > Di for some i i . De\ufb01nition 1: A rate-distortion tuple (R1, R2, D1, D2) is achievable, if there is a family of (n, R(n) 1 , R(n) 2 , D1, D2, \u01eb(n))-successive re\ufb01nement codes where lim n\u2192\u221eR(n) i = Ri for i \u2208{1, 2}, (9) lim n\u2192\u221e\u01eb(n) = 0. (10) The achievable rate-distortion region is known: Theorem 3 ([4]): Consider a discrete memoryless source Xn with law P. The rate-distortion tuple (R1, R2, D1, D2) is achievable if and only if there is a joint law PX, \u02c6 X1, \u02c6 X2 of random variables (X, \u02c6 X1, \u02c6 X2) (where X is distributed according to P) such that I(X; \u02c6 X1) \u2264R1 (11) I(X; \u02c6 X1, \u02c6 X2) \u2264R2 (12) \f5 E h di(X, \u02c6 Xi) i \u2264Di for i \u2208{1, 2}. (13) In some cases, we can achieve the optimum rates at both decoders simultaneously: De\ufb01nition 2: For i \u2208{1, 2}, let Ri(P, Di) denote the rate-distortion function of the source P when the distortion measure is di(\u00b7, \u00b7) and the distortion level is Di. If (R1(P, D1), R2(P, D2), D1, D2) is achievable, then we say the source is successively re\ufb01nable at (D1, D2). Furthermore, if the source is successively re\ufb01nable at (D1, D2) for all non-degenerate D1, D2 (i.e., for which R1(P, D1) < R2(P, D2)), then we say the source is successively re\ufb01nable. A necessary and suf\ufb01cient condition for successive re\ufb01nability is known. Theorem 4 ([1], [3]): A source P is successively re\ufb01nable at (D1, D2) if and only if there exists a conditional distribution P \u02c6 X1, \u02c6 X2|X such that X \u2212\u02c6 X2 \u2212\u02c6 X1 forms a Markov chain and Ri(P, Di) = I(X; \u02c6 Xi) (14) E h di(X, \u02c6 Xi) i \u2264Di (15) for i \u2208{1, 2}. The condition in the theorem holds for the cases of a Gaussian source under quadratic distortion and for any discrete memoryless sources under Hamming distortion. Note that the successive re\ufb01nability is not shared by all sources and distortion measures. For instance, symmetric Gaussian mixtures under quadratic distortion are not successively re\ufb01nable [17]. The above results of successive re\ufb01nability can be generalized to the case of k decoders. Note that we can also de\ufb01ne successive re\ufb01nability using R(P, D1, D2) where R(P, D1, D2) is the minimum rate R2 such that (R1(P, D1), R2, D1, D2) is achievable. Using Theorem 3, we can characterize R(P, D1, D2), R(P, D1, D2) = inf P \u02c6 X1, \u02c6 X2|X: E[d1(X, \u02c6 X1)]\u2264D1, E[d2(X, \u02c6 X2)]\u2264D2, I(X; \u02c6 X1)\u2264R1(P,D1) I(X; \u02c6 X1, \u02c6 X2). (16) De\ufb01nition 2 implies that the source is successively re\ufb01nable at (D1, D2) if and only if R(P, D1, D2) = R2(P, D2). III. PROBLEM SETTING We consider the successive re\ufb01nement problem with two decoders. Let Xn = (X1, \u00b7 \u00b7 \u00b7 , Xn) be i.i.d. with law P, where the source alphabet is X. An encoder f (n) = \u0010 f (n) 1 , f (n) 2 \u0011 maps a source sequence to a pair of messages, f (n) 1 : X n \u2192{1, \u00b7 \u00b7 \u00b7 , M1} (17) f (n) 2 : X n \u2192{1, \u00b7 \u00b7 \u00b7 , M2}. (18) The \ufb01rst decoder receives only the output of f (n) 1 (Xn), and therefore we say that its rate is R1 = (1/n) log M1. The second decoder receives the output of both functions, so its rate is R2 = (1/n) log M1M2. Decoder 1 employs a decoder g(n) 1 : {1, \u00b7 \u00b7 \u00b7 , M1} \u2192\u02c6 X n 1 and decoder 2 employs a decoder g(n) 2 : {1, \u00b7 \u00b7 \u00b7 , M1}\u00d7 {1, \u00b7 \u00b7 \u00b7 , M2} \u2192\u02c6 X n 2 , where \u02c6 X1 and \u02c6 X2 are the reconstruction alphabets for each decoder. Decoder i has its own \f6 distortion measure di : X \u00d7 \u02c6 Xi \u2192[0, \u221e) with a target distortion Di. Both d1 and d2 are symbol by symbol distortion measures which induce block distortion measures by di(xn, \u02c6 xn i ) = 1 n n X j=1 di(xj, \u02c6 xi,j) (19) for all i \u2208{1, 2}, xn \u2208X n, \u02c6 xn 1 \u2208\u02c6 X n 1 and \u02c6 xn 2 \u2208\u02c6 X n 2 . The setting is described in Figure 1. Xn Enc Dec 1 Dec 2 \u02c6 Xn 1 \u02c6 Xn 2 m1 m2 Fig. 1. Successive Re\ufb01nement De\ufb01nition 3: We say that (n, M1, M2, D1, D2, \u01eb1, \u01eb2) is achievable if there exists an encoder-decoder pair that satis\ufb01es Pr h d1(Xn, g(n) 1 (f (n) 1 (Xn))) > D1 i \u2264\u01eb1 (20) Pr h d2(Xn, g(n) 2 (f (n) 1 (Xn), f (n) 2 (Xn))) > D2 i \u2264\u01eb2, (21) and such a code is called a (n, M1, M2, D1, D2, \u01eb1, \u01eb2)-code. Note that we consider the two error events separately, unlike in the de\ufb01nition of a successive re\ufb01nement code in Section II-B. Our goal is to characterize the achievable (n, M1, M2, D1, D2, \u01eb1, \u01eb2) region in general. Motivated by successive re\ufb01nability, we de\ufb01ne strong successive re\ufb01nability as follows. De\ufb01nition 4: The source is strongly successively re\ufb01nable at (D1, D2, \u01eb1, \u01eb2) if (n, M1, M2, D1, D2, \u01eb1, \u01eb2) is achievable for some M1, M2 satisfying 1 n log M1 = R1(P, D1) + r V1(P, D1) n Q\u22121(\u01eb1) + o \u0012 1 \u221an \u0013 (22) 1 n log M1M2 = R2(P, D2) + r V2(P, D2) n Q\u22121(\u01eb2) + o \u0012 1 \u221an \u0013 (23) where Ri(P, Di) and Vi(P, Di) are the point-to-point rate-distortion function and the source dispersion for the i-th decoder. Furthermore, if the source is strongly successively re\ufb01nable at (D1, D2, \u01eb1, \u01eb2) for all non-degenerate D1, D2, \u01eb1, \u01eb2 (i.e., RP,D1,\u01eb(n) < RP,D2,\u01eb(n)), then we say the source is strongly successively re\ufb01nable. While standard successive re\ufb01nability implies that the successive re\ufb01nement structure does not cause any loss in the compression rate (asymptotically), strong successive re\ufb01nability implies that we also do not lose from the dispersion point of view. Note that in order to verify that a source is strongly successively re\ufb01nable, it is suf\ufb01cient to \ufb01nd an achievability scheme since the converse will follow from the converse in point-to-point source coding. \f7 IV. MAIN RESULTS Our results in this section pertain to discrete memoryless sources under general distortion, as well as Gaussian sources under quadratic distortion. The results are given here, with proofs in Section V. A. Discrete Memoryless Source Let Xn be i.i.d. with distribution P and the distortion measures be d1 : X \u00d7 \u02c6 X1 \u2192[0, \u221e) and d2 : X \u00d7 \u02c6 X2 \u2192 [0, \u221e). We assume that the alphabets X, \u02c6 X1 and \u02c6 X2 are \ufb01nite, and therefore distortion measures d1 and d2 are bounded by some constant dM. We further assume that P(x) > 0 for all x \u2208X since one can remove the source symbol from X that has zero probability. Then, the following theorem provides the achievable rates including the second order term: Theorem 5 (Achievability for Discrete Memoryless Source): Assume that both R1(P, D1) and R(P, D1, D2) are continuously twice differentiable with respect to D1, D2, and the elements of P in some neighborhood of (P, D1, D2). Then, there exists an (n, M1, M2, D1, D2, \u01eb1, \u01eb2)-code such that 1 n log M1 = R1(P, D1) + r V1(P, D1) n Q\u22121(\u01eb1) + O \u0012log n n \u0013 (24) 1 n log M1M2 = R(P, D1, D2) + r V (P, D1, D2) n Q\u22121(\u01eb2) + O \u0012log n n \u0013 (25) where V1(P, D1) \u225cVAR [R\u2032 1(X, D1)] (26) = X x\u2208X P(x)(R\u2032 1(x, D1))2 \u2212 \"X x\u2208X P(x)R\u2032 1(x, D1) #2 (27) V (P, D1, D2) \u225cVAR [R\u2032(X, D1, D2)] (28) = X x\u2208X P(x)(R\u2032(x, D1, D2))2 \u2212 \"X x\u2208X P(x)R\u2032(x, D1, D2) #2 . (29) Similarly to Theorem 1, R\u2032 1(x, D1) is the derivative of R1(P, D1) with respect to the probability P(x) and R\u2032(x, D1, D2) is the derivative3 of R(P, D1, D2) with respect to the probability P(x): R\u2032 1(x, D1) \u225c \u0014\u2202R1(Q, D1) \u2202Q(x) \u0015 Q=P (30) R\u2032(x, D1, D2) \u225c \u0014\u2202R(Q, D1, D2) \u2202Q(x) \u0015 Q=P . (31) By applying the above theorem to the special case where R( \u02dc P, D1, D2) = R2( \u02dc P, D2) for all \u02dc P in some neighborhood of P, we get the following corollary. 3Similar to the de\ufb01nition of R\u2032(x, D), we can de\ufb01ne R\u2032(x, D1, D2) using an extension. Then, R\u2032(x, D1, D2) and V (P, D1, D2) are well and uniquely-de\ufb01ned as well, where details are given in Appendix A. \f8 Corollary 6: Suppose Ri(P, Di) is continuously twice differentiable with respect to Di and the elements of P in some neighborhood of (P, Di) for i \u2208{1, 2}. If all sources \u02dc P in some neighborhood of P are successively re\ufb01nable at D1, D2, then there exists an (n, M1, M2, D1, D2, \u01eb1, \u01eb2) code such that 1 n log M1 = R1(P, D1) + r V1(P, D1) n Q\u22121(\u01eb1) + O \u0012log n n \u0013 (32) 1 n log M1M2 = R2(P, D2) + r V2(P, D2) n Q\u22121(\u01eb2) + O \u0012log n n \u0013 , (33) i.e., the source P is strongly successively re\ufb01nable at (D1, D2, \u01eb1, \u01eb2). The corollary is because R( \u02dc P, D1, D2) = R2( \u02dc P, D2) for all \u02dc P in some neighborhood of P implies that their derivatives at (P, D1, D2) coincide, i.e., \u0014\u2202R2(Q, D2) \u2202Q(x) \u0015 Q=P = \u0014\u2202R(Q, D1, D2) \u2202Q(x) \u0015 Q=P . (34) Since the source dispersion is the variance of the derivatives, we have V (P, D1, D2) = V2(P, D2). Remark 1: Any discrete memoryless source under Hamming distortion measure is successively re\ufb01nable. Therefore, Corollary 6 implies that any discrete memoryless source under Hamming distortion is also strongly successively re\ufb01nable provided R(P, D) is appropriately differentiable. Note that the size of the set {D : R(P, D) is not differentiable} is at most |X| [18]. B. Gaussian Memoryless Source Let Xn be an i.i.d. N(0, \u03c32) source, and suppose the distortion measure is quadratic (at both decoders). Theorem 7 (Achievability for Gaussian Memoryless Source): The memoryless Gaussian source under quadratic distortion is strongly successively re\ufb01nable, i.e., for \u03c32 > D1 > D2, there exists an (n, M1, M2, D1, D2, \u01eb1, \u01eb2) code such that 1 n log M1 = 1 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb1) + O \u0012log n n \u0013 (35) 1 n log M1M2 = 1 2 log \u03c32 D2 + r 1 2nQ\u22121(\u01eb2) + O \u0012log n n \u0013 . (36) V. PROOF A. Method of Types Our proofs for \ufb01nite alphabet sources rely heavily on the method of types [19]. In this section, we brie\ufb02y review its notation and results that we use. Without loss of generality we assume X = {1, 2, . . ., rx}. For any sequence xn \u2208X n, let N(a|xn) be the number of symbol a \u2208X in the sequence xn. Let the type of a sequence xn be an rx dimensional vector Pxn = (N(1|xn)/n, N(2|xn)/n, . . . , N(rx|xn)/n). Then, denote Pn(X) be the set of all types on X n, i.e., Pn(X) = {Pxn | xn \u2208X n}. The size of the set Pn(X) is at most polynomial in n, more precisely, |Pn(X)| \u2264(n + 1)rx. (37) \f9 For given type P, de\ufb01ne type class of P by TP = {xn \u2208X n | Pxn = P}. (38) We can also de\ufb01ne type class Txn = {\u02dc xn \u2208X n | P\u02dc xn = Pxn} using a sequence xn \u2208X n. We can bound the size of type class. 1 (n + 1)rx exp (nH(P)) \u2264|TP | \u2264exp (nH(P)) (39) where H(P) denote an entropy of random variable with law P. We further consider the conditional types. Let Y be a set of alphabet where we also assume Y = {1, 2, . . ., ry} to be \ufb01nite. Consider a stochastic kernel W : X \u2192Y. We say that yn \u2208Yn has conditional type W given xn \u2208X n if N(a, b|xn, yn) = N(a|xn)W(b|a). (40) Then, we can de\ufb01ne conditional type class of W given xn \u2208X n by TW (xn) = {yn \u2208Yn | yn has conditional type W given xn}. (41) We can also bound a size of conditional type class. For sequence xn \u2208X n with type P, and for conditional type W, we have 1 (n + 1)rxry exp (nH(P|W)) \u2264|TW (xn)| \u2264exp (nH(P|W)) . (42) H(P|W) denotes a conditional entropy of U given V where (U, V ) are random variables with a joint law P \u00d7 W. B. Proof of Theorem 5 A key tool used in the proof is a re\ufb01ned version of the type covering lemma [19]. We say a set B is D-covering a set A if for all a \u2208A, there exists an element b \u2208B such that d(a, b) \u2264D. In the successive re\ufb01nement setting, we need to cover a set in a successive manner. De\ufb01nition 5: Let d1 : A \u00d7 B \u2192[0, \u221e) and d2 : A \u00d7 C \u2192[0, \u221e) be distortion measures. Consider sets A \u2282A, B \u2282B and Cb \u2282C for all b \u2208B. We say B and {Cb}b\u2208B successively (D1, D2)-cover a set A, if for all a \u2208A, there exist b \u2208B and c \u2208Cb such that d1(a, b) \u2264D1 and d2(a, c) \u2264D2. The following lemma provides an upper bound of minimum size of sets that successively (D1, D2)-cover a type class TP . Lemma 8 (Re\ufb01ned Covering Lemma): For \ufb01xed n, let P \u2208Pn(X) be a type on X where P(x) > 3/n for all x \u2208X. Suppose \u2225\u2207R(P, D1, D2)\u2225is bounded in some neighborhood of (D1, D2) where \u2207R(P, D1, D2) = \u0012 \u2202 \u2202D1 R(P, D1, D2), \u2202 \u2202D2 R(P, D1, D2) \u0013 . (43) Then for D1, D2 \u2208(0, dM), there exist sets B1 \u2282\u02c6 X n 1 and B2(\u02c6 xn 1 ) \u2282\u02c6 X n 2 for each \u02c6 xn 1 \u2208B1 where B1 and {B2(\u02c6 xn 1 )}\u02c6 xn 1 \u2208B1 successively (D1, D2)-cover TP with following properties: \f10 \u2022 The size of B1 is upper bounded: 1 n log |B1| \u2264R1(P, D1) + k1 log n n . (44) \u2022 For all \u02c6 xn 1 \u2208B1, the size of B2(\u02c6 xn 1 ) is also bounded: 1 n log (|B1| \u00b7 |B2(\u02c6 xn 1 )|) \u2264R(P, D1, D2) + k2 log n n , (45) where k1 and k2 are universal constants, i.e., do not depend on the distribution P or n. The proof of Lemma 8 is given in Appendix B. The following corollary provides a successive re\ufb01nement scheme using B1 and {B2(\u02c6 xn 1 )}\u02c6 xn 1 \u2208B1 from Lemma 8. Corollary 9: For length of sequence n and type Q \u2208Pn(X), let \u02dc R satisfy \u02dc R \u2265R1(Q, D) + k1log n/n. Then, there exists a coding scheme for TQ such that \u2022 Encoding functions are fQ,1 : TQ \u2192{1, . . . , MQ,1} and fQ,2 : TQ \u2192{1, . . . , MQ,2}. \u2022 Decoder 1 and Decoder 2 employ gQ,1 : {1, . . . , MQ,1} \u2192\u02c6 X n 1 (46) gQ,2 : {1, . . . , MQ,1} \u00d7 {1, . . . , MQ,2} \u2192\u02c6 X n 2 (47) respectively. \u2022 For all xn \u2208TQ, encoding and decoding functions satisfy d1 (xn, gQ,1(fQ,1(xn))) \u2264D1 (48) d2 (xn, gQ,2(fQ,1(xn), fQ,2(xn))) \u2264D2. (49) \u2022 The number of messages are bounded: \u02dc R \u22641 n log MQ,1 \u2264\u02dc R + log n n (50) 1 n log MQ,1MQ,2 \u2264R(Q, D1, D2) + (k2 + 1)log n n . (51) The proof of Corollary 9 is given in Appendix C. Let us now describe the achievability scheme. Similar to the idea from [6], we will consider the four cases according to the type Q of the input sequence xn. For each case, the encoding will be done in a different manner. Before specifying four cases, we need to de\ufb01ne \u2206R1 and \u2206R2. Let \u2206R1 be the in\ufb01mal value such that the probability of {R1(PXn, D1) > R1(P, D1) + \u2206R1} is smaller than \u01eb1, and \u2206R2 be the in\ufb01mal value such that the probability of {R(PXn, D1, D2) > R(P, D1, D2) + \u2206R2} is smaller than \u01eb2. Recall that PXn denotes the type of Xn. The error occurs at decoder 1 if and only if R1(PXn, D1) > R1(P, D1) + \u2206R1, and therefore probability of error at decoder 1 is less than \u01eb1. Similarly, the error occurs at decoder 2 if and only if R(PXn, D1, D2) > R(P, D1, D2) + \u2206R2, and therefore probability of error at decoder 2 is less than \u01eb2. The following lemma bounds \u2206R1 and \u2206R2. \f11 Lemma 10: \u2206R1 = r V1(P, D1) n Q\u22121(\u01eb1) + O \u0012log n n \u0013 (52) \u2206R2 = r V (P, D1, D2) n Q\u22121(\u01eb2) + O \u0012log n n \u0013 . (53) The proof follows directly from [16, Lemma 3]. We are ready to de\ufb01ne four cases based on the type of the source sequence as well as corresponding encoding schemes. 1) Q \u2208A(0,0) \u225c{Q \u2208Pn(X) : R1(Q, D1) \u2212R1(P, D1) \u2264\u2206R1, R(Q, D1, D2) \u2212R(P, D1, D2) \u2264\u2206R2}. In this case, both decoders decode successfully. Since R(Q, D1) \u2264R(P, D1) + \u2206R1, by Corollary 9, there exist encoding and decoding functions fQ,1, fQ,2, gQ,1, gQ,2 such that d1 (xn, gQ,1(fQ,1(xn))) \u2264D1 (54) d2 (xn, gQ,2(fQ,1(xn), fQ,2(xn))) \u2264D2 (55) for all xn \u2208TQ and R1(P, D1) + \u2206R1 + k1 log n n \u22641 n log M (0,0) Q,1 \u2264R1(P, D1) + \u2206R1 + (k1 + 1)log n n (56) 1 n log M (0,0) Q,1 M (0,0) Q,2 \u2264R(Q, D1, D2) + (k2 + 1)log n n . (57) We emphasize that we have R1(P, D1) instead of R1(Q, D1) in (56). This is because we need to aggregate the codewords at the end of the proof. More precisely, we have to \ufb01x the number of codewords for decoder 1 in order to bound the number of codewords only for decoder 2. 2) Q \u2208A(0,1) \u225c{Q \u2208Pn(X) : R1(Q, D1) \u2212R1(P, D1) \u2264\u2206R1, R(Q, D1, D2) \u2212R(P, D1, D2) > \u2206R2}. For those Q, the encoder only D1 covers TQ. Thus, decoder 1 will decode successfully and decoder 2 will declare an error. In this case, we do not need a message for decoder 2 and we can think of M (0,1) 2 = 1. For decoder 1, by Theorem 1, we can \ufb01nd encoding and decoding functions f (0,1) : X n \u2192{1, . . . , M (0,1) 1 } and g(0,1) : {1, . . ., M (0,1) 1 } \u2192\u02c6 X n 1 such that d1(xn, g(0,1)(f (0,1)(xn))) \u2264D1 (58) for all Q \u2208A(0,1) and xn \u2208TQ where 1 n log M (0,1) 1 = R1(P, D1) + \u2206R1 + O \u0012log n n \u0013 . (59) 3) Q \u2208A(1,0) \u225c{Q \u2208Pn(X) : R1(Q, D1) \u2212R1(P, D1) > \u2206R1, R(Q, D1, D2) \u2212R(P, D1, D2) \u2264\u2206R2}. In this case, the encoder only D2 covers TQ. Thus, decoder 2 will decode successfully and decoder 1 will declare an error. In this case, we do not need a message for decoder 1. However, because of the structure of successive re\ufb01nement code, we need to reformulate the point-to-point code for the second decoder into the \f12 form of successive re\ufb01nement code. More precisely, we can \ufb01nd functions \u02dc f (1,0) Q : X n \u2192{1, . . . , \u02dc M (1,0) Q,2 } and \u02dc g(1,0) Q : {1, . . ., \u02dc M (1,0) Q,2 } \u2192\u02c6 Xn 2 such that d2(xn, \u02dc g(1,0) Q ( \u02dc f (1,0) Q (xn))) \u2264D2 (60) for all xn \u2208TQ where 1 n log \u02dc M (1,0) Q,2 \u2264R2(Q, D2) + k1 log n n . (61) Let M (1,0) Q,1 and M (1,0) Q,2 be R1(P, D1) + \u2206R1 + k1 log n n \u22641 n log M (1,0) Q,1 \u2264R1(P, D1) + \u2206R1 + (k1 + 1)log n n (62) 1 n log M (1,0) Q,1 M (1,0) Q,2 \u22641 n log \u02dc M (1,0) Q,2 + log n n . (63) For simplicity, we neglect the fact that the number of messages are integers since it will increase the rate by at most log n/n bits/symbol. Let h be a one to one mapping from {1, . . . , M (1,0) Q,1 } \u00d7 {1, . . . , M (1,0) Q,2 } to {1, . . ., \u02dc M (1,0) Q,2 }. Then, we can de\ufb01ne encoding and decoding functions f (1,0) Q,1 : X n \u2192{1, . . ., M (1,0) Q,1 }, f (1,0) Q,2 : X n \u2192{1, . . . , M (1,0) Q,2 }, and g(1,0) Q,2 : {1, . . . , M (1,0) Q,1 } \u00d7 {1, . . . , M (1,0) Q,2 } \u2192\u02c6 X n 2 where \u0010 f (1,0) Q,1 (xn), f (1,0) Q,1 (xn) \u0011 =h\u22121 \u0010 \u02dc f (1,0) Q (xn) \u0011 (64) g(1,0) Q,2 (m1, m2) =\u02dc g(1,0) Q (h(m1, m2)). (65) Note that the \ufb01rst message is useless for decoder 1, but we do not care since it will declare an error anyway. On the other hand, decoder 2 will decode both m1 and m2 successfully. 4) Q \u2208A(1,1) \u225c{Q \u2208Pn(X) : R1(Q, D1) \u2212R1(P, D1) > \u2206R1, R(Q, D1, D2) \u2212R(P, D1, D2) > \u2206R2}. The encoder sends nothing and the both decoder will declare errors. We can assume M (1,1) 1 = M (1,1) 2 = 1. Finally, we merge all encoding functions together. Given source sequence xn, the encoder describes a type of sequence as a part of the \ufb01rst message using |X| log(n + 1) bits. This affects at most O(log n/n) bits/symbol in rates. Based on the type of sequence, it employs an encoding function accordingly, as described above. Since the decoder also knows the type of the sequence, it can employ the corresponding decoding function. Since all M (0,0) Q,1 , M (0,1) 1 , M (1,0) Q,1 have the same upper bound, we can bound M1: 1 n log M1 \u2264R1(P, D1) + \u2206R1 + (k1 + 1)log n n + |X| log(n + 1) n (66) \u2264R1(P, D1) + r V1(P, D1) n Q\u22121(\u01eb1) + O \u0012log n n \u0013 . (67) Similarly, we can show that 1 n log M1M2 \u2264R(P, D1, D2) + r V (P, D1, D2) n Q\u22121(\u01eb2) + O \u0012log n n \u0013 . (68) This concludes the proof. \f13 C. Proof of Theorem 7 Instead of type covering arguments that we used in the previous section, we use the result of sphere covering for Gaussian sources. Theorem 11 ([20]): There is an absolute constant ks such that, if R > 1 and n \u22659, any n-dimensional spheres of radius R can be covered by less than ksn5/2Rn spheres of radius 1. For simplicity, we refer to the sphere of radius r by r-ball and denote by B(xn, r) = {\u02dc xn : d(xn, \u02dc xn) \u2264r2}, the set of points in the sphere centered at xn with radius r. The above theorem immediately implies the following corollary. Corollary 12: For n \u22659 and R1 > R2 > 0, we can \ufb01nd a set C \u2282Rn of size M that satis\ufb01es: \u2022 For all xn \u2208B(0, R1), there is an element \u02c6 xn \u2208C such that xn \u2208B(\u02c6 xn, R2). \u2022 The size of the set M is upper bounded by 1 n log M \u22641 2 log R1 R2 + 5 2 log n n + O \u0012 1 n \u0013 . (69) Let r1 and r2 be radius of the balls such that Pr \u0002 X2 1 + \u00b7 \u00b7 \u00b7 + X2 n > r2 1 \u0003 = \u01eb1 and Pr \u0002 X2 1 + \u00b7 \u00b7 \u00b7 + X2 n > r2 2 \u0003 = \u01eb2. First, consider the case \u01eb1 < \u01eb2. It is clear that Q\u22121(\u01eb1) > Q\u22121(\u01eb2) and r1 > r2. We can further divide this case into the following three cases, 1) Xn \u2208B(0, r2), i.e., X2 1 + \u00b7 \u00b7 \u00b7 + X2 n \u2264r2 2. In this case, we design a code such that both decoders can decode successfully. Let C(0,0) 1 \u2282Rn be the set that satis\ufb01es: \u2022 \f \f \fC(0,0) 1 \f \f \f = M (0,0) 1 \u2022 B(0, r2) \u2282S \u02c6 xn\u2208C(0,0) 1 B(\u02c6 xn, \u221anD1) \u2022 1 n log M (0,0) 1 \u22641 2 log \u03c32 D1 + q 1 2nQ\u22121(\u01eb2) + O \u0010 log n n \u0011 which implies that there are M (0,0) 1 number of \u221anD1-balls that covers the r2-ball. Upper bound on M (0,0) 1 can be found similarly to the proof of Theorem 2. Since Q\u22121(\u01eb1) > Q\u22121(\u01eb2), it is clear that 1 n log M (0,0) 1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb1) + O \u0012log n n \u0013 . (70) Similarly, we can cover a \u221anD1-ball with M (0,0) 2 number of \u221anD2-balls. In other words, there exists C(0,0) 2 \u2282Rn that satis\ufb01es: \u2022 \f \f \fC(0,0) 2 \f \f \f = M (0,0) 2 \u2022 B(0, \u221anD1) \u2282S \u02c6 xn\u2208C(0,0) 2 B(\u02c6 xn, \u221anD2) \u2022 1 n log M (0,0) 2 \u22641 2 log D1 D2 + O \u0010 log n n \u0011 where upper bound on M (0,0) 2 is because of Corollary 12. Thus, if xn \u2208B(0, r2), then we can \ufb01nd \u02c6 xn 1 \u2208C(0,0) 1 such that xn \u2208B(\u02c6 xn 1, \u221anD1) which implies (1/n) \u2225xn \u2212\u02c6 xn 1 \u22252 2 \u2264D1. Furthermore, since xn \u2212\u02c6 xn 1 \u2208B(0, \u221anD1), we can \ufb01nd \u02dc xn \u2208C(0,0) 2 such that xn \u2212\u02c6 xn 1 \u2208B(\u02dc xn, \u221anD2) which implies (1/n) \u2225xn \u2212\u02c6 xn 1 \u2212\u02dc xn\u22252 2 \u2264D2. Finally, we can take \u02c6 xn 2 = \u02c6 xn 1 + \u02dc xn, and we get (1/n) \u2225xn \u2212\u02c6 xn 2 \u22252 2 \u2264D2. \f14 2) Xn \u2208B(0, r1) but Xn / \u2208B(0, r2), i.e., r2 2 < X2 1 + \u00b7 \u00b7 \u00b7 + X2 n \u2264r2 1. We will only send a message to decoder 1, and decoder 2 will declare an error. We can cover r1-ball with M (0,1) 1 number of \u221anD1-balls where 1 n log M (0,1) 1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb1) + O \u0012log n n \u0013 . (71) Therefore, there exists C(0,1) 1 that satis\ufb01es: \u2022 \f \f \fC(0,1) 1 \f \f \f = M (0,1) 1 \u2022 B(0, r2) \u2282S \u02c6 xn\u2208C(0,1) 1 B(\u02c6 xn, \u221anD1) \u2022 1 n log M (0,1) 1 \u22641 2 log \u03c32 D1 + q 1 2nQ\u22121(\u01eb1) + O \u0010 log n n \u0011 . We can think M (0,1) 2 to be one. 3) Xn / \u2208B(0, r1) and Xn / \u2208B(0, r2), i.e., r2 1 < X2 1 + \u00b7 \u00b7 \u00b7 + X2 n. The encoder does not send any messages and both decoder will declare an error. We can think both M (0,2) 1 and M (0,2) 2 to be one. Finally, we employ the codebook C1 = C(0,0) 1 \u222aC(0,1) 1 \u222aC(0,2) 1 and the same for C2 where |C1| = M1 and |C2| = M2. Then, we can see that 1 n log M1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb1) + O \u0012log n n \u0013 (72) 1 n log M1M2 \u22641 2 log \u03c32 D2 + r 1 2nQ\u22121(\u01eb2) + O \u0012log n n \u0013 . (73) Similarly, we can consider the case \u01eb1 \u2265\u01eb2. In this case, it is clear that Q\u22121(\u01eb1) \u2264Q\u22121(\u01eb2) and r1 \u2264r2. We can further divide the case into the following three cases, 1) Xn \u2208B(0, r1), i.e., X2 1 + \u00b7 \u00b7 \u00b7 + X2 n \u2264r2 1. In this case, both decoders can decode successfully. We can \ufb01nd M (1,0) 1 number of \u221anD1-balls that covers r1-ball where 1 n log M (1,0) 1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb1) + O \u0012log n n \u0013 . (74) Similar to previous cases, we can de\ufb01ne C(1,0) 1 to be a set of the ball centers. Also, we can cover \u221anD1-ball with M (1,0) 2 number of \u221anD2-balls where 1 n log M (1,0) 2 \u22641 2 log D1 D2 + O \u0012log n n \u0013 . (75) Since Q\u22121(\u01eb1) \u2264Q\u22121(\u01eb2), it is clear that 1 n log M (1,0) 1 M (1,0) 2 \u22641 2 log \u03c32 D2 + r 1 2nQ\u22121(\u01eb2) + O \u0012log n n \u0013 . (76) 2) Xn \u2208B(0, r2) but Xn / \u2208B(0, r1), i.e., r2 1 < X2 1 + \u00b7 \u00b7 \u00b7 + X2 n \u2264r2 2. We will only send a message to decoder 2, and decoder 1 will declare an error. We can cover r2-ball with \u02dc M (1,1) number of \u221anD2-balls where 1 n log \u02dc M (1,1) \u22641 2 log \u03c32 D2 + r 1 2nQ\u22121(\u01eb2) + O \u0012log n n \u0013 . (77) \f15 Similar to the proof of Theorem 5, we can split the message \u02dc m(1,1) \u2208{1, . . ., \u02dc M (1,1)} into (m(1,1) 1 , m(1,1) 2 ) \u2208 {1, . . . , M (1,1) 1 } \u00d7 {1, . . . , M (1,1) 2 } such that M (1,1) 1 M (1,1) 2 = \u02dc M (1,1) (78) 1 n log M (1,1) 1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb2) + O \u0012log n n \u0013 (79) 1 n log M (1,1) 2 \u22641 2 log D1 D2 + O \u0012log n n \u0013 . (80) Recall that the decoder 1 does not care about the reconstruction of the source, where, on the other hand, decoder 2 will get both M (1,1) 1 and M (1,1) 2 and will be able to reconstruct the source based on \u02dc M (1,1). 3) Xn / \u2208B(0, r1) and Xn / \u2208B(0, r2), i.e., r2 2 < X2 1 + \u00b7 \u00b7 \u00b7 + X2 n. We will not send any messages and both decoder will declare an error. We can think both M (1,2) 1 and M (1,2) 2 to be one. Similar to the case of \u01eb1 < \u01eb2, we can combine the codebooks and get 1 n log M1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb1) + O \u0012log n n \u0013 (81) 1 n log M1M2 \u22641 2 log \u03c32 D2 + r 1 2nQ\u22121(\u01eb2) + O \u0012log n n \u0013 . (82) This concludes the proof. Remark 2: If we have \u01eb1 = \u01eb2 = \u01eb, radius r1 and r2 are the same and the proof can be simpli\ufb01ed. In this case, an error will occur at both decoders if and only if X2 1 +\u00b7 \u00b7 \u00b7+X2 n > r2 where r = r1 = r2. Since both decoders share the same error events, encoding can be done successively in a simple manner and we do not have to consider the case of message splitting. More precisely, given codebook {( \u02c6 Xn 1 (i), \u02c6 Xn 2 (j)) : 1 \u2264i \u2264M1, 1 \u2264j \u2264M2}, the encoder \ufb01nds i such that (1/n) \r \r \rXn \u2212\u02c6 Xn 1 (i) \r \r \r 2 2 \u2264D1 and then \ufb01nds j such that (1/n) \r \r \rXn \u2212\u02c6 Xn 1 (i) \u2212\u02c6 Xn 2 (j) \r \r \r 2 2 \u2264D2. This is the key idea of Section VI where we use the successive re\ufb01nement technique to construct a point-to-point source coding scheme with low complexity. VI. LAYERED CODES We considered the successive re\ufb01nement problem with two decoders so far. In this section, we show that the idea of successive re\ufb01nement is also useful for point to point lossy compression where we have one encoder and one decoder. The intuition is that successive re\ufb01nement coding provides a tree structure for a coding scheme which allows low encoding complexity. More precisely, if the source is successively re\ufb01nable, we can add L \u22121 virtual mid-stage decoders and employ a successive re\ufb01nement scheme for L decoders without any (asymptotic) performance loss. For \ufb01xed L, this is a simple extension of successive re\ufb01nement, however, we also provide a result for L = Ln growing with n. Since the number of decoders L corresponds to the level of tree and larger L leads to lower complexity of the scheme, we have a great advantage in terms of complexity by taking growing L = Ln. Note that the tree structured vector quantization (TVSQ) has been extensively studied, and also has a successive approximation property. For example, in [21], Effros et al. combined pruned TVSQ with a universal noiseless coder \f16 which enables progressive transmission of sources. While this approach guarantees optimality at zero distortion, it cannot achieve the rate-distortion function in general. The precise problem description is the following. Let n be the block length of the coding scheme. The codebook consists of L sub-codebooks (C(n) 1 , C(n) 2 , \u00b7 \u00b7 \u00b7 , C(n) L ) and each sub-codebook consists of Mi codewords for 1 \u2264i \u2264L. We consider the following encoding scheme which we call layered coding: \u2022 Find c1 \u2208C(n) 1 that minimizes some function \u03c81(xn, c1). \u2022 For i \u22652, given c1, \u00b7 \u00b7 \u00b7 , ci\u22121, \ufb01nd ci \u2208C(n) i that minimizes \u03c8i(xn, c1, c2, \u00b7 \u00b7 \u00b7 , ci\u22121), where \u03c81, ..., \u03c8L are simple functions that depend on the speci\ufb01c implementation of the scheme. One can think of (c1, . . . , ci) as messages for an i-th (virtual) decoder. The compressed representation of the source consists of a length L vector (m1, \u00b7 \u00b7 \u00b7 , mL) which indicates the index of codeword from each sub-codebook. Note that the total number of codewords is M1 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ML and the rate of the scheme is R = PL i=1 1 n log Mi. Once the decoder receives the message, it reconstructs \u02c6 Xn = \u03c6(m1, \u00b7 \u00b7 \u00b7 , mL) with some function \u03c6. De\ufb01nition 6: An (n, L, {M1, \u00b7 \u00b7 \u00b7 , ML}, D, \u01eb)-layered code is a coding scheme with L sub-codebooks where the size of the i-th sub-codebook is Mi, and the probability of excess distortion Pr h d(Xn, \u02c6 Xn) > D i is at most \u01eb. Note that the de\ufb01nition of the layered code is exactly equal to that of the successive re\ufb01nement code except the fact that the layered coding scheme only considers the distortion at the last decoder. A. Layered Coding Schemes We show the existence of layered coding schemes for a Gaussian source under quadratic distortion and for a binary source under Hamming distortion. For \ufb01xed L, it is easy to have a layered coding scheme, since sources are successively re\ufb01nable in both cases and we can apply the successive re\ufb01nement schemes. In this section, we generalize the result even further in two aspects. First, we consider how fast the coding rate can converge to the rate-distortion function, and provide an achievable rate including a dispersion term. Then, we allow L to be a function of block length n, and provide a layered coding scheme for L = Ln growing with n. Our next theorem shows an existence of a rate-distortion achieving layered coding scheme for given n and L. Theorem 13: For i.i.d. Gaussian sources under quadratic distortion and i.i.d. binary sources under Hamming distortion, there exists a (n, L, {M1, . . . , ML}, D, \u01eb)-layered code such that L X i=1 1 n log Mi \u2264R(D) + r V (D) n Q\u22121(\u01eb) + L\u02dc k log n n + O \u0012log n n \u0013 (83) for some constant \u02dc k where the O (log n/n) term does not depend on D or L. The proof and discussion of Theorem 13 are given in Section VI-A1 and Section VI-A2. Note that L\u02dc k log n/n is also in the class of O(log n/n) for constant L, however, we will also consider the case where L = Ln grows with n. We would like to point out that the last O(log n/n) remains the same even when L = Ln increases as n grows. 1) Gaussian source under quadratic distortion: For Gaussian source under quadratic distortion, we can generalize Theorem 7 to the case of multiple decoders. As we mentioned in Remark 2, we choose all \u01ebi to be equal to \u01eb. \f17 Lemma 14: Let a source be i.i.d. Gaussian N(0, \u03c32) under quadratic distortion. For all L, there exists a (n, L, {M1, \u00b7 \u00b7 \u00b7 , ML}, D, \u01eb)-layered code such that 1 n log M1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb) + O \u0012log n n \u0013 (84) 1 n log Mi \u22641 2 log Di\u22121 Di + 3log n n for 2 \u2264i \u2264L (85) for any D1 > D2 > \u00b7 \u00b7 \u00b7 > DL = D where the O (log n/n) term depends on \u01eb but not on L or the Di values. The choice of \u03c8i and \u03c6 will be speci\ufb01ed in the proof. The fact that the O (log n/n) term is not dependent on the speci\ufb01c choice of Di\u2019s and L is important in cases we consider later where L and Di vary with n. Proof: Consider the successive re\ufb01nement problem with target distortions D1 > \u00b7 \u00b7 \u00b7 > DL = D and target excess distortion probabilities \u01eb1 = \u00b7 \u00b7 \u00b7 = \u01ebL = \u01eb. Given sub-codebooks C1, . . . , CL, the basic idea of the scheme is as as shown in Algorithm 1. Note that the input of the algorithm is a given sequence xn and the set of sub-codebooks C1, . . . , CL where the output is the collection of sub-codewords cm1, . . . , cmL. Algorithm 1 Encoding Scheme. Set D1 > D2 > \u00b7 \u00b7 \u00b7 > DL = D, and let x(0) = xn. for i = 1 to L do Find a codeword cmi \u2208Ci such that \r \rx(i\u22121) \u2212cmi \r \r2 2 \u2264nDi. If there is no such codeword, declare an error. Let x(i) = x(i\u22121) \u2212cmi. end for We construct sub-codebooks based on Corollary 12. Let r be a radius such that Pr \u0002 X2 1 + \u00b7 \u00b7 \u00b7 + X2 n > r2\u0003 = \u01eb. Similar to the proof of Theorem 7, we can \ufb01nd M1 number of \u221anD1-balls that covers the r-ball where 1 n log M1 \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb) + O \u0012log n n \u0013 . (86) Again, the term O (log n/n) only depends on \u01eb where we provide the details in Appendix D. Then, for i \u22652, we can cover p nDi\u22121-ball with Mi number of \u221anDi-balls where 1 n log Mi \u22641 2 log Di\u22121 Di + 3log n n . (87) The i-th sub-codebook Ci is a set of centers of \u221anDi-balls, and therefore |Ci| = Mi. Suppose the encoder found cm1, \u00b7 \u00b7 \u00b7 , cmi\u22121 successfully, which implies \r \rcm1 + \u00b7 \u00b7 \u00b7 + cmi\u22121 \u2212xn\r \r2 2 \u2264nDi\u22121. In other words, xn is in the ball with radius p nDi\u22121 where the center of the ball is at cm1 + \u00b7 \u00b7 \u00b7 + cmi\u22121. Then, by construction, we can always \ufb01nd cmi \u2208Ci such that \u03c8i(xn, cm1, . . . , cmi) = \u2225cm1 + \u00b7 \u00b7 \u00b7 + cmi \u2212xn\u22252 2 \u2264nDi. We can repeat the same procedure L times and \ufb01nd (m1, m2, . . . , mL). The error occurs if and only if the event X2 1 + \u00b7 \u00b7 \u00b7 + X2 n > r2 happens at the beginning, and therefore the excess distortion probability is \u01eb. The reconstruction at the decoder will be \u03c6(cm1, . . . , cmL) = cm1 + \u00b7 \u00b7 \u00b7 + cmL. \f18 The overall rate of Lemma 14 can be bounded by L X i=1 1 n log Mi (88) \u22641 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb) + O \u0012log n n \u0013 + L X i=2 \u00141 2 log Di\u22121 Di + 3log n n \u0015 (89) = 1 2 log \u03c32 D + r 1 2nQ\u22121(\u01eb) + 3(L \u22121)log n n + O \u0012log n n \u0013 . (90) 2) Binary source under Hamming distortion: The next lemma provides a similar result for a binary source under Hamming distortion. Lemma 15: Let the source be i.i.d. Bern(p) and the distortion be measured by Hamming distortion function, where the target distortion is D. For large enough n, there is a (n, L, {M1, \u00b7 \u00b7 \u00b7 , ML}, D, \u01eb)-layered code for all L and D1 > D2 > \u00b7 \u00b7 \u00b7 > DL = D such that 1 n log M1 \u2264h2(p) \u2212h2(D1) + r V (p, D) n Q\u22121(\u01eb) + O \u0012log n n \u0013 (91) 1 n log Mi \u2264h2(Di\u22121) \u2212h2(Di) + k3 log n n , for 2 \u2264i \u2264L (92) where O (log n/n) only depends on \u01eb, we denote dispersion of Bern(p) source with V (p, D) = p(1 \u2212p) log2((1 \u2212 p)/p), and a binary entropy function with h2(p) = \u2212p log p \u2212(1 \u2212p) log(1 \u2212p) and k3 is a constant that does not depend on any of the variables. Proof: Similar to the proof of Lemma 14, we can consider the successive re\ufb01nement problem with target distortions D1 > \u00b7 \u00b7 \u00b7 > DL = D and target excess distortion probabilities \u01eb1 = \u00b7 \u00b7 \u00b7 = \u01ebL = \u01eb. The basic idea of coding is very similar to the Gaussian case. The difference is that we use Hamming instead of l2 balls, and therefore we need Lemma 8 instead of Corollary 12. A Hamming ball with radius r is de\ufb01ned by BH(r) \u2206 = {yn \u2208{0, 1}n : n X i=1 yi \u2264r}. (93) Given sub-codebooks C1, . . . , CL, the basic idea of the achievability scheme is the following: Algorithm 2 Enoding Scheme. Set D1 > D2 > \u00b7 \u00b7 \u00b7 > DL = D, and let x(0) = xn. for i = 1 to L do Find the codeword cmi \u2208Ci such that d(x(i\u22121), cmi) \u2264Di. If there is no such codeword, declare an error. Let x(i) = x(i\u22121) \u2295cmi. end for Similar to Algorithm 1, the input of the algorithm is a given sequence xn and the set of sub-codebooks C1, . . . , CL where the output is the collection of sub-codewords cm1, . . . , cmL. \f19 In the \ufb01rst stage, similar to [16, Theorem 1], we can \ufb01nd a sub-codebook C1 with size M1 such that the excess distortion probability is smaller than \u01eb and 1 n log M1 \u2264h(p) \u2212h(D1) + r V (p, D1) n Q\u22121(\u01eb) + O \u0012log n n \u0013 . (94) Similar to the Gaussian case, the term O (log n/n) only depends on \u01eb, where the detail is provided in Appendix E. For i \u22652 and the given type Q, Lemma 8 implies that there is MQ,i Hamming balls with radius nDi that covers all sequences of type Q where 1 n log MQ,i \u2264R(Q, Di) + k1 log n n (95) =h(Q(1)) \u2212h(Di) + k1 log n n . (96) Let CQ,i be a set of centers of Hamming balls with radius nDi, and therefore |CQ,i| = MQ,i. The i-th sub-codebook Ci is union of CQ,i\u2019s for all type Q \u2208T (Di\u22121, Di) \u225c{Q \u2208Pn(X) : Di < Q(1) \u2264Di\u22121} and zero codeword (0, 0, 0, \u00b7 \u00b7 \u00b7 , 0), i.e., Ci = {(0, . . . , 0)} \u222a [ Q\u2208T (Di\u22121,Di) CQ,i. (97) Then, we have 1 n log Mi = 1 n log |Ci| (98) \u22641 n log \uf8eb \uf8ed1 + X Q\u2208T (Di\u22121,Di) |CQ,i| \uf8f6 \uf8f8 (99) \u22641 n log \u0012 1 + (n + 1) max Q\u2208T (Di\u22121,Di) MQ,i \u0013 (100) \u2264h(Di\u22121) \u2212h(Di) + (k1 + 1)log n n (101) where (100) is because |T (Di\u22121, Di)| \u2264nDi\u22121 \u2212nDi + 1. We can set k3 \u2206 = k1 + 1. Suppose the encoder could \ufb01nd cm1, \u00b7 \u00b7 \u00b7 , cmi\u22121 successfully which implies d(cm1 \u2295\u00b7 \u00b7 \u00b7 \u2295cmi\u22121, xn) \u2264nDi\u22121. In other words, xn is in the Hamming ball with radius nDi\u22121 where the center of ball is at cm1 \u2295\u00b7 \u00b7 \u00b7 \u2295cmi\u22121. Then, by construction, we can always \ufb01nd cmi \u2208Ci such that \u03c8i(cm1, . . . , cmi) = d(cm1 \u2295\u00b7 \u00b7 \u00b7 \u2295cmi, xn) \u2264nDi. We can repeat the same procedure L times and \ufb01nd (m1, m2, . . . , mL). The error occurs if and only if the \ufb01rst sub-codebook fails to cover the source xn at the beginning, and therefore the excess distortion probability is \u01eb. The reconstruction at the decoder will be \u03c6(cm1, . . . , cmL) = cm1 \u2295\u00b7 \u00b7 \u00b7\u2295cmL. Remark 3: We would like to point out that Lemma 15 is limited to memoryless binary sources while Theorem 5 holds for any discrete memoryless sources. The main difference is the operation between source symbols. More precisely, in Lemma 15, the source is encoded and then the \u201cerror\u201d sequence (modulo 2 difference) is encoded again. Note that Hamming distortion is closely related to this operation. However, It is hard to generalize this idea to non-binary sources because there are no corresponding differences when the distortion measure is arbitrary. The \f20 modulo |X| difference could work, but it is complex to analyse even when the distortion measure is still Hamming distortion. The overall rate of Lemma 15 can be bounded by L X i=1 1 n log Mi \u2264h2(p) \u2212h2(D1) + r V (p, D1) n Q\u22121(\u01eb) + O \u0012log n n \u0013 + L X i=2 \u0014 h2(Di\u22121) \u2212h2(Di) + k3 log n n \u0015 (102) =h2(p) \u2212h2(D) + r V (p, D1) n Q\u22121(\u01eb) + k3(L \u22121)log n n + O \u0012log n n \u0013 . (103) B. Discussion 1) Rate-Distortion Trade-Off: In both (90) and (103), it is obvious that the choice of L has an important role. For simplicity, we only consider the case where M1 = M2 = \u00b7 \u00b7 \u00b7 = ML = M, and we neglect the fact that the number of messages M is an integer. We can \ufb01nd M and D1 > D2 > \u00b7 \u00b7 \u00b7 > DL which satisfy (84) and (85) (or (91) and (92)) with equality. For example, in the Gaussian case, we can \ufb01nd M and D1, \u00b7 \u00b7 \u00b7 , DL sequentially: 1 n log M =1 2 log \u03c32 D1 + r 1 2nQ\u22121(\u01eb) + O \u0012log n n \u0013 (104) 1 n log M =1 2 log Di\u22121 Di + 3log n n for 2 \u2264i \u2264L, (105) Clearly, the number of possible reconstructions is M L = enR and the rate of the scheme is R = (1/n)L log M. On the other hand, the complexity is of order M \u00d7 L since the encoder is searching a right codeword over M sub-codewords at each stage. Thus, for \ufb01xed rate R, we can say that the coding complexity (or size of codebooks) scales with L exp (nR/L) which is a decreasing function of L. This shows that larger L provides a lower complexity of the scheme. It is worth emphasizing that we can set L = Ln to be increasing with n. This is because the bounds in both corollaries hold uniformly for all L. On the other hand, in both corollaries, the overall rate can be bounded by R(D) + r V (D) n Q\u22121(\u01eb) + \u02dc kLn log n n + O \u0012log n n \u0013 (106) for some constant \u02dc k, where we denote by R(D) and V (D) the rate-distortion function and the source dispersion. However, the optimum rate is given by R(D) + r V (D) n Q\u22121(\u01eb) + O \u0012log n n \u0013 . (107) We can see that there is a penalty term \u02dc kLn(log n/n) because of using layered coding. If Ln is growing too fast with n in order to achieve low-complexity of the scheme, then the rate penalty term Ln(log n/n) can be too large and we may lose (second-order) rate optimality. This shows the trade-off between the rate and complexity of the scheme. Consider the following two examples, which are valid for both the Gaussian and binary cases. \u2022 If Ln = L is constant, then the scheme achieves the rate-distortion and the dispersion as well, but the complexity is exponential (albeit with a smaller exponent). \f21 \u2022 If Ln(log n/n) \u21920 as n \u2192\u221e, we can achieve the rate-distortion function. For example, if Ln = n/log2 n+1, then the achieved rate is R = R(D) + O \u0012 1 log n \u0013 , (108) i.e., the scheme achieves the rate-distortion function as n increases, while the coding complexity is of order n log2 nnR log n. Note that the excess distortion probability \u01eb is \ufb01xed. We would like to point out that the rate is near polynomial in n. \u2022 If Ln(log n/\u221an) \u21920 as n \u2192\u221e, we can achieve the source dispersion. For example, if Ln = \u221an/log2 n+1, then the achieved rate is R =R(D) + r V (D) n Q\u22121(\u01eb) + O \u0012 1 \u221an log n \u0013 . (109) Note that R \u2212R(D) is inversely proportional to \u221an with coef\ufb01cient p V (D)Q\u22121(\u01eb), in other words, layered coding can achieve the second order optimum rate. On the other hand, coding complexity is of order (\u221an/log2 n)n \u221anR log n which is better than the original exponential complexity. 2) Generalized Successive Re\ufb01nability: We would like to emphasize another interesting feature of layered coding. Layered coding can be viewed as a successive re\ufb01nement scheme with L decoders. Since our result allows L = Ln to be increasing with n, this can be viewed as another generalized version of successive re\ufb01nement. If the source is either binary or Gaussian and limn\u2192\u221eLn(log n/n) = 0, the source is successively re\ufb01nable with in\ufb01nitely many decoders, where the rate increment is negligible. For comparison, in the classical successive re\ufb01nement result, the number of decoders is not increasing and the rate increment between neighboring decoders is strictly positive. In [13], this property is termed in\ufb01nitesimal successive re\ufb01nability, and the results here establish that Gaussian and binary sources are in\ufb01nitesimally successively re\ufb01nable sources (under the relevant distortion criteria). Moreover, if we further assume limn\u2192\u221eLn log n \u221an = 0, each decoder can achieve the optimum distortion including dispersion term. In this case, we can say that the binary and Gaussian sources are strongly in\ufb01nitesimally successively re\ufb01nable sources. In [13], the authors also pointed out that in\ufb01nitesimal successive re\ufb01nability yields another interesting property called ratelessness. Consider a binary or Gaussian source with limn\u2192\u221eL log n n = 0, where the decoder received the \ufb01rst few fraction of messages, i.e., (m1, m2, . . . , m\u03b1L) for some 0 < \u03b1 < 1. Based on the proof of Lemma 14 and Lemma 14, the decoder will still be able to reconstruct the source sequence with distortion D(\u03b1R) which is the minimum achievable distortion at rate \u03b1R. If we have limn\u2192\u221eLn log n \u221an = 0, an even stronger ratelessness property can be established. In this case, the decoder can achieve the optimum distortion including dispersion terms. VII." + }, + { + "url": "http://arxiv.org/abs/1406.6730v2", + "title": "Rateless Lossy Compression via the Extremes", + "abstract": "We begin by presenting a simple lossy compressor operating at near-zero rate:\nThe encoder merely describes the indices of the few maximal source components,\nwhile the decoder's reconstruction is a natural estimate of the source\ncomponents based on this information. This scheme turns out to be near-optimal\nfor the memoryless Gaussian source in the sense of achieving the zero-rate\nslope of its distortion-rate function. Motivated by this finding, we then\npropose a scheme comprised of iterating the above lossy compressor on an\nappropriately transformed version of the difference between the source and its\nreconstruction from the previous iteration. The proposed scheme achieves the\nrate distortion function of the Gaussian memoryless source (under squared error\ndistortion) when employed on any finite-variance ergodic source. It further\npossesses desirable properties we respectively refer to as infinitesimal\nsuccessive refinability, ratelessness, and complete separability. Its storage\nand computation requirements are of order no more than $\\frac{n^2}{\\log^{\\beta}\nn}$ per source symbol for $\\beta>0$ at both the encoder and decoder. Though the\ndetails of its derivation, construction, and analysis differ considerably, we\ndiscuss similarities between the proposed scheme and the recently introduced\nSparse Regression Codes (SPARC) of Venkataramanan et al.", + "authors": "Albert No, Tsachy Weissman", + "published": "2014-06-25", + "updated": "2016-03-08", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION Consider an independent and identically distributed (i.i.d.) standard Gaussian source Xn = (X1, X2, . . . , Xn). It is well known [1] that the maximum value concentrates on \u221a2 log n, i.e., max1\u2264i\u2264n Xi \u2248\u221a2 log n. This fact suggests a simple lossy source coding scheme for the Gaussian source under quadratic distortion. The encoder sends the index of the maximum value and the decoder reconstructs \u02c6 Xn according to \u02c6 Xi = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 \u221a2 log n if Xi is the maximum \u2212 \u221a2 log n n\u22121 otherwise. (1) For the meager log n nats that it requires, this simple scheme achieves essentially optimum distortion (in a sense made concrete in Section II) and has obviously modest storage and computational requirements. We can generalize The material in this paper has been presented in part at the 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton). This work was supported by the NSF Center for Science of Information under Grant Agreement CCF-0939370. arXiv:1406.6730v2 [cs.IT] 8 Mar 2016 \f2 this scheme by describing the indices of the kn largest values, and the scheme still achieves optimum distortion for its operating rate. Note that this scheme can be considered a special case of a permutation code [2], where the encoder sends a rough ordering of the source. It can perform as well as the best entropy-constrained scalar quantizer (ECSQ) but cannot achieve the optimum distortion-rate function at general positive rates [3]. In [2], the authors mentioned the kn = 1 case explicitly as being asymptotically optimum under the expected distortion criterion. Our focus is more on the excess distortion probability than the expected distortion. Furthermore, we establish a more general result where kn grows sub-linearly in n. We generalize this idea to a scheme we refer to as Coding with Random Orthogonal Matrices (CROM), which achieves the distortion-rate function at all rates. Let A be a random n by n matrix uniformly drawn from the set of all n by n orthogonal matrices, i.e., for any n-dimensional vector Y n, the random vector AY n is uniformly distributed on the sphere with radius \u2225Y n\u2225. Since a random vector uniformly distributed on a high-dimensional sphere is close in distribution to an i.i.d. Gaussian random vector, we can expect the behavior of A(Xn \u2212\u02c6 Xn) to be similar to that of an i.i.d. Gaussian random vector. Therefore, we can apply the above scheme again to describe a lossy version of it, using another log n nats, and so on. In this paper, we show that this iterative scheme achieves the Gaussian rate distortion function for any \ufb01nite-variance ergodic source under quadratic distortion, while enjoying additional properties such as a strong notion of successive re\ufb01nability and polynomial complexity. One nice property of CROM is ratelessness. Similar to the rateless codes in the channel coding setting, CROM is able to reconstruct a source with partial messages while the optimum distortion for that rate is achieved. More precisely, suppose the decoder received \ufb01rst fraction \u03bd of the messages for some 0 < \u03bd < 1, then it can reconstruct a source with a distortion DG(\u03bdR). Thanks to the ratelessness, the encoder does not have to determine the rate ahead of encoding. However, unlike in many rateless channel coding settings, CROM requires that the bits observed are the \ufb01rst fraction \u03bd bits, rather than that number of bits gleaned from any set of locations along the stream. Much work has been dedicated to reducing the complexity of rate-distortion codes (cf. [4], [5], [6] and references therein). In particular, Venkataramanan et al. proposed the sparse regression code (SPARC) that achieves the Gaussian distortion-rate function with low complexity [7], [8]. SPARC and CROM have similarities, which we discuss in detail. The paper is organized as follows. In Section II, we present the simple zero-rate scheme and the sense in which it is optimal for Gaussian sources. CROM is described, along with some of its properties and performance guarantees, in Section III. We compare our scheme with SPARC in Section IV. We test CROM via simulation in Section V. We also discuss dual channel coding results in Section VI. Section VII provides proofs of our main results and we conclude the paper in Section VIII. Notation: Both Xn and X denote an n-dimensional random vector (X1, X2, . . . , Xn). We let X(i) denote the i-th largest element of Xn. We denote an n by n random orthogonal matrix by A, and a non-random orthogonal matrix by A. We denote the distortion rate-function of the memoryless standard Gaussian source by DG(R). Finally, we use nats instead of bits and log denotes logarithm to the natural base unless speci\ufb01ed otherwise. \f3 II. OPTIMUM ZERO-RATE GAUSSIAN SOURCE CODING SCHEME In this section, we propose a simple zero-rate lossy compressor which is essentially optimal for the i.i.d. standard Gaussian source under quadratic distortion. Before that, let us be more rigorous regarding our notion of \u201czero-rate optimum source coding\u201d for a Gaussian source under squared error distortion. Consider a scheme using a number of nats for the lossy description of the source which is sub-linear in the block length n, i.e., the rate Rn of the scheme converges to zero. Suppose the scheme achieves a distortion Dn(\u03f5), where the target excess distortion probability is \u03f5, i.e., Pr \u0014 1 n \r \r \rXn \u2212\u02c6 Xn\r \r \r 2 > Dn(\u03f5) \u0015 < \u03f5. (2) We further de\ufb01ne D(n, 0, \u03f5) to be the minimum distortion achievable over all possible strictly zero-rate schemes when the target excess distortion probability is \u03f5. Following lemma shows that the best reconstruction is the all zero vector 0 = (0, 0, . . . , 0) for the i.i.d. standard Gaussian source under squared error distortion. Lemma 1. Let Xn be the i.i.d. standard Gaussian source. Then, for any xn \u2208Rn and D > 0, the following inequality holds. Pr h \u2225Xn \u2212xn\u22252 > D i \u2265Pr h \u2225Xn\u22252 > D i . (3) Proof: Since Xn has spherically symmetric distribution, namely AXn is also i.i.d. standard Gaussian for any orthogonal matrix A, Pr h \u2225Xn \u2212xn\u22252 > D i only depends on \u2225xn\u2225. Let \u2225xn\u2225= a, then Pr h \u2225Xn \u2212xn\u22252 > D i =Pr \u0002 (X1 \u2212a)2 + Pn i=2 X2 i > D \u0003 (4) =E \u0002 Pr \u0002 (X1 \u2212a)2 > D \u2212Pn i=2 X2 i \u0003 \f \f X2, X3, . . . , Xn \u0003 (5) \u2265E \u0002 Pr \u0002 X2 1 > D \u2212Pn i=2 X2 i \u0003 \f \f X2, X3, . . . , Xn \u0003 (6) =Pr h \u2225Xn\u22252 > D i . (7) Therefore, D(n, 0, \u03f5) \u2206 = inf \u001a D : Pr \u0014 1 n \u2225Xn\u22252 > D \u0015 < \u03f5. \u001b . (8) It is not hard to show that D(n, 0, \u03f5) = 1 + r 2 nQ\u22121(\u03f5) + O \u0012 1 n \u0013 . (9) Finally, we say that a sequence of zero-rate schemes achieves the zero-rate optimum if lim n\u2192\u221e Dn(\u03f5) \u2212D(n, 0, \u03f5) Rn = D\u2032 G(0) (10) for all \u03f5 > 0, where D\u2032 G(0) = \u22122 is the slope of the Gaussian distortion-rate function at zero rate. Equivalently, Dn(\u03f5) = 1 \u22122Rn + r 2 nQ\u22121(\u03f5) + o (Rn) . (11) \f4 This de\ufb01nition is reminiscent of the \ufb01nite block length result in lossy compression [9], [10], where the authors showed the minimum distortion D(n, R, \u03f5) among all possible schemes for given rate R, target excess distortion probability \u03f5, and block length n is D(n, R, \u03f5) = DG(R) + r 2 nQ\u22121(\u03f5) + O \u0012log n n \u0013 . (12) Recall that DG(R) denotes the Gaussian distortion-rate function of memoryless standard Gaussian source. We are now ready to propose the simple zero-rate optimum source coding scheme. Let Xn = (X1, X2, . . . , Xn) be an i.i.d. standard normal random process. The encoder simply sends the index of the maximum value, m = arg max1\u2264i\u2264n Xi, and the decoder reconstructs \u02c6 Xn as \u02c6 Xi = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 \u03b1n if i = m \u2212\u03b1n n\u22121 otherwise, (13) where \u03b1n > 0 is naturally chosen as E \u0002 X(1) \u0003 \u2248\u221a2 log n. Note that the encoder only describes the index of the maximum entry but not its value. This scheme works because the unsent value of the maximum entry concentrates on the speci\ufb01c value near \u221a2 log n, i.e., max1\u2264i\u2264n Xi \u2248\u221a2 log n, which is a well-known fact from extreme value theory [1]. The rate of this scheme is Rn = log n n nats per symbol, and it is not hard to show that the distortion is reduced by 2 log n n (plus lower order terms), which is twice the rate we are using. Therefore, it is natural to suspect that such a scheme is zero-rate optimum. We can generalize this scheme to send more than one index: The encoder sends the indices of the kn largest values of Xn, and the decoder reconstructs \u02c6 Xn as \u02c6 Xi = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 \u03b1n if Xi is one of the kn largest values of Xn \u2212kn\u03b1n n\u2212kn otherwise. (14) Here we will choose kn = \u2308log\u03b2 n\u2309for some \u03b2 > 0 and \u03b1n to be roughly the expected value of the kn-th largest value of Xn, i.e., \u03b1n \u2248E \u0002 X(kn) \u0003 . Clearly this scheme has rate Rn = 1 n log \u0000 n kn \u0001 where limn\u2192\u221eRn = 0. The following theorem shows that this scheme is optimal at zero rate. Theorem 2. For any \u03b2 \u22650 and kn = \u2308log\u03b2 n\u2309, there is an \u03b1n > 0 such that the above scheme achieves the zero-rate optimum. More precisely, for any \u03f5 > 0, the scheme achieves Pr \u0014 1 n \r \r \rXn \u2212\u02c6 Xn\r \r \r 2 > Dn \u0015 \u2264\u03f5, (15) where Dn = 1 \u22122Rn + r 2 nQ\u22121(\u03f5) + O \u0012kn log log n n \u0013 . (16) Since Rn = O \u0010 kn log n n \u0011 , we can say that the above scheme is zero-rate optimum. \f5 The proof is given in Section VII-B. We note that the encoding and decoding can be done in almost linear time. Moreover, we do not need to store an entire codebook, but only the single real number \u03b1n needs to be stored. Remark 1. Note that Verd` u [11] also considered the slope of the rate-distortion function at Dmax as a counterpart to the capacity per unit cost. However, our requirements for zero-rate optimum scheme is more stronger since we incorporates the second order (or dispersion) term q 2 nQ\u22121(\u03f5). Remark 2. The above scheme only describes the index of the largest element. However, the encoder can send indices of both the maximum and the minimum, which is also the zero-rate optimum. Note that the minimum value will be close to \u2212\u221a2 log n, and therefore we can expect the similar behavior. III. CODING WITH RANDOM ORTHOGONAL MATRICES A. Preliminaries Before presenting the scheme, we brie\ufb02y review some key ingredients: random orthogonal matrices and spherical distributions. Let O(n) be the set of all n by n orthogonal matrices. We write A \u223cUnif(O(n)) to denote that A is a random n by n orthogonal matrix uniformly drawn from O(n). This uniform distribution is with respect to Haar measure, cf. [12]. More precisely, the random matrix A is uniformly distributed on O(n) if and only if B \u00d7 A has the same distribution with A for any orthogonal matrix B \u2208O(n). QR decomposition of random matrix with i.i.d. Gaussian entries provides a uniformly distributed random orthogonal matrix. There is a more ef\ufb01cient methods called subgroup algorithm to generate such matrices [13], [14]. Now, let us recall the de\ufb01nition of a radially symmetric random vector and its relation with uniform random orthogonal matrices. De\ufb01nition 1. An n-dimensional random vector Xn has a spherical distribution if and only if Xn and AXn has the same distribution for all orthogonal matrices A \u2208O(n). One nice property of a spherically distributed random vector Xn is that its characteristic function is radially symmetric [15], i.e., \u03c6(t) = E \u0002 exp(itT Xn) \u0003 = g(\u2225t\u2225) for some g(\u00b7). Therefore, it is enough to consider the norm \u2225Xn\u22252 2 for a spherically distributed random vector Xn. It is clear that an i.i.d. Gaussian random vector has a spherical distribution. The following lemma shows how to symmetrize a vector with a uniform random orthogonal matrix. Lemma 3. Suppose A is a uniform random orthogonal matrix on O(n). For any random vector Xn, the random vector AXn has a spherical distribution. The lemma is direct consequence of the respective de\ufb01nitions of a uniform random orthogonal matrix and a spherical distribution. \f6 B. Coding with Random Orthogonal Matrices For notational convenience, de\ufb01ne gk : Rn \u2192{0, 1}n to be the function that \ufb01nds the k largest values of the input. If there is an ambiguity, the function picks the smallest index \ufb01rst. Speci\ufb01cally, if zn = gk(xn), then zi = 1 if and only if xi is one of the k largest entries of xn and zi = 0 otherwise. Let A1, A2, . . . , ALn+1 \u2208O(n) be orthogonal matrices, \u03b11, \u03b12, . . . , \u03b1Ln be scalars, and assume that kn is a positive integer smaller than n. We are now ready to describe the iterative scheme. Algorithm 1 CROM Set X(1) = A1Xn. for i = 1 to Ln do Let m(i) = gkn(X(i)). Let U(i) = (U (i) 1 , U (i) 2 , \u00b7 \u00b7 \u00b7 , U (i) n ) where U (i) j = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 q n\u2212kn nkn if m(i) j = 1 \u2212 q kn n(n\u2212kn) otherwise. (17) Let X(i+1) = Ai+1(X(i) \u2212\u03b1iU(i)). end for Send (m(1), m(2), . . . , m(Ln)). The unit vector U(i) indicates the kn largest values of X(i), and \u03b1i\u2019s are scaling factors which depend on the norm of X(i) and will be speci\ufb01ed later. Since A\u22121 i+1 = AT i+1, the inverse of the recursion is X(i) = AT i+1X(i+1) +\u03b1iU(i) for all i. This implies Xn = \u03b11AT 1 U(1) + \u03b12AT 1 AT 2 U(2) + \u00b7 \u00b7 \u00b7 + \u03b1i(AT 1 \u00b7 \u00b7 \u00b7 AT i )U(i) + (AT 1 \u00b7 \u00b7 \u00b7 AT i+1)X(i+1). (18) Therefore, when the decoder receives (m(1), m(2), . . . , m(i)) for some i \u2264Ln, it outputs the reconstruction \u02c6 X(i) = \u03b11AT 1 U(1) + \u03b12AT 1 AT 2 U(2) + \u00b7 \u00b7 \u00b7 + \u03b1i(AT 1 \u00b7 \u00b7 \u00b7 AT i )U(i). (19) The decoder can sequentially generate reconstructions using the relation \u02c6 X(i+1) = \u02c6 X(i) + \u03b1i(AT 1 \u00b7 \u00b7 \u00b7 AT i+1)U(i+1). Note that the decoder can compute \u02c6 X(i) ef\ufb01ciently according to \u02c6 X(i) = AT 1 \u0010 \u03b11U(1) + AT 2 \u0010 \u03b12U(2) + \u00b7 \u00b7 \u00b7 + \u03b1iAT i U(i)\u0011\u0011 . (20) Since we need log \u0000 n kn \u0001 nats to store (send) m(i), rate R corresponds to Ln = nR log( n kn) number of iterations. We are ready to state our main theorem asserting that Algorithm 1 achieves the Gaussian distortion-rate function. Theorem 4. Suppose Xn is emitted by an ergodic source of marginal second moment \u03c32. For any \u03b2 \u22650, let kn = \u2308(log n)\u03b2\u2309and suppose the rate is R > 0. If we take \u03b1i = r n\u03c32 \u0010 1 \u2212e\u2212 2 Ln R\u0011 \u0010 e\u2212i\u22121 Ln R + e i\u22121 Ln R\u03b3n \u0011 \u0010 e\u2212i\u22121 Ln R \u2212e i\u22121 Ln R\u03b3n \u0011 , (21) \f7 and small enough scalar \u03b3n \u2261\u03b3 > 0, there exists orthogonal matrices A1, \u00b7 \u00b7 \u00b7 , ALn+1 \u2208O(n) such that Algorithm 1 satis\ufb01es lim n\u2192\u221ePr \u0014 1 n \r \r \rXn \u2212\u02c6 X(i)\r \r \r 2 > \u03c32 \u0010 e\u2212 i Ln R + e i Ln R\u03b3n \u00112 for some 0 \u2264i \u2264Ln \u0015 = 0. (22) Recall that (22) holds for any small enough \u03b3n \u2261\u03b3 > 0 for any ergodic Xn. If we have stronger assumptions that Xn is i.i.d. distributed with E \u0002 |X1|3\u0003 < \u221e, then we can \ufb01nd vanishing \u03b3n = O \u0010 log log n log n \u0011 that satis\ufb01es (22). The proof of Theorem 4 is given in Section VII-C with full details regarding the choice of \u03b3n. Remark 3. Theorem 4 implies that (22) holds for any \ufb01xed \u03b2. In terms of complexity, large \u03b2 is preferred since it implies small number of iteration which results in lower complexity. On the other hand, our result relies on the concentration of kn = (log n)\u03b2 largest values of n i.i.d. Gaussian random vector. If \u03b2 is too big, then the kn largest values may deviate too much. We will see the trade-off with simulation results in Section V. C. Discussion 1) Role of Orthogonal Matrices: It is known that an i.i.d. Gaussian random vector has a spherical distribution and the variance of its norm is very small. Therefore, if a random vector Xn has a spherical distribution and the variance of its norm is small enough, Xn can be thought of as an approximately i.i.d. Gaussian random vector. In the proof of CROM, we employ a randomization argument. Speci\ufb01cally, we assume that A1, A2, . . . , Ai+1 are drawn i.i.d. Unif(O(n)) and show that equation (22) holds when the probability is averaged over this ensemble of random matrices. The source at i-th iteration X(i) = Ai(X(i\u22121) \u2212\u03b1i\u22121U(i\u22121)) has spherical distribution by Lemma 3, and we can therefore expect X(i) to be a near Gaussian source, where we indirectly show that the norm of X(i) has small variance. This shows that multiplying by uniformly distributed random matrices can be thought of as a way to not only symmetrize but also Gaussianize the random vector so that we can apply the idea of Theorem 2 iteratively. Note that the conditional distribution of AXn is no longer similar to Gaussian when the matrix A is known to both the encoder and the decoder. However, in the proof, we implicitly showed that the maximum element of AXn is very close to \u221a2 log n with high probability as if it is i.i.d. Gaussian random vector. A similar idea can be found in the work of Asnani et al. [16]. The authors showed that any coding scheme for a Gaussian network source coding problem can be adapted to perform well for other network source coding problems that are not necessarily Gaussian but have the same covariances. The key idea of the paper is applying an orthogonal transformation to the sources which basically \u201cGaussianizes\u201d them so that the coding scheme for Gaussian sources are applicable in the transform domain. 2) Storage and Computational Complexity: Unlike the zero-rate scheme of Section II, this scheme requires the storage of matrices (and scalars). Since Ln = \u0016 nR log( n kn) \u0017 = O \u0010 n (log n)\u03b2+1 \u0011 , both the encoder and decoder must keep O \u0010 n3 log\u03b2+1 n \u0011 real values to store matrices A1, A2, . . . , ALn. In terms of computation, the encoder \ufb01nds the kn largest entries of an n dimensional vector and performs a matrix-vector multiplication for each iteration. The \f8 dominant cost is O(n2), the cost of matrix-vector multiplication. Therefore, the overall computational complexity is of order O \u0010 n3 log\u03b2+1 n \u0011 . Instead of storing A1, A2, . . . , ALn, it is also possible to store random seeds at both encoder and decoder to generate them. In this case, the CROM requires O(1) storage space. However, generating a uniform random orthogonal matrix takes O(n3) [13], and therefore the overall computational complexity will be of order O \u0010 n4 log\u03b2+1 n \u0011 . 3) In\ufb01nitesimal Successive Re\ufb01nability: Suppose the decoder gets only the \ufb01rst i messages (m(1), m(2), \u00b7 \u00b7 \u00b7 , m(i)). Note it needs to have seen only the \ufb01rst n i Ln R nats for that. With this partial message set, the decoder is able to reconstruct \u02c6 X(i) which achieves a distortion \u03c32 \u0010 e\u2212 i Ln R + e i Ln R\u03b3n \u00112 , (23) where the theorem guarantees e i Ln R\u03b3n is arbitrarily negligible for large enough n. In other words, the decoder essentially achieves a distortion \u03c32e\u22122 i Ln R, which is the Gaussian distortion-rate function at rate i Ln R. Evidently, CROM can be viewed as a successive re\ufb01nement coding scheme with Ln stages. Since we have a growing number of stages (in n), the rate increment at each stage is negligible (i.e., sub-linear number of additional nats per stage) and this is a key difference from classical successive re\ufb01nement problems where the number of stages is \ufb01xed. Note that Theorem 4 implies that the probability of excess distortion beyond the relevant point on the distortion-rate curve at any of the successive re\ufb01nement stages is negligible. Therefore, if the source is i.i.d. Gaussian, our coding scheme simultaneously achieves every point on the optimum distortion-rate curve. This in\ufb01nitesimal successive re\ufb01nability can be considered a strengthened version of successive re\ufb01nement. In other words, to implement and operate CROM, the value of the rate R need not be known or set in advance, a point we will expound in Section III-C4. In [17], the similar property called \u201cincremental re\ufb01nements\u201d was discussed. The paper discovered a new limiting behavior of additive rate-distortion function at zero-rate, and proposed a re\ufb01nement idea. However, additive ratedistortion function is a mutual information between the input and the output of the Gaussian test channel, where it is not clear how to achieve it. On the other hand, we proposed a concrete scheme that achieves rate-distortion function. 4) (Near) Ratelessness: In the channel coding setting, it is well-known that rateless coding schemes, including Raptor codes, achieve the capacity of erasure channels. In this setting, the rate R does not have to be speci\ufb01ed in advance, and the receiver is able to decode a message upon observing suf\ufb01ciently many packets (or bits), regardless of their order. As we have discussed above, CROM has a similar property in that a rate R does not need to be speci\ufb01ed in advance of the code design. This is because R Ln is a function of n only, and therefore \u03b1i\u2019s are independent to R. Furthemore, we will see in the proof that \u03b3n depends only on n. If the source is i.i.d. N(0, \u03c32), the decoder can achieve a distortion DG(\u03bdR) upon observing fraction \u03bd of the message bits. This is similar to a rateless code in channel coding because the decoder can achieve the optimum as soon as it collects suf\ufb01ciently many of the message bits. However, the CROM decoder needs its observed bits to be a contiguous sequence at the beginning of the message bit stream while it is enough to have any combination of channel output observations in \f9 the rateless channel coding setting. Note that our scheme can be considered as a progressive coder where \u201cprogressive\u201d refers to the re\ufb01nability. However, it is often the case that the re\ufb01nement layer of progressive code is often useless without the base layer, where re\ufb01nement layers of CROM are useful by themselves. More precisely, the decoder can have the following reconstruction based only on m(i1), . . . , m(il), \u02c6 X = l X j=1 \u03b1ij(AT 1 \u00b7 \u00b7 \u00b7 AT ij)U(ij) (24) where with m(1), . . . , m(il) the reconstruction would be \u02c6 X(il) = il X j=1 \u03b1j(AT 1 \u00b7 \u00b7 \u00b7 AT j )U(j). (25) 5) Complete Separability: In the classical separation scheme, the source encoder must know the channel capacity C in order to design the source coding scheme with rate R(D) < C where the source encoder often does not have this prior knowledge. However, if the source is Gaussian, the proposed scheme achieves the optimum distortion without channel information. Let C0 be a suf\ufb01ciently large constant and say the encoder uses the proposed scheme with rate R = C0. When the decoder receives the \ufb01rst C/C0 fraction of message bits and performs the reconstruction, we achieve the distortion D that satis\ufb01es RG(D) = C due to the in\ufb01nitesimal successive re\ufb01nability. Since we can achieve the optimum performance using a simple scheme while the source encoder is blind to the capacity of the link, we can call this property complete separability. Xn Encoder Relay C1 Decoder C2 \u02c6 Xn Fig. 1: Relay Network Another interesting example is a relay network without a direct link, as described in Figure 1, where the source is i.i.d. Gaussian. Both the links from the encoder to the relay node and the relay node to the decoder are noiseless with capacity C1 and C2 respectively, when we assume that C1 > C2. If the encoder knows the capacity of both links, then the problem is equivalent to the successive re\ufb01nement problem. However, consider the case where the encoder only knows C1. If the encoder is optimized only for the \ufb01rst link, the relay node has to decode the whole message and compress it again with rate C2. However, if we use CROM, the relay node can simply send the \ufb01rst C2 C1 fraction of messages to the decoder and the decoder will be able to have optimal reconstruction with respect to its own link capacity. 6) Convergence Rate: After the i-th iteration, the decoder can achieve a distortion \u03c32 \u0010 e\u2212 i Ln R + \u03b3ne i Ln R\u00112 = \u03c32 \u0010 e\u22122 i Ln R + 2\u03b3n + e2 i Ln R\u03b32 n \u0011 (26) \u2264\u03c32 \u0010 e\u22122 i Ln R + 2\u03b3n + e2R\u03b32 n \u0011 . (27) \f10 Recall that the Gaussian distortion-rate function at rate i Ln R is \u03c32 exp \u0010 \u22122 i Ln R \u0011 , and therefore the gap between the achieved distortion and DG \u0010 i Ln R \u0011 is uniformly bounded by 2\u03c32\u03b3n + \u03c32e2R\u03b32 n at all stages. Note that if the source is i.i.d. with bounded E \u0002 |X1|3\u0003 , we can choose vanishing \u03b3n = O \u0010 log log n log n \u0011 such that the probability of error decays on the order of O \u0010 1 log n \u0011 . IV. COMPARISON TO SPARC Recall that CROM can be viewed as a nonzero-rate generalization of the zero-rate scheme introduced in Section II. On the other hand, SPARC implements the idea of describing a codeword with a linear combination of subcodewords. Though the derivations of these two schemes were based on different ideas, they share several similarities. In this section, we outline the similarities and differences. A. Sparse Linear Regression Codes Let us brie\ufb02y review SPARC. Let Xn be the \ufb01rst n components of an ergodic source with mean 0 and variance 1. De\ufb01ne L sub-codebooks C1, C2, . . . , CL, where each sub-codebook has M sub-codewords. Sub-codewords are generated independently according to the standard normal distribution. Parameters M and L are chosen to be M L = enR, where R is the rate of the scheme, and de\ufb01ne constants c1, c2, . . . , cL appropriately. Then, the following algorithm exhibits the main structure of the sparse linear regression code (SPARC), which was presented in [7] and shown to achieve the Gaussian distortion-rate function for any ergodic source (under appropriate choice of parameters). Algorithm 2 SPARC Set X(1) = Xn. for i = 1 to L do Let U(i) = argmax U n\u2208Ci < X(i), U n > and m(i) be the index of U(i). Let X(i+1) = X(i) \u2212ciU(i). end for Send (m(1), m(2), . . . , m(Ln)). Note that there is another version of SPARC [8] where encoding is not done sequentially but is done by exhaustive search. Since we are focusing on ef\ufb01cient lossy compressors, we only consider the SPARC described in Algorithm 2 throughout the paper. B. Main Differences In SPARC, the codebook consists of L sub-codebooks where each sub-codebook has M codewords. Our proposed iterative scheme is similar to SPARC with L = nR log n and M = n; \ufb01nding the sub-codeword that achieves the \f11 maximum inner product can be viewed as \ufb01nding the maximum entries after multiplying the matrix in our iterative scheme. There are, however, two main differences. The \ufb01rst is that our scheme \ufb01nds the kn largest values at each iteration. This implies that one iteration of our proposed encoding scheme is equivalent to kn iterations of SPARC\u2019s encoding. In Section III-C2, we have seen that CROM requires O \u0010 n2 log\u03b2+1 n \u0011 operations per symbol, for an arbitrarily chosen \u03b2 > 0. The gap between the distortion and DG(R) is log log n log n . In SPARC, the gap between the distortion and DG(R) is log log M log M . In order to calibrate with CROM, we can set M = n. However, ML operation per symbol is required for SPARC encoding where M L = enR, and therefore the number of operations for SPARC is O \u0010 n2 log n \u0011 . Thus, SPRAC requires log\u03b2 n times more operations. The same relation holds when we consider the storage complexity. CROM requires to store O \u0010 n3 log\u03b2+1 n \u0011 real numbers, where the SPARC encoder and decoder have to store O \u0010 n3 log n \u0011 real numbers. The second difference is the structure of the sub-codebook. The columns of orthogonal matrix are orthogonal to each other, and this implies that CROM is similar to SPARC with structured sub-codewords. For example, if kn = 1, all sub-codewords of CROM are orthogonal to each other, where SPARC draws sub-codewords according to i.i.d. Gaussian. C. Key Lemma As we discussed in Section IV-B, sub-codewords in CROM is drawn from the surface of the sphere while subcodewords in SPARC are drawn according to the i.i.d. Gaussian distribution. Under this difference, we would like to introduce some dualities. For example, consider the following lemma used in the proof of SPARC. Lemma 5. [7, Lemma 1] Let Z1, . . . , ZN be independent random vectors with i.i.d. standard Gaussian elements. Then for any random vector B supported on the n dimensional unit sphere and independent of the Zi\u2019s, the inner products {< Zi, B >}N i=1 are i.i.d. standard Gaussian random variables that are independent of B. On the other hand, recall Lemma 3, which asserts that any random vector multiplied by uniform random orthogonal matrix has a spherical distribution. D. Successive Re\ufb01nability That SPARC possesses the successive re\ufb01nability property was brie\ufb02y mentioned by the authors, however, the main theorem in [7] only guarantees that the probability of error at the end of the process will vanish. On the other hand, we have seen that CROM has uniform convergence rates, uniformly and simultaneously on all points on the rate distortion curve, in Section III-C6. V. SIMULATION RESULTS In this section, we test CROM via simulations on sources with \u03c32 = 1. We choose \u03b1i = r n \u0010 1 \u2212e\u2212 2 Ln R\u0011 e\u2212i\u22121 Ln R. (28) \f12 Note that parameters are not optimized for the expected distortion, so there might be a better choice of \u03b1i. All results are averaged over 100 random trials. First, We compare the performance of CROM and SPARC in Figure 2a. We choose i.i.d. standard Gaussian source Xn where n = 256. We simulated for M = 128, 256, 512 for SPARC. Note that the complexity of SPARC is higher when M is large. We let kn = 1 for CROM which corresponds to M = 256 case of SPARC. Note that the performance of CROM is similar to the performance of SPARC with M = 256. As we discussed in Remark 3, the complexity of CROM decreases when kn is large, however, the performance will be worse when kn is large. Figure 2b shows trade-off between the small and the large kn. In order to simulate CROM with higher n, we use structured orthogonal matrices to reduce the storage and computational complexity. Note that any orthogonal matrix is a product of n(n\u22121) 2 Givens rotations which are matrices of the form \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 \u00b7 \u00b7 \u00b7 0 \u00b7 \u00b7 \u00b7 0 \u00b7 \u00b7 \u00b7 0 . . . ... . . . . . . . . . 0 \u00b7 \u00b7 \u00b7 cos \u03b8 \u00b7 \u00b7 \u00b7 \u2212sin \u03b8 \u00b7 \u00b7 \u00b7 0 . . . . . . . . . . . . 0 \u00b7 \u00b7 \u00b7 sin \u03b8 \u00b7 \u00b7 \u00b7 cos \u03b8 \u00b7 \u00b7 \u00b7 0 . . . . . . . . . ... . . . 0 \u00b7 \u00b7 \u00b7 0 \u00b7 \u00b7 \u00b7 0 \u00b7 \u00b7 \u00b7 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . (29) This suggests to construct sparse orthogonal matrices using Givens rotations as a building block. Suppose n be the power of 2, i.e., n = 2s. We recursively de\ufb01ne the sparse orthogonal matrices A(s) r for 1 \u2264r \u2264s. A(s) r (\u03b81, . . . , \u03b8n/2) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8eb \uf8ec \uf8ec \uf8ed A(s\u22121) r\u22121 (\u03b81, . . . , \u03b8n/4) 0 0 A(s\u22121) r\u22121 (\u03b8n/2+1, . . . , \u03b8n/2) \uf8f6 \uf8f7 \uf8f7 \uf8f8 if r > 1 \uf8eb \uf8ec \uf8ec \uf8ed diag(cos \u03b81, . . . , cos \u03b8n/2) diag(\u2212sin \u03b81, . . . , \u2212sin \u03b8n/2) diag(sin \u03b81, . . . , sin \u03b8n/2) diag(cos \u03b81, . . . , cos \u03b8n/2) \uf8f6 \uf8f7 \uf8f7 \uf8f8 if r = 1 , (30) where diag(x1, . . . , xn) is a diagonal matrix with entries x1, . . . , xn. The following matrices (31), (32), (33) show \f13 three types of sparse orthogonal matrices when n = 8. A(3) 1 (\u03b81, \u03b82, \u03b83, \u03b84) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed cos \u03b81 0 0 0 \u2212sin \u03b81 0 0 0 0 cos \u03b82 0 0 0 \u2212sin \u03b82 0 0 0 0 cos \u03b83 0 0 0 \u2212sin \u03b83 0 0 0 0 cos \u03b84 0 0 0 \u2212sin \u03b84 sin \u03b81 0 0 0 cos \u03b81 0 0 0 0 sin \u03b82 0 0 0 cos \u03b82 0 0 0 0 sin \u03b83 0 0 0 cos \u03b83 0 0 0 0 sin \u03b84 0 0 0 cos \u03b84 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (31) A(3) 2 (\u03b81, \u03b82, \u03b83, \u03b84) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed cos \u03b81 0 \u2212sin \u03b81 0 0 0 0 0 0 cos \u03b82 0 \u2212sin \u03b82 0 0 0 0 sin \u03b81 0 cos \u03b81 0 0 0 0 0 0 sin \u03b82 0 cos \u03b82 0 0 0 0 0 0 0 0 cos \u03b83 0 \u2212sin \u03b83 0 0 0 0 0 0 cos \u03b84 0 \u2212sin \u03b84 0 0 0 0 sin \u03b83 0 cos \u03b83 0 0 0 0 0 0 sin \u03b84 0 cos \u03b84 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (32) A(3) 3 (\u03b81, \u03b82, \u03b83, \u03b84) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed cos \u03b81 \u2212sin \u03b81 0 0 0 0 0 0 sin \u03b81 cos \u03b81 0 0 0 0 0 0 0 0 cos \u03b82 \u2212sin \u03b82 0 0 0 0 0 0 sin \u03b82 cos \u03b82 0 0 0 0 0 0 0 0 cos \u03b83 \u2212sin \u03b83 0 0 0 0 0 0 sin \u03b83 cos \u03b83 0 0 0 0 0 0 0 0 cos \u03b84 \u2212sin \u03b84 0 0 0 0 0 0 sin \u03b84 cos \u03b84 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . (33) Each matrix A(r) s is a product of n/2 Givens rotations. Therefore, the product of log n consecutive sparse orthogonal matrices is equivalent to the product of n log n 2 Givens rotations. If we draw angles uniformly randomly, the product is expected to have similar distribution to uniform random orthogonal matrix. Since each row has exactly two non-zero elements, the matrix multiplication requires O(n) operations. Also, the storage complexity is O(n). Another well-known orthogonal matrix is discrete cosine transform matrix of type-II (DCT-II). We can use Fast Fourier Transform (FFT) algorithm to multiply DCT matrix ef\ufb01ciently. Also, DCT matrix requires O(1) of storage space. Instead of original CROM with uniform random orthogonal matrices, we propose two modi\ufb01ed version of CROM using the above structured orthogonal matrices. First, at i-th iteration, we choose A(s) r (\u03b8(s) r,1, . . . , \u03b8(s) r,n/2) where i \u2261r (mod s), and \u03b8(s) r,1, . . . , \u03b8(s) r,n/2 are uniformly sampled from [0, 2\u03c0]. The second approach is using ADCT \u00d7 A(s) r (\u03b8(s) r,1, . . . , \u03b8(s) r,n/2) where ADCT denotes the DCT-II matrix. Figure 2c shows performances of two modi\ufb01ed algorithms when n = 1024 and kn = 1. Note that the performance of sparse orthogonal matrices is worse than \f14 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Optimum CROM SPARC M=128 SPARC M=256 (a) Distortion-rate curves of CROM and SPARC where n = 256. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Optimum k=1 k=3 k=5 (b) Distortion-rate curves for k = 1, 3, 5 where n = 1024. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Optimum Orthogonal Sparse + DCTII Sparse (c) Distortion-rate curves for different matrix constructions where n = 1024. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Optimum n=65536 n=1024 n=256 (d) Distortion-rate curves for n = 256, 1024, 65536 where k = 1. Fig. 2: Distortion-rate curves of CROM and SPARC. x-axis shows the rate in nats, and the y-axis represents the average distortion. uniformly generated orthogonal matrices, on the other hand, the performance of sparse orthogonal matrices with DCT-II matrix is comparable to those of uniform orthogonal matrices. Since modi\ufb01ed CROM has lower complexity, we can test CROM with larger n. Figure 2d shows the distortionrate curve of the second approach with sparse orthogonal matrices and the DCT-II matrix where n = 65536 and k = 1. Compare to the simulation result of n = 1024 with uniform random orthogonal matrices, its distortion-rate curve shows better performance. VI. CHANNEL CODING DUAL In [18], we can \ufb01nd a dual result in the Gaussian channel coding problem. In this section, we brie\ufb02y review the idea of [18] (with slightly changed notation). Consider the AWGN channel Yi = Xi + Zi where Zn is an i.i.d. standard normal random vector. Suppose the number of messages is n, i.e., the rate of the scheme is Rn = log n n nats per channel use. Based on message m \u2208{1, 2, . . . , n}, the encoder simply sends Xn where Xm = (1+\u03f5n)\u221a2 log n and Xi = \u2212(1+\u03f5n) \u221a2 log n n\u22121 if i \u0338= m. Then, the decoder \ufb01nds the index of the maximum value of Y n and recovers the message, i.e., \u02c6 m = arg max1\u2264i\u2264n Yi. The average power that the encoder uses is Pn = 2(1 + \u03f5n)2 1 n\u22121 log n. We will specify \u03f5n such that limn\u2192\u221e\u03f5n = 0. \f15 Before considering the probability of error P (n) e , let us introduce the following useful lemma. Lemma 6. Let Zn be an i.i.d. standard normal random vector, then Pr \u0014 max 1\u2264i\u2264n Zi > p 2 log n \u0015 \u2264 1 \u221alog n. (34) Proof: Pr \u0002 max1\u2264i\u2264n Zi > \u221a2 log n \u0003 =1 \u2212\u03a6( p 2 log n)n (35) =1 \u2212(1 \u2212Q(a))n (36) \u2264nQ(a) (37) \u2264n 1 \u221a2 log n 1 \u221a 2\u03c0 exp \u0012 \u22122 log n 2 \u0013 (38) \u2264 1 \u221alog n, (39) where \u03a6(x) is a standard normal cumulative distribution function and Q(x) = 1 \u2212\u03a6(x). We used the fact that Q(x) \u22641 xf(x) where f(x) is a probability density function of standard normal random variable. Now we are ready to bound P (n) e . Without loss of generality, we can assume that m = 1. P (n) e =Pr [Y1 < max2\u2264i\u2264n Yi] (40) =Pr h (1 + \u03f5n)\u221a2 log n + Z1 < \u2212(1 + \u03f5n) \u221a2 log n n\u22121 + max2\u2264i\u2264n Zi i (41) =Pr h n n\u22121(1 + \u03f5n)\u221a2 log n + Z1 < max2\u2264i\u2264n Zi i (42) \u2264Pr \u0002\u221a2 log n < max2\u2264i\u2264n Zi \u0003 + Pr \u00141 + n\u03f5n n \u22121 p 2 log n + Z1 < 0 \u0015 (43) \u2264 1 \u221alog n + Pr \u00141 + n\u03f5n n \u22121 p 2 log n + Z1 < 0 \u0015 . (44) If we choose \u03f5n such that 1+n\u03f5n n\u22121 = (log n)\u22121/3, then 1+n\u03f5n n\u22121 \u221a2 log n goes to in\ufb01nity as n grows. Therefore, lim n\u2192\u221eP (n) e = 0. (45) Since Pn converges to zero as n grows, we can approximate the capacity by C(Pn) = 1 2 log(1 + Pn) \u2248Pn 2 = (1 + \u03f5n)2 log n n\u22121 . It is clear that Rn C(Pn) converges to one as n grows, i.e., lim n\u2192\u221e Rn C(Pn) = 1. (46) This is reminiscent of the de\ufb01nition of a zero-rate optimal scheme in the source coding problem. We can say that this scheme is zero-rate optimal in the channel coding setting. We further note that the encoding and decoding can be done in almost linear time, and essentially no extra information needs to be stored. However, unlike CROM, we could not \ufb01nd an iterative scheme building on this zero-rate one that achieves reliable communication at a positive rate. The main challenge is that the tail behavior on the left side is very different from the right side. In the source coding problem, a small maximum value (which corresponds to the left tail) yields an \f16 error, while it is a large maximum value (which corresponds to the right tail) that yields an error in the channel coding problem. More precisely, the cumulative distribution function of the maximum of Gaussian random variables converges to exp (\u2212e\u2212x) with normalizing constants. This function decays double-exponentially as x decreases, which allows a small cumulative error for our iterative scheme CROM. However, exp (\u2212e\u2212x) converges to one only exponentially as x grows. Therefore, in the similar channel coding scheme, the cumulative error does not remain negligible when we employ the scheme iteratively. We believe that for similar reasons a channel coding analog of SPARC with ef\ufb01cient encoding would not work. Note that Erez et al. discussed rateless coding for Gaussian channels [19]. The goal of the paper \u201cRateless Coding for Gaussian Channels seems design a channel code where the transmitter can be blind to the channel gain and the variance of the noise. Note that the proposed rateless code requires the base code that achieves the capacity. On the other hand, we would like to design a concrete coding scheme that achieves the channel capacity when the channel information is known. VII. PROOFS A. Extreme Value of Gaussian Random Variables Before providing proofs, consider the following lemma which shows the probabilistic bound of Z(i) when Zn is an i.i.d. standard normal random vector. Lemma 7. Let \u03f5 > 0. If positive integers n and i satisfy 0 \u2264 1 n\u2212i+1 log ni\u22121 \u03f5 \u22641, then Pr h Z(i) < \u03a6\u22121 \u0010 1 \u2212 1 n\u2212i+1 log ni\u22121 \u03f5 \u0011i \u2264\u03f5, (47) where \u03a6(x) = R x \u2212\u221e 1 \u221a 2\u03c0e\u2212z2 2 dz is a standard normal cumulative distribution function. Proof: Since \u03a6(Z1), \u03a6(Z2), . . . , \u03a6(Zn) are i.i.d. uniform random variables, \u03a6(Z(i)) can be considered as the i-th largest value of an n dimensional i.i.d. uniform random vector. The probability density function of \u03a6(Z(i)) is n! (n\u2212i)!(i\u22121)!xn\u2212i(1 \u2212x)i\u22121. Therefore, Pr h Z(i) < \u03a6\u22121 \u0010 1 \u2212 1 n\u2212i+1 log ni\u22121 \u03f5 \u0011i =Pr h \u03a6 \u0000Z(i) \u0001 < 1 \u2212 1 n\u2212i+1 log ni\u22121 \u03f5 i (48) = Z 1\u2212 1 n\u2212i+1 log ni\u22121 \u03f5 0 n! (n \u2212i)!(i \u22121)!xn\u2212i(1 \u2212x)i\u22121dx (49) \u2264 Z 1\u2212 1 n\u2212i+1 log ni\u22121 \u03f5 0 n! (n \u2212i)!(i \u22121)!xn\u2212idx (50) = n! (n \u2212i + 1)!(i \u22121)! \u0012 1 \u2212 1 n \u2212i + 1 log ni\u22121 \u03f5 \u0013n\u2212i+1 (51) \u2264ni\u22121 exp \u0012 \u2212log ni\u22121 \u03f5 \u0013 (52) =\u03f5. (53) This concludes the proof. \f17 B. Proof of Theorem 2 In the proof, we use \u03b1 = \u03b1n for simplicity. By the de\ufb01nition of \u02c6 Xn, we have \r \r \rXn \u2212\u02c6 Xn\r \r \r 2 = \u2225Xn\u22252 + 2kn\u03b1 n \u2212kn n X i=1 Xi \u2212 2n\u03b1 n \u2212kn kn X i=1 X(i) + nkn n \u2212kn \u03b12. (54) Let \u03b3n and \u03b4n be positive real numbers where we specify their values later. Then, Pr \u0014\r \r \rXn \u2212\u02c6 Xn\r \r \r 2 > n(1 + \u03b3n \u2212\u03b4n) \u0015 = Pr \" \u2225Xn\u22252 + 2kn\u03b1 n \u2212kn n X i=1 Xi \u2212 2n\u03b1 n \u2212kn kn X i=1 X(i) + nkn n \u2212kn \u03b12 > n(1 + \u03b3n \u2212\u03b4n) # (55) \u2264Pr h \u2225Xn\u22252 > n(1 + \u03b3n) i + Pr \" 2kn\u03b1 n \u2212kn n X i=1 Xi \u2212 2n\u03b1 n \u2212kn kn X i=1 X(i) + nkn n \u2212kn \u03b12 > \u2212n\u03b4n # . (56) Consider the \ufb01rst term of (56). Let \u03b3n = q 2 nQ\u22121 \u0010 \u03f5 \u221215 \u221an \u22122 n \u0011 , then we have Pr h \u2225Xn\u22252 > n(1 + \u03b3n) i \u2264Q \u0012rn 2 \u03b3n \u0013 + 15 \u221an (57) = \u03f5 \u22122 n. (58) In (57), we used Berry-Esseen theorem [20]: sup x \f \f \f \fPr \u0014Pn i=1(X2 i \u22121) \u03c3\u221an > x \u0015 \u2212Q(x) \f \f \f \f < \u03c1 \u221an, (59) where \u03c32 = E \u0002 X4\u0003 \u2212E \u0002 X2\u00032 = 2 and \u03c1 = E \u0002 X6 1 \u0003 = 15. Consider the second term of (56). Pr h 2kn\u03b1 n\u2212kn Pn i=1 Xi \u2212 2n\u03b1 n\u2212kn Pkn i=1 X(i) + nkn n\u2212kn \u03b12 > \u2212n\u03b4n i = Pr \" 1 n n X i=1 Xi \u22121 kn kn X i=1 X(i) + \u03b1 2 > \u2212n \u2212kn 2kn\u03b1 \u03b4n # (60) = Pr \" 1 kn kn X i=1 X(i) \u22121 n n X i=1 Xi < \u03b1 2 + n \u2212kn 2kn\u03b1 \u03b4n # . (61) Let \u03b1 = q n\u2212kn kn \u03b4n = pn \u2212qn, where pn =\u03a6\u22121 \u0012 1 \u2212 1 n \u2212kn + 1 log nkn \u0013 (62) =Q\u22121 \u0012 kn n \u2212kn + 1 log n \u0013 (63) qn = 1 \u221anQ\u22121 \u0012 1 n \u0013 . (64) Then, we have Pr h 2kn\u03b1 n\u2212kn Pn i=1 Xi \u2212 2n\u03b1 n\u2212kn Pkn i=1 X(i) + nkn n\u2212kn \u03b12 > \u2212n\u03b4n i \f18 = Pr \" 1 kn kn X i=1 X(i) \u22121 n n X i=1 Xi < pn \u2212qn # (65) \u2264Pr \" 1 kn kn X i=1 X(i) < pn # + Pr \" 1 n n X i=1 Xi > qn # (66) \u2264Pr \u0002 X(kn) < pn \u0003 + Pr \" 1 n n X i=1 Xi > qn # . (67) By Lemma 7, Pr \u0002 X(kn) < pn \u0003 \u22641 n. Since 1 n Pn i=1 Xi has a Gaussian distribution with zero mean and variance 1 n, Pr \u0002 1 n Pn i=1 Xi > qn \u0003 = 1 n. (68) Therefore, Pr h 2kn\u03b1 n\u2212kn Pn i=1 Xi \u2212 2n\u03b1 n\u2212kn Pkn i=1 X(i) + nkn n\u2212kn \u03b12 > \u2212n\u03b4n i \u22642 n. (69) With (58), we have Pr \u0014\r \r \rXn \u2212\u02c6 Xn\r \r \r 2 > n(1 + \u03b3n \u2212\u03b4n) \u0015 \u2264\u03f5. (70) Now, let consider the bound on 1 + \u03b3n \u2212\u03b4n. It is clear that the inequality q 2 log n\u2212kn+1 kn log3 n 1+2 log n\u2212kn+1 kn log3 n 1 \u221a 2\u03c0 log2 n > 1 holds for large enough n, and therefore Q s 2 log n \u2212kn + 1 kn log3 n ! \u2265 q 2 log n\u2212kn+1 kn log3 n 1 + 2 log n\u2212kn+1 kn log3 n 1 \u221a 2\u03c0 kn log3 n n \u2212kn + 1 (71) \u2265kn log n n \u2212kn + 1, (72) which implies pn = Q\u22121 \u0012 kn n \u2212kn + 1 log n \u0013 \u2265 s 2 log n \u2212kn + 1 kn log3 n . (73) On the other hand, it is not hard to show that qn = 1 \u221anQ\u22121 \u0012 1 n \u0013 \u2264 r 2 log n 2 n . (74) Now, we are ready to bound Dn = 1 + \u03b3n \u2212\u03b4n. Since Rn = 1 n log \u0000 n kn \u0001 , we have Dn =1 + \u03b3n \u2212 kn n \u2212kn (pn \u2212qn)2 (75) =1 + r 2 nQ\u22121 \u0012 \u03f5 \u221215 \u221an \u22122 n \u0013 \u2212 kn n \u2212kn \u0012 Q\u22121 \u0012 kn n \u2212kn + 1 log n \u0013 \u2212 1 \u221anQ\u22121 \u0012 1 n \u0013\u00132 (76) \u22641 + r 2 nQ\u22121(\u03f5) + O \u0012 1 n \u0013 \u2212 kn n \u2212kn s 2 log n \u2212kn + 1 kn log3 n \u2212 r 2 log n 2 n !2 (77) =1 + r 2 nQ\u22121(\u03f5) \u22122Rn + O \u0012kn log log n n \u0013 . (78) This concludes the proof. \f19 C. Proof of Theorem 4 Throughout the proof, we will let \u03c32 = 1 and use L instead of Ln for simplicity. Also, instead of choosing speci\ufb01c orthogonal matrices A1, . . . , AL+1, we employ a randomization argument. More precisely, we assume that A1, A2, . . . , Ai+1 are drawn i.i.d. Unif(O(n)) and show that equation (22) holds when the probability is averaged over this ensemble of random matrices. Let Si = \r \rX(i)\r \r and X(i) = SiB(i) where B(i) is uniformly distributed on the n-dimensional unit sphere and independent to Si. Since we draw random matrices independently, random variables B(1), . . . , B(L+1) are also independent. Recall (18) and (19), we have \r \r \rXn \u2212\u02c6 X(i)\r \r \r 2 = \r \rX(i+1)\r \r2 = S2 i+1, and this implies that the distortion after the i-th iteration coincides with S2 i+1 divided by n. We further let \u02dc S be a chi-distributed random variable with degrees of freedom n and independent to all B(i)\u2019s, i.e., \u02dc S2 \u223c\u03c72(n). Using union bound, we can obtain an upper bound on the excess distortion probability. Pr h 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n for some 0 \u2264i \u2264L i \u2264Pr \u0014 1 \u221anS1 > 1 + \u03b3n or 1 \u221an \u02dc S > p 1 + \u03b3n \u0015 + L X i=1 Pr \u0014 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n, 1 \u221anSj+1 \u2264e\u2212j L R + e j L R\u03b3n for all j < i, and 1 \u221an \u02dc S \u2264 p 1 + \u03b3n \u0015 (79) \u2264Pr \u0014 1 \u221anS1 > 1 + \u03b3n or 1 \u221an \u02dc S > p 1 + \u03b3n \u0015 + L X i=1 Pr \u0014 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n, 1 \u221anSi \u2264e\u2212i\u22121 L R + e i\u22121 L R\u03b3n, and 1 \u221an \u02dc S \u2264 p 1 + \u03b3n \u0015 (80) From the de\ufb01nition of X(i+1), we have S2 i+1 = \r \r \rX(i+1)\r \r \r 2 (81) = \r \r \rX(i) \u2212\u03b1iU(i)\r \r \r 2 (82) = \r \r \rX(i)\r \r \r 2 + \u03b12 i \u22122\u03b1i \u2212 s kn n(n \u2212kn)1 + r n (n \u2212kn)kn m(i) !T X(i), (83) where \u0000m(i)\u0001T X(i) is a sum of kn largest value of X(i). Let Vi = \u0012 \u2212 q kn n(n\u2212kn)1 + q n (n\u2212kn)kn m(i) \u0013T B(i), then Vi and Si are independent. We can now rewrite (83) as S2 i+1 = S2 i + \u03b12 i \u22122\u03b1iSiVi. (84) It is not hard to show that S2 i + \u03b12 i \u22122\u03b1iSiVi is an increasing function in Si when 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n and 1 \u221anSi \u2264e\u2212i\u22121 L R + e i\u22121 L R\u03b3n. Therefore, S2 i+1 = S2 i + \u03b12 i \u22122\u03b1iSiVi \u2264n \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u00112 + \u03b12 i \u22122\u221an\u03b1i \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u0011 Vi, (85) which is equivalent to n \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u00112 + \u03b12 i \u2212n \u0010 e\u2212i L R + e i L R\u03b3n \u00112 2\u221an\u03b1i \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u0011 > Vi. (86) \f20 This implies Pr h 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n, 1 \u221anSi \u2264e\u2212i\u22121 L R + e i\u22121 L R\u03b3n, and 1 \u221an \u02dc S \u2264\u221a1 + \u03b3n i = Pr \uf8ee \uf8ef \uf8f0 n \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u00112 + \u03b12 i \u2212n \u0010 e\u2212i L R + e i L R\u03b3n \u00112 2\u221an\u03b1i \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u0011 > Vi and 1 \u221an \u02dc S \u2264 p 1 + \u03b3n \uf8f9 \uf8fa \uf8fb. (87) Recall that we took \u03b1i = r n \u0010 1 \u2212e\u22122 L R\u0011 \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u0011 \u0010 e\u2212i\u22121 L R \u2212e i\u22121 L R\u03b3n \u0011 , (88) and it can be easily shown that n \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u00112 + \u03b12 i \u2212n \u0010 e\u2212i L R + e i L R\u03b3n \u00112 2\u221an\u03b1i \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u0011 = 2\u03b12 i 2\u221an\u03b1i \u0010 e\u2212i\u22121 L R + e i\u22121 L R\u03b3n \u0011 (89) \u2264 s\u0010 1 \u2212e\u22122 L R\u0011 e\u2212i\u22121 L R \u2212e i\u22121 L R\u03b3n e\u2212i\u22121 L R + e i\u22121 L R\u03b3n (90) \u2264 r\u0010 1 \u2212e\u22122 L R\u0011 1 \u2212\u03b3n 1 + \u03b3n . (91) Thus, we have Pr h 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n, and 1 \u221anSi \u2264e\u2212i\u22121 L R + e i\u22121 L R\u03b3n, and 1 \u221an \u02dc S \u2264\u221a1 + \u03b3n i \u2264Pr \u0014r\u0010 1 \u2212e\u22122 L R\u0011 1 \u2212\u03b3n 1 + \u03b3n > Vi and 1 \u221an \u02dc S \u2264 p 1 + \u03b3n \u0015 (92) \u2264Pr \"r n \u0010 1 \u2212e\u22122 L R\u0011 (1 \u2212\u03b3n) > \u02dc SVi # (93) \u2264Pr \"r 2nR L (1 \u2212\u03b3n) > \u02dc SVi # . (94) Since B(i) is uniformly distributed on a unit sphere and it is independent of \u02dc S, we have \u02dc SB(i) (d) = Z where Z is an n dimensional i.i.d. standard normal random vector. Furthermore, \u02dc SVi (d) \u2261 \u2212 s kn n(n \u2212kn)1 + r n (n \u2212kn)kn m(1) !T Z (95) = r nkn n \u2212kn 1 kn kn X i=1 Z(i) \u22121 n n X i=1 Zi ! . (96) If we have \u03b3n \u22651 \u2212(pn \u2212qn)2 L 2nR nkn n\u2212kn , where pn and qn were de\ufb01ned as (63) and (64), then we can apply the similar technique from the proof of Theorem 2. I.e., Pr h 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n, and 1 \u221anSi \u2264e\u2212i\u22121 L R + e i\u22121 L R\u03b3n, and 1 \u221an \u02dc S \u2264\u221a1 + \u03b3n i \u2264Pr \"r 2nR L (1 \u2212\u03b3n) > \u02dc SVi # (97) \f21 \u2264Pr \" pn \u2212qn > 1 kn kn X i=1 Z(i) \u22121 n n X i=1 Zi # (98) \u22642 n. (99) Recall that pn \u2265 q 2 log n\u2212kn+1 kn log3 n and qn \u2264 q 2 log n 2 n . Therefore, it is easy to check that 1 \u2212(pn \u2212qn)2 L 2nR nkn n \u2212kn \u22641 \u2212 s 2 log n \u2212kn + 1 kn log3 n \u2212 r 2 log n 2 n !2 kn 2 log \u0000n k \u0001 (100) = O \u0012log log n log n \u0013 . (101) Firstly, if \u03b3n is equal to any constant \u03b3 > 0, due to the stationarity of the source, we have lim n\u2192\u221ePr \u0014 1 \u221anS1 > 1 + \u03b3n or 1 \u221an \u02dc S > p 1 + \u03b3n \u0015 = 0. (102) Therefore, lim n\u2192\u221ePr \u0014 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n for some 0 \u2264i \u2264L \u0015 = lim n\u2192\u221ePr \u0014 1 \u221anS1 > 1 + \u03b3n or 1 \u221an \u02dc S > p 1 + \u03b3n \u0015 + lim n\u2192\u221e 2 nLn (103) = 0. (104) Suppose the source is i.i.d. distributed with E \u0002 |X1|3\u0003 < \u221e, then we can let \u03b3n = O \u0010 log log n log n \u0011 such that \u03b3n \u22651 \u2212 s 2 log n \u2212kn + 1 kn log3 n \u2212 r 2 log n 2 n !2 kn 2 log \u0000n k \u0001 (105) and still have lim n\u2192\u221ePr \u0014 1 \u221anS1 > 1 + \u03b3n or 1 \u221an \u02dc S > p 1 + \u03b3n \u0015 = 0. (106) We would like to point out that the right hand side of (105) is independent to the choice of R. Finally, it is clear that lim n\u2192\u221ePr \u0014 1 \u221anSi+1 > e\u2212i L R + e i L R\u03b3n for some 0 \u2264i \u2264L \u0015 = lim n\u2192\u221ePr \u0014 1 \u221anS1 > 1 + \u03b3n or 1 \u221an \u02dc S > p 1 + \u03b3n \u0015 + lim n\u2192\u221e 2 nLn (107) = 0. (108) This concludes the proof. VIII." + } + ], + "Ernest K. Ryu": [ + { + "url": "http://arxiv.org/abs/1905.10899v3", + "title": "ODE Analysis of Stochastic Gradient Methods with Optimism and Anchoring for Minimax Problems", + "abstract": "Despite remarkable empirical success, the training dynamics of generative\nadversarial networks (GAN), which involves solving a minimax game using\nstochastic gradients, is still poorly understood. In this work, we analyze\nlast-iterate convergence of simultaneous gradient descent (simGD) and its\nvariants under the assumption of convex-concavity, guided by a continuous-time\nanalysis with differential equations. First, we show that simGD, as is,\nconverges with stochastic sub-gradients under strict convexity in the primal\nvariable. Second, we generalize optimistic simGD to accommodate an optimism\nrate separate from the learning rate and show its convergence with full\ngradients. Finally, we present anchored simGD, a new method, and show\nconvergence with stochastic subgradients.", + "authors": "Ernest K. Ryu, Kun Yuan, Wotao Yin", + "published": "2019-05-26", + "updated": "2020-10-12", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG", + "stat.ML" + ], + "main_content": "Introduction Training of generative adversarial networks (GAN) [21], solving a minimax game using stochastic gradients, is known to be dif\ufb01cult. Despite the remarkable empirical success of GANs, further understanding the global training dynamics empirically and theoretically is considered a major open problem [20, 52, 39, 37, 48]. The local training dynamics of GANs and minimax games are understood reasonably well. Several works have analyzed convergence assuming the loss functions have linear gradients and assuming the training uses full (deterministic) gradients. Although the linear gradient assumption is reasonable for local analysis (even though the loss functions may not be continuously differentiable due to ReLU activation functions) such results say very little about global convergence. Although the full gradient assumption is reasonable when the learning rate is small, such results say very little about how the randomness affects the training. This work investigates global convergence of simultaneous gradient descent (simGD) and its variants for zero-sum games with a convex-concave cost using using stochastic subgradients. We speci\ufb01cally study convergence of the last iterates as opposed to the averaged iterates. Organization. Section 2 presents convergence of simGD with stochastic subgradients under strict convexity in the primal variable. The goal is to establish a minimal suf\ufb01cient condition of global convergence for simGD without modi\ufb01cations. Section 3 presents a generalization of optimistic simGD [11], which allows an optimism rate separate from the learning rate. We prove the generalized optimistic simGD using full gradients converges, and experimentally demonstrate that the optimism rate must be tuned separately from the learning rate when using stochastic gradients. However, it is unclear whether optimistic simGD is theoretically compatible with stochastic gradients. Section 4 presents anchored simGD, a new method, and presents its convergence with stochastic subgradients. The presentation and analyses of Sections 2, 3, and 4 are guided by continuous-time Preprint. Under review. arXiv:1905.10899v3 [cs.LG] 12 Oct 2020 \f\ufb01rst-order ordinary differential equations (ODE). In particular, we interpret optimism and anchoring as discretizations of certain regularized dynamics. Contribution. The contribution of this work is in Theorems 1, 2, 3, and 4, the convergence analyses of the discrete algorithms, and the insight provided by the continuous-time ODE analyses. We do not present the continuous-time analyses as rigorous mathematical theorems to avoid discussing the existence and uniqueness of solutions to the differential equations. The strongest contribution of this work is the ODE analysis of Section 4.1, which is based on a nonpositive Lyapunov function, and Theorem 4, which is the \ufb01rst result establishing last-iterate convergence for convex-concave cost functions using stochastic subgradients without assuming strict convexity or analogous assumptions. Prior work. When the goal is to improve the training of GANs, there are several independent directions, such as designing better architectures, choosing good loss functions, or adding appropriate regularizers [52, 2, 60, 1, 22, 66, 58, 37, 38, 41]. In this work, we accept these factors as a given and focus on how to train (optimize) the model effectively. Optimism is a simple modi\ufb01cation to remedy the cycling behavior of simGD, which can occur even under the bilinear convex-concave setup [11, 12, 13, 36, 18, 30, 42, 49]. These prior work assume the gradients are linear and use full gradients. Although the recent name \u2018optimism\u2019 originates from its use in online optimization [9, 53, 54, 63], the idea dates back to Popov\u2019s work in the 1980s [51] and has been studied independently in the mathematical programming community [35, 32, 34, 33, 10, 55]. We note that there are other mechanisms similar to optimism and anchoring such as \u201cprediction\u201d [68], \u201cnegative momentum\u201d [19], and \u201cextragradient\u201d [27, 65, 8]. In this work, we focus on optimism and anchoring. Classical literature on minimax games analyze convergence of the Polyak-averaged iterates (which assigns less weight to newer iterates) when solving convex-concave saddle point problems using stochastic subgradients [7, 47, 46, 26, 18]. For GANs, however, last iterates or exponentially averaged iterates [69] (which assigns more weight to newer iterates) are used in practice. Therefore, the classical work with Polyak averaging do not fully explain the empirical success of GANs. We point out that we are not the \ufb01rst to utilize classical techniques for analyzing the training of GANs and minimax games. In particular, the stochastic approximation technique [24, 16], control theoretic techniques [24, 45], ideas from variational inequalities and monotone operator theory [17, 18], and continuous-time ODE analysis [24, 10] have been utilized for analyzing GANs and minimax games. 2 Stochastic simultaneous subgradient descent Consider the cost function L : Rm \u00d7 Rn \u2192R and the minimax game minx maxu L(x, u). We say (x\u22c6, u\u22c6) \u2208Rm \u00d7 Rn is a solution to the minimax game or a saddle point of L if L(x\u22c6, u) \u2264L(x\u22c6, u\u22c6) \u2264L(x, u\u22c6), \u2200x \u2208Rm, u \u2208Rn. We assume L is convex-concave and has a saddle point. (A0) By convex-concave, we mean L(x, u) is a convex function in x for \ufb01xed u and a concave function in u for \ufb01xed x. De\ufb01ne G(x, u) = \u0014 \u2202xL(x, u) \u2202u(\u2212L(x, u)) \u0015 , where \u2202x and \u2202u respectively denote the subdifferential with respect to x and u. For simplicity, write z = (x, u) \u2208Rm+n and G(z) = G(x, u). Note that 0 \u2208G(z) if and only if z is a saddle point. Since L is convex-concave, the operator G is monotone [57]: (g1 \u2212g2)T (z1 \u2212z2) \u22650 (1) \u2200g1 \u2208G(z1), g2 \u2208G(z2), z1, z2 \u2208Rm+n. Let g(z; \u03c9) be a stochastic subgradient oracle, i.e., E\u03c9g(z; \u03c9) \u2208G(z) for all z \u2208Rm+n, where \u03c9 is a random variable. Consider Simultaneous Stochastic Sub-Gradient Descent zk+1 = zk \u2212\u03b1kg(zk; \u03c9k) (SSSGD) 2 \fz(0) z* z(t) z(0) z* z(t) Figure 1: z(t) with \u02d9 z(t) = \u2212G(z(t)). (Left) L(x, u) = xu. All points satisfy G(z)T (z \u2212z\u22c6) = 0 so \u2225z(t) \u2212z\u22c6\u2225does not decrease and z(t) forms a cycle. (Right) L(x, u) = 0.2x2 + xu. The dashed line denotes where G(z)T (z \u2212z\u22c6) = 0, but it is visually clear that z\u22c6= 0 is the only cluster point. for k = 0, 1, . . . , where z0 \u2208Rm+n is a starting point, \u03b10, \u03b11, . . . are positive learning rates, and \u03c90, \u03c91, . . . are IID random variables. (We read SSSGD as \u201ctriple-SGD\u201d.) In this section, we provide convergence of SSSGD when L(x, u) is strictly convex in x. 2.1 Continuous-time illustration To understand the asymptotic dynamics of the stochastic discrete-time system, we consider a corresponding deterministic continuous-time system. For simplicity, assume G is single-valued and smooth. Consider \u02d9 z(t) = \u2212g(t), g(t) = G(z(t)) with an initial value z(0) = z0. (We introduce g(t) for notational simplicity.) Let z\u22c6be a saddle point, i.e., G(z\u22c6) = 0. Then z(t) does not move away from z\u22c6: d dt 1 2\u2225z(t) \u2212z\u22c6\u22252 = \u2212g(t)T (z(t) \u2212z\u22c6) \u22640, where we used (1). However, there is no mechanism forcing z(t) to converge to a solution. Consider the two examples L0(x, u) = xu and L\u03c1(x, u) = (\u03c1/2)x2 + xu with G0(x, u) = \u0014 0 1 \u22121 0 \u0015 \u0014 x u \u0015 , G\u03c1(x, u) = \u0014 \u03c1 1 \u22121 0 \u0015 \u0014 x u \u0015 (2) where x \u2208R and u \u2208R and \u03c1 > 0. Note that L0 is the canonical counter example that also arises as the Dirac-GAN [37]. See Figure 1. The classical LaSalle\u2013Krasnovskii invariance principle [28, 29] states (paraphrased) if z\u221eis a cluster point of z(t), then the dynamics starting at z\u221ewill have a constant distance to z\u22c6. On the left of Figure 1, we can see \u2225z(t) \u2212z\u22c6\u22252 is constant as d dt 1 2\u2225z(t) \u2212z\u22c6\u22252 = 0 for all t. On the right of Figure 1, we can see that although d dt 1 2\u2225z(t) \u2212z\u22c6\u22252 = 0 when z(t) = (0, u) for u \u0338= 0 (the dotted line) this 0 derivative is temporary as z(t) will soon move past the dotted line. Therefore, z(t) can maintain a constant constant distance to z\u22c6only if it starts at 0, and 0 is the only cluster point of z(t). 2.2 Discrete-time convergence analysis Consider the further assumptions \u221e X k=0 \u03b1k = \u221e, \u221e X k=0 \u03b12 k < \u221e (A1) E\u03c91,\u03c92\u2225g(z1; \u03c91) \u2212g(z2; \u03c92)\u22252 \u2264R2 1\u2225z1 \u2212z2\u22252 + R2 2 \u2200z1, z2 \u2208Rm+n, (A2) where \u03c91 and \u03c92 are independent random variables and R1 \u22650 and R2 \u22650. These assumptions are standard in the sense that analogous assumptions are used in convex minimization to establish almost sure convergence of stochastic gradient descent. Theorem 1. Assume (A0), (A1), and (A2). Furthermore, assume L(x, u) is strictly convex in x for all u. Then SSSGD converges in the sense of zk a.s. \u2192z\u22c6where z\u22c6is a saddle point of L. 3 \fWe can alternatively assume L(x, u) is strictly concave in u for all x and obtain the same result. The proof uses the stochastic approximation technique of [16]. We show that the discrete-time process converges (in an appropriate topology) to a continuous-time trajectory satisfying a differential inclusion and use the LaSalle\u2013Krasnovskii invariance principle to argue that cluster points are solutions. Related prior work. Theorem 3.1 of [36] considers the more general mirror descent setup and proves convergence under the assumption of \u201cstrict coherence\u201d, which is analogous to the stronger assumption of strict convex-concavity in both x and u. 3 Simultaneous GD with optimism Consider the setup where L is continuously differentiable and we access full (deterministic) gradients G(x, u) = \u0014 \u2207xL(x, u) \u2212\u2207uL(x, u) \u0015 . Consider Optimistic Simultaneous Gradient Descent zk+1 = zk \u2212\u03b1G(zk) \u2212\u03b2(G(zk) \u2212G(zk\u22121)) (SimGD-O) for k \u22650, where z0 \u2208Rm+n is a starting point, z\u22121 = z0, \u03b1 > 0 is learning rate, and \u03b2 > 0 is the optimism rate. Optimism is a modi\ufb01cation to simGD that remedies the cycling behavior; for the bilinear example L0 of (2), simGD (case \u03b2 = 0) diverges while SimGD-O with appropriate \u03b2 > 0 converges. In this section, we provide a continuous-time interpretation of SimGD-O as a regularized dynamics and provide convergence for the deterministic setup. 3.1 Continuous-time illustration Consider the continuous-time dynamics \u02d9 z(t) = \u2212\u03b1g(t) \u2212\u03b2 \u02d9 g(t), g(t) = G(z(t)). The discretization \u02d9 z(t) \u2248zk+1 \u2212zk and \u02d9 g(t) \u2248G(zk) \u2212G(zk\u22121) yields SimGD-O. We discuss how this system arises as a certain regularized dynamics and derive the convergence rate \u2225g(t)\u22252 \u2264O(1/t). Regularized gradient mapping. The Moreau\u2013Yosida [43, 70] regularization of G with parameter \u03b2 > 0 is G\u03b2 = \u03b2\u22121(I \u2212(I + \u03b2G)\u22121). To clarify, I : Rm+n \u2192Rm+n is the identity mapping and (I + \u03b2G)\u22121 is the inverse (as a function) of I + \u03b2G, which is well-de\ufb01ned by Minty\u2019s theorem [40]. It is straightforward to verify that G\u03b2(z) = 0 if and only if G(z) = 0, i.e., G\u03b2 and G share the same equilibrium points. For small \u03b2, we can think of G\u03b2 as an approximation G that is better-behaved. Speci\ufb01cally, G is merely monotone (satis\ufb01es (1)), but G\u03b2 is furthermore \u03b2-cocoercive, i.e., (G\u03b2(z1) \u2212G\u03b2(z2))T (z1 \u2212z2) \u2265\u03b2\u2225G\u03b2(z1) \u2212G \u03b2 (z2)\u22252 \u2200z1, z2 \u2208Rm+n. (3) Regularized dynamics. Consider the regularized dynamics \u02d9 \u03b6(t) = \u2212\u03b1G\u03b2(\u03b6(t)). Reparameterize the dynamics \u02d9 \u03b6(t) = \u2212\u03b1G\u03b2(\u03b6(t)) with z(t) = (I + \u03b2G)\u22121(\u03b6(t)) and g(t) = G(z(t)) to get \u03b6(t) = z(t) + \u03b2g(t) and \u02d9 z(t) + \u03b2 \u02d9 g(t) = \u02d9 \u03b6(t) = \u2212\u03b1 \u03b2 (\u03b6(t) \u2212z(t)) = \u2212\u03b1g(t). This gives us \u02d9 z(t) = \u2212\u03b1g(t) \u2212\u03b2 \u02d9 g(t). 4 \fRate of convergence. We now derive a rate of convergence. Let z\u22c6satisfy G(z\u22c6) = 0 (and therefore G\u03b2(z\u22c6) = 0). Then d dt 1 2\u2225\u03b6(t) \u2212z\u22c6\u22252 = (\u03b6(t) \u2212z\u22c6)T \u02d9 \u03b6(t) = \u2212\u03b1(\u03b6(t) \u2212z\u22c6)T G\u03b2(\u03b6(t)) \u2264\u2212\u03b1\u03b2\u2225G\u03b2(\u03b6(t))\u22252, where we use cocoercivity, (3). This translates to d dt 1 2\u2225z(t) + \u03b2g(t) \u2212z\u22c6\u22252 \u2264\u2212\u03b1\u03b2\u2225g(t)\u22252. (4) The quantity \u2225g(t)\u22252 is nonincreasing since d dt 1 2\u2225g(t)\u22252 = \u22121 \u03b1 \u02d9 \u03b6(t)T \u02d9 g(t) = \u22121 \u03b1 lim h\u21920 1 h2 (\u03b6(t + h) \u2212\u03b6(t))T (G\u03b2(\u03b6(t + h)) \u2212G\u03b2(\u03b6(t))) \u2264\u2212\u03b2 \u03b1 lim h\u21920 1 h2 \u2225G\u03b2(\u03b6(t + h)) \u2212G\u03b2(\u03b6(t))\u2225 = \u2212\u03b2 \u03b1\u2225\u02d9 g(t)\u22252 \u22640, where we use cocoercivity, (3). Finally, integrating (4) on both sides gives us 1 2\u2225z(t) + \u03b2g(t) \u2212z\u22c6\u22252 \u22121 2\u2225z(0) + \u03b2g(0) \u2212z\u22c6\u22252 \u2264\u2212\u03b1\u03b2 Z t 0 \u2225g(s)\u22252 ds \u2264\u2212\u03b1\u03b2t\u2225g(t)\u22252 \u2225g(t)\u22252 \u2264 1 2\u03b1\u03b2t\u2225z(0) + \u03b2g(0) \u2212z\u22c6\u22252. Related prior work. Attouch et al. [3, 4] \ufb01rst used the Moreau\u2013Yosida regularization in continuoustime dynamics (but not for analyzing optimism) and [10] interpreted a forward-backward-forwardtype method as a discretization of continuous-time dynamics with the Douglas\u2013Rachford operator. [11] interprets optimism as augmenting \u201cfollow the regularized leader\u201d with the (optimistic) prediction that the next gradient will be the same as the current gradient in online learning setup. [49] interprets optimism as \u201ccentripetal acceleration\u201d but does not provide a formal analysis with differential equations. 3.2 Discrete-time convergenece analysis The discrete-time method SimGD-O converges under the assumption L is differentiable and \u2207L is R-Lipschitz continuous. (A3) Theorem 2. Assume (A0) and (A3). If 0 < \u03b1 < 2\u03b2(1 \u22122\u03b2R), then SimGD-O converges in the sense of min i=0,...,k \u2225G(zk)\u22252 \u2264 2 + 2\u03b22R2 \u03b1(2\u03b2 \u2212\u03b1 \u22124\u03b22R)k \u2225z0 + \u03b2G(z0) \u2212z\u22c6\u22252. Furthermore, zk \u2192z\u22c6, where z\u22c6is a saddle point of L. The proof can be considered a discretization of the continuous-time analysis. Corollary 1. In the setup of Theorem 2, the choice \u03b1 = 1/(8R) and \u03b2 = 2\u03b1 yields min i=0,...,k \u2225G(zk)\u22252 \u2264136R2 k \u2225z0 + \u03b2G(z0) \u2212z\u22c6\u22252 \u2264289R2 k \u2225z0 \u2212z\u22c6\u22252 . The parameter choices of Corollary 1 are almost optimal. The optimal choices that minimize the bound of Theorem 2 are \u03b1 = 0.124897/R and \u03b2 = 1.94431\u03b1; they provide a factor of 135.771, a very small improvement over the factor 136. 5 \fFigure 2: Plot of \u2225zk \u2212z\u22c6\u22252 vs. iteration count for simGD-OS (left) and SSSGD-A (right) with \u03b1k = 1/kp and \u03b2k = 1/kq. We use L0 of (2) and Gaussian random noise. The shaded region denotes \u00b1 standard error. For simGD-OS, we see that neither q = 0 nor q = p leads to convergence. Rather, q must satisfy 0 < q < p so that the learning rate diminishes faster than the optimism rate. Continuous vs. discrete analysis. The discrete-time analysis of SimGD-O of Theorem 2 bounds the squared gradient norm of the best iterate, while the continuous-time analysis bounds the squared gradient norm of the \u201clast iterate\u201d (at terminal time). The discrepancy comes from the fact that while we have monotonic decrease of \u2225g(t)\u2225in continuous-time, we have no analogous monotonicity condition on \u2225gk\u2225in discrete-time. To the best of our knowledge, there is no result establishing a O(1/k) rate on the squared gradient norm of the last iterate for SimGD-O or the related \u201cextragradient method\u201d [27]. Theorem 3 is the \ufb01rst result showing a rate close to O(1/k) on the last literate. Related prior work. [49] show convergence of simGD-O for \u03b1 \u0338= \u03b2 and bilinear L. [34] and [10] show convergence of simGD-O for \u03b1 = \u03b2 and convex-concave L. Theorem 2 establishes convergence for \u03b1 \u0338= \u03b2 and convex-concave L and presents an explicit rate. 3.3 Dif\ufb01culty with stochastic gradients Training in machine learning usually relies on stochastic gradients, rather than full gradients. We can consider a stochastic variation of SimGD-O: zk+1 = zk \u2212\u03b1kg(zk; \u03c9k) \u2212\u03b2k(g(zk; \u03c9k) \u2212g(zk\u22121; \u03c9k\u22121)) (SimGD-OS) with learning rate \u03b1k and optimism rate \u03b2k. Figure 2 presents experiments of SimGD-OS on a simple bilinear problem. The choice \u03b2k = \u03b1k where \u03b1k \u21920 does not lead to convergence. Discretizing \u02d9 z(t) = \u2212\u03b1g(t)\u2212\u03b2 \u02d9 g(t) with a diminishing step hk leads to the choice \u03b1k = \u03b1hk and \u03b2k = \u03b2, but this choice does not lead to convergence either. Rather, it is necessary to tune \u03b1k and \u03b2k separately as in Theorem 2 to obtain convergence and dynamics appear to be sensitive to the choice of \u03b1k and \u03b2k. In particular, both \u03b1k and \u03b2k must diminish and \u03b1k must diminish faster than \u03b2k. One explanation of this dif\ufb01culty is that the \ufb01nite difference approximation \u03b1\u22121 k (g(zk; \u03c9k) \u2212g(zk\u22121; \u03c9k\u22121)) \u2248\u02d9 g(t) is unreliable when using stochastic gradients. Whether the observed convergence holds generally in the nonlinear convex-concave setup and whether optimism is compatible with subgradients is unclear. This motivates anchoring of the following section which is provably compatible with stochastic subgradients. Related prior work. [18] show averaged iterates of SimGD-OS converge if iterates are projected onto a compact set. [36] show almost sure convergence of SimGD-OS under strict convex-concavity (and more generally under \u201cstrict coherence\u201d). However, such analyses do not provide a compelling reason to use optimism since SimGD without optimism already converges under these setups. 4 Simultaneous GD with anchoring Consider setup of Section 3. We propose Anchored Simultaneous Gradient Descent zk+1 = zk \u2212 1 \u2212p (k + 1)p G(zk) + (1 \u2212p)\u03b3 k + 1 (z0 \u2212zk) (SimGD-A) 6 \ffor k \u22650, where z0 \u2208Rm+n is a starting point, p \u2208(1/2, 1), and \u03b3 > 0 is the anchor rate. In this section, we provide a continuous-time illustration of SimGD-A and provide convergence for both the deterministic and stochastic setups. 4.1 Continuous-time illustration Consider the continuous-time dynamics \u02d9 z(t) = \u2212g(t) + \u03b3 t (z0 \u2212z(t)), g(t) = G(z(t)). for t \u22650, where \u03b3 \u22651 and z(0) = z0. We will derive the convergence rate \u2225g(t)\u22252 \u2264O(1/t2). Discretizing the continuous-time ODE with diminishing steps (1 \u2212p)/(k + 1)p leads to SimGD-A. Rate of convergence. First note 0 \u22641 h2 \u27e8z(t + h) \u2212z(t), g(t + h) \u2212g(t)\u27e9\u2192\u27e8\u02d9 z(t), \u02d9 g(t)\u27e9 as h \u21920. Using this, we have d dt 1 2 \u2225\u02d9 z(t)\u22252 = \u2212 D \u02d9 z(t), \u02d9 g(t) + \u03b3 t \u02d9 z(t) + \u03b3 t2 (z0 \u2212z(t)) E = \u2212\u27e8\u02d9 z(t), \u02d9 g(t)\u27e9\u2212\u03b3 t \u2225\u02d9 z(t)\u22252 + \u03b3 t2 \u27e8z(t) \u2212z0, \u02d9 z\u27e9 \u2264\u2212\u03b3 t \u2225\u02d9 z(t)\u22252 + \u03b3 t2 \u27e8z(t) \u2212z0, \u02d9 z\u27e9. Using \u03b3 \u22651, we have d dt 1 2 \u2225\u02d9 z(t)\u22252 + 1 t \u2225\u02d9 z(t)\u22252 \u2264\u03b3 t2 \u27e8z(t) \u2212z0, \u02d9 z\u27e9. Multiplying by t2 and integrating both sides gives us t2 2 \u2225\u02d9 z(t)\u22252 \u2264\u03b3 2 \u2225z(t) \u2212z0\u22252. Reorganizing, we get t2 2 \u2225g(t)\u22252 \u2212\u03b3t\u27e8g(t), z0 \u2212z(t)\u27e9+ \u03b32 2 \u2225z(t) \u2212z0\u22252 \u2264\u03b3 2 \u2225z(t) \u2212z0\u22252 Using \u03b3 \u22651, the monotonicity inequality, and Young\u2019s inequality, we get \u2225g(t)\u22252 \u22642\u03b3 t \u27e8g(t), z0 \u2212z(t)\u27e9\u22642\u03b3 t \u27e8g(t), z0 \u2212z\u22c6\u27e9 \u22641 2\u2225g(t)\u22252 + 2\u03b32 t2 \u2225z0 \u2212z\u22c6\u22252 and conclude \u2225g(t)\u22252 \u22644\u03b32 t2 \u2225z0 \u2212z\u22c6\u22252. Interestingly, anchoring leads to a faster rate O(1/t2) compared to the rate O(1/t) of optimism in continuous time. The discretized method, however, is not faster than O(1/k). We further discuss this difference later. 7 \fAnother interpretation: Nonpositive \u201cLyapunov\u201d function. We reorganize the ODE analysis to resemble the Su\u2013Boyd\u2013Candes Lyapunov analysis of Nesterov acceleration [62]. De\ufb01ne V (t) = t2\u2225g(t)\u22252 + 2\u03b3t\u27e8g(t), z(t) \u2212z0\u27e9+ \u03b3(\u03b3 \u22121)\u2225z(t) \u2212z0\u22252. Taking the derivative and plugging in \u02d9 z(t) = \u2212g(t) + \u03b3 t (z0 \u2212z(t)), we get \u02d9 V (t) = \u22122t2\u27e8\u02d9 g(t), \u02d9 z(t)\u27e9\u22122t(\u03b3 \u22121)\u2225\u02d9 z(t)\u22252 \u2264\u22122t(\u03b3 \u22121)\u2225\u02d9 z(t)\u22252 \u22640, where the equality of the \ufb01rst line can be veri\ufb01ed through straightforward, albeit somewhat tedious, calculations and the inequality of the second line follows from monotonicity as discussed in the main analysis. Note that V (0) = 0. Since V (t) is monotonically nonincreasing, V (t) is a nonincreasing nonpositive \u201cLyapunov\u201d function. Finally, we translate the bound V (t) \u22640 into a bound on the size of the gradient: 0 \u2265V (0) \u2265V (t) \u2265t2\u2225g(t)\u22252 + 2\u03b3t\u27e8g(t), z(t) \u2212z0\u27e9 \u2265t2\u2225g(t)\u22252 + 2\u03b3t\u27e8g(t), z\u22c6\u2212z0\u27e9 \u2265t2\u2225g(t)\u22252 \u2212t2 2 \u2225g(t)\u22252 \u2212\u03b32 2 \u2225z\u22c6\u2212z0\u22252 \u2265t2 2 \u2225g(t)\u22252 \u2212\u03b32 2 \u2225z\u22c6\u2212z0\u22252, where the third and fourth lines follow from the monotonicity and Young\u2019s inequality. Reorganizing, we obtain 4\u03b32 t2 \u2225z\u22c6\u2212z0\u22252 \u2265\u2225g(t)\u22252. V (t) is a somewhat unusual \u201cLyapunov\u201d function. Often, Lyapunov functions are nonnegative and are interpreted as the \u201cenergy\u201d of the system. In our analysis, V (t) is nonpositive and it is unclear if an insightful physical analogy can be drawn. Related prior work. Anchoring was inspired by Halpern\u2019s method [23, 67, 31] and James\u2013Stein estimator [61, 25]; these methods pull/shrink the iterates/estimator towards a speci\ufb01ed point z0. 4.2 Discrete-time convergenece analysis We now present convergence results with anchoring. In Theorem 3, we use deterministic gradients, and in Theorem 4, we use stochastic subgradients. Theorem 3. Assume (A0) and (A3). If p \u2208(1/2, 1) and \u03b3 \u22652, then SimGD-A converges in the sense of \u2225G(zk)\u22252 \u2264O \u0012 1 k2\u22122p \u0013 . The proof can be considered a discretization of the continuous-time analysis. Consider the setup of Section 2. We propose Anchored Simultaneous Stochastic SubGradient Descent zk+1 = zk \u2212 1 \u2212p (k + 1)p g(zk; \u03c9k) + (1 \u2212p)\u03b3 (k + 1)1\u2212\u03b5 (z0 \u2212zk) (SSSGD-A) (The small \u03b5 > 0 is introduced for the proof of Theorem 4.) Theorem 4. Assume (A0) and (A2). If p \u2208(1/2, 1), \u03b5 \u2208(0, 1/2), and \u03b3 > 0, then SSSGD-A converges in the sense of zk L2 \u2192z\u22c6, where z\u22c6is a saddle point. (To clarify, we do not assume L is differentiable.) 8 \fFurther discussion. There is a discrepancy in the rate between the continuous time analysis O(1/t2) and Thoerem 3\u2019s discrete time rate O(1/k2\u22122p) for p \u2208(1/2, 1), which is slightly slower than O(1/k). In discretizing the continuous-time calculations to obtain a discrete proof, errors accumulate and prevent the rate from being better than O(1/k). This is not an artifact of the proof. Simple tests on bilinear examples show divergence when p < 1/2. The small \u03b5 > 0 in Theorem 4 is introduced to make the proof work, we believe this is an artifact of the proof. In particular, we conjecture that Lemma 17 holds with o(s/\u03c4) rather than O(s/\u03c4), and, if so, it is possible to establish convergence with \u03b5 = 0. In Figure 2, it seems that that the choice \u03b5 = 0 and p = 2/3 is optimal for SSSGD-A. While we do not have a theoretical explanation for this, we point out that this is not surprising as p = 2/3 is known to be optimal in stochastic convex minimization [44, 64]. 5" + }, + { + "url": "http://arxiv.org/abs/1905.05406v1", + "title": "Plug-and-Play Methods Provably Converge with Properly Trained Denoisers", + "abstract": "Plug-and-play (PnP) is a non-convex framework that integrates modern\ndenoising priors, such as BM3D or deep learning-based denoisers, into ADMM or\nother proximal algorithms. An advantage of PnP is that one can use pre-trained\ndenoisers when there is not sufficient data for end-to-end training. Although\nPnP has been recently studied extensively with great empirical success,\ntheoretical analysis addressing even the most basic question of convergence has\nbeen insufficient. In this paper, we theoretically establish convergence of\nPnP-FBS and PnP-ADMM, without using diminishing stepsizes, under a certain\nLipschitz condition on the denoisers. We then propose real spectral\nnormalization, a technique for training deep learning-based denoisers to\nsatisfy the proposed Lipschitz condition. Finally, we present experimental\nresults validating the theory.", + "authors": "Ernest K. Ryu, Jialin Liu, Sicheng Wang, Xiaohan Chen, Zhangyang Wang, Wotao Yin", + "published": "2019-05-14", + "updated": "2019-05-14", + "primary_cat": "cs.CV", + "cats": [ + "cs.CV", + "eess.IV" + ], + "main_content": "Introduction Many modern image processing algorithms recover or denoise an image through the optimization problem minimize x\u2208Rd f(x) + \u03b3g(x), where the optimization variable x \u2208Rd represents the image, f(x) measures data \ufb01delity, g(x) measures noisiness or complexity of the image, and \u03b3 \u22650 is a parameter representing the relative importance between f and g. Total variation denoising, inpainting, and compressed sensing fall under this setup. A priori knowledge of the image, such as that the image should have small noise, is encoded in g(x). So g(x) is small if x has small noise or complexity. A posteriori knowledge of the image, such as noisy or partial 1Department of Mathematics, University of California, Los Angeles, USA 2Department of Computer Science and Engineering, Texas A&M University, USA. Correspondence to: Wotao Yin . Proceedings of the 36 th International Conference on Machine Learning, Long Beach, California, PMLR 97, 2019. Copyright 2019 by the author(s). measurements of the image, is encoded in f(x). So f(x) is small if x agrees with the measurements. First-order iterative methods are often used to solve such optimization problems, and ADMM is one such method: xk+1 = argmin x\u2208Rd \b \u03c32g(x) + (1/2)\u2225x \u2212(yk \u2212uk)\u22252\t yk+1 = argmin y\u2208Rd \b \u03b1f(y) + (1/2)\u2225y \u2212(xk+1 + uk)\u22252\t uk+1 = uk + xk+1 \u2212yk+1 with \u03c32 = \u03b1\u03b3. Given a function h on Rd and \u03b1 > 0, de\ufb01ne the proximal operator of h as Prox\u03b1h(z) = argmin x\u2208Rd \b \u03b1h(x) + (1/2)\u2225x \u2212z\u22252\t , which is well-de\ufb01ned if h is proper, closed, and convex. Now we can equivalently write ADMM as xk+1 = Prox\u03c32g(yk \u2212uk) yk+1 = Prox\u03b1f(xk+1 + uk) uk+1 = uk + xk+1 \u2212yk+1. We can interpret the subroutine Prox\u03c32g : Rd \u2192Rd as a denoiser, i.e., Prox\u03c32g : noisy image 7\u2192less noisy image (For example, if \u03c3 is the noise level and g(x) is the total variation (TV) norm, then Prox\u03c32g is the standard Rudin\u2013 Osher\u2013Fatemi (ROF) model (Rudin et al., 1992).) We can think of Prox\u03b1f : Rd \u2192Rd as a mapping enforcing consistency with measured data, i.e., Prox\u03b1f : less consistent 7\u2192more consistent with data More precisely speaking, for any x \u2208Rd we have g(Prox\u03c32g(x)) \u2264g(x), f(Prox\u03b1f(x)) \u2264f(x). However, some state-of-the-art image denoisers with great empirical performance do not originate from optimization problems. Such examples include non-local means (NLM) (Buades et al., 2005), Block-matching and 3D \ufb01ltering (BM3D) (Dabov et al., 2007), and convolutional neural arXiv:1905.05406v1 [cs.CV] 14 May 2019 \fPlug-and-Play Methods Provably Converge with Properly Trained Denoisers networks (CNN) (Zhang et al., 2017a). Nevertheless, such a denoiser H\u03c3 : Rd \u2192Rd still has the interpretation H\u03c3 : noisy image 7\u2192less noisy image where \u03c3 \u22650 is a noise parameter. Larger values of \u03c3 correspond to more aggressive denoising. Is it possible to use such denoisers for a broader range of imaging problems, even though we cannot directly set up an optimization problem? To address this question, (Venkatakrishnan et al., 2013) proposed Plug-and-Play ADMM (PnPADMM), which simply replaces the proximal operator Prox\u03c32g with the denoiser H\u03c3: xk+1 = H\u03c3(yk \u2212uk) yk+1 = Prox\u03b1f(xk+1 + uk) uk+1 = uk + xk+1 \u2212yk+1. Surprisingly and remarkably, this ad-hoc method exhibited great empirical success, and spurred much follow-up work. Contribution of this paper. The empirical success of Plug-and-Play (PnP) naturally leads us to ask theoretical questions: When does PnP converge and what denoisers can we use? Past theoretical analysis has been insuf\ufb01cient. The main contribution of this work is the convergence analyses of PnP methods. We study two Plug-and-play methods, Plug-and-play forward-backward splitting (PNP-FBS) and PNP-ADMM. For the analysis, we assume the denoiser H\u03c3 satis\ufb01es a certain Lipschitz condition, formally de\ufb01ned as Assumption (A). Roughly speaking, the condition corresponds to the denoiser H\u03c3 being close to the identity map, which is reasonable when the denoising parameter \u03c3 is small. In particular, we do not assume that H\u03c3 is nonexpansive or differentiable since most denoisers do not have such properties. Under the assumption, we show that the PnP methods are contractive iterations. We then propose real spectral normalization (realSN), a technique based on (Miyato et al., 2018) for more accurately constraining deep learning-based denoisers in their training to satisfy the proposed Lipschitz condition. Finally, we present experimental results validating our theory. Code used for experiments is available at: https://github. com/uclaopt/Provable_Plug_and_Play/ 1.1. Prior work Plug-and-play: Practice. The \ufb01rst PnP method was the Plug-and-play ADMM proposed in (Venkatakrishnan et al., 2013). Since then, other schemes such as the primal-dual method (Heide et al., 2014; Meinhardt et al., 2017; Ono, 2017), ADMM with increasing penalty parameter (Brifman et al., 2016; Chan et al., 2017), generalized approximate message passing (Metzler et al., 2016), Newton iteration (Buzzard et al., 2018), Fast Iterative ShrinkageThresholding Algorithm (Kamilov et al., 2017; Sun et al., 2018b), (stochastic) forward-backward splitting (Sun et al., 2019; 2018a;b), and alternating minimization (Dong et al., 2018) have been combined with the PnP technique. PnP method reported empirical success on a large variety of imaging applications: bright \ufb01eld electron tomography (Sreehari et al., 2016), camera image processing (Heide et al., 2014), compression-artifact reduction (Dar et al., 2016), compressive imaging (Teodoro et al., 2016), deblurring (Teodoro et al., 2016; Rond et al., 2016; Wang & Chan, 2017), electron microscopy (Sreehari et al., 2017), Gaussian denoising (Buzzard et al., 2018; Dong et al., 2018), nonlinear inverse scattering (Kamilov et al., 2017), Poisson denoising (Rond et al., 2016), single-photon imaging (Chan et al., 2017), super-resolution (Brifman et al., 2016; Sreehari et al., 2016; Chan et al., 2017), diffraction tomography (Sun et al., 2019), Fourier ptychographic microscopy (Sun et al., 2018b), low-dose CT imaging (Venkatakrishnan et al., 2013; He et al., 2018; Ye et al., 2018; Lyu et al., 2019), hyperspectral sharpening (Teodoro et al., 2017; 2019), inpainting (Chan, 2019; Tirer & Giryes, 2019), and superresolution (Dong et al., 2018). A wide range of denoisers have been used for the PnP framework. BM3D has been used the most (Heide et al., 2014; Dar et al., 2016; Rond et al., 2016; Sreehari et al., 2016; Chan et al., 2017; Kamilov et al., 2017; Ono, 2017; Wang & Chan, 2017), but other denoisers such as sparse representation (Brifman et al., 2016), non-local means (Venkatakrishnan et al., 2013; Heide et al., 2014; Sreehari et al., 2016; 2017; Chan, 2019), Gaussian mixture model (Teodoro et al., 2016; 2017; Shi & Feng, 2018; Teodoro et al., 2019), Patch-based Wiener \ufb01ltering (Venkatakrishnan et al., 2013), nuclear norm minimization (Kamilov et al., 2017), deep learningbased denoisers (Meinhardt et al., 2017; He et al., 2018; Ye et al., 2018; Tirer & Giryes, 2019) and deep projection model based on generative adversarial networks (Chang et al., 2017) have also been considered. Plug-and-play: Theory. Compared to the empirical success, much less progress was made on the theoretical aspects of PnP optimization. (Chan et al., 2017) analyzed convergence with a bounded denoiser assumption, establishing convergence using an increasing penalty parameter. (Buzzard et al., 2018) provided an interpretation of \ufb01xed points via \u201cconsensus equilibrium\u201d. (Sreehari et al., 2016; Sun et al., 2019; Teodoro et al., 2017; Chan, 2019; Teodoro et al., 2019) proved convergence of PNP-ADMM and PNPFBS with the assumption that the denoiser is (averaged) nonexpansive by viewing the methods to be \ufb01xed-point iterations. The nonexpansiveness assumption is not met with most denoisers as is, but (Chan, 2019) proposed modi\ufb01cations to the non-local means and Gaussian mixture model denoisers, which make them into linear \ufb01lters, to enforce \fPlug-and-Play Methods Provably Converge with Properly Trained Denoisers nonexpansiveness. (Dong et al., 2018) presented a proof that relies on the existence of a certain Lyapunov function that is monotonic under H\u03c3, which holds only for simple H\u03c3. (Tirer & Giryes, 2019) analyzed a variant of PnP, but did not establish local convergence since their key assumption is only expected to be satis\ufb01ed \u201cin early iterations\u201d. Other PnP-type methods. There are other lines of works that incorporate modern denoisers into model-based optimization methods. The plug-in idea with half quadratic splitting, as opposed to ADMM, was discussed (Zoran & Weiss, 2011) and this approach was carried out with deep learningbased denoisers in (Zhang et al., 2017b). (Danielyan et al., 2012; Egiazarian & Katkovnik, 2015) use the notion of Nash equilibrium to propose a scheme similar to PnP. (Danielyan et al., 2010) proposed an augmented Lagrangian method similar to PnP. (Romano et al., 2017; Reehorst & Schniter, 2019) presented Regularization by Denoising (RED), which uses the (nonconvex) regularizer xT (x \u2212H\u03c3(x)) given a denoiser H\u03c3, and use denoiser evaluations in its iterations. (Fletcher et al., 2018) applies the plug-in approach to vector approximate message passing. (Yang et al., 2016; Fan et al., 2019) replaced both the proximal operator enforcing data \ufb01delity and the denoiser with two neural networks and performed end-to-end training. Broadly, there are more works that incorporate model-based optimization with deep learning (Chen et al., 2018; Liu et al., 2019). Image denoising using deep learning. Deep learningbased denoising methods have become state-of-the-art. (Zhang et al., 2017a) proposed an effective denoising network called DnCNN, which adopted batch normalization (Ioffe & Szegedy, 2015) and ReLU (Krizhevsky et al., 2012) into the residual learning (He et al., 2016). Other represenative deep denoising models include the deep convolutional encoder-decoder with symmetric skip connection (Mao et al., 2016), N 3Net (Pl\u00a8 otz & Roth, 2018), and MWCNN (Liu et al., 2018). The recent FFDNet (Zhang et al., 2018) handles spatially varying Gaussian noise. Regularizing Lipschitz continuity. Lipschitz continuity and its variants have started to receive attention as a means for regularizing deep classi\ufb01ers (Bartlett et al., 2017; Bansal et al., 2018; Oberman & Calder, 2018) and GANs (Miyato et al., 2018; Brock et al., 2019). Regularizing Lipschitz continuity stabilizes training, improves the \ufb01nal performance, and enhances robustness to adversarial attacks (Weng et al., 2018; Qian & Wegman, 2019). Speci\ufb01cally, (Miyato et al., 2018) proposed to normalize all weights to be of unit spectral norms to thereby constrain the Lipschitz constant of the overall network to be no more than one. 2. PNP-FBS/ADMM and their \ufb01xed points We now present the PnP methods we investigate in this work. We quickly note that although PNP-FBS and PNP-ADMM are distinct methods, they share the same \ufb01xed points by Remark 3.1 of (Meinhardt et al., 2017) and Proposition 3 of (Sun et al., 2019). We call the method xk+1 = H\u03c3(I \u2212\u03b1\u2207f)(xk) (PNP-FBS) for any \u03b1 > 0, plug-and-play forward-backward splitting (PNP-FBS) or plug-and-play proximal gradient method. We interpret PNP-FBS as a \ufb01xed-point iteration, and we say x\u22c6is a \ufb01xed point of PNP-FBS if x\u22c6= H\u03c3(I \u2212\u03b1\u2207f)(x\u22c6). Fixed points of PNP-FBS have a simple, albeit non-rigorous, interpretation. An image denoising algorithm must trade off the two goals of making the image agree with measurements and making the image less noisy. PNP-FBS applies I\u2212\u03b1\u2207f and H\u03c3, each promoting such objectives, repeatedly in an alternating fashion. If PNP-FBS converges to a \ufb01xed point, we can expect the limit to represent a compromise. We call the method xk+1 = H\u03c3(yk \u2212uk) yk+1 = Prox\u03b1f(xk+1 + uk) (PNP-ADMM) uk+1 = uk + xk+1 \u2212yk+1 for any \u03b1 > 0, plug-and-play alternating directions method of multipliers (PNP-ADMM). We interpret PNP-ADMM as a \ufb01xed-point iteration, and we say (x\u22c6, u\u22c6) is a \ufb01xed point of PNP-ADMM if x\u22c6= H\u03c3(x\u22c6\u2212u\u22c6) x\u22c6= Prox\u03b1f(x\u22c6+ u\u22c6). If we let yk = x\u22c6and uk = u\u22c6in (PNP-ADMM), then we get xk+1 = yk+1 = x\u22c6and uk+1 = uk = u\u22c6. We call the method xk+1/2 = Prox\u03b1f(zk) xk+1 = H\u03c3(2xk+1/2 \u2212zk) (PNP-DRS) zk+1 = zk + xk+1 \u2212xk+1/2 plug-and-play Douglas\u2013Rachford splitting (PNP-DRS). We interpret PNP-DRS as a \ufb01xed-point iteration, and we say z\u22c6 is a \ufb01xed point of PNP-DRS if x\u22c6= Prox\u03b1f(z\u22c6) x\u22c6= H\u03c3(2x\u22c6\u2212z\u22c6). PNP-ADMM and PNP-DRS are equivalent. Although this is not surprising as the equivalence between convex ADMM and DRS is well known, we show the steps establishing equivalence in the supplementary document. \fPlug-and-Play Methods Provably Converge with Properly Trained Denoisers We introduce PNP-DRS as an analytical tool for analyzing PNP-ADMM. It is straightforward to verify that PNP-DRS can be written as zk+1 = T(zk), where T = 1 2I + 1 2(2H\u03c3 \u2212I)(2Prox\u03b1f \u2212I). We use this form to analyze the convergence of PNP-DRS and translate the result to PNP-ADMM. 3. Convergence via contraction We now present conditions that ensure the PnP methods are contractive and thereby convergent. If we assume 2H\u03c3 \u2212I is nonexpansive, standard tools of monotone operator theory tell us that PnP-ADMM converges. However, this assumption is too strong. Chan et al. presented a counter example demonstrating that 2H\u03c3 \u2212I is not nonexpansive for the NLM denoiser (Chan et al., 2017). Rather, we assume H\u03c3 : Rd \u2192Rd satis\ufb01es \u2225(H\u03c3 \u2212I)(x) \u2212(H\u03c3 \u2212I)(y)\u22252 \u2264\u03b52\u2225x \u2212y\u22252 (A) for all x, y \u2208Rd for some \u03b5 \u22650. Since \u03c3 controls the strength of the denoising, we can expect H\u03c3 to be close to identity for small \u03c3. If so , Assumption (A) is reasonable. Under this assumption, we show that the PNP-FBS and PNP-DRS iterations are contractive in the sense that we can express the iterations as xk+1 = T(xk), where T : Rd \u2192Rd satis\ufb01es \u2225T(x) \u2212T(y)\u2225\u2264\u03b4\u2225x \u2212y\u2225 for all x, y \u2208Rd for some \u03b4 < 1. We call \u03b4 the contraction factor. If x\u22c6satis\ufb01es T(x\u22c6) = x\u22c6, i.e., x\u22c6is a \ufb01xed point, then xk \u2192x\u22c6geometrically by the classical Banach contraction principle. Theorem 1 (Convergence of PNP-FBS). Assume H\u03c3 satis\ufb01es assumption (A) for some \u03b5 \u22650. Assume f is \u00b5-strongly convex, f is differentiable, and \u2207f is L-Lipschitz. Then T = H\u03c3(I \u2212\u03b1\u2207f) satis\ufb01es \u2225T(x)\u2212T(y)\u2225\u2264max{|1\u2212\u03b1\u00b5|, |1\u2212\u03b1L|}(1+\u03b5)\u2225x\u2212y\u2225 for all x, y \u2208Rd. The coef\ufb01cient is less than 1 if 1 \u00b5(1 + 1/\u03b5) < \u03b1 < 2 L \u2212 1 L(1 + 1/\u03b5). Such an \u03b1 exists if \u03b5 < 2\u00b5/(L \u2212\u00b5). Theorem 2 (Convergence of PNP-DRS). Assume H\u03c3 satis\ufb01es assumption (A) for some \u03b5 \u22650. Assume f is \u00b5-strongly convex and differentiable. Then T = 1 2I + 1 2(2H\u03c3 \u2212I)(2Prox\u03b1f \u2212I) satis\ufb01es \u2225T(x) \u2212T(y)\u2225\u22641 + \u03b5 + \u03b5\u03b1\u00b5 + 2\u03b52\u03b1\u00b5 1 + \u03b1\u00b5 + 2\u03b5\u03b1\u00b5 \u2225x \u2212y\u2225 for all x, y \u2208Rd. The coef\ufb01cient is less than 1 if \u03b5 (1 + \u03b5 \u22122\u03b52)\u00b5 < \u03b1, \u03b5 < 1. Corollary 3 (Convergence of PNP-ADMM). Assume H\u03c3 satis\ufb01es assumption (A) for some \u03b5 \u2208[0, 1). Assume f is \u00b5-strongly convex. Then PNP-ADMM converges for \u03b5 (1 + \u03b5 \u22122\u03b52)\u00b5 < \u03b1. Proof. This follows from Theorem 2 and the equivalence of PNP-DRS and PNP-ADMM. For PNP-FBS, we assume f is \u00b5-strongly convex and \u2207f is L-Lipschitz. For PNP-DRS and PNP-ADMM, we assume f is \u00b5-strongly convex. These are standard assumptions that are satis\ufb01ed in application such as image denoising/deblurring and single photon imaging. Strong convexity, however, does exclude a few applications such as compressed sensing, sparse interpolation, and super-resolution. PNP-FBS and PNP-ADMM are distinct methods for \ufb01nding the same set of \ufb01xed points. Sometimes, PNP-FBS is easier to implement since it only requires the computation of \u2207f rather than Prox\u03b1f. On the other hand, PNP-ADMM has better convergence properties as demonstrated theoretically by Theorems 1 and 2 and empirically by our experiments. The proof of Theorem 2 relies on the notion of \u201cnegatively averaged\u201d operators of (Giselsson, 2017). It is straightforward to modify Theorems 1 and 2 to establish local convergence when Assumption (A) holds locally. Theorem 2 can be generalized to the case when f is strongly convex but non-differentiable using the notion of subgradients. Recently, (Fletcher et al., 2018) proved convergence of \u201cplug-and-play\u201d vector approximate message passing, a method similar to ADMM, assuming Lipschitz continuity of the denoiser. Although the method, the proof technique, and the notion of convergence are different from ours, the similarities are noteworthy. 4. Real spectral normalization: enforcing Assumption (A) We now present real spectral normalization, a technique for training denoisers to satisfy Assumption (A) and connect the practical implementations to the theory of Section 3. \fPlug-and-Play Methods Provably Converge with Properly Trained Denoisers 4.1. Deep learning denoisers: SimpleCNN and DnCNN We use a deep denoising model called DnCNN (Zhang et al., 2017a), which learns the residual mapping with a 17-layer CNN and reports state-of-the-art results on natural image denoising. Given a noisy observation y = x + e, where x is the clean image and e is noise, the residual mapping R outputs the noise, i.e., R(y) = e so that y \u2212R(y) is the clean recovery. Learning the residual mapping is a popular approach in deep learning-based image restoration. We also construct a simple convolutional encoder-decoder model for denoising and call it SimpleCNN. SimpleCNN consists of 4 convolutional layers, with ReLU and meansquare-error (MSE) loss and does not utilize any pooling or (batch) normalization. We remark that realSN and the theory of this work is applicable to other deep denoisers. We use SimpleCNN to show that realSN is applicable to any CNN denoiser. 4.2. Lipschitz constrained deep denoising Denote the denoiser (SimpleCNN or DnCNN) as H(y) = y \u2212R(y), where y is the noisy input and R is the residual mapping, i.e., R(y) = y \u2212H(y) = (I \u2212H)(y). Enforcing Assumption (A) is equivalent to constraining the Lipschitz constant of R(y). We propose a variant of the spectral normalization (SN) (Miyato et al., 2018) for this. Spectral normalization. (Miyato et al., 2018) proposed to normalize the spectral norm of each layer-wise weight (with ReLU non-linearity) to one. Provided that we use 1-Lipschitz nonlinearities (such as ReLU), the Lipschitz constant of a layer is upper-bounded by the spectral norm of its weight, and the Lipschitz constant of the full network is bounded by the product of spectral norms of all layers (Gouk et al., 2018). To avoid the prohibitive cost of singular value decomposition (SVD) every iteration, SN approximately computes the largest singular values of weights using a small number of power iterations. Given the weight matrix Wl \u2208Rm\u00d7n of the l-th layer, vectors ul \u2208Rm, vl \u2208Rm are initialized randomly and maintained in the memory to estimate the leading \ufb01rst left and right singular vector of Wl respectively. During each forward pass of the network, SN is applied to all layers 1 \u2264l \u2264L following the two-step routine: 1. Apply one step of the power method to update ul, vl: vl \u2190W T l ul / \u2225W T l ul\u22252, ul \u2190Wlvl / \u2225Wlvl\u22252 2. Normalize Wl with the estimated spectral norm: Wl \u2190Wl/\u03c3(Wl), where \u03c3(Wl) = uT l Wlvl While the basic methodology of SN suits our goal, the SN in (Miyato et al., 2018) uses a convenient but inexact implementation for convolutional layers. A convolutional layer is represented by a four-dimensional kernel Kl of shape (Cout, Cin, h, w), where h, w are kernel\u2019s height and width. SN reshapes Kl into a two-dimensional matrix \u02dc Kl of shape (Cout, Cin \u00d7 h \u00d7 w) and regards \u02dc Kl as the matrix Wl above. This relaxation underestimates the true spectral norm of the convolutional operator (Corollary 1 of (Tsuzuku et al., 2018)) given by \u03c3(Kl) = max x\u0338=0 \u2225Kl \u2217x\u22252/\u2225x\u22252, where x is the input to the convolutional layer and \u2217is the convolutional operator. This issue is not hypothetical. When we trained SimpleCNN with SN, the spectral norms of the layers were 3.01, 2.96, 2.82, and 1.31, i.e., SN failed to control the Lipschitz constant below 1. Real spectral normalization. We propose an improvement to SN for convolutional1 layers, called the real spectral normalization (realSN), to more accurately constrain the network\u2019s Lipschitz constant and thereby enforce Assumption (A). In realSN, we directly consider the convolutional linear operator Kl : RCin\u00d7h\u00d7w \u2192RCout\u00d7h\u00d7w, where h, w are input\u2019s height and width, instead of reshaping the convolution kernel Kl into a matrix. The power iteration also requies the conjugate (transpose) operator K\u2217 l . It can be shown that K\u2217 l is another convolutional operator with a kernel that is a rotated version of the forward convolutional kernel; the \ufb01rst two dimensions are permuted and the last two dimensions are rotated by 180 degrees (Liu et al., 2019). Instead of two vectors ul, vl as in SN, realSN maintains Ul \u2208RCout\u00d7h\u00d7w and Vl \u2208RCin\u00d7h\u00d7w to estimate the leading left and right singular vectors respectively. During each forward pass of the neural network, realSN conducts: 1. Apply one step of the power method with operator Kl: Vl \u2190K\u2217 l (Ul) / \u2225K\u2217 l (Ul)\u22252, Ul \u2190Kl(Vl) / \u2225Kl(Vl)\u22252. 2. Normalize the convolutional kernel Kl with estimated spectral norm: Kl \u2190Kl/\u03c3(Kl), where \u03c3(Kl) = \u27e8Ul, Kl(Vl)\u27e9 By replacing \u03c3(Kl) with \u03c3(Kl)/cl, realSN can constrain the Lipschitz constant to any upper bound C = QL l=1 cl. Using the highly ef\ufb01cient convolution computation in modern deep learning frameworks, realSN can be implemented simply and ef\ufb01ciently. Speci\ufb01cally, realSN introduces three additional one-sample convolution operations for each layer in each training step. When we used a batch size of 128, the extra computational cost of the additional operations is mild. 1We use stride 1 and zero-pad with width 1 for convolutions. \fPlug-and-Play Methods Provably Converge with Properly Trained Denoisers =1.198 0.95 1 1.05 1.1 1.15 1.2 (a) BM3D =0.96 0.86 0.88 0.9 0.92 0.94 0.96 (b) SimpleCNN =0.758 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 (c) RealSN-SimpleCNN =0.484 0.43 0.44 0.45 0.46 0.47 0.48 (d) DnCNN =0.464 0.4 0.41 0.42 0.43 0.44 0.45 0.46 (e) RealSN-DnCNN Figure 1. Histograms for experimentally verifying Assumption (A). The x-axis represents values of \u2225(I\u2212H\u03c3)(x)\u2212(I\u2212H\u03c3)(y)\u2225/\u2225x\u2212y\u2225 and the y-axis represents the frequency. The vertical red bar corresponds to the maximum value. 4.3. Implementation details We refer to SimpleCNN and DnCNN regularized by realSN as RealSN-SimpleCNN and RealSN-DnCNN, respectively. We train them in the setting of Gaussian denoising with known \ufb01xed noise levels \u03c3 = 5, 15, 40. We used \u03c3 = 5, 15 for CS-MRI and single photon imaging, and \u03c3 = 40 for Poisson denoising. The regularized denoisers are trained to have Lipschitz constant (no more than) 1. The training data consists of images from the BSD500 dataset, divided into 40 \u00d7 40 patches. The CNN weights were initialized in the same way as (Zhang et al., 2017a). We train all networks using the ADAM optimizer for 50 epochs, with a mini-batch size of 128. The learning rate was 10\u22123 in the \ufb01rst 25 epochs, then decreased to 10\u22124. On an Nvidia GTX 1080 Ti, DnCNN took 4.08 hours and realSN-DnCNN took 5.17 hours to train, so the added cost of realSN is mild. 5. Poisson denoising: validating the theory Consider the Poisson denoising problem, where given a true image xtrue \u2208Rd, we observe independent Poisson random variables yi \u223cPoisson((xtrue)i), so yi \u2208N, for i = 1, . . . , d. For details and motivation for this problem setup, see (Rond et al., 2016). For the objective function f(x), we use the negative loglikelihood given by f(x) = Pd i=1 \u2113(xi; yi), where \u2113(x; y) = \uf8f1 \uf8f2 \uf8f3 \u2212y log(x) + x for y > 0, x > 0 0 for y = 0, x \u22650 \u221e otherwise. We can compute Prox\u03b1f elementwise with Prox\u03b1f(x) = (1/2) \u0010 x \u2212\u03b1 + p (x \u2212\u03b1)2 + 4\u03b1y \u0011 . The gradient of f is given by \u2202f/\u2202xi = \u2212yi/xi + 1 for xi > 0 for i = 1, . . . , d. We set \u2202f/\u2202xi = 0 when xi = 0, although, strictly speaking, \u2202f/\u2202xi is unde\ufb01ned when yi > 0 and xi = 0. This does not seem to cause any problems in the experiments. Since we force the denoisers to output nonnegative pixel values, PNP-FBS never needs to evaluate \u2202f/\u2202xi for negative xi. For H\u03c3, we choose BM3D, SimpleCNN with and without realSN, and DnCNN with and without realSN. Note that these denoisers are designed or trained for the purpose of Gaussian denoising, and here we integrate them into the PnP frameworks for Poisson denoising. We scale the image so that the peak value of the image, the maximum mean of the Poisson random variables, is 1. The y-variable was initialized to the noisy image for PnP-FBS and PnP-ADMM, and the u-variable was initialized to 0 for PnP-ADMM. We use the test set of 13 images in (Chan et al., 2017). Convergence. We \ufb01rst examine which denoisers satisfy Assumption (A) with small \u03b5. In Figure 1, we run PnP iterations of Poisson denoising on a single image (\ufb02ag of (Rond et al., 2016)) with different models, calculate \u2225(I \u2212 H\u03c3)(x) \u2212(I \u2212H\u03c3)(y)\u2225/\u2225x \u2212y\u2225between the iterates and the limit, and plot the histogram. The maximum value of a histogram, marked by a vertical red bar, lower-bounds the \u03b5 of Assumption (A). Remember that Corollary 3 requires \u03b5 < 1 to ensure convergence of PnP-ADMM. Figure 1(a) proves that BM3D violates this assumption. Figures 1(b) and 1(c) and Figures 1(d) and 1(e) respectively illustrate that RealSN indeed improves (reduces) \u03b5 for SimpleCNN and DnCNN. Figure 2 experimentally validates Theorems 1 and 2, by examining the average (geometric mean) contraction factor (de\ufb01ned in Section 3) of PnP-FBS and ADMM2 iterations over a range of step sizes. Figure 2 qualitatively shows that PnP-ADMM exhibits more stable convergence than PnPFBS. Theorem 1 ensures PnP-FBS is a contraction when \u03b1 is within an interval and Theorem 2 ensures PnP-ADMM is a contraction when \u03b1 is large enough. We roughly observe this behavior for the denoisers trained with RealSN. Table 1. Average PSNR performance (in dB) on Poisson denoising (peak = 1) on the testing set in (Chan et al., 2017) BM3D RealSN-DnCNN RealSN-SimpleCNN PNP-ADMM 23.4617 23.5873 18.7890 PNP-FBS 18.5835 22.2154 22.7280 Empirical performance. Our theory only concerns convergence and says nothing about the recovery performance of the output the methods converge to. We empirically verify that the PnP methods with RealSN, for which we analyzed convergence, yield competitive denoising results. 2We compute the contraction factor of the equivalent PnP-DRS. \fPlug-and-Play Methods Provably Converge with Properly Trained Denoisers (a) PnP-FBS (b) PNP-ADMM Figure 2. Average contraction factor of 500 iterations for the Poisson denoising experiment. The x-axis represents the value of \u03b1 and y-axis represents the contraction factor. Although lower means faster convergence, a smoother curve means the method is easier to tune and has more stable convergence. We \ufb01x \u03b1 = 0.1 for all denoisers in PNP-ADMM, and \u03b1 = 0.0125 in PNP-FBS. For deep learning-based denoisers, we choose \u03c3 = 40/255. For BM3D, we choose \u03c3 = \u221a\u03b3\u03b1 as suggested in (Rond et al., 2016) and use \u03b3 = 1. Table 1 compares the PnP methods with BM3D, RealSNDnCNN, and RealSN-SimpleCNN plugged in. In both PnP methods, one of the two denoisers using RealSN, for which we have theory, outperforms BM3D. It is interesting to obverse that the PnP performance does not necessarily hinge on the strength of the plugged in denoiser and that different PnP methods favor different denoisers. For example, RealSN-SimpleCNN surpasses the much more sophisticated RealSN-DnCNN under PnP-FBS. However, RealSN-DnCNN leads to better, and overall best, denoising performance when plugged into PnP-ADMM. 6. More applications We now apply PnP on two imaging problemsand show that RealSN improves the reconstruction of PnP.3 3Code for our experiments in Sections 5 and 6 is available at https://github.com/uclaopt/Provable_Plug_ and_Play/ Single photon imaging. Consider single photon imaging with quanta image sensors (QIS) (Fossum, 2011; Chan & Lu, 2014; Elgendy & Chan, 2016) with the model z = 1(y \u22651), y \u223cPoisson(\u03b1sgGxtrue) where xtrue \u2208Rd is the underlying image, G : Rd \u2192RdK duplicates each pixel to K pixels, \u03b1sg \u2208R is sensor gain, K is the oversampling rate, z \u2208{0, 1}dK is the observed binary photons. We want to recover xtrue from z. The likelihood function is f(x) = n X j=1 \u2212K0 j log(e\u2212\u03b1sgxj/K)\u2212K1 j log(1\u2212e\u2212\u03b1sgxj/K), where K1 j = PK i=1 z(j\u22121)K+i is the number of ones in the j-th unit pixel, K0 j = PK i=1 1\u2212z(j\u22121)K+i is the number of zeros in the j-th unit pixel. The gradient of f(x) is given by \u2202f/\u2202xj = (\u03b1sg/K)(K0 j \u2212K1 j /(e\u03b1sgxj/K \u22121)) and the proximal operator of f is given in (Chan & Lu, 2014). We compare PnP-ADMM and PnP-FBS respectively with the denoisers BM3D, RealSN-DnCNN, and RealSNSimpleCNN. We take \u03b1sg = K = 8. The y-variable was initialized to K1 for PnP-FBS and PnP-ADMM, and the u-variable was initialized to 0 for PnP-ADMM. All deep denoisers used in this experiment were trained with \ufb01xed noise level \u03c3 = 15. We report the PSNRs achieved at the 50th iteration, the 100th iteration, and the best PSNR values achieved within the \ufb01rst 100 iterations. Table 2 reports the average PSNR results on the 13 images used in (Chan et al., 2017). Experiments indicate that PnPADMM methods constantly yields higher PNSR than the PnP-FBS counterparts using the same denoiser. The best overall PSNR is achieved using PnP-ADMM with RealSNDnCNN, which shows nearly 1dB improvement over the result obtained with BM3D. We also observe that deep denoisers with RealSN make PnP converges more stably. Compressed sensing MRI. Magnetic resonance imaging (MRI) is a widely-used imaging technique with a slow data acquisition. Compressed sensing MRI (CS-MRI) accelerates MRI by acquiring less data through downsampling. PnP is useful in medical imaging as we do not have a large amount of data for end-to-end training: we train the denoiser H\u03c3 on natural images, and then \u201cplug\u201d it into the PnP framework to be applied to medical images. CS-MRI is described mathematically as y = Fpxtrue + \u03b5e, where xtrue \u2208Cd is the underlying image, Fp : Cd \u2192Ck is the linear measurement model, y \u2208Ck is the measured data, and \u03b5e \u223cN(0, \u03c3eIk) is measurement noise. We want to recover xtrue from y. The objective function is f(x) = (1/2)\u2225y \u2212Fpx\u22252. \fPlug-and-Play Methods Provably Converge with Properly Trained Denoisers Table 2. Average PSNR (in dB) of single photon imaging task on the test set of (Chan et al., 2017) PnP-ADMM, \u03b1 = 0.01 Average PSNR BM3D RealSNRealSNDnCNN SimpleCNN Iteration 50 30.0034 31.0032 29.2154 Iteration 100 30.0014 31.0032 29.2151 Best Overall 30.0474 31.0431 29.2155 PnP-FBS, \u03b1 = 0.005 Average PSNR BM3D RealSNRealSNDnCNN SimpleCNN Iteration 50 28.7933 27.9617 29.0062 Iteration 100 29.0510 27.9887 29.0517 Best Overall 29.5327 28.4065 29.3563 Table 3. CS-MRI results (30% sample with additive Gaussian noise \u03c3e = 15) in PSNR (dB). Sampling approach Random Radial Cartesian Image Brain Bust Brain Bust Brain Bust Zero-\ufb01lling 9.58 7.00 9.29 6.19 8.65 6.01 TV (Lustig et al., 2005) 16.92 15.31 15.61 14.22 12.77 11.72 RecRF (Yang et al., 2010) 16.98 15.37 16.04 14.65 12.78 11.75 BM3D-MRI (Eksioglu, 2016) 17.31 13.90 16.95 13.72 14.43 12.35 PnP-FBS BM3D 19.09 16.36 18.10 15.67 14.37 12.99 DnCNN 19.59 16.49 18.92 15.99 14.76 14.09 RealSN-DnCNN 19.82 16.60 18.96 16.09 14.82 14.25 SimpleCNN 15.58 12.19 15.06 12.02 12.78 10.80 RealSN-SimpleCNN 17.65 14.98 16.52 14.26 13.02 11.49 PnP-ADMM BM3D 19.61 17.23 18.94 16.70 14.91 13.98 DnCNN 19.86 17.05 19.00 16.64 14.86 14.14 RealSN-DnCNN 19.91 17.09 19.08 16.68 15.11 14.16 SimpleCNN 16.68 12.56 16.83 13.47 13.03 11.17 RealSN-SimpleCNN 17.77 14.89 17.00 14.47 12.73 11.88 The gradient of f(x) is given in (Liu et al., 2016) and the proximal operator of f(x) is given in (Eksioglu, 2016). We use BM3D, SimpleCNN and DnCNN, and their variants by RealSN for the PnP denoiser H\u03c3. We take Fp as the Fourier k-domain subsampling (partial Fourier operator). We tested random, radial, and Cartesian sampling (Eksioglu, 2016) with a sampling rate of 30%. The noise level \u03c3e is taken as 15/255. We compare PnP frameworks with zero-\ufb01lling, totalvariation (TV) (Lustig et al., 2005), RecRF (Yang et al., 2010), and BM3D-MRI (Eksioglu, 2016) 4. The parameters are taken as follows. For TV, the regularization parameter \u03bb is taken as the best one from {a \u00d7 10b, a \u2208 {1, 2, 5}, b \u2208{\u22125, \u22124, \u22123, \u22122, \u22121, 0, 1}}. For RecRF, the two parameters \u03bb, \u00b5 are both taken from the above sets and the best results are reported. For BM3D-MRI, we set the \u201c\ufb01nal noise level (the noise level in the last iteration)\u201d as 2\u03c3e, which is suggested in their MATLAB library. For PnP methods with H\u03c3 as BM3D, we set \u03c3 = 2\u03c3e, take \u03b1 \u2208{0.1, 0.2, 0.5, 1, 2, 5} and report the best results. 4Some recent deep-learning based methods (Yang et al., 2016; Kulkarni et al., 2016; Metzler et al., 2017; Zhang & Ghanem, 2018) are not compared here because we assume we do not have enough medical images for training. For PNP-ADMM with H\u03c3 as deep denoisers, we take \u03c3 = \u03c3e = 15/255 and \u03b1 = 2.0 uniformly for all the cases. For PNP-FBS with H\u03c3 as deep denoisers, we take \u03c3 = \u03c3e/3 = 5/255 and \u03b1 = 0.4 uniformly. All deep denoisers are trained on BSD500 (Martin et al., 2001), a natural image data set; no medical image is used in training. The y-variable was initialized to the zero-\ufb01lled solution for PnP-FBS and PnP-ADMM, and the u-variable was initialized to 0 for PnP-ADMM. Table 3 reports our results on CS-MRI, from which we can con\ufb01rm the effectiveness of PnP frameworks. Moreover, using RealSN-DnCNN seems to the clear winner over all. We also observe that PnPADMM generally outperforms PnP-FBS when using the same denoiser, which supports Theorems 1 and 2. 7." + }, + { + "url": "http://arxiv.org/abs/1712.10279v2", + "title": "Vector and Matrix Optimal Mass Transport: Theory, Algorithm, and Applications", + "abstract": "In many applications such as color image processing, data has more than one\npiece of information associated with each spatial coordinate, and in such cases\nthe classical optimal mass transport (OMT) must be generalized to handle\nvector-valued or matrix-valued densities. In this paper, we discuss the vector\nand matrix optimal mass transport and present three contributions. We first\npresent a rigorous mathematical formulation for these setups and provide\nanalytical results including existence of solutions and strong duality. Next,\nwe present a simple, scalable, and parallelizable methods to solve the vector\nand matrix-OMT problems. Finally, we implement the proposed methods on a CUDA\nGPU and present experiments and applications.", + "authors": "Ernest K. Ryu, Yongxin Chen, Wuchen Li, Stanley Osher", + "published": "2017-12-29", + "updated": "2018-06-17", + "primary_cat": "math.OC", + "cats": [ + "math.OC", + "cs.SY", + "math.FA", + "65K10, 65K05, 90C25" + ], + "main_content": "Introduction Optimal mass transport (OMT) is a subject with a long history. Started by Monge [41] and developed by many great mathematicians [34, 9, 28, 39, 33, 5, 46], the subject now has incredibly rich theory and applications. It has found numerous applications in di\ufb00erent areas such as partial di\ufb00erential equations, probability theory, physics, economics, image processing, and control [26, 31, 55, 42, 17, 18, 13]. See also [50, 56, 2] and references therein. However, in many applications such as color image processing, there is more than one piece of information associated with each spatial coordinate, and such data can be interpreted as vector-valued or matrix-valued densities. As the classical optimal mass transport works with scalar probability densities, such applications require a new notion of mass transport. For this purpose, Chen et al. [21, 20, 16, 14] recently developed a framework for vector-valued and matrix-valued optimal mass transport. See also [44, 43, 45, 27, 48, 58] for other di\ufb00erent frameworks. For vector-valued OMT, potential applications include color image processing, multi-modality medical imaging, and image processing involving textures. For matrix-valued OMT, we have di\ufb00usion tensor imaging, multivariate spectral analysis, and stress tensor analysis. Several mathematical aspects of vector and matrix-valued OMT were not addressed in the previous work [21, 20, 16]. As the \ufb01rst contribution of this paper, we present duality and existence results of the continuous vector and matrix-valued OMT problems along with rigorous problem formulations. Although the classical theory of OMT is very rich, only recently has there been much attention to numerical methods to compute the OMT. Several recent work proposed algorithms to solve the L2 OMT [3, 24, 7, 30, 6, 15, 19, 29] and the L1 OMT [36, 53]. As the second contribution of this paper, we present \ufb01rst-order primal-dual methods to solve the vector and matrix-valued OMT problems. The methods simultaneously solve for both the primal and dual solutions (hence a primal-dual method) and are scalable as they are \ufb01rst-order methods. We also discuss the convergence of the methods. As the third contribution of this paper, we implement the proposed method on a CUDA GPU and present several applications. The proposed algorithms\u2019 simple structure allows us to e\ufb00ectively utilize the computing capability of the CUDA architecture, and we demonstrate this through our experiments. We furthermore release the code for scienti\ufb01c reproducibility. The rest of the paper is structured as follows. In Section 2 we give a quick review of the classic OMT theory, which allows us to present the later sections in an analogous manner and thereby outline the similarities and di\ufb00erences. In Section 3 and Section 4 we present the vector and matrix-valued OMT problems 1 arXiv:1712.10279v2 [math.OC] 17 Jun 2018 \fand state a few theoretical results. In Section 5, we present and prove the analytical results. In Section 6, we discuss preliminaries we need for Section 7, where we present the algorithm. In Section 8, we present the experiments and applications. 2 Optimal mass transport Let \u2126\u2282Rd be a closed, convex, compact domain. Let \u03bb0 and \u03bb1 be nonnegative densities supported on \u2126 with unit mass, i.e., R \u2126\u03bb0(x) dx = R \u2126\u03bb1(x) dx = 1. Let \u2225\u00b7 \u2225denote any norm on Rd. In 1781, Monge posed the optimal mass transport (OMT) problem, which solves minimize T Z \u2126 \u2225x \u2212T(x)\u2225\u03bb0(x) dx. (1) The optimization variable T : \u2126\u2192\u2126is smooth, one-to-one, and transfers \u03bb0(x) to \u03bb1(x). The optimization problem (1) is nonlinear and nonconvex. In 1940, Kantorovich relaxed (1) into a linear (convex) optimization problem: S(\u03bb0, \u03bb1) = \uf8eb \uf8ec \uf8ec \uf8ed minimize \u03c0 R \u2126\u00d7\u2126\u2225x \u2212y\u2225\u03c0(x, y) dxdy subject to \u03c0(x, y) \u22650 R \u2126\u03c0(x, y) dy = \u03bb0(x) R \u2126\u03c0(x, y) dx = \u03bb1(y). \uf8f6 \uf8f7 \uf8f7 \uf8f8 (2) The optimization variable \u03c0 is a joint nonnegative measure on \u2126\u00d7 \u2126having \u03bb0(x) and \u03bb1(y) as marginals. To clarify, S(\u03bb0, \u03bb1) denotes the optimal value of (2). 2.1 Scalar optimal mass transport The theory of optimal transport [28, 56, 57] remarkably points out that (2) is equivalent to the following \ufb02ux minimization problem: S(\u03bb0, \u03bb1) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed minimize u R \u2126\u2225u(x)\u2225dx subject to divx(u)(x) = \u03bb0(x) \u2212\u03bb1(x) u(x)T n(x) = 0, for all ( x \u2208\u2202\u2126, n(x) normal to \u2202\u2126, \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (3) where u = (u1, . . . , ud) : \u2126\u2192Rd is the optimization variable and divx denote the (spatial) divergence operator. Although (3) and (2) are mathematically equivalent, (3) is much more computationally e\ufb00ective as its optimization variable u is much smaller when discretized. It is worth mentioning that OMT in formulation (3) is very close to the problems in compressed sensing. Its objective function is homogeneous degree one and the constraint is linear. It can be observed that for characterizing the OMT, divergence operator in (3) play the key roles. Later on, we extend the de\ufb01nition of L1 OMT problem by extending these di\ufb00erential operators into a general meaning. The optimization problem (3) has the following dual problem: S(\u03bb0, \u03bb1) = maximize \u03c6 R \u2126\u03c6(x)(\u03bb1(x) \u2212\u03bb0(x)) dx subject to \u2225\u2207x\u03c6(x)\u2225\u2217\u22641 for all x \u2208\u2126, ! (4) where the optimization and \u03c6 : \u2126\u2192R is a function. We write \u2225\u00b7 \u2225\u2217for the dual norm of \u2225\u00b7 \u2225. It is well-known that, strong duality holds between (3) and (4) in the sense that the minimized and maximized objective values are equal [56]. Therefore, we take either (3) or (4) as the de\ufb01nition of S. Rigorous de\ufb01nitions of the optimization problems (3) or (4) are somewhat technical. We skip this discussion, as scalar optimal mass transport is standard. In Section 5, we rigorously discuss the vectorOMT problems, so any rigorous discussion of the scalar-OMT problems can be inferred as a special case. 2 \f1 2 3 4 c1 c2 c3 c4 c5 Figure 1: Example graph with k = 4 nodes and \u2113= 5 edges. To make cj the cost of traversing edge j, the edge weight is de\ufb01ned to be 1/c2 j for j = 1, . . . , \u2113. 2.2 Theoretical properties The algorithm we present in Section 7 is a primal-dual algorithm and, as such, \ufb01nds solutions to both Problems (3) and (4). This is well-de\ufb01ned as both the primal and dual problems have solutions [56]. Write R+ for the set of nonnegative real numbers. Write P(\u2126, R) for the space of nonnegative densities supported on \u2126with unit mass. We can use S(\u03bb0, \u03bb1) as a distance measure between \u03bb0, \u03bb1 \u2208P(\u2126, R). The value S : P(\u2126, R) \u00d7 P(\u2126, R) \u2192R+ de\ufb01nes a metric on P(\u2126, R) [56]. 3 Vector optimal mass transport Next we discuss the vector-valued optimal transport, proposed recently in [20]. The basic idea is to combine scalar optimal mass transport with network \ufb02ow problems [1]. Color image processing is the simplest application to think of for this setup. On an RGB color image, there are 3 values associated with each pixel. To generalize scalar-OMT to this setup, the cost for changing colors is de\ufb01ned with a graph. The edges represent the cost of changing one channel/color to another. 3.1 Gradient and divergence on graphs Consider a connected, positively weighted, undirected graph G with k nodes and \u2113edges. To de\ufb01ne an incidence matrix for G, we say an edge {i, j} points from i to j, i.e., i \u2192j, if i < j. This choice is arbitrary and does not a\ufb00ect the \ufb01nal result. With this edge orientation, the incidence matrix D \u2208Rk\u00d7\u2113is Die = \uf8f1 \uf8f2 \uf8f3 +1 if edge e = {i, j} for some node j > i \u22121 if edge e = {j, i} for some node j < i 0 otherwise. For example, the incidence matrix of the graph of Figure 1 is D = \uf8ee \uf8ef \uf8ef \uf8f0 1 1 0 1 0 \u22121 0 1 0 0 0 \u22121 \u22121 0 1 0 0 0 \u22121 \u22121 \uf8f9 \uf8fa \uf8fa \uf8fb. Write \u2206G = \u2212D diag{1/c2 1, \u00b7 \u00b7 \u00b7 , 1/c2 \u2113}DT for the (negative) graph Laplacian, where 1/c2 1, \u00b7 \u00b7 \u00b7 , 1/c2 \u2113are the edge weights. The edge weights are de\ufb01ned so that and cj represents the cost of traversing edge j for j = 1, . . . , \u2113. We de\ufb01ne the gradient operator on G as \u2207Gx = diag{1/c1, \u00b7 \u00b7 \u00b7 , 1/c\u2113}DT x and the divergence operator as divGy = \u2212D diag{1/c1, \u00b7 \u00b7 \u00b7 , 1/c\u2113}y. So the Laplacian can be rewritten as \u2206G = divG\u2207G. Note that divG = \u2212\u2207\u2217 G, where \u2207\u2217 G is the adjoint of \u2207G. This is in analogy with the usual spatial gradient and divergence operators [38, 22, 23, 20]. The edge weights of a graph G should be considered modeling parameters. In color image processing, for example, the edge weights of G represent the cost of changing one color to another, and they should be tuned to make results visually look best. 3 \f3.2 Vector optimal mass transport We say \u21c0 \u03bb : \u2126\u2192Rk + is a nonnegative vector-valued density with unit mass if \u21c0 \u03bb(x) = \uf8ee \uf8ef \uf8f0 \u03bb1(x) . . . \u03bbk(x) \uf8f9 \uf8fa \uf8fb, Z \u2126 k X i=1 \u03bbi(x) dx = 1. Assume \u21c0 \u03bb0 and \u21c0 \u03bb1 are nonnegative vector-valued densities supported on \u2126with unit mass. We de\ufb01ne the optimal mass transport between vector-valued densities as V ( \u21c0 \u03bb0, \u21c0 \u03bb1) = \uf8eb \uf8ec \uf8ed minimize \u21c0 u, \u21c0 w R \u2126\u2225 \u21c0 u(x)\u2225u + \u03b1\u2225 \u21c0 w(x)\u2225w dx subject to divx( \u21c0 u)(x) + divG( \u21c0 w(x)) = \u21c0 \u03bb0(x) \u2212 \u21c0 \u03bb1(x) \u21c0 u satis\ufb01es zero-\ufb02ux b.c. \uf8f6 \uf8f7 \uf8f8 (5) where \u21c0 u : \u2126\u2192Rk\u00d7d and \u21c0 w : \u2126\u2192R\u2113are the optimization variables, \u03b1 > 0 is a parameter, and \u2225\u00b7 \u2225u is a norm on Rk\u00d7d and \u2225\u00b7 \u2225w is a norm on R\u2113. The parameter \u03b1 represents the relative importance of the two \ufb02ux terms \u21c0 u and \u21c0 w. We write \u21c0 u = \uf8ee \uf8ef \uf8f0 uT 1 . . . uT k \uf8f9 \uf8fa \uf8fb \u21c0 w = \uf8ee \uf8ef \uf8f0 w1 . . . w\u2113 \uf8f9 \uf8fa \uf8fb divx( \u21c0 u) = \uf8ee \uf8ef \uf8f0 divx(u1) . . . divx(uk) \uf8f9 \uf8fa \uf8fb. We call divx the spatial divergence operator. The zero-\ufb02ux boundary condition is ui(x)T n(x) = 0, for all ( x \u2208\u2202\u2126, n(x) normal to \u2202\u2126. for i = 1, . . . , k. Note that \u21c0 w has no boundary conditions. The optimization problem (5) has the following dual problem: V ( \u21c0 \u03bb0, \u21c0 \u03bb1) = \uf8eb \uf8ec \uf8ec \uf8ed maximize \u21c0 \u03c6 R \u2126\u27e8 \u21c0 \u03c6(x), \u21c0 \u03bb1(x) \u2212 \u21c0 \u03bb0(x)\u27e9dx subject to \u2225\u2207x \u21c0 \u03c6(x)\u2225u\u2217\u22641 \u2225\u2207G \u21c0 \u03c6(x)\u2225w\u2217\u2264\u03b1 for all x \u2208\u2126, \uf8f6 \uf8f7 \uf8f7 \uf8f8 (6) where the optimization variable \u21c0 \u03c6 : \u2126\u2192Rk is a function. We write \u2225\u00b7 \u2225u\u2217and \u2225\u00b7 \u2225w\u2217for the dual norms of \u2225\u00b7 \u2225u and \u2225\u00b7 \u2225w, respectively. As stated in Theorem 1, strong duality holds between (5) and (6) in the sense that the minimized and maximized objective values are equal. Therefore, we take either (5) or (6) as the de\ufb01nition of V . In Section 5, we rigorously de\ufb01ne the primal and dual problems and prove Theorem 1. 3.3 Theoretical properties The algorithm we present in Section 7 is a primal-dual algorithm and, as such, \ufb01nds solutions to both Problems (5) and (6). This is well-de\ufb01ned as both the primal and dual problems have solutions. Theorem 1. The (in\ufb01nite dimensional) primal and dual optimization problems (5) and (6) have solutions, and their optimal values are the same, i.e., strong duality holds. Write P(\u2126, Rk) for the space of nonnegative vector-valued densities supported on \u2126with unit mass. We can use V ( \u21c0 \u03bb0, \u21c0 \u03bb1) as a distance measure between \u21c0 \u03bb0, \u21c0 \u03bb1 \u2208P(\u2126, Rk). Theorem 2. V : P(\u2126, Rk) \u00d7 P(\u2126, Rk) \u2192R+ de\ufb01nes a metric on P(\u2126, Rk). 4 \f4 Quantum gradient operator and matrix optimal mass transport We closely follow the treatment in [21]. In particular, we de\ufb01ne a notion of gradient on the space of Hermitian matrices and its dual, i.e., the (negative) divergence. Some applications of matrix-OMT, such as di\ufb00usion tensor imaging, have real-valued data while some applications, such as multivariate spectral analysis, have complex-valued data [54]. To accommodate the wide range of applications, we develop the matrix-OMT with complex-valued matrices. Write C, H, and S for the set of k \u00d7 k complex, Hermitian, and skew-Hermitian matrices respectively. We write H+ for the set of k \u00d7 k positive semide\ufb01nite Hermitian matrices, i.e., M \u2208H+ if v\u2217Mv \u22650 for all v \u2208Ck. Write tr for the trace, i.e. for any M \u2208H, we have tr(M) = Pk i=1 Mii. Write CN for the block-column concatenation of N matrices in C, i.e., Z \u2208CN if Z = [Z\u2217 1 \u00b7 \u00b7 \u00b7 Z\u2217 N]\u2217and Z1, . . . , ZN \u2208C. De\ufb01ne HN and SN likewise. For X, Y \u2208C, we use the Hilbert-Schmidt inner product \u27e8X, Y \u27e9= Re tr(XY \u2217) = k X i=1 k X j=1 (ReXijReYij + ImXijImYij). (This is the standard inner product when we view C as the real vector space R2n2.) For X \u2208C, we use the norm \u2225X\u22252 = (\u27e8X, X\u27e9)1/2. For X, Y \u2208CN, we use the inner product \u27e8X, Y\u27e9= PN s=1\u27e8Xs, Ys\u27e9. 4.1 Quantum gradient and divergence operators We de\ufb01ne the gradient operator, given a L = [L1, \u00b7 \u00b7 \u00b7 , L\u2113]\u2217\u2208H\u2113, as \u2207L : H \u2192S\u2113, X 7\u2192 \uf8ee \uf8ef \uf8f0 L1X \u2212XL1 . . . L\u2113X \u2212XL\u2113 \uf8f9 \uf8fa \uf8fb. De\ufb01ne the divergence operator as divL : S\u2113\u2192H, Z = \uf8ee \uf8ef \uf8f0 Z1 . . . Z\u2113 \uf8f9 \uf8fa \uf8fb7\u2192 \u2113 X s=1 \u2212LsZs + ZsLs. Note that divL = \u2212\u2207\u2217 L, where \u2207\u2217 L is the adjoint of \u2207L. This is in analogy with the usual spatial gradient and divergence operators. Write \u2206L = divL\u2207L. A standing assumption throughout, is that the null space of \u2207L, denoted by ker(\u2207L), contains only scalar multiples of the identity matrix I. The choice of L a\ufb00ects \u2207L and, in turn, a\ufb00ects the induced metric. This de\ufb01nition of gradient operator is inspired by the Lindblad equation in Quantum Mechanics [21]. There has been some work on similar matrix optimal transport theories which focus on the applications in physics [11, 40]. However, how to appropriately choose L to \ufb01t a speci\ufb01c engineering application is still a topic of open research, and there is currently no established standard choice. 4.2 Matrix optimal mass transport We say \u039b : \u2126\u2192H+ is a nonnegative matrix-valued density with unit mass if Z \u2126 tr(\u039b(x)) dx = 1. Assume \u039b0 and \u039b1 are nonnegative matrix-valued densities supported on \u2126with unit mass. We de\ufb01ne the optimal mass transport between matrix-valued densities as M(\u039b0, \u039b1) = \uf8eb \uf8ec \uf8ed minimize U,W R \u2126\u2225U(x)\u2225u + \u03b1\u2225W(x)\u2225w dx subject to divx(U)(x) + divL(W(x)) = \u039b0(x) \u2212\u039b1(x) U satis\ufb01es zero-\ufb02ux b.c. \uf8f6 \uf8f7 \uf8f8 (7) 5 \fwhere U : \u2126\u2192Hd and W : \u2126\u2192S\u2113are the optimization variables, \u03b1 > 0 is a parameter, and \u2225\u00b7 \u2225u is a norm on Hd and \u2225\u00b7 \u2225w is a norm on S\u2113. The parameter \u03b1 represents the relative importance of the two \ufb02ux terms U and W. We write U = \uf8ee \uf8ef \uf8f0 U1 . . . Ud \uf8f9 \uf8fa \uf8fb W = \uf8ee \uf8ef \uf8f0 W1 . . . W\u2113 \uf8f9 \uf8fa \uf8fb uij = \uf8ee \uf8ef \uf8f0 (U1)ij . . . (Ud)ij \uf8f9 \uf8fa \uf8fb. We de\ufb01ne the spatial divergence as divx(U) = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 divx(u11) divx(u12) \u00b7 \u00b7 \u00b7 divx(u1k) divx(u12) ... . . . . . . ... . . . divx(u1k) divx(u2k) \u00b7 \u00b7 \u00b7 divx(uk,k) \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb . The zero-\ufb02ux boundary condition is uij(x)T n(x) = 0, for all ( x \u2208\u2202\u2126, n(x) normal to \u2202\u2126. for i, j = 1, . . . , k. Note that W has no boundary conditions [16]. The optimization problem (7) has the following dual problem: M(\u039b0, \u039b1) = \uf8eb \uf8ed maximize \u03a6 R \u2126\u27e8\u03a6(x), \u039b1(x) \u2212\u039b0(x)\u27e9dx subject to \u2225\u2207x\u03a6(x)\u2225u\u2217\u22641 \u2225\u2207L\u03a6(x)\u2225w\u2217\u2264\u03b1 for all x \u2208\u2126, \uf8f6 \uf8f8 (8) where the optimization variable \u03a6 : \u2126\u2192H is a function. We write \u2225\u00b7 \u2225u\u2217and \u2225\u00b7 \u2225w\u2217for the dual norms of \u2225\u00b7 \u2225u and \u2225\u00b7 \u2225w, respectively. As stated in Theorem 3, strong duality holds between (7) and (8) in the sense that the minimized and maximized objective values are equal. Therefore we take either (7) or (8) as the de\ufb01nition of M. In Section 5.6, we rigorously de\ufb01ne the primal and dual problems. To prove the results on matrixOMT (Theorems 3 and 4) one can take the arguments of Section 5 that prove the results on vector-OMT (Theorems 1 and 2) and change only the notation. We therefore simply point this out, instead of repeating same argument. 4.3 Theoretical properties The algorithm we present in Section 7 is a primal-dual algorithm and, as such, \ufb01nds solutions to both Problems (7) and (8). This is well-de\ufb01ned as both the primal and dual problems have solutions. Theorem 3. The (in\ufb01nite dimensional) primal and dual optimization problems (7) and (8) have solutions, and their optimal values are the same, i.e., strong duality holds. Write P(\u2126, H+) for the space of nonnegative matrix-valued densities supported on \u2126with unit mass. We can use M(\u039b0, \u039b1) as a distance measure between \u039b0, \u039b1 \u2208P(\u2126, , H+). Theorem 4. M : P(\u2126, H+) \u00d7 P(\u2126, H+) \u2192R+ de\ufb01nes a metric on P(\u2126, H+). 5 Duality proof In this section, we establish the theoretical results. For notational simplicity, we only prove the results for the vector-OMT primal and dual problems (5) and (6). Analogous results for the matrix-OMT primal and dual problems (7) and (8) follow from the same logic. 6 \fAlthough the classical scalar-OMT literature is very rich, standard techniques for proving scalar-OMT duality do not simply apply to our setup. For example, Villani\u2019s proof of strong duality, presented as Theorem 1.3 of [56], relies on and works with the linear optimization formulation (2). However, our vector and matrix-OMT formulations directly generalize the \ufb02ux formulation (3) and do not have formulations analogous to (2). We need a direct approach to analyze duality between the \ufb02ux formulation and the dual (with one function variable), and we provide this in this section. We further assume \u2126\u2282Rd has a piecewise smooth boundary. Write \u2126\u25e6and \u2202\u2126for the interior and boundary of \u2126. For simplicity, assume \u2126as full a\ufb03ne dimensions, i.e., \u2126\u25e6= \u2126. The rigorous form of the dual problem (6) is maximize \u21c0 \u03c6\u2208W 1,\u221e(\u2126,Rk) R \u2126\u27e8 \u21c0 \u03c6(x), \u21c0 \u03bb1(x) \u2212 \u21c0 \u03bb0(x)\u27e9dx subject to ess supx\u2208\u2126\u2225\u2207x \u21c0 \u03c6(x)\u2225u\u2217\u22641 supx\u2208\u2126\u2225\u2207G \u21c0 \u03c6(x)\u2225w\u2217\u2264\u03b1, (9) where W 1,\u221e(\u2126, Rk) is the standard Sobolev space of functions from \u2126to Rk with bounded weak gradients. That (9) has a solution directly follows from the Arzel` a-Ascoli Theorem. To rigorously de\ufb01ne the primal problem (5) requires more de\ufb01nitions, and we do so later as (11). 5.1 Fenchel-Rockafellar duality Let L : X \u2192Y be a continuous linear map between locally convex topological vector spaces X and Y , and let f : X \u2192R \u222a{\u221e} and g : Y \u2192R \u222a{\u221e} be lower-semicontinuous convex functions. Write d\u22c6= sup x\u2208X {\u2212f(x) \u2212g(Lx)} p\u22c6= inf y\u2217\u2208Y \u2217{f \u2217(L\u2217y\u2217) + g\u2217(\u2212y\u2217)}, where f \u2217(x\u2217) = sup x\u2208X {\u27e8x\u2217, x\u27e9\u2212f(x)} g\u2217(y\u2217) = sup y\u2208Y {\u27e8y\u2217, y\u27e9\u2212g(y)}. In this framework of Fenchel-Rockafellar duality, d\u22c6\u2264p\u22c6, i.e., weak duality, holds unconditionally, and this is not di\ufb03cult to prove. On the other hand, d\u22c6= p\u22c6, i.e., strong duality, requires additional assumptions and is more di\ufb03cult to prove. The following theorem does this with a condition we can use. Theorem 5. [Theorem 17 and 18 [51]] If there is an x \u2208X such that f(x) < \u221eand g is bounded above in a neighborhood of Lx, then p\u22c6= d\u22c6. Furthermore, if p\u22c6= d\u22c6< \u221e, the in\ufb01mum of infy\u2217\u2208Y \u2217{f \u2217(L\u2217y\u2217) + g\u2217(\u2212y\u2217)} is attained. 5.2 Spaces Throughout this section, \u2225\u00b7 \u22251, \u2225\u00b7 \u22252, and \u2225\u00b7 \u22253 denote unspeci\ufb01ed \ufb01nite dimensional norms. As all \ufb01nite dimensional norms are equivalent, we do not bother to precisely specify which norms they are. De\ufb01ne C(\u2126, Rk) = \u001a \u21c0 \u03c6 : \u2126\u2192Rk \f \f \f \u21c0 \u03c6 is continuous, \u2225 \u21c0 \u03c6\u2225\u221e= max x\u2208\u2126\u2225 \u21c0 \u03c6(x)\u22251 < \u221e \u001b . Then C(\u2126, Rk) is a Banach space equipped with the norm \u2225\u00b7 \u2225\u221e. We de\ufb01ne C(\u2126, Rk\u00d7d) likewise. If \u21c0 \u03c6 \u2208C(\u2126, Rk) is continuously di\ufb00erentiable, \u2207x \u21c0 \u03c6 is de\ufb01ned on \u2126\u25e6. We say \u2207x \u21c0 \u03c6 has a continuous extension to \u2126, if there is a g \u2208C(\u2126, Rk\u00d7d) such that g|\u2126\u25e6= \u2207x \u21c0 \u03c6. De\ufb01ne C1(\u2126, Rk) = n\u21c0 \u03c6 : Rd \u2192Rk \f \f \u21c0 \u03c6 is continuously di\ufb00erentiable on \u2126\u25e6, \u2207x \u21c0 \u03c6 has a continuous extension to \u2126, \u2225 \u21c0 \u03c6\u2225\u221e,\u221e= max x\u2208\u2126\u2225 \u21c0 \u03c6(x)\u22252 + sup x\u2208\u2126 \u2225\u2207x \u21c0 \u03c6(x)\u22253 < \u221e o . 7 \fThen C1(\u2126, Rk) is a Banach space equipped with the norm \u2225\u00b7 \u2225\u221e,\u221e. Write M(\u2126, Rk) for the space of Rk-valued signed \ufb01nite Borel measures on \u2126, and de\ufb01ne M(\u2126, Rk\u00d7d) likewise. Write (C(\u2126, Rk\u00d7d))\u2217, (C1(\u2126, Rk))\u2217for the topological dual of C(\u2126, Rk\u00d7d), C1(\u2126, Rk), respectively. The standard Riesz-Markov theorem tells us that (C(\u2126, Rk\u00d7d))\u2217= M(\u2126, Rk\u00d7d). Fully characterizing (C1(\u2126, Rk))\u2217is hard, but we do not need to do so. Instead, we only use the following simple fact. Any g \u2208M(\u2126, Rk) de\ufb01nes the bounded linear map \u21c0 \u03c6 7\u2192 R \u2126\u27e8 \u21c0 \u03c6(x), g(dx)\u27e9for any \u21c0 \u03c6 \u2208C1(\u2126, Rk). In other words, M(\u2126, Rk\u00d7d) \u2282(C1(\u2126, Rk))\u2217with the appropriate identi\ufb01cation. 5.3 Operators We rede\ufb01ne \u2207x : C1(\u2126, Rk) \u2192C(\u2126, Rk\u00d7d) so that \u2207x \u21c0 \u03c6 is the continuous extension of the usual \u2207x \u21c0 \u03c6 to all of \u2126. This makes \u2207x a bounded linear operator. De\ufb01ne the dual (adjoint) operator \u2207\u2217 x : M(\u2126, Rk\u00d7d) \u2192 (C1(\u2126, Rk))\u2217by Z \u2126 \u27e8 \u21c0 \u03c6(x), (\u2207\u2217 x \u21c0 u)(dx)\u27e9= Z \u2126 \u27e8(\u2207x \u21c0 \u03c6)(x), \u21c0 u(dx)\u27e9 for any \u21c0 \u03c6 \u2208C1(\u2126, Rk) and \u21c0 u \u2208M(\u2126, Rk\u00d7d). De\ufb01ne the \u2207G (which is simply a multiplication by a Rk\u00d7\u2113matrix) as \u2207G : C1(\u2126, Rk) \u2192C(\u2126, R\u2113). Since C1(\u2126, R\u2113) \u2282C(\u2126, R\u2113), there is nothing wrong with de\ufb01ning the range of \u2207G to be C(\u2126, R\u2113), and this still makes \u2207G a bounded linear operator. De\ufb01ne the dual (adjoint) operator \u2207\u2217 G : M(\u2126, R\u2113) \u2192(C1(\u2126, Rk))\u2217 by identifying \u2207\u2217 G with the transpose of the matrix that de\ufb01nes \u2207G. Since \u2207\u2217 G is simply multiplication by a matrix, we can further say \u2207\u2217 G : M(\u2126, R\u2113) \u2192M(\u2126, Rk) \u2282(C1(\u2126, Rk))\u2217. We write divG = \u2212\u2207\u2217 G. 5.4 Zero-\ufb02ux boundary condition Let \u21c0 m : \u2126\u2192Rk\u00d7d a smooth function. Then integration by parts tells us that Z \u2126 \u27e8\u2207x \u21c0 \u03d5(x), \u21c0 m(x)\u27e9dx = \u2212 Z \u2126 \u27e8 \u21c0 \u03d5(x), divx \u21c0 m(x)\u27e9dx holds for all smooth \u21c0 \u03d5 : \u2126\u2192Rk if and only if \u21c0 m(x) satis\ufb01es the zero-\ufb02ux boundary condition, i.e., \u21c0 m(x)n(x) = 0 for all x \u2208\u2202\u2126, where n(x) denotes the normal vector at x. Here divx denotes the usual (spatial) divergence. To put it di\ufb00erently, \u2207\u2217 x = \u2212divx holds when the zero-\ufb02ux boundary condition holds. We generalize this notion to measures. We say \u21c0 u \u2208M(\u2126, Rk\u00d7d) satis\ufb01es the zero-\ufb02ux boundary condition in the weak sense if there is a \u21c0 g \u2208M(\u2126, Rk) \u2282(C1(\u2126, Rk))\u2217such that Z \u2126 \u27e8\u2207x \u21c0 \u03c6(x), \u21c0 u(dx)\u27e9= \u2212 Z \u2126 \u27e8 \u21c0 \u03c6(x), \u21c0 g(dx)\u27e9 holds for all \u21c0 \u03c6 \u2208C1(\u2126, Rk). In other words, \u21c0 u satis\ufb01es the zero-\ufb02ux boundary condition if \u2207\u2217 x \u21c0 u \u2208M(\u2126, Rk) \u2282 (C1(\u2126, Rk))\u2217. In this case, we write divx( \u21c0 u) = \u21c0 g and divx( \u21c0 u) = \u2212\u2207\u2217 x( \u21c0 u). This de\ufb01nition is often used in elasticity theory. 5.5 Duality To establish duality, we view the dual problem (6) as the primal problem and obtain the primal problem (5) as the dual of the dual. We do this because the dual of C(\u2126, Rk\u00d7d) is known, while the dual of M(\u2126, Rk\u00d7d) is di\ufb03cult to characterize. Consider the problem maximize \u21c0 \u03c6\u2208C1(\u2126,Rk) R \u2126\u27e8 \u21c0 \u03c6(x), \u21c0 \u03bb1(x) \u2212 \u21c0 \u03bb0(x)\u27e9dx subject to \u2225\u2207x \u21c0 \u03c6(x)\u2225u\u2217\u22641 \u2225\u2207G \u21c0 \u03c6(x)\u2225w\u2217\u2264\u03b1 for all x \u2208\u2126, (10) 8 \fwhich is equivalent to (9). De\ufb01ne L : C1(\u2126, Rk) \u2192C(\u2126, Rk\u00d7d) \u00d7 C(\u2126, R\u2113) \u21c0 \u03c6 7\u2192(\u2207x \u21c0 \u03c6, \u2207G \u21c0 \u03c6) and g(a, b) = \u001a 0 if \u2225a(x)\u2225u\u2217\u22641, \u2225b(x)\u2225w\u2217\u2264\u03b1 for all x \u2208\u2126 \u221e otherwise. Rewrite (10) as maximize \u21c0 \u03c6\u2208C1(\u2126,Rk) R \u2126\u27e8 \u21c0 \u03c6(x), \u21c0 \u03bb1(x) \u2212 \u21c0 \u03bb0(x)\u27e9dx \u2212g(L \u21c0 \u03c6), and consider its Fenchel-Rockafellar dual minimize \u21c0 u\u2208M(\u2126,Rk\u00d7d) \u21c0 w\u2208M(\u2126,Rk) R \u2126\u2225 \u21c0 u(x)\u2225u + \u03b1\u2225 \u21c0 w(x)\u2225w dx subject to \u2212\u2207\u2217 x \u21c0 u \u2212\u2207\u2217 G \u21c0 w = \u21c0 \u03bb0 \u2212 \u21c0 \u03bb1 as members of (C1(\u2126, Rk))\u2217. The constraint \u2212\u2207\u2217 x \u21c0 u = \u21c0 \u03bb0 \u2212 \u21c0 \u03bb1 \u2212divG \u21c0 w \u2208M(\u2126, Rk) \u2282(C1(\u2126, Rk))\u2217 implies \u2212\u2207\u2217 x \u21c0 u \u2208M(\u2126, Rk), i.e., \u21c0 u satis\ufb01es the zero-\ufb02ux boundary condition. We now state the rigorous form of the primal problem (5) minimize \u21c0 u\u2208M(\u2126,Rk\u00d7d) \u21c0 w\u2208M(\u2126,Rk) R \u2126\u2225 \u21c0 u(x)\u2225u + \u03b1\u2225 \u21c0 w(x)\u2225w dx subject to divx \u21c0 u + divG \u21c0 w = \u21c0 \u03bb0 \u2212 \u21c0 \u03bb1 as members of M(\u2126, Rk) \u21c0 u satis\ufb01es zero-\ufb02ux b.c in the weak sense. (11) The point \u21c0 \u03c6 = 0 satis\ufb01es the assumption of Theorem 5. Furthermore, it is easy to verify that the optimal value of the dual problem (6) is bounded. This implies strong duality, (11) is feasible, and (11) has a solution. For V ( \u21c0 \u03bb0, \u21c0 \u03bb1) to be a metric, the 4 metric axioms must be satis\ufb01ed. These are relatively straightforward to prove, and interested readers can \ufb01nd the argument in [16]. However, one more implicit axiom (not shown in previous works) must be shown: V ( \u21c0 \u03bb0, \u21c0 \u03bb1) < \u221efor all \u21c0 \u03bb0, \u21c0 \u03bb1 \u2208P(\u2126, Rk). This follows from the fact that (11) has a solution. This completes the proof that V : P(\u2126, Rk) \u00d7 P(\u2126, Rk) \u2192R+ de\ufb01nes a metric. 5.6 Formal setup of Matrix-OMT The rigorous form of the primal problem (7) is minimize U\u2208M(\u2126,Hd) W\u2208M(\u2126,S\u2113) R \u2126\u2225U(x)\u2225u + \u03b1\u2225W(x)\u2225w dx subject to divx(U) + divLW = \u039b0 \u2212\u039b1 as members of M(\u2126, H) U satis\ufb01es zero-\ufb02ux b.c in the weak sense, and the rigorous form of the dual problem (8) is maximize \u03a6\u2208W 1,\u221e(\u2126,H) R \u2126\u27e8\u03a6(x), \u039b1(x) \u2212\u039b0(x)\u27e9dx subject to \u2225\u2207x\u03a6(x)\u2225u\u2217\u22641 \u2225\u2207L\u03a6(x)\u2225w\u2217\u2264\u03b1 for all x \u2208\u2126, where the measure and Sobolev spaces are de\ufb01ned similarly. Theorems 3 and 4 follow from the exact same reasoning as that of the vector counterparts. 9 \f6 Algorithmic preliminaries Consider the Lagrangian for the vector optimal transport problems (5) and its dual (6) L( \u21c0 u, \u21c0 w, \u21c0 \u03c6) = Z \u2126 \u2225 \u21c0 u(x)\u2225u + \u03b1\u2225 \u21c0 w(x)\u2225w dx + Z \u2126 \u27e8 \u21c0 \u03c6(x), divx( \u21c0 u)(x) + divG( \u21c0 w(x)) \u2212 \u21c0 \u03bb0(x) + \u21c0 \u03bb1(x)\u27e9dx, (12) which is convex with respect to \u21c0 u and \u21c0 w and concave with respect to \u03c6. Finding a saddle point of (12) is equivalent to solving (5) and (6), when the primal problem (5) has a solution, the dual problem (6) has a solution, and the optimal values of (5) and (6) are equal. See [4, Theorem 7.1], [37, Theorem 2], or any reference on standard convex analysis such as [51] for further discussion on this point. To solve the optimal transport problems, we discretize the continuous problems and apply PDHG method, which we soon describe, to solve the discretized convex-concave saddle point problem. 6.1 PDHG method Consider the convex-concave saddle function L(x, y, z) = f(x) + g(y) + \u27e8Ax + By, z\u27e9\u2212h(z), where f, g, and h are (closed and proper) convex functions and x \u2208Rn, y \u2208Rm, z \u2208Rl, A \u2208Rl\u00d7n, and B \u2208Rl\u00d7m. Note L is convex in x and y and concave in z. Assume L has a saddle point and step sizes \u00b5, \u03bd, \u03c4 > 0 satisfy 1 > \u03c4\u00b5\u03bbmax(AT A) + \u03c4\u03bd\u03bbmax(BT B). Write \u2225\u00b7 \u22252 for the standard Euclidean norm. Then the method xk+1 = argmin x\u2208Rn \u001a L(x, yk, zk) + 1 2\u00b5\u2225x \u2212xk\u22252 2 \u001b yk+1 = argmin y\u2208Rm \u001a L(xk, y, zk) + 1 2\u03bd \u2225y \u2212yk\u22252 2 \u001b (13) zk+1 = argmax z\u2208Rl \u001a L(2xk+1 \u2212xk, 2yk+1 \u2212yk, z) \u22121 2\u03c4 \u2225z \u2212zk\u22252 2 \u001b converges to a saddle point. This method is called the Primal-Dual Hybrid Gradient (PDHG) method or the (preconditioned) Chambolle-Pock method [25, 12, 49]. PDHG can be interpreted as a proximal point method under a certain metric [32]. The quantity Rk = 1 \u00b5\u2225xk+1 \u2212xk\u22252 2 + 1 \u00b5\u2225yk+1 \u2212yk\u22252 2 + 1 \u03c4 \u2225zk+1 \u2212zk\u22252 2 \u22122\u27e8zk+1 \u2212zk, A(xk+1 \u2212xk) + B(yk+1 \u2212yk)\u27e9. is the \ufb01xed-point residual of the non-expansive mapping de\ufb01ned by the proximal point method. Therefore Rk = 0 if and only if (xk, yk, zk) is a saddle point of L, and Rk decreases monotonically to 0, cf., review paper [52]. We can use Rk as a measure of progress and as a termination criterion. 6.2 Shrink operators As the subproblems of PDHG (13) are optimization problems themselves, PDHG is most e\ufb00ective when these subproblems have closed-form solutions. The problem de\ufb01nitions of scalar, vector, and matrix-OMT involve norms. For some, but not all, choices of norms, the \u201cshrink\u201d operators shrink(x0; \u00b5) = argmin x\u2208Rn \b \u00b5\u2225x\u2225+ (1/2)\u2225x \u2212x0\u22252 2 \t 10 \fhave closed-form solutions. Therefore, when possible, it is useful to choose such norms for computational e\ufb03ciency. Readers familiar with the compressed sensing or proximal methods literature may be familiar with this notion. For the vector-OMT, we focus on norms \u2225 \u21c0 u\u22252 2 = d X s=1 \u2225us\u22252 2 \u2225 \u21c0 u\u22251,2 = d X s=1 \u2225us\u22252 \u2225 \u21c0 u\u22251 = k X s=1 \u2225us\u22251 for \u21c0 u \u2208Rk\u00d7d and \u2225 \u21c0 w\u22252 2 = \u2113 X s=1 (ws)2 \u2225 \u21c0 w\u22251 = \u2113 X s=1 |ws| for \u21c0 w \u2208R\u2113. The shrink operators of these norms have closed-form solutions. For the matrix-OMT, we focus on norms \u2225U\u22252 2 = d X s=1 k X i,j=1 |(Us)ij|2 \u2225U\u22251 = d X s=1 k X i,j=1 |(Us)ij| \u2225U\u22251,nuc = d X s=1 \u2225Us\u2225nuc for U \u2208Hd and \u2225\u00b7 \u22252, \u2225\u00b7 \u22251, and \u2225\u00b7 \u22251,nuc for W \u2208S\u2113, which are de\ufb01ned likewise. The nuclear norm \u2225\u00b7 \u2225nuc is the sum of the singular values. The shrink operators of these norms have closed-form solutions. We provide further information and details on shrink operators in the appendix. 7 Algorithms We now present simple and parallelizable algorithms for the OMT problems. These algorithms are, in particular, very well-suited for GPU computing. In Section 7 and 8 we use an n \u00d7 n discretization of the 2D domain [0, 1] \u00d7 [0, 1] to obtain approximate solutions to the continuous problems. For simplicity of notation, we use the same symbol to denote the discretized variables and their continuous counterparts. Whether we are referring to the continuous variable or its discretization should be clear from context. As mentioned in Section 6, these methods are the PDHG method applied to discretizations of the continuous problems. In the implementation, it is important to get the discretization at the boundary correct in order to respect the zero-\ufb02ux boundary conditions. For interested readers, the details are provided in the appendix. Instead of detailing the somewhat repetitive derivations of the algorithms in full, we simply show the key steps and arguments for the \u21c0 u update of vector-OMT. The other steps follow from similar logic. When we discretize the primal and dual vector-OMT problems and apply PDHG to the discretized Lagrangian form of (12), we get \u21c0 uk+1 = argmin \u21c0 u\u2208Rn\u00d7n\u00d7k\u00d7d \uf8f1 \uf8f2 \uf8f3 X ij \u2225 \u21c0 uij\u2225u + \u27e8 \u21c0 \u03c6ij, (divxu)ij\u27e9+ 1 2\u00b5\u2225 \u21c0 uij \u2212 \u21c0 uk ij\u22252 2 \uf8fc \uf8fd \uf8fe = argmin \u21c0 u\u2208Rn\u00d7n\u00d7k\u00d7d \uf8f1 \uf8f2 \uf8f3 X ij \u00b5\u2225 \u21c0 uij\u2225u \u2212\u00b5\u27e8(\u2207x \u21c0 \u03c6)ij, uij\u27e9+ (1/2)\u2225 \u21c0 uij \u2212 \u21c0 uk ij\u22252 2 \uf8fc \uf8fd \uf8fe. Since the minimization splits over the i, j indices, we write \u21c0 uk+1 ij = argmin \u21c0 uij\u2208Rk\u00d7d n \u00b5\u2225 \u21c0 uij\u2225u \u2212\u00b5\u27e8(\u2207x \u21c0 \u03c6)ij, uij\u27e9+ (1/2)\u2225 \u21c0 uij \u2212 \u21c0 uk ij\u22252 2 o = argmin \u21c0 uij\u2208Rk\u00d7d n \u00b5\u2225 \u21c0 uij\u2225u + (1/2)\u2225 \u21c0 uij \u2212( \u21c0 uk ij + \u00b5(\u2207x \u21c0 \u03c6)ij)\u22252 2 o = shrink( \u21c0 uk ij + \u00b5(\u2207x \u21c0 \u03c6)ij; \u00b5). 11 \fAt the boundary, these manipulations need special care. When we incorporate ghost cells in our discretization, these seemingly cavalier manipulations are also correct on the boundary. We further explain the ghost cells and discretization in the appendix. 7.1 Scalar-OMT algorithm The scalar-OMT algorithm can be viewed as a special case of vector-OMT or matrix-OMT algorithms. This scalar-OMT algorithm was presented in [36], but we restate it here for completeness. First-order Method for S-OMT Input: Problem data \u03bb0, \u03bb1 Initial guesses u0, \u03c60 and step sizes \u00b5, \u03c4 Output: Optimal u\u22c6and \u03c6\u22c6 for k = 1, 2, \u00b7 \u00b7 \u00b7 (Iterate until convergence) uk+1 ij = shrink(uk ij + \u00b5(\u2207\u03a6k)ij, \u00b5) for i, j = 1, . . . , n \u03c6k+1 ij = \u03c6k ij + \u03c4(divx(2uk+1 \u2212uk)ij + \u03bb1 ij \u2212\u03bb0 ij) for i, j = 1, . . . , n end This method converges for step sizes \u00b5, \u03c4 > 0 that satisfy 1 > \u03c4\u00b5\u03bbmax(\u2212\u2206x). For the particular setup of \u2126= [0, 1]\u00d7[0, 1] and \u2206x = 1/(n\u22121), the bound \u03bbmax(\u2212\u2206x) \u22648/(\u2206x)2 = 8(n\u22121)2 is known. In our experiments, we use \u00b5 = 1/(16\u03c4(n \u22121)2), a choice that ensures convergence for any \u03c4 > 0. We tune \u03c4 for the fastest convergence. 7.2 Vector-OMT algorithm Write shrinku and shrinkw for the shrink operators with respect to \u2225\u00b7 \u2225u and \u2225\u00b7 \u2225w. The vector-OMT algorithm is as follows: First-order Method for V-OMT Input: Problem data G, \u21c0 \u03bb0, \u21c0 \u03bb1, \u03b1 Initial guesses \u21c0 u0, \u21c0 w0, \u21c0 \u03c60 and step sizes \u00b5, \u03bd, \u03c4 Output: Optimal \u21c0 u\u22c6, \u21c0 w\u22c6, and \u21c0 \u03c6\u22c6 for k = 1, 2, \u00b7 \u00b7 \u00b7 (Iterate until convergence) \u21c0 uk+1 ij = shrinku( \u21c0 uk ij + \u00b5(\u2207 \u21c0 \u03c6k)ij, \u00b5) for i, j = 1, . . . , n \u21c0 wk+1 ij = shrinkw( \u21c0 wk ij + \u03bd(\u2207G \u21c0 \u03c6k ij), \u03b1\u03bd) for i, j = 1, . . . , n \u21c0 \u03c6k+1 ij = \u21c0 \u03c6k ij + \u03c4(divx(2 \u21c0 uk+1 \u2212 \u21c0 uk)ij + divG(2 \u21c0 wk+1 \u2212 \u21c0 wk)ij + \u21c0 \u03bb1 ij \u2212 \u21c0 \u03bb0 ij) for i, j = 1, . . . , n end This method converges for step sizes \u00b5, \u03bd, \u03c4 > 0 that satisfy 1 > \u03c4\u00b5\u03bbmax(\u2212\u2206x) + \u03c4\u03bd\u03bbmax(\u2212\u2206G). For the particular setup of \u2126= [0, 1]\u00d7[0, 1] and \u2206x = 1/(n\u22121), the bound \u03bbmax(\u2212\u2206x) \u22648(n\u22121)2 is known. Given a graph G, we can compute \u03bbmax(\u2212\u2206G) with a standard eigenvalue routine. In our experiments, we use \u00b5 = 1/(32\u03c4(n \u22121)2) and \u03bd = 1/(4\u03c4\u03bbmax(\u2212\u2206G)), a choice that ensures convergence for any \u03c4 > 0. We tune \u03c4 for the fastest convergence. 12 \f7.3 Matrix-OMT algorithm Write shrinku and shrinkw for the shrink operators with respect to \u2225\u00b7 \u2225u and \u2225\u00b7 \u2225w. The matrix-OMT algorithm is as follows: First-order Method for M-OMT Input: Problem data L, \u039b0, \u039b1, \u03b1, Initial guesses U0, W0, \u03a60 and step sizes \u00b5, \u03bd, \u03c4 Output: Optimal U\u22c6, W\u22c6, and \u03a6\u22c6 for k = 1, 2, \u00b7 \u00b7 \u00b7 (Iterate until convergence) Uk+1 ij = shrinku(Uk ij + \u00b5(\u2207\u03a6k)ij, \u00b5) for i, j = 1, . . . , n Wk+1 ij = shrinkw(Wk ij + \u03bd(\u2207L\u03a6k ij), \u03b1\u03bd) for i, j = 1, . . . , n \u03a6k+1 ij = \u03a6k ij + \u03c4(divx(2Uk+1 \u2212Uk)ij + divL(2Wk+1 \u2212Wk)ij + \u039b1 ij \u2212\u039b0 ij) for i, j = 1, . . . , n end This method converges for step sizes \u00b5, \u03bd, \u03c4 > 0 that satisfy 1 > \u03c4\u00b5\u03bbmax(\u2212\u2206x) + \u03c4\u03bd\u03bbmax(\u2212\u2206L). For the particular setup of \u2126= [0, 1] \u00d7 [0, 1] and \u2206x = 1/(n \u22121), the bound \u03bbmax(\u2212\u2206x) \u22648(n \u22121)2 is known. Given L, we can compute the value of \u03bbmax(\u2212\u2206L) by explicitly forming a k2 \u00d7 k2 matrix representing the linear operator \u2212\u2206L and applying a standard eigenvalue routine. In our experiments, we use \u00b5 = 1/(32\u03c4(n \u22121)2) and \u03bd = 1/(4\u03c4\u03bbmax(\u2212\u2206L)), a choice that ensures convergence for any \u03c4 > 0. We tune \u03c4 for the fastest convergence. 7.4 Parallelization For the vector-OMT, the computation for the \u21c0 u, \u21c0 w, and \u21c0 \u03c6 updates splits over the indices (i, j), i.e., the computation splits pixel-by-pixel. Furthermore, the \u21c0 u and \u21c0 w updates can be done in parallel. Parallel processors handing jobs split over (i, j) must be synchronized before and after the \u21c0 \u03c6 update. The same is true for the scalar-OMT and matrix-OMT. This highly parallel and regular algorithmic structure makes the proposed algorithms very well suited for CUDA GPUs. We demonstrate this through our experiments. 8 Examples In this section, we provide example applications of vector and matrix-OMT with numerical tests to demonstrate the e\ufb00ectiveness of our algorithms. As mentioned in the introduction, potential applications of vector and matrix-OMT are broad. Here, we discuss two of the simplest applications. We implemented the algorithm on C++ CUDA and ran it on a Nvidia GPU. For convenience, we MEXed this code, i.e., the code is made available as a Matlab function. For scienti\ufb01c reproducibility, we release this code. 8.1 Color images Color images in RGB format is one of the more immediate examples of vector-valued densities. At each spatial position of a 2D color image, the color is represented as a combination of the three basic colors red (R), green (G), and blue (B). We allow any basic color to change to another basic color with cost c1 for R to G, c2 for R to B, and c3 for G to B. So the graph G as described in Section 3.1 has 3 nodes and 3 edges. Consider the two color images on the domain \u2126= [0, 1] \u00d7 [0, 1] with 256 \u00d7 256 discretization shown in Figures 2a and 2b. The initial and target densities \u21c0 \u03bb0 and \u21c0 \u03bb1 both have three disks at the same location, 13 \f(a) \u21c0 \u03bb0 (b) \u21c0 \u03bb1 (c) Velocity \ufb01eld u with c1 = 1, c2 = 1, c3 = 1, \u03b1 = 1 and \u2225 \u21c0 u\u22251,2, \u2225 \u21c0 w\u22251. V ( \u21c0 \u03bb0, \u21c0 \u03bb1) = 0.57. (d) Velocity \ufb01eld u with c1 = 1, c2 = 1, c3 = 1, \u03b1 = 1 and \u2225 \u21c0 u\u22251, \u2225 \u21c0 w\u22251. V ( \u21c0 \u03bb0, \u21c0 \u03bb1) = 0.67. Figure 2: Color image vector-OMT example. but with di\ufb00erent colors. The optimal \ufb02ux depends on the choice of norms. Figures 2c and 2d, show \ufb02uxes optimal with respect to di\ufb00erent norms. Whether it is optimal to spatially transport the colors or to change the colors depends on the parameters c1, c2, c3, \u03b1 as well as the norms \u2225\u00b7 \u2225u and \u2225\u00b7 \u2225w. With the parameters of Figure 3a it is optimal to spatially transport the colors, while with the parameters of Figure 3b it is optimal to change the colors. Finally, we test our algorithm on the setup of Figure 2c with grid sizes 32 \u00d7 32, 64 \u00d7 64, 128 \u00d7 128, and 256 \u00d7 256. Table 1 shows the number of iterations and runtime tested on a Nvidia Titan Xp GPU required 14 \f(a) c1 = 1, c2 = 1, c3 = 1, \u03b1 = 10. (b) c1 = 1, c2 = 1, c3 = 1, \u03b1 = 0.1. Figure 3: Color image vector-OMT example, with a more complicated shape and norms \u2225 \u21c0 u\u22251,2, \u2225 \u21c0 w\u22252. Grid Size Iteration count Time per-iter Iteration time \u03c4 V ( \u21c0 \u03bb0, \u21c0 \u03bb1) 32 \u00d7 32 0.5 \u00d7 104 27\u00b5s 0.14s 1 0.57 64 \u00d7 64 0.5 \u00d7 104 27\u00b5s 0.14s 1 0.57 128 \u00d7 128 2 \u00d7 104 23\u00b5s 0.47s 3 0.57 256 \u00d7 256 2 \u00d7 104 68\u00b5s 1.36s 3 0.57 Table 1: Computation cost for vector-OMT as a function of grid size. to achieve a 10\u22123 precision, measured as the ratio between the duality gap and primal objective value. 8.2 Di\ufb00usion tensor imaging Di\ufb00usion tensor imaging (DTI) is a technique used in magnetic resonance imaging. DTI captures orientations and intensities of brain \ufb01bers at each spatial position as ellipsoids and gives us a matrix-valued density. Therefore, the metric de\ufb01ned by the matrix-OMT provides a natural way to compare the di\ufb00erences between brain di\ufb00usion images. In Figure 4 we visualize di\ufb00usion tensor images by using colors to indicate di\ufb00erent orientations of the tensors at each voxel. In this paper, we present simple 2D examples as a proof of concept and leave actual 3D imaging for a topic of future work. Consider three synthetic matrix-valued densities \u039b0, \u039b1, and \u039b2 in Figure 5. The densities \u039b0 and \u039b1 have mass at the same spatial location, but the ellipsoids have di\ufb00erent shapes. The densities \u039b0 and \u039b2 have the same ellipsoids at di\ufb00erent spatial locations. We compute the distance between \u039b0, \u039b1, and \u039b2 for di\ufb00erent parameters \u03b1 and \ufb01xed L = [L1, L2]\u2217with L1 = \uf8ee \uf8f0 1 0 0 0 2 0 0 0 0 \uf8f9 \uf8fb, L2 = \uf8ee \uf8f0 1 1 1 1 0 0 1 0 0 \uf8f9 \uf8fb. Table 2, shows the results with \u2225U\u22252 and \u2225W\u22251 and grid size 128 \u00d7 128. As we can see, whether \u039b0 is more \u201csimilar\u201d to \u039b1 or \u039b2, i.e., whether M(\u03bb0, \u03bb1) < M(\u03bb0, \u03bb2) or M(\u03bb0, \u03bb1) > M(\u03bb0, \u03bb2), depends on whether the cost on U, spatial transport, is higher than the cost on W, changing the ellipsoids. 15 \fFigure 4: Example of 2D di\ufb00usion tensor images. (a) \u039b0 (b) \u039b1 (c) \u039b2 Figure 5: Synthetic matrix-valued distributions. Again, we test our algorithm on the setup of Figure 5 with \u03b1 = 1 on grid sizes 32\u00d732, 64\u00d764, 128\u00d7128, and 256 \u00d7 256. Table 3 shows the number of iterations and runtime tested on a Nvidia Titan Xp GPU required to achieve a precision of 10\u22123, measured as the ratio between the duality gap and primal objective value. 8.3 Discussion The best choice of the norms for the vector and matrix-OMT problem depends on the application. In color image processing, for example, we recommend to use \u2225\u00b7 \u22251,2 for the \u2225\u00b7 \u2225u norm (which penalizes transport over space) since it respects the Euclidean geometry in the spatial domain and the penalties on di\ufb00erent colors are decoupled. On the other hand, the choice for \u2225\u00b7\u2225w norm (which penalizes change in color) seems to matter less. As another example, the nuclear norm is relevant to the problem of Michell trusses in structural mechanics [8]. The GPU implementation provides signi\ufb01cant acceleration. For reference, we compared a serial CPU implementation against a parallel CUDA implementation for the scalar-OMT problem. When we run the method on a 1024 \u00d7 1024 image, the serial implementation took 157.8ms per iteration while the CUDA implementation took 1.27ms per iteration. For computational problems with very simple parallelizable structures, 50-fold to 150-fold speedup is common. An Intel Core i7 CPU X990 @ 3.47GHz and Nvidia 16 \fParameters M(\u039b0, \u039b1) M(\u039b0, \u039b2) M(\u039b1, \u039b2) \u03b1 = 10 2.71 0.27 3.37 \u03b1 = 3 0.81 0.27 1.08 \u03b1 = 1 0.27 0.27 0.54 \u03b1 = 0.3 0.081 0.27 0.35 \u03b1 = 0.1 0.027 0.27 0.29 Table 2: Distances between the three images \u039b0, \u039b1 and \u039b2. Grid Size Iteration count Time per-iter Iteration time \u03c4 M(\u039b0, \u039b1) 32 \u00d7 32 1 \u00d7 104 39\u00b5s 0.39s 10 0.27 64 \u00d7 64 1.5 \u00d7 104 39\u00b5s 0.59s 10 0.27 128 \u00d7 128 2 \u00d7 104 85\u00b5s 1.70s 30 0.27 256 \u00d7 256 4 \u00d7 104 330\u00b5s 13.2s 60 0.27 Table 3: Computation cost for matrix-OMT as a function of grid size. Titan Xp GPU were used for this experiment. 9" + }, + { + "url": "http://arxiv.org/abs/1708.06908v2", + "title": "Proximal-Proximal-Gradient Method", + "abstract": "In this paper, we present the proximal-proximal-gradient method (PPG), a\nnovel optimization method that is simple to implement and simple to\nparallelize. PPG generalizes the proximal-gradient method and ADMM and is\napplicable to minimization problems written as a sum of many differentiable and\nmany non-differentiable convex functions. The non-differentiable functions can\nbe coupled. We furthermore present a related stochastic variation, which we\ncall stochastic PPG (S-PPG). S-PPG can be interpreted as a generalization of\nFinito and MISO over to the sum of many coupled non-differentiable convex\nfunctions. We present many applications that can benefit from PPG and S-PPG and\nprove convergence for both methods. A key strength of PPG and S-PPG is,\ncompared to existing methods, its ability to directly handle a large sum of\nnon-differentiable non-separable functions with a constant stepsize independent\nof the number of functions. Such non-diminishing stepsizes allows them to be\nfast.", + "authors": "Ernest K. Ryu, Wotao Yin", + "published": "2017-08-23", + "updated": "2017-10-18", + "primary_cat": "math.OC", + "cats": [ + "math.OC" + ], + "main_content": "Introduction In the past decade, \ufb01rst-order methods like the proximal-gradient method and ADMM have enjoyed wide popularity due to their broad applicability, simplicity, and good empirical performance on problems with large data sizes. However, there are many optimization problems such existing simple \ufb01rst-order methods cannot directly handle. Without a simple and scalable method to solve them such optimization problems have been excluded from machine learning and statistical modeling. In this paper we present the proximal-proximal-gradient method (PPG), a novel method that expands the class of problems that one can solve with a simple and scalable \ufb01rst-order method. Consider the optimization problem minimize r(x) + 1 n n X i=1 (fi(x) + gi(x)), (1) where x \u2208Rd is the optimization variable, f1, . . . , fn, g1, . . . , gn, and r are convex, closed, and proper functions from Rd to R\u222a{\u221e}. Furthermore, assume 1 arXiv:1708.06908v2 [math.OC] 18 Oct 2017 \ff1, . . . , fn are di\ufb00erentiable. We call the method xk+1/2 = prox\u03b1r 1 n n X i=1 zk i ! xk+1 i = prox\u03b1gi \u0010 2xk+1/2 \u2212zk i \u2212\u03b1\u2207fi(xk+1/2) \u0011 zk+1 i = zk i + xk+1 i \u2212xk+1/2, (PPG) the proximal-proximal-gradient method (PPG). The xk+1 i and zk+1 i updates are performed for all i = 1, . . . , n and \u03b1 > 0 is a stepsize parameter. To clarify, x, x1, . . . , xn and z1, . . . , zn are all vectors in Rd (xi is not a component of x), xk+1 1 , . . . , xk+1 n and xk+1/2 approximates the solution to Problem (1). Throughout this paper we write proxh for the proximal operator with respect to the function h, de\ufb01ned as proxh(x0) = argmin x \u001a h(x) + 1 2\u2225x \u2212x0\u22252 2 \u001b for a function h : Rd \u2192R \u222a{\u221e}. When h is the zero function, proxh is the identity operator. When h is convex, closed, and proper, the minimizer that de\ufb01nes proxh exists and is unique [39]. For many interesting functions h, the proximal operator proxh has a closed or semi-closed form solution and is computationally easy to evaluate [11, 44]. We loosely say such functions are proximable. In general, the proximal-gradient method or ADMM cannot directly solve optimization problems expressed in the form of (1). When f1, . . . , fn are not proximable, ADMM either doesn\u2019t apply or must run another optimization algorithm to evaluate the proximal operators at each iteration. When n \u22652 and g1, . . . , gn are nondi\ufb00erentiable nonseparable, so g1 + \u00b7 \u00b7 \u00b7 + gn is not proximable (although each individual g1, . . . , gn is proximable). Hence, proximal-gradient doesn\u2019t apply. One possible approach to solving (1) is to smooth the non-smooth parts and applying a (stochastic) gradient method. Sometimes, however, keeping nonsmooth part is essential. For example, it is the non-smoothness of total variation penalty that induces sharp edges in image processing. In these situations (PPG) is particularly useful as it can handle a large sum of smooth and non-smooth terms directly without smoothing. Distributed PPG. To understand the algorithmic structure of the method, it is helpful to see how (PPG) is well-suited for a distributed computing network. See Figure 1, which illustrates a parameter server computing model with a master node and n worker nodes. At each iteration, the workers send their zk i to the parameter server, the parameter server computes xk+1/2 and broadcasts it to the workers, and each worker i computes zk+1 i with access to fi, gi and zk i for i = 1, . . . , n. The workers maintain their private copy of zk i and do not directly communicate with each other. 2 \fParameter server Worker 1 (zk 1) Worker 2 (zk 2) \u00b7 \u00b7 \u00b7 Worker n (zk n) Figure 1: When (PPG) is implemented on a parameter server computing model, the worker nodes communicate (synchronously) with the parameter server but not directly with each other. In other words, a distributed implementation performs step 1 of (PPG), the xk+1/2 update, with an all-reduce operation. It performs step 2 and step 3, the xk+1 i and zk+1 i updates, in parallel. Time complexity, space complexity, and parallelization. At each iteration, (PPG) evaluates the proximal operators with respect to r and g1, . . . , gn and computes the gradients of f1, . . . , fn. So based on the iteration cost alone, we can predict (PPG) to be useful when r and g1, . . . , gn are individually proximable and the gradients of f1, . . . , fn are easy to evaluate. If, furthermore, the number of iterations required to reach necessary accuracy is not exorbitant, (PPG) is actually useful. Let\u2019s say the computational costs of evaluating prox\u03b1r is cr, prox\u03b1gi is cg for i = 1, . . . , n, and \u2207fi is cf for i = 1, . . . , n. Then the time complexity of (PPG) is O(nd+cr +ncg +ncf) per iteration (recall x \u2208Rd). As discussed, this cost can be reduced with parallelization. Computing xk+1/2 involves computing an average (a reduce operation), computing prox\u03b1r, and a broadcast, and computing zk+1 i for i = 1, . . . , n is embarrassingly parallel. The space complexity of (PPG) is O(nd) since it must store zk 1, . . . zk n. (xk 1, . . . , xk n need not be stored.) When the problem has sparse structure, the computation time and storage can be further reduced. For example, if fi + gi does not depend on (xi)1 for some i, then (zi)1 and (xi)1 can be eliminated from the algorithm since (zi)k+1 1 = (xk+1/2)1. The storage requirement of O(nd) is fundamentally di\ufb03cult to improve upon due to the n non-di\ufb00erentiable terms. Consider the case where r = f1 = \u00b7 \u00b7 \u00b7 = fn = 0: minimize g1(x) + \u00b7 \u00b7 \u00b7 + gn(x). If g1, . . . , gn were di\ufb00erentiable, then \u2207g1(x\u2217)+\u00b7 \u00b7 \u00b7+\u2207gn(x\u2217) = 0 would certify x\u2217is optimal. However, we allow g1, . . . , gn to be non-di\ufb00erentiable, so one must \ufb01nd a particular set of subgradients ui \u2208\u2202gi(x\u2217) for i = 1, . . . , n such that u1 + \u00b7 \u00b7 \u00b7 + un = 0 to certify x\u2217is optimal. The choices of subgradients, u1, . . . , un, depend on each other and cannot be found independently. In other words, certifying optimality requires O(nd) information, and that is what PPG uses. For comparison, ADMM also uses O(nd) storage when used to minimize 3 \fa sum of n non-smooth functions. Stochastic PPG. Each iteration of (PPG) updates zk+1 i for all i = 1, . . . , n, which takes at least O(nd) time per iteration. In Section 3 we present the method (S-PPG) which can be considered a stochastic variation of (PPG). Each iteration of (S-PPG) only updates one zk+1 i for some i and therefore can take as little as O(d) time per iteration. When compared epoch by epoch, (S-PPG) can be faster than (PPG). Convergence. Assume Problem (1) has a solution (not necessarily unique) and meets a certain regularity condition. Furthermore, assume each fi in has L-Lipschitz gradient for i = 1, . . . , n, so \u2225\u2207fi(x) \u2212\u2207fi(y)\u22252 \u2264L\u2225x \u2212y\u22252 for all x, y \u2208Rd and i = 1, . . . , n. Then (PPG) converges to a solution of Problem (1) for 0 < \u03b1 < 3/(2L). In particular, we do not need strong convexity to establish convegence. In section 4 we discuss convergence in more detail. In particular, we prove both that the iterates converge to a solution and that the objective values converge to the optimal value. Contribution of this work. The methods of this paper, (PPG) and (S-PPG), are novel methods that can directly handle a sum of many di\ufb00erentiable and nondi\ufb00erentiable but proximable functions. To solve such problems, exising \ufb01rstorder methods like ADMM must evaluate proximal operators of di\ufb00erentiable functions, which may hinder computational performance if said operator has no closed form solution. Furthermore, the simplicity of our methods allows simple and e\ufb03cient parallelization, a point we discuss brie\ufb02y. The theoretical analysis of (PPG) and (S-PPG), especially that of (S-PPG), is novel. As we discuss later, (S-PPG) can be interpreted as a generalization to varianced reduced gradient methods like Finito/MISO or SAGA. The techniques we use to analyze (S-PPG) is di\ufb00erent from those used to analyze other varianced reduced gradient methods, and we show more general (albeit not faster) convergence guarantees. In particular, we establish almost sure convergence of iterates, and we do so without any strong convexity assumptions. To the best of our knowledge, the existing varianced reduced gradient method literature does not prove such results. Finally, our method is the \ufb01rst work to establish a clear connection between operator splitting methods and varianced reduced gradient methods. As the name implies, existing varianced reduced gradient methods view the method as improving, by reducing variance, stochastic gradient methdos. Our work identi\ufb01es Finito/MISO as stochastic block-coordinate update applied to an operator splitting method. It is this observation that allows us to analyze (S-PPG), which generalizes Finito/MISO to handle a sum of non-di\ufb00erentiable but proximable functions. 4 \f2 Relationship to other methods In this section, we discuss how (PPG) generalizes certain known methods. To clarify, PPG is a proper generalization of these existing methods and cannot be analyzed as a special case of one of these methods. Proximal-gradient. When gi = 0 for i = 1, . . . , n in (1), (PPG) simpli\ufb01es to xk+1 = prox\u03b1r xk \u2212\u03b1 n n X i=1 \u2207fi(xk) ! . This method is the called proximal-gradient method or forward-backward splitting [45, 13]. ADMM. When f = r = 0 in (1), (PPG) simpli\ufb01es to xk+1 i = argmin x \u001a gi(x) \u2212(yk i )T x + 1 2\u03b1\u2225x \u2212xk\u22252 2 \u001b yk+1 i = yk i + \u03b1(\u00af xk+1 \u2212xk+1 i ) where \u00af xk+1 = 1 n n X i=1 xk+1 i . This is also an instance of ADMM [21, 20]. See \u00a77.1 of [8]. Generalized forward-backward splitting. When r = 0 and f1 = f2 = \u00b7 \u00b7 \u00b7 = fn = f in (1), (PPG) simpli\ufb01es to zk+1 i = zk i \u2212xk + prox\u03b1gi \u00002xk \u2212zk i \u2212\u03b1\u2207f(xk) \u0001 xk+1 = 1 n n X i=1 zk+1 i . This is an instance of generalized forward-backward splitting [46]. Davis-Yin splitting. When n = 1 in (1), (PPG) reduces to Davis-Yin splitting, and much of convergence analysis for (PPG) is inspired by that of DavisYin splitting [15]. However, (PPG) is more general and parallelizable. Furthermore, (1) has a stochastic variation (S-PPG). 3 Stochastic PPG Each iteration of (PPG) updates zk+1 i for all i = 1, . . . , n, which requires at least O(nd) time (without parallelization or sparse structure). Often, the data that speci\ufb01es Problem (1) is of size O(nd), and, roughly speaking, (PPG) must process the entire dataset every iteration. This cost of O(nd) time per iteration may be ine\ufb03cient in certain applications. 5 \fThe following method, which we call the stochastic proximal-proximal-gradient method (S-PPG), overcomes this issue: xk+1/2 = prox\u03b1r 1 n n X i=1 zk i ! i(k) \u223cUniform({1, . . . , n}) xk+1 i(k) = prox\u03b1gi(k) \u0010 2xk+1/2 \u2212zk i(k) \u2212\u03b1\u2207fi(k)(xk+1/2) \u0011 zk+1 j = ( zk i(k) + xk+1 i(k) \u2212xk+1/2 for j = i(k), zk j for j \u0338= i(k). (S-PPG) At each iteration, only zk+1 i(k) is updated, where the index i(k) is chosen uniformly at random from {1, . . . , n}. We can interpret (S-PPG) as a stochastic or coordinate update version of (PPG). Time and space complexity. The space requirement of (S-PPG) is no different from that of (PPG); both methods use O(nd) space to store zk 1, . . . , zk n. However, the cost per iteration of (S-PPG) can be as low as O(d). This is achieved with the following simple trick: maintain the quantity \u00af zk = 1 n n X i=1 zk i , and update it with \u00af zk+1 = \u00af zk + (1/n)(xk+1 i(k) \u2212xk+1/2). Application to big-data problems. Consider a big-data problem setup where the data that describes Problem (1) is stored on a hard drive, but is too large to \ufb01t in a system\u2019s memory. Under this setup, an optimization algorithm that goes through the entire dataset every iteration is likely impractical. (S-PPG) can handle this setup e\ufb00ectively by keeping the data and the zk i iterates on the hard drive as illustrated in Figure 3. At iteration k, (S-PPG) selects index i(k), reads block i(k) containing fi(k), gi(k), and zk i(k) from the hard drive, performs computation, updates \u00af zk+1, and writes zk+1 i(k) back to block i(k). The \u00af zk iterate is used and updated every iteration and therefore should be stored in memory. Interpretation as a variance reduced gradient method. Remarkably (S-PPG) converges to a solution with a \ufb01xed value of \u03b1 (which is independent of n). Many traditional and modern stochastic optimization methods require their step sizes to diminish to 0, theoretically and empirically, and this limits their rates of convergence. On the other hand, several modern \u201cvariance reduced gradient\u201d methods take advantage of a \ufb01nite sum structure similar to that of (1) and achieve a 6 \fCPU/Memory, \u00af zk f1, g1, zk 1 f2, g2, zk 2 \u00b7 \u00b7 \u00b7 fn, gn, zk n Figure 2: An illustration of (S-PPG) applied to a big-data problem. The bottom blocks represent n blocks of data stored on the hard drive. The CPU accesses only one of the n blocks per iteration. faster rate with a constant step size. In fact, these methods achieve superior performance compared to full gradient updates. Such methods include Finito, MISO, SVRG, SAG, SAGA, and SDCA [31, 34, 26, 69, 54, 16, 17, 41, 63, 35, 29, 52, 53, 55]. In fact, (S-PPG) directly generalizes Finito and MISO. When g1 = \u00b7 \u00b7 \u00b7 = gn = 0 and r = 0 in (1), we can rewrite (S-PPG) as wk = 1 n n X i=1 zk i i(k) \u223cUniform({1, . . . , n}) \u03c6k+1 i(k) = wk zk+1 i(k) = \u03c6k+1 i(k) \u2212\u03b1\u2207fi(k)(\u03c6k+1 i(k) ) zk+1 j = zk j for j \u0338= i(k), which is Finito and an instance of MISO [34, 17]. Therefore it is appropriate to view (S-PPG) as a variance reduced gradient method as opposed to a stochastic gradient method. Of course, (S-PPG) is more general as it can handle a sum of non-smooth terms as well. Comparison to SAGA. (S-PPG) is more general than SAGA [16] as it can directly handle a sum of many non-di\ufb00erentiable functions. However, when g1 = \u00b7 \u00b7 \u00b7 = gn = 0 in (1), one can use SAGA, and it is interesting to compare (S-PPG) with SAGA under this scenario. The di\ufb00erence in storage requirement is small. (S-PPG) must store at least nd numbers while SAGA must store at least (n + 1)d. This is because SAGA stores the current iterate and n gradients, while (S-PPG) only stores z1, . . . , zn. On the other hand, there is a di\ufb00erence in memory access. At each iteration (S-PPG) reads and updates \u00af z while SAGA reads and updates the current iterate and average gradient. So (S-PPG) reads n fewer numbers and writes n fewer numbers per iteration compared to SAGA. Both SAGA and (S-PPG) read and update information corresponding to a randomly chosen index, and the memory access for this is comparable. 7 \fComparison to stochastic proximal iteration. When f1 = \u00b7 \u00b7 \u00b7 = fn = 0 and r = 0 in (1), the optimization problem is minimize 1 n n X i=1 gi(x). A stochastic method that can solve this problem is stochastic proximal iteration: i(k) \u223cUniform({1, . . . , n}) xk+1 = prox\u03b1kgi(k)(xk), where \u03b1k is a appropriately decreasing step size. Stochastic proximal iteration has been studied under many names such as stochastic proximal point, incremental stochastic gradient, and implicit stochastic gradient [30, 28, 38, 5, 60, 49, 61, 6, 51, 62, 59]. Stochastic proximal iteration requires the step size \u03b1k to be diminishing, whereas (S-PPG) converges with a constant step size. As mentioned, optimization methods with diminishing step size tend to have slower rates, which we can observe in the numerical experiments. We experimentally compare (PPG) and (S-PPG) to stochastic proximal iteration in Section 6. Communication e\ufb03cient implementation. One way to implement (S-PPG) on a distributed computing network so that communication between nodes are minimized is to have nodes update and randomly pass around the z variable. See Figure 3. Each iteration, the current node updates z and passes it to another randomly selected node. Every neighbor and the current node is chosen with probability 1/n. The communication cost of this implementation of (S-PPG) is O(d) per iteration. When the number of iterations required for convergence is not large, this method is communication e\ufb03cient. For recent work on communication e\ufb03cient optimization methods, see [71, 40, 65, 24, 56, 1, 70]. Convergence. Assume Problem (1) has a solution (not necessarily unique) and meets a certain regularity condition. Furthermore, assume each fi in has LLipschitz continuous gradient for i = 1, . . . , n, so \u2225\u2207fi(x) \u2212\u2207f(y)\u2225\u2264L\u2225x \u2212y\u2225 for all x, y \u2208Rd and i = 1, . . . , n. Then (S-PPG) converges to a solution of Problem (1) for 0 < \u03b1 < 3/(2L). In particular, we do not assume strong convexity to establish convergence, whereas many of the mentioned variance reduced gradient methods do. 4 Convergence In this section, we present and discuss the convergence of (PPG) and (S-PPG). For this section, we introduce some new notation. We write x = (x1, . . . , xn), and we use other boldface letters like \u03bd and z in a similar manner. We use the 8 \ff3, g3, z3 current z f4, g4, z4 past z f5, g5, z5 past z f6, g6, z6 past z f1, g1, z1 past z f2, g2, z2 past z Figure 3: Distributed implementation of (S-PPG) without synchronization. Node 3 has the current copy of z, and will pass it to another randomly selected node. bar notation for \u00af z = (z1 + \u00b7 \u00b7 \u00b7 + zn)/n. We write \u00af f(x) = (f1(x) + \u00b7 \u00b7 \u00b7 + fn(x))/n \u00af g(x) = (g1(x) + \u00b7 \u00b7 \u00b7 + gn(x))/n, and, with some abuse of notation, we write \u00af f(x) = (f1(x1) + \u00b7 \u00b7 \u00b7 + fn(xn))/n \u00af g(x) = (g1(x1) + \u00b7 \u00b7 \u00b7 + gn(xn))/n. Note that fi and gi for depend xi instead of a common x. The main problem (1) is equivalent to minimize r(x) + \u00af f(x) + \u00af g(x) (2) subject to x \u2212xi = 0, i = 1, . . . , n, (3) where x and x = (x1, . . . , xn) are the optimization variables. Convex duality tells us that under certain regularity conditions x\u22c6is a solution of Problem (1) if and only if (x\u22c6, x\u22c6, \u03bd\u22c6) is a saddle point of the Lagrangian L(x, x, \u03bd) = r(x) + \u00af f(x) + \u00af g(x) + 1 n n X i=1 \u03bdi(xi \u2212x), (4) where \u03bd\u22c6= (\u03bd\u22c6 1, . . . , \u03bd\u22c6 n) is a dual solution, and x\u22c6= (x\u22c6, . . . , x\u22c6). We simply assume the Lagrangian (4) has a saddle point. This is not a stringent requirement and is merely assumed to avoid pathologies. 9 \fDe\ufb01ne the mapping p(z) as (p(z))i = (1/\u03b1)(x \u2212x\u2032 i) for i = 1, . . . , n, where x = prox\u03b1r(\u00af z), (5) x\u2032 i = prox\u03b1gi (2x \u2212zi \u2212\u03b1\u2207fi(x)) . With this notation, we can express (PPG) as zk+1 = zk \u2212\u03b1p(zk), and we can say z is a \ufb01xed point of (PPG) and (S-PPG) if and only if p(z) = 0. Lemma 1. z\u22c6= (z\u22c6 1, . . . , z\u22c6 n) is a \ufb01xed point of (PPG) and (S-PPG) if and only if z\u22c6 i = x\u22c6+ \u03b1\u03bd\u22c6 i for i = 1, . . . , n, where x\u22c6and \u03bd\u22c6 1, . . . , \u03bd\u22c6 n are primal and dual solutions. In particular, we can recover the x\u22c6as x\u22c6= prox\u03b1r((1/n)(z\u22c6 1 + \u00b7 \u00b7 \u00b7 + z\u22c6 n)). This lemma provides us insight as to why the method converges. Let\u2019s write zk i = xk i + \u03b1\u03bdk i . Then the updates can be written as xk+1/2 = argmin x \u001a r(x) + (\u00af \u03bdk i )T (\u00af xk i \u2212x) + 1 2\u03b1\u2225x \u2212\u00af xk i \u22252 2 \u001b and xk+1 i = argmin x n f(xk+1/2) + (\u2207f(xk+1/2))T (x \u2212xk+1/2) + g(x) + (x \u22122xk+1/2 + xk i )T \u03bdk i + 1 2\u03b1\u2225x \u22122xk+1/2 + xk i \u22252 2 o . Since (x\u22c6, x\u22c6, \u03bd\u22c6) is a saddle point of the Lagrangian, we can see that z\u22c6 i = x\u22c6+ \u03b1\u03bd\u22c6 i is a \ufb01xed point of (PPG). 4.1 Deterministic analysis. We examine the convergence results for (PPG). Theorem 1. Assume f1, . . . , fn are di\ufb00erentiable and have L-Lipschitz continuous gradients. Assume the Lagrangian (4) has a saddle point and 0 < \u03b1 < 3/(2L). Then the sequence \u2225p(zk)\u22252 \u21920 monotonically with rate \u2225p(zk)\u22252 \u2264O(1/ \u221a k). Furthermore zk \u2192z\u22c6, xk+1/2 \u2192x\u22c6, and xk+1 i \u2192x\u22c6for all i = 1, . . . , n, where z\u22c6is a \ufb01xed point of (PPG) and x\u22c6is a solution of (1). Theorem 1 should be understood as two related but separate results. The \ufb01rst result states p(zk) \u21920 and provides a rate. Since p(z) = 0 implies prox\u03b1r(\u00af z) is a solution, the rate does quantify progress. The second result 10 \fstates that the iterates of (PPG) converge but with no guarantee of rate (just like gradient descent without strong convexity). To obtain a more direct measure of progress, de\ufb01ne Ek = r(xk+1/2) + \u00af f(xk+1/2) + \u00af g(xk+1) \u2212 \u0000r(x\u22c6) + \u00af f(x\u22c6) + \u00af g(x\u22c6) \u0001 . Ek is almost like the suboptimality of iterates, but not quite, as the point where \u00af g is evaluated at is di\ufb00erent from the point where r and \u00af f is evaluated at. In fact, Ek is not necessarily positive. Nevertheless, we can show a rate on |Ek|. Theorem 2. Under the setting of Theorem 1, |Ek| \u2264O(1/ \u221a k). De\ufb01ne a similar quantity ek = \u0000r(xk+1/2) + \u00af f(xk+1/2) + \u00af g(xk+1/2) \u0001 \u2212 \u0000r(x\u22c6) + \u00af g(x\u22c6) + \u00af f(x\u22c6) \u0001 . While ek truly measures suboptimality of xk+1/2, it is possible for ek = \u221efor all k = 1, 2, . . . because r and g are possibly nonsmooth and valued \u221eat some points. We need an additional assumption for ek to be a meaningful quantity. Corollary 1. Assume the setting of Theorem 1. Further assume \u00af g(x) is Lipschitz continuous with parameter Lg. Then 0 \u2264ek \u2264|Ek| + Lg\u2225p(zk)\u2225 and ek = O(1/ \u221a k). The proof of Corollary 1 follows immediately from combining Theorems 1 and 2 with ek\u2019s and Lg\u2019s de\ufb01nitions. The 1/ \u221a k rates for Theorem 2 and Corollary 1 can be improved to the 1/k rates by using the ergodic iterates: xk+1/2 erg = 1 k k X j=1 xj+1/2, xk+1 erg = 1 k k X j=1 xj+1. With these ergodic iterates, we de\ufb01ne Ek erg = \u0000r(xk+1/2 erg ) + \u00af f(xk+1/2 erg ) + \u00af g(xk+1 erg ) \u0001 \u2212 \u0000r(x\u22c6) + \u00af g(x\u22c6) + \u00af f(x\u22c6) \u0001 , ek erg = \u0000r(xk+1/2 erg ) + \u00af f(xk+1/2 erg ) + \u00af g(xk+1/2 erg ) \u0001 \u2212 \u0000r(x\u22c6) + \u00af g(x\u22c6) + \u00af f(x\u22c6) \u0001 . Theorem 3. Assume the setting of Theorem 1. Then |Ek erg| \u2264O(1/k) Further assume \u00af g(x) is Lipschitz continuous with parameter Lg. Then ek erg \u2264O(1/k). 11 \fFinally, under rather strong conditions on the problems, linear convergence can also be shown. Theorem 4. Assume the setting of Theorem 1. Furthermore, assume \u00af g is di\ufb00erentiable with Lipschitz continuous gradient. If one (or more) of r, \u00af g, or \u00af f is strongly convex, then (PPG) converges linearly in the sense that \u2225zk \u2212z\u22c6\u22252 2 \u2264O(e\u2212Ck) for some C > 0. Consequently, |Ek| and ek also converge linearly. 4.2 Stochastic analysis. As it turns out, the condition that guarantees (S-PPG) converges is the same as that of (PPG). In particular, there is not step size reduction! Theorem 5. Apply the same assumptions in Theorem 1. That is, assume f1, . . . , fn are di\ufb00erentiable and have L-Lipschitz continuous gradients, and assume the Lagrangian (4) has a saddle point and 0 < \u03b1 < 3/(2L). Then the sequence \u2225p(zk)\u22252 \u21920 with probability one at the rate min i=0,...,k E\u2225p(zi)\u22252 2 \u2264O(1/k). Furthermore zk \u2192z\u22c6, xk+1/2 \u2192x\u22c6, and xk+1 i \u2192x\u22c6for all i = 1, . . . , n with probability one. The expected objective rates of (S-PPG) also match those of (PPG). Theorem 6. Under the same setting of Theorem 5, we have E|Ek| \u2264O(1/ \u221a k) and |Ek erg| \u2264O(1/k). (6) Further assume \u00af g(x) is Lipschitz continuous with parameter Lg. Then Eek erg \u2264O(1/k). Due to space limitation, we state without proof that, under the setting of Theorem 4, (S-PPG) yields linearly convergent E\u2225zk \u2212z\u22c6\u22252, E|Ek|, and E|ek|. 5 Applications of PPG To utilize (PPG), a given optimization problem often needs to be recast into the form of (1). In this section, we show some techniques for this while presenting some interesting applications. All examples presented in this section are naturally posed as minimize r(x) + 1 n n X i=1 fi(x) + 1 m m X j=1 gj(x), (7) 12 \fwhere n \u0338= m. There is more than one way to recast Problem (7) into the form of Problem (1). Among these options, the most symmetric one, loosely speaking, is minimize r(x) + 1 mn n X i=1 m X j=1 \u0000fi(x) + gj(x) \u0001 , which leads to the method xk+1/2 = prox\u03b1r \uf8eb \uf8ed1 mn n X i=1 m X j=1 zk ij \uf8f6 \uf8f8 xk+1 ij = prox\u03b1gj \u0010 2xk+1/2 \u2212zk ij \u2212\u03b1\u2207fi(xk+1/2) \u0011 zk+1 ij = zk ij + xk+1 ij \u2212xk+1/2. In general, the product mn can be quite large, and if so, this approach is likely impractical. In many examples, however, mn is not large as since n = 1 or m is small. Another option, feasible when neither n nor m is small, is minimize r(x) + 1 m + n \uf8eb \uf8ed n X i=1 ((m + n)/n)fi(x) + m X j=1 ((m + n)/m)gj(x) \uf8f6 \uf8f8, which leads to the method xk+1/2 = prox\u03b1r \uf8eb \uf8ed 1 n + m \uf8eb \uf8ed n X i=1 yk i + m X j=1 zk j \uf8f6 \uf8f8 \uf8f6 \uf8f8 yk+1 i = xk+1/2 \u2212\u03b1((m + n)/n)\u2207fi(xk+1/2) zk+1 j = zk j \u2212xk+1/2 + prox\u03b1((m+n)/m)gj \u0010 2xk+1/2 \u2212zk j \u0011 . Overlapping group lasso. Let G be a collection of groups of indices. So G \u2286 {1, 2, . . . , d} for each G \u2208G. The groups can be overlapping, i.e., G1 \u2229G2 \u0338= \u2205 is possible for G1, G2 \u2208G and G1 \u0338= G2. We let xG \u2208R|G| denote a subvector corresponding to the indices of G \u2208G, where x \u2208Rd is the whole vector. So the entries xi for i \u2208G form the vector xG. The overlapping group lasso problem is minimize 1 2\u2225Ax \u2212b\u22252 2 + \u03bb1 X G\u2208G \u2225xG\u22252, where x \u2208Rd is the optimization variable, A \u2208Rm\u00d7d and b \u2208Rm are problem data, and \u03bb1 > 0 is a regularization parameter. As it is, the regularizer (the second term) is not proximable when the groups overlap. 13 \fPartition the collection of groups G into n non-overlapping collections. So G is a disjoint union of G1, . . . , Gn, and if G1, G2 \u2208Gi and G1 \u0338= G2 then G1 \u2229G2 = \u2205for i = 1, . . . , n. With some abuse of notation, we write Gc i = {i \u2208{1, . . . , d} | i / \u2208G for all G \u2208Gi} Now we recast the problem into the form of (1) minimize 1 2\u2225Ax \u2212b\u22252 2 + 1 n n X i=1 X G\u2208Gi \u03bb2\u2225xG\u22252 ! , where \u03bb2 = n\u03bb1. The regularizer (the second term) is now a sum of n proximable terms. For example, we can have a setup with d = 42, n = 3, G = G1 \u222aG2 \u222aG3, and G1 = {{1, . . . , 9}, {10, . . . , 18}, {19, . . . , 27}, {28, . . . , 36}} G2 = {{4, . . . , 12}, {13, . . . , 21}, {22, . . . , 30}, {31, . . . , 39}} G3 = {{7, . . . , 15}, {16, . . . , 24}, {25, . . . , 33}, {34, . . . , 42}} The groups within Gi do not overlap for each i = 1, 2, 3, and Gc 2 = {1, 2, 3, 40, 41, 42}. We view the \ufb01rst term as the r term, the second term as the sum of g1, . . . , gn, and f1 = \u00b7 \u00b7 \u00b7 = fn = 0 in the notation of (1), and apply (PPG): xk+1/2 = (I + \u03b1AT A)\u22121 \u03b1AT b + 1 n n X i=1 zk i ! zk+1/2 i = 2xk+1/2 \u2212zk i (xk+1 i )G = u\u03b1\u03bb2 \u0010 (zk+1/2 i )G \u0011 for G \u2208Gi (xk+1 i )j = (zk+1/2 i )j for j \u2208Gc i zk+1 i = zk i + xk+1 i \u2212xk+1/2, where the indices i implicitly run through i = 1, . . . , n, and u\u03b1, de\ufb01ned in the Section 7, is the vector soft-thresholding operator. To reduce the cost of computing xk+1/2, we can precompute and store the Cholesky factorization of the positive de\ufb01nite matrix I + \u03b1AT A (which costs O(m2d + d3)) and the matrix-vector product AT b (which costs O(md)). This cost is paid upfront once, and the subsequent iterations can be done in O(d2+dn) time. For recent work on overlapping group lasso, see [68, 72, 27, 36, 32, 67, 25, 10, 7, 64]. Low-rank and sparse matrix completion. Consider the setup where we partially observe a matrix M \u2208Rd1\u00d7d2 on the set of indices \u2126. More precisely, we observe Mij for (i, j) \u2208\u2126while Mij for (i, j) / \u2208\u2126are unknown. We assume M has a low-rank plus sparse structure, i.e., M = Ltrue + Strue with Ltrue is 14 \flow-rank and Strue is sparse. Here Ltrue models the true underlying structure while Strue models outliers. Furthermore, let\u2019s assume 0 \u2264Mij \u22641 for all (i, j). The goal is to estimate the unobserved entries of M, i.e., Mij for (i, j) / \u2208\u2126. To estimate M, we solve the following regularized regression minimize \u03bb1\u2225L\u2225\u2217+ \u03bb2\u2225S\u22251 + X (i,j)\u2208\u2126 \u2113(Sij + Lij \u2212Mij) subject to 0 \u2264S + L \u22641, where S, L \u2208Rd1\u00d7d2 are the optimization variables, the constraint 0 \u2264S + L \u2264 1 applies element-wise, and \u03bb1, \u03bb2 > 0 are regularization parameters. The constraint is proximable by Lemma 3, and we can use (PPG) either when \u2113 is di\ufb00erentiable or when \u2113+ I[0,1] is proximable. We view n = 1, the \ufb01rst term as the r term, the second term as the g1 term, and the last term as the f1 term in the notation of (1), and apply (PPG): Lk+1/2 = t\u03b1 \u0000Zk\u0001 Sk+1/2 ij = s\u03b1 \u0000Y k ij \u0001 for all (i, j) Zk+1/2 ij = ( 2Lk+1/2 ij \u2212Zk ij for (i, j) / \u2208\u2126 2Lk+1/2 ij \u2212Zk ij \u2212\u03b1(Lk+1/2 ij + Sk+1/2 ij \u2212Mij) for (i, j) \u2208\u2126 Y k+1/2 ij = ( 2Sk+1/2 ij \u2212Y k ij for (i, j) / \u2208\u2126 2Sk+1/2 ij \u2212Y k ij \u2212\u03b1(Lk+1/2 ij + Sk+1/2 ij \u2212Mij) for (i, j) \u2208\u2126 Ak+1 = Zk+1/2 + Y k+1/2 Bk+1 = Zk+1/2 \u2212Y k+1/2 Lk+1 ij = 1 2 \u0000\u03a0[0,1](Ak+1 ij ) + Bk+1 ij \u0001 for all (i, j) Sk+1 ij = 1 2 \u0000\u03a0[0,1](Ak+1 ij ) \u2212Bk+1 ij \u0001 for all (i, j) Zk+1 = Zk + Lk+1 \u2212Lk+1/2 Y k+1 = Y k + Sk+1 \u2212Sk+1/2. t\u03b1 and s\u03b1, de\ufb01ned in the Section 7, are respectively the matrix and scalar softthresholding operators. \u03a0[0,1] is the projection onto the interval [0, 1], and the Lk+1 and Sk+1 updates follow from Lemma 3. The only non-trivial operation for this method is computing the SVD to evaluate t\u03b1 \u0000Zk\u0001 . All other operations are elementary and embarrassingly parallel. For a discussion on low-rank + sparse factorization, see [9]. Regression with fused lasso. Consider the problem setup where we have Axtrue = b and we observe A and b. Furthermore, the coordinates of xtrue are ordered in a meaningful way and we know a priori that |xi+1 \u2212xi| \u2264\u03b5 for i = 1, . . . , d \u22121 and some \u03b5 > 0. \ufb01nally, we also know that xtrue is sparse. 15 \fTo estimate x, we solve the fused lasso problem minimize \u03bb\u2225x\u22251 + 1 n n X i=1 \u2113i(x) subject to |xi+1 \u2212xi| \u2264\u03b5, i = 1, . . . , d \u22121 where \u2113i(x) = (1/2)(aT i x \u2212yi)2 and x \u2208Rd is the optimization variable. We recast the problem into the form of (1) minimize \u03bb\u2225x\u22251 + 1 2n n X i=1 (\u2113i(x) + go(x)) + n X i=1 (\u2113i(x) + ge(x)) ! where go(x) = X i=1,3,5,... I[\u2212\u03b5,\u03b5](xi+1 \u2212xi) ge(x) = X i=2,4,6,... I[\u2212\u03b5,\u03b5](xi+1 \u2212xi) and I[\u2212\u03b5,\u03b5](x) = \u001a 0 if |x| \u2264\u03b5 \u221e otherwise. Since go and ge are proximable by Lemma 3, we can apply (PPG). For recent work on fused lasso, see [57, 47, 58, 23, 33, 42, 66, 73]. Network lasso. Consider the problem setup where we have an undirected graph G = (E, V ). Each node v \u2208V has a parameter to estimate xv \u2208Rd and an associated loss function \u2113v. Furthermore, we know that neighbors of G have similar parameters in the sense that \u2225xu \u2212xv\u22252 is small if {u, v} \u2208E and that xv is sparse for each v \u2208V . Under this model, we solve the network lasso problem minimize X v\u2208V \u03bb1\u2225xv\u22251 + \u2113v(xv) + X {u,v}\u2208E \u03bb2\u2225xu \u2212xv\u22252, where xv for all v \u2208V are the optimization variables, and \u2113v(xv) for all v \u2208V a di\ufb00erentiable loss function, and \u03bb1, \u03bb2 > 0 are regularization parameters [22]. Say the G has an edge coloring E1, . . . , EC. So E1, . . . , EC partitions E such that if {u, v} \u2208Ec then {u, v\u2032} / \u2208Ec for any v\u2032 \u0338= v and c = 1, . . . , C. Figure 4 illustrates this de\ufb01nition. (The chromatic index \u03c7\u2032(G) is the smallest possible 16 \fFigure 4: An edge coloring of the hypercube graph Q3. value of C, but C need not be \u03c7\u2032(G).) With the edge coloring, we recast the problem into the form of (1) minimize X v\u2208V \u03bb1\u2225xv\u22251 + 1 C C X c=1 \uf8eb \uf8edX v\u2208V \u2113v(xv) + X {u,v}\u2208Ec \u03bb3\u2225xu \u2212xv\u22252 \uf8f6 \uf8f8, where \u03bb3 = C\u03bb2. We view the \u21131 regularizer as the r term, the loss functions as the f term, and the summation over Ec as the g terms in the notation of (1), and apply (PPG): xk+1/2 v = s\u03b1\u03bb1 1 C C X c=1 zk cv ! zk+1/2 cv = 2xk+1/2 v \u2212zk cv \u2212\u03b1\u2207\u2113v(xk+1/2 v ) sk c = zk+1/2 cu + zk+1/2 cv dk c = zk+1/2 cu \u2212zk+1/2 cv xk+1 cu = sk c + u\u03b1\u03bb3 \u0000dk c \u0001 for {u, v} \u2208Ec xk+1 cv = sk c \u2212u\u03b1\u03bb3 \u0000dk c \u0001 for {u, v} \u2208Ec xk+1 cv = zk+1/2 cv for {v, u\u2032} / \u2208Ec for all u\u2032 \u2208V zk+1 cv = zk cv + xk+1 cv \u2212xk+1/2 v , where the colors c implicitly run through c = 1, . . . , C unless speci\ufb01ed otherwise and the nodes v implicitly run through all v \u2208V . Here s\u03b1 and u\u03b1, de\ufb01ned in the Section 7, are respectively the scalar and vector soft-thresholding operators. Although this algorithm, as stated, seemingly maintains C copies of xv we can actually simplify it so that v maintains min{deg(v) + 1, C} copies of xv for all v \u2208V . Since 2|E|/|V | is the average degree of G, storage requirement is O(|V | + |E|) when simpli\ufb01ed. Let each node have a set N \u2286{1, . . . , C} such that c \u2208N if there is a neighbor connected through an edge with color c. Write N c = {1, . . . , C}\\N. 17 \fWith this notation, we can rewrite the algorithm in a simpler, vertex-centric manner: FOR EACH node x1/2 = s\u03b1\u03bb1 1 C |N c|zc\u2032 + X c\u2208N zk c !! FOR EACH color c \u2208S z1/2 c = 2x1/2 \u2212zc \u2212\u03b1\u2207f(x1/2) Through edge with color c, send z1/2 c and receive z\u20321/2 c s = z1/2 c + z\u20321/2 c d = z1/2 c \u2212z\u20321/2 c xc = (1/2)(s + u\u03b1\u03bb2 (d)) zc = zc + xc \u2212x1/2, IF N c \u0338= \u2205 zc\u2032 = x1/2 \u2212\u03b1\u2207f(x1/2) SVM. We solve the standard (primal) support vector machine setup [14] minimize \u03bb 2 \u2225x\u22252 2 + 1 n n X i=1 gi(x), (8) where x \u2208Rd is the optimization variable. The problem data is embedded in gi(x) = max{1 \u2212yiaT i x, 0} where \u03bb > 0 is a regularization parameter and ai \u2208Rd, bi \u2208R and yi \u2208{\u22121, +1} are problem data for i = 1, . . . , n. Applying Lemma 2 and working out the details, we get a closed-form solution for the proximal operator: prox\u03b1gi(x0) = x0 + \u03a0[0,\u03b1] \u00121 \u2212yiaT i x0 \u2225ai\u22252 2 \u0013 yiai for i = 1, . . . , n. We view r = (\u03bb/2)\u2225x\u22252 2 and f = 0 in the notation of (1), and apply (PPG): xk+1/2 = 1 1 + \u03b1\u03bb 1 n n X i=1 zk i \u03b2k i = yi\u03a0[0,\u03b1] \u00121 \u2212yiaT i (2xk+1/2 \u2212zk i ) \u2225ai\u22252 2 \u0013 zk+1 i = xk+1/2 + \u03b2iai, where the indices i implicitly run through i = 1, . . . , n. 18 \fGeneralized linear model. In the setting of generalized linear models, the maximum likelihood estimator is the solution of the optimization problem minimize 1 n n X i=1 \u0000A(xT i \u03b2) \u2212TixT i \u03b2 \u0001 , where \u03b2 \u2208Rd is the optimization variable, xi \u2208Rd and Ti \u2208R for i = 1, . . . , n are problem data, and A is a convex function on R. We view r = 0, f = 0, and gi(\u03b2) = A(xT i \u03b2) \u2212TixT i \u03b2 in the notation of (1), and apply (PPG): \u03b2k+1 = 1 n n X i=1 zk i zk+1 i = zk i + prox\u03b1gi(2\u03b2k \u2212zk i ) \u2212\u03b2k where the indices i implicitly run from i = 1, . . . , n. For an introduction on generalized linear models, see [37]. Network Utility Maximization. In the problem of network utility maximization, one solves the optimization problem minimize (1/n) Pn i=1 fi(xi) subject to xi \u2208Xi i = 1, . . . , n Aixi \u2264y i = 1, . . . , n y \u2208Y, where f1, . . . , fn are functions, A1, . . . , An are matrices, X1, . . . , Xn, Y are sets, and x1, . . . , xn, y are the optimization variables (For convenience, we convert the maximization problem into a minimization problem.) For a comprehensive discussion on network utility maximization, see [43]. This optimization problem is equivalent to the master problem minimize 1 n n X i=1 gi(y) + IY (y) where we de\ufb01ne the subproblems gi(y) = inf {fi(xi) | xi \u2208Xi, Aixi \u2264y} , for i = 1, . . . , n. The master problem only involves the variable y, and xi is the variable in the ith subproblem. For each i = 1, . . . , n, the function gi may not be di\ufb00erentiable, but it is convex if fi and Xi are convex. If Y is convex, then the equivalent problem is convex. 19 \f0 50 100 150 200 250 Iterations 10-8 10-4 100 Square distance to solution Overlapping Group Lasso ADMM PPG SPPG Figure 5: The error \u2225xk+1/2 \u2212x\u22c6\u22252 vs. iteration for overlapping group lasso example. We view f = 0 and r = IY in the notation of (1), and apply (PPG): xk+1/2 = \u03a0Y 1 n n X i=1 zk i ! yk+1 i = argmin xi\u2208Xi Aixi\u2264y y\u2208Y \u001a \u03b1fi(xi) + 1 2\u2225y \u2212(2xk+1/2 \u2212zk i )\u22252 2 \u001b zk+1 i = zk i + yk+1 i \u2212yk+1/2. Network utility maximization is often performed on a distributed computing network, and if so the optimization problem for evaluating the proximal operators can be solved in a distributed, parallel fashion. 6 Experiments In this section, we present numerical experiments on two applications discussed in Section 5. The \ufb01rst experiment is small and is meant to serve as a proof of concept. The second experiment is more serious; the problem size is large and we compare the performance with existing methods. For the sake of scienti\ufb01c reproducibility, we provide the code used to generate these experiments. 20 \f100 101 102 103 104 Epochs 10-8 10-6 10-4 10-2 100 Square distance to solution SVM PPG S-PPG SPI Figure 6: The error \u2225xk+1/2 \u2212x\u22c6\u22252 vs. iteration for the SVM example. For both experiments, we observe linear convergence. This is a pleasant surprise, as the theory presented in Section 4 and proved in Section 7 only guarantee a O(1/k) rate. Overlapping group lasso. The problem size of this setup is m = 300, d = 42, n = 3. The groups are as described in Section 5. The dominant cost per iteration of (PPG) is evaluating prox\u03b1r which takes O(d2) time with the precomputed factorization. Since the cost per iteration of (S-PPG) is no cheaper than that of (PPG), there is no reason to use (S-PPG). We compare the performance of (PPG) to consensus ADMM (cf. \u00a77.1 of [8]). Both methods use the same computational subroutines and therefore have essentially the same computational cost per iteration. We show the results in Figure 5. SVM. The problem size of this setup is n = 217 = 131, 072 and d = 512. The synthetic dataset A and y are randomly genearated and the regularization parameter \u03bb = 0.1 is used. So the problem data A consists of 64 \u00d7 220 numbers and requires 500MB storage to store in double-precision \ufb02oating-point format. First, we compare the performance of (PPG) and (S-PPG) to the stochastic proximal iteration with diminishing step size \u03b1k = C/k in Figure 6. For all three methods, the parameters were roughly tuned for optimal performance. Next, we compare the (PPG) and (S-PPG) to a state-of-the-art SVM solver 21 \fMethod Run time Objective value LIBLINEAR 8.47s 3.7699 CPU (PPG) 33.9s (30 iterations) 3.7364 CPU (S-PPG) 37.2s (30 epochs) 3.7364 CUDA (PPG) 0.68s (30 iterations) 3.7364 Table 1: Run time as a function of grid size LIBLINEAR [19]. Since LIBLINEAR is based on a second order method while (PPG) and (S-PPG) are \ufb01rst-order methods, comparing the number of iterations is not very meaningful. Rather we compare the wall-clock time these methods take to reach an equivalent level of accuracy. To compare the quality of the solutions, we use the objective value of the problem (8). The CPU code for (PPG) and (S-PPG) are written in C++. The code is serial, not heaviliy optimized, and does not utilize BLAS (Basic Linear Algebra Subprograms) libraries or any SIMD instructions. On the other hand, LIBLINEAR is heavily optimized and does utilize BLAS. We also implemented (PPG) on CUDA and ran it on a GPU. The algorithmic structure of (PPG) is particularly well-suited for CUDA, especially when the problem size is large. Roughly speaking, the xk+1/2 update requires a reduce operation, which can be done e\ufb00ectively on CUDA. The zk i updates are embarassingly parallel, and can be done very e\ufb00ectively on CUDA. The threads must globally synchronize twice per iteration: once before computing the average of zk i for i = 1, . . . , n and once after xk+1/2 has been computed. Generally speaking, global synchronization on a GPU is expensive, but we have empirically veri\ufb01ed that the computational bottleneck is in the other operations, not the synchronization, when the problem size is reasonably large. Table 1 shows the results. We see that CPU implementation of (PPG) and (S-PPG) are competitive with LIBLINEAR, and could even be faster than LIBLINEAR if the code is further optimized. On the other hand, the CUDA implementation of (PPG) clearly outperforms LIBLINEAR. (PPG) and (S-PPG) were run until the objective values were good as that of LIBLINEAR. These expeirments were run on an Intel Core i7-990 GPU and a GeForce GTX TITAN X GPU. 7 Convergence proofs We say a function f is closed convex and proper if its epigraph \b (x, \u03b1) | x \u2208Rd, |f(x)| < \u221e, f(x) \u2264\u03b1 \t is a closed subset of Rd+1, f is convex, and f(x) = \u2212\u221enowhere and f(x) < \u221e for some x. 22 \fA closed convex and proper function f has L-Lipschitz continuous gradient if f is di\ufb00erentiable everywhere and \u2225\u2207f(x) \u2212\u2207f(y)\u22252 \u2264L\u2225x \u2212y\u22252 for all x, y \u2208Rd. This holds if and only if a closed convex and proper function f satis\ufb01es \u2225\u2207f(x) \u2212\u2207f(y)\u22252 2 \u2264L(\u2207f(x) \u2212\u2207f(y))T (x \u2212y) for all x, y \u2208Rd, which is known as the Baillon-Haddad Theorem [2]. See [3, 4, 50] for a discussion on this. Proximal operators are \ufb01rmly non-expansive. This means for any closed convex and proper function f and x, y \u2208Rd, \u2225proxf(x) \u2212proxf(y)\u22252 2 \u2264(proxf(x) \u2212proxf(y))T (x \u2212y). By applying Cauchy-Schwartz inequality, we can see that \ufb01rmly non-expansive operators are non-expansive. Some proximable functions. As discussed, many standard references like [11, 44] provide a list proximable functions. Here we discuss the few we use. An optimization problem of the form minimize f(x) subject to x \u2208C, where x is the optimization variable and C is a constraint set, can be transformed into the equivalent optimization problem minimize f(x) + IC(x). The indicator function IC is de\ufb01ned as IC(x) = \u001a 0 for x \u2208C \u221e otherwise, and the proximal operator with respect to IC is prox\u03b1IC(x) = \u03a0C(x) where \u03a0C is the projection onto C for any \u03b1 > 0. So IC is proximable if the projection onto C is easy to evaluate. The following results are well known. The proximal operator with respect to r(x) = |x| is called the scalar soft-thresholding operator prox\u03bbr(x) = s\u03bb(x) = \uf8f1 \uf8f2 \uf8f3 x + \u03bb x < \u2212\u03bb 0 \u2212\u03bb \u2264x \u2264\u03bb x \u2212\u03bb x > \u03bb. 23 \fThe proximal operator with respect to r(x) = \u2225x\u22252 is called the vector softthresholding operator prox\u03bbr(x) = u\u03bb(x) = \u001a max{1 \u2212\u03bb/\u2225x\u22252, 0}x for x \u0338= 0 0 otherwise. The proximal operator with respect to r(M) = \u2225M\u2225\u2217is called the matrix soft-thresholding operator prox\u03bbr(M) = t\u03bb(M) = {Us\u03bb(\u03a3)V T | U\u03a3V T = M is the SVD}, where s\u03bb(\u03a3) is applied element-wise to the diagonals. Lemma 2. Assume g(x) = f(aT x) where a \u2208Rd and f is a closed, convex, and proper function on R. Then the proxg can be evaluated by solving a onedimensional optimization problem. Proof. By examining the optimization problem that de\ufb01nes proxg proxg(x0) = argmin x \u001a f(aT x) + 1 2\u2225x \u2212x0\u22252 2 \u001b we see that solution must be of the form x0 + \u03b2a. So proxg(x0) = x0 + \u03b2a, \u03b2 = argmin \u03b2 \u001a f(aT x0 + \u03b2\u2225a\u22252 2) + \u2225a\u22252 2 2 \u03b22 \u001b . Lemma 3. Let g(x1, x2, . . . , xn) = f(a1x1 + a2x2 + \u00b7 \u00b7 \u00b7 + anxn) where x1, . . . , xn \u2208Rd, a \u2208Rn, a \u0338= 0, and f : R \u2192R \u222a{\u221e} is closed, convex, and proper. Then we can compute proxg with w = prox\u2225a\u22252 2f(a1\u03be1 + a2\u03be2 + \u00b7 \u00b7 \u00b7 + an\u03ben) v = 1 \u2225a\u22252 2 (a1\u03be1 + a2\u03be2 + \u00b7 \u00b7 \u00b7 + an\u03ben \u2212w) proxg(\u03be1, \u03be2, . . . , \u03ben) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed \u03be1 \u2212a1v \u03be2 \u2212a2v . . . \u03ben \u2212anv \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8. Proof. The optimality conditions of proxg(\u03be1, \u03be2, . . . , \u03ben) gives us 0 \u2208a2 1v + a1(x1 \u2212\u03be1) . . . . . . 0 \u2208a2 nv + an(xn \u2212\u03ben) 24 \ffor some v \u2208\u2202f(a1x1 + \u00b7 \u00b7 \u00b7 + anxn). Summing this we get 0 \u2208\u2225a\u22252 2v + (a1x1 + \u00b7 \u00b7 \u00b7 + anxn) \u2212(a1\u03be1 + \u00b7 \u00b7 \u00b7 + an\u03ben) and with w = a1x1 + \u00b7 \u00b7 \u00b7 + anxn we have w = prox\u2225a\u22252 2f(a1\u03be1 + a2\u03be2 + \u00b7 \u00b7 \u00b7 + an\u03ben). The expression for v and proxg(\u03be1, \u03be2, . . . , \u03ben) follows from reorganizing the equations. So if g(x, y) = f(x + y) then proxg(x0, y0) = 1 2 \u0012x0 \u2212y0 + prox2f(x0 + y0) y0 \u2212x0 + prox2f(x0 + y0) \u0013 . If g(x, y) = f(x \u2212y) then proxg(x0, y0) = 1 2 \u0012x0 + y0 + prox2f(x0 \u2212y0) x0 + y0 \u2212prox2f(x0 \u2212y0) \u0013 . 7.1 Deterministic analysis Let h be a closed, convex, and proper function on Rd. When x = prox\u03b1h(x0), we have \u03b1u + x = x0 with u \u2208\u2202h(x0). To simplify the notation, we write \u03b1 \u02dc \u2207h(x0) + x = x0 where \u02dc \u2207h(x0) \u2208\u2202h(x0). So is \u02dc \u2207h(x0) a subgradient of h at x0, and which subgradient \u02dc \u2207h(x0) is referring to depends on the context. (This notation is convenient yet potentially sloppy, but we promise to not commit any fallacy of equivocation.) Proof of Lemma 1. Assume z\u22c6is a \ufb01xed point of (PPG) or (S-PPG). Then p(z\u22c6) = 0. Then we have x\u22c6= x\u2032\u22c6 i for i = 1, . . . , n where x\u22c6and x\u2032\u22c6 1 , . . . , x\u2032\u22c6 n are as de\ufb01ned in (5). So 0 = \u03b1 \u02dc \u2207r(x\u22c6) + x\u22c6\u2212\u00af z 0 = \u03b1 \u02dc \u2207g1(x\u22c6) \u2212x\u22c6+ z\u22c6 1 + \u03b1\u2207f1(x\u22c6) . . . . . . 0 = \u03b1 \u02dc \u2207gn(x\u22c6) \u2212x\u22c6+ z\u22c6 n + \u03b1\u2207fn(x\u22c6). 25 \fAdding these up and dividing by n appropriately gives us 0 = \u02dc \u2207r(x\u22c6) + \u02dc \u2207\u00af g(x\u22c6) + \u2207\u00af f(x\u22c6), so x\u22c6is a solution of Problem (1). Reorganize the de\ufb01nition of x in (5) to get x\u22c6= argmin x \u001a r(x) \u22121 \u03b1(\u00af z\u22c6\u2212x\u22c6)T x + 1 2\u03b1\u2225x \u2212x\u22c6\u22252 2 \u001b . So x\u22c6minimizes L(\u00b7, x\u22c6, (1/\u03b1)(z\u22c6\u2212x\u22c6)), where L is de\ufb01ned in (4). Reorganize the de\ufb01nition of x\u2032 in (5) to get x\u22c6= argmin x \u001a f(x\u22c6) + (\u2207f(x\u22c6))T (x \u2212x\u22c6) + g(x) + 1 \u03b1(z\u22c6 i \u2212x\u22c6)T x + 1 2\u03b1\u2225x \u2212x\u22c6\u22252 2 \u001b . So x\u22c6minimizes L(x\u22c6, \u00b7, (1/\u03b1)(z\u22c6\u2212x\u22c6)). So \u03bd\u22c6= (1/\u03b1)(z\u22c6\u2212x\u22c6) is a dual solution of Problem (1). The argument works in the other direction as well. If we assume (x\u22c6, \u03bd\u22c6) is a primal dual solution of Problem (1), we can show that z\u22c6= x\u22c6+ \u03b1\u03bd\u22c6is a \ufb01xed point of (PPG) by following a similar line of logic. Before we proceed to the main proofs, we introduce more notation. De\ufb01ne the function \u00af r(x) = 1 n n X i=1 r(xi) + IC(x), where C = {(x1, . . . , xn) | x1 = \u00b7 \u00b7 \u00b7 = xn}. So IC(x) = \u001a 0 for x1 = \u00b7 \u00b7 \u00b7 = xn \u221e otherwise. As before, we write \u00af f(x) = 1 n n X i=1 fi(xi), \u00af g(x) = 1 n n X i=1 gi(xi). With this new notation, we can recast Problem (1) into minimize \u00af r(x) + \u00af f(x) + \u00af g(x), where x \u2208Rdn is the optimization variable. We can also rewrite the de\ufb01nition of p as the following three-step process, which starts from z, produces intermediate points x, x\u2032, and yields p(z): x = prox\u03b1\u00af r(z) (9a) x\u2032 = prox\u03b1\u00af g \u00002x \u2212z \u2212\u03b1\u2207\u00af f(x) \u0001 (9b) p(z) = (1/\u03b1)(x \u2212x\u2032). (9c) 26 \fNote that x1, . . . , xn in x out of (9a) are identical due to IC. This is the same p as the p de\ufb01ned in (5); we\u2019re just using the new notation. We treat the boldface variables as vectors in Rdn. So the inner product between boldface variables is xT z = n X i=1 xT i zi and the gradient of \u00af f(x) is \u2207\u00af f(x) = 1 n \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 \u2207f1(x1) \u2207f2(x2) . . . \u2207fn(xn) \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb. We use \u02dc \u2207g(x) and \u02dc \u2207r(x) in the same manner as before. Lemma 4. Let z and \u02dc z be any points in Rdn. Then \u03b1\u2225p(z) \u2212p(\u02dc z)\u22252 \u2264(p(z) \u2212p(\u02dc z))T (z \u2212\u02dc z) \u2212(\u2207\u00af f(x) \u2212\u00af f(\u02dc x))T (x\u2032 \u2212\u02dc x\u2032) where \u02dc x and \u02dc x\u2032 are obtained by applying (9a) and then (9b), respectively, to \u02dc z instead of z. Proof. This Lemma is similar to Lemma 3.3 of [15]. We reproduce the proof using this paper\u2019s notation. \u2225z \u2212\u03b1p(z) \u2212\u02dc z + \u03b1p(\u02dc z)\u22252 2 (a) = \u2225z \u2212x \u2212\u02dc z + \u02dc x\u22252 2 + \u2225x\u2032 \u2212\u02dc x\u2032\u22252 2 + 2(z \u2212x \u2212\u02dc z + \u02dc x)T (x\u2032 \u2212\u02dc x\u2032) (b) \u2264(z \u2212x \u2212\u02dc z + \u02dc x)T (z \u2212\u02dc z) + (x\u2032 \u2212\u02dc x\u2032)T (2x \u2212z \u2212\u03b1\u2207f(x) \u22122\u02dc x + \u02dc z + \u03b1\u2207f(\u02dc x)) + 2(z \u2212x \u2212\u02dc z + \u02dc x)T (x\u2032 \u2212\u02dc x\u2032) = (z \u2212\u03b1p(z) \u2212\u02dc z + \u03b1p(\u02dc z))T (z \u2212\u02dc z) \u2212\u03b1(\u2207f(x) \u2212\u2207f(\u02dc x))T (x\u2032 \u2212\u02dc x\u2032), where (a) is due to \u03b1p(z) = x \u2212x\u2032 and \u03b1p(\u02dc z) = \u02dc x \u2212\u02dc x\u2032, (b) follows from the fact that the two mappings: z 7\u2192z \u2212x = z \u2212prox\u03b1\u00af r(z) and 2x \u2212z \u2212\u03b1\u2207\u00af f(x) 7\u2192x\u2032 = prox\u03b1\u00af g \u00002x \u2212z \u2212\u03b1\u2207\u00af f(x) \u0001 are both \ufb01rmly non-expansive. By expanding the \u2225z \u2212\u03b1p(z) \u2212\u02dc z + \u03b1p(\u02dc z)\u22252 2 and cancellation, we obtain \u2225\u03b1p(z) \u2212\u03b1p(\u02dc z)\u22252 2 \u2264(\u03b1p(z) \u2212\u03b1p(\u02dc z))T (z \u2212\u02dc z) \u2212\u03b1(\u2207f(x) \u2212\u2207f(\u02dc x))T (x\u2032 \u2212\u02dc x\u2032), which proves the lemma by dividing both sides by \u03b1. 27 \fLemma 5. Let z\u22c6be any \ufb01xed point of (PPG), i.e., p(z\u22c6) = 0. Then \u2225zk \u2212z\u22c6\u22252 2 \u2264\u2225z0 \u2212z\u22c6\u22252 2 (10) for all k = 0, 1, . . . . Moreover, we have \u221e X k=0 \u2225p(zk)\u22252 < \u221e (11) \u221e X k=0 \u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6)\u22252 < \u221e. (12) Finally, \u2225p(zk)\u22252 monotonically decreases. Proof. With the Baillon-Haddad theorem and Young\u2019s inequality on (a) we get \u2212(\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x))T (x\u2032 \u2212\u02dc x\u2032) (13) = \u2212(\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x))T (x \u2212\u03b1p(z) \u2212\u02dc x + \u03b1p(\u02dc z)) (14) = \u2212(\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x))T (x \u2212\u02dc x) + \u03b1(\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x))T (p(z) \u2212p(\u02dc z)) (15) (a) \u2264\u22121 L\u2225\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x)\u22252 + 3 4L\u2225\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x)\u22252 + L\u03b12 3 \u2225p(z) \u2212p(\u02dc z)\u22252 (16) = L\u03b12 3 \u2225p(z) \u2212p(\u02dc z)\u22252 \u2212 1 4L\u2225\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x)\u22252. (17) Combining Lemma 4 and equation (17), we obtain \u2212(p(z) \u2212p(\u02dc z))T (z \u2212\u02dc z) \u2264\u03b1( \u03b1L 3 \u22121)\u2225p(z) \u2212p(\u02dc z)\u22252 \u2212 1 4L\u2225\u2207\u00af f(x) \u2212\u2207\u00af f(\u02dc x)\u22252. (18) Applying (18) separately with (z, \u02dc z) = (zk, z\u22c6) and (zk, zk+1), we get, respectively, \u2225zk+1 \u2212z\u22c6\u22252 = \u2225zk \u2212z\u22c6\u22252 + \u03b12\u2225p(zk)\u22252 \u22122\u03b1(zk \u2212z\u22c6)T p(zk) \u2264\u2225zk \u2212z\u22c6\u22252 \u2212\u03b12(1 \u22122\u03b1L 3 )\u2225p(zk)\u22252 \u2212\u03b1 2L\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6)\u22252 (19) and \u2225p(zk+1)\u22252 = \u2225p(zk)\u22252 + \u2225p(zk+1) \u2212p(zk)\u22252 \u22122(p(zk+1) \u2212p(zk))T 1 \u03b1(zk+1 \u2212zk) \u2264\u2225p(zk)\u22252 \u2212(1 \u22122\u03b1L 3 )\u2225p(zk+1) \u2212p(zk)\u22252. (20) (In (17) we can use a di\ufb00erent parameter for Young\u2019s inequality, and improve inequalities (19) and (20) to allow \u03b1 < 2/L, which is better than \u03b1 < 3/(2L). In fact, \u03b1 < 2/L is su\ufb03cient for convergence in Theorem 1. However, we use the current version because we need the last term of inequality (19) for proving Theorem 3.) Summing (19) through k = 0, 1, . . . give us the summability result. Inequality (20) states that \u2225p(zk)\u22252 monotonically decreases. 28 \fProof of Theorem 1. Lemma 5 already states that \u2225p(zk)\u22252 2 decreases monotonically. Using inequality (11) of Lemma 5, we get \u2225p(zk)\u22252 2 = min i=0,1,...,k \u2225p(zi)\u22252 2 \u22641 k \u221e X i=0 \u2225p(zi)\u22252 2 = C/k = O(1/k) for some \ufb01nite constant C. (The rate O(1/k) can be improved to o(1/k) using, say, Lemma 1.2 of [18], but we present the simpler argument.) By (10) of Lemma 5, zk is a bounded sequence and will have a limit point, which we call z\u221e. Lemma 4 also implies p is a continuous function. Since p(zk) \u21920 and p is continuous, the limit point z\u221emust satisfy p(z\u221e) = 0. Applying inequality (10) with z\u22c6= z\u221etells us that \u2225zk \u2212z\u221e\u22252 2 \u21920, i.e., the entire sequence converges. Since prox\u03b1r is a continuous function xk+1/2 = prox\u03b1r(\u00af zk) \u2192prox\u03b1r(\u00af z\u22c6) = x\u22c6. With this same argument, we also conclude that xk i \u2192x\u22c6for all i = 1, . . . , n. Proof of Theorem 2. A convex function h satis\ufb01es the inequality h(x) \u2212h(\u02dc x) \u2264( \u02dc \u2207h(x))T (x \u2212\u02dc x) for any x and \u02dc x (so long as a subgradient \u02dc \u2207h(x) exists). Applying this inequality we get Ek \u2264( \u02dc \u2207\u00af r(xk+1/2) + \u2207\u00af f(xk+1/2))T (xk+1/2 \u2212x\u22c6) + ( \u02dc \u2207\u00af g(xk+1))T (xk+1 \u2212x\u22c6) = ( \u02dc \u2207\u00af r(xk+1/2) + \u2207\u00af f(xk+1/2) + \u02dc \u2207\u00af g(xk+1))T (xk+1/2 \u2212x\u22c6) \u2212( \u02dc \u2207\u00af g(xk+1))T (\u03b1p(zk)) = (xk+1/2 \u2212\u03b1 \u02dc \u2207\u00af g(xk+1) \u2212x\u22c6)T p(zk) (21) = ((zk+1 \u2212z\u22c6) + \u03b1( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(xk+1/2)))T p(zk) (22) = (zk+1 \u2212z\u22c6)T p(zk) + \u03b1( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(xk+1/2))T p(zk). (23) For the second equality, we used p(zk) = \u02dc \u2207\u00af r(xk+1/2) + \u2207\u00af f(xk+1/2) + \u02dc \u2207\u00af g(xk+1). and combined terms. For the third equality, we used x\u22c6= z\u22c6\u2212\u03b1 \u02dc \u2207\u00af r(x\u22c6) and xk+1/2\u2212\u03b1\u2207\u00af f(xk+1/2) \u2212\u03b1 \u02dc \u2207\u00af g(xk+1) = zk \u2212\u03b1 \u02dc \u2207\u00af r(xk+1/2) \u2212\u03b1\u2207\u00af f(xk+1/2) \u2212\u03b1 \u02dc \u2207\u00af g(xk+1) = zk \u2212\u03b1p(zk) = zk+1. Likewise we have Ek \u2265( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6))T (xk+1/2 \u2212x\u22c6) + ( \u02dc \u2207\u00af g(x\u22c6))T (xk+1 \u2212x\u22c6) = (xk+1 \u2212x\u22c6)T p(z\u22c6) + ( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6))T \u03b1p(zk) (24) = ( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6))T \u03b1p(zk). (25) 29 \fHere we use p(z\u22c6) = \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6) + \u02dc \u2207\u00af r(x\u22c6) = 0 Theorem 1 states that the sequences z1, z2, . . . and x1+1/2, x2+1/2, . . . both converge and therefore are bounded and that \u2225p(zk)\u22252 = O(1/ \u221a k). Combining this with the bounds on Ek gives us |Ek| \u2264O(1/ \u221a k). Proof of Theorem 3. By Jensen\u2019s inequality, we have Ek erg \u22641 k k X i=1 Ei. (26) Continuing the last line of (23), we get Ek \u2264(zk+1 \u2212z\u22c6)T p(zk) + \u03b1( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(xk+1/2))T p(zk) (27) = 1 2\u03b1\u2225zk \u2212z\u22c6\u22252 \u2212 1 2\u03b1\u2225zk+1 \u2212z\u22c6\u22252 \u2212\u03b1 2 \u2225p(zk)\u22252 + \u03b1( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(xk+1/2))T p(zk). (28) Combining (26) and (27) and after telescopic cancellation, Ek erg \u2264 1 2\u03b1k \u2225z1 \u2212z\u22c6\u22252 + 1 k k X i=1 \u03b1( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(xi+1/2))T p(zi) (29) = O(1/k) + 1 k \u03b1( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6))T k X i=1 p(zi) + 1 k \u03b1(\u2207\u00af f(xi) \u2212\u2207\u00af f(x\u22c6))T k X i=1 p(zi) (30) \u2264O(1/k) + 1 k\u03b1\u2225zk+1 \u2212z1\u2225\u2225\u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6)\u2225+ 1 2k k X i=1 \u0000\u03b12\u2225p(zi)\u22252 + \u2225\u2207\u00af f(xi) \u2212\u2207\u00af f(x\u22c6)\u22252\u0001 (31) = O(1/k), (32) where the last line holds due to the boundedness of zk and Lemma 5. With a similar argument as in (25), we get Ek erg \u2265( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6))T (xk+1/2 erg \u2212xk+1 erg ) = 1 k k X i=1 \u03b1p(zk) !T ( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6)) (33) = 1 k(zk \u2212z0)T ( \u02dc \u2207\u00af r(x\u22c6) + \u2207\u00af f(x\u22c6)) = O(1/k), (34) where have once again used the boundedness of (zk)k. Furthermore, since |Ek erg \u2212ek erg| \u2264Lg\u2225xk+1 erg \u2212xk+1/2 erg \u2225= Lg\u22251 k(zk+1 \u2212z1)\u2225= O(1/k), (35) we have ek erg = O(1/k). 30 \f7.2 Stochastic analysis Proof of Theorem 5. To express the updates of (S-PPG) we introduce the following notation: p(zk)[i] = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 . . . p(zk)i . . . 0 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb . With this notation, we can express the iterates of (S-PPG) as zk+1 = zk \u2212\u03b1p(zk)[i(k)], (36) and we also have conditional expectations Ekp(zk)[i(k)] = 1 np(zk), Ek\u2225p(zk)[i(k)]\u22252 = 1 n\u2225p(zk)\u22252. Here, we let E denote the expectation over all random variables i(1), i(2), . . ., and Ek denote the expectation over i(k) conditioned on i(1), i(2), . . . , i(k \u22121). The convergence of this algorithm has been recently analyzed in [12] when i(k) is chosen at random. Below, we adapt its proof to our setting with new rate results. Note that Lemma 4 and inequality (17) remain valid, as they are not tied to a speci\ufb01c sequence of random samples. Hence, similar to (19), we have \u2225zk+1 \u2212z\u2217\u22252 = \u2225zk \u2212z\u2217\u22252 + \u03b12\u2225p(zk)[i(k)]\u22252 \u22122\u03b1(zk \u2212z\u2217)T p(zk)[i(k)]. (37) We take the conditional expectation to get Ek\u2225zk+1 \u2212z\u2217\u22252 = \u2225zk \u2212z\u2217\u22252 + \u03b12Ek\u2225p(zk)[i(k)]\u22252 \u22122\u03b1(zk \u2212z\u2217)T Ekp(zk)[i(k)] (38) = \u2225zk \u2212z\u2217\u22252 + \u03b12 n \u2225p(zk)\u22252 \u22122\u03b1 n (zk \u2212z\u2217)T p(zk) (39) \u2264\u2225zk \u2212z\u2217\u22252 \u2212\u03b12 n (1 \u22122\u03b1L 3 )\u2225p(zk)\u22252 \u2212 \u03b1 2Ln\u2225\u2207\u00af f(xk) \u2212\u2207\u00af f(x\u2217)\u22252. (40) By the same reasoning as before, we have min 0\u2264i\u2264k E\u2225p(zk)\u22252 \u2264O(1/k). By (40), the sequence \u0000\u2225zk \u2212z\u2217\u22252\u0001 k\u22650 is a nonnegative supermartingale. Applying Theorem 1 of [48] to (40) yields the following three properties, which hold with probability one for every \ufb01xed point z\u22c6: 31 \f1. the squared \ufb01xed-point residual sequence is summable, that is, \u221e X k=0 \u2225p(zk)\u22252 < \u221e, (41) 2. \u2225zk \u2212z\u2217\u22252 converges to a nonnegative random number, and 3. (zk)k\u22650 is bounded. To proceed as before, however, we need \u2225zk \u2212z\u2217\u22252 to converge to a nonnegative random number (not necessarily 0) for all \ufb01xed points z\u2217with probability one. For each \ufb01xed point z\u2217, the previous argument states that there is a measure one event set1, \u2126(z\u2217), such that \u2225zk \u2212z\u2217\u22252 converges for all (zk)k\u22650 taken from \u2126(z\u2217). Note that \u2126(z\u2217) depends on z\u2217because we must select z\u2217to form (40) \ufb01rst. Since the number of \ufb01xed points (unless there is only one) is uncountable, \u2229\ufb01xed point z\u2217\u2126(z\u2217) may not be measure one. Indeed, it is measure one as we now argue. Let Z\u2217be the set of \ufb01xed points. Since Rdn is separable (i.e., containing a countable, dense subset), Z\u2217has a countable dense subset, which we write as {z\u2217 1, z\u2217 2, . . . }. By countability, \u2126c = \u2229i=1,2,...\u2126(z\u2217 i ) is a measure one event set, and it is de\ufb01ned independently of the choice of z\u2217. Next we show that \u2225zk\u2212z\u2217\u22252 to converge to a nonnegative random number (not necessarily 0) for all \ufb01xed points z\u2217with probability one by \u2126c, that is, limk \u2225zk\u2212z\u2217\u2225exists for all z\u2217\u2208Z\u2217 and all (zk)k\u22650 \u2208\u2126c. Now consider any z\u2217\u2208Z\u2217and (zk)k\u22650 \u2208\u2126c. Then for any \u03b5 > 0, there is a z\u2217 i such that \u2225z\u2217\u2212z\u2217 i \u2225\u2264\u03b5. By the triangle inequality, we can bound \u2225zk \u2212z\u2217\u2225 as \u2225zk \u2212z\u2217\u2225\u2264\u2225zk \u2212z\u2217 i \u2225+ \u2225z\u2217 i \u2212z\u2217\u2225\u2264\u2225zk \u2212z\u2217 i \u2225+ \u03b5, \u2225zk \u2212z\u2217\u2225\u2265\u2225zk \u2212z\u2217 i \u2225\u2212\u2225z\u2217 i \u2212z\u2217\u2225\u2265\u2225zk \u2212z\u2217 i \u2225\u2212\u03b5. Since \u2225zk \u2212z\u2217 i \u2225converges, we have we have lim sup k \u2225zk \u2212z\u2217\u2225\u2264\u03b5, lim inf k \u2225zk \u2212z\u2217\u2225\u2265\u2212\u03b5. As \u03b5 > 0 is arbitrary, lim infk \u2225zk \u2212z\u2217\u2225= lim supk \u2225zk \u2212z\u2217\u2225. So, limk \u2225zk \u2212z\u2217 i \u2225 exists. Finally, we can proceed with the same argument as in the proof of Theorem 1, and conclude that zk \u2192z\u2217, xk+1/2 \u2192x\u2217, and xk+1 \u2192x\u2217on the measure one set \u2126c. 1Each event is a randomly realized sequence of iterates (zk)k\u22650 in S-PPG. 32 \fProof of Theorem 6. This proof focuses on the treatments that are di\ufb00erent from the deterministic analysis. We only go through the steps of estimating the upper bound of E(Ek), skipping the similar treatment to obtain the lower bound and other rates. We reuse a part of (23) but avoid replacing zk \u2212\u03b1p(zk) by zk+1 because of (36): Ek \u2264(xk+1/2 \u2212\u03b1 \u02dc \u2207\u00af g(xk+1) \u2212x\u2217)T p(zk) (42) = ((zk \u2212\u03b1p(zk) \u2212z\u2217) + \u03b1( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(xk+1/2)))T p(zk) (43) = (zk \u2212\u03b1p(zk) \u2212z\u2217)T p(zk) + \u03b1( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(xk+1/2))T p(zk) (44) = (zk \u2212\u03b1p(zk) \u2212z\u2217)T p(zk) + \u03b1( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(x\u2217))T p(zk) (45) + \u03b1(\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217))T p(zk). (46) By the Cauchy-Schwarz inequality, E(Ek) \u2264E\u2225zk \u2212\u03b1p(zk) \u2212z\u2217\u2225\u00b7 E\u2225p(zk)\u2225+ \u03b1\u2225\u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(x\u2217)\u2225\u00b7 E\u2225p(zk)\u2225 (47) + \u03b1E\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u2225\u00b7 E\u2225p(zk)\u2225 (48) \u2264 \u0012q E\u2225zk \u2212\u03b1p(zk) \u2212z\u2217\u22252 + \u03b1\u2225\u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(x\u2217)\u2225 (49) + \u03b1 q E\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252 \u0013q E\u2225p(zk)\u22252. (50) Here we have p E\u2225zk \u2212\u03b1p(zk) \u2212z\u2217\u22252 \u2264 p \u2225z0 \u2212z\u2217\u22252 since, similar to (19), \u2225zk \u2212\u03b1p(zk) \u2212z\u2217\u22252 = \u2225zk \u2212z\u2217\u22252 + \u03b12\u2225p(zk)\u22252 \u22122\u03b1(zk \u2212z\u2217)T p(zk) \u2264\u2225zk \u2212z\u2217\u22252 \u2212\u03b12(1 \u22122\u03b1L 3 )\u2225p(zk)\u22252 \u2212\u03b1 2L\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252 \u2264\u2225zk \u2212z\u2217\u22252 (51) and, by (40), E\u2225zk \u2212z\u2217\u22252 \u2264E\u2225zk\u22121 \u2212z\u2217\u22252 \u2264\u00b7 \u00b7 \u00b7 \u2264\u2225z0 \u2212z\u2217\u22252. The next term in (50), \u03b1\u2225\u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(x\u2217)\u2225, is a constant. For the third term, from \u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252 (a) \u2264L2\u2225xk+1/2 \u2212x\u2217\u22252 (b) \u2264L2\u2225zk \u2212z\u2217\u22252, where (a) is due to Lipschitz continuity and (b) due to nonexpansiveness of the proximal mapping, it follows that \u03b1 p E\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252 \u2264\u03b1L p \u2225z0 \u2212z\u2217\u22252. Since E\u2225p(zk)\u22252 = O(1/k), we immediately have E(Ek) \u2264O(1/ \u221a k). Similarly, we can also show \u2212E(Ek) \u2264O(1/ \u221a k). Therefore, E|Ek| = O(1/ \u221a k). By extending the previous analysis of |Ek erg| and ek erg to E|Ek erg| and E(ek erg), respectively, along the same line of arguments, it is straightforward to show E|Ek erg| = O(1/k) and E(ek erg) = O(1/k). 33 \f7.3 Linear convergence analysis We \ufb01rst review some de\ufb01nitions and inequities. Let h be a closed convex proper function. We let \u00b5h \u22650 be the strong-convexity constant of h, where \u00b5h > 0 when h is strongly convex and \u00b5h = 0 otherwise. When h is di\ufb00erentiable and \u2207h is Lipschitz continuous, we de\ufb01ne (1/\u03b2h) be the Lipschitz constant of \u2207h. When h is either non-di\ufb00erentiable or di\ufb00erentiable but \u2207h is not Lipschitz, we de\ufb01ne \u03b2h = 0. Under these de\ufb01nitions, we have h(y)\u2212h(x) \u2265\u27e8\u02dc \u2207h(x), y \u2212x\u27e9+ 1 2 max \b \u00b5h\u2225x \u2212y\u22252, \u03b2h\u2225\u02dc \u2207h(x) \u2212\u02dc \u2207f(y)\u22252\t | {z } Sh(x,y) , (52) for any points x, y where the subgradients \u02dc \u2207h(x), \u02dc \u2207h(y) exist. Note that Sh(x, y) = Sh(y, x). For the three convex functions \u00af r, \u00af f, \u00af g, we introduce their parameters \u00b5\u00af r, \u03b2\u00af r, \u00b5 \u00af f, \u03b2 \u00af f, \u00b5\u00af g, \u03b2\u00af g, as well as the combination S(z, z\u2217) = S\u00af r(x, x\u2217) + S \u00af f(x, x\u2217) + S\u00af g(x\u2032, x\u2217). (53) As we assume each fi has L-Lipschitz gradient, we set \u03b2 \u00af f = 1/L. Since \u00af r includes the indicator function IC, which is non-di\ufb00erentiable, we set \u03b2\u00af r = 0. The values of remaining parameters \u00b5\u00af r, \u00b5 \u00af f, \u00b5\u00af g, \u03b2\u00af g \u22650 are kept unspeci\ufb01ed. Applying (52) to each pair of the three functions in E = \u0000\u00af r(x) + \u00af f(x) + \u00af g(x\u2032) \u0001 \u2212 \u0000\u00af r(x\u2217) + \u00af f(x\u2217) + \u00af g(x\u2217) \u0001 yields E \u2264( \u02dc \u2207\u00af r(x) + \u2207\u00af f(x))T (x \u2212x\u2217) + ( \u02dc \u2207\u00af g(x\u2032))T (x\u2032 \u2212x\u2217) \u2212S(z, z\u2217), (54) E \u2265( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(x\u2217))T (x \u2212x\u2217) + ( \u02dc \u2207\u00af g(x\u2217))T (x\u2032 \u2212x\u2217) + S(z, z\u2217). (55) In this fashion, both the upper and lower bounds on Ek, which we previously derive, are tightened by S(zk, z\u2217). In particular, we can tightened (28) and (25) as Ek \u2264 1 2\u03b1\u2225zk \u2212z\u2217\u22252 \u2212 1 2\u03b1\u2225zk+1 \u2212z\u2217\u22252 \u2212\u03b1 2 \u2225p(zk)\u22252 + \u03b1( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(xk+1/2))T p(zk) \u2212S(zk, z\u2217), (56) Ek \u2265( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(x\u2217))T \u03b1p(zk) + S(zk, z\u2217), (57) where the two terms involving S(zk, z\u2217) are newly added. Combining the upper and lower bounds of Ek yields 1 2\u03b1\u2225zk+1 \u2212z\u2217\u22252 \u2264 1 2\u03b1\u2225zk \u2212z\u2217\u22252 \u2212Q, (58) 34 \fwhere Q = \u2212\u03b1( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(xk+1/2))T p(zk) + \u03b1( \u02dc \u2207\u00af r(x\u2217) + \u2207\u00af f(x\u2217))T p(zk) + \u03b1 2 \u2225p(zk)\u22252 + 2S(zk, z\u2217) (59) = \u2212\u03b1(\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217))T p(zk) + \u03b1 2 \u2225p(zk)\u22252 + 2S(zk, z\u2217) (60) = \u0010 \u2212\u03b1(\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217))T p(zk) + \u03b1 2 \u2225p(zk)\u22252 + 2S \u00af f(zk, z\u2217) \u0011 + 2S\u00af r(zk, z\u2217) + 2S\u00af g(zk, z\u2217) (61) \u2265c1 \u0010 \u2225p(zk)\u22252 + \u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252\u0011 + \u00b5 \u00af f 2 \u2225xk+1/2 \u2212x\u22c6\u22252 + 2S\u00af r(zk, z\u2217) + 2S\u00af g(zk, z\u2217), (62) where c1 > 0 is a constant and the inequality follows from the Young\u2019s inequality: \u03b1(\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217))T p(zk) \u2264\u03b1 4 \u2225p(zk)\u22252 + \u03b1\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252 and \u03b1 < \u03b2 \u00af f. Later on, in three di\ufb00erent cases, we will show Q \u2265C\u2225zk \u2212z\u2217\u22252. (63) Hence, by substituting (63) into (58), we obtain the Q-linear (or quotient-linear) convergence relation \u2225zk+1 \u2212z\u2217\u2225\u2264 \u221a 1 \u22122\u03b1C\u2225zk \u2212z\u2217\u2225, (64) from which it is easy to further derive the Q-linear convergence results for |Ek| and ek. Case 1. Assume \u00af g is both strongly convex and has Lipschitz gradient, i.e., \u00b5\u00af g, \u03b2\u00af g > 0, (and \u00af f still has Lipschitz gradient). In this case, Q = c1 \u0010 \u2225p(zk)\u22252 + \u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252\u0011 + 2S\u00af g(zk, z\u2217) (65) \u2265c1 \u0010 \u2225p(zk)\u22252 + \u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252\u0011 + \u00b5\u00af g\u2225xk+1 \u2212x\u22c6\u22252. (66) (Here we use \u2207\u00af g instead of \u02dc \u2207\u00af g since \u00af g is di\ufb00erentiable.) By the identities zk = xk+1 \u2212\u03b1 \u0000\u2207\u00af g(xk+1) + \u2207\u00af f(xk+1/2) \u0001 + 2\u03b1p(zk), (67) z\u2217= x\u22c6 \u2212\u03b1 \u0000\u2207\u00af g(x\u22c6) + \u2207\u00af f(x\u22c6) \u0001 , (68) the triangle inequality, and \u2225\u2207\u00af g(xk+1) \u2212\u2207\u00af g(x\u22c6)\u2225\u2264(1/\u03b2\u00af g)\u2225xk+1 \u2212x\u22c6\u2225, we get \u2225zk \u2212z\u2217\u22252 = \r \r(xk+1 \u2212x\u22c6) \u2212\u03b1 \u0000\u2207\u00af g(xk+1) \u2212\u2207\u00af g(x\u22c6) \u0001 \u2212\u03b1 \u0000\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6) \u0001 + 2\u03b1p(zk) \r \r2 (69) \u22643 \r \r(xk+1 \u2212x\u22c6) \u2212\u03b1 \u0000\u2207\u00af g(xk+1) \u2212\u2207\u00af g(x\u22c6) \u0001\r \r2 + 3\u03b12\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6)\u22252 + 12\u03b12\u2225p(zk)\u22252 (70) \u22643(1 + \u03b1 \u03b2\u00af g )2\u2225xk+1 \u2212x\u22c6\u22252 + 3\u03b12\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6)\u22252 + 12\u03b12\u2225p(zk)\u22252. (71) 35 \fSince (71) is bounded by (66) up to a constant factor, we have established (63) for this case. Case 2. \u00af f is strongly convex and \u00af g has Lipschitz gradient, i.e., \u00b5 \u00af f, \u03b2\u00af g > 0 (and \u00af f still has Lipschitz gradient). In this case, Q = c1 \u0010 \u2225p(zk)\u22252 + \u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252\u0011 + \u00b5 \u00af f\u2225xk+1/2 \u2212x\u2217\u22252. (72) By the identities zk = xk+1/2 \u2212\u03b1 \u0000\u2207\u00af g(xk+1) + \u2207\u00af f(xk+1/2) \u0001 + \u03b1p(zk), (73) z\u2217= x\u22c6 \u2212\u03b1 \u0000\u2207\u00af g(x\u22c6) + \u2207\u00af f(x\u22c6) \u0001 , (74) the triangle inequality, and \u2225\u2207\u00af g(xk+1) \u2212\u2207\u00af g(x\u22c6)\u2225\u2264(1/\u03b2\u00af g)\u2225xk+1 \u2212x\u22c6\u2225, we get \u2225zk \u2212z\u2217\u22252 = \r \r(xk+1/2 \u2212x\u22c6) \u2212\u03b1 \u0000\u2207\u00af g(xk+1) \u2212\u2207\u00af g(x\u22c6) \u0001 \u2212\u03b1 \u0000\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6) \u0001 + \u03b1p(zk) \r \r2 (75) \u22644\u2225xk+1/2 \u2212x\u22c6\u22252 + 4\u03b12\u2225\u2207\u00af g(xk+1) \u2212\u2207\u00af g(x\u22c6)\u22252 + 4\u03b12\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6)\u22252 + 4\u03b12\u2225p(zk)\u22252 (76) \u22644\u2225xk+1/2 \u2212x\u22c6\u22252 + 4 \u03b12 \u03b22 \u00af g \u2225xk+1 \u2212x\u22c6\u22252 + 4\u03b12\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6)\u22252 + 4\u03b12\u2225p(zk)\u22252 (77) \u22644(1 + 2\u03b12 \u03b22 \u00af g )\u2225xk+1/2 \u2212x\u22c6\u22252 + 4\u03b12\u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u22c6)\u22252 + 4\u03b12(1 + 2\u03b12 \u03b22 \u00af g )\u2225p(zk)\u22252, (78) where the last line follows from \u2225xk+1 \u2212x\u22c6\u22252 = \u2225xk+1/2 \u2212x\u22c6\u2212\u03b1p(zk)\u22252 \u22642\u2225xk+1/2 \u2212x\u22c6\u22252 + 2\u03b12\u2225p(zk)\u22252. Since (78) is bounded by (72) up to a constant factor, we have established (63) for this case. Case 3. \u00af r is strongly convex and \u00af g has Lipschitz gradient, in short, \u00b5\u00af r\u03b2\u00af g > 0, (and \u00af f still has Lipschitz gradient). In this case, Q = c1 \u0010 \u2225p(zk)\u22252 + \u2225\u2207\u00af f(xk+1/2) \u2212\u2207\u00af f(x\u2217)\u22252\u0011 + \u00b5\u00af r\u2225xk+1/2 \u2212x\u2217\u22252. (79) We still use (78), which is bounded by (79) up to a constant factor, we have established (63) for this case. 8" + }, + { + "url": "http://arxiv.org/abs/1412.4845v2", + "title": "Adaptive Importance Sampling via Stochastic Convex Programming", + "abstract": "We show that the variance of the Monte Carlo estimator that is importance\nsampled from an exponential family is a convex function of the natural\nparameter of the distribution. With this insight, we propose an adaptive\nimportance sampling algorithm that simultaneously improves the choice of\nsampling distribution while accumulating a Monte Carlo estimate. Exploiting\nconvexity, we prove that the method's unbiased estimator has variance that is\nasymptotically optimal over the exponential family.", + "authors": "Ernest K. Ryu, Stephen P. Boyd", + "published": "2014-12-16", + "updated": "2015-01-09", + "primary_cat": "stat.ME", + "cats": [ + "stat.ME", + "math.OC", + "stat.CO", + "62L20, 65C05, 90C15, 90C25" + ], + "main_content": "Introduction Consider the problem of approximating the expected value (or integral) I = E\u03c6(X) = Z \u03c6(x)f(x) dx, where X \u223cf is a random variable on Rk and \u03c6 : Rk \u2192R. The standard Monte Carlo method estimates I by taking independent identically distributed (IID) samples X1, X2, . . . , Xn \u223cf and using \u02c6 IMC n = 1 n n X i=1 \u03c6(Xi). This estimator is unbiased, i.e., E\u02c6 IMC n = I, and has variance Var(\u02c6 IMC n ) = 1 nVarX\u223cf [\u03c6(X)] = 1 n \u0012Z \u03c62(x)f(x) dx \u2212I2 \u0013 . To reduce the variance of the estimator, the standard \ufb01gure of merit, one can use importance sampling: choose a sampling (importance) distribution \u02dc f satisfying \u02dc f(x) > 0 whenever 1 \f\u03c6(x)f(x) \u0338= 0, take IID samples X1, X2, . . . , Xn \u223c\u02dc f (as opposed to sampling from f, the nominal distribution) and use \u02c6 IIS n = 1 n n X i=1 \u03c6(Xi)f(Xi) \u02dc f(Xi) . Again, the estimator is unbaised, i.e., E\u02c6 IIS n = I, and has variance Var(\u02c6 IIS n ) = 1 nVarX\u223c\u02dc f \u0014\u03c6(X)f(X) \u02dc f(X) \u0015 = 1 n \u0012Z \u03c62(x)f 2(x) \u02dc f(x) dx \u2212I2 \u0013 . When \u02dc f = f, importance sampling reduces to standard Monte Carlo. Choosing \u02dc f wisely can reduce the variance, but this can be di\ufb03cult in general. One approach is to use a priori information on the integrand \u03c6(x)f(x) to manually \ufb01nd an appropriate sampling distribution \u02dc f, perhaps through several informal iterations [14,20,24,30,34,36,37]. Another approach is to automate the process of \ufb01nding the sampling distribution through adaptive importance sampling. In adaptive importance sampling, one adaptively improves the sampling distribution while simultaneously accumulating the estimate for I. A particular form of importance sampling generates a sequence of sampling distributions \u02dc f1, \u02dc f2, . . . and a series of samples X1 \u223c\u02dc f1, X2 \u223c\u02dc f2, . . . and forms the estimate \u02c6 IAIS n = 1 n n X i=1 \u03c6(Xi) f(Xi) \u02dc fi(Xi) . At each iteration n, the sampling distribution \u02dc fn, which is itself random, is adaptively determined based on the past data, \u02dc fn, . . . , \u02dc fn\u22121 and X1, . . . , Xn\u22121. Again, \u02c6 IAIS n is unbiased, i.e., E\u02c6 IAIS n = I, and Var(\u02c6 IAIS n ) = 1 n2 n X i=1 E \u02dc fiVarXi\u223c\u02dc fi \u0014\u03c6(Xi)f(Xi) \u02dc fi(Xi) \u0015 = 1 n2 n X i=1 E \u02dc fi \u0012Z \u03c62(x)f 2(x) \u02dc f 2(x) dx \u2212I2 \u0013 , where E \u02dc fi denotes the expectation over the random sampling distribution \u02dc fi. Again, when \u02dc fi = \u02dc f for all i, adaptive importance sampling reduces to standard (non-adaptive) importance sampling. Now determining how to choose \u02dc fn at each iteration fully speci\ufb01es the method. In this paper, we propose an instance of adaptive importance sampling, which we call Convex Adaptive Monte Carlo (Convex AdaMC). First, we choose an exponential family of distributions F as the set of candidate sampling distributions. De\ufb01ne T : Rk \u2192Rp and h : Rk \u2192R+. Then our density function is f\u03b8(x) = exp \u0000\u03b8TT(x) \u2212A(\u03b8) \u0001 h(x), where A : Rp \u2192R \u222a{\u221e}, de\ufb01ned as A(\u03b8) = log Z exp(\u03b8T T(x))h(x) dx, 2 \fserves as a normalizing factor. (When A(\u03b8) = \u221e, we de\ufb01ne f\u03b8 = 0 and remember that this does not de\ufb01ne a distribution.) Finally, let \u0398 \u2286Rp be a convex set, and our exponential family is F = {f\u03b8 | \u03b8 \u2208\u0398}, where \u03b8 is called the natural parameter of F. Note that the choice of T, h, and \u0398 fully speci\ufb01es our family F. Next, de\ufb01ne V : Rp \u2192R\u222a{\u221e} to be the per-sample variance of the importance sampled estimator with sampling distribution f\u03b8, V (\u03b8) = VarX\u223cf\u03b8 \u0014\u03c6(X)f(X) f\u03b8(X) \u0015 = Z \u03c62(x)f 2(x) f\u03b8(x) dx \u2212I2. (So the importance sampled estimator using n IID samples from f\u03b8 has variance V (\u03b8)/n.) A natural approach is to solve minimize V (\u03b8) subject to \u03b8 \u2208\u0398, (1) where \u03b8 is the optimization variable, as this will give us the best sampling distribution among F to importance sample from. We write V \u22c6to denote the optimal value, i.e., the optimal per-sample variance over the family. The \ufb01rst key insight of this paper is that V is a convex function, a consequence of F being an exponential family. Roughly speaking, one can e\ufb03ciently \ufb01nd a global minimum of a convex functions through standard methods if one can compute the function value and its gradient [22]. This fact, however, is not directly applicable to our setting as evaluating V (\u03b8) or \u2207V (\u03b8) for any given \u03b8 is in general as hard as evaluating I itself. The second key insight is that we can minimize the convex function V through a standard algorithm of stochastic optimization, stochastic gradient descent, while simuiltaneously accumulating an estimate for I. Because of convexity, we can prove theoretical guarantees. In Convex AdaMC, we generate a sequence of sampling distribution parameters \u03b81, \u03b82, . . . and a series of samples X1 \u223cf\u03b81, X2 \u223cf\u03b82, . . ., with which we form the estimate \u02c6 IAMC n = 1 n n X i=1 \u03c6(Xi) f(Xi) f\u03b8i(Xi). Again, \u02c6 IAMC n is unbiased, i.e., E\u02c6 IAMC n = I. Furthermore, we show that Var(\u02c6 IAMC n ) = 1 nV \u22c6+ O \u0012 1 n3/2 \u0013 = 1 n \u0012 V \u22c6+ O \u0012 1 n1/2 \u0013\u0013 . (2) This shows that the per-sample variance of Convex AdaMC converges to the optimal per sample variance over our family F; i.e., our estimator Convex AdaMC asymptotically performs as well as any (adaptive or non-adaptive) importance sampling estimator using sampling distributions from F. In particular, Convex AdaMC does not su\ufb00er from becoming trapped in (non-optimal) local minima, a problem other adaptive importance sampling methods can have. 3 \f2 Convexity of the variance Let\u2019s establish a few important properties of our variance function V (\u03b8) = Z \u03c62(x)f 2(x) exp(A(\u03b8) \u2212\u03b8T T(x))h(x) dx \u2212I2. When A(\u03b8) = \u221e, we de\ufb01ne V (\u03b8) = \u221e. Not only is this de\ufb01nition natural but is also convenient since V (\u03b8) = \u221enow indicates that \u03b8 is invalid either because the variance is in\ufb01nite or because \u03b8 doesn\u2019t de\ufb01ne a sampling distribution. Recall that a function V : Rp \u2192R \u222a{\u221e} is convex if V (\u03b7\u03b81 + (1 \u2212\u03b7)\u03b82) \u2264\u03b7V (\u03b81) + (1 \u2212\u03b7)V (\u03b82) holds for any \u03b7 \u2208[0, 1] and \u03b81, \u03b82 \u2208\u0398. Convexity is important because it allows us to prove a theoretical guarantee. Theorem 1. The variance of the importance sampling estimator V (\u03b8) is a convex function of \u03b8, the natural parameter of the exponential family. Proof. We \ufb01rst show A(\u03b8) is convex. By H\u00a8 older\u2019s inequality, we have exp(A(\u03b7\u03b81 + (1 \u2212\u03b7)\u03b82)) = Z exp((\u03b7\u03b81 + (1 \u2212\u03b7)\u03b82)TT(x))h(x) dx \u2264 \u0012Z exp(\u03b8T 1 T(x))h(x) dx \u0013\u03b7 \u0012Z exp(\u03b8T 2 T(x))h(x) dx \u00131\u2212\u03b7 , and by taking the log on both sides we get A(\u03b7\u03b81 + (1 \u2212\u03b7)\u03b82) \u2264\u03b7A(\u03b81) + (1 \u2212\u03b7)A(\u03b82). Since exp(\u00b7) is an increasing convex function and A(\u03b8) \u2212\u03b8TT(x) is convex in \u03b8, the composition exp(A(\u03b8) \u2212\u03b8TT(x)) is convex in \u03b8. Finally, V (\u03b8) is convex as it is an integral of the convex functions exp(A(\u03b8) \u2212\u03b8TT(x))h(x) over x; see [16, \u00a7B.2], [31, \u00a75], or [5, \u00a73.2]. We note in passing that log V (\u03b8) is also a convex function of \u03b8, which is a stronger statement than convexity of V (\u03b8). This fact, however, is not useful for us since we do not have a simple way to obtain a stochastic gradient for log V (\u03b8), whereas, as we will see later, we do for V (\u03b8). As we will see soon, stochastic gradient descent hinges on evaluating the derivative of V under the integral. The following lemma is a consequence of Theorem 2.7.1 of [18]. Lemma 1. V is di\ufb00erentiable and its gradient can be evaluated under the integral on int {\u03b8 | V (\u03b8) < \u221e}, where int denotes the interior. In particular, we have \u2207V (\u03b8) = Z \u2207\u03b8 \u03c62(x)f 2(x) f\u03b8(x) dx = Z (\u2207A(\u03b8) \u2212T(x))\u03c62(x)f 2(x) f 2 \u03b8 (x) f\u03b8(x) dx = EX\u223cf\u03b8 \u0014 (\u2207A(\u03b8) \u2212T(X))\u03c62(X)f 2(X) f 2 \u03b8 (X) \u0015 . 4 \fSo when we take a sample X \u223cf\u03b8, the random vector g = (\u2207A(\u03b8) \u2212T(X))\u03c62(X)f 2(X) f 2 \u03b8 (X) satis\ufb01es Eg = \u2207V (\u03b8). 3 The method Stochastic gradient descent is a standard method for solving minimize V (\u03b8) subject to \u03b8 \u2208\u0398, using the algorithm \u03b8n+1 = \u03a0(\u03b8n \u2212\u03b1ngn), where \u03a0 is (Euclidean) projection onto \u0398, the step size \u03b1n > 0 is an appropriately chosen sequence, and the stochastic gradient gn is a random variable satisfying E [gn | \u03b8n] = \u2207V (\u03b8n). The intuition is that \u2212gn, although noisy, generally points towards a descent direction of V at \u03b8n, and therefore each step reduces the function value of V in expectation [17,27,29,35]. Our algorithm, which we call Convex AdaMC, is Xn \u223c f\u03b8n \u02c6 IAMC n = 1 n n X i=1 \u03c6(Xi)f(Xi) f\u03b8i(Xi) gn = (\u2207A(\u03b8n) \u2212T(Xn))\u03c62(Xn)f 2(Xn) f 2 \u03b8n(Xn) \u03b8n+1 = \u03a0 \u0012 \u03b8n \u2212C \u221angn \u0013 , where C > 0 and \u03b81 \u2208\u0398. As mentioned in the introduction, the estimator \u02c6 IAMC n is unbiased and has variance given by (2) under a technical condition to be presented in \u00a74. We can view Convex AdaMC as an adaptive importance sampling method where the third and fourth line of the algorithm updates the sampling distribution. Alternatively, we can view Convex AdaMC as stochastic gradient descent on the convex function V with an additional step, the second line of the algorithm, that accumulates the estimate of I but does not otherwise a\ufb00ect the iteration. The computational cost of Convex AdaMC is cheap, of course, if all of its operations are cheap. This is the case if \u03c6 and f are functions we can easily evaluate, if our family of distributions, F, is one of the well-known exponential families, and if \u0398 is a set we can easily project onto. 5 \f4 Analysis Before we present our convergence results, we discuss the choice of \u0398. For any convex domain \u0398, our variance function V is convex and minimizing V (\u03b8) over \u03b8 \u2208\u0398 is a mathematically well-de\ufb01ned problem. However, for our method to be well-de\ufb01ned and for the proof of convergence to work out, we need further restrictions on \u0398. De\ufb01ne K(\u03b8) = EX\u223cf\u03b8 \u0014\u03c64(X)f 4(X) f 4 \u03b8 (X) \u0015 = Z \u03c64(x)f 4(x) f 3 \u03b8 (x) dx. We require that \u0398 is convex and compact and that \u0398 \u2286int {\u03b8 | K(\u03b8) < \u221e}. In other words, \u0398 must be a convex compact subset of the interior of the set of natural parameters for which their importance sampled estimates have \ufb01nite 4th moment. Since {\u03b8 | K(\u03b8) < \u221e} \u2286{\u03b8 | V (\u03b8) < \u221e} \u2286{\u03b8 | A(\u03b8) < \u221e} it follows that any \u03b8 \u2208\u0398 de\ufb01nes a sampling distribution f\u03b8 that produces an importance sampled estimate of \ufb01nite variance. Theorem 2. Assume \u0398 \u2286int {\u03b8 | K(\u03b8) < \u221e} is nonempty, convex, and compact. De\ufb01ne D = max\u03b81,\u03b82\u2208\u0398 \u2225\u03b81 \u2212\u03b82\u22252 and G2 = sup \u03b8\u2208\u0398 EX\u223cf\u03b8 \r \r \r \r(\u2207A(\u03b8) \u2212T(X))\u03c62(X)f 2(X) f 2 \u03b8 (X) \r \r \r \r 2 2 . Then G < \u221e, and \u02c6 IAMC n , the unbiased estimator of Convex AdaMC, satis\ufb01es 1 nV \u2217\u2264Var(\u02c6 IAMC n ) \u22641 nV \u2217+ \u0012D2 2C + CG2 \u0013 1 n3/2. Proof. We defer the proof of G < \u221eto the appendix. Since the conditional dependency of our sequences \u03b81, \u03b82, . . . and X1, X2, . . . is \u03b81 \u03b82 \u03b83 \u03b84 X1 X2 X3 \u00b7 \u00b7 \u00b7 Xi is independent of the entire past conditioned on \u03b8i for all i. With this insight, we have E\u02c6 IAMC n = 1 n n X i=1 E\u03c6(Xi)f(Xi) f\u03b8i(Xi) = 1 n n X i=1 E \u0014 E \u0014\u03c6(Xi)f(Xi) f\u03b8i(Xi) | \u03b8i \u0015\u0015 = 1 n n X i=1 E [I] = I 6 \fand Var(\u02c6 IAMC n ) = E(\u02c6 IAMC n \u2212I)2 = 1 n2 n X i=1 E \u0012\u03c6(Xi)f(Xi) f\u03b8i(Xi) \u2212I \u00132 + 2 n2 X 1\u2264i 0, then Jnn D \u2192N (0, V \u22c6). Since n X i=1 \u03c32 ni = 1 n n X i=1 V (\u03b8i), and since, by Theorem 2, we have 1 n n X i=1 (EV (\u03b8i) \u2212V \u22c6) = E \f \f \f \f \f 1 n n X i=1 V (\u03b8i) \u2212V \u22c6 \f \f \f \f \f = O(1/\u221an) \u21920, i.e., Pn i=1 \u03c32 ni converges to V \u22c6in L1, we have n X i=1 \u03c32 ni P \u2192V \u2217. Finally, since \u0398 \u2286{\u03b8 | K(\u03b8) < \u221e} is a compact set and K(\u03b8) is a continuous function by Lemma 2, we have B = sup \u03b8\u2208\u0398 K(\u03b8) < \u221e and we conclude n X i=1 EY 2 niI{|Yni|\u2265\u03b5} = 1 n n X i=1 E\u03c62(Xi)f 2(Xi) f 2 \u03b8i(Xi) In \u03c62(Xi)f2(Xi)/f2 \u03b8i(Xi)\u2265n\u03b52 o \u2264 1 n2\u03b52 n X i=1 E\u03c64(Xi)f 4(Xi) f 4 \u03b8i(Xi) \u2264B n\u03b52 \u21920. Since this proves the conditions we need, applying the martingale CLT completes the proof. 14" + } + ], + "Changho Suh": [ + { + "url": "http://arxiv.org/abs/1709.00675v1", + "title": "Two-Way Interference Channel Capacity: How to Have the Cake and Eat it Too", + "abstract": "Two-way communication is prevalent and its fundamental limits are first\nstudied in the point-to-point setting by Shannon [1]. One natural extension is\na two-way interference channel (IC) with four independent messages: two\nassociated with each direction of communication. In this work, we explore a\ndeterministic two-way IC which captures key properties of the wireless Gaussian\nchannel. Our main contribution lies in the complete capacity region\ncharacterization of the two-way IC (w.r.t. the forward and backward sum-rate\npair) via a new achievable scheme and a new converse. One surprising\nconsequence of this result is that not only we can get an interaction gain over\nthe one-way non-feedback capacities, we can sometimes get all the way to\nperfect feedback capacities in both directions simultaneously. In addition, our\nnovel outer bound characterizes channel regimes in which interaction has no\nbearing on capacity.", + "authors": "Changho Suh, Jaewoong Cho, David Tse", + "published": "2017-09-03", + "updated": "2017-09-03", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION Two-way communication, where two nodes want to communicate data to each other, is prevalent. The \ufb01rst study of such two-way channels was done by Shannon [1] in the setting of point-to-point memoryless channels. When the point-to-point channels in the two directions are orthogonal (such as when the two directions are allocated different time slots or different frequency bands, or when the transmitted signal can be canceled perfectly as in full-duplex C. Suh and J. Cho are with the School of Electrical Engineering at Korea Advanced Institute of Science and Technology, South Korea (Email: {chsuh, cjw2525}@kaist.ac.kr) D. Tse is with the Electrical Engineering Department at Stanford University, CA, USA (Email: dntse@stanford.edu). DRAFT \f2 fwd msg 1 fwd msg 2 bwd msg 1 bwd msg 2 Fig. 1. Two interfering two-way communication links, consisting of two IC\u2019s, one in each direction. The IC\u2019s are orthogonal to each other and do not necessarily have the same channel gains. communication), the problem is not interesting as feedback does not increase point-to-point capacity. Hence, communication in one direction cannot increase the capacity of the other direction and no interaction gain is possible. One can achieve no more than the one-way capacity in each direction. The situation changes in network scenarios where feedback can increase capacity. In these scenarios, communication in one direction can potentially increase the capacity of the other direction by providing feedback in addition to communicating data. One scenario of particular interest is the setting of the two-way interference channel (two-way IC), modeling two interfering two-way communication links (Fig. 1). Not only is this scenario common in wireless communication networks, it has also been demonstrated that feedback provides a signi\ufb01cant gain for communication over (one-way) IC\u2019s [2], [3], [4]. In particular, [3] reveals that the feedback gain can be unbounded, i.e., the gap between the feedback and non-feedback capacities can be arbitrarily large for certain channel parameters. This suggests the potential of signi\ufb01cant interaction gain in two-way IC\u2019s. On the other hand, the feedback result [3] assumes a dedicated in\ufb01nite-capacity feedback link. In the two-way setting, any feedback needs to be transmitted through a backward IC, which also needs to carry its own backward data traf\ufb01c. The question is when we take in consideration the competition with the backward traf\ufb01c, whether there is still any net interaction gain through feedback? To answer this question, [5] investigated a two-way IC under the linear deterministic model [6], which approximates a Gaussian channel. A scheme is proposed to demonstrate a net interaction DRAFT \f3 interaction gain interaction gain interaction gain no interaction gain interaction gain perfect feedback perfect feedback no feedback gain no interaction gain Fig. 2. When can one have the cake and eat it too? The plot is over two channel parameters of the deterministic model, \u03b1 and \u02dc \u03b1, where \u03b1 is the ratio of the interference-to-noise ratio (in dB) to the signal-to-noise ratio (in dB) of the IC in the forward direction and \u02dc \u03b1 is the corresponding quantity of the IC in the backward direction. The parameter \u03b3 is the ratio of the backward signal-to-noise ratio (in dB) to the forward signal-to-noise ratio (in dB), and is \ufb01xed to be a value between 1 and 4. White region: feedback does not increase capacity in either direction and thus interaction is not useful. Purple: feedback does increase capacity but interaction cannot provide such increase. Light blue: feedback can be provided through interaction and there is a net interaction gain. Dark blue: interaction is so ef\ufb01cient that one can achieve perfect feedback capacity simultaneously in both directions. This implies that one can obtain the maximal feedback gain without any sacri\ufb01ce for feedback transmission (have the cake and eat it too). gain, i.e., one can simultaneously achieve better than the non-feedback capacities in both directions. While an outer bound is also derived, it has a gap to the lower bound. Hence, there has been limited understanding on the maximal gain that can be reaped by feedback. In particular, whether or not one can get all the way to perfect feedback capacities in both directions has been unanswered. Recently Cheng-Devroye [7] derived an outer bound, but it does not give a proper answer as the result assumes a partial interaction scenario in which interaction is enabled only at two nodes, while no interaction is permitted at the other two nodes. In this work, we settle this open problem and completely characterize the capacity region of the deterministic two-way IC via a new capacity-achieving transmission scheme as well as a novel outer bound. For simplicity, we assume the IC in each direction is symmetrical between the two users; however the IC\u2019s in the two directions are not necessarily the same (for example, they DRAFT \f4 may use different frequency bands). For some channel gains, the new scheme simultaneously achieves the perfect feedback sum-capacities of the IC\u2019s in both directions. This occurs even when feedback offers gains in both directions and thus feedback must be explicitly or implicitly carried over each IC while sending the traf\ufb01c in its own direction. Fig. 2 shows for what channel gains this happens. In the new scheme, feedback allows the exploitation of the following as side information: (i) past received signals; (ii) users\u2019 own messages; (iii) even the future information via retrospective decoding (to be detailed later; see Remark 3 in particular). While the \ufb01rst two were already shown to offer a feedback gain in literature, the third is newly exploited. It turns out this new exploitation leads us to achieve the perfect feedback capacities in both directions, which can never be done by the prior schemes [3], [4], [5]. Our new outer bound leads to the characterization of channel regimes in which interaction provides no gain in capacity. The bound is neither cutset nor more sophisticated bounds such as genie-aided bounds [8], [2], [3], [9], [10], [11], [14] and the generalized network sharing bound [12]. We employ a notion called triple mutual information, also known as interaction information [13]. In particular, we exploit one key property of the notion, commutativity, to derive the bound. II. MODEL Fig. 3 describes a two-way deterministic IC where user k wants to send its own message Wk to user \u02dc k, while user \u02dc k wishes to send its own message \u02dc Wk to user k, k = 1, 2. We assume that (W1, W2, \u02dc W1, \u02dc W2) are independent and uniformly distributed. For simplicity, we consider a setting where both forward and backward ICs are symmetric but not necessarily the same. In the forward IC, n and m indicate the number of signal bit levels for direct and cross links respectively. The corresponding values in the backward IC are denoted by (\u02dc n, \u02dc m). Let Xk \u2208Fmax(n,m) 2 be user k\u2019s transmitted signal and Vk \u2208Fm 2 be a part of Xk visible to user \u02dc j(\u0338= \u02dc k). Similarly let \u02dc Xk be user \u02dc k\u2019s transmitted signal and \u02dc Vk be a part of \u02dc Xk visible to user j(\u0338= k). The deterministic model abstracts broadcast and superposition of signals in the wireless Gaussian channel. See [6] for explicit details. A signal bit level observed by both users is broadcasted. If multiple signal levels arrive at the same signal level at a user, we assume modulo-2-addition. The encoded signal Xki of user k at time i is a function of its own message and past received signals: Xki = fki(Wk, \u02dc Y i\u22121 k ). DRAFT \f5 m n \u02dc n \u02dc m \u02dc Y i\u22121 1 \u02dc Y i\u22121 2 X2i X1i Y1i Y2i Enc 1 Enc 2 Dec 1 Dec 2 \u02dc X1i \u02dc X2i \u02dc Y1i \u02dc Y2i Dec \u02dc 1 Dec \u02dc 2 Enc \u02dc 1 Enc \u02dc 2 Y i\u22121 1 Y i\u22121 2 User 1 User 2 User 1 User 2 User User User User Fig. 3. A two-way deterministic interference channel (IC). We de\ufb01ne \u02dc Y i\u22121 k := { \u02dc Ykt}i\u22121 t=1 where \u02dc Ykt denotes user k\u2019s received signal at time t, offered through the backward IC. Similarly the encoded signal \u02dc Xki of user \u02dc k at time i is a function of its own message and past received signals: \u02dc Xki = \u02dc fki( \u02dc Wk, Y i\u22121 k ). A rate tuple (R1, R2, \u02dc R1, \u02dc R2) is said to be achievable if there exists a family of codebooks and encoder/decoder functions such that the decoding error probabilities go to zero as code length N tends to in\ufb01nity. For simplicity, we focus on a sum-rate pair regarding the forward and bacward ICs: (R, \u02dc R) := (R1 + R2, \u02dc R1 + \u02dc R2).1 The capacity region is de\ufb01ned as the closure of the set of achievable sumrate pairs: C = closure{(R, \u02dc R) : (R1, R2, \u02dc R1, \u02dc R2) \u2208Chigh} where Chigh denotes the one w.r.t. the high-dimensional rate tuple. 1The extension to the four-rate tuple case is not that challenging although it requires a complicated yet tedious analysis. Given our results (to be presented soon) and the tradeoff w.r.t. (R1, R2) (or ( \u02dc R1, \u02dc R2)) already characterized in [3], the extension does not provide any additional insights. Hence, here we consider a simpler sum-rate pair setting. DRAFT \f6 III. MAIN RESULTS Our main contribution lies in characterization of the capacity region of the two-way IC, formally stated below. Theorem 1 (Capacity region): The capacity region C of the two-way IC is the set of (R, \u02dc R) such that R \u2264max(2n \u2212m, m) =: Cpf (1) \u02dc R \u2264max(2\u02dc n \u2212\u02dc m, \u02dc m) =: \u02dc Cpf (2) R + \u02dc R \u22642(n + \u02dc n) (3) R + \u02dc R \u22642 max(n \u2212m, m) + 2 max(\u02dc n \u2212\u02dc m, \u02dc m) (4) where Cpf and \u02dc Cpf indicate the perfect feedback sum-capacities of the forward and backward IC\u2019s, respectively [3]. Proof: The achievability proof relies on two novel transmission schemes. In particular, we highlight key features of the second scheme that we call retrospective decoding which plays a crucial role to achieve perfect feedback capacities in both directions. The \ufb01rst feature is that it consists of two stages, each comprising a suf\ufb01ciently large number L of time slots. The second feature is that in the second stage, feedback-aided successive re\ufb01nement w.r.t. the fresh symbols sent in the \ufb01rst stage occurs in a retrospective manner: the fresh symbol sent in time i of stage I is re\ufb01ned in time 2L + 2 \u2212i of stage II where 1 \u2264i \u2264L. See Section IV for the detailed proof. For the converse proof, we \ufb01rst note that the \ufb01rst two bounds (1) and (2) match the perfectfeedback bound [3], [14], [5]. So one can prove them with a simple modi\ufb01cation to the proof in the references. The third bound is due to cutset: R1 + \u02dc R2 \u2264n + \u02dc n and R2 + \u02dc R1 \u2264n + \u02dc n. Our contribution lies in the derivation of the last bound. See Section V-B for the proof. We state two baselines for comparison to our main result. Baseline 1 ([8], [15]): The capacity region Cno for the non-interactive scenario is the set of (R, \u02dc R) such that R \u2264min {2 max(n \u2212m, m), max(2n \u2212m, m), 2n} =: Cno \u02dc R \u2264min {2 max(\u02dc n \u2212\u02dc m, \u02dc m), max(2\u02dc n \u2212\u02dc m, \u02dc m), 2\u02dc n} =: \u02dc Cno. Baseline 2 ([3]): The capacity region for the perfect feedback scenario is Cpf = {(R, \u02dc R) : R \u2264Cpf, \u02dc R \u2264\u02dc Cpf}. DRAFT \f7 With Theorem 1 and Baseline 1, one can readily see that feedback gain (in terms of capacity region) occurs as long as (\u03b1 / \u2208[ 2 3, 2], \u02dc \u03b1 / \u2208[ 2 3, 2]), where \u03b1 := m n and \u02dc \u03b1 := \u02dc m \u02dc n . A careful inspection reveals that there are channel regimes in which one can enhance Cno (or \u02dc Cno) without sacri\ufb01cing the other counterpart. This implies a net interaction gain. De\ufb01nition 1 (Interaction gain): We say that an interaction gain occurs if one can achieve (R, \u02dc R) = (Cno + \u03b4, \u02dc Cno + \u02dc \u03b4) for some \u03b4 \u22650 and \u02dc \u03b4 \u22650 such that max(\u03b4, \u02dc \u03b4) > 0. A tedious yet straightforward calculation with this de\ufb01nition leads us to identify channel regimes which exhibit an interaction gain, marked in light blue in Fig. 2. We also \ufb01nd the regimes in which feedback does increase capacity but interaction cannot provide such increase, meaning that whenever \u03b4 > 0, \u02dc \u03b4 must be \u2212\u03b4 and vice versa. These are (\u03b1 \u22642 3, \u02dc \u03b1 \u22642 3) and (\u03b1 \u22652, \u02dc \u03b1 \u22652) marked in purple in Fig. 2. The cutset bound (3) proves this for (\u03b1 \u22652, \u02dc \u03b1 \u22652). The regime of (\u03b1 \u22642 3, \u02dc \u03b1 \u22642 3) has been open as to whether both \u03b4 and \u02dc \u03b4 can be non-negative. Our novel bound (4) cracks the open regime, demonstrating that there is no interaction gain in the regime. Achieving perfect feedback capacities: One interesting observation is that there are channel regimes in which both \u03b4 and \u02dc \u03b4 can be strictly positive. This is unexpected because it implies that not only feedback does not sacri\ufb01ce one transmission for the other, it can actually improve both simultaneously. More interestingly, \u03b4 and \u02dc \u03b4 can reach up to the maximal feedback gains, re\ufb02ected in Cpf \u2212Cno and \u02dc Cpf \u2212\u02dc Cno. The dark blue regimes in Fig. 2 indicate such channel regimes when 1 \u2264\u03b3 := \u02dc n n \u22644. Note that such regimes depend on \u03b3. The amount of feedback that one can send is limited by available resources offered by the backward (or forward) IC. Hence, the feedback gain can be saturated depending on availability of the resources, which is affected by the channel asymmetry parameter \u03b3. One point to note here is that for any \u03b3, there always exists a non-empty set of (\u03b1, \u02dc \u03b1) in which perfect feedback capacities can be achieved. Corollary 1 stated below exhibits all of such channel regimes. Corollary 1: Consider a case in which feedback helps in both ICs: Cpf > Cno and \u02dc Cpf > \u02dc Cno. In this case, the channel regimes in which C = Cpf are: (I) \u03b1 < 2/3, \u02dc \u03b1 > 2, Cpf \u2212Cno \u22642 \u02dc m \u2212\u02dc Cpf, \u02dc Cpf \u2212\u02dc Cno \u22642n \u2212Cpf; (II) \u02dc \u03b1 < 2/3, \u03b1 > 2, \u02dc Cpf \u2212\u02dc Cno \u22642m \u2212Cpf, Cpf \u2212Cno \u22642\u02dc n \u2212\u02dc Cpf. Proof: A tedious yet straightforward calculation with Theorem 1 completes the proof. DRAFT \f8 Remark 1 (Why the Perfect Feedback Regimes?): When \u03b1 < 2/3 and \u02dc \u03b1 > 2, 2 \u02dc m indicates the total number of resource levels at the receivers in the backward channel. Hence, one can interpret 2 \u02dc m \u2212\u02dc Cpf as the remaining resource levels (resource holes) that can potentially be utilized to aid forward transmission. It turns out feedback can maximize resource utilization by \ufb01lling up the resource holes under-utilized in the non-interactive case. Note that Cpf \u2212Cno represents the amount of feedback that needs to be sent for achieving Cpf. Hence, the condition Cpf \u2212Cno \u22642 \u02dc m \u2212\u02dc Cpf (similarly \u02dc Cpf \u2212\u02dc Cno \u22642n \u2212Cpf) in Corollary 1 implies that as long as we have enough resource holes, we can get all the way to perfect feedback capacity. We will later provide an intuition as to why feedback can do so while describing our achievability; see Remark 3 in particular. \u25a1 IV. ACHIEVABILITY PROOF OF THEOREM 1 We \ufb01rst illustrate new transmission schemes via two toy examples in which the key ingredients of our achievability idea are well presented. Once the description of the schemes is done via the examples, we will then outline the proof for generalization while leaving a detailed proof for arbitrary channel parameters in Appendix A. A. Example 1: (n, m) = (2, 1), (\u02dc n, \u02dc m) = (1, 2) See Fig. 4 for the channel structure of the example. The claimed rate region in this example reads {(R, \u02dc R) : R \u2264Cpf = 3, \u02dc R \u2264\u02dc Cno = 2}. This is the case in which one can achieve Cpf while maintaining \u02dc Cno. We introduce a new transmission scheme (that we call Scheme 1) to achieve the claimed rate region. Perfect feedback scheme: A perfect feedback scheme was presented in [3]. Here we consider a different scheme which allows us to resolve the tension between feedback and independent messages when translated into a two-way scenario. The scheme operates in two stages. See Fig. 4. In stage I, four fresh symbols ((A, a) from user 1 and (B, b) from user 2) are transmitted. The scheme in [3] feeds a \u2295B back to user 1, so that user 1 can decode B which turns out to help re\ufb01ning the corrupted symbol a in stage II. On the other hand, here we send a\u2295B back to user 2. This way, user 2 can get a by removing its own symbol B. Similarly user 1 can get b. Now in stage II, user 2 intends to re-send b on top, as the b is corrupted due to A in stage I. But here a challenge arises. The challenge is that the b causes interference to user \u02dc 1 at the bottom level. DRAFT \f9 User 1 User 2 User User User 1 User 2 User User stage I stage I stage II stage II get get interference neutralization Fig. 4. A perfect feedback scheme for (n, m) = (2, 1) where Cpf = 3 (top); a nonfeedback scheme for (\u02dc n, \u02dc m) = (1, 2) where \u02dc Cpf = \u02dc Cno = 2 (bottom). But here the symbol b obtained via feedback at user 1 can play a role. The idea of interference neutralization [16] comes into play. User 1 sending the b on bottom enables neutralizing the interference. This then allows user 1 to transmit another fresh symbol, say A\u2032, without being interfered. Similarly user 2 can carry B\u2032 interference-free. This way, we send 6 symbols during two time slots, thus achieving Cpf = 3. As for the backward IC, we employ a nonfeedback scheme in [15]. User \u02dc 1 and \u02dc 2 send ( \u02dc A, \u02dc B) on top levels. This yields \u02dc Cno = 2. We are now ready to illustrate our achievability. Like the perfect feedback scheme, it still operates in two stages and the operation of stage I remains unchanged. A new idea comes in feedback strategy. Recall that a \u2295B is the one that is desired to be fed back to user 2. But the a \u2295B has a con\ufb02ict with transmission of \u02dc A. It seems an explicit selection needs to be made between the two competing transmissions. But it turns out the two transmissions come without the con\ufb02ict. The idea is to combine the XORing scheme introduced in network coding literature [17] with interference neutralization [16]. See Fig. 5. User \u02dc 1 simply sends the XOR of DRAFT \f10 stage 1 stage I stage II stage II get get User 1 User 2 User User interference alignment & neutralization User 1 User 2 User User desired to be fed back interference neutralization XORing has has Fig. 5. XORing with interferene neutralization for feedback strategy; Employing interference alignment and neutralization for re\ufb01nement of the past corrupted symbols. a\u2295B and \u02dc A on top. User 1 can then extract \u02dc A\u2295B by using its own symbol a as side information. But it is still interfered with by B. Here a key observation is that B is also available at user \u02dc 2 it was received cleanly at the top level in stage I. User \u02dc 2 sending the B on bottom enables user 1 to achieve interference neutralization at the bottom level, thereby decoding \u02dc A of interest. Now consider user 2 side. User 2 can exploit B to obtain a \u2295\u02dc A. Note that a \u2295\u02dc A is not the same as a wanted by user 2 in the perfect feedback scheme. Nonetheless a \u2295\u02dc A can serve the same role as a and this will be clearer soon. Similarly, user \u02dc 2 sending \u02dc B \u2295(b \u2295A) on top while user \u02dc 1 sending A (already delivered via the forward IC) on bottom, user 2 can decode \u02dc B of interest and user 1 can get b \u2295\u02dc B. Now in stage II, we take a similar approach as in the perfect feedback case. User 2 intends to re-send b on top. Recall in the perfect feedback scheme that user 1 sent the fedback symbol b on bottom, in order to remove the interference caused to user \u02dc 1. But the situation is different here. User 1 has b \u2295\u02dc B instead. It turns out this can also play the same role. The idea is to DRAFT \f11 use interference alignment and neutralization [18], [19], [16]. User 1 sends b \u2295\u02dc B on bottom. Here \u02dc B seems to cause interference to user \u02dc 1. But this can be canceled as \u02dc B is already decoded at user 2 see the bottom level at user 2 in the backward channel. User 2 sending b \u2295\u02dc B on top enables interference neutralization. This allows user 1 to send another fresh symbol A\u2032 on bottom interference-free. Note that b \u2295\u02dc B can be viewed as the aligned interference w.r.t. A. Similarly user 1 sending a \u2295\u02dc A on top and user 2 sending B\u2032 \u2295(a \u2295\u02dc A) on bottom, user \u02dc 1 and \u02dc 2 can decode a and B\u2032 respectively. This way, we achieve Cpf = 3 as in the perfect feedback case while maintaining \u02dc Cno = 2. Hence, the claimed rate region is achieved. \u25a1 Remark 2 (Exploiting Side Information): Note in Fig. 5 (bottom) that the two backward symbols ( \u02dc A, \u02dc B) and the two feedback signals (a\u2295B, b\u2295A) can be transmitted through 2-bit-capacity backward IC. This is because each user can cancel the seemingly interfering information by exploiting what has been received and its own symbols as side information. The side information allows the backward IC to have an effectively larger capacity, thus yielding a gain. This gain equalizes feedback cost, which in turn enables feedback to come for free in the end. The nature of the gain offered by side information coincides with that of the two-way relay channel [20] and many other examples [21], [22], [3], [23], [24]. \u25a1 B. Example 2: (n, m) = (2, 1), (\u02dc n, \u02dc m) = (0, 1) Scheme 1 is intended for the regimes in which feedback provides a gain only in one direction, e.g., Cpf > Cno and \u02dc Cpf = \u02dc Cno. For the regimes feedback helps in both directions, we develop another transmission scheme (that we call Scheme 2) which enables us to get sometimes all the way to perfect feedback capacities. In this section, we illustrate the scheme via Example 2 in which (Cpf = 3 > 2 = Cno, \u02dc Cpf = 1 > 0 = \u02dc Cno) and one can achieve (R, \u02dc R) = (Cpf, \u02dc Cpf). See Fig. 6 for the channel structure of the example. Our scheme operates in two stages. But one noticeable distinction is that each stage comprises a suf\ufb01ciently large number of time slots. Speci\ufb01cally stage I consists of L time slots, while stage II uses L + 1 time slots. It turns out our scheme ensures transmission of 6L forward symbols and 2L backward symbols, thus yielding: (R, \u02dc R) = \u0012 6L 2L + 1, 2L 2L + 1 \u0013 \u2212 \u2192(3, 1) = (Cpf, \u02dc Cpf). as L \u2192\u221e. Here are details. DRAFT \f12 time 1 time 1 time 2 time 2 XORing XORing desired to be fed back User 1 User 2 User 1 User 2 User User User User time L time L tension has time i: XORing has time i: Fig. 6. Stage I: Employ L time slots. The operation in each time slot is similar to stage I\u2019s operation in the perfect feedback case. We simply forward the XOR of a feedback signal and a new independent symbol. Here we see the tension between them. Before describing details, let us review the perfect feedback scheme of the backward IC [3] which takes a relaying idea. User \u02dc 1 delivers a backward symbol, say \u02dc a, to user 1 via the feedbackassisted path: user \u02dc 1 \u2192user 2 \u2192feedback \u2192user \u02dc 2 \u2192user 1. Similarly user \u02dc 2 sends \u02dc b to user 2. This yields \u02dc Cpf = 1. Stage I: We employ L time slots. In each time slot, we mimick the perfect feedback scheme although we have the tension between feedback and independent message transmissions. Time 1: Four fresh symbols are transmitted over the forward IC. User \u02dc 1 then extracts the one that is desired to be fed back: a1 \u2295B1. Next we send the XOR of a1 \u2295B1 and a backward symbol, say \u02dc a1. Similarly user \u02dc 2 transmits (b1 \u2295A1) \u2295\u02dc b1. User 1 then gets b1 \u2295\u02dc b1 using its own symbol A1. Similarly user 2 gets a1 \u2295\u02dc a1. Time 2: User 1 superimposes b1 \u2295\u02dc b1 with another new symbol, say a2, sending the XOR on bottom. On top is another fresh symbol A2 transmitted. Similarly user 2 sends (B2, b2\u2295(a1\u2295\u02dc a1)). User \u02dc 1 transmits (a2 \u2295b1 \u2295\u02dc b1 \u2295B2) \u2295\u02dc a2. Similarly user \u02dc 2 sends (b2 \u2295a1 \u2295\u02dc a1 \u2295A2) \u2295\u02dc b2. User 1 then gets b2 \u2295\u02dc a1 \u2295\u02dc b2 by using its own signal a1 \u2295A2. Similarly user 2 obtains a2 \u2295\u02dc b1 \u2295\u02dc a2. Repeating the above, one can readily verify that at time i \u2208{2, \u00b7 \u00b7 \u00b7 , L}, user 1 and 2 get bi \u2295\u02dc ai\u22121 \u2295\u02dc bi\u22121 and ai \u2295\u02dc bi\u22121 \u2295\u02dc ai\u22121 respectively; similarly user \u02dc 1 and \u02dc 2 get ai \u2295bi\u22121 \u2295\u02dc bi\u22121 \u2295Bi and bi \u2295ai\u22121 \u2295\u02dc ai\u22121 \u2295Ai on bottom, respectively. See Fig. 6. DRAFT \f13 time L+1 time L+1 time L+2 time L+2 time 2L+1 has has (from time L) (from time 1) has has (from time L) time 2L+1 interference alignment & neutralization (from time L) (from time 1) (from time 2L) (from time 2L) User 1 User 2 User 1 User 2 User User User User (from time L) has Fig. 7. Stage II: Time L + 1 aims at decoding (\u02dc aL,\u02dc bL). At time L + 1 + i, given (\u02dc aL+1\u2212i,\u02dc bL+1\u2212i) (decoded in time L + i), we decode (aL+1\u2212i, bL+1\u2212i) which in turn helping decoding (\u02dc aL\u2212i,\u02dc bL\u2212i). We iterate this from i = 1 to i = L. Stage II: We employ L+1 time slots. We perform re\ufb01nement w.r.t. the fresh symbols sent in stage I. The novel feature here is that the successive re\ufb01nement occurs in a retrospective manner: the fresh symbol sent at time i is re\ufb01ned at time 2L + 2 \u2212i in stage II where 1 \u2264i \u2264L. Here one key point to emphasize is that the re\ufb01ned symbol in stage II acts as side information, which in turn helps re\ufb01ning other past symbols in later time. In the example, the decoding order reads: (\u02dc aL,\u02dc bL) \u2192(aL, bL) \u2192\u00b7 \u00b7 \u00b7 \u2192(\u02dc a1,\u02dc b1) \u2192(a1, b1). (5) Time L+1: User 1 sends bL\u2295\u02dc aL\u22121\u2295\u02dc bL (received at time L) on bottom. It turns out this acts as ignition for re\ufb01ning all the corrupted symbols in the past. Similarly user 2 sends aL \u2295\u02dc bL\u22121 \u2295\u02dc aL on bottom. User \u02dc 1 can then obtain bL \u2295\u02dc bL which would be forwarded to user 2. User 2 can then decode \u02dc bL of interest. Similarly \u02dc aL is delivered to user 1. Time L + 2: The decoded symbols (\u02dc aL,\u02dc bL) turn out to play a key role to re\ufb01ne past forward transmission. Remember that bL sent by user 2 at time L in stage I was corrupted. User 2 retransmits the bL on top as in the perfect feedback case. But here the problem is that the situation is different from that in the perfect feedback case where bL was available at user 1 and helped nulling interference. Note that bL is not available here. Instead user 1 has an interfered version: bL \u2295\u02dc bL \u2295\u02dc aL\u22121. Nonetheless we can effectively do the same as in the perfect feedback case. DRAFT \f14 User 1 sends bL \u2295\u02dc bL \u2295\u02dc aL\u22121 on bottom. Clearly the neutralization is not perfect as it contains \u02dc bL. Here the idea is to exploit the \u02dc bL as side information to enable interference alignment and neutralization [18], [19], [16]. Note that user 2 can exploit the knowledge of \u02dc bL to construct the aligned interference bL \u2295\u02dc bL. Sending the bL \u2295\u02dc bL on top, user 2 can completely neutralize the interference as in the perfect feedback case. This enables user 1 to deliver A\u2032 1 interference-free on bottom. Similarly we can deliver (aL, B\u2032 1). On the other hand, exploiting aL (decoded right before) as side information, user \u02dc 1 can extract bL\u22121 \u2295\u02dc bL\u22121 \u2295BL from the one received at time L. Sending this then allows user 2 to decode \u02dc bL\u22121. Similarly \u02dc aL\u22121 can be decoded at user 1. Time L+3 \u223cTime 2L+1: We repeat the same as before. At time L+1+i where 2 \u2264i \u2264L, exploiting (\u02dc aL+1\u2212i,\u02dc bL+1\u2212i) decoded in time L + i, we decode (aL+1\u2212i, bL+1\u2212i), which in turn helps decoding (\u02dc aL\u2212i,\u02dc bL\u2212i). Now let us compute an achievable rate. In stage I, we sent (4L, 2L) fresh forward and backward symbols. In stage II, we sent only 2L fresh forward symbols. This yields the desired rate in the limit of L \u2192\u221e. Remark 3 (Exploiting Future Symbols as Side Information): Note in Fig. 6 the two types of tension: (1) forward-symbol feedback vs. backward symbols; (2) the other counterpart. As illustrated in Fig. 7, our scheme leads us to resolve both tensions. This then enables us to fully utilize the remaining resource level 2 \u02dc m \u2212\u02dc Cpf = 1 for sending the forward-symbol feedback of Cpf \u2212Cno = 1, thereby achieving Cpf. Similarly we can \ufb01ll up the resource holes 2n \u2212Cpf = 1 with the backward-symbol feedback of \u02dc Cpf \u2212\u02dc Cno = 1. This comes from the fact that our feedback scheme exploits the following as side information: (i) past received signals; (ii) users\u2019 own symbols; (iii) partially decoded symbols. While the \ufb01rst two were already shown to be bene\ufb01cial in the prior works [3], [5] (as well as in Example 1), the third type of information is the newly exploited one which turns out to yield the strong interaction gain. One can view this as future information. Recall the decoding order (5). When decoding (\u02dc aL\u22121,\u02dc bL\u22121), we exploited (aL, bL) (future symbols w.r.t. (\u02dc aL\u22121,\u02dc bL\u22121)) as side information. A conventional belief is that feedback allows us to know only about the past. In contrast, we discover a new viewpoint on the role of feedback. Feedback enables exploiting future information as well via retrospective decoding. \u25a1 DRAFT \f15 no feedback gain (no need to consider) Fig. 8. Regimes to check for achievability proof. By symmetry, it suf\ufb01ces to consider (R1), (R2), (R3), (R4), (R5). C. Proof Outline We categorize regimes depending on the values of channel parameters. Notice that C = Cno when (\u03b1 \u2208[ 2 3, 2], \u02dc \u03b1 \u2208[ 2 3, 2]). Also by symmetry, it suf\ufb01ces to consider only \ufb01ve regimes see Fig. 8: (R1) \u03b1 > 2, \u02dc \u03b1 > 2; (R2) \u03b1 \u2208(0, 2/3), \u02dc \u03b1 \u2208(0, 2/3); (R3) \u03b1 > 2, \u02dc \u03b1 \u2208[2/3, 2]; (R4) \u03b1 \u2208(0, 2/3), \u02dc \u03b1 \u2208[2/3, 2]; (R5) \u03b1 \u2208(0, 2/3), \u02dc \u03b1 > 2. As \ufb01gured out in Fig. 2, (R1) and (R2) are the ones in which there is no interaction gain. The proof builds only upon the perfect feedback scheme [3]. One thing to note here is that there are many subcases depending on whether or not available resources offered by a channel are enough to achieve the perfect feedback bound. Hence, a tedious yet careful analysis is required to cover all such subcases. On the other hand, (R3) and (R4) are the ones in which there is an interaction gain but only in one direction. So in this case, the nonfeedback scheme suf\ufb01ces for the backward DRAFT \f16 (Scheme 1) (Scheme 2) Fig. 9. Achievaility for (n, m) = (4, 2), (\u02dc n, \u02dc m) = (1, 3) via network decomposition. IC while a non-trivial scheme needs to be employed for the forward IC. It turns out Scheme 1 illustrated in Example 1 plays a key role in proving the claimed achievable region. (R5) is the one in which there is an interaction gain and sometimes one can get to perfect feedback capacities. We fully utilize the ideas presented in Scheme 1 and Scheme 2 to prove the claimed rate region. One key feature to emphasize is that the idea of network decomposition developed in [25] is utilized to provide a conceptually simpler proof for generalization. Here we illustrate the network decomposition idea via Example 3, while leaving a detailed proof in Appendix A. Example 3: (n, m) = (4, 2), (\u02dc n, \u02dc m) = (1, 3): Network decomposition relies on graph coloring. See Fig. 9. For the forward IC, we assign a color (say green) to level 1 and the levels connected to level 1. The green-colored graph then represents a subchannel, say (n(1), m(1)) = (2, 1), which has no overlap with the remaining uncolored subchannel (n(2), m(2)) = (2, 1). Following the notation in [25], we represent this by: (4, 2) \u2212 \u2192(2, 1) \u00d7 (2, 1). Similarly the backward channel can be decomposed as: (1, 3) \u2212 \u2192(1, 2) \u00d7 (0, 1). We then pair up one forward-subchannel (2, 1) and one backward-subchannel, say (1, 2), and apply Scheme 1 for the pair as in Fig. 5. This gives (R(1), \u02dc R(1)) = (3, 2). For the remaining pair of (2, 1) and (0, 1), we perform Scheme 2 DRAFT \f17 independently. This yields (R(2), \u02dc R(2)) = (3, 1). Combining these two achieves the desired rate region: {(R, \u02dc R) : R \u2264Cpf = 6, \u02dc R \u2264\u02dc Cpf = 3}. \u25a1 V. CONVERSE PROOF OF THEOREM 1 The \ufb01rst two (1) and (2) are the perfect-feedback bounds [3], [14], [5]. So the proof is immediate via a slight modi\ufb01cation. The third bound (3) is cutset: R1 + \u02dc R2 \u2264n + \u02dc n and R2 + \u02dc R1 \u2264n + \u02dc n. The last is a new bound. For completeness, we will provide detailed proof for the cutset and perfect feedback bounds in the subsequent section. We will then derive the new bound in Section V-B. A. Proof of the Cutset & Perfect Feedback Bound Proof of (3): Starting with Fano\u2019s inequality, we get N(R1 + \u02dc R2 \u2212\u01ebN) \u2264I(W1, \u02dc W2; Y N 1 , \u02dc W1, \u02dc Y N 2 , W2) (a) = X H(Y1i, \u02dc Y2i| \u02dc W1, W2, Y i\u22121 1 , \u02dc Y i\u22121 2 , X2i) (b) = X H(Y1i| \u02dc W1, W2, Y i\u22121 1 , \u02dc Y i\u22121 2 , X2i) + X H( \u02dc Y2i| \u02dc W1, W2, Y i 1, \u02dc Y i\u22121 2 , X2i, \u02dc X1i) (c) \u2264 X H(Y1i|X2i) + X H( \u02dc Y2i| \u02dc X1i) (d) \u2264N(n + \u02dc n) where (a) follows from the fact that (W1, \u02dc W2) is independent of (W2, \u02dc W1), and X2i is a function of (W2, \u02dc Y i\u22121 2 ); (b) follows from the fact that \u02dc X1i is a function of ( \u02dc W1, Y i\u22121 1 ); (c) follows from the fact that conditioning reduces entropy; (d) follows from the fact that the right-hand-side is maximized when (X1, X2, \u02dc X1, \u02dc X2) are uniformly distributed and independent. Similarly one can show N(R2 + \u02dc R1 \u2212\u01ebN) \u2264N(n + \u02dc n). If (R1, R2, \u02dc R1, \u02dc R2) is achievable, then \u01ebN \u21920 as N tends to in\ufb01nity. Therefore, we get the desired bound. DRAFT \f18 Proof of (1): Starting with Fano\u2019s inequality, we get N(R1 + R2 \u2212\u01ebN) (a) \u2264I(W1; Y N 1 | \u02dc W1, W2, \u02dc W2) + I(W2; Y N 2 | \u02dc W2, \u02dc W1) = H(Y N 1 | \u02dc W1, W2, \u02dc W2) + H(Y N 2 | \u02dc W2, \u02dc W1) \u2212 n H(Y N 1 , Y N 2 | \u02dc W1, W2, \u02dc W2) \u2212H(Y N 1 | \u02dc W1, \u02dc W2, W2, Y N 2 ) o = H(Y N 1 | \u02dc W1, \u02dc W2, W2, Y N 2 ) \u2212H(Y N 2 | \u02dc W1, W2, \u02dc W2, Y N 1 ) + H(Y N 2 | \u02dc W2, \u02dc W1) \u2264H(Y N 1 | \u02dc W1, \u02dc W2, W2, Y N 2 ) + H(Y N 2 | \u02dc W2, \u02dc W1) (b) = X H(Y1i| \u02dc W1, \u02dc W2, W2, Y N 2 , Y i\u22121 1 , \u02dc Xi 1, \u02dc X2i, \u02dc Y i 2, X2i, V1i) + H(Y N 2 | \u02dc W2, \u02dc W1) (c) \u2264 X H(Y1i|V1i, X2i) + X H(Y2i) \u2264N \b (n \u2212m)+ + max(n, m) \t = N max(2n \u2212m, m) where (a) follows from the independence of (W1, W2, \u02dc W1, \u02dc W2); (b) follows from the fact that \u02dc Xi 1 is a function of ( \u02dc W1, Y i\u22121 1 ), X2i is a function of (W2, \u02dc Y i\u22121 2 ), and V1i is a function of (X2i, Y2i); (c) follows from the fact that conditioning reduces entropy. This completes the proof. B. Proof of a Novel Outer Bound The proof hinges upon several lemmas stated below. The proof is streamlined with the help of a key notion, called triple mutual information (or interaction information [13]), which is de\ufb01ned as I(X; Y ; Z) := I(X; Y ) \u2212I(X; Y |Z). (6) It turns out that the commutative property of the notion plays a crucial role in deriving several key steps in the proof: I(X; Y ; Z) = I(X; Z; Y ) = \u00b7 \u00b7 \u00b7 = I(Z; Y ; X). (7) DRAFT \f19 Using this notion and starting with Fano\u2019s inequality, we get N(R1 + R2 \u2212\u01ebN) \u2264I(W1; Y N 1 , \u02dc W1) + I(W2; Y N 2 , \u02dc W2) \u2264I(W1; Y N 1 , V N 1 | \u02dc W1) + I(W2; Y N 2 , V N 2 | \u02dc W2) = X n I(W1; Y1i, V1i| \u02dc W1, Y i\u22121 1 , V i\u22121 1 ) + I(W2; Y2i, V2i| \u02dc W2, Y i\u22121 2 , V i\u22121 2 ) o = X n I(V1i; W1| \u02dc W1, Y i\u22121 1 , V i\u22121 1 ) + I(Y1i; W1| \u02dc W1, Y i\u22121 1 , V i 1) +I(V2i; W2| \u02dc W2, Y i\u22121 2 , V i\u22121 2 ) + I(Y2i; W2| \u02dc W2, Y i\u22121 2 , V i 2) o (a) = X n I(Y1i; W1, W2, \u02dc W2| \u02dc W1, Y i\u22121 1 , V i 1) + I(Y2i; W2, W1, \u02dc W1| \u02dc W2, Y i\u22121 2 , V i 2) +I(V1i; W1| \u02dc W1, Y i\u22121 1 , V i\u22121 1 ) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, Y i\u22121 1 , V i 1) +I(V2i; W2| \u02dc W2, Y i\u22121 2 , V i\u22121 2 ) \u2212I(Y2i; W1, \u02dc W1|W2, \u02dc W2, Y i\u22121 2 , V i 2) o \u2264 X {H(Y1i|V1i) + H(Y2i|V2i) +I(V1i; W1| \u02dc W1, Y i\u22121 1 , V i\u22121 1 ) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, Y i\u22121 1 , V i 1) +I(V2i; W2| \u02dc W2, Y i\u22121 2 , V i\u22121 2 ) \u2212I(Y2i; W1, \u02dc W1|W2, \u02dc W2, Y i\u22121 2 , V i 2) o where (a) follows from a chain rule. By symmetry, we get: N( \u02dc R1 + \u02dc R2 \u2212\u01ebN) \u2264 X n H( \u02dc Y1i| \u02dc V1i) + H( \u02dc Y2i| \u02dc V2i) +I( \u02dc V1i; \u02dc W1|W1, \u02dc Y i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc Y1i; W2, \u02dc W2|W1, \u02dc W1, \u02dc Y i\u22121 1 , \u02dc V i 1) +I( \u02dc V2i; \u02dc W2|W2, \u02dc Y i\u22121 2 , \u02dc V i\u22121 2 ) \u2212I( \u02dc Y2i; W1, \u02dc W1|W2, \u02dc W2, \u02dc Y i\u22121 2 , \u02dc V i 2) o . Now adding the above two and using Lemma 1 stated below, we get: N(R1 + R2 + \u02dc R1 + \u02dc R2 \u2212\u01ebN) \u2264 X n H(Y1i|V1i) + H(Y2i|V2i) + H( \u02dc Y1i| \u02dc V1i) + H( \u02dc Y2i| \u02dc V2i) o \u22642N max(n \u2212m, m) + 2N max(\u02dc n \u2212\u02dc m, \u02dc m). Hence, we get the desired bound. DRAFT \f20 Lemma 1: X n I(V1i; W1| \u02dc W1, Y i\u22121 1 , V i\u22121 1 ) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, Y i\u22121 1 , V i 1) +I(V2i; W2| \u02dc W2, Y i\u22121 2 , V i\u22121 2 ) \u2212I(Y2i; W1, \u02dc W1|W2, \u02dc W2, Y i\u22121 2 , V i 2) +I( \u02dc V1i; \u02dc W1|W1, \u02dc Y i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc Y1i; \u02dc W2, W2| \u02dc W1, W1, \u02dc Y i\u22121 1 , \u02dc V i 1) +I( \u02dc V2i; \u02dc W2|W2, \u02dc Y i\u22121 2 , \u02dc V i\u22121 2 ) \u2212I( \u02dc Y2i; \u02dc W1, W1| \u02dc W2, W2, \u02dc Y i\u22121 2 , \u02dc V i 2) o \u22640 C. Proof of Lemma 1 First consider: (1st and 2nd terms in summation of LHS) (a) = X n I(V1i; W1| \u02dc W1, Y i\u22121 1 , V i\u22121 1 ) \u2212I(Y1i; W2, \u02dc W2, \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1) (b) = X n I(V1i, \u02dc V1i; W1| \u02dc W1, Y i\u22121 1 , V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1, \u02dc V i 1) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, \u02dc Y i 1, Y i\u22121 1 ) o (c) = X n I(V1i, \u02dc V1i; W1| \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(V1i, \u02dc V1i; W1; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1, \u02dc V i 1) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, \u02dc Y i 1, Y i\u22121 1 ) o where (a) follows from the fact that \u02dc Y i 1 is a function of (W1, \u02dc W1, W2, \u02dc W2); (b) follows from the fact that \u02dc V i 1 is a function of ( \u02dc W1, Y i\u22121 1 ); and (c) is due to the de\ufb01nition of triple mutual information (6). Using Lemma 2 stated at the end of this section, we get: (1st and 2nd terms in summation of LHS) \u2264 X n I(V1i, \u02dc V1i; W1| \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) + I( \u02dc Y1i; Y i\u22121 1 |W1, \u02dc W1, \u02dc Y i\u22121 1 , \u02dc V i 1) \u2212I( \u02dc V1i; W1, \u02dc Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, \u02dc Y i 1, Y i\u22121 1 ) o . DRAFT \f21 Now combining this with the 5th and 6th terms in summation of LHS gives: (1st, 2nd, 5th and 6th terms of LHS in the claimed bound) (a) \u2264 X n I(V1i, \u02dc V1i; W1| \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) + I( \u02dc Y1i; Y i\u22121 1 |W1, \u02dc W1, \u02dc Y i\u22121 1 , \u02dc V i 1) \u2212I( \u02dc V1i; W1, \u02dc Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, \u02dc Y i 1, Y i\u22121 1 ) o + X n I( \u02dc V1i; \u02dc W1|W1, \u02dc Y i\u22121 1 , V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc Y1i; \u02dc W2, W2, Y i\u22121 1 | \u02dc W1, W1, \u02dc Y i\u22121 1 , \u02dc V i 1) o (b) \u2264 X n I(V1i, \u02dc V1i; W1| \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc V1i; W1, \u02dc Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; W2, \u02dc W2|W1, \u02dc W1, \u02dc Y i 1, Y i\u22121 1 ) o + X n I( \u02dc V1i; \u02dc W1, W1, \u02dc Y i\u22121 1 |V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc Y1i; \u02dc W2, W2| \u02dc W1, W1, \u02dc Y i\u22121 1 , Y i\u22121 1 ) o (c) \u2264 X n I(V1i, \u02dc V1i; W1| \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i, \u02dc Y1i; W2, \u02dc W2|W1, \u02dc W1, \u02dc Y i\u22121 1 , Y i\u22121 1 ) + I(V1i, \u02dc V1i; \u02dc W1|V i\u22121 1 , \u02dc V i\u22121 1 ) o (d) = I(V N 1 , \u02dc V N 1 ; W1, \u02dc W1) \u2212I(Y N 1 , \u02dc Y N 1 , V N 2 , \u02dc V N 2 ; W2, \u02dc W2|W1, \u02dc W1) \u2264I(V N 1 , \u02dc V N 1 ; W1, \u02dc W1) \u2212I(V N 2 , \u02dc V N 2 ; W2, \u02dc W2|W1, \u02dc W1) where (a) follows from the fact that V i\u22121 1 and Y i\u22121 1 are functions of (W1, \u02dc Y i\u22121 1 ) and (W1, W2, \u02dc W1, \u02dc W2), respectively; (b) follows from a chain rule (applied on the last term) and the non-negativity of mutual information; (c) follows from a chain rule (combining the 2nd and 4th terms; also combining the 3rd and 5th terms) and the non-negativity of mutual information; (d) follows from a chain rule (combining the 1st and 3rd terms) and the fact that (V N 2 , \u02dc V N 2 ) is a function of (W1, \u02dc W1, Y N 1 , \u02dc Y N 1 ). Applying the same to the 3rd, 4th, 7th and 8th terms in summation of LHS, we get: (LHS in the claimed bound) \u2264I(V N 1 , \u02dc V N 1 ; W1, \u02dc W1) \u2212I(V N 2 , \u02dc V N 2 ; W2, \u02dc W2|W1, \u02dc W1) + I(V N 2 , \u02dc V N 2 ; W2, \u02dc W2) \u2212I(V N 1 , \u02dc V N 1 ; W1, \u02dc W1|W2, \u02dc W2) \u2264I(W2, \u02dc W2, V N 1 , \u02dc V N 1 ; W1, \u02dc W1) \u2212I(V N 2 , \u02dc V N 2 ; W2, \u02dc W2|W1, \u02dc W1) + I(W1, \u02dc W1, V N 2 , \u02dc V N 2 ; W2, \u02dc W2) \u2212I(V N 1 , \u02dc V N 1 ; W1, \u02dc W1|W2, \u02dc W2) = 0. This completes the proof. DRAFT \f22 Lemma 2: \u2212 X n I(V1i, \u02dc V1i; W1; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) + I(Y1i; \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1, \u02dc V i 1) o \u2264 X n I( \u02dc Y1i; Y i\u22121 1 |W1, \u02dc W1, \u02dc Y i\u22121 1 , \u02dc V i 1) \u2212I( \u02dc V1i; W1, \u02dc Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) o . Proof: See Section V-D. D. Proof of Lemma 2 \u2212 X n I(V1i, \u02dc V1i; W1; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) + I(Y1i; \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1, \u02dc V i 1) o (a) = X n I(V1i, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(V1i, \u02dc V1i; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1, \u02dc V i 1) o (b) \u2264 X n I(V1i, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc V1i; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1, \u02dc V i 1) o (c) = X n I( \u02dc Y i 1, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc Y i 1; Y i\u22121 1 |W1, \u02dc W1, V i 1, \u02dc V i 1) \u2212I( \u02dc V1i; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I(Y1i; \u02dc Y i 1|W1, \u02dc W1, Y i\u22121 1 , V i 1, \u02dc V i 1) o (d) = X n I( \u02dc Y i 1, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc Y i 1; Y i 1|W1, \u02dc W1, V i 1, \u02dc V i 1) \u2212I( \u02dc V1i; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) o (e) = X n I( \u02dc Y1i, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, \u02dc Y i\u22121 1 , V i\u22121 1 , \u02dc V i\u22121 1 ) + I( \u02dc Y i\u22121 1 ; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc Y i 1; Y i 1|W1, \u02dc W1, V i 1, \u02dc V i 1) \u2212I( \u02dc V1i; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) o (f) \u2264 X n I( \u02dc Y1i, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, \u02dc Y i\u22121 1 , V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212I( \u02dc V1i; Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) o (g) = X n I( \u02dc Y1i; Y i\u22121 1 |W1, \u02dc W1, \u02dc Y i\u22121 1 , V i\u22121 1 , \u02dc V i 1) \u2212I( \u02dc V1i; Y i\u22121 1 ; W1, \u02dc Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) o (h) = X n I( \u02dc Y1i; Y i\u22121 1 |W1, \u02dc W1, \u02dc Y i\u22121 1 , \u02dc V i 1) \u2212I( \u02dc V1i; W1, \u02dc Y i\u22121 1 | \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) o where (a) follows from the de\ufb01nition of triple mutual information (6); (b) follows from the non-negativity of mutual information; (c) follows from a chain rule and the fact that V1i is a DRAFT \f23 \u2026 \u2026 User 1 User 2 User 1 User 2 User 1 User 2 User 1 User 2 User 1 User 2 User 1 User 2 User 1 User 2 User \u02dc 1 User \u02dc 2 User 1 User 2 User \u02dc 1 User \u02dc 2 User \u02dc 1 User \u02dc 2 User \u02dc 1 User \u02dc 2 User \u02dc 1 User \u02dc 2 User \u02dc 1 User \u02dc 2 User \u02dc 1 User \u02dc 2 User \u02dc 1 User \u02dc 2 Fig. 10. Two-way parallel ICs. The rich diversity on channel gains across many parallel subchannels can often occur in broadband systems. function of (W1, \u02dc Y i 1) (see below) X I(V1i, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) = X I(V1i, \u02dc Y i 1, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212 X I( \u02dc Y i 1; Y i\u22121 1 |W1, \u02dc W1, V i 1, \u02dc V i 1) = X I( \u02dc Y i 1, \u02dc V1i; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) \u2212 X I( \u02dc Y i 1; Y i\u22121 1 |W1, \u02dc W1, V i 1, \u02dc V i 1); (d) follows from a chain rule (combining the 2nd and 4th terms); (e) follow from a chain rule (applying to the 1st term); (f) follows from N X i=1 I( \u02dc Y i\u22121 1 ; Y i\u22121 1 |W1, \u02dc W1, V i\u22121 1 , \u02dc V i\u22121 1 ) = N\u22121 X i=0 I( \u02dc Y i 1; Y i 1|W1, \u02dc W1, V i 1, \u02dc V i 1) \u2264 N X i=1 I( \u02dc Y i 1; Y i 1|W1, \u02dc W1, V i 1, \u02dc V i 1); (g) follows from a chain rule and the de\ufb01nition of triple mutual information; (h) follows from the fact that V i\u22121 1 and \u02dc V1i are a function of (W1, \u02dc Y i\u22122 1 ) and ( \u02dc W1, Y i\u22121 1 ), respectively. VI. DISCUSSION A. System Implication As suggested in Fig. 2, an interaction gain occurs when forward and backward ICs are somewhat different. This asymmetry occurs naturally in FDD systems where the forward and DRAFT \f24 backward channels are on completely different bands. Even in TDD systems, the asymmetry can occur since the forward and backward channels can be on different subcarriers or different coherent time. Also one can create this asymmetry by opportunistically pairing subbands for the forward and backward transmissions. While this asymmetry is not likely to occur in narrowband systems, it can often occur in broadband systems where there are a multitude of subchannels with a wide dynamic range of channel gains. For example, in 4G-LTE and 5G systems, one can easily expect rich diversity on channel gains, since an operating bandwidth of the systems is much larger than coherence bandwidth of typical wireless channels (around the order of 0.1 MHz). Fig. 10 illustrates an example which can represent such scenario where there are a variety of parallel subchannels. Our results suggest that pairs of (FW(1), BACK(2)) and (FW(2), BACK(1)), for instance, can provide a signi\ufb01cant gain with interaction. Another interesting observation is that even though forward-and-backward parallel ICs are identical, there exist many pairs of forwardbackward subchannels that can yield capacity improvements. In tomorrow\u2019s communication systems, a broader system bandwidth is expected to support a variety of multimedia services. Hence, it is believed that our results will provide detailed guidelines as to how to design future communication systems. In this paper, we investigate the bene\ufb01t of interaction for a full duplex system. This is only for illustrative purpose. As suggested in Remarks 2 and 3, the nature of the interaction gain comes from exploiting the past received signals, partially decoded symbols and users\u2019 own information as side information. This nature is not limited to the full duplex system. So one can readily see that the interaction gain carries over to the half duplex system. While the detailed capacity region of the half-duplex system is distinct, the channel regimes in which feedback offers a gain remain unchanged. In other words, we have the same picture as in Fig. 2. B. Translation to the Gaussian Channel The deterministic-channel achievability proposed in this work gives insights into an achievable scheme in the Gaussian channel. This is inspired by several observations that can be made from Scheme 1 (see Example 1 in Fig. 5) and Scheme 2 (see Example 2 in Figs. 6 and 7). (Extracting feedback signals \u2192quantize-&-binning): Note in Fig. 5 that the fedback signal a\u2295B at user \u02dc 1 can be interpreted as a quantized version of the received signal (A, a\u2295B) at the DRAFT \f25 level below the clean signal A. This motivates the use of quantize-and-binning [6], [26] in the Gaussian channel. There are two points to make. The \ufb01rst is that the binning scheme [27] can be employed solely without quantization in this example. Binning the received signal (A, a\u2295B) may construct a linear combination of the two components: a \u2295B \u2295A. The distinction is then user \u02dc 1 feeds back \u02dc A \u2295(a \u2295B \u2295A) instead. Nonetheless, user 1 can still get \u02dc A of interest, as A is also known. On the other hand, user 2 obtains a\u2295A\u2295\u02dc A instead of a\u2295\u02dc A. This is not an issue either. We can still achieve interference alignment and neutralization in forward transmission at stage II. User 1 sending a \u2295A \u2295\u02dc A on top (with the help of the decoded symbol \u02dc A) and user 2 sending B\u2032 \u2295(a \u2295A \u2295\u02dc A), user \u02dc 2 can still obtain B\u2032 interference-free. Also user \u02dc 1 can get a with the help of A which has already been received in stage I. The second point to note is that the binning-only approach might not work properly for other channel parameter regimes. This is because mixing all the equations may include some undesirable symbols that prevent the optimal transmission. In that case, both quantization and binning are desired to be employed with a careful choice of a quantization level, set to exclude undesirable symbols. (XORing with interference neutralization \u2192superposition with dirty paper coding): Observe in Fig. 5 that the fedback signal a \u2295B is XORed with a backward symbol \u02dc A. This motivates the use of superposition coding in the Gaussian channel. On the other hand, user \u02dc 2 sends B on bottom for interference neutralization. To this end, we employ quantization scheme for extracting B from the received signal (B, b \u2295A) and utilize dirty paper coding [28] for nulling. (Interference alignment and neutralization \u2192structured coding): Note at the second stage in Fig. 5 that user 2 computes the XOR of b (its own symbol) and \u02dc B (decoded from the received signal in backward transmission) and then sends the XOR on a proper level (the top level) for nulling. This motivates the use of structured coding [29], as computation needs to be made across appropriate symbols and the computed signal should be placed in a structured location for nulling. (Retrospective decoding): To be the best of our knowledge, this is a novel feature that has never been introduced in network information theory literature. Hence, it requires a new achievability technique which includes a careful decoding order as well as sets proper symbols to decode for each time slot. Also note that decoded symbols in an intermediate time slot are part of the entire symbols. See Fig. 7 for instance. Here aL (a decoded symbol in time L + 2) is part of the entire symbols (AL, aL) sent in time L. Hence, this scheme needs to be properly combined DRAFT \f26 with Han-Kobayashi message splitting [30]. C. Uni\ufb01ed Achievability The noisy network coding [26] together with Han-Kobayashi message splitting is a fairly generic scheme that yields reasonably good performances for a variety of multi-user channels. It implements many achievablility techniques such as quantize-and-binning and superposition coding. However, it has a room for improvement as it does not incorporate dirty paper coding and structured coding. An effort has been made by Nazer-Gastpar [29] for implementing structured codes. But this approach still has a room for improvement, as it does not allow for the key operation that appears in our achievability: retrospective decoding. As suggested in Example 2, the key operation seems required for achieving the optimal performance. There seems no way to achieve the perfect feedback bound without an intermediate decoding of partial symbols which admits a carefully-designed backward ordering. One future work of interest is to develop a generic achievable scheme that can be applied to general discrete memoryless networks as well as uni\ufb01es all of the techniques mentioned earlier: (1) quantize-and-binning; (2) superposition coding (or block Markov coding); (3) structured coding; (4) Han-Kobayashi message-splitting; (5) retrospective decoding. This development is expected to open the door to characterizing and/or approximating many of interesting interference networks. D. Uni\ufb01ed Converse In this work, we develop a new converse technique which well captures the tension between feedback and independent message transmissions. Hence, unlike the prior upper bounds such as cutset [27], genie-aided bounds [8], [2], [3], [9], [10], [11], [14], generalized network sharing bounds [12], it gives rise to the tight capacity characterization of interactive multi-user channels like the two-way IC. Encouragingly, our novel bound (4) subsumes the following bounds as special cases: the nonfeedback-case counterpart R1 + R2 \u2264H(Y1|V1) + H(Y2|V2) [8]; the ratelimited-feedback-case counterpart R1 +R2 \u2264H(Y1|V1)+H(Y2|V2)+Cbitpipe FB1 +Cbitpipe FB2 [4]. Here Cbitpipe FBi denotes the capacity of the bit-piped feedback link that connects user \u02dc i to user i. One future work of interest is to extend this bound to arbitrary discrete memoryless networks in which many nodes interact with each other. DRAFT \f27 E. Role of Interaction in General Networks This work focuses on an interference channel setting in which each user wishes to deliver its own message to its counterpart. As mentioned earlier, the nature of interaction gain is not limited to this particular setting. So it would be interesting to explore the role of interaction for a variety of different settings. While initial efforts along this research direction have been made for a multicast channel setting [31], function computation settings [32], [33], and multi-hop network settings [34], an explicit comparison between non-interactive vs interactive scenarios was not made yet. One research direction of interest is to investigate the capacity regions of such channels, thereby discovering two-way scenarios in which one can achieve a huge interaction gain. VII." + }, + { + "url": "http://arxiv.org/abs/1602.04567v1", + "title": "Adversarial Top-$K$ Ranking", + "abstract": "We study the top-$K$ ranking problem where the goal is to recover the set of\ntop-$K$ ranked items out of a large collection of items based on partially\nrevealed preferences. We consider an adversarial crowdsourced setting where\nthere are two population sets, and pairwise comparison samples drawn from one\nof the populations follow the standard Bradley-Terry-Luce model (i.e., the\nchance of item $i$ beating item $j$ is proportional to the relative score of\nitem $i$ to item $j$), while in the other population, the corresponding chance\nis inversely proportional to the relative score. When the relative size of the\ntwo populations is known, we characterize the minimax limit on the sample size\nrequired (up to a constant) for reliably identifying the top-$K$ items, and\ndemonstrate how it scales with the relative size. Moreover, by leveraging a\ntensor decomposition method for disambiguating mixture distributions, we extend\nour result to the more realistic scenario in which the relative population size\nis unknown, thus establishing an upper bound on the fundamental limit of the\nsample size for recovering the top-$K$ set.", + "authors": "Changho Suh, Vincent Y. F. Tan, Renbo Zhao", + "published": "2016-02-15", + "updated": "2016-02-15", + "primary_cat": "cs.IR", + "cats": [ + "cs.IR", + "cs.IT", + "cs.LG", + "math.IT", + "stat.ML" + ], + "main_content": "INTRODUCTION Ranking is one of the fundamental problems that has proved crucial in a wide variety of contexts\u2014social choice [1], [2], web search and information retrieval [3], recommendation systems [4], ranking individuals by group comparisons [5] and crowdsourcing [6], to name a few. Due to its wide applicability, a large volume of work on ranking has been done. The two main paradigms in the literature include spectral ranking algorithms [3], [7], [8] and maximum likelihood estimation (MLE) [9]. While these ranking schemes yield reasonably good estimates which are faithful globally w.r.t. the latent preferences (i.e., low \u21132 loss), it is not necessarily guaranteed that this results in optimal ranking accuracy. Accurate ranking has more to do with how well the ordering of the estimates matches that of the true preferences (a discrete/combinatorial optimization problem), and less to do with how well we can estimate the true preferences (a continuous optimization problem). In applications, a ranking algorithm that outputs a total ordering of all the items is not only overkill, but it also unnecessarily increases complexity. Often, we pay attention to only a few signi\ufb01cant items. Thus, recent work such as that by Chen and Suh [10] studied the top-K identi\ufb01cation task. Here, one aims to recover a correct set of top-ranked items only. This work characterized the minimax limit on the sample size required (i.e., the sample complexity) for reliable top-K ranking, assuming the Bradley-Terry-Luce (BTL) model [11], [12]. While this result is concerned with practical issues, there are still limitations when modeling other realistic scenarios. The BTL model considered in [10] assumes that the quality of pairwise comparison information which forms the basis of the model is the same across annotators. In reality (e.g., crowdsourced settings), however, the quality of the information can vary signi\ufb01cantly across different annotators. For instance, there may be a nonnegligible fraction of spammers who provide answers in an adversarial manner. In the context of adversarial web search [13], web contents can be maliciously manipulated by spammers for commercial, social, or political bene\ufb01ts in a robust manner. Alternatively, there may exist false information such as false voting in social networks and fake ratings in recommendation systems [14]. As an initial effort to address this challenge, we investigate a so-called adversarial BTL model, which postulates the existence of two sets of populations\u2014the faithful and adversarial populations, each of which has proportion C. Suh is with the School of Electrical Engineering at Korea Advanced Institute of Science and Technology (email: chsuh@kaist.ac.kr). V. Y. F. Tan is with the Department of Electrical and Computer Engineering and the Department of Mathematics, National University of Singapore. (email: vtan@nus.edu.sg). R. Zhao is with the Department of Electrical and Computer Engineering, National University of Singapore. (email: elezren@nus.edu.sg). C. Suh is supported by a gift from Samsung. V. Y. F. Tan and R. Zhao gratefully acknowledge \ufb01nancial support from the National University of Singapore (NUS) under the NUS Young Investigator Award R-263-000-B37-133. \f2 \u03b7 and 1 \u2212\u03b7 respectively. Speci\ufb01cally we consider a BTL-based pairwise comparison model in which there exist latent variables indicating ground-truth preference scores of items. In this model, it is assumed that comparison samples drawn from the faithful population follow the standard BTL model (the probability of item i beating item j is proportional to item i\u2019s relative score to item j), and those of the adversarial population act in an \u201copposite\u201d manner, i.e., the probability of i beating j is inversely proportional to the relative score. See Fig. 1. A. Main contributions We seek to characterize the fundamental limits on the sample size required for top-K ranking, and to develop computationally ef\ufb01cient ranking algorithms. There are two main contributions in this paper. Building upon RankCentrality [7] and SpectralMLE [10], we develop a ranking algorithm to characterize the minimax limit required for top-K ranking, up to constant factors, for the \u03b7-known scenario. We also show the minimax optimality of our ranking scheme by proving a converse or impossibility result that applies to any ranking algorithm using information-theoretic methods. As a result, we \ufb01nd that the sample complexity is inversely proportional to (2\u03b7 \u22121)2, which suggests that less distinct the population sizes, the larger the sample complexity. We also demonstrate that our result recovers that of the \u03b7 = 1 case in [10], so the work contained herein is a strict generalization of that in [10]. The second contribution is to establish an upper bound on the sample complexity for the more practically-relevant scenario where \u03b7 is unknown. A novel procedure based on tensor decomposition approaches in Jain-Oh [15] and Anandkumar et al. [16] is proposed to \ufb01rst obtain an estimate of the parameter \u03b7 that is in a neighborhood of \u03b7, i.e., we seek to obtain an \u03b5-globally optimal solution. This is usually not guaranteed by traditional iterative methods such as Expectation Maximization [17]. Subsequently, the estimate is then used in the ranking algorithm that assumes knowledge of \u03b7. We demonstrate that this algorithm leads to an order-wise worse sample complexity relative to the \u03b7-known case. Our theoretical analyses suggest that the degradation is unavoidable if we employ this natural two-step procedure. B. Related work The most relevant related works are those by Chen and Suh [10], Negahban et al. [7], and Chen et al. [6]. Chen and Suh [10] focused on top-K identi\ufb01cation under the standard BTL model, and derived an \u2113\u221eerror bound on preference scores which is intimately related to top-K ranking accuracy. Negahban et al. [7] considered the same comparison model and derived an \u21132 error bound. A key distinction in our work is that we consider a different measurement model in which there are two population sets, although the \u2113\u221eand \u21132 norm error analyses in [7], [10] play crucial roles in determining the sample complexity. The statistical model introduced by Chen et al. [6] attempts to represent crowdsourced settings and forms the basis of our adversarial comparison model. We note that no theoretical analysis of the sample complexity is available in [6] or other related works on crowdsourced rankings [18]\u2013[20]. For example, Kim et al. [20] employed variational EM-based algorithms to estimate the latent scores; global optimality guarantees for such algorithms are dif\ufb01cult to establish. Jain and Oh [15] developed a tensor decomposition method [16] for learning the parameters of a mixture model [21]\u2013[23] that includes our model as a special case. We specialize their model and relevant results to our setting for determining the accuracy of the estimated \u03b7. This allows us to establish an upper bound on the sample complexity when \u03b7 is unknown. Recently, Shah and Wainwright [24] showed that a simple counting method [25] achieves order-wise optimal sample complexity for top-K ranking under a general comparison model which includes, as special cases, a variety of parametric ranking models including the one under consideration in this paper (the BTL model). However, the authors made assumptions on the statistics of the pairwise comparisons which are different from that in our model. Hence, their result is not directly applicable to our setting. C. Notations We provide a brief summary of the notations used throughout the paper. Let [n] represent {1, 2, \u00b7 \u00b7 \u00b7 , n}. We denote by \u2225w\u2225, \u2225w\u22251, \u2225w\u2225\u221ethe \u21132 norm, \u21131 norm, and \u2113\u221enorm of w, respectively. Additionally, for any two sequences f(n) and g(n), f(n) \u2273g(n) or f(n) = \u2126(g(n)) mean that there exists a (universal) constant c such \f3 FAITHFUL POPULATION ADVERSARIAL POPULATION A B \u03b7 1 \u2212\u03b7 score vector w Y := \u0001 Y (\u2113) ij \u0002 (i, j) \u2208E pairwise sample 1 2 3 4 top-K ranking \u02c6 SK Y (\u2113) ij \u223cBern \u0003 wi wi + wj \u0004 \u2113\u2208A w.p. \u03b7 Y (\u2113) ij \u223cBern \u0003 wj wi + wj \u0004 \u2113\u2208B w.p. 1 \u2212\u03b7 \u03c8(\u00b7) Fig. 1. Adversarial top-K ranking given samples Y = {Y (\u2113) ij } where (i, j) \u2208E and E is the edge set of an Erd\u02dd os-R\u00b4 enyi random graph. that f(n) \u2265cg(n); f(n) \u2272g(n) or f(n) = O(g(n)) mean that there exists a constant c such that f(n) \u2264cg(n); and f(n) \u224dg(n) or f(n) = \u0398(g(n)) mean that there exist constants c1 and c2 such that c1g(n) \u2264f(n) \u2264c2g(n). The notation poly(n) denotes a sequence in O(nc) for some c > 0. II. PROBLEM SETUP We now describe the model which we will analyze subsequently. We assume that the observations used to learn the rankings are in the form of a limited number of pairwise comparisons over n items. In an attempt to re\ufb02ect the adversarial crowdsourced setting of our interest in which there are two population sets\u2014the faithful and adversarial sets\u2014we adopt a comparison model introduced by Chen et al. [6]. This is a generalization of the BTL model [11], [12]. We delve into the details of the components of the model. Preference scores: As in the standard BTL model, this model postulates the existence of a ground-truth preference score vector w = (w1, w2, . . . , wn) \u2208Rn +. Each wi represents the underlying preference score of item i. Without loss of generality, we assume that the scores are in non-increasing order: w1 \u2265w2 \u2265. . . \u2265wn > 0. (1) It is assumed that the dynamic range of the score vector is \ufb01xed irrespective of n: wi \u2208[wmin, wmax], \u2200i \u2208[n], (2) for some positive constants wmin and wmax. In fact, the case in which the ratio wmax wmin grows with n can be readily translated into the above setting by \ufb01rst separating out those items with vanishing scores (e.g., via a simple voting method like Borda count [25], [26]). Comparison graph: Let G := ([n], E) be the comparison graph such that items i and j are compared by an annotator if the node pair (i, j) belongs to the edge set E. We will assume throughout that the edge set E is drawn in accordance to the Erd\u02dd os-R\u00b4 enyi (ER) model G \u223cGn,p. That is node pair (i, j) appears independently of any other node pair with an observation probability p \u2208(0, 1). Pairwise comparisons: For each edge (i, j) \u2208E, we observe L comparisons between i and j. Each outcome, indexed by \u2113\u2208[L] and denoted by Y (\u2113) ij , is drawn from a mixture of Bernoulli distributions weighted by an unknown parameter \u03b7 \u2208(1/2, 1]. The \u2113-th observation of edge (i, j) has distribution Bern( wi wi+wj ) with probability \u03b7 and distribution Bern( wj wi+wj ) with probability 1 \u2212\u03b7. Hence, Y (\u2113) ij \u223cBern \u0012 \u03b7 wi wi + wj + (1 \u2212\u03b7) wj wi + wj \u0013 . (3) See Fig. 1. When \u03b7 = 1/2, all the observations are fair coin tosses. In this case, no information can be gleaned about the rankings. Thus we exclude this degenerate setting from our study. The case of \u03b7 \u2208[0, 1/2) is equivalent \f4 to the \u201cmirrored\u201d case of 1 \u2212\u03b7 \u2208(1/2, 1] where we \ufb02ip 0\u2019s to 1\u2019s and 1\u2019s to 0\u2019s. So without loss of generality, we assume that \u03b7 \u2208(1/2, 1]. We allow \u03b7 to depend on n. Conditioned on the graph G, the Y (\u2113) ij \u2019s are independent and identically distributed across all \u2113\u2019s, each according to the distribution of (3). The collection of suf\ufb01cient statistics is Yij := 1 L L X \u2113=1 Y (\u2113) ij , \u2200(i, j) \u2208E. (4) The per-edge number of samples L is measure of the quality of the measurements. We let Yi := {Yij}j:(i,j)\u2208E, \u20d7 Yij := {Y (\u2113) ij : \u2113\u2208[L]} and Y := {Yij}(i,j)\u2208E be various statistics of the available data. Performance metric: We are interested in recovering the top-K ranked items in the collection of n items from the data Y . We denote the true set of top-K ranked items by SK which, by our ordering assumption, is the set [K]. We would like to design a ranking scheme \u03c8 : {0, 1}|E|\u00d7L \u2192 \u0000[n] K \u0001 that maps from the available measurements to a set of K indices. Given a ranking scheme \u03c8, the performance metric we consider is the probability of error Pe(\u03c8) := Pr [\u03c8(Y ) \u0338= SK] . (5) We consider the fundamental admissible region Rw of (p, L) pairs in which top-K ranking is feasible for a given w, i.e., Pe(\u03c8) can be arbitrarily small for large enough n. In particular, we are interested in the sample complexity S\u03b4 := inf p\u2208[0,1],L\u2208Z+ sup a\u2208\u2126\u03b4 \u001a\u0012n 2 \u0013 pL : (p, L) \u2208Ra \u001b , (6) where \u2126\u03b4 := {a \u2208Rn : (aK \u2212aK+1)/amax \u2265\u03b4}. Here we consider a minimax scenario in which, given a score estimator, nature can behave in an adversarial manner, and so she chooses the worst preference score vector that maximizes the probability of error under the constraint that the normalized score separation between the K-th and (K + 1)-th items is at least \u03b4. Note that \u0000n 2 \u0001 p is the expected number of edges of the ER graph so \u0000n 2 \u0001 pL is the expected number of pairwise samples drawn from the model of our interest. III. MAIN RESULTS As suggested in [10], a crucial parameter for successful top-K ranking is the separation between the two items near the decision boundary, \u2206K := wK \u2212wK+1 wmax . (7) The sample complexity depends on w and K only through \u2206K\u2014more precisely, it decreases as \u2206K increases. Our contribution is to identify relationships between \u03b7 and the sample complexity when \u03b7 is known and unknown. We will see that the sample complexity increases as \u2206K decreases. This is intuitively true as \u2206K captures how distinguishable the top-K set is from the rest of the items. We assume that the graph G is drawn from the ER model Gn,p with edge appearance probability p. We require p to satisfy p > log n n . (8) From random graph theory, this implies that the graph is connected with high probability. If the graph were not connected, rankings cannot be inferred [9]. We start by considering the \u03b7-known scenario in which key ingredients for ranking algorithms and analysis can be easily digested, as well as which forms the basis for the \u03b7-unknown setting. Theorem 1 (Known \u03b7). Suppose that \u03b7 is known and G \u223cGn,p. Also assume that L = O(poly(n)) and Lnp \u2265 c0 (2\u03b7\u22121)2 log n. Then with probability \u22651 \u2212c1n\u2212c2, the set of top-K set can be identi\ufb01ed exactly provided L \u2265c3 log n (2\u03b7 \u22121)2np\u22062 K . (9) \f5 Conversely, for a \ufb01xed \u01eb \u2208(0, 1 2), if L \u2264c4 (1 \u2212\u01eb) log n (2\u03b7 \u22121)2np\u22062 K (10) holds, then for any top-K ranking scheme \u03c8, there exists a preference vector w with separation \u2206K such that Pe(\u03c8) \u2265\u01eb. Here, and in the following, ci > 0, i \u2208{0, 1, . . . , 4} are \ufb01nite universal constants. Proof: See Section IV for the algorithm and a sketch of the achievability proof (suf\ufb01ciency). The proof of the converse (impossibility part) can be found in Section V. This theorem asserts that the sample complexity scales as S\u2206K \u224d n log n (2\u03b7 \u22121)2 \u22062 K . (11) This result recovers that for the faithful scenario where \u03b7 = 1 in [10]. When \u03b7 \u22121 2 is uniformly bounded above 0, we achieve the same order-wise sample complexity. This suggests that the ranking performance is not substantially worsened if the sizes of the two populations are suf\ufb01ciently distinct. For the challenging scenario in which \u03b7 \u22481 2, the sample complexity depends on how \u03b7 \u22121 2 scales with n. Indeed, this dependence is quadratic. This theoretical result will be validated by experimental results in Section VII. Several other remarks are in order. No computational barrier: Our proposed algorithm is based primarily upon two popular ranking algorithms: spectral methods and MLE, both of which enjoy nearly-linear time complexity in our ranking problem context. Hence, the information-theoretic limit promised by (11) can be achieved by a computationally ef\ufb01cient algorithm. Implication of the minimax lower bound: The minimax lower bound continues to hold when \u03b7 is unknown, since we can only do better for the \u03b7-known scenario, and hence the lower bound is also a lower bound in the \u03b7-unknown scenario. Another adversarial scenario: Our results readily generalize to another adversarial scenario in which samples drawn from the adversarial population are completely noisy, i.e., they follow the distribution Bern(1 2). With a slight modi\ufb01cation of our proof techniques, one can easily verify that the sample complexity is on the order of n log n \u03b72\u22062 K if \u03b7 is known. This will be evident after we describe the algorithm in Section IV. Theorem 2 (Unknown \u03b7). Suppose that \u03b7 is unknown and G \u223cGn,p. Also assume that L = O(poly(n)) and Lnp \u2265 c0 (2\u03b7\u22121)4 log2 n. Then with probability \u22651 \u2212c1n\u2212c2, the top-K set can be identi\ufb01ed exactly provided L \u2265c3 log2 n (2\u03b7 \u22121)4np\u22064 K . (12) Proof: See Section VI for the key ideas in the proof. This theorem implies that the sample complexity satis\ufb01es S\u2206K \u2272 n log2 n (2\u03b7 \u22121)4 \u22064 K . (13) This bound is worse than (11)\u2014the inverse dependence on (2\u03b7 \u22121)2\u22062 K is now an inverse dependence on (2\u03b7 \u2212 1)4\u22064 K. This is because our algorithm involves estimating \u03b7, incurring some loss. Whether this loss is fundamentally unavoidable (i.e., whether the algorithm is order-wise optimal or not) is open. See detailed discussions in Section VIII. Moreover, since the estimation of \u03b7 is based on tensor decompositions with polynomial-time complexity, our algorithm for the \u03b7-unknown case is also, in principle, computationally ef\ufb01cient. Note that minimax lower bound in (11) also serves as a lower bound in the \u03b7-unknown scenario. IV. ALGORITHM AND ACHIEVABILITY PROOF OF THEOREM 1 A. Algorithm Description Inspired by the consistency between the preference scores w and ranking under the BTL model, our scheme also adopts a two-step approach where w is \ufb01rst estimated and then the top-K set is returned. Recently a top-K ranking algorithm SpectralMLE [10] has been developed for the faithful scenario and it is shown to have order-wise optimal sample complexity. The algorithm yields a small \u2113\u221eloss of the score vector w \f6 SPECTRAL MLE Y shifting \u02dc Y RANK CENTRALITY w(0) point-wise MLE \u02c6 w return top-K items \u03c8(Y ) Fig. 2. Ranking algorithm for the \u03b7-known scenario: (1) shifting the empirical mean of pairwise measurements to get \u02dc Yij = Yij\u2212(1\u2212\u03b7) 2\u03b7\u22121 , which converges to wi wi+wj as L \u2192\u221e; (2) performing SpectralMLE [10] seeded by \u02dc Y to obtain a score estimate \u02c6 w; (3) return a ranking based on the estimate \u02c6 w. Our analysis reveals that the \u2113\u221enorm bound w.r.t. \u02c6 w satis\ufb01es \u2225\u02c6 w \u2212w\u2225\u221e\u2272 1 2\u03b7\u22121 q log n npL , which in turn ensures Pe \u21920 under \u2206K \u227f 1 2\u03b7\u22121 q log n npL . which ensures a small point-wise estimate error. Establishing a key relationship between the \u2113\u221enorm error and top-K ranking accuracy, Chen and Suh [10] then identify an order-wise tight bound on the \u2113\u221enorm error required for top-K ranking, thereby characterizing the sample complexity. Our ranking algorithm builds on SpectralMLE, which proceeds in two stages: (1) an appropriate initialization that concentrates around the ground truth in an \u21132 sense, which can be obtained via spectral methods [3], [7], [8]; (2) a sequence of T iterative updates sharpening the estimates in a point-wise manner using MLE. We observe that RankCentrality [7] can be employed as a spectral method in the \ufb01rst stage. In fact, RankCentrality exploits the fact that the empirical mean Yij converges to the relative score wi wi+wj as L \u2192\u221e. This motivates the use of the empirical mean for constructing the transition probability from j to i of a Markov chain. Note that the detailed balance equation \u03c0i wj wi+wj = \u03c0j wi wi+wj that holds as L \u2192\u221ewill enforce that the stationary distribution of the Markov chain is identical to w up to some constant scaling. Hence, the stationary distribution is expected to serve as a reasonably good global score estimate. However, in our problem setting where \u03b7 is not necessarily 1, the empirical mean does not converge to the relative score, instead it behaves as Yij L\u2192\u221e \u2212 \u2192 \u03b7 wi wi + wj + (1 \u2212\u03b7) wj wi + wj . (14) Note, however, that the limit is linear in the desired relative score and \u03b7, implying that knowledge of \u03b7 leads to the relative score. A natural idea then arises. We construct a shifted version of the empirical mean: \u02dc Yij := Yij \u2212(1 \u2212\u03b7) 2\u03b7 \u22121 L\u2192\u221e \u2212 \u2192 wi wi + wj , (15) and take this as an input to RankCentrality. This then forms a Markov chain that yields a stationary distribution that is proportional to w as L \u2192\u221eand hence a good estimate of the ground-truth score vector when L is large. This serves as a good initial estimate to the second stage of SpectralMLE as it guarantees a small point-wise error. A formal and more detailed description of the procedure is summarized in Algorithm 1. For completeness, we also include the procedure of RankCentrality in Algorithm 2. Here we emphasize two distinctions w.r.t. the second stage of SpectralMLE. First, the computation of the pointwise MLE w.r.t. say, item i, requires knowledge of \u03b7: L \u0000\u03c4, w(t) \\i ; Yi \u0001 = Y j:(i,j)\u2208E \"\u0012 \u03b7 \u03c4 \u03c4 + w(t) j + (1 \u2212\u03b7) w(t) j \u03c4 + w(t) j \u0013Yij\u0012 \u03b7 w(t) j \u03c4 + w(t) j + (1 \u2212\u03b7) \u03c4 \u03c4 + w(t) j \u00131\u2212Yij# . (16) Here, L(\u03c4, w(t) \\i ; Yi) is the pro\ufb01le likelihood of the preference score vector [w(t) 1 , \u00b7 \u00b7 \u00b7 , w(t) i\u22121, \u03c4, w(t) i+1, \u00b7 \u00b7 \u00b7 , w(t) n ] where w(t) indicates the preference score estimate in the t-th iteration, w(t) \\i denotes the score estimate excluding the i-th component, and Yi is the data available at node i. The second difference is the use of a different threshold \u03bet which incorporates the effect of \u03b7: \u03bet := c 2\u03b7 \u22121 (s log n npL + 1 2t s log n pL \u2212 s log n npL !) , (17) \f7 Algorithm 1 Adversarial top-K ranking for the \u03b7-known scenario Input: The average comparison outcome Yij for all (i, j) \u2208E; the score range [wmin, wmax]. Partition E randomly into two sets Einit and Eiter each containing 1 2 |E| edges. Denote by Y init i (resp. Y iter i ) the components of Yi obtained over Einit (resp. Eiter). Compute the shifted version of the average comparison output: \u02dc Yij = Yij\u2212(1\u2212\u03b7) 2\u03b7\u22121 . Denote by \u02dc Y init i the components of \u02dc Yi obtained over Einit Initialize w(0) to be the estimate computed by Rank Centrality on \u02dc Y init i (1 \u2264i \u2264n). Successive Re\ufb01nement: for t = 0 : T do 1) Compute the coordinate-wise MLE wmle i \u2190arg max\u03c4 L \u0010 \u03c4, w(t) \\i ; Y iter i \u0011 where L is the likelihood function de\ufb01ned in (16). 2) For each 1 \u2264i \u2264n, set w(t+1) i \u2190 ( wmle i , |wmle i \u2212w(t) i | > \u03bet; w(t) i , else, where \u03bet is the replacement threshold de\ufb01ned in (17). Output the indices of the K largest components of w(T). Algorithm 2 Rank Centrality [7] Input: The shifted average comparison outcome \u02dc Yij for all (i, j) \u2208Eiter. Compute the transition matrix \u02c6 P = [\u02c6 pij]1\u2264i,j\u2264n such that for (i, j) \u2208Eiter \u02c6 pij = ( \u02dc Yji dmax , if i \u0338= j; 1 \u2212 1 dmax P k:(i,k)\u2208Eiter \u02dc Yki, if i = j. where dmax is the maximum out-degrees of vertices in Eiter. Output the stationary distribution of \u02c6 P. where c > 0 is a constant. This threshold is used to decide whether w(t+1) i should be set to be the pointwise MLE wmle i in (22) (if |wmle i \u2212w(t) i | > \u03bet) or remains as w(t) i (otherwise). The design of \u03bet is based on (1) the \u2113\u221eloss incurred in the \ufb01rst stage; and (2) a desirable \u2113\u221eloss that we intend to achieve at the end of the second stage. Since these two values are different, \u03bet needs to be adapted accordingly. Notice that the computation of \u03bet requires knowledge of \u03b7. The two modi\ufb01cations in (16) and (17) result in a more complicated analysis vis-` a-vis Chen and Suh [10]. B. Achievability Proof of Theorem 1 Let \u02c6 w be the \ufb01nal estimate w(T) in the second stage. We carefully analyze the \u2113\u221eloss of the w vector, showing that under the conditions in Theorem 1 \u2225\u02c6 w \u2212w\u2225\u221e\u2264 c1 2\u03b7 \u22121 s log n npL . (18) holds with probability exceeding 1 \u2212c2n\u2212c3. This bound together with the following observation completes the proof. Observe that if wK \u2212wK+1 \u2265 c4 2\u03b7\u22121 q log n npL , then for a top-K item 1 \u2264i \u2264K and a non-top-K item \f8 j \u2265K + 1, \u02c6 wi \u2212\u02c6 wj \u2265wi \u2212wj \u2212|wi \u2212\u02c6 wi| \u2212|wj \u2212\u02c6 wj| (19) \u2265wK \u2212wK+1 \u22122\u2225\u02c6 w \u2212w\u2225\u221e> 0. (20) This implies that our ranking algorithm outputs the top-K ranked items as desired. Hence, as long as wK \u2212wK+1 \u227f 1 2\u03b7\u22121 q log n npL holds (coinciding with the claimed bound in Theorem 1), we can guarantee perfect top-K ranking, which completes the proof of Theorem 1. The remaining part is the proof of (18). The proof builds upon the analysis made in [10], which demonstrates the relationship between \u2225w(0)\u2212w\u2225 \u2225w\u2225 and \u2225w(T) \u2212w\u2225\u221e. We establish a new relationship for the arbitrary \u03b7 case, formally stated in the following lemma. We will then use this to prove (18). Lemma 1. Fix \u03b4, \u03be > 0. Consider \u02c6 wub such that it is independent of G and satis\ufb01es \u2225\u02c6 wub \u2212w\u2225 \u2225w\u2225 \u2264\u03b4 and \u2225\u02c6 wub \u2212w\u2225\u221e\u2264\u03be. (21) Consider an estimate of the score vector \u02c6 w such that | \u02c6 wi \u2212wi| \u2264| \u02c6 wub i \u2212wi| for all i \u2208[n]. Let wmle i := arg max \u03c4 L(\u03c4, \u02c6 w\\i; Yi). (22) Then, the pointwise error |wmle i \u2212wi| \u2264c0 max ( \u03b4 + log n np \u00b7 \u03be, c1 2\u03b7 \u22121 s log n npL ) (23) holds with probability at least 1 \u2212c2n\u2212c3. Proof: The relationship in the faithful scenario \u03b7 = 1, which was proved in [10], means that the point-wise MLE wmle i is close to the ground truth wi in a component-wise manner, once an initial estimate \u02c6 w is accurate enough. Unlike the faithful scenario, in our setting, we have (in general) noisier measurements Yi due to the effect of \u03b7. Nonetheless this lemma reveals that the relationship for the case of \u03b7 = 1 is almost the same as that for an arbitrary \u03b7 case only with a slight modi\ufb01cation. This implies that a small point-wise loss is still guaranteed as long as we start from a reasonably good estimate. Here the only difference in the relationship is that the multiplication term of 1 2\u03b7\u22121 additionally applies in the upper bound of (23). See Appendix A for the proof. Obviously the accuracy of the point-wise MLE re\ufb02ected in the \u2113\u221eerror depends crucially on an initial error \u2225w(0) \u2212w\u2225. In fact, Lemma 1 leads to the claimed bound (18) once the initial estimation error is properly chosen as follows: \u2225w(0) \u2212w\u2225 \u2225w\u2225 \u2272 1 2\u03b7 \u22121 s log n npL . (24) Here we demonstrate that the desired initial estimation error can indeed be achieved in our problem setting, formally stated in Lemma 2 (see below). On the other hand, adapting the analysis in [10], one can verify that with the replacement threshold \u03bet de\ufb01ned in (17), the \u21132 loss is monotonically decreasing in an order-wise sense, i.e., \u2225w(t) \u2212w\u2225 \u2225w\u2225 \u2272\u2225w(0) \u2212w\u2225 \u2225w\u2225 . (25) We are now ready to prove (18) when L = O(poly(n)) and \u2225w(t) \u2212w\u2225 \u2225w\u2225 \u224d\u03b4 \u224d 1 2\u03b7 \u22121 s log n npL . (26) Lemma 1 asserts that in this regime, the point-wise MLE wmle is expected to satisfy \u2225wmle \u2212w\u2225\u221e\u2272\u2225w(t) \u2212w\u2225 \u2225w\u2225 + log n np \u2225w(t) \u2212w\u2225\u221e. (27) \f9 Using the analysis in [10], one can show that the choice of \u03bet in (17) enables us to detect outliers (where an estimation error is large) and drag down the corresponding point-wise error, thereby ensuring that \u2225w(t+1)\u2212w\u2225\u221e\u224d \u2225wmle \u2212w\u2225\u221e. This together with the fact that \u2225w(t) \u2212w\u2225 \u2225w\u2225 \u2272\u2225w(0) \u2212w\u2225 \u2225w\u2225 \u2272 1 2\u03b7 \u22121 s log n npL (28) (see (26) above and Lemma 2) gives \u2225w(t+1) \u2212w\u2225\u221e\u2272 1 2\u03b7 \u22121 s log n npL + log n np \u2225w(t) \u2212w\u2225\u221e. (29) A straightforward computation with this recursion yields (18) if log n np is suf\ufb01ciently small (e.g., p > 2 log n n ) and T, the number of iterations in the second stage of SpectralMLE, is suf\ufb01ciently large (e.g., T = O(log n)). Lemma 2. Let L = O(poly(n)) and Lnp \u2265 c0 (2\u03b7\u22121)2 log n. Let w(0) be an initial estimate: an output of RankCentrality [7] when seeded by \u02dc Y := {\u02dc Yij}(i,j)\u2208E. Then, \u2225w \u2212w(0)\u2225 \u2225w\u2225 \u2264 c1 2\u03b7 \u22121 s log n npL (30) holds with probability exceeding 1 \u2212c2n\u2212c3. Proof: Here we provide only a sketch of the proof, leaving details to Appendix B. The proof builds upon the analysis structured by Lemma 2 in Negahban et al. [7], which bounds the deviation of the Markov chain w.r.t. the transition matrix \u02c6 P after t steps: \u2225\u02c6 pt \u2212w\u2225 \u2225w\u2225 \u2264\u03c1t\u2225\u02c6 p0 \u2212w\u2225 \u2225w\u2225 rwmax wmin + 1 1 \u2212\u03c1\u2225\u2206\u2225 rwmax wmin (31) where \u02c6 pt denotes the distribution w.r.t. \u02c6 P at time t seeded by an arbitrary initial distribution \u02c6 p0, the matrix \u2206:= \u02c6 P \u2212P, indicates the \ufb02uctuation of the transition probability matrix1 around its mean P := E[ \u02c6 P], and \u03c1 := \u03bbmax + \u2225\u2206\u2225 q wmax wmin . Here \u03bbmax = max{\u03bb2, \u2212\u03bbn} and \u03bbi indicates the i-th eigenvalue of P. Unlike the faithful scenario \u03b7 = 1, in the arbitrary \u03b7 case, the bound on \u2225\u2206\u2225depends on \u03b7: \u2225\u2206\u2225\u2272 1 2\u03b7 \u22121 s log n npL , (32) which will be proved in Lemma B by using various concentration bounds (e.g., Hoeffding and Tropp [27]). Adapting the analysis in [7], one can easily verify that \u03c1 < 1 under one of the conditions in Theorem 1 that Lnp \u227f log n (2\u03b7\u22121)2 . Applying the bound on \u2225\u2206\u2225and \u03c1 < 1 to (31) gives the claimed bound, which completes the proof. V. CONVERSE PROOF OF THEOREM 1 As in Chen and Suh\u2019s work [10], by Fano\u2019s inequality, we see that it suf\ufb01ces for us to upper bound the mutual information between a set of appropriately chosen rankings M of cardinality M := min{K, n \u2212K} + 1. More speci\ufb01cally, let \u03c3 : [n] \u2192[n] represent a permutation over [n]. We also denote by \u03c3(i) and \u03c3([K]) the corresponding index of the i-th ranked item and the index set of all top-K items, respectively. We subsequently impose a uniform prior over M as follows: If K < n/2 then Pr(\u03c3([K]) = S) = 1 M for S = {2, . . . , K} \u222a{i}, i = 1, K + 1, . . . , n (33) and if K \u2265n/2, then Pr(\u03c3([K]) = S) = 1 M for S = {1, . . . , K + 1} \\ {i}, i = 1, . . . , K + 1. (34) 1The notation \u2206= \u02c6 P \u2212P, a matrix, should not be confused with the scalar normalized score separation \u2206K, de\ufb01ned in (7). \f10 In words, each alternative hypothesis is generated by swapping only two indices of the hypothesis (ranking) obeying \u03c3([K]) = [K]. Clearly, the original minimax error probability is lower bounded by the corresponding error probability of this reduced ensemble. Let the set of observations for the edge (i, j) \u2208E be denoted as \u20d7 Yij := {Y (\u2113) ij : \u2113\u2208[L]}. We also \ufb01nd it convenient to introduce an erased version of the observations Z = {\u20d7 Zij : i, j \u2208[n]} which is related to the true observations Y := {\u20d7 Yij : (i, j) \u2208E} as follows, \u20d7 Zij = \u001a \u20d7 Yij (i, j) \u2208E e (i, j) / \u2208E . (35) Here e is an erasure symbol. Let \u03c3, a chance variable, be a uniformly distributed ranking in M (the ensemble of rankings created in (33)\u2013(34)). Let P\u20d7 Yij|\u03c3j be the distribution of the observations given that the ranking is \u03c3j \u2208M where j \u2208[M] and a similar notation is used for when \u20d7 Yij is replaced by \u20d7 Zij. Now, by the convexity of the relative entropy and the fact that the rankings are uniform, the mutual information can be bounded as I(\u03c3; Z) \u2264 1 M2 X \u03c31,\u03c32\u2208M D \u0000PZ|\u03c31 \r \rPZ|\u03c32 \u0001 (36) = 1 M2 X \u03c31,\u03c32\u2208M X i\u0338=j D \u0010 P\u20d7 Zij|\u03c31 \r \rP\u20d7 Zij|\u03c32 \u0011 (37) = p M2 X \u03c31,\u03c32\u2208M X i\u0338=j D \u0010 P\u20d7 Yij|\u03c31 \r \rP\u20d7 Yij|\u03c32 \u0011 (38) = p M2 X \u03c31,\u03c32\u2208M X i\u0338=j L X \u2113=1 D \u0010 PY (\u2113) ij |\u03c31 \r \rPY (\u2113) ij |\u03c32 \u0011 . (39) Assume that under ranking \u03c31, the score vector is w := (w1, . . . , wn) and under ranking \u03c32, the score vector is w\u2032 := (w\u03c0(1), . . . , w\u03c0(n)) for some \ufb01xed permutation \u03c0 : [n] \u2192[n]. By using the statistical model described in Section II, we know that D \u0010 PY (\u2113) ij |\u03c31 \r \rPY (\u2113) ij |\u03c32 \u0011 = D \u0012 \u03b7 wi wi + wj + (1 \u2212\u03b7) wj wi + wj \r \r \r\u03b7 w\u03c0(i) w\u03c0(i) + w\u03c0(j) + (1 \u2212\u03b7) w\u03c0(j) w\u03c0(i) + w\u03c0(j) \u0013 (40) where D(\u03b1\u2225\u03b2) := \u03b1 log \u03b1 \u03b2 + (1 \u2212\u03b1) log 1\u2212\u03b1 1\u2212\u03b2 is the binary relative entropy. For brevity, write a := wi wi + wj , and b := w\u03c0(i) w\u03c0(i) + w\u03c0(j) . (41) Furthermore, we note that the chi-squared divergence is an upper bound for the relative entropy between two distributions P = {Pi}i\u2208X and Q = {Qi}i\u2208X on the same (countable) alphabet X (see e.g. [28, Lemma 6.3]), i.e., D(P\u2225Q) \u2264\u03c72(P\u2225Q) := X i\u2208X (Pi \u2212Qi)2 Qi . (42) We also use the notation \u03c72(\u03b1\u2225\u03b2) to denote the binary chi-squared divergence similarly to the binary relative entropy. Now, we may bound (40) using the following computation D \u0000\u03b7a + (1 \u2212\u03b7)(1 \u2212a) \r \r\u03b7b + (1 \u2212\u03b7)(1 \u2212b) \u0001 \u2264\u03c72 \u0000\u03b7a + (1 \u2212\u03b7)(1 \u2212a) \r \r\u03b7b + (1 \u2212\u03b7)(1 \u2212b) \u0001 (43) = (2\u03b7 \u22121)2(a \u2212b)2 \u0000(2\u03b7 \u22121)b + (1 \u2212\u03b7) \u0001\u0000\u03b7 \u2212(2\u03b7 \u22121)b \u0001 (44) Now |a \u2212b| \u2264 wK wK + wK+1 \u2212 wK+1 wK + wK+1 \u2264wmax 2wmin \u2206K. (45) Hence, if we consider the case where \u03b7 = (1/2)+ (which is the regime of interest), uniting (44) and (45) we obtain D \u0000\u03b7a + (1 \u2212\u03b7)(1 \u2212a) \r \r\u03b7b + (1 \u2212\u03b7)(1 \u2212b) \u0001 \u2272(2\u03b7 \u22121)2\u22062 K. (46) \f11 By construction of the hypotheses in (33)\u2013(34), conditional on any two distinct rankings \u03c31, \u03c32 \u2208M, the distributions of \u20d7 Yij (namely P\u20d7 Yij|\u03c31 and P\u20d7 Yij|\u03c32) are different over at most 2n locations so X i\u0338=j L X l=1 D \u0010 PY (\u2113) ij |\u03c31 \r \rPY (\u2113) ij |\u03c32 \u0011 \u2272nL(2\u03b7 \u22121)2\u22062 K. (47) Thus, plugging this into the bound on the mutual information in (39), we obtain I(\u03c3; Z) \u2272pnL(2\u03b7 \u22121)2\u22062 K. (48) Plugging this into Fano\u2019s inequality, and using the fact that M \u2264n/2 (from M = min{K, n \u2212K} + 1), we obtain Pe(\u03c8) \u22651 \u2212I(\u03c3; Z) log M \u2212 1 log M (49) \u22651 \u2212I(\u03c3; Z) log(n/2) \u2212 1 log(n/2). (50) Thus, if S = \u0000n 2 \u0001 pL \u2264c2(1\u2212\u01eb) log n (2\u03b7\u22121)2\u22062 K for some small enough but positive c2, we see that Pe(\u03c8) \u2265\u01eb. (51) Since this is independent of the decoder \u03c8, the converse part is proved. VI. ALGORITHM AND PROOF OF THEOREM 2 A. Algorithm Description The proof of Theorem 2 follows by combining the results of Jain and Oh [15] with the analysis for the case when \u03b7 is known in Theorem 1. Jain and Oh were interested in disambiguating a mixture distribution from samples. This corresponds to our model in (3). They showed using tensor decomposition methods that it is possible to \ufb01nd a globally optimal solution for the mixture weight \u03b7 using a computationally ef\ufb01cient algorithm. They also provided an \u21132 bound on the error of the distributions but as mentioned, we are more interested in controlling the \u2113\u221eerror so we estimate w separately. The use of the \u21132 bound in [15] leads to a worse sample complexity for top-K ranking. Thus, in the \ufb01rst step, we will use the method in [15] to estimate \u03b7 given the data samples (pairwise comparisons) Y . The estimate is denoted as \u02c6 \u03b7. It turns out that one can specialize the result in [15] with suitably parametrized \u201cdistribution vectors\u201d \u03c00 := \u0014 . . . wi wi + wj wj wi + wj wi\u2032 wi\u2032 + wj\u2032 wj\u2032 wi\u2032 + wj\u2032 . . . \u0015T (52) and \u03c01 := 12|E| \u2212\u03c00 \u2208R2|E| and where in (52), (i, j) runs through all values in E. Hence, we are in fact applying [15] to a more restrictive setting where the two probability distributions represented by \u03c00 and \u03c01 are \u201ccoupled\u201d but this does not preclude the application of the results in [15]. In fact, this assumption makes the calculation of relevant parameters (in Lemma 6) easier. The relevant second and third moments are M2 := \u03b7\u03c00 \u2297\u03c00 + (1 \u2212\u03b7)\u03c01 \u2297\u03c01, (53) M3 := \u03b7\u03c00 \u2297\u03c00 \u2297\u03c00 + (1 \u2212\u03b7)\u03c01 \u2297\u03c01 \u2297\u03c01, (54) where \u03c0j \u2297\u03c0j \u2208R(2|E|)\u00d7(2|E|) is the outer product and \u03c0j \u2297\u03c0j \u2297\u03c0j \u2208R(2|E|)\u00d7(2|E|)\u00d7(2|E|) is the 3-fold tensor outer product. If one has the exact M2 and M3, we can obtain the mixture weight \u03b7 exactly. The intuition as to why tensor methods are applicable to problems involving latent variables has been well-documented (e.g. [16]). Essentially, the secondand third-moments contained in M2 and M3 provide suf\ufb01cient statistics for identifying and hence estimating all the parameters of an appropriately-de\ufb01ned model with latent variables (whereas second-order information contained in M2 is, in general, not suf\ufb01cient for reconstructing the parameters). Thus, the problem boils down to analyzing the precision of \u03b7 when we only have access to empirical versions of M2 and M3 formed from pairwise comparisons in G. As shown in Lemma 5 to follow, there is a tradeoff between the sample size per edge L and the quality of the estimate of \u03b7. Hence, this causes a degradation to the overall sample complexity re\ufb02ected in Theorem 2. \f12 Y shifting \u02dc Y RANK CENTRALITY w(0) point-wise MLE \u02c6 w return top-K items \u03c8(Y ) TENSOR METHOD \u02c6 \u03b7 SPECTRAL MLE Fig. 3. Ranking algorithm for the unknown \u03b7 scenario. The key distinction relative to the known \u03b7 case is that we estimate \u03b7 based on the tensor decomposition method [15], [16] and the estimate \u02c6 \u03b7 is employed for shifting Y and performing the point-wise MLE. This method allows us to get \u2225\u02c6 w \u2212w\u2225\u221e\u2272 1 2\u03b7\u22121 4 q log2 n npL , which ensures that Pe \u21920 under \u2206K \u227f 1 2\u03b7\u22121 4 q log2 n npL . Algorithm 3 Estimating mixing coef\ufb01cient \u03b7 [15] Input: The collection of observed pairwise comparisons Y Split Y evenly into two subsets of samples Y (1) and Y (2) Estimate the second-order moment matrix M2 in (53) based on Y (1) using Algorithm 2 (MatrixAltMin) in [15] Estimate a third-order statistic G (de\ufb01ned in [15, Theorem 1]) based on (M2, M3, Y (2)) using Algorithm 3 (TensorLS) in [15] Compute the \ufb01rst eigenvalue \u03bb1 of G using the robust power method in [16] Return the estimated mixing coef\ufb01cient \u02c6 \u03b7 = \u03bb\u22122 1 In the second step, we plug the estimate \u02c6 \u03b7 into the algorithm for the \u03b7-known case by shifting the observations Y similarly to (15) but with \u02c6 \u03b7 instead of \u03b7. See Fig. 3. However, here there are a couple of important distinctions relative to the case where \u03b7 is known exactly. First, the likelihood function L(\u00b7) in (16) needs to be modi\ufb01ed since it is a function of \u03b7 in which now we only have its estimate \u02c6 \u03b7. Second, since the guarantee on the \u2113\u221eloss of the preference score vector w is different (and in fact worse), we need to design the threshold \u03bet differently from (17). We call the modi\ufb01ed threshold \u02c6 \u03bet, to be de\ufb01ned precisely in (58). B. Proof of Theorem 2 As in Section IV-B, the crux is to analyze the \u2113\u221eloss of the w vector. We show that \u2225\u02c6 w \u2212w\u2225\u221e\u2264 c0 2\u03b7 \u22121 4 s log2 n npL (55) holds with probability \u22651 \u2212c1n\u2212c2. To guarantee accurate top-K ranking, we then follow the same argument as in (19)\u2013(20). We lower bound \u2225\u02c6 w \u2212w\u2225\u221ein (55) by \u2206K and solve for L. Thus, it suf\ufb01ces to show (55) under the conditions of Theorem 2. The proof of (55) follows from several lemmata, two of which we present in this section. These are the analogues of Lemmas 1 and 2 for the \u03b7-known case. Once we have these two lemmata, the strategy to proving (55) is almost the same as that in the \u03b7-known setting in Section IV-B so we omit the details. The \ufb01rst lemma concerns the relationship between the normalized \u21132 error and the \u2113\u221eerror when we do not have access to the true mixture weight \u03b7, but only an estimate of it given via Algorithm 3. \f13 Lemma 3. Consider \u02c6 wub such that it is independent of G and satis\ufb01es (21). Consider \u02c6 w such that | \u02c6 wi \u2212wi| \u2264 | \u02c6 wub i \u2212wi| for all i \u2208[n]. Now de\ufb01ne wmle i := arg max \u03c4 \u02c6 L(\u03c4, \u02c6 w\\i; Yi), (56) where \u02c6 L(\u00b7) is the surrogate likelihood (cf. (16)) constructed with \u02c6 \u03b7 in place of \u03b7. Then, for all i, the same pointwise MLE bound in (23) holds with probability \u22651 \u2212c0n\u2212c1. Proof: The proof parallels that of Lemma 1 but is more technical. We analyze the \ufb01delity of the estimate \u02c6 \u03b7 relative to \u03b7 as a function of L (Lemma 5). This requires the specialization of Jain and Oh [15] to our setting. By proving several continuity statements, we show that the estimated normalized log-likelihood (NLL) 1 L log \u02c6 L(\u00b7) is uniformly close to the true NLL 1 L log L(\u00b7) w.h.p. This leads us to prove (23), which is the same as the \u03b7-known case. The details are deferred to Appendix C. Similarly to the case where \u03b7 is known, we need to subsequently control the initial error \u2225w(0) \u2212w\u2225. For the \u03b7-known case, this is done in Lemma 2 so the following lemma is an analogue of Lemma 2. Lemma 4. Assume the conditions of Theorem 2 hold. Let w(0) be an initial estimate, i.e., an output of RankCentrality when seeded by \u02dc Y which consists of the shifted observations with \u02c6 \u03b7 in place of \u03b7 (cf. (15)). Then, \u2225w \u2212w(0)\u2225 \u2225w\u2225 \u2264 c0 2\u03b7 \u22121 4 s log2 n npL (57) holds with probability \u22651 \u2212c1n\u2212c2. Proof: See Section VI-C for a sketch of the proof and Appendix D for a detailed calculation of an upper bound on the spectral norm of the \ufb02uctuation matrix, which is a key ingredient of the proof of Lemma 4. We remark that (57) is worse than its \u03b7-known counterpart in (30). In particular, there is now a fourth root inverse dependence on L (compared to a square root inverse dependence), which means we potentially need many more observations to drive the normalized \u21132 error \u2225w\u2212w(0)\u2225 \u2225w\u2225 down to the same level. This loss is present because there is a penalty incurred in estimating \u03b7 via the tensor decomposition approach, especially when \u03b7 is close to 1/2. In the analysis, we need to control the Lipschitz constants of functions such as t 7\u2192 1 2t\u22121 and t 7\u2192 1\u2212t 2t\u22121 (see e.g. (15)). Such functions behave badly near 1/2. In particular, the gradient diverges as t \u21931/2. We have endeavored to optimize (57) so that it is as tight as possible, at least using the proposed methods. Using Lemmas 3 and 4 and invoking a similar argument as in the \u03b7-known scenario, we can now to prove (55). One key distinction here lies in the choice of the threshold: \u02c6 \u03bet := c 2\u02c6 \u03b7 \u22121 \uf8f1 \uf8f2 \uf8f3 4 s log2 n npL + 1 2t \uf8eb \uf8ed4 s n log2 n pL \u2212 4 s log2 n npL \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe. (58) The rationale behind this choice, which is different from (17), is that it drives the initial \u2113\u221eloss (associated to the initial \u21132 loss in Lemma 4) to approach the desired \u2113\u221eloss in (55). Taking this choice, which we optimized, and adapting the analysis in [10] with Lemma 3, one can verify that the \u2113\u221eloss is monotonically decreasing in an orderwise sense: \u2225w(t)\u2212w\u2225 \u2225w\u2225 \u2272\u2225w(0)\u2212w\u2225 \u2225w\u2225 similarly to (25). By applying Lemma 3 to the regime where L = O(poly(n)) and \u2225w(t) \u2212w\u2225 \u2225w\u2225 \u224d\u03b4 \u224d 1 2\u03b7 \u22121 4 s log2 n npL , (59) we get \u2225wmle \u2212w\u2225\u221e\u2272\u2225w(t) \u2212w\u2225 \u2225w\u2225 + log n np \u2225w(t) \u2212w\u2225\u221e. (60) As in the \u03b7-known setting, one can show that the replacement threshold \u02c6 \u03bet leads to \u2225wmle \u2212w\u2225\u221e\u224d\u2225w(t) \u2212w\u2225\u221e. This together with Lemma 4 gives \u2225w(t+1) \u2212w\u2225\u221e\u2272 1 2\u03b7 \u22121 4 s log2 n npL + log n np \u2225w(t) \u2212w\u2225\u221e. (61) \f14 A straightforward computation with this recursion yields the claimed bound as long as log n np is suf\ufb01ciently small (e.g., p > 2 log n n ) and T is suf\ufb01ciently large (e.g., T = O(log n)). This completes the proof of (55). C. Proof Sketch of Lemma 4 The proof of Lemma 4 relies on the \ufb01delity of the estimate \u02c6 \u03b7 as a function of L when we use the tensor decomposition approach by Jain and Oh [15] on the problem at hand. Lemma 5 (Fidelity of \u03b7 estimate). If the number of observations per observed node pair L satis\ufb01es L \u227f1 \u03b52 log n \u03b4 , , (62) then the estimate \u02c6 \u03b7 is \u03b5-close to the true value \u03b7 with probability exceeding 1 \u2212\u03b4. Proof: The complete proof using Theorem 3 and Lemma 6 is provided in Section VI-D. We take \u03b4 = n\u2212c0 (for some constant c0 > 0) in the sequel so (62) reduces to L \u227f1 \u03b52 log n. A major contribution in the present paper is to \ufb01nd a \u201csweet spot\u201d for \u03b5; if it is chosen too small, \u2225\u02c6 w \u2212w\u2225\u221eis reduced (improving the estimation error) but L increases (worsening the overall sample complexity). Conversely, if \u03b5 is chosen to be too large, the requirement on L in (62) is relaxed, but \u2225\u02c6 w \u2212w\u2225\u221eincreases and hence, the overall sample complexity grows (worsens) eventually. The estimate in (62) is reminiscent of a Chernoff-Hoeffding bound estimate of the sample size per edge L required to ensure that the average of i.i.d. random variables is \u03b5-close to its mean with probability \u22651 \u2212\u03b4. However, the justi\ufb01cation is more involved and requires specializing Theorem 3 (to follow) to our setting. Now, we denote the difference matrix \u2206:= \u02c6 P \u2212P in which \u02c6 \u03b7 is used in place of \u03b7 as \u02c6 \u2206. Now using Lemma 5, several continuity arguments, and some concentration inequalities, we are able to establish that \u2225\u02c6 \u2206\u2225\u2272 1 2\u03b7 \u22121 4 s log2 n npL (63) with probability \u22651\u2212c1n\u2212c2. The inequality (63) is proved in Appendix D. Now similarly to the proof of Lemma 1, \u03c1 < 1 under the conditions of Theorem 2. Applying the bound on the spectral norm of \u2225\u02c6 \u2206\u2225in (63) to (31) (which continues to hold in the \u03b7-unknown setting) completes the proof of Lemma 4. D. Proof of Lemma 5 To prove Lemma 5, we specialize the non-asymptotic bound on the recovery of parameters in a mixture model in [15] to our setting; cf. (52). Before stating this, we introduce a few notations. Let the singular value decomposition of M2, de\ufb01ned in (53), be written as M2 = U\u03a3V T where \u03a3 = diag(\u03c31(M2), \u03c32(M2)) and U \u2208R(2|E|)\u00d72 the matrix consisting of the left-singular vectors, is further decomposed as U = \u0002 ((U (1))T (U (2))T . . . (U (|E|))T \u0003T . (64) Each submatrix U (k) \u2208R2\u00d72 where k denotes a node pair. We say that M2 is \u02dc \u00b5-block-incoherent if the operator norms for all |E| blocks of U, namely U (k), are upper bounded as \u2225U (k)\u22252 \u2264\u02dc \u00b5 s 2 |E|, \u2200k \u2208E. (65) For M2, the smallest block-incoherent constant \u02dc \u00b5 is known as the block-incoherence of M2. We denote this as \u00b5(M2) := inf{\u02dc \u00b5 : M2 is \u02dc \u00b5-block-incoherent}. Theorem 3 (Jain and Oh [15]). Fix any \u03b5, \u03b4 > 0. There exists a polynomial-time algorithm in |E|, 1 \u03b5 and log 1 \u03b4 (Algorithm 1 in [15]) such that if |E| \u227f\u03c31(M2)4.5\u00b5(M2) \u03c32(M2)4.5 (66) \f15 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.6 \u03b7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Success rate \u2206K = 0.1 \u2206K = 0.2 \u2206K = 0.3 \u2206K = 0.4 \u2206K = 0.5 (a) 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 \u03b7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Success rate (b) Fig. 4. Success rates across \u03b7 for (a) \u03b7 close to 1/2 and (b) \u03b7 close to 1. and for a large enough (per-edge) sample size L satisfying L \u227f \u00b5(M2)\u03c31(M2)6|E|3 min{\u03b7, 1 \u2212\u03b7}\u03c32(M2)9 \u00b7 log(n/\u03b4) \u03b52 , (67) the estimate of the mixture weight \u02c6 \u03b7 is \u03b5-close to the true mixture weight \u03b7 with probability exceeding 1 \u2212\u03b4. It remains to estimate the scalings of \u03c31(M2), \u03c32(M2) and \u00b5(M2). These require calculations based on \u03c00, \u03c01 and M2 and are summarized in the following crucial lemma. Lemma 6. For a \ufb01xed sequence of graphs with |E| edges, \u03c3i(M2) = \u0398(|E|), i = 1, 2, (68) \u00b5(M2) = \u0398(1). (69) Proof: The proof of this lemma can be found in Appendix E. It hinges on the fact that \u2225\u03c00\u22252 = \u2225\u03c01\u22252, as the populations have \u201cpermuted\u201d preference scores. Now the proof of Lemma 5 is immediate upon substituting (68) into (66)\u2013(67). We then notice that |E| = \u0398(n2p) = \u03c9(1) with high probability so (66) is readily satis\ufb01ed. Also \u00b5(M2)\u03c31(M2)6|E|3 min{\u03b7,1\u2212\u03b7}\u03c32(M2)9 = \u0398(1) so we recover (62) as desired. VII. EXPERIMENTAL RESULTS For the case where \u03b7 is known, a number of experiments on synthetic data were conducted to validate Theorem 1. We \ufb01rst state parameter settings common to all experiments. The total number of items is n = 1000 and the number of ranked items K = 10. In the pointwise MLE step in Algorithm 1, we set the number of iterations T = \u2308log n\u2309and c = 1 in the formula for the threshold \u03bet in (17). The observation probability of each edge of the Erd\u02dd os-R\u00b4 enyi graph is p = 6 log n n . The latent scores are uniformly generated from the dynamic range [0.5, 1]. Each (empirical) success rate is averaged over 1000 Monte Carlo trials. We \ufb01rst examine the relations between success rates and \u03b7 for various values of the normalized separation of the scores \u2206K \u2208{0.1, 0.2, . . . , 0.5}. Here we consider two different scenarios, one being such that \u03b7 is close to 1/2 and the other being such that \u03b7 is close to 1. We set the number of samples per edge, L = 1000 for the \ufb01rst case and L = 10 for the second. This is because when \u03b7 is small, more data samples are needed to achieve non-negligible success rates. The results for these two scenarios are shown in Figs. 4(a) and 4(b) respectively. For both cases, when L is \ufb01xed, we observe as \u03b7 increases, the success rates increase accordingly. However, the effect of \u03b7 on success rates is more prominent when \u03b7 is close to 1/2. This is in accordance to (11) in Theorem 1 since \f16 0 2 4 6 8 10 12 14 16 18 Snorm 0 0.2 0.4 0.6 0.8 1 Success rate \u03b7 = 0.6 \u03b7 = 0.7 \u03b7 = 0.8 \u03b7 = 0.9 \u03b7 = 1.0 Fig. 5. Success rates across normalized sample size Snorm. 1/(2\u03b7 \u22121)2 has sharp decrease (as \u03b7 increases) near 1/2 and a gentler decrease near 1. Also, success rates increase when \u2206K increases. This again corroborates (11) which says that the sample complexity is proportional to 1/\u22062 K. Next we examine the relations between success rates and normalized sample size Snorm := S\u2206K (n log n)/[(2\u03b7 \u22121)2\u22062 K], (70) for \u03b7 \u2208{0.6, 0.7, . . . , 1}. We \ufb01x \u2206K = 0.4 in this case. The results are shown in Fig. 5. We observe the relations between success rates and Snorm are almost the same for all \u03b7\u2019s so the implied constant factor in \u224dnotation in (11) depends very weakly on \u03b7 (if at all). Finally we numerically examine the relation between the sample complexity and \u03b7. We \ufb01x \u2206K = 0.4 and focus on the regime where \u03b7 is close to 1/2. For each \u03b7, we use the bisection method to approximately \ufb01nd the minimum sample size per edge \u02c6 L that achieves a high success rate qth = 0.99. Speci\ufb01cally, the bisection procedure terminates when the empirical success rate \u02c6 q corresponding to \u02c6 L satis\ufb01es |\u02c6 q \u2212qth| < \u01eb, where \u01eb is set to 5 \u00d7 10\u22123. We repeat such a procedure 10 times to get an average result \u02c6 Lave. We also compute the resulting standard deviation and observe that it is small across the 10 independent runs. De\ufb01ne the expected minimum total sample size \u02c6 S := \u0012n 2 \u0013 p\u02c6 Lave. (71) To illustrate the explicit dependence of \u02c6 S on \u03b7, we further normalize \u02c6 S to \u02c6 Snorm := \u02c6 S (n log n)/\u22062 K , (72) thus isolating the dependence of minimum total sample size on \u03b7 only. We then \ufb01t a curve C/(2\u03b7 \u22121)2 to \u02c6 Snorm, where C is chosen to best \ufb01t the points by optimizing a least-squares-like objective function. The empirical results (mean and one standard deviation) together with the \ufb01tted curve are shown in Fig. 6. We observe \u02c6 Snorm depends on \u03b7 via 1/(2\u03b7 \u22121)2 almost perfectly up to a constant. This corroborates our theoretical result in (11), i.e., the reciprocal dependence of the sample complexity on (2\u03b7 \u22121)2. For the case where \u03b7 is not known, the storage costs turn out to be prohibitive even for a moderate number of items n. Hence, we leave the implementation of the algorithm for the \u03b7-unknown case to future work. It is likely that one may need to formulate the ranking problem in an online manner [29] or resort to online methods for performing tensor decompositions [30]\u2013[32]. \f17 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.6 \u03b7 0 0.5 1 1.5 2 \u02c6 Snorm \u00d7104 C (2\u03b7\u22121)2 Fig. 6. Normalized empirical sample size \u02c6 Snorm for \u03b7 close to 1/2. VIII." + }, + { + "url": "http://arxiv.org/abs/1005.3338v1", + "title": "Feedback Capacity of the Gaussian Interference Channel to within 2 Bits", + "abstract": "We characterize the capacity region to within 2 bits/s/Hz and the symmetric\ncapacity to within 1 bit/s/Hz for the two-user Gaussian interference channel\n(IC) with feedback. We develop achievable schemes and derive a new outer bound\nto arrive at this conclusion. One consequence of the result is that feedback\nprovides multiplicative gain, i.e., the gain becomes arbitrarily large for\ncertain channel parameters. It is a surprising result because feedback has been\nso far known to provide no gain in memoryless point-to-point channels and only\nbounded additive gain in multiple access channels. The gain comes from using\nfeedback to maximize resource utilization, thereby enabling more efficient\nresource sharing between the interfering users. The result makes use of a\ndeterministic model to provide insights into the Gaussian channel. This\ndeterministic model is a special case of El Gamal-Costa deterministic model and\nas a side-generalization, we establish the exact feedback capacity region of\nthis general class of deterministic ICs.", + "authors": "Changho Suh, David Tse", + "published": "2010-05-18", + "updated": "2010-05-18", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION Shannon showed that feedback does not increase capacity in memoryless point-to-point channels [1]. On the other hand, feedback can indeed increase capacity in memory channels such as colored Gaussian noise channels. While it can provide multiplicative gain especially in the low SNR regime, the gain is bounded, i.e., feedback can provide a capacity increase of at most one October 29, 2018 DRAFT \f2 bit [2], [3]. In the multiple access channel (MAC), however, Gaarder and Wolf [4] showed that feedback could increase capacity although the channel is memoryless. Inspired by this result, Ozarow [5] found the feedback capacity region for the two-user Gaussian MAC. Ozarow\u2019s result reveals that feedback provides only additive power gain. The reason for the bounded power gain is that in the MAC, transmitters cooperation induced by feedback can at most boost signal power via aligning signal directions. Boosting signal power provides a capacity increase of a constant number of bits. A question arises: will feedback help signi\ufb01cantly in other channels where a receiver wants to decode only desired message in the presence of interference? To answer this question, we focus on the simple two-user Gaussian interference channel (IC) where each receiver wants to decode the message only from its corresponding transmitter. We \ufb01rst make progress on the symmetric capacity. Gaining insights from a deterministic model [6] and Alamouti\u2019s scheme [7], we develop a simple two-staged achievable scheme. We then derive a new outer bound to show that the proposed scheme achieves the symmetric capacity to within one bit for all values of the channel parameters. An interesting consequence of this result is that feedback can provide multiplicative gain in interference channels. This can be shown from the generalized degrees-of-freedom (g.d.o.f.) in Fig. 1. The notion was de\ufb01ned in [8] as d(\u03b1) \u225c lim SNR,INR\u2192\u221e Csym(SNR, INR) log SNR , (1) where \u03b1 (x-axis) indicates the ratio of INR to SNR in dB scale: \u03b1 \u225c log INR log SNR. Notice that in certain weak interference regimes (0 \u2264\u03b1 \u2264 2 3) and in the very strong interference regime (\u03b1 \u22652), feedback gain becomes arbitrarily large as SNR and INR go to in\ufb01nity. For instance, when \u03b1 = 1 2, the gap between the non-feedback and the feedback capacity becomes unbounded with the increase of SNR and INR, i.e., CFB sym \u2212CNO sym \u2212 \u21921 4 log SNR \u2212 \u2192\u221e. (2) Observing the ratio of the feedback to the non-feedback capacity in the high SNR regime, one can see that feedback provides multiplicative gain (50% gain for \u03b1 = 1 2): CFB sym CNO sym \u21921.5. Moreover, we generalize the result to characterize the feedback capacity region to within 2 bits per user for all values of the channel parameters. Unlike the symmetric case, we develop an October 29, 2018 DRAFT \f3 \u0001 \u0002 \u0000\u0003 \u0004\u0005 \u0006 \u0007 \b \t \u000b \f \r \u000e \u000f \u0010 \u0011 \u0012 \u0013 \u0014 \u0015 \u0016 \u0017 \u0018 \u0019 \u001a \u001b \u001c \u001d Fig. 1. The generalized degrees-of-freedom of the Gaussian interference channel (IC) with feedback. For certain weak interference regimes (0 \u2264\u03b1 \u2264 2 3) and for the very strong interference regime (\u03b1 \u22652), the gap between the non-feedback and the feedback capacity becomes arbitrarily large as SNR and INR go to in\ufb01nity. This implies that feedback can provide unbounded gain. in\ufb01nite-staged achievable scheme that employs three techniques: (i) block Markov encoding [9], [10]; (ii) backward decoding [11]; and (iii) Han-Kobayashi message splitting [12]. This result shows interesting contrast with the non-feedback capacity result. In the non-feedback case, it has been shown that the capacity region is described by \ufb01ve types of inequalities including the bounds for R1 + 2R2 and 2R1 + R2 [12], [8]. On the other hand, our result shows that the feedback capacity region requires only three types of inequalities without the R1 + 2R2 and 2R1 + R2 bounds. We also develop new interpretation, what we call a resource hole interpretation, to provide qualitative insights as to where feedback gain comes from. We \ufb01nd that the gain comes from using feedback to maximize resource utilization, thereby enabling more ef\ufb01cient resource sharing between the interfering users. Also the ef\ufb01cient resource utilization due to feedback turns out to deactivate the 2R1 + R2 bound. Our results make use of a deterministic model [6] to provide insights into the Gaussian channel. This deterministic model is a special case of El Gamal-Costa model [13]. As a sidegeneralization, we establish the exact feedback capacity region of this general class of determinOctober 29, 2018 DRAFT \f4 istic ICs. From this result, one can infer an approximate feedback capacity region of two-user Gaussian MIMO ICs, as Teletar and Tse [14] did in the non-feedback case. Interference channels with feedback have received previous attention [15], [16], [17], [18]. Kramer [15], [16] developed a feedback strategy in the Gaussian IC; Kramer and Gastpar [17] derived an outer bound. However, the gap between the inner and outer bounds becomes arbitrarily large with the increase of SNR and INR.1 Jiang-Xin-Garg [18] found an achievable region in the discrete memoryless IC with feedback, based on block Markov encoding [9] and binning. However, their scheme involves three auxiliary random variables and therefore requires further optimization. Also no outer bounds are provided. We propose explicit achievable schemes and derive a new tighter outer bound to characterize the capacity region to within 2 bits and the symmetric capacity to within 1 bit universally. Subsequent to our work, Prabhakaran and Viswanath [19] have found an interesting connection between our feedback problem and the conferencing encoder problem. Making such a connection, they have independently characterized the sum feedback capacity to within 19 bits/s/Hz. II. MODEL Fig. 2 describes the two-user Gaussian IC with feedback. Without loss of generality, we normalize signal power and noise power to 1, i.e., Pk = 1, Zk \u223cCN(0, 1), \u2200k = 1, 2. Hence, the signal-to-noise ratio and the interference-to-noise ratio can be de\ufb01ned to capture channel gains: SNR1 \u225c|g11|2, SNR2 \u225c|g22|2, INR12 \u225c|g12|2, INR21 \u225c|g21|2. (3) There are two independent and uniformly distributed sources, Wk \u2208{1, 2, \u00b7 \u00b7 \u00b7 , mk} , \u2200k = 1, 2. Due to feedback, the encoded signal Xki of user k at time i is a function of its own message and past output sequences: Xki = f i k \u0000Wk, Yk1, \u00b7 \u00b7 \u00b7 , Yk(i\u22121) \u0001 = f i k \u0000Wk, Y i\u22121 k \u0001 , (4) where we use shorthand notation Y i\u22121 k to indicate the sequence up to i\u22121. A rate pair (R1, R2) is achievable if there exists a family of codebook pairs with codewords (satisfying power 1Although this strategy can be arbitrarily far from optimality, a careful analysis reveals that it can also provide multiplicative feedback gain. See Fig. 13 for this. October 29, 2018 DRAFT \f5 Enc1 Enc2 Dec1 Dec2 W1 W2 \u02c6 W1 \u02c6 W2 X1 X2 Delay Y1 Y2 Z1 \u223cCN(0, 1) Z2 \u223cCN(0, 1) Delay g11 g22 g12 g21 Fig. 2. The Gaussian interference channel (IC) with feedback constraints) and decoding functions such that the average decoding error probabilities go to zero as block length N goes to in\ufb01nity. The capacity region C is the closure of the set of the achievable rate pairs. III. SYMMETRIC CAPACITY TO WITHIN ONE BIT We start with a symmetric channel setting where |g11| = |g22| = |gd| and |g12| = |g21| = |gc|: SNR \u225cSNR1 = SNR2, INR \u225cINR12 = INR21. (5) Not only is this symmetric case simple, it also provides the key ingredients to both the achievable scheme and outer bound needed for the characterization of the capacity region. Furthermore, this case provides enough qualitative insights as to where feedback gain comes from. Hence, we \ufb01rst focus on the symmetric channel. Keep in mind however that our proposed scheme for a symmetric rate is different from that for a rate region: in the symmetric case, the scheme employs only two stages (or blocks), while an in\ufb01nite number of stages are used in the general case. We will address this in Section IV. The symmetric capacity is de\ufb01ned by Csym = sup {R : (R, R) \u2208C} , (6) October 29, 2018 DRAFT \f6 \u221220 0 20 40 60 80 \u221220 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 SNR (dB) INR (dB) Gap (bits/s/Hz) Fig. 3. The gap between our inner and upper bounds. The gap is upper-bounded by exactly one bit. The worst-case gap occurs when SNR \u2248INR and these values go to in\ufb01nity. In the strong interference regime, the gap vanishes with the increase of SNR and INR, while in the weak interference regime, the gap does not, e.g., the gap is around 0.5 bits for \u03b1 = 1 2. where C is the capacity region. Theorem 1: We can achieve a symmetric rate of Rsym = max 1 2 log (1 + INR) , 1 2 log (1 + SNR + INR)2 \u2212 SNR 1+INR 1 + 2INR !! . (7) The symmetric capacity is upper-bounded by Csym = 1 2 sup 0\u2264\u03c1\u22641 \u0014 log \u0012 1 + (1 \u2212\u03c12)SNR 1 + (1 \u2212\u03c12)INR \u0013 + log \u0010 1 + SNR + INR + 2\u03c1 \u221a SNR \u00b7 INR \u0011\u0015 . (8) For all channel parameters of SNR and INR, we can achieve all rates R up to Csym \u22121, i.e., Csym \u22121 \u2264R \u2264Csym. (9) A. Deterministic Model As a stepping stone towards the Gaussian IC, we use an intermediate model: a linear deterministic model [6], illustrated in Fig. 4. This model is useful in the non-feedback Gaussian IC: October 29, 2018 DRAFT \f7 Tx 1 Tx 2 Rx 1 Rx 2 Delay Delay n11 n21 n22 n12 noise level Fig. 4. The deterministic IC with feedback it was shown in [20] that the deterministic IC can approximate the Gaussian IC to within a constant number of bits irrespective of the channel parameter values. Our approach is to \ufb01rst develop insights from this model and then translate them to the Gaussian channel. The connection with the Gaussian channel is as follows. The deterministic IC is characterized by four values: n11, n12, n21 and n22 where nij indicates the number of signal bit levels (or resource levels) from transmitter i to receiver j. These values correspond to channel gains in dB scale, i.e., \u2200i \u0338= j, nii = \u230alog SNRi\u230b, nij = \u230alog INRij\u230b. (10) In the symmetric channel, n \u225cn11 = n22 and m \u225cn12 = n21. Signal bit levels at a receiver can be mapped to the binary streams of a signal above the noise level. Upper signal levels correspond to most signi\ufb01cant bits and lower signal levels correspond to least signi\ufb01cant bits. A signal bit level observed by both the receivers above the noise level is broadcasted. If multiple signal levels arrive at the same signal level at a receiver, we assume a modulo-2-addition. B. Achievable Scheme for the Deterministic IC Strong Interference Regime (m \u2265n): We explain the scheme through the simple example of \u03b1 := m n = 3, illustrated in Fig. 5. Note that each receiver can see only one signal level from its corresponding transmitter. Therefore, in the non-feedback case, each transmitter can send only October 29, 2018 DRAFT \f8 Time 1 Time 2 Delay Delay Time 1 Time 2 Tx 1 Tx 2 Rx 1 Rx 2 a1 a2 a3 b1 b2 b3 decode(b1, b2, b3) decode(a1, a2, a3) a1 a2 a3 \u2295b1 \u2295a1 b1 b2 b3 a1 a2 a3 \u2295b1 \u2295a1 b1 b2 b3 b1 b2 b3 a1 a2 a3 Fig. 5. An achievable scheme for the deterministic IC: strong interference regime \u03b1 := m n = 3. 1 bit through the top signal level. However, feedback can create a better alternative path, e.g., [transmitter1 \u2192receiver2 \u2192feedback \u2192transmitter2 \u2192receiver1]. This alternative path enables to increase the non-feedback rate. The feedback scheme consists of two stages. In the \ufb01rst stage, transmitters 1 and 2 send independent binary symbols (a1, a2, a3) and (b1, b2, b3), respectively. Each receiver defers decoding to the second stage. In the second stage, using feedback, each transmitter decodes information of the other user: transmitters 1 and 2 decode (b1, b2, b3) and (a1, a2, a3), respectively. Each transmitter then sends the other user\u2019s information. Each receiver gathers the received bits sent during the two stages: the six linearly independent equations containing the six unknown symbols. As a result, each receiver can solve the linear equations to decode its desired bits. Notice that the second stage was used for re\ufb01ning all the bits sent previously, without sending additional information. Therefore, the symmetric rate is 3 2 in this example. Notice the 50% improvement from the non-feedback rate of 1. We can easily extend the scheme to arbitrary (n, m). In the \ufb01rst stage, each transmitter sends m bits using all the signal levels. Using two stages, these m bits can be decoded with the help of feedback. Thus, we can achieve: Rsym = m 2 . (11) Remark 1: The gain in the strong interference regime comes from the fact that feedback provides a better alternative path through the two cross links. The cross links relay the other user\u2019s information through feedback. We can also explain this gain using a resource hole interpretation. October 29, 2018 DRAFT \f9 Notice that in the non-feedback case, each transmitter can send only 1 bit through the top level and therefore there is a resource hole (in the second level) at each receiver. However, with feedback, all of the resource levels at the two receivers can be \ufb01lled up. Feedback maximizes resource utilization by providing a better alternative path. This concept coincides with correlation routing in [15]. On the other hand, in the weak interference regime, there is no better alternative path, since the cross links are weaker than the direct links. Nevertheless, it turns out that feedback gain can also be obtained in this regime. Weak Interference Regime (m \u2264n): Let us start by examining the scheme in the nonfeedback case. Unlike the strong interference regime, only part of information is visible to the other receiver in the weak interference regime. Hence, information can be split into two parts [12]: common m bits (visible to the other receiver) and private (n \u2212m) bits (invisible to the other receiver). Notice that using common levels causes interference to the other receiver. Sending 1 bit through a common level consumes a total of 2 levels at the two receivers (say $2), while using a private level costs only $1. Because of this, a reasonable achievable scheme is to follow the two steps sequentially: (i) sending all of the cheap (n \u2212m) private bits on the lower levels; (ii) sending some number of common bits on the upper levels. The number of common bits is decided depending on m and n. Consider the simple example of \u03b1 = m n = 1 2, illustrated in Fig. 6 (a). First transmitters 1 and 2 use the cheap private signal levels, respectively. Once the bottom levels are used, however using the top levels is precluded due to a con\ufb02ict with the private bits already sent, thus each transmitter can send only one bit. Observe the two resource holes on the top levels at the two receivers. We \ufb01nd that feedback helps \ufb01ll up all of these resource holes to improve the performance. The scheme uses two stages. As for the private levels, the same procedure is applied as that in the non-feedback case. How to use the common levels is key to the scheme. In the \ufb01rst stage, transmitters 1 and 2 send private bits a2 and b2 on the bottom levels, respectively. Now transmitter 1 squeezes one more bit a1 on its top level. While a1 is received cleanly at receiver 1, it causes interference at receiver 2. Feedback can however resolve this con\ufb02ict. In the second stage, with feedback transmitter 2 can decode the common bit a1 of the other user. As for the bottom levels, transmitters 1 and 2 send new private bits a3 and b3, respectively. The idea now is that transmitter 2 sends the other user\u2019s October 29, 2018 DRAFT \f10 b1 a1 b1 b3 b2 b1 b1 b3 \u2295b1 b1 a1 a3 a2 a2 \u2295b1 a1 a1 a1 b2 \u2295a1 a3 \u2295a1 decode decode (b) A feedback scheme Tx 1 Tx 2 Rx 1 Rx 2 Delay Delay b2 a2 (a) A non-feedback scheme Tx 1 Tx 2 Rx 1 Rx 2 b2 a2 private resource hole (cost $ 1) common (cost $ 2) Fig. 6. Achievable schemes for the weak interference regime, e.g., \u03b1 = m n = 1 2. common bit a1 on its top level. This transmission allows receiver 2 to re\ufb01ne the corrupted bit b2 from b2 \u2295a1 without causing interference to receiver 1, since receiver 1 already had the side information of a1 from the previous broadcasting. We paid $2 for the earlier transmission of a1, but now we can get a rebate of $1. Similarly, with feedback, transmitter 2 can squeeze one more bit b1 on its top level without causing interference. Therefore, we can achieve the symmetric rate of 3 2 in this example, i.e., the 50% improvement from the non-feedback rate of 1. This scheme can be easily generalized to arbitrary (n, m). In the \ufb01rst stage, each transmitter sends m bits on the upper levels and (n \u2212m) bits on the lower levels. In the second stage, each transmitter forwards the m bits of the other user on the upper levels and sends new (n \u2212m) private bits on the lower levels. Then, each receiver can decode all of the n bits sent in the \ufb01rst stage and new (n \u2212m) private bits sent in the second stage. Therefore, we can achieve: Rsym = n + (n \u2212m) 2 = n \u2212m 2 . (12) Remark 2 (A Resource Hole Interpretation): Observe that all the resource levels are fully packed after applying the feedback scheme. Thus, feedback maximizes resource utilization to improve the performance signi\ufb01cantly. Remark 3 (Exploiting Side Information): Another interpretation can be made to explain this gain. Recall that in the non-feedback case, the broadcast nature of the wireless medium precludes us from using the top level for one user when we are already using the bottom level for the other October 29, 2018 DRAFT \f11 user. In contrast, if feedback is allowed, the top level can be used to improve the non-feedback rate. Suppose that transmitters 1 and 2 send a1 and b1 through their top levels, respectively. Receivers 1 and 2 then get the clean bits a1 and b1, respectively. With feedback, in the second stage, these bits (a1, b1) can be exploited as side information to re\ufb01ne the corrupted bits. For example, with feedback transmitter 1 decodes the other user\u2019s bit b1 and forwards it through the top level. This transmission allows receiver 1 to re\ufb01ne the corrupted bit a2 from a2 \u2295b1 without causing interference to receiver 2, since receiver 2 already had the side information of b1 from the previous broadcasting. We exploited the side information with the help of feedback to re\ufb01ne the corrupted bit without causing interference. The exploitation of side information was also observed and pointed out in network coding examples such as the butter\ufb02y network [21], two-way relay channels [22], general wireless networks [23], and broadcast erasure channels with feedback [24]. C. Optimality of the Achievable Scheme for the Deterministic IC Now a natural question arises: is the scheme optimal? In this section, using the resource hole interpretation, we provide a positive conjecture on the optimality. Later in Section V, we will provide a rigorous proof to settle this conjecture. From W to V Curve: Fig. 7 shows (i) the symmetric feedback rate (11), (12) of the achievable scheme (representing the \u201cV\u201d curve); (ii) the non-feedback capacity [20] (representing the \u201cW\u201d curve). Using the resource hole interpretation, we will provide intuition as to how we can go from the W curve to the V curve with feedback. Observe that the total number of resource levels and transmission cost depend on (n, m). Speci\ufb01cally, suppose that the two senders employ the same transmission strategy to achieve the symmetric rate: using x private and y common levels. We then get: # of resource levels at each receiver = max(n, m), transmission cost = 1 \u00d7 x + 2 \u00d7 y. (13) Here notice that using a private level costs 1 level, while using a common level costs 2 levels. Now observe that as \u03b1 = m n grows: for 0 \u2264\u03b1 \u22641, transmission cost increases; for \u03b1 \u22651, the number of resource levels increases. Since all the resource levels are fully utilized with feedback, this observation implies that with feedback a total number of transmission bits must decrease October 29, 2018 DRAFT \f12 1 2 1 2 Rsym n \u03b1 \u0001 m n feedback cost increases resource increases no feedback 1 1 2 Fig. 7. Symmetric feedback rate (11), (12) for the deterministic IC. Feedback maximizes resource utilization while it cannot reduce cost. The \u201cV\u201d curve is obtained when all of the resource levels are fully packed with feedback. This shows the optimality of the feedback scheme. when 0 \u2264\u03b1 \u22641 (inversely proportional to transmission cost) and must increase when \u03b1 \u22651 (proportional to the number of resource levels). This is re\ufb02ected in the V curve. In contrast, in the non-feedback case, for some range of \u03b1, resource levels are not fully utilized, as shown in the \u03b1 = 1 2 example of Fig. 6 (a). This is re\ufb02ected in the W curve. Why We Cannot Go Beyond the V Curve: While feedback maximizes resource utilization to \ufb01ll up all of the resource holes, it cannot reduce transmission cost. To see this, consider the example in Fig. 6 (b). Observe that even with feedback, a common bit still has to consume two levels at the two receivers. For example, the common bit a1 needs to occupy the top level at receiver 1 in time 1; and the top level at receiver 2 in time 2. In time 1, while a1 is received cleanly at receiver 1, it interferes with the private bit b2. In order to re\ufb01ne b2, receiver 2 needs to get a1 cleanly and therefore needs to reserve one resource level for a1. Thus, in order not to interfere with the private bit b1, the common bit a1 needs to consume a total of the two resource levels at the two receivers. As mentioned earlier, assuming that transmission cost is not reduced, a total number of transmission bits is re\ufb02ected in the V curve. As a result, we cannot go beyond the \u201cV\u201d curve with feedback, showing the optimality of the achievable scheme. Later in Section V, we will prove this rigorously. October 29, 2018 DRAFT \f13 gd gc Z1 Z2 \u02c6 XN 2 XN\u2217 2 XN 1 \u02c6 XN 1 Y N,(1) 1 \u2212XN\u2217 1 Delay Delay gc gd XN 2 Y N,(2) 1 Y N,(1) 2 Y N,(2) 2 Block 1 Block 2 Block 1 Block 2 decode decode Fig. 8. An Alamouti-based achievable scheme for the Gaussian IC: strong interference regime D. An Achievable Scheme for the Gaussian IC Let us go back to the Gaussian channel. We will translate the deterministic IC scheme to the Gaussian IC. Let us \ufb01rst consider the strong interference regime. Strong Interference Regime (INR \u2265SNR): The structure of the transmitted signals in Fig. 5 sheds some light on the Gaussian channel. Observe that in the second stage, each transmitter sends the other user\u2019s information sent in the \ufb01rst stage. This reminds us of Alamouti\u2019s scheme [7]. The beauty of Alamouti\u2019s scheme is that received signals can be designed to be orthogonal during two time slots, although the signals in the \ufb01rst time slot are sent without any coding. This was exploited and pointed out in distributed space-time codes [25]. With Alamouti\u2019s scheme, transmitters are able to encode their messages so that received signals are orthogonal. Orthogonality between the two different signals guarantees complete removal of the interfering signal. In accordance with the deterministic IC example, the scheme uses two stages (or blocks). In the \ufb01rst stage, transmitters 1 and 2 send codewords XN 1 and XN 2 with rates R1 and R2, respectively. In the second stage, using feedback, transmitters 1 and 2 decode XN 2 and XN 1 , October 29, 2018 DRAFT \f14 respectively. This can be decoded if R1, R2 \u22641 2 log (1 + INR) bits/s/Hz. (14) We are now ready to apply Alamouti\u2019s scheme. Transmitters 1 and 2 send XN\u2217 2 and \u2212XN\u2217 1 , respectively. Receiver 1 can then gather the two received signals: for 1 \u2264i \u2264N, \uf8ee \uf8f0Y (1) 1i Y (2)\u2217 1i \uf8f9 \uf8fb= \uf8ee \uf8f0 gd gc \u2212g\u2217 c g\u2217 d \uf8f9 \uf8fb \uf8ee \uf8f0X1i X2i \uf8f9 \uf8fb+ \uf8ee \uf8f0Z(1) 1i Z(2)\u2217 1i \uf8f9 \uf8fb. (15) To extract X1i, it multiplies the row vector orthogonal to the vector associated with X2i and therefore we get: h g\u2217 d \u2212gc i \uf8ee \uf8f0Y (1) 1i Y (2)\u2217 1i \uf8f9 \uf8fb= (|gd|2 + |gc|2)X1i + g\u2217 dZ(1) 2i \u2212gcZ(2)\u2217 1i . (16) The codeword XN 1 can be decoded if R1 \u22641 2 log (1 + SNR + INR) bits/s/Hz. (17) Similar operations are done at receiver 2. Since (17) is implied by (14), we get the desired result: the left term in (7). Weak Interference Regime (INR \u2264SNR): Unlike the strong interference regime, in the weak interference regime, there are two types of information: common and private information. A natural idea is to apply Alamouti\u2019s scheme only for common information and newly add private information. It was shown in [26] that this scheme can approximate the symmetric capacity to within \u22481.7 bits/s/Hz. However, the scheme can be improved to reduce the gap further. Unlike the deterministic IC, in the Gaussian IC, private signals have some effects, i.e., these private signals cannot be completely ignored. Notice that the scheme includes decode-andforward operation at the transmitters after receiving the feedback. And so when each transmitter decodes the other user\u2019s common message while treating the other user\u2019s private signals as noise, private signals can incur performance loss. This can be avoided by instead performing amplify-and-forward: with feedback, the transmitters get the interference plus noise and then forward it subject to the power constraints. This transmission allows each receiver to re\ufb01ne its corrupted signal sent in the previous time, without October 29, 2018 DRAFT \f15 causing signi\ufb01cant interference.2 Importantly, notice that this scheme does not require messagesplitting. Even without splitting messages, we can re\ufb01ne the corrupted signals (see Appendix A to understand this better). Therefore, there is no loss due to private signals. Speci\ufb01cally, the scheme uses two stages. In the \ufb01rst stage, each transmitter k sends codeword XN k with rate Rk. In the second stage, with feedback transmitter 1 gets the interference plus noise: SN 2 = gcXN 2 + Z(1),N 1 . (18) Now the complex conjugate technique based on Alamouti\u2019s scheme is applied to make XN 1 and SN 2 well separable. Transmitters 1 and 2 send SN\u2217 2 \u221a1+INR and \u2212 SN\u2217 1 \u221a1+INR, respectively, where \u221a 1 + INR is a normalization factor to meet the power constraint. Under Gaussian input distribution, we can compute the rate under MMSE demodulation: 1 2I(X1i; Y (1) 1i , Y (2) 2i ). Straightforward calculations give the desired result: the right term in (7). See Appendix A for detailed computations. Remark 4 (Amplify-and-Forward Reduces the Gap Further): As mentioned earlier, unlike the decode-and-forward scheme, the amplify-and-forward scheme does not require messagesplitting, thereby removing the effect of private signals. This improves the performance to reduce the gap further. E. An Outer Bound Due to the overlap with the outer bound for the capacity region, we defer the proof to Theorem 3 in Section IV-B. F. One-Bit Gap to the Symmetric Capacity Using the symmetric rate of (7) and the outer bound of (8), we get: 2In Appendix A, we provide intuition behind this scheme. October 29, 2018 DRAFT \f16 2( \u00af Csym \u2212Rsym) (a) \u2264log \u0012 1 + SNR 1 + INR \u0013 + log \u0010 1 + SNR + INR + 2 \u221a SNR \u00b7 INR \u0011 \u2212log (1 + SNR + INR)2 \u2212 SNR 1+INR 1 + 2INR ! = log \u00121 + SNR + INR 1 + INR \u00b7 \u0010 1 + SNR + INR + 2 \u221a SNR \u00b7 INR \u0011\u0013 + log \u0012 (1 + 2INR)(1 + INR) (1 + SNR + INR)2(1 + INR) \u2212SNR \u0013 = log 1 + SNR + INR + 2 \u221a SNR \u00b7 INR 1 + SNR + INR \u00b7 1 + 2INR 1 + INR \u2212 SNR (1+SNR+INR)2 ! (b) \u2264log \uf8eb \uf8ed2 \u00b7 2 \u0010 1 + INR \u2212 SNR (1+SNR+INR)2 \u0011 \u22121 + 2SNR (1+SNR+INR)2 1 + INR \u2212 SNR (1+SNR+INR)2 \uf8f6 \uf8f8 = log 2 \u00b7 ( 2 \u2212 1 \u2212 2SNR (1+SNR+INR)2 1 + INR \u2212 SNR (1+SNR+INR)2 !)! (c) \u2264log 4 = 2, (19) where (a) follows from choosing trivial maximum and minimum values of the outer bound (8) and the lower bound (7), respectively; (b) follows from 1 + SNR + INR + 2 \u221a SNR \u00b7 INR \u2264 2(1 + SNR + INR); and (c) follows from (1 + SNR + INR)2 \u22652SNR and SNR (1+SNR+INR)2 \u22641. Fig. 3 shows a numerical result for the gap between the inner and outer bounds. Notice that the gap is upper-bounded by exactly one bit. The worst-case gap occurs when SNR \u2248INR and these values go to in\ufb01nity. Also note that in the strong interference regime, the gap approaches 0 with the increase of SNR and INR, while in the weak interference regime, the gap does not vanish. For example, when \u03b1 = 1 2, the gap is around 0.5 bits. Remark 5 (Why does a 1-bit gap occur?): Observe in Figs. 8 and 15 that the transmitted signals of the two senders are uncorrelated in our scheme. The scheme completely loses power gain (also called beamforming gain). On the other hand, when deriving the outer bound of (8), we allow for arbitrary correlation between the transmitters. Thus, the 1-bit gap is based on the outer bound. In the actual system, correlation is in-between and therefore one can expect that an actual gap to the capacity is less than 1 bit. Beamforming gain is important only when SNR and INR are quite close, i.e., \u03b1 \u22481. This is October 29, 2018 DRAFT \f17 because when \u03b1 = 1, the interference channel is equivalent to the multiple access channel where Ozarow\u2019s scheme [5] and Kramer\u2019s scheme [15] (that capture beamforming gain) are optimal. In fact, the capacity theorem in [16] has shown that Kramer\u2019s scheme is optimal for one speci\ufb01c case of INR = SNR \u2212 \u221a 2SNR, although it is arbitrarily far from optimality for the other cases. This observation implies that our proposed scheme can be improved further. IV. CAPACITY REGION TO WITHIN 2 BITS A. An Achievable Rate Region We have developed an achievable scheme meant for the symmetric rate and provided a resource hole interpretation. For the case of the capacity region, we \ufb01nd that while this interpretation can also be useful, the two-staged scheme cannot be applied. A new achievable scheme needs to be developed for the region characterization. To see this, let us consider a deterministic IC example in Fig. 9 where an in\ufb01nite number of stages need to be employed to achieve a corner point of (2, 1) with feedback. Observe that R1 R2 a2 a3 a2 a1 a1 a1 a4 a5 a6 b1 b1 \u2295a1 a3 b2 b3 b2 \u2295a3 b3 \u2295a5 a5 a3 a1 a4 \u2295a1 a3 a6 \u2295a3 a1 a3 Time 1 Time 2 Time 3 decode Time 1 Time 2 Time 3 2 1 Tx 1 Tx 2 Rx 1 Rx 2 Delay Delay Fig. 9. A deterministic IC example where an in\ufb01nite number of stages need to be employed to achieve the rate pair of (2, 1) with feedback. to guarantee R1 = 2, transmitter 1 needs to send 2 bits every time slot. Once transmitter 1 sends (a1, a2), transmitter 2 cannot use its top level since the transmission causes interference to receiver 1. It can use only the bottom level to send information. This transmission however suffers from interference: receiver 2 gets the interfered signal b1 \u2295a1. We will show that this October 29, 2018 DRAFT \f18 corrupted bit can be re\ufb01ned with feedback. In time 2, transmitter 2 can decode a1 with feedback. In an effort to achieve the rate pair of (2, 1), transmitter 1 sends (a3, a4) and transmitter 2 sends b2 on the bottom level. Now apply the same idea used in the symmetric case: transmitter 2 sends the other user\u2019s information a1 on the top level. This transmission allows receiver 2 to re\ufb01ne the corrupted signal b1 without causing interference to receiver 1, since receiver 1 already had a1 as side information. Notice that during the two time slots, receiver 1 can decode 4 bits (2 bits/time), while receiver 2 can decode 1 bits (0.5 bits/time). The point (2, 1) is not achieved yet due to unavoidable loss occurred in time 1. This loss, however, can be amortized by iterating the same operation. As this example shows, the previous two-staged scheme needs to be modi\ufb01ed so as to incorporate an in\ufb01nite number of stages. Let us apply this idea to the Gaussian channel. The use of an in\ufb01nite number of stages motivates the need for employing block Markov encoding [9], [10]. Similar to the symmetric case, we can now think of two possible schemes: (1) decode-and-forward (with message-splitting); and (2) amplify-and-forward (without message-splitting). As pointed out in Remark 4, in the Gaussian channel, private signals cannot be completely ignored, thereby incurring performance loss, thus the amplify-and-forward scheme without message-splitting has better performance. However, it requires heavy computations to compute the rate region, so we focus on the decodeand-forward scheme, although it induces a larger gap. As for a decoding operation, we employ backward decoding [11], [27]. Here is the outline of our scheme. We employ block Markov encoding with a total size B of blocks. In block 1, each transmitter splits its own message into common and private parts and then sends a codeword superimposing the common and private messages. For power splitting, we adapt the idea of the simpli\ufb01ed Han-Kobayashi scheme [8] where private power is set such that a private signal is seen below the noise level at the other receiver. In block 2, with feedback, each transmitter decodes the other user\u2019s common message (sent in block 1) while treating the other user\u2019s private signal as noise. Two common messages are then available at the transmitter: (1) its own message; and (2) the other user\u2019s message decoded with the help of feedback. Conditioned on these two common messages, each transmitter generates new common and private messages. It then sends the corresponding codeword. Each transmitter repeats this procedure until block B\u22121. In the last block B, to facilitate backward decoding, each transmitter sends the predetermined common message and a new private message. Each receiver waits until total B blocks have been October 29, 2018 DRAFT \f19 received and then performs backward decoding. We will show that this scheme enables us to obtain an achievable rate region that approximates the capacity region. Theorem 2: The feedback capacity region includes the set R of (R1, R2) such that R1 \u2264log \u0010 1 + SNR1 + INR21 + 2\u03c1 p SNR1 \u00b7 INR21 \u0011 \u22121 (20) R1 \u2264log (1 + (1 \u2212\u03c1)INR12) + log \u0012 2 + SNR1 INR12 \u0013 \u22122 (21) R2 \u2264log \u0010 1 + SNR2 + INR12 + 2\u03c1 p SNR2 \u00b7 INR12 \u0011 \u22121 (22) R2 \u2264log (1 + (1 \u2212\u03c1)INR21) + log \u0012 2 + SNR2 INR21 \u0013 \u22122 (23) R1 + R2 \u2264log \u0012 2 + SNR1 INR12 \u0013 + log \u0010 1 + SNR2 + INR12 + 2\u03c1 p SNR2 \u00b7 INR12 \u0011 \u22122 (24) R1 + R2 \u2264log \u0012 2 + SNR2 INR21 \u0013 + log \u0010 1 + SNR1 + INR21 + 2\u03c1 p SNR1 \u00b7 INR21 \u0011 \u22122 (25) for 0 \u2264\u03c1 \u22641. Proof: Our achievable scheme is generic, not limited to the Gaussian IC. We therefore characterize an achievable rate region for discrete memoryless ICs and then choose an appropriate joint distribution to obtain the desired result. In fact, this generic scheme can also be applied to El Gamal-Costa deterministic IC (to be described in Section V). Lemma 1: The feedback capacity region of the two-user discrete memoryless IC includes the set of (R1, R2) such that R1 \u2264I(U, U2, X1; Y1) (26) R1 \u2264I(U1; Y2|U, X2) + I(X1; Y1|U1, U2, U) (27) R2 \u2264I(U, U1, X2; Y2) (28) R2 \u2264I(U2; Y1|U, X1) + I(X2; Y2|U1, U2, U) (29) R1 + R2 \u2264I(X1; Y1|U1, U2, U) + I(U, U1, X2; Y2) (30) R1 + R2 \u2264I(X2; Y2|U1, U2, U) + I(U, U2, X1; Y1), (31) over all joint distributions p(u)p(u1|u)p(u2|u)p(x1|u1, u)p(x2|u2, u). Proof: See Appendix B. October 29, 2018 DRAFT \f20 Now we will choose the following Gaussian input distribution to complete the proof: \u2200k = 1, 2, U \u223cCN (0, \u03c1); Uk \u223cCN (0, \u03bbck); Xpk \u223cCN (0, \u03bbpk), (32) where Xk = U + Uk + Xkp; \u03bbck and \u03bbpk indicate the powers allocated to the common and private message of transmitter k, respectively; and (U, Uk, Xkp)\u2019s are independent. By symmetry, it suf\ufb01ces to prove (20), (21) and (24). To prove (20), consider I(U, U2, X1; Y1) = h(Y1) \u2212h(Y1|U, U2, X1). Note |KY1|X1,U2,U| = 1 + \u03bbp2INR21. (33) As mentioned earlier, for power splitting, we adapt the idea of the simpli\ufb01ed Han-Kobayashi scheme [8]. We set private power such that the private signal appears below the noise level at the other receiver. This idea mimics that of the deterministic IC example where the private bit is below the noise level so that it is invisible. The remaining power is assigned to the common message. Speci\ufb01cally, we set: \u03bbp2 = min \u0012 1 INR21 , 1 \u0013 , \u03bbc2 = 1 \u2212\u03bbp2, (34) This choice gives I(U, U2, X1; Y1) = log \u0010 1 + SNR1 + INR21 + 2\u03c1 p SNR1 \u00b7 INR21 \u0011 \u22121, (35) which proves (20). With the same power setting, we can compute: I(U1; Y2|U, X2) = log (1 + (1 \u2212\u03c1)INR12) \u22121, (36) I(X1; Y1|U, U1, U2) = log \u0012 2 + SNR1 INR12 \u0013 \u22121. (37) This proves (21). Lastly, by (35) and (37), we prove (24). Remark 6 (Three Types of Inequalities): In the non-feedback case, it is shown in [8] that an approximate capacity region is characterized by \ufb01ve types of inequalities including the bounds for 2R1+R2 and R1+2R2. In contrast, in the feedback case, our achievable rate region is described by only three types of inequalities.3 In Section VI-B, we will provide qualitative insights as to why the 2R1 + R2 bound is missing with feedback. 3It is still unknown whether or not the exact feedback capacity region includes only three types of inequalities. October 29, 2018 DRAFT \f21 Remark 7 (Connection to Related Work [27]): Our achievable scheme is essentially the same as the scheme introduced by Tuninetti [27] in a sense that the three techniques (message-splitting, block Markov encoding and backward decoding) are jointly employed.4 However, the scheme in [27] uses \ufb01ve auxiliary random variables requiring further optimization. On the other hand, we obtain an explicit rate region by reducing those \ufb01ve auxiliary random variables into three and then choosing a joint input distribution appropriately. B. An Outer Bound Region Theorem 3: The feedback capacity region is included by the set C of (R1, R2) such that R1 \u2264log \u0010 1 + SNR1 + INR21 + 2\u03c1 p SNR1 \u00b7 INR21 \u0011 (38) R1 \u2264log \u00001 + (1 \u2212\u03c12)INR12 \u0001 + log \u0012 1 + (1 \u2212\u03c12)SNR1 1 + (1 \u2212\u03c12)INR12 \u0013 (39) R2 \u2264log \u0010 1 + SNR2 + INR12 + 2\u03c1 p SNR2 \u00b7 INR12 \u0011 (40) R2 \u2264log \u00001 + (1 \u2212\u03c12)INR21 \u0001 + log \u0012 1 + (1 \u2212\u03c12)SNR2 1 + (1 \u2212\u03c12)INR21 \u0013 (41) R1 + R2 \u2264log \u0012 1 + (1 \u2212\u03c12)SNR1 1 + (1 \u2212\u03c12)INR12 \u0013 + log \u0010 1 + SNR2 + INR12 + 2\u03c1 p SNR2 \u00b7 INR12 \u0011 (42) R1 + R2 \u2264log \u0012 1 + (1 \u2212\u03c12)SNR2 1 + (1 \u2212\u03c12)INR21 \u0013 + log \u0010 1 + SNR1 + INR21 + 2\u03c1 p SNR1 \u00b7 INR21 \u0011 (43) for 0 \u2264\u03c1 \u22641. Proof: By symmetry, it suf\ufb01ces to prove the bounds of (38), (39) and (42). The bounds of (38) and (39) are nothing but cutset bounds. Hence, proving the non-cutset bound of (42) is the main focus of this proof. Also recall that this non-cutset bound is used to obtain the outer bound of (8) for the symmetric capacity in Theorem 1. We go through the proof of (38) and (39). We then focus on the proof of (42), where we will also provide insights as to the proof idea. Proof of (38): Starting with Fano\u2019s inequality, we get: N(R1 \u2212\u01ebN) \u2264I(W1; Y N 1 ) (a) \u2264 X [h(Y1i) \u2212h(Z1i)], 4The author in [27] considers a different context: the conferencing encoder problem. However, Prabhakaran and Viswanath [19] have made an interesting connection between the feedback problem and the conferencing encoder problem. See [19] for details. October 29, 2018 DRAFT \f22 where (a) follows from the fact that conditioning reduces entropy. Assume that X1 and X2 have covariance \u03c1, i.e., E[X1X\u2217 2] = \u03c1. Then, we get: h(Y1) \u2264log 2\u03c0e \u0010 1 + SNR1 + INR21 + 2|\u03c1| p SNR1 \u00b7 INR21 \u0011 . (44) If (R1, R2) is achievable, then \u01ebN \u21920 as N \u2192\u221e. Therefore, we get the desired bound: R1 \u2264h(Y1) \u2212h(Z1) \u2264log \u0010 1 + SNR1 + INR21 + 2|\u03c1| p SNR1 \u00b7 INR21 \u0011 . (45) Proof of (39): Starting with Fano\u2019s inequality, we get: N(R1 \u2212\u01ebN) \u2264I(W1; Y N 1 , Y N 2 , W2) (a) = X [h(Y1i, Y2i|W2, Y i\u22121 1 , Y i\u22121 2 ) \u2212h(Z1i) \u2212h(Z2i)] (b) = X [h(Y1i, Y2i|W2, Y i\u22121 1 , Y i\u22121 2 , Xi 2) \u2212h(Z1i) \u2212h(Z2i)] (c) = X [h(Y2i|W2, Y i\u22121 1 , Y i\u22121 2 , Xi 2) \u2212h(Z2i)] + X [h(Y1i|W2, Y i\u22121 1 , Y i\u22121 2 , Xi 2, Y2i, Si 1) \u2212h(Z1i)] (d) \u2264 X [h(Y2i|X2i) \u2212h(Z2i) + h(Y1i|X2i, S1i) \u2212h(Z1i)] where (a) follows from the fact that W1 is independent from W2 and h(Y N 1 , Y N 2 |W1, W2) = h(Y N 1 , SN 1 |W1, W2) = P[h(Z1i) + h(Z2i)] (see Claim 1); (b) follows from the fact that Xi 2 is a function of (W2, Y i\u22121 2 ); (c) follows from the fact that Si 1 is a function of (Y i 2, Xi 2); (d) follows from the fact that conditioning reduces entropy. Hence, we get the desired result: R1 \u2264h(Y2|X2) \u2212h(Z2) + h(Y1|X2, S1) \u2212h(Z1) (a) \u2264log \u00001 + (1 \u2212|\u03c1|2)INR12 \u0001 + log \u0012 1 + (1 \u2212|\u03c1|2)SNR1 1 + (1 \u2212|\u03c1|2)INR12 \u0013 where (a) follows from the fact that h(Y2|X2) \u2264log 2\u03c0e \u00001 + (1 \u2212|\u03c1|2)INR12 \u0001 , (46) h(Y1|X2, S1) \u2264log 2\u03c0e \u0012 1 + (1 \u2212|\u03c1|2)SNR1 1 + (1 \u2212|\u03c1|2)INR12 \u0013 . (47) The inequality of (47) is obtained as follows. Given (X2, S1), the variance of Y1 is upper-bounded by Var [Y1|X2, S1] \u2264KY1 \u2212KY1(X2,S1)K\u22121 (X2,S1)K\u2217 Y1(X2,S1), October 29, 2018 DRAFT \f23 where KY1 = E \u0002 |Y1|2\u0003 = 1 + SNR1 + INR21 + \u03c1g\u2217 11g21 + \u03c1\u2217g11g\u2217 21, KY1(X2,S1) = E [Y1[X\u2217 2, S\u2217 1]] = [\u03c1g11 + g21, g\u2217 12g11 + \u03c1\u2217g21g\u2217 12] , K(X2,S1) = E \uf8ee \uf8f0 \uf8ee \uf8f0|X2|2 X2S\u2217 1 X\u2217 2S1 |S1|2 \uf8f9 \uf8fb \uf8f9 \uf8fb= \uf8ee \uf8f0 1 \u03c1\u2217g\u2217 12 \u03c1g12 1 + INR12 \uf8f9 \uf8fb. (48) By further calculation, we can get (47). Proof of (42): The proof idea is based on the genie-aided argument [13]. However, \ufb01nding an appropriate genie is not simple since there are many possible combinations of the random variables. The deterministic IC example in Fig. 6 (b) gives insights into this. Note that providing a1 and (b1, b2, b3) to receiver 1 does not increase the rate R1, i.e., these are useless gifts. This may motivate us to choose a genie as (g12X1, W2). However, in the Gaussian channel, providing g12X1 is equivalent to providing X1. This is of course too much information, inducing a loose upper bound. Inspired by the technique in [8], we instead consider a noisy version of g12X1: S1 = g12X1 + Z2. (49) Intuition behind this is that we cut off g12X1 at the noise level. Indeed this matches intuition in the deterministic IC. This genie together with W2 turns out to lead to the desired tight upper bound. On top of the genie-aided argument, we need more techniques. In the feedback problem, the functional relationship between the random variables is more complicated and thus needs be well explored. We identify several functional relationships through Claims 1, 2 and 3, which turn out to play a signi\ufb01cant role in proving the non-cutset bound of (42). October 29, 2018 DRAFT \f24 Starting with Fano\u2019s inequality, we get: N(R1 + R2 \u2212\u01ebN) \u2264I(W1; Y N 1 ) + I(W2; Y N 2 ) (a) \u2264I(W1; Y N 1 , SN 1 , W2) + I(W2; Y N 2 ) (b) = h(Y N 1 , SN 1 |W2) \u2212h(Y N 1 , SN 1 |W1, W2) + I(W2; Y N 2 ) (c) = h(Y N 1 , SN 1 |W2) \u2212 X [h(Z1i) + h(Z2i)] + I(W2; Y N 2 ) (d) = h(Y N 1 |SN 1 , W2) \u2212 X h(Z1i) + h(Y N 2 ) \u2212 X h(Z2i) (e) = h(Y N 1 |SN 1 , W2, XN 2 ) \u2212 X h(Z1i) + h(Y N 2 ) \u2212 X h(Z2i) (f) \u2264 N X i=1 [h(Y1i|S1i, X2i) \u2212h(Z1i) + h(Y2i) \u2212h(Z2i)] where (a) follows from the fact that adding information increases mutual information (providing a genie); (b) follows from the independence of W1 and W2; (c) follows from h(Y N 1 , SN 1 |W1, W2) = P [h(Z1i) + h(Z2i)] (see Claim 1); (d) follows from h(SN 1 |W2) = h(Y N 2 |W2) (see Claim 2); (e) follows from the fact that XN 2 is a function of (W2, SN\u22121 1 ) (see Claim 3); (f) follows from the fact that conditioning reduces entropy. Hence, we get R1 + R2 \u2264h(Y1|S1, X2) \u2212h(Z1) + h(Y2) \u2212h(Z2). Note that h(Y2) \u2264log 2\u03c0e \u0010 1 + SNR2 + INR12 + 2|\u03c1| p SNR2 \u00b7 INR12 \u0011 . (50) From (47) and (50), we get the desired upper bound. Claim 1: h(Y N 1 , SN 1 |W1, W2) = P [h(Z1i) + h(Z2i)] . Proof: h(Y N 1 , SN 1 |W1, W2) = X h(Y1i, S1i|W1, W2, Y i\u22121 1 , Si\u22121 1 ) (a) = X h(Y1i, S1i|W1, W2, Y i\u22121 1 , Si\u22121 1 , X1i, X2i) (b) = X h(Z1i, Z2i|W1, W2, Y i\u22121 1 , Si\u22121 1 , X1i, X2i) (c) = X [h(Z1i) + h(Z2i)] , where (a) follows from the fact that X1i is a function of (W1, Y i\u22121 1 ) and X2i is a function of (W2, Si\u22121 1 ) (by Claim 3); (b) follows from the fact that Y1i = g11X1i + g21X2i + Z1i and October 29, 2018 DRAFT \f25 S1i = g12X1i+Z2i; (c) follows from the memoryless property of the channel and the independence assumption of Z1i and Z2i. Claim 2: h(SN 1 |W2) = h(Y N 2 |W2). Proof: h(Y N 2 |W2) = X h(Y2i|Y i\u22121 2 , W2) (a) = X h(S1i|Y i\u22121 2 , W2) (b) = X h(S1i|Y i\u22121 2 , W2, Xi 2, Si\u22121 1 ) (c) = X h(S1i|W2, Si\u22121 1 ) = h(SN 1 |W2), where (a) follows from the fact that Y2i is a function of (X2i, S1i) and X2i is a function of (W2, Y i\u22121 2 ); (b) follows from the fact that Xi 2 is a function of (W2, Y i\u22121 2 ) and Si\u22121 1 is a function of (Y i\u22121 2 , Xi\u22121 2 ); (c) follows from the fact that Y i\u22121 2 is a function of (Xi\u22121 2 , Si\u22121 1 ) and Xi 2 is a function of (W2, Si\u22121 1 ) (by Claim 3). Claim 3: For all i \u22651, Xi 1 is a function of (W1, Si\u22121 2 ) and Xi 2 is a function of (W2, Si\u22121 1 ). Proof: By symmetry, it is enough to prove only one. Notice that Xi 2 is a function of (W2, Y i\u22121 2 ) and Y i\u22121 2 is a function of (Xi\u22121 2 , Si\u22121 1 ). Hence, Xi 2 is a function of (W2, Xi\u22121 2 , Si\u22121 1 ). Iterating the same argument, we conclude that Xi 2 is a function of (W2, X21, Si\u22121 1 ). Since X21 depends only on W2, we complete the proof. C. 2-Bit Gap to the Capacity Region Theorem 4: The gap between the inner and upper bound regions (given in Theorems 2 and 3) is at most 2 bits/s/Hz/user: R \u2286C \u2286C \u2286R \u2295([0, 2] \u00d7 [0, 2]) . (51) Proof: The proof is immediate by Theorem 2 and 3. We de\ufb01ne \u03b41 to be the difference between min {(38), (39)} and min {(20), (21)}. Similarly, we de\ufb01ne \u03b42 and \u03b412. Straightforward computation gives \u03b41 \u2264max \u001a 1, log \u0012 1 + SNR1 1 + INR12 \u0013 \u2212log \u0012 2 + SNR1 INR12 \u0013 + 2 \u001b \u22642. Similarly, we get \u03b42 \u22642 and \u03b412 \u22642. This completes the proof. October 29, 2018 DRAFT \f26 Remark 8 (Why does a 2-bit gap occur?): The achievable scheme meant for the capacity region involves message-splitting. As mentioned in Remark 4, message-splitting incurs some loss in the process of decoding the common message while treating private signals as noise. Accounting for the effect of private signals, the effective noise power becomes double, thus incurring a 1-bit gap. The other 1-bit gap comes from a relay structure of the feedback IC. To see this, consider an extreme case where user 2\u2019s rate is completely ignored. In this case, we can view the [transmitter2, receiver2] communication pair as a single relay which only helps the [transmitter1, receiver1] communication pair. It has been shown in [6] that for this single relay Gaussian channel, the worst-case gap between the best known inner bound [9] and the outer bound is 1 bit/s/Hz. This incurs the other 1-bit gap. This 2-bit gap is based on the outer bound region in Theorem 3, which allows for arbitrary correlation between the transmitters. So, one can expect that an actual gap to the capacity region is less than 2 bits. Remark 9 (Reducing the gap): As discussed, the amplify-and-forward scheme has the potential to reduce the gap. However, due to the inherent relay structure, reducing the gap into a less-than-one bit is challenging. As long as no signi\ufb01cant progress is made on the single relay Gaussian channel, one cannot easily reduce the gap further. Remark 10 (Comparison with the two-staged scheme): Specializing to the symmetric rate, it can be shown that the in\ufb01nite-staged scheme in Theorem 2 can achieve the symmetric capacity to within 1 bit. Coincidentally, this gap matches the gap result of the two-staged scheme in Theorem 1. However, the 1-bit gap comes from different reasons. In the in\ufb01nite-staged scheme, the 1-bit gap comes from message-splitting. In contrast, in the two-staged scheme, the gap is due to lack of beamforming gain. One needs to come up with a new technique that well combines these two schemes to reduce the gap into a less-than-one bit. V. THE FEEDBACK CAPACITY REGION OF EL GAMAL-COSTA MODEL We have so far made use of the linear deterministic IC to provide insights into approximating the feedback capacity region of the Gaussian IC. The deterministic IC is a special case of El Gamal-Costa deterministic IC [13]. In this section, we establish the exact feedback capacity region for this general class of deterministic ICs. Fig. 10 (a) illustrates El Gamal-Costa deterministic IC with feedback. The key condition of October 29, 2018 DRAFT \f27 g2 f1 f2 X1 X2 V1 V2 Y1 Y2 g1 W1 W2 \u02c6 W1 \u02c6 W2 Delay Delay Enc 1 Enc 2 Dec 1 Dec 2 Fig. 10. El Gamal-Costa deterministic IC with feedback this model is given by H(V2|Y1, X1) = 0, H(V1|Y2, X2) = 0, (52) where Vk is a part of Xk (k = 1, 2), visible to the other receiver. This implies that in any working system where X1 and X2 are decodable at receivers 1 and 2 respectively, V1 and V2 are completely determined at receivers 2 and 1, respectively, i.e., these are common signals. Theorem 5: The feedback capacity region of El Gamal-Costa deterministic IC is the set of (R1, R2) such that R1 \u2264min {H(Y1), H(Y2|X2, U) + H(Y1|V1, V2, U)} R2 \u2264min {H(Y2), H(Y1|X1, U) + H(Y2|V1, V2, U)} R1 + R2 \u2264min {H(Y1|V1, V2, U) + H(Y2), H(Y2|V2, V1, U) + H(Y1)} for some joint distribution p(u, x1, x2) = p(u)p(x1|u)p(x2|u). Here U is a discrete random variable which takes on values in the set U where |U| \u2264min(|V1||V2|, |Y1|, |Y2|) + 3. Proof: Achievability proof is straightforward by Lemma 1. Let Uk = Vk, \u2200k. Fix a joint distribution p(u)p(u1|u)p(u2|u)p(x1|u1, u)p(x2|u2, u). We now write a joint distribution p(u, x1, x2, u1, u2) October 29, 2018 DRAFT \f28 in two different ways: p(u, x1, x2, u1, u2) = p(u)p(x1|u)p(x2|u)\u03b4(u1 \u2212g1(x1))\u03b4(u2 \u2212g2(x2) = p(u)p(u1|u)p(u2|u)p(x1|u1, u)p(x2|u2, u) (53) where \u03b4(\u00b7) indicates the Kronecker delta function. This gives p(x1|u) := p(x1|u1, u)p(u1|u) \u03b4(u1 \u2212g1(x1)) p(x2|u) := p(x2|u2, u)p(u2|u) \u03b4(u2 \u2212g2(x2)) (54) Now we can generate a joint distribution p(u)p(x1|u)p(x2|u). Hence, we complete the achievability proof. See Appendix C for converse proof. As a by-product, we obtain the feedback capacity region of the linear deterministic IC. Corollary 1: The feedback capacity region of the linear deterministic IC is the set of (R1, R2) such that R1 \u2264min {max(n11, n12), max(n11, n21)} R2 \u2264min {max(n22, n21), max(n22, n12)} R1 + R2 \u2264min \b max(n22, n12) + (n11 \u2212n12)+, max(n11, n21) + (n22 \u2212n21)+\t . Proof: The proof is straightforward by Theorem 5. The capacity region is achieved when U is constant; and X1 and X2 are independent and uniformly distributed. VI. ROLE OF FEEDBACK Recall in Fig. 1 that feedback gain is bounded for 0 \u2264\u03b1 \u22642 3 in terms of the symmetric rate. So a natural question arises: is feedback gain marginal also from a perspective of the capacity region? With the help of Corollary 1, we show that feedback can provide multiplicative gain even in this regime. We also answer another interesting question posed in Section IV: why is the 2R1 + R2 bound missing with feedback? October 29, 2018 DRAFT \f29 Fig. 11. Feedback capacity region of the linear deterministic IC A. Feedback Gain from a Capacity Region Perspective Fig. 11 shows the feedback capacity region of the linear deterministic IC under the symmetric channel setting: n = n11 = n22 and m = n12 = n21. Interestingly, while for 2 3 \u2264\u03b1 \u22642, the symmetric capacity does not improve with feedback, the feedback capacity region is enlarged even for this regime. B. Why is 2R1 + R2 Bound Missing with Feedback? Consider an example where 2R1 + R2 bound is active in the non-feedback case. Fig. 12 (a) shows an example where a corner point of (3, 0) can be achieved. Observe that at the two receivers, the \ufb01ve signal levels are consumed out of the six signal levels. There is one resource hole. This resource hole is closely related to the 2R1+R2 bound, which will be shown in Fig. 12 (b). Suppose the 2R1 + R2 bound is active. This implies that if R1 is reduced by 1 bit, then R2 should be increased by 2 bits. To decrease R1 by 1 bit, suppose that transmitter 1 sends no information on the second signal level. We then see the two empty signal levels at the two October 29, 2018 DRAFT \f30 R1 R2 Tx 1 Tx 2 Rx 1 Rx 2 3 3 a2 a3 a2 a1 a1 a3 a1 a2 Resource Hole 2R1 + R2 \u22646 (a) A resource hole vs. 2R1 + R2 bound R1 R2 Tx 1 Tx 2 Rx 1 Rx 2 3 3 2R1 + R2 \u22646 a3 a1 a1 a3 a1 Must be a resource hole. (b) Whenever 2R1 + R2 is active, there is a resource hole. R1 R2 a2 a3 a2 Tx 1 Tx 2 Rx 1 Rx 2 a1 a1 a4 a5 a6 a3 b1 a1 a4 a6 a2 a2 a2 b2 a8 a9 a7 a9 a4 a5 a5 a5 a7 a7 3 3 1 b1 \u2295a2 a8 \u2295a5 b2 \u2295a5 a5 \u2295a2 Time 1 Time 2 Time 3 decode Time 1 Time 2 Time 3 b3 \u2295a8 b3 Delay Delay (c) Feedback \ufb01lls up all resource holes to maximize resource utilization Fig. 12. Relationship between a resource hole and 2R1 + R2 bound. The 2R1 + R2 bound is missing with feedback. receivers (marked as the gray balls): one at the second level at receiver 1; the other at the bottom level at receiver 2. Now transmitter 2 can send 1 bit on the bottom level to increase R2 by 1 bit (marked as the thick red line). Also it allows transmitter 2 to send one more bit on the top level. This implies that the top level at receiver 2 must be a resource hole in the previous case. This observation combined with the following observation can give an answer to the question. Fig. 12 (c) shows the feedback role that it \ufb01lls up all the resource holes to maximize resource utilization. We employ the same feedback strategy used in Fig. 9 to obtain the result in Fig. 12 (c). Notice that with feedback, all of the resource holes are \ufb01lled up except a hole in the \ufb01rst stage, which can be amortized by employing an in\ufb01nite number of stages. Therefore, we can now see why the 2R1 + R2 bound is missing with feedback. October 29, 2018 DRAFT \f31 VII. DISCUSSION A. Comparison to Related Work [15], [16], [17] For the symmetric Gaussian IC, Kramer [15], [16] developed a feedback strategy based on Schalkwijk-Kailath scheme [28] and Ozarow\u2019s scheme [5]. Due to lack of closed-form rateformula for the scheme, we cannot see how Kramer\u2019s scheme is close to our symmetric rate in Theorem 1. To see this, we compute the generalized degrees-of-freedom of Kramer\u2019s scheme. Lemma 2: The generalized degrees-of-freedom of Kramer\u2019s scheme is given by d(\u03b1) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 \u2212\u03b1, 0 \u2264\u03b1 < 1 3; 3\u2212\u03b1 4 , 1 3 \u2264\u03b1 < 1; 1+\u03b1 4 , \u03b1 \u22651. (55) Proof: See Appendix D. \u001e \u001f ! \" # $ % & Fig. 13. Generalized degrees-of-freedom comparison 0 0.5 1 1.5 2 5 10 15 20 25 30 log(INR)/log(SNR) Rsym (bits/s/Hz) SNR = 40 dB Gastpar\u2212Kramer Outer Bound New Outer Bound Proposed Scheme Kramer Scheme Fig. 14. The symmetric rate comparison Note in Fig. 13 that Kramer\u2019s scheme can be arbitrarily far from optimality, i.e., it has an unbounded gap to the symmetric capacity for all values of \u03b1 except \u03b1 = 1. We also plot the symmetric rate for \ufb01nite channel parameters as shown in Fig. 14. Notice that Kramer\u2019s scheme is very close to the outer bounds only when INR is similar to SNR. In fact, the capacity theorem in [16] says that they match each other at INR = SNR \u2212 \u221a 2SNR. However, if INR is quite different from SNR, it becomes far away from the outer bounds. Also note that our new bound is much tighter than Gastpar-Kramer\u2019s outer bounds in [15], [17]. October 29, 2018 DRAFT \f32 B. Closing the Gap Less than 1-bit gap to the symmetric capacity: Fig. 14 implies that our achievable scheme can be improved especially when \u03b1 \u22481 where beamforming gain plays a signi\ufb01cant role. As mentioned earlier, our two-staged scheme completely loses beamforming gain. In contrast, Kramer\u2019s scheme captures the beamforming gain. As discussed in Remark 10, one may develop a uni\ufb01ed scheme that beats both the schemes for all channel parameters to reduce the worst-case gap. Less than 2-bit gap to the capacity region: As mentioned in Remark 8, a 2-bit gap to the feedback capacity region can be improved up to a 1-bit gap. The idea is to remove message splitting. Recall that the Alamouti-based amplify-and-forward scheme in Theorem 1 improves the performance by removing message splitting. Translating the same idea to the characterization of the capacity region is needed for the improvement. A noisy binary expansion model in Fig. 16 may give insights into this. C. Extension to Gaussian MIMO ICs with Feedback The feedback capacity result for El Gamal-Costa model can be extended to Teletar-Tse IC [14] where in Fig. 10, fk\u2019s are deterministic functions satisfying El Gamal-Costa condition (52) while gk\u2019s follow arbitrary probability distributions. Once extended, one can infer an approximate feedback capacity region of the two-user Gaussian MIMO IC, as [14] did in the non-feedback case. VIII." + }, + { + "url": "http://arxiv.org/abs/1004.4663v1", + "title": "On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage", + "abstract": "The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes\nhas recently motivated a new class of codes, called Regenerating Codes, that\noptimally trade off storage cost for repair bandwidth. In this paper, we\naddress bandwidth-optimal (n,k,d) Exact-Repair MDS codes, which allow for any\nfailed node to be repaired exactly with access to arbitrary d survivor nodes,\nwhere k<=d<=n-1. We show the existence of Exact-Repair MDS codes that achieve\nminimum repair bandwidth (matching the cutset lower bound) for arbitrary\nadmissible (n,k,d), i.e., k 1 2 or k \u2264d < 2k \u22121? The work in [4] sheds some light on this case: speci\ufb01cally, it was shown that under scalar linear codes, when either k n > 1 2 + 2 n or k + 1 \u2264d \u2264max(k + 1, 2k \u22124), there is a price for exact repair. What if non-linear or vector linear codes are used? The tightness of the optimal tradeoff of (1) under these assumptions has remained open. In this paper, we show that using vector linear codes, the optimal tradeoff of (1) can be indeed attained for all admissible values of (n, k, d), i.e., k < n and k \u2264d \u2264n\u22121. That is if we are willing to deal with arbitrarily small subsymbols, then Exact-Repair MDS codes can come with no loss of optimality over functional-repair MDS codes. Note that we will use this de\ufb01nition of admissibility throughout the paper. Our achievable scheme builds on the concept of interference alignment, which was introduced 1In [1], Dimakis-Godfrey-Wu-Wainwright-Ramchandran translated the regenerating-codes problem into a multicast communication problem where random-network-coding-based schemes require a huge \ufb01eld size especially for large networks. In storage problems, the \ufb01eld size issue is further aggravated by the need to support a dynamically expanding network size due to the need for continual repair. 2In scalar linear codes, symbols are not allowed to be split into arbitrarily small subsymbols as with vector linear codes. October 4, 2018 DRAFT \f4 in the context of wireless communication networks [6], [7]. In particular, the interference alignment scheme in [7] that permits an arbitrarily large number of symbol extensions (i.e., vector linear codes) forms the basis of our results here. The results in [4] say that under scalar linear codes, the case of either k n > 1 2+ 2 n or k+1 \u2264d \u2264max(k+1, 2k\u22124) induces more constraints than the available number of design variables. This parallels the problem encountered by Cadambe and Jafar in [7] in the conceptually similar but physically different context of wireless interference channels. Cadambe and Jafar resolve this issue in [7] using the idea of symbol-extension, which is analogous to the idea of vector linear codes for the distributed storage repair problem studied here. Building on the connection described in [5] between the wireless interference and the distributed storage repair problems, we leverage the scheme introduced in [7] to the repair problem, showing the existence of Exact-Repair MDS codes that achieve minimum repair bandwidth (matching the cutset lower bound) for all admissible values of (n, k, d). II. INTERFERENCE ALIGNMENT FOR EXACT-REPAIR MDS CODES Linear network coding [8], [9] (that allows multiple messages to be linearly combined at network nodes) has been established recently as a useful tool for addressing interference issues even in wireline networks where all the communication links are orthogonal and non-interfering. This attribute was \ufb01rst observed in [10], where it was shown that interference alignment could be exploited for storage networks, speci\ufb01cally for Exact-Repair MDS codes having small k (k = 2). However, generalizing interference alignment to large values of k (even k = 3) proves to be challenging, as we describe in the sequel. In order to appreciate this better, let us \ufb01rst review the scheme of [10] that was applied to the exact repair problem. We will then address the dif\ufb01culty of extending interference alignment for larger systems and describe how to address this in Section III. A. Review of (4, 2) Exact-Repair MDS Codes [10] Fig. 1 illustrates an interference alignment scheme for a (4, 2) Exact-Repair MDS code de\ufb01ned over GF(5). First one can easily check the MDS property of the code, i.e., all the source \ufb01les can be reconstructed from any k(= 2) nodes out of n(= 4) nodes. As an illustration, let us see how failed node 1 (storing (a1, a2)) can be exactly repaired. We assume that the degree d is 3, October 4, 2018 DRAFT \f5 a1 + b1 1 1 b2 b1 1 1 a2 a1 a1 a2 b1 b2 a1 a2 b1 + b2 1 2a1 + a2 + b1 + b2 2a2 + b2 2a1 + b1 a2 + b2 1 a1 + 2a2 + b1 + b2 A source \ufb01le Encoded packets How to repair? Interference alignment failed node 1 node 2 node 3 node 4 (parity node 1) (parity node 2) Fig. 1. Interference alignment for a (4, 2) Exact-Repair MDS code de\ufb01ned over GF(5) [10]. Designing appropriate projection vectors, we can align interference space of (b1, b2) into one-dimensional linear space spanned by [1, 1]t. As a result, we can successfully decode 2 desired unknowns (a1, a2) from 3 equations containing 4 unknowns (a1, a2, b1, b2). and a source \ufb01le size M is 4. The cutset bound (1) then gives the fundamental limits of: storage cost \u03b1 = 2; and repair-bandwidth-per-link \u03b2 := \u03b3 d = 1. The example illustrated in Fig. 1 shows that the parameter set described above is achievable using interference alignment. Here is a summary of the scheme. Recall that the bandwidthper-link is \u03b2 = 1 and we use a scalar linear code, i.e., each symbol has unit capacity and cannot be split into arbitrarily small subsymbols. Hence, each survivor node uses a projection vector to project its data into a scalar. Choosing appropriate projection vectors, we get the equations as shown in Fig. 1: (b1 + b2); a1 + 2a2 + (b1 + b2); 2a1 + a2 + (b1 + b2). Observe that the undesired signals (b1, b2) (interference) are aligned onto an 1-dimensional linear subspace, thereby achieving interference alignment. Therefore, we can successfully decode (a1, a2) with three equations although there are four unknowns. Similarly, we can repair (b1, b2) when it has failed. For parity node repair, a remapping technique is introduced. The idea is to de\ufb01ne parity node October 4, 2018 DRAFT \f6 A1B\u22121 1 v\u03b11 A2B\u22121 1 v\u03b11 rank \u0001\u0002 A1B\u22121 1 v\u03b11 A2B\u22121 2 v\u03b11 \u0003\u0004 = 2 v\u03b11 B1v\u03b12 B2v\u03b13 v\u03b12 = B\u22121 1 v\u03b11 v\u03b13 = B\u22121 2 v\u03b11 v\u03b11 atA1 + btB1 atA2 + btB2 at bt v\u03b12 v\u03b13 Interference alignment Projection Vectors node 1 node 2 node 3 node 4 (parity node 1) (parity node 2) \uf8ee \uf8f0 vt \u03b11b (A1v\u03b12)ta + (B1v\u03b12)tb (A2v\u03b13)ta + (B2v\u03b13)tb \uf8f9 \uf8fb= \uf8ee \uf8f0 0 (A1v\u03b12)t (A2v\u03b13)t \uf8f9 \uf8fba + \uf8ee \uf8f0 vt \u03b11 (B1v\u03b12)t (B2v\u03b13)t \uf8f9 \uf8fbb Fig. 2. Geometric interpretation of interference alignment. The blue solid-line and red dashed-line vectors indicate linear subspaces with respect to \u201ca\u201d and \u201cb\u201d, respectively. The choice of v\u03b12 = B\u22121 1 v\u03b11 and v\u03b13 = B\u22121 2 v\u03b11 enables interference alignment. For the speci\ufb01c example of Fig. 1, the corresponding encoding matrices are A1 = [1, 0; 0, 2], B1 = [1, 0; 0, 1]. A2 = [2, 0; 0, 1], B2 = [1, 0; 0, 1]. symbols with new variables as follows: Node 3: a\u2032 1 := a1 + b1; a\u2032 2 := 2a2 + b2; Node 4: b\u2032 1 := 2a1 + b1; b\u2032 2 := a2 + b2. We can then rewrite (a1, a2) and (b1, b2) with respect to (a\u2032 1, a\u2032 2) and (b\u2032 1, b\u2032 2). In terms of prime notation, parity nodes turn into systematic nodes and vice versa. With this remapping, one can easily design projection vectors for exact repair of parity nodes. B. Geometric Interpretation Using matrix notation, we provide geometric interpretation of interference alignment for the same example in Fig. 1. Let a = (a1, a2)t and b = (b1, b2)t be 2-dimensional information-unit vectors, where (\u00b7)t indicates a transpose. Let Ai and Bi be 2-by-2 encoding submatrices for parity node i (i = 1, 2). Finally we de\ufb01ne 2-dimensional projection vectors v\u03b1i\u2019s (i = 1, 2, 3). Let us consider exact repair of systematic node 1. By connecting to three nodes, we get: btv\u03b11; at(A1v\u03b12) + bt(B1v\u03b12); at(A2v\u03b13) + bt(B2v\u03b13). Recall the goal of decoding 2 desired October 4, 2018 DRAFT \f7 unknowns out of 3 equations including 4 unknowns. To achieve this goal, we need: rank \uf8eb \uf8ed \uf8ee \uf8f0(A1v\u03b12)t (A2v\u03b13)t \uf8f9 \uf8fb \uf8f6 \uf8f8= 2; rank \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 vt \u03b11 (B1v\u03b12)t (B2v\u03b13)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8= 1. (2) The second condition can be met by setting v\u03b12 = B\u22121 1 v\u03b11 and v\u03b13 = B\u22121 2 v\u03b11. This choice forces the interference space to be collapsed into a one-dimensional linear subspace, thereby achieving interference alignment. With this setting, the \ufb01rst condition now becomes rank \u0000\u0002 A1B\u22121 1 v\u03b11 A2B\u22121 2 v\u03b11 \u0003\u0001 = 2. (3) It can be easily veri\ufb01ed that the choice of Ai\u2019s and Bi\u2019s given in Figs. 1 and 2 guarantees the above condition. When the node 2 fails, we get a similar condition: rank \u0000\u0002 B1A\u22121 1 v\u03b21 B2A\u22121 2 v\u03b21 \u0003\u0001 = 2, (4) where v\u03b2i\u2019s denote projection vectors for node 2 repair. This condition also holds under the given choice of encoding matrices. With this remapping, one can easily design projection vectors for exact repair of parity nodes. C. Connection with Interference Channels in Communication Problems Observe the three equations shown in Fig. 2: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 0 (A1v\u03b12)t (A2v\u03b13)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fba | {z } desired signals + \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 vt \u03b11 (B1v\u03b12)t (B2v\u03b13)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fbb | {z } interference . Separating into two parts, we can view this problem as a wireless communication problem, wherein a subset of the information is desired to be decoded in the presence of interference. Note that for each term (e.g., A1v\u03b12), the matrix A1 and vector v\u03b12 correspond to channel matrix and transmission vector in wireless communication problems, respectively. There are, however, signi\ufb01cant differences. In the wireless communication problem, the channel matrices are provided by nature and therefore not controllable. The transmission strategy alone (vector variables) can be controlled for achieving interference alignment. On the other hand, in our storage repair problems, both matrices and vectors are controllable, i.e., projection October 4, 2018 DRAFT \f8 vectors and encoding matrices can be arbitrarily designed, resulting in more \ufb02exibility. However, our storage repair problem comes with unparalleled challenges due to the MDS requirement and the multiple failure con\ufb01gurations. These induce multiple interference alignment constraints that need to be simultaneously satis\ufb01ed. What makes this dif\ufb01cult is that the encoding matrices, once designed, must be the same for all repair con\ufb01gurations. This is particularly acute for large values of k (even k = 3), as the number of possible failure con\ufb01gurations increases with n (which increases with k). III. A PROPOSED FRAMEWORK FOR EXACT-REPAIR MDS CODES We propose a conceptual framework based on vector linear codes to address the exact repair problem. As described earlier, this framework is based on that of interference alignment for wireless channels in [7]. We leverage the connection between the two problems to develop ExactRepair MDS codes that are optimal in repair bandwidth for all admissible values of (n, k, d). Our framework consists of four components: (1) developing a code structure for exact repair of systematic nodes based on the vector linear codes; (2) drawing a dual structure between the systematic and parity node repair; (3) guaranteeing the MDS-code property; (4) providing a probabilistic guarantee of the existence of the code for a large enough alphabet size. In particular, the diagonal structure of single-antenna wireless channels (exploited in [7]) forms the basis of the structure of encoding submatrices of our codes. The framework covers all admissible values of (n, k, d). This contrasts the scalar-linear code based framework in [5] which covers the case of k n \u22641 2 and d \u22652k \u22121, but which provides deterministic codes with small alphabet size and guaranteed zero error. Furthermore, addressing different code parameters in the case of k n \u22641 2 and d \u22652k \u22121 requires speci\ufb01c attention, such as the design of puncturing codes introduced in [4]. See [5] for details. In contrast, here we target only the existence of exact-repair codes without specifying constructions. This allows for a simpler characterization of the solution space for the entire range of admissible repair code parameters. In order to convey the concepts in a clear and concise manner, we \ufb01rst focus on the simplest example which does not belong to the framework in [5]: (6, 3, 4) Exact-Repair MDS codes. This example is a representative of the general case of k < n and k \u2264d \u2264n\u22121, with the generalization following in a straightforward way from this example. This will be discussed in Section IV. October 4, 2018 DRAFT \f9 v\u03b11 v\u03b12 v\u03b13 v\u03b14 Interference alignment v\u03b13 = B\u22121 1 v\u03b11 v\u03b14 = B\u22121 2 v\u03b11 at bt ct (C1B\u22121 1 )v\u03b11 (C2B\u22121 2 )v\u03b11 v\u03b12 Goal: rank=2 rank=1 rank=1 atA1 + btB1 + ctC1 atA2 + btB2 + ctC2 atA3 + btB3 + ctC3 \uf8ee \uf8ef \uf8ef \uf8f0 0 0 (A1v\u03b13)t (A2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fba + \uf8ee \uf8ef \uf8ef \uf8f0 vt \u03b11 0 (B1v\u03b13)t (B2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fbb + \uf8ee \uf8ef \uf8ef \uf8f0 0 vt \u03b12 (C1v\u03b13)t (C2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fbc Fig. 3. Dif\ufb01culty of achieving interference alignment simultaneously when using scalar linear codes A. Systematic Node Repair For k \u22653 (more-than-two interfering information units), achieving interference alignment for exact repair turns out to be signi\ufb01cantly more complex than the k = 2 case. Fig. 3 illustrates this dif\ufb01culty through the example of repairing node 1 for a (6, 3, 4) code. In accordance with the (4, 2) code example in Figs. 1 and 2, we choose M = 6 so that repair-bandwidth-per-link has unit capacity (\u03b2 := \u03b3 d = 1). By the optimal tradeoff of (1), this gives \u03b1 = 2. Suppose that we use scalar linear codes, i.e., each symbol has unit capacity and cannot be chopped up into arbitrarily smaller chunks. We de\ufb01ne a = (a1, a2)t, b = (b1, b2)t and c = (c1, c2)t. We de\ufb01ne 2-by-2 encoding submatrices of Ai, Bi and Ci (for i = 1, 2, 3); and 2-dimensional projection vectors v\u03b1i\u2019s. Suppose that survivor nodes (2, 3, 4, 5) are connected for exact repair of node 1. We then get the 4 (= d) equations: \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 0 (A1v\u03b13)t (A2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb a + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 vt \u03b11 0 (B1v\u03b13)t (B2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb b + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 vt \u03b12 (C1v\u03b13)t (C2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb c. October 4, 2018 DRAFT \f10 a1 a2 Scalar linear code 1 a1 Vector linear code a2 aB aB+1 aB+2 a2B 1 B symbol subsymbol Fig. 4. Illustration of the idea of vector linear codes through storage node 1 in the (6, 3, 4) code example. In scalar linear codes, symbols are not allowed to be split. On the other hand, vector linear codes allow to split symbols into arbitrarily small subsymbols. In this example, node 1 stores \u03b1 = 2 symbols, each of which has unit capacity. In vector linear codes, this unitcapacity symbol can be split into subsymbols with arbitrarily small capacity. For example, we can split each symbol into B number of subsymbols, so each subsymbol has 1 B capacity. In order to successfully recover the desired signal components of \u201ca\u201d, the matrices associated with b and c should have rank 1, respectively, while the matrix associated with a should have full rank of 3. In accordance with the (4, 2) code example in Fig. 2, if one were to set v\u03b13 = B\u22121 1 v\u03b11 and v\u03b14 = B\u22121 2 v\u03b11, then it is possible to achieve interference alignment with respect to b. However, this choice also speci\ufb01es the interference space of c. If the Bi\u2019s and Ci\u2019s are not designed judiciously, interference alignment is not guaranteed for c. Hence, it is not evident how to achieve interference alignment at the same time. In order to address the challenge of simultaneous interference alignment, we invoke the idea of symbol extension introduced in [7], which is equivalent to the concept of vector linear codes in the storage repair problem. Fig. 4 illustrates the idea of vector linear codes through storage node 1 in the (6, 3, 4) code example. While scalar linear codes do not allow symbol splitting, vector linear codes permit the splitting of symbols into arbitrarily small subsymbols. In this example, each node stores \u03b1 = 2 symbols, each of which has unit capacity. In vector linear codes, this unit-capacity symbol is allowed to be split into subsymbols with arbitrary small capacity. In this example, we split each symbol into B number of subsymbols, so each subsymbol has 1 B capacity. October 4, 2018 DRAFT \f11 at bt ct V Goal: atA1 + btB1 + ctC1 atA2 + btB2 + ctC2 atA3 + btB3 + ctC3 \uf8ee \uf8ef \uf8ef \uf8f0 0 0 (A1V)t (A2V)t \uf8f9 \uf8fa \uf8fa \uf8fba + \uf8ee \uf8ef \uf8ef \uf8f0 \u00af Vt 0 (B1V)t (B2V)t \uf8f9 \uf8fa \uf8fa \uf8fbb + \uf8ee \uf8ef \uf8ef \uf8f0 0 \u00af Vt (C1V)t (C2V)t \uf8f9 \uf8fa \uf8fa \uf8fbc \u00af V \u03b3 = 2 (m+1)4 m4 + 2 m4 m4 \u2212 \u21924 V = {(Be1 1 Be2 2 Ce3 1 Ce4 2 ) w : e1, e2, e3, e4 \u2208{1, \u00b7 \u00b7 \u00b7 , m}} V := [v1, \u00b7 \u00b7 \u00b7 , vm4] \u00af V := [\u00af v1, \u00b7 \u00b7 \u00b7 , \u00af v(m+1)4] m \u2192\u221e rank = 2m4 rank \u2264(m + 1)4 rank \u2264(m + 1)4 \u00af V = {(Be1 1 Be2 2 Ce3 1 Ce4 2 ) w : e1, e2, e3, e4 \u2208{1, \u00b7 \u00b7 \u00b7 , m + 1}} a = \uf8ee \uf8ef \uf8f0 a1 . . . a2m4 \uf8f9 \uf8fa \uf8fb Fig. 5. Illustration of exact repair of systematic node 1 for (6, 3, 4) Exact-Repair MDS codes. We split each symbol into B = mN number of subsymbols, where m is an arbitrarily large positive integer and the exponent N is equal to 4 and is carefully chosen depending on code parameters, i.e., N = (k \u22121)(d \u2212k + 1) = 4. This corresponds to the total number of encoding submatrices involved in the connection except for those associated with desired signals. Note that each subsymbol has 1 m4 capacity. The maximum \ufb01le size (based on the optimal tradeoff of (1)) is M = 6 units, inducing a storage cost \u03b1 = 2 units. Hence, each storage contains 2m4 number of subsymbols and the size of encoding submatrices is 2m4-by-2m4. We consider diagonal encoding submatrices. A failed node is exactly repaired by having systematic and parity survivor nodes project their data onto linear subspaces spanned by column vectors of \u00af V := [\u00af v1, \u00b7 \u00b7 \u00b7 , \u00af v(m+1)4] and V := [v1, \u00b7 \u00b7 \u00b7 , vm4], respectively. Here \u00af vi \u2208\u00af V and vi \u2208V. Notice that B1vi, B2vi, C1vi, C2vi \u2208\u00af V, \u2200i = 1, \u00b7 \u00b7 \u00b7 , m4. Hence, the matrix associated with interference b has rank of at most (m + 1)4 instead of 2m4. Similarly the matrix associated with interference c has rank of at most (m + 1)4. This enables simultaneous interference alignment as m \u2192\u221e. On the other hand, rank[A1V, A2V] = 2m4 with probability 1, providing probabilistic guarantee of decodability of desired signals. Finally, notice that total repair bandwidth \u03b3 = 2 (m+1)4 m4 + 2 \u00b7 1 approaches the cutset lower bound of 4 units as m goes to in\ufb01nity. Therefore, we can ensure exact repair of systematic node 1 with minimum repair bandwidth matching the cutset lower bound. This idea of vector linear codes is key to interference alignment for the storage repair problem. Fig. 5 illustrates exact repair of systematic node 1 for (6, 3, 4) Exact-Repair MDS codes. Using vector linear codes, we split each symbol into B = mN number of subsymbols, where m is an arbitrarily large positive integer and the exponent N is carefully chosen depending on code October 4, 2018 DRAFT \f12 parameters. Speci\ufb01cally, N = (k \u22121)(d \u2212k + 1). (5) This choice of N and the form of B = mN are closely related to the scheme to be described in the sequel. In this example, N = 4. The maximum \ufb01le size (based on the cutset bound of (1)) is M = 6 units, inducing a storage cost \u03b1 = 2 units. Since each subsymbol has 1 m4 capacity, each storage contains \u03b1m4(= 2m4) number of subsymbols, e.g., at = (a1, \u00b7 \u00b7 \u00b7 , a2m4), where ai indicates a subsymbol. Note that the size of encoding submatrices (Ai, Bi, Ci) is 2m4-by-2m4. We consider diagonal encoding submatrices. As pointed out in [7], the diagonal matrix structure ensures commutativity and this property provides the key to the interference alignment scheme (to be described shortly): Ai = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 \u03b1i,1 0 \u00b7 \u00b7 \u00b7 0 0 \u03b1i,2 \u00b7 \u00b7 \u00b7 0 . . . . . . ... . . . 0 \u00b7 \u00b7 \u00b7 0 \u03b1i,2m4 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb (commutative property holds). (6) A failed node 1 is exactly repaired through the following steps. Suppose without loss of generality that survivor nodes (2, 3, 4, 5) are used for exact repair of node 1, i.e., k \u22121 = 2 systematic nodes and d \u2212k + 1 = 2 parity nodes. One can alternatively use 1 systematic node and 3 parity nodes for repair instead. This does not fundamentally alter the analysis, and will be covered in Remark 1 in the next section. For the time being, assume the above con\ufb01guration for the connection: k \u22121 systematic nodes and d \u2212k + 1 parity nodes. Parity survivor nodes project their data using the following projection matrix: V := [v1, \u00b7 \u00b7 \u00b7 , vm4] \u2208F2m4\u00d7m4 q , (7) where vi \u2208V. The set V is de\ufb01ned as: V := {(Be1 1 Be2 2 Ce3 1 Ce4 2 ) w : e1, e2, e3, e4 \u2208{1, \u00b7 \u00b7 \u00b7 , m}} , (8) where w = [1, \u00b7 \u00b7 \u00b7 , 1]t. Note that |V| \u2264m4. The vector vi maps to a different sequence of October 4, 2018 DRAFT \f13 (e1, e2, e3, e4). For example, we can map: v1 = B1B2C1C2w, v2 = B2 1B2C1C2w, v3 = B3 1B2C1C2w, . . . vm4\u22121 = Bm 1 Bm 2 Cm 1 Cm\u22121 2 w, vm4 = Bm 1 Bm 2 Cm 1 Cm 2 w. (9) Let us consider the equations downloaded from parity node 1 and 2 (node 4 and 5): at(A1V) + bt(B1V) + ct(C1V); at(A2V) + bt(B2V) + ct(C2V). (10) Note that by (8), any column vector in [B1V, B2V] or [C1V, C2V] is an element of \u00af V de\ufb01ned as: \u00af V := {(Be1 1 Be2 2 Ce3 1 Ce4 2 ) w : e1, e2, e3, e4 \u2208{1, \u00b7 \u00b7 \u00b7 , m + 1}} . (11) This implies that rank[B1V, B2V] \u2264(m + 1)4 and rank[C1V, C2V] \u2264(m + 1)4. This allows for simultaneous interference alignment. Systematic survivor nodes project their data using the following projection matrix: \u00af V := [\u00af v1, \u00b7 \u00b7 \u00b7 , \u00af v(m+1)4] \u2208F2m4\u00d7(m+1)4 q , (12) where \u00af vi \u2208\u00af V. We also map \u00af vi to a difference sequence of (e1, e2, e3, e4) as in (9). We can then guarantee that: span[B1V, B2V] \u2282span[ \u00af V] span[C1V, C2V] \u2282span[ \u00af V]. (13) Hence, we can completely get rid of any interference. Now let us analyze the decodability of the desired signal vector. To successfully recover a, we need: rank[A1V, A2V] = 2m4. (14) Using standard arguments based on the technique in [7] and Schwartz-Zippel lemma [11], we can ensure the condition of (14) with probability 1 for a suf\ufb01ciently large \ufb01eld size q. October 4, 2018 DRAFT \f14 Speci\ufb01cally, we randomly and uniformly choose each diagonal element (non-zero) of all of the encoding submatrices in Fq. We then compute the determinant of [A1V, A2V] by adapting the technique in [7]. Using Schwartz-Zippel lemma [11], we can then show that the probability that the polynomial of the determinant is identically zero goes to zero for a suf\ufb01ciently large \ufb01eld size. The proof is tedious and therefore we omit details. See [7], [11] for details. We now validate that total repair bandwidth is: \u03b3 = (k \u22121)(m + 1)4 m4 + (d \u2212k + 1) \u00b7 m4 m4 = 2(m + 1)4 m4 + 2 \u00b7 1 \u2212 \u21924 units. (15) The \ufb01rst equality is because each subsymbol has capacity of 1 m4 and we use projection matrix \u00af V \u2208F2m4\u00d7(m+1)4 q and V \u2208F2m4\u00d7m4 q when connecting to systematic nodes and parity nodes, respectively. Note that as m goes to in\ufb01nity, total repair bandwidth approaches minimum repair bandwidth matching the cutset lower bound of (1). B. Dual Relationship between Systematic and Parity Node Repair We will show that parity nodes can be repaired by drawing a dual relationship with systematic nodes. The procedure has two steps. The \ufb01rst is to remap parity nodes with a\u2032, b\u2032, and c\u2032, respectively: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 a\u2032 b\u2032 c\u2032 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 At 1 Bt 1 Ct 1 At 2 Bt 2 Ct 2 At 3 Bt 3 Ct 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 a b c \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb. Systematic nodes can then be rewritten in terms of the prime notations: at = a\u2032tA\u2032 1 + b\u2032tB\u2032 1 + c\u2032tC\u2032 1, bt = a\u2032tA\u2032 2 + b\u2032tB\u2032 2 + c\u2032tC\u2032 2, ct = a\u2032tA\u2032 3 + b\u2032tB\u2032 3 + c\u2032tC\u2032 3, (16) where the newly mapped encoding matrices (A\u2032 i, B\u2032 i, Ci)\u2019s are de\ufb01ned as: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A\u2032 1 A\u2032 2 A\u2032 3 B\u2032 1 B\u2032 2 B\u2032 3 C\u2032 1 C\u2032 2 C\u2032 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A1 A2 A3 B1 B2 B3 C1 C2 C3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \u22121 . (17) October 4, 2018 DRAFT \f15 a\u2032tA\u2032 1 + b\u2032tB\u2032 1 + c\u2032tC\u2032 1 a\u2032tA\u2032 3 + b\u2032tB\u2032 3 + c\u2032tC\u2032 3 b\u2032t c\u2032t a\u2032t \uf8ee \uf8ef \uf8ef \uf8f0 0 0 (A\u2032 1V\u2032)t (A\u2032 2V\u2032)t \uf8f9 \uf8fa \uf8fa \uf8fba\u2032 + \uf8ee \uf8ef \uf8ef \uf8f0 \u00af V\u2032t 0 (B\u2032 1V\u2032)t (B\u2032 2V\u2032)t \uf8f9 \uf8fa \uf8fa \uf8fbb\u2032 + \uf8ee \uf8ef \uf8ef \uf8f0 0 \u00af V\u2032t (C\u2032 1V\u2032)t (C\u2032 2V\u2032)t \uf8f9 \uf8fa \uf8fa \uf8fbc\u2032 a\u2032tA\u2032 2 + b\u2032tB\u2032 2 + c\u2032tC\u2032 2 V\u2032 \u00af V\u2032 \u00af V\u2032 V\u2032 Goal: rank = 2m4 rank \u2264(m + 1)4 rank \u2264(m + 1)4 V\u2032 = \u0007\b B\u2032e1 1 B\u2032e2 2 C\u2032e3 1 C\u2032e4 2 \t w : e1, e2, e3, e4 \u2208{1, \u00b7 \u00b7 \u00b7 , m} \u00af V\u2032 = \u0007\b B\u2032e1 1 B\u2032e2 2 C\u2032e3 1 C\u2032e4 2 \t w : e1, e2, e3, e4 \u2208{1, \u00b7 \u00b7 \u00b7 , m + 1} Fig. 6. Illustration of exact repair of parity node 1 for (6, 3, 4) Exact-Repair MDS codes. Notice that A\u2032 i\u2019s, B\u2032 i\u2019s, and C\u2032 i\u2019s are also diagonal matrices, since these matrices are functions of diagonal matrices Ai\u2019s, Bi\u2019s, and Ci\u2019s. Survivor nodes 1 and 2 project their data onto linear subspaces spanned by column vectors of V\u2032 := [v\u2032 1, \u00b7 \u00b7 \u00b7 , v\u2032 m4]. Here v\u2032 i \u2208V\u2032. Notice that B\u2032 1v\u2032 i, B\u2032 2v\u2032 i, C\u2032 1v\u2032 i, C\u2032 2v\u2032 i \u2208\u00af V, \u2200i = 1, \u00b7 \u00b7 \u00b7 , m4. Hence, the matrix associated with interference b\u2032 has rank of at most (m+1)4 instead of 2m4. Similarly the matrix associated with interference c\u2032 has rank of at most (m + 1)4. This enables simultaneous interference alignment as m \u2192\u221e. Survival nodes 5 and 6 project their data using \u00af V := [\u00af v1, \u00b7 \u00b7 \u00b7 , \u00af v(m+1)4] where \u00af vi \u2208\u00af V. We can then clean out any interference. On the other hand, it is guaranteed that rank[A\u2032 1V\u2032, A\u2032 2V\u2032] = 2m4 with probability 1, guaranteeing of decodability of desired signals with probability 1. As in Section III-A, we consider random construction of the code, i.e., each diagonal element in each encoding submatrix is i.i.d. uniformly drawn from Fq \\ {0}. Then, for a suf\ufb01ciently large \ufb01eld size, the above composite matrix has non-zero determinant with probability 1 (again due to Schwartz-Zippel lemma). With this remapping, one can now dualize the relationship between systematic and parity node repair. Speci\ufb01cally, if all of the A\u2032 i\u2019s, B\u2032 i\u2019s, and C\u2032 i\u2019s are diagonal matrices, then exact repair of the parity nodes becomes transparent, as illustrated in Fig. 6. Indeed A\u2032 i\u2019s, B\u2032 i\u2019s, and C\u2032 i\u2019s are diagonal matrices, since these matrices are functions of diagonal matrices Ai\u2019s, Bi\u2019s, and Ci\u2019s. Therefore, following the same procedure in Section III-A, we can guarantee exact repair of parity nodes with probability 1. Remark 1 (Connecting to arbitrary d nodes suf\ufb01ce for exact repair): In Section III-A, we considered the only one connection con\ufb01guration for exact repair: connecting to k \u22121 systematic nodes and d\u2212k+1 parity nodes. We now consider other connection con\ufb01gurations. For example, consider the case when node 1 fails, as shown in Fig. 5. Suppose we connect to nodes (2, 4, 5, 6) for exact repair of node 1: 1 systematic node and 3 parity nodes. The idea is to remap one parity October 4, 2018 DRAFT \f16 node to make it look like a systematic node. We then virtually connect to 2 systematic and 2 parity nodes. Speci\ufb01cally, we can remap node 6 with c\u2032t: c\u2032t = atA3 + btB3 + ctC3 (18) We can then rewrite node 4 and 5 in terms of a, b and c\u2032 and therefore we virtually have connection with 2 systematic and 2 parity nodes. Note that corresponding encoding submatrices after remappring are still diagonal matrices. Hence, we can apply the same procedures as those in Section III-A. C. The MDS-Code Property The third part of our framework is to guarantee the MDS-code property. Consider all four possibilities corresponding to the Data Collector (DC) contacting (1) 3 systematic nodes; (2) 3 parity nodes; (3) 1 systematic and 2 parity nodes; (4) 1 systematic and 2 parity nodes. The \ufb01rst is a trivial case. The second case has been already veri\ufb01ed in the process of forming the dual structure. The third case requires the invertibility of all of each encoding submatrix. In this case, it is obvious since encoding submatrix is diagonal and each element is non-zero. The last case is also easy to check. Consider a speci\ufb01c example where the DC connects to nodes 3, 4 and 5. In this case, we \ufb01rst recover c from node 3 and subtract the terms associated with c from nodes 4 and 5. We then get: h at bt i \uf8ee \uf8f0A1 A2 B1 B2 \uf8f9 \uf8fb. (19) Again, using the technique in [7] and Schwartz-Zippel lemma, for a suf\ufb01ciently large \ufb01eld size, this composite matrix has non-zero determinant with probability 1. D. Existence of Codes As mentioned several times, for a suf\ufb01ciently large \ufb01eld size, a random construction for encoding submatrices suf\ufb01ces to guarantee exact repair of all nodes and MDS-code property with probability 1. Hence, we obtain the following theorem. Lemma 1 ((6, 3, 4) Exact-Repair MDS Codes): There exist vector linear Exact-Repair MDS codes that achieve the minimum repair bandwidth corresponding to the cutset bound of (1), allowing for any failed node to be exactly repaired with access to any arbitrary d = 4 survivor October 4, 2018 DRAFT \f17 nodes, provided storage symbols can be split into a suf\ufb01ciently large number of subsymbols, and the \ufb01eld size can be made suf\ufb01ciently large. IV. GENERALIZATION As one can easily see, the interference alignment technique described in Section III-A can be generalized to all admissible values of (n, k, d), i.e., k < n and k \u2264d \u2264n \u22121. Theorem 1 ((n, k, d) Exact-Repair MDS Codes): There exist vector linear Exact-Repair MDS codes that achieve the minimum repair bandwidth corresponding to the cutset bound of (1), allowing for any failed node to be exactly repaired with access to any arbitrary d survivor nodes, where k \u2264d \u2264n \u22121, provided storage symbols can be split into a suf\ufb01ciently large number of subsymbols, and the \ufb01eld size can be made suf\ufb01ciently large. Proof: In the interests of conceptual simplicity, and to parallel the analysis of the (6, 4, 3) example described earlier, we provide only a sketch of the proof for the general case. This can be formalized to be precise at the cost of much heavier notational clutter, which we consciously avoid. Systematic Node Repair: Let G(i) l indicate an encoding submatrix for parity node i, associated with information unit l, where 1 \u2264i \u2264n \u2212k and 1 \u2264l \u2264k. Let wl be lth information-unit vector. Without loss of generality, consider exact repair of systematic node 1. Using vector linear codes, we split each symbol into B = mN number of subsymbols, where m is an arbitrarily large positive integer and the exponent N is given by N = (k \u22121)(d \u2212k + 1). (20) The maximum \ufb01le size (based on the cutset bound of (1)) is M = k(d\u2212k +1) units, inducing a storage cost \u03b1 = d \u2212k + 1 units. Since each subsymbol has 1 mN capacity, each storage contains \u03b1mN(= (d \u2212k + 1)mN) number of subsymbols. Note that the size of encoding submatrices is \u03b1mN-by-\u03b1m4. A failed node 1 is exactly repaired through the following steps. Suppose without loss of generality that we connect k \u22121 systematic nodes and \ufb01rst d \u2212k + 1 parity nodes3. Parity 3As mentioned earlier, we can convert the other connection con\ufb01gurations into this particular con\ufb01guration with the remapping technique October 4, 2018 DRAFT \f18 survivor nodes project their data using the following projection matrix: V := [v1, \u00b7 \u00b7 \u00b7 , vmN] \u2208F\u03b1mN \u00d7mN q , (21) where vi \u2208V. The set V is de\ufb01ned as: V := ( Y i=1,\u00b7\u00b7\u00b7 ,d\u2212k+1,l=2,\u00b7\u00b7\u00b7,k h G(i) l iei,l w : ei,l \u2208{1, \u00b7 \u00b7 \u00b7 , m} ) , (22) where w = [1, \u00b7 \u00b7 \u00b7 , 1]t. Note that |V| \u2264mN. Let us consider the equations downloaded from parity nodes: wt 1(G(1) 1 V) + wt 2(G(1) 2 V) + \u00b7 \u00b7 \u00b7 + wt k(G(1) k V); . . . wt 1(G(d\u2212k+1) 1 V) + wt 2(G(d\u2212k+1) 2 V) + \u00b7 \u00b7 \u00b7 + wt k(G(d\u2212k+1) k V). (23) Note that by (22), for l \u0338= 1, any column vector in [G(1) l V, \u00b7 \u00b7 \u00b7 , G(d\u2212k+1) l V] is an element of \u00af V de\ufb01ned as: \u00af V := ( Y i=1,\u00b7\u00b7\u00b7 ,d\u2212k+1,l=2,\u00b7\u00b7\u00b7,k h G(i) l iei,l w : ei,l \u2208{1, \u00b7 \u00b7 \u00b7 , m + 1} ) , (24) This implies that for l \u0338= 1, rank[G(1) l V, \u00b7 \u00b7 \u00b7 , G(d\u2212k+1) l V] \u2264(m + 1)N. This allows for simultaneous interference alignment. Systematic survivor nodes project their data using the following projection matrix: \u00af V := [\u00af v1, \u00b7 \u00b7 \u00b7 , \u00af v(m+1)N ] \u2208F\u03b1mN \u00d7(m+1)N q , (25) where \u00af vi \u2208\u00af V. We can then guarantee that for l \u0338= 1: span[G(1) l V, \u00b7 \u00b7 \u00b7 , G(d\u2212k+1) l V] \u2282span[ \u00af V]. (26) Hence, we can clean out any interference. Now let us consider the decodability of desired signals. To successfully recover w1, we need: rank[G(1) 1 V, \u00b7 \u00b7 \u00b7 , G(d\u2212k+1) 1 V] = (d \u2212k + 1)mN = \u03b1mN. (27) Using the technique in [7] and Schwartz-Zippel lemma [11], we can ensure the (14) with probability 1 for a suf\ufb01ciently large \ufb01eld size. October 4, 2018 DRAFT \f19 Finally we validate that total repair bandwidth is: \u03b3 = (k \u22121)(m + 1)N mN + (d \u2212k + 1) \u00b7 mN mN \u2212 \u2192d. (28) Note that as m goes to in\ufb01nity, total repair bandwidth approaches minimum repair bandwidth matching the cutset lower bound of (1). Parity Node Repair: As discussed in Section III-B, we can draw a dual structure by remapping parity nodes with primed new notations. The key observation is that newly mapped encoding submatrices are still diagonal matrices. Hence, we can apply the same technique used in systematic node repair. MDS-Code Property: We check the invertibility of a composite matrix when a Data Collector connects to i systematic nodes and k \u2212i parity nodes for i = 0, \u00b7 \u00b7 \u00b7 , k. As mentioned earlier, for a suf\ufb01ciently large \ufb01eld size, the composite matrix has non-zero determinant with probability 1. V." + }, + { + "url": "http://arxiv.org/abs/1003.3707v2", + "title": "Downlink Interference Alignment", + "abstract": "We develop an interference alignment (IA) technique for a downlink cellular\nsystem. In the uplink, IA schemes need channel-state-information exchange\nacross base-stations of different cells, but our downlink IA technique requires\nfeedback only within a cell. As a result, the proposed scheme can be\nimplemented with a few changes to an existing cellular system where the\nfeedback mechanism (within a cell) is already being considered for supporting\nmulti-user MIMO. Not only is our proposed scheme implementable with little\neffort, it can in fact provide substantial gain especially when interference\nfrom a dominant interferer is significantly stronger than the remaining\ninterference: it is shown that in the two-isolated cell layout, our scheme\nprovides four-fold gain in throughput performance over a standard multi-user\nMIMO technique. We show through simulations that our technique provides\nrespectable gain under a more realistic scenario: it gives approximately 20%\ngain for a 19 hexagonal wrap-around-cell layout. Furthermore, we show that our\nscheme has the potential to provide substantial gain for macro-pico cellular\nnetworks where pico-users can be significantly interfered with by the nearby\nmacro-BS.", + "authors": "Changho Suh, Minnie Ho, David Tse", + "published": "2010-03-19", + "updated": "2010-05-26", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION One of the key performance metrics in the design of cellular systems is that of cell-edge spectral ef\ufb01ciency. As a result, fourth-generation (4G) cellular systems, such as WiMAX [1] and 3GPP-LTE [2], require at least a doubling in cell-edge throughput over previous 3G systems [2]. Given the disparity between average and cell-edge spectral ef\ufb01ciencies (ratios of about 4:1) [1], the desire to improve cell-edge throughput performance is likely to continue. Since the throughput of cell-edge users is greatly limited by the presence of co-channel interference from other cells, developing an intelligent interference management scheme is the key to improving cell-edge throughput. One interesting recent development, called interference alignment (IA) [3], [4], manages interference by aligning multiple interference signals in a signal subspace with dimension smaller than the number of interferers. While most of the work on IA [4], [5], [6] has focused on K point-to-point interfering links, it has also been shown in [7], [8], [9] that IA can be used to improve the cell-edge user throughput in a cellular network. Especially, it was shown in [7] that near interference-free throughput performance can be achieved in the cellular network. While IA promises substantial theoretical gain in cellular networks, it comes with challenges in implementation. First, the uplink IA scheme in [7] requires extensive channel-state-information (CSI) to be exchanged over the backhaul between base-stations (BSs) of different cells. A second challenge comes from realistic cellular environments that involve multiple unaligned out-of-cell interferers. Lastly, the integration of IA with other system issues, such as scheduling, needs to be addressed. We propose a new IA technique for downlink cellular systems that addresses many of these practical concerns. Unlike the uplink IA, our downlink IA scheme requires feedback only within a cell. As a consequence, our technique can be implemented with small changes to existing 4G standards where the within-a-cell feedback mechanism is already being considered for supporting multi-user MIMO. Our proposed technique builds on the idea of the IA technique in [7] that aims for a two-isolated cell layout and can thus cancel interference only from one neighboring BS. We observe that the IA technique in [7] may give up the opportunity of providing matched\ufb01ltered gain (also called beam-forming gain in the case of multiple antennas) in the presence of a large number of interferers. Our new technique balances these two scenarios, inspired by the DRAFT May 27, 2010 \f3 idea of the standard MMSE receiver that uni\ufb01es a zero-forcing receiver (optimum in the high SNR regime) and a matched \ufb01lter (optimum in the low SNR regime). Through simulations, we show that our scheme provides approximately 55% and 20% gain in cell-edge throughput performance for a linear cell layout and 19 hexagonal wrap-around-cell layout, respectively, as compared to a standard multi-user MIMO technique. We also \ufb01nd that our scheme has the potential to provide signi\ufb01cant performance for heterogeneous networks [10], e.g., macro-pico cellular networks where a dominant interference can be much stronger than the residual interference. For instance, pico-users can be signi\ufb01cantly interfered with by the nearby macro-BS, as compared to the aggregated remaining BSs. We show that for these networks our scheme can give around 30% to 200% gain over the standard technique. Furthermore, our scheme is easily combined with a widely-employed opportunistic scheduler [11] for signi\ufb01cant multi-user-diversity gain. II. INTERFERENCE ALIGNMENT A. Review of Uplink IA We begin by reviewing uplink IA in [7]. Fig. 1 illustrates an example for the case of two isolated cells \u03b1 and \u03b2. Suppose that each cell has K users and each user (e.g., user k in cell \u03b1) sends one symbol (or stream) along a transmitted vector v\u03b1k \u2208CM. Each user can generate multiple dimensions by using subcarriers (in an OFDM system), antennas, or both: M = (# of subcarriers) \u00d7 (# of antennas). (1) In this paper, we assume that each BS has the same number of dimensions: the M-by-M symmetric con\ufb01guration. The asymmetric case will be discussed in Section VI. The idea of interference alignment is to design the transmitted vectors so that they are aligned onto a onedimensional linear subspace at the other BS. Due to the randomness in wireless channels, the transmitted vectors are likely to be linearly independent at the desired BS. Note that for M = K + 1, the desired signals span a K-dimensional linear space while the interference signals only occupy a one-dimensional subspace. Hence, each BS can recover K desired symbols using K + 1 dimensions. The performance in the interference-limited regime can be captured by a notion of degreesof-freedom (dof). Here, dof per cell = K K+1, so as K gets large, we can asymptotically achieve May 27, 2010 DRAFT \f4 Cell \u03b1 Cell \u03b2 1 2 K . . . 1 2 . . . K BS \u03b1 BS \u03b2 G\u03b21 G\u03b21 G\u03b21 dof(per cell) = K K + 1 \u2212 \u21921 v\u03b11 v\u03b12 v\u03b1K v\u03b21 v\u03b22 v\u03b2K Fig. 1. Uplink interference alignment. Interference-free degrees-of-freedom can be asymptotically achieved with an increase in K. However, this scheme requires exchange of cross-channel information over the backhaul between BSs of different cells. interference-free dof = 1. On the other hand, one implementation challenge comes from the overhead of exchanging CSI needed for enabling the IA technique. The IA scheme requires each user to know its cross-channel information to the other BS. While in a time-division-multiplexing system, channels can be estimated using reciprocity, in a frequency-division-multiplexing system, backhaul cooperation is required to convey such channel knowledge. Fig. 1 shows a route to obtain the CSI of G\u03b21: BS \u03b2 \u2192backhaul \u2192BS \u03b1 \u2192feedback \u2192user 1 of cell \u03b1. Here G\u03b21 \u2208C(K+1)\u00d7(K+1) indicates the cross-channel from user 1 of cell \u03b1 to BS \u03b2. On the contrary, in the downlink, we show that IA can be applied without backhaul cooperation. B. Downlink Interference Alignment Fig. 2 illustrates an example of downlink IA where there are two users in each cell. The uplinkdownlink duality says that the dof of the uplink is the same as that of the downlink. Hence, dof per cell = K K+1 = 2 3. To achieve this, each BS needs to send two symbols (streams) over three dimensions. The idea is similar to that of the uplink IA in a sense that two dimensions are used for transmitting desired signals and the remaining one dimension is reserved for interference signals. However, the method of interference alignment is different. DRAFT May 27, 2010 \f5 (3-by-2 matrix) 1 2 1 2 BS \u03b1 BS \u03b2 \u03b1 P P \u03b2 u\u2217 \u03b11 u\u2217 \u03b12 u\u2217 \u03b21 u\u2217 \u03b22 v\u03b21 v\u03b11 v\u03b12 v\u03b22 u\u2217 \u03b12H\u03b12P u\u2217 \u03b11H\u03b11P u\u2217 \u03b21H\u03b21P u\u2217 \u03b22H\u03b22P ZF precoder u\u03b11 G\u03b21P Interference alignment between out-of-cell and intra-cell interference ZF precoder H\u03b11Pv\u03b11 H\u03b11Pv\u03b12 Fig. 2. Downlink interference alignment. Interference alignment is achieved between out-of-cell and intra-cell interference vectors at multiple users at the same time. Unlike the uplink IA, our downlink IA scheme does not require backhaul cooperation. We \ufb01rst set a 3-by-2 precoder matrix P at BS \u03b1 and BS \u03b2, respectively. This spreads two data streams over three-dimensional resources. Next, each user, such as user 1 in cell \u03b1, estimates the interference G\u03b21P using pilots or a preamble. User 1 then generates a vector u\u03b11 that lies in the null-space of the G\u03b21P. Since the G\u03b21P is of dimension 3-by-2, such a vector u\u03b11 always exists, and when applied to the received signal, it will null out the out-of-cell interference. Note that the receive vector u\u03b11 does not guarantee the cancellation of intra-cell interference from user 2 in the same cell \u03b1. This is accomplished as follows. In cell \u03b1, each user feeds back its equivalent channel u\u2217 \u03b1kH\u03b1kP (obtained after applying the receive vector) to its own BS \u03b1, where H\u03b1k \u2208C3\u00d73 indicates the direct-channel from BS \u03b1 to user k in the cell. BS \u03b1 then applies an additional zero-forcing precoder formed by the pseudo-inverse of the composite matrix [u\u2217 \u03b11H\u03b11P; u\u2217 \u03b12H\u03b12P]. This zero-forcing precoder guarantees user 2\u2019s transmitted signal H\u03b11Pv\u03b12 to lie in the interference space G\u03b21P. Note that u\u2217 \u03b11(H\u03b11Pv\u03b12) = 0. A series of operations enables interference alignment. Let us call this scheme zero-forcing IA. To see this, let us observe the interference plane of user 1 in cell \u03b1. Note that there are three May 27, 2010 DRAFT \f6 interference vectors: two out-of-cell interference vectors and one intra-cell interference vector. These three vectors are aligned onto a two-dimensional linear subspace. Interference alignment is achieved between out-of-cell and intra-cell interference signals to save one dimension. Similarly, user 2 in the cell can save one dimension. Hence, two dimensions can be saved in total by sacri\ufb01cing only one dimension. If the number of users is K, each cell can save K dimensions by sacri\ufb01cing one dimension. The loss will become negligible with the increase of K, as was seen in the uplink IA. While the downlink dof is the same as that of the uplink, the way interference is aligned is quite different. Note in Fig. 1 that in uplink IA, interference alignment is achieved among out-of-cell interference vectors only. On the other hand, in downlink IA, interference alignment is achieved between out-of-cell and intra-cell interference vectors at multiple users at the same time. Feedback Mechanism: Note two key system aspects of the technique. First, the exchange of cross-channel information between BSs or between users in different cells is not needed. Each BS can \ufb01x precoder P, independent of channel gains. Each user can then specify the space orthogonal to the out-of-cell interference signal space. This enables the user to design a zero-forcing receive vector without knowing the actually transmitted vectors. Each user then feeds back the equivalent channel u\u03b1kH\u03b1kP and the BS forms the zero-forcing transmit vectors only with the feedback of the equivalent channels. Hence, the scheme requires only within-acell feedback mechanism. This is in stark contrast to the uplink IA which requires backhaul cooperation between different BSs. Secondly, while feedback is required from the user to the BS, this feedback is the same as the feedback used for standard multi-user MIMO techniques. The only difference is that in downlink IA, two cascaded precoders are used and the receive vector of each user is chosen as a null vector of out-of-cell interference signal space. As a result, the scheme can be implemented with little change to an existing cellular system supporting multi-user MIMO. C. Performance and Limitations Fig. 3 shows the sum-rate performance for downlink zero-forcing IA in a two-isolated cell layout where M = 4 (e.g., a 4-by-4 antenna con\ufb01guration). As a baseline scheme, we use a matched \ufb01lter receiver: one of the standard multi-user MIMO techniques [12], [13]. The scheme DRAFT May 27, 2010 \f7 \u221220 \u221210 0 10 20 30 0 5 10 15 20 25 30 35 Transmit SNR (dB) Sum Rate (bits/s/Hz) Total Number of Users = 10 300% improvement Interference Alignment (IA) Matched filtering (baseline) Fig. 3. Performance of downlink interference alignment for a two-isolated cell layout with a 4-by-4 antenna con\ufb01guration (M = 4). An opportunistic scheduler is employed to choose a set of 3 users out of 10 such that the sum rate is maximized. uses the dominant left-singular vector of the direct-channel as a receive vector. We assume a zeroforcing vector at the transmitter to null out intra-cell interference. Nulling intra-cell interference is important as its power has the same order as the desired signal power. uMF \u03b1k = a maximum left-singular vector of H\u03b1k, (2) vZF \u03b1k = kth normalized column of H(HH\u2217)\u22121, H := \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 . . . uMF\u2217 \u03b1k H\u03b1k . . . \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb. (3) Note that the matched \ufb01lter receiver maximizes beam-forming gain but it ignores the interference signal space. Also notice that the receive-and-transmit vectors are interconnected, i.e., a receiver vector can be updated as a function of a transmit vector and vice versa. One way to compute the transmit-and-receive vectors is to employ an iterative algorithm [12], [13]. We call this scheme iterative matched \ufb01ltering. See Appendix A for further details. In Fig. 3, we assume no iteration for fair comparison of CSI overhead. An opportunistic scheduler [11] is employed to choose a May 27, 2010 DRAFT \f8 Fig. 4. Different layouts in a downlink cellular system. A parameter \u03b3 indicates the relative strength of the interference power from a dominant interferer to the remaining interference power (summed from the other BSs). set of 3 users out of 10 such that the sum rate is maximized. We also consider uncoordinated schedulers, i.e., scheduling information is not exchanged between different BSs. One can clearly see that the zero-forcing IA provides signi\ufb01cant (asymptotically optimum for large SNR) performance gain for the two-isolated-cell case, as there are no residual interferers. However, for realistic multi-cellular environments, the performance may not be very good due to the remaining interferers. In order to take multi-cellular environments into account, we introduce a parameter \u03b3 that captures the relative strength of the interference power from a dominant interferer to the remaining interference power (summed from the other BSs): \u03b3 := INRrem INRdom , (4) where INRdom and INRrem denote the ratios of the dominant and aggregate interference power over the noise power, respectively. Note that by adapting \u03b3, one can cover arbitrary mobile location and cellular layouts. While, at one extreme (\u03b3 = 0), the zero-forcing IA provides signi\ufb01cant performance, at the other extreme (\u03b3 \u226b1), the scheme may not be good as it completely loses receive beam-forming gain (the zero-forcing IA receiver is independent of the direct-channel since it depends only on the interference space). In this case, one can expect that matched \ufb01ltering will perform much DRAFT May 27, 2010 \f9 better than the IA scheme. This motivates the need for developing a new IA technique that can balance the degrees-of-freedom gain with the matched-\ufb01ltered power gain depending on the value of \u03b3. III. PROPOSED NEW IA SCHEME The zero-forcing IA and matched \ufb01ltering schemes remind us of a conventional zero-forcing receiver and a matched \ufb01lter receiver in a point-to-point channel with colored noise. So it is natural to think of a uni\ufb01ed technique like the standard MMSE receiver. However, in our cellular context, a straightforward design of an MMSE receiver usually requires the knowledge of transmitted vectors from the other cell. Moreover, a chicken-and-egg problem arises between different cells, due to the interconnection of the transmit-and-receive vector pairs. In order to decouple the vector design between cells, we consider uncoordinated systems, i.e., transmit vector information is not exchanged between different cells. Under this assumption, a goal is to mimic an MMSE receiver. The idea is to color an interference signal space by using two cascaded precoders, one of which is a \ufb01xed precoder \u00af P located at the front-end. With the \ufb01xed precoder, we can color the interference space, to some extent, to be independent of actually transmitted vectors. To see this, consider the covariance matrix of interference-plus-noise: \u03a6k = (1 + INRrem)I + SNR S (G\u03b2k \u00af PB\u03b2B\u2217 \u03b2 \u00af P\u2217G\u2217 \u03b2k), (5) where S is the total number of streams assigned to the scheduled users in the cell (S \u2264M) and B\u03b2 indicates the zero-forcing precoder of a dominant interferer (BS \u03b2): B\u03b2 = [v\u03b21, \u00b7 \u00b7 \u00b7 , v\u03b2S] \u2208 CM\u00d7S. Assume that the aggregate interference except the dominant interference is white Gaussian1. Without loss of generality, we assume that Gaussian noise power is normalized to 1. Assume the total transmission power is equally allocated to each stream. We control the coloredness of interference signals by differently weighting the last (M \u2212S) columns of \u00af P with a parameter \u03ba (0 \u2264\u03ba \u22641): \u00af P = [f1, \u00b7 \u00b7 \u00b7 , fS, \u03bafS+1, \u00b7 \u00b7 \u00b7 , \u03bafM] \u2208CM\u00d7M, (6) 1To be more accurate, we may consider two or three dominant interferers for an actual realization. See Section VI for details. May 27, 2010 DRAFT \f10 where [f1, \u00b7 \u00b7 \u00b7 , fM] is a unitary matrix. Since we consider uncoordinated systems, B\u03b2 is unknown. Hence, we use the expectation of the covariance matrix over B\u03b2: \u00af \u03a6k := E[\u03a6k] = (1 + INRrem)I + SNR S (G\u03b1k \u00af P\u00af P\u2217G\u2217 \u03b1k), (7) where we assume that each entry of B\u03b2 is i.i.d. CN (0, 1 S). Two extreme cases give insights into designing \u03ba. When the residual interference is negligible, i.e., \u03b3 \u226a1, the scheme should mimic the zero-forcing IA, so \u00af P should be rank-de\ufb01cient, i.e., \u03ba = 0. Note in this case that the null space of the interference signals can be speci\ufb01ed, independent of B\u03b2. As a result, the expected covariance matrix acts as the actual covariance matrix to induce the solution of the zero-forcing IA. At the other extreme (\u03b3 \u226b1), the scheme should mimic matched \ufb01ltering. This motivates us to choose a unitary matrix \u00af P. One way for smoothly sweeping between the two cases is to set: \u03ba = min (\u221a\u03b3, 1) . (8) Note that for \u03b3 \u226a1, \u03ba \u22480 and for \u03b3 \u226b1, \u03ba is saturated as 1. Considering system aspects, however, the \u03ba needs to be carefully chosen. In the above choice, the \u03ba varies with mobile location, since INRrem is a function of mobile location. This can be undesirable because it requires frequent adaptation of BS precoder which supports users from the cell center to the cell edge. Therefore, we propose to \ufb01x \u03ba. For example, we can \ufb01x \u03ba based on the case of SNR = 20 dB, a cell-edge mobile location, and a \ufb01xed network layout, e.g., \u03ba \u22480.34 for the linear cell layout and \u03ba \u22480.64 for the 19 hexagonal wrap-around cell layout (See Fig. 4). With \u00af \u03a6k, we then use the standard formula of an MMSE receiver. Similar to the iterative matched \ufb01ltering technique, we also employ an iterative approach to compute transmit-andreceive vector pairs. 1) (Intialization): Each user initializes its receive vector as follows: \u2200k \u2208{1, \u00b7 \u00b7 \u00b7 , K}, u(0) \u03b1k = normalization n \u00af \u03a6\u22121 k H\u03b1k \u00af Pv(0) \u03b1k o , (9) where we set v(0) \u03b1k as a maximum eigenvector of \u00af P\u2217H\u2217 \u03b1k \u00af \u03a6\u22121 k H\u03b1k \u00af P to initially maximize beam-forming gain. Each user then feeds back the equivalent channel u(0)\u2217 \u03b1k H\u03b1k \u00af P to its DRAFT May 27, 2010 \f11 own BS. With this feedback information, the BS computes zero-forcing transmit vectors: \u2200k v(1) \u03b1k = kth normalized column of H(1)\u2217(H(1)H(1)\u2217)\u22121, where H(1) := \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 . . . u(0)\u2217 \u03b1k H\u03b1k \u00af P . . . \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb (10) . 2) (Opportunistic Scheduling): The BS \ufb01nds A\u2217such that A\u2217= arg max A\u2208K X k\u2208A log 1 + SNR S ||u(0)\u2217 \u03b1k H\u03b1kv(1) \u03b1k||2 1 + INRrem ! , where K is a collection of subsets \u2282{1, \u00b7 \u00b7 \u00b7 , K} that has cardinality |K| = \u0000K S \u0001 . 3) (Iteration): For A\u2217, we iterate the following. The BS informs each user of v(i) \u03b1k via precoded pilots. Each user updates the receive vector as follows: u(i) \u03b1k = normalization n \u00af \u03a6\u22121 k H\u03b1k \u00af Pv(i) \u03b1k o . Each user then feeds back the updated equivalent channel to its own BS. With this feedback information, the BS computes zero-forcing transmit vectors v(i+1) \u03b1k . Remarks: Although users can see out-of-cell interference, the scheduler at BS cannot compute it. Hence, we assume that the scheduler makes a decision assuming no dominant interference. Note that the denominator inside the logarithmic term contains only noise and residual interference. To reduce CSI overhead, we assume that a scheduler decision is made before the iteration step. In practice, we may not prefer to iterate, since it requires more feedback information. Note that the feedback overhead is exactly the same as that of iterative matched-\ufb01ltering (baseline). The only difference is that we use the \ufb01xed precoder \u00af P and the MMSE-like receiver employing the \u00af \u03a6k. This requires very little change to an existing cellular system supporting multi-user MIMO. May 27, 2010 DRAFT \f12 \u221220 \u221210 0 10 20 30 40 0 1 2 3 4 5 6 7 Transmit SNR (dB) Sum Rate (bits/s/Hz) 19 Cells; Total Number of Users (per cell) = 10 Proposed Unified Scheme (ITER=1) Proposed Unified Scheme (no ITER) Zero\u2212Forcing IA Matched Filtering (ITER=1) Matched Filtering (no ITER) 20% 20% (a) 0 1 2 3 4 5 3 3.5 4 4.5 5 5.5 6 6.5 7 Number of Iteration Sum Rate (bits/s/Hz) SNR = 20 dB; K=10 (per cell) Proposed Unified Schene Proposed Unified Schene (Random Initialization) Zero\u2212Forcing IA Matched Filtering (b) Fig. 5. The sum-rate performance for a 19 hexagonal cell layout where the number K of users per cell is 10 and the number S of streams is 3: (a) as a function of transmit SNR; (b) as a function of the number of iteration. IV. SIMULATION RESULTS Through simulations, we evaluate the performance of the proposed scheme for downlink cellular systems. We consider one of the possible antenna con\ufb01gurations in the 4G standards [1], [2]: 4 transmit and 4 receive antennas. To minimize the change to the existing 4G systems, we consider using only antennas for the multiple dimensions, i.e., M = 4. We focus on three different cellular layouts, illustrated in Fig. 4. We consider a speci\ufb01c mobile location (the midpoint between two adjacent cells), as the cell-edge throughput performance is of our main interest. We use the standard ITU-Ped path-loss model, with i.i.d. Rayleigh fading components for each of the antenna. Fig. 5 shows the throughput performance for 19 hexagonal cellular systems where \u03b3 \u22480.4. We consider K = 10 and S = 3. We \ufb01nd through simulations that using three streams provides the best performance for a practical number of users per cell (around 10). See Appendix B for further details. Note that the zero-forcing IA scheme is worse than the matched \ufb01ltering (baseline). This implies that when \u03b3 \u22480.4 (residual interference is not negligible), boosting power gain gives better performance than mitigating dominant out-of-cell interference. However, the proposed uni\ufb01ed IA technique outperforms both of them for all regimes. It gives approximately 20% throughput gain when SNR = 20 dB. We also investigate the convergence of the proposed scheme. Note in Fig. 5(b) that the DRAFT May 27, 2010 \f13 \u221220 \u221210 0 10 20 30 40 0 2 4 6 8 10 Transmit SNR (dB) Sum Rate (bits/s/Hz) Linear Cells; Total Number of Users (per cell) = 10 Proposed Unified Scheme (ITER=1) Proposed Unified Scheme (no ITER) Zero\u2212Forcing IA Matched Filtering (ITER=1) Matched Filtering (no ITER) 55% 55% Fig. 6. The sum-rate performance for a linear cell layout as a function of SNR. We consider K = 10 (per cell) and S = 3. proposed scheme converges to the limits very fast, i.e., even one iteration is enough to derive most of the asymptotic performance gain. This means that additional iterations provide marginal gain, while requiring a larger overhead of CSI feedback. Another observation is that the converged limits of the proposed technique is invariant to the initial values of transmit-and-receive vectors. Note that random initialization induces the same limits as that of our carefully chosen initial values, but it requires more iterations to achieve the limits. Therefore, initial values need to be carefully chosen to minimize the overhead of CSI feedback. Fig. 6 shows throughput performance for a linear cell layout. In this case, the residual interference is signi\ufb01cantly reduced at \u03b3 \u22480.1, so mitigating dominant out-of-cell interference improves the performance more signi\ufb01cantly than beam-forming does. The gain of the proposed scheme over the matched \ufb01ltering is signi\ufb01cant, i.e., approximately 55% in the high SNR regime of interest. Notice that a crossover point between the zero-forcing IA and the matched \ufb01ltering occurs at around SNR = 0 dB. The bene\ufb01t of the zero-forcing IA is substantial. V. MACRO-PICO CELLULAR NETWORKS We have observed that our scheme shows promise especially when dominant interference is much stronger than the remaining interference, i.e., \u03b3 \u226a1. Such scenario occurs often in May 27, 2010 DRAFT \f14 Fig. 7. Macro-pico cellular networks. The pico-user can see signi\ufb01cant interference from the nearby macro-BS. The interference problem can be further aggravated when the pico-BS is close to the nearby macro-BS (small d) and the power levels of the two BSs are quite different. heterogeneous networks [10] which use a mix of macro, pico, femto, and relay BSs to enable \ufb02exible and low-cost deployment. In this section, we focus on a scenario of the macro-pico cell deployment, illustrated in Fig. 7. As shown in the \ufb01gure, suppose that pico-BS is deployed at a distance d from the nearby macro-BS and a user is connected to the pico-BS. The pico-user can then see signi\ufb01cant interference from the nearby macro-BS, and this interference can be much stronger than the aggregated interference from the remaining macro-BSs, especially when d is small. The interference problem can be further aggravated due to range extension techniques2 [10] and the disparity between the transmit power levels of the macro-BS and the pico-BS. This motivates the need for intelligent interference management techniques. We show that our IA scheme can resolve this problem to 2Range extension extends the footprint of pico-cells by allowing more users to connect even if users do not see the pico-BS as the strongest downlink received power. The purpose for this is to better utilize cell-splitting and maximize cell of\ufb02oading gain. DRAFT May 27, 2010 \f15 \u221220 \u221210 0 10 20 30 40 2 4 6 8 10 12 14 16 18 20 22 SNR (dB) Sum Rate (bits/s/Hz) Total Number of Users = 10 Proposed Unified Scheme Zero\u2212Forcing IA Matched Filtering 150% (a) \u221220 \u221210 0 10 20 30 40 0 1 2 3 4 5 6 7 8 9 SNR (dB) Sum Rate (bits/s/Hz) Total Number of Users = 10 Proposed Unified Scheme Zero\u2212Forcing IA Matched Filtering 28% (b) Fig. 8. The sum-rate performance for macro-pico cell layout (on top of 19 wrap-around macro cells): (a) d R = 0.5; (b) d R = 1. The number K of users per cell is 10; the number S of streams is 3; and no iteration is performed. provide substantial gain. To show this, we evaluate the throughput performance of pico-users in the simple scenario shown in Fig. 7. We assume the 19 hexagonal wrap-around cellular layout, and on top of it we deploy one pico-BS. Based on [10], we consider the power levels of 46 dBm and 30 dBm for the macro-BS and the pico-BS, respectively, so the difference is 16 dB. Consistent with previous simulation setups, we consider a speci\ufb01c mobile location where the downlink received power from the pico-BS is the same as that from the nearby macro-BS. Due to the disparity of the power levels, the pico-users are closer to the pico-BS. We assume a 4-by-4 antenna con\ufb01guration where M = 4. Fig. 8 shows the throughput performance of the pico-users as a function of SNR. We assume that K = 10, S = 3 and no iterations. We employ the opportunistic scheduler to choose the best 3 users out of 10. Fig. 8 (a) considers the case of d R = 0.5 where pico-users are signi\ufb01cantly interfered with by the nearby macro-BS. In this case, as one can expect, our IA scheme provides signi\ufb01cant gain of 150% over the matched \ufb01ltering, similar to the two-isolated cell case. In Fig. 8 (b), we also consider the case of d R = 1 where the minimum gain of our scheme is expected. Even in this worst case, our proposed scheme gives approximately 28% gain over the matched \ufb01ltering. Recall that in this simulation we consider the speci\ufb01c mobile location where the downlink May 27, 2010 DRAFT \f16 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 d/R Sum Rate (bits/s/Hz) SNR (per antenna) = 20 dB Proposed Unified Scheme Zero\u2212Forcing IA Resource Partitioning Matched Filtering 25% 225% 20% Fig. 9. Comparison to resource partitioning. The sum-rate performance as a function of d R for SNR = 20 dB. received power from the two BSs are the same. In fact, this is a conservative case. As mentioned earlier, the use of the range extension technique expands the footprint of pico-cells and therefore aggravates the interference problem. One can expect a larger gain of our IA scheme when range extension is employed. Comparison to Resource Partitioning: In this scenario, as an alternative to our IA scheme, one may consider resource partitioning to resolve the interference problem. This is because unlike the conventional macro cellular networks containing many neighboring cells, this macropico network scenario has a fewer number of dominant interferers, thus making resource coordination simpler. For example, we can use a frequency reuse of 1 2 for the scenario in Fig. 7. However, resource partitioning requires explicit coordination of frequency resources which can increase the control channel overhead. On the contrary, our IA scheme does not require explicit coordination, as it adapts only the number of streams under frequency reuse of 1. In addition to this implementation advantage, our scheme shows respectable gain over resource partitioning. Fig. 9 shows the throughput performance as a function of d R when SNR = 20 dB and K = 10. We use S = 3 for the IA schemes and the matched \ufb01ltering, while for resource partitioning we optimize the number of streams to plot the best performance curve. Notice that our scheme gives approximately 20% gain for d R = 0.5. The smaller ratio of d R, the larger the gain, while for large d R, the gain becomes marginal. DRAFT May 27, 2010 \f17 VI. EXTENSION A. Asymmetric Antenna Con\ufb01guration As one natural extension, we consider asymmetric antenna con\ufb01guration where the BSs are equipped with more antennas. A slight modi\ufb01cation of our technique can cover this case. Consider M-by-N antenna con\ufb01guration where M > N. Compared to the symmetric case, the only difference is that the number of streams is limited by the number N of receive antennas, i.e., S \u2264N. Other operations remain the same. Speci\ufb01c operations are as follows. Each BS sets the precoder \u00af P as follows: \u00af P = [f1, \u00b7 \u00b7 \u00b7 , fS, \u03bafS+1, \u00b7 \u00b7 \u00b7 , \u03bafM] \u2208CM\u00d7M, (11) where 0 \u2264\u03ba \u22641. Notice that S \u2264N. Each user computes the expected covariance matrix by averaging over the transmitted signals from the other cell and then applies the standard MMSE formula for a receive vector. The BS then computes the zero-forcing transmit vectors with the feedback information. These steps can then be iterated. While our technique can be extended to any antenna con\ufb01guration, interpretation needs to be carefully made for some cases. For example, consider 4-by-2 antenna con\ufb01guration in a two-cell layout. Our scheme allows each BS to send one stream out of two and therefore each user sees only one interference vector from the other cell. This induces no interference alignment. Even in this con\ufb01guration, however, interference alignment can be achieved if multiple subcarriers are incorporated. This will be discussed in the following section. B. Using Subcarriers Recall in our simulations that only antennas are employed to generate multiple dimensions. We can also increase M by using multiple subcarriers, thereby improving performance as the dimension reserved for interference signals becomes negligible with the increase of M. For example, we can create 8-by-4 con\ufb01guration by using two subcarriers in a 4-by-2 antenna con\ufb01guration. Interestingly, unlike the 4-by-2 con\ufb01guration, this 8-by-4 con\ufb01guration enables interference alignment. To see this, consider a two-cell layout where each cell has three users. Our scheme allows each BS to transmit three streams out of four and thus each user sees \ufb01ve interfering vectors in total: three out-of-cell and two intra-cell interfering vectors. Notice the \ufb01ve interfering May 27, 2010 DRAFT \f18 vectors are aligned onto a three dimensional linear subspace, thereby achieving interference alignment. C. Open-Loop Multi-User MIMO Since the feedback mechanism of our scheme is the same as that of standard multi-user MIMO techniques, any CSI feedback reduction scheme used for standard techniques can also be applied to our proposed scheme. For example, an open-loop multi-user MIMO technique can be easily applied to our scheme. Our scheme has only two differences: (1) each BS employs two cascaded precoders, including a \ufb01xed precoder \u00af P; (2) each user employs an MMSE-like receiver using \u00af \u03a6k. D. Multiple Interferers Our IA technique removes the interference from a single dominant interferer. A slight modi\ufb01cation can be made to cope with multiple dominant interferers. For example, consider a 19 hexagonal cell layout in Fig. 4 and suppose that mobiles are located at the middle point of three neighboring BSs. In this case, mobiles see two dominant interferers. One simple way is to take multiple dominant interferers into account in the process of computing the expected covariance matrix. Speci\ufb01cally, we can use: \u00af \u03a6k := E \u0014 (1 + INRrem)I + SNR S G\u03b2k \u00af PB\u03b2B\u2217 \u03b2 \u00af P\u2217G\u2217 \u03b2k + SNR S G\u03b3k \u00af PB\u03b3B\u2217 \u03b3 \u00af P\u2217G\u2217 \u03b3k \u0015 = (1 + INRrem)I + SNR S G\u03b2k \u00af P\u00af P\u2217G\u2217 \u03b2k + SNR S G\u03b3k \u00af P\u00af P\u2217G\u2217 \u03b3k, (12) where G\u03b2k denotes cross-link channel from BS \u03b2 to user k in cell \u03b1 and B\u03b2 indicates the zero-forcing precoder of BS \u03b2, and we use similar notation (G\u03b3k, B\u03b3) for cell \u03b3. We further assume that each entry of B\u03b2 and B\u03b3 is i.i.d. CN (0, 1 S). E. Optimization of \u03ba Our proposed scheme employs a parameter \u03ba in constructing the precoder \u00af P. We have considered one particular choice of (8), and simulation results are based on this choice. However, the performance can be improved by optimizing \u03ba. It could be future work to \ufb01nd the optimum \u03ba for different cellular layouts. DRAFT May 27, 2010 \f19 VII." + }, + { + "url": "http://arxiv.org/abs/1001.0107v2", + "title": "Exact Regeneration Codes for Distributed Storage Repair Using Interference Alignment", + "abstract": "The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes\nhas recently motivated a new class of codes, called Regenerating Codes, that\noptimally trade off storage cost for repair bandwidth. On one end of this\nspectrum of Regenerating Codes are Minimum Storage Regenerating (MSR) codes\nthat can match the minimum storage cost of MDS codes while also significantly\nreducing repair bandwidth. In this paper, we describe Exact-MSR codes which\nallow for any failed nodes (whether they are systematic or parity nodes) to be\nregenerated exactly rather than only functionally or information-equivalently.\nWe show that Exact-MSR codes come with no loss of optimality with respect to\nrandom-network-coding based MSR codes (matching the cutset-based lower bound on\nrepair bandwidth) for the cases of: (a) k/n <= 1/2; and (b) k <= 3. Our\nconstructive approach is based on interference alignment techniques, and,\nunlike the previous class of random-network-coding based approaches, we provide\nexplicit and deterministic coding schemes that require a finite-field size of\nat most 2(n-k).", + "authors": "Changho Suh, Kannan Ramchandran", + "published": "2009-12-31", + "updated": "2010-04-15", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION In distributed storage systems, maximum distance separable (MDS) erasure codes are wellknown coding schemes that can offer maximum reliability for a given storage overhead. For an (n, k) MDS code for storage, a source \ufb01le of size M bits is divided equally into k units (of size M k bits each), and these k data units are expanded into n encoded units, and stored at n October 29, 2018 DRAFT \f2 nodes. The code guarantees that a user or Data Collector (DC) can reconstruct the source \ufb01le by connecting to any arbitrary k nodes. In other words, any (n \u2212k) node failures can be tolerated with a minimum storage cost of M k at each of n nodes. While MDS codes are optimal in terms of reliability versus storage overhead, they come with a signi\ufb01cant maintenance overhead when it comes to repairing failed encoded nodes to restore the MDS system-wide property. Speci\ufb01cally, consider failure of a single encoded node and the cost needed to restore this node. It can be shown that this repair incurs an aggregate cost of M bits of information from k nodes. Since each encoded unit contains only M k bits of information, this represents a k-fold inef\ufb01ciency with respect to the repair bandwidth. This challenge has motivated a new class of coding schemes, called Regenerating Codes [1], [2], which target the information-theoretic optimal tradeoff between storage cost and repair bandwidth. On one end of this spectrum of Regenerating Codes are Minimum Storage Regenerating (MSR) codes that can match the minimum storage cost of MDS codes while also signi\ufb01cantly reducing repair bandwidth. As shown in [1], [2], the fundamental tradeoff between bandwidth and storage depends on the number of nodes that are connected to repair a failed node, simply called the degee d where k \u2264d \u2264n \u22121. The optimal tradeoff is characterized by (\u03b1, \u03b3) = \u0012M k , M k \u00b7 d d \u2212k + 1, \u0013 , (1) where \u03b1 and \u03b3 denote the optimal storage cost and repair bandwidth, respectively for repairing a single failed node, while retaining the MDS-code property for the user. Note that this code requires the same minimal storage cost (of size M k ) as that of conventional MDS codes, while substantially reducing repair bandwidth by a factor of k(d\u2212k+1) d (e.g., for (n, k, d) = (31, 6, 30), there is a 5x bandwidth reduction). In this paper, without loss of generality, we normalize the repair-bandwidth-per-link ( \u03b3 d) to be 1, making M = k(d \u2212k + 1). One can partition a whole \ufb01le into smaller chunks so that each has a size of k(d \u2212k + 1)1. While MSR codes enjoy substantial bene\ufb01ts over MDS codes, they come with some limitations in construction. Speci\ufb01cally, the achievable schemes in [1], [2] that meet the optimal tradeoff bound of (1) restore failed nodes in a functional manner only, using a random-network-coding based framework. This means that the replacement nodes maintain the MDS-code property (that 1In practice, the order of a \ufb01le size is of 103 (Kb) \u223c109 (Gb). Hence, it is reasonable to consider this arbitrary size of the chunk. October 29, 2018 DRAFT \f3 any k out of n nodes can allow for the data to be reconstructed) but do not exactly replicate the information content of the failed nodes. Mere functional repair can be limiting. First, in many applications of interest, there is a need to maintain the code in systematic form, i.e., where the user data in the form of k information units are exactly stored at k nodes and parity information (mixtures of k information units) are stored at the remaining (n\u2212k) nodes. Secondly, under functional repair, additional overhead information needs to be exchanged for continually updating repairing-and-decoding rules whenever a failure occurs. This can signi\ufb01cantly increase system overhead. A third problem is that the randomnetwork-coding based solution of [1] can require a huge \ufb01nite-\ufb01eld size, which can signi\ufb01cantly increase the computational complexity of encoding-and-decoding2. Lastly, functional repair is undesirable in storage security applications in the face of eavesdroppers. In this case, information leakage occurs continually due to the dynamics of repairing-and-decoding rules that can be potentially observed by eavesdroppers [3]. These drawbacks motivate the need for exact repair of failed nodes. This leads to the following question: is there a price for attaining the optimal tradeoff of (1) with the extra constraint of exact repair? The work in [4] sheds some light on this question: speci\ufb01cally, it was shown that under scalar linear codes3, when k n > 1 2 + 2 n, there is a price for exact repair. For large n, this case boils down to k n > 1 2, i.e., redundancy less than two. Now what about for k n \u22641 2? This paper resolves this open problem and shows that it is indeed possible to attain the optimal tradeoff of (1) for the case of k n \u22641 2 (and d \u22652k \u22121), while also guaranteeing exact repair. Furthermore, we show that for the special case of k \u22643, there is no price for exact repair, regardless of the value of n. The interesting special case in this class is the (5, 3) Exact-MSR code4, which is not covered by the \ufb01rst case of k n \u22641 2. Our achievable scheme builds on the concept of interference alignment, which was introduced 2In [1], Dimakis-Godfrey-Wu-Wainwright-Ramchandran translated the regenerating-codes problem into a multicast communication problem where random-network-coding-based schemes require a huge \ufb01eld size especially for large networks. In storage problems, the \ufb01eld size issue is further aggravated by the need to support a dynamically expanding network size due to the need for continual repair. 3In scalar linear codes, symbols are not allowed to be split into arbitrarily small sub-symbols as with vector linear codes. This is equivalent to having large block-lengths in the classical setting. Under non-linear and vector linear codes, whether or not the optimal tradeoff can be achieved for this regime remains open. 4Independently, Cullina-Dimakis-Ho in [5] found (5, 3) E-MSR codes de\ufb01ned over GF(3), based on a search algorithm. October 29, 2018 DRAFT \f4 in the context of wireless communication networks [6], [7]. The idea of interference alignment is to align multiple interference signals in a signal subspace whose dimension is smaller than the number of interferers. Speci\ufb01cally, consider the following setup where a decoder has to decode one desired signal which is linearly interfered with by two separate undesired signals. How many linear equations (relating to the number of channel uses) does the decoder need to recover its desired input signal? As the aggregate signal dimension spanned by desired and undesired signals is at most three, the decoder can naively recover its signal of interest with access to three linearly independent equations in the three unknown signals. However, as the decoder is interested in only one of the three signals, it can decode its desired unknown signal even if it has access to only two equations, provided the two undesired signals are judiciously aligned in a 1-dimensional subspace. See [6], [7], [8] for details. We will show in the sequel how this concept relates intimately to our repair problem. At a high level, the connection comes from our repair problem involving recovery of a subset (related to the subspace spanned by a failed node) of the overall aggregate signal space (related to the entire user data dimension). There are, however, signi\ufb01cant differences some bene\ufb01cial and some detrimental. On the positive side, while in the wireless problem, the equations are provided by nature (in the form of channel gain coef\ufb01cients), in our repair problem, the coef\ufb01cients of the equations are man-made choices, representing a part of the overall design space. On the \ufb02ip side, however, the MDS requirement of our storage code and the multiple failure con\ufb01gurations that need to be simultaneously addressed with a single code design generate multiple interference alignment constraints that need to be simultaneously satis\ufb01ed. This is particularly acute for a large value of k, as the number of possible failure con\ufb01gurations increases with n (which increases with k). Finally, another difference comes from the \ufb01nite-\ufb01eld constraint of our repair problem. We propose a common-eigenvector based conceptual framework (explained in Section IV) that covers all possible failure con\ufb01gurations. Based on this framework, we develop an interference alignment design technique for exact repair. We also propose another interference alignment scheme for a (5, 3) code5, which in turn shows the optimality of the cutset bound (1) for the case k \u22643. As in [4], our coding schemes are deterministic and require a \ufb01eld size of at most 5The \ufb01nite-\ufb01eld nature of the problem makes this challenging. October 29, 2018 DRAFT \f5 Functional repair Partially exact repair Exact repair Fig. 1. Repair models for distributed storage systems. In exact repair, the failed nodes are exactly regenerated, thus restoring lost encoded fragments with their exact replicas. In functional repair, the requirement is relaxed: the newly generated node can contain different data from that of the failed node as long as the repaired system maintains the MDS-code property. In partially exact repair, only systematic nodes are repaired exactly, while parity nodes are repaired only functionally. 2(n \u2212k). This is in stark contrast to the random-network-coding based solutions [1]. II. CONNECTION TO RELATED WORK As stated earlier, Regenerating Codes, which cover an entire spectrum of optimal tradeoffs between repair bandwidth and storage cost, were introduced in [1], [2]. As discussed, MSR codes occupy one end of this spectrum corresponding to minimum storage. At the other end of the spectrum live Minimum Bandwidth Regenerating (MBR) codes corresponding to minimum repair bandwidth. The optimal tradeoffs described in [1], [2] are based on random-networkcoding based approaches, which guarantee only functional repair. The topic of exact repair codes has received attention in the recent literature [9], [10], [4], [5], [11]. Wu and Dimakis in [9] showed that the MSR point (1) can be attained for the cases of: k = 2 and k = n\u22121. Rashmi-Shah-Kumar-Ramchandran in [10] showed that for d = n\u22121, the optimal MBR point can be achieved with a deterministic scheme requiring a small \ufb01nite-\ufb01eld size and zero repair-coding-cost. Subsequently, Shah-Rashmi-Kumar-Ramchandran in [4] developed partially exact codes for the MSR point corresponding to k n \u22641 2 + 2 n, where exact repair is limited to the systematic component of the code. See Fig. 1. Finding the fundamental limits under exact repair of all nodes (including parity) remained an open problem. A key contribution of this paper is to resolve this open problem by showing that E-MSR codes come with no extra cost over the optimal tradeoff of (1) for the case of k n \u22641 2 (and d \u22652k \u22121). For the most general case, October 29, 2018 DRAFT \f6 \ufb01nding the fundamental limits under exact repair constraints for all values of (n, k, d) remains an open problem. The constructive framework proposed in [4] forms the inspiration for our proposed solution in this paper. Indeed, we show that the code introduced in [4] for exact repair of only the systematic nodes can also be used to repair the non-systematic (parity) node failures exactly provided repair construction schemes are appropriately designed. This design for ensuring exact repair of all nodes is challenging and had remained an open problem: resolving this for the case of k n \u22641 2 (and d \u22652k \u22121) is a key contribution of this work. Another contribution of our work is the systematic development of a generalized family of code structures (of which the code structure of [4] is a special case), together with the associated optimal repair construction schemes. This generalized family of codes provides conceptual insights into the structure of solutions for the exact repair problem, while also opening up a much larger constructive design space of solutions. III. INTERFERENCE ALIGNMENT FOR DISTRIBUTED STORAGE REPAIR Linear network coding [12], [13] (that allows multiple messages to be linearly combined at network nodes) has been established recently as a useful tool for addressing interference issues even in wireline networks where all the communication links are orthogonal and non-interfering. This attribute was \ufb01rst observed in [9], where it was shown that interference alignment could be exploited for storage networks, speci\ufb01cally for minimum storage regenerating (MSR) codes having small k (k = 2). However, generalizing interference alignment to large values of k (even k = 3) proves to be challenging, as we describe in the sequel. In order to appreciate this better, let us \ufb01rst review the scheme of [9] that was applied to the exact repair problem. We will then address the dif\ufb01culty of extending interference alignment for larger systems and describe how to address this in Section IV. A. Review of (4, 2) E-MSR Codes [9] Fig. 2 illustrates an interference alignment scheme for a (4, 2) MDS code de\ufb01ned over GF(5). First one can easily check the MDS property of the code, i.e., all the source \ufb01les can be reconstructed from any k(= 2) nodes out of n(= 4) nodes. Let us see how failed node 1 (storing (a1, a2)) can be exactly repaired. We assume that the degree d (the number of storage October 29, 2018 DRAFT \f7 a1 + b1 1 1 b2 b1 1 1 a2 a1 a1 a2 b1 b2 a1 a2 b1 + b2 1 2a1 + a2 + b1 + b2 2a2 + b2 2a1 + b1 a2 + b2 1 a1 + 2a2 + b1 + b2 A source \ufb01le Encoded packets How to repair? Interference alignment failed node 1 node 2 node 3 node 4 (parity node 1) (parity node 2) Fig. 2. Interference alignment for a (4, 2) E-MSR code de\ufb01ned over GF(5) [9]. Choosing appropriate projection weights, we can align interference space of (b1, b2) into one-dimensional linear space spanned by [1, 1]t. As a result, we can successfully decode 2 desired unknowns (a1, a2) from 3 equations containing 4 unknowns (a1, a2, b1, b2). nodes connected to repair a failed node) is 3, and a source \ufb01le size M is 4. The cutset bound (1) then gives the fundamental limits of: storage cost \u03b1 = 2; and repair-bandwidth-per-link \u03b3 d = 1. The example illustrated in Fig. 2 shows that the parameter set described above is achievable using interference alignment. Here is a summary of the scheme. First notice that since the bandwidth-per-link is 1, two symbols in each storage node are projected into a scalar variable with projection weights. Choosing appropriate weights, we get the equations as shown in Fig. 2: (b1 + b2); a1 + 2a2 + (b1 + b2); 2a1 + a2 + (b1 + b2). Observe that the undesired signals (b1, b2) (interference) are aligned onto an 1-dimensional linear subspace, thereby achieving interference alignment. Therefore, we can successfully decode (a1, a2) with three equations although there are four unknowns. Similarly, we can repair (b1, b2) when it has failed. B. Matrix Notation We introduce matrix notation that provides geometric interpretation of interference alignment and is useful for generalization. Let a = (a1, a2)t and b = (b1, b2)t be 2-dimensional informationunit vectors, where (\u00b7)t indicates a transpose. Let Ai and Bi be 2-by-2 encoding matrices for parity node i (i = 1, 2), which contain encoding coef\ufb01cients for the linear combination of the October 29, 2018 DRAFT \f8 A1B\u22121 1 v\u03b11 A2B\u22121 1 v\u03b11 rank \u0001\u0002 A1B\u22121 1 v\u03b11 A2B\u22121 2 v\u03b11 \u0003\u0004 = 2 v\u03b11 B1v\u03b12 B2v\u03b13 v\u03b12 = B\u22121 1 v\u03b11 v\u03b13 = B\u22121 2 v\u03b11 v\u03b11 atA1 + btB1 atA2 + btB2 at bt v\u03b12 v\u03b13 Interference alignment Projection Vectors node 1 node 2 node 3 node 4 (parity node 1) (parity node 2) \uf8ee \uf8f0 vt \u03b11b (A1v\u03b12)ta + (B1v\u03b12)tb (A2v\u03b13)ta + (B2v\u03b13)tb \uf8f9 \uf8fb= \uf8ee \uf8f0 0 (A1v\u03b12)t (A2v\u03b13)t \uf8f9 \uf8fba + \uf8ee \uf8f0 vt \u03b11 (B1v\u03b12)t (B2v\u03b13)t \uf8f9 \uf8fbb Fig. 3. Geometric interpretation of interference alignment. The blue solid-line and red dashed-line vectors indicate linear subspaces with respect to \u201ca\u201d and \u201cb\u201d, respectively. The choice of v\u03b12 = B\u22121 1 v\u03b11 and v\u03b13 = B\u22121 2 v\u03b11 enables interference alignment. For the speci\ufb01c example of Fig. 2, the corresponding encoding matrices are A1 = [1, 0; 0, 2], B1 = [1, 0; 0, 1]. A2 = [2, 0; 0, 1], B2 = [1, 0; 0, 1]. components of \u201ca\u201d and \u201cb\u201d. For example, parity node 1 stores information in the form of atA1 + btB1, as shown in Fig. 3. The encoding matrices for systematic nodes are not explicitly de\ufb01ned since those are trivially inferred. Finally we de\ufb01ne 2-dimensional projection vectors v\u03b1i\u2019s (i = 1, 2, 3). Let us consider exact repair of systematic node 1. By connecting to three nodes, we get: btv\u03b11; at(A1v\u03b12) + bt(B1v\u03b12); at(A2v\u03b13) + bt(B2v\u03b13). Recall the goal, which is to decode 2 desired unknowns out of 3 equations including 4 unknowns. To achieve this goal, we need: rank \uf8eb \uf8ed \uf8ee \uf8f0(A1v\u03b12)t (A2v\u03b13)t \uf8f9 \uf8fb \uf8f6 \uf8f8= 2; rank \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 vt \u03b11 (B1v\u03b12)t (B2v\u03b13)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8= 1. (2) The second condition can be met by setting v\u03b12 = B\u22121 1 v\u03b11 and v\u03b13 = B\u22121 2 v\u03b11. This choice forces the interference space to be collapsed into a one-dimensional linear subspace, thereby achieving interference alignment. With this setting, the \ufb01rst condition now becomes rank \u0000\u0002 A1B\u22121 1 v\u03b11 A2B\u22121 2 v\u03b11 \u0003\u0001 = 2. (3) It can be easily veri\ufb01ed that the choice of Ai\u2019s and Bi\u2019s given in Figs. 2 and 3 guarantees the above condition. When the node 2 fails, we get a similar condition: rank \u0000\u0002 B1A\u22121 1 v\u03b21 B2A\u22121 2 v\u03b21 \u0003\u0001 = 2, (4) October 29, 2018 DRAFT \f9 where v\u03b2i\u2019s denote projection vectors for node 2 repair. This condition also holds under the given choice of encoding matrices. C. Connection with Interference Channels in Communication Problems Observe the three equations shown in Fig. 3: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 0 (A1v\u03b12)t (A2v\u03b13)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fba | {z } desired signals + \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 vt \u03b11 (B1v\u03b12)t (B2v\u03b13)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fbb | {z } interference . Separating into two parts, we can view this problem as a wireless communication problem, wherein a subset of the information is desired to be decoded in the presence of interference. Note that for each term (e.g., A1v\u03b12), the matrix A1 and vector v\u03b12 correspond to channel matrix and transmission vector in wireless communication problems, respectively. There are, however, signi\ufb01cant differences. In the wireless communication problem, the channel matrices are provided by nature and therefore not controllable. The transmission strategy alone (vector variables) can be controlled for achieving interference alignment. On the other hand, in our storage repair problems, both matrices and vectors are controllable, i.e., projection vectors and encoding matrices can be arbitrarily designed, resulting in more \ufb02exibility. However, our storage repair problem comes with unparalleled challenges due to the MDS requirement and the multiple failure con\ufb01gurations. These induce multiple interference alignment constraints that need to be simultaneously satis\ufb01ed. What makes this dif\ufb01cult is that the encoding matrices, once designed, must be the same for all repair con\ufb01gurations. This is particularly acute for large values of k (even k = 3), as the number of possible failure con\ufb01gurations increases with n (which increases with k). IV. A PROPOSED FRAMEWORK FOR EXACT-MSR CODES We propose a common-eigenvector based conceptual framework to address the exact repair problem. This framework draws its inspiration from the work in [4] which guarantees the exact repair of systematic nodes, while satisfying the MDS code property, but which does not provide exact repair of failed parity nodes. In providing a solution for the exact repair of all nodes, we propose here a generalized family of codes (of which the code in [4] is a special case). October 29, 2018 DRAFT \f10 v\u03b11 v\u03b12 v\u03b13 v\u03b14 v\u03b15 Interference alignment v\u03b13 = B\u22121 1 v\u03b11 v\u03b14 = B\u22121 2 v\u03b11 v\u03b15 = B\u22121 3 v\u03b11 at bt ct (C1B\u22121 1 )v\u03b11 (C2B\u22121 2 )v\u03b11 (C3B\u22121 3 )v\u03b11 v\u03b12 Goal: rank=3 rank=1 rank=1 atA1 + btB1 + ctC1 atA2 + btB2 + ctC2 atA3 + btB3 + ctC3 \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 0 (A1v\u03b13)t (A2v\u03b14)t (A3v\u03b15)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb a + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 vt \u03b11 0 (B1v\u03b13)t (B2v\u03b14)t (B3v\u03b15)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb b + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 vt \u03b12 (C1v\u03b13)t (C2v\u03b14)t (C3v\u03b15)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb c Fig. 4. Dif\ufb01culty of achieving interference alignment simultaneously. This both provides insights into the structure of codes for exact repair of all nodes, as well as opens up a much larger design space for constructive solutions. Speci\ufb01cally, we propose a common-eigenvector based approach building on a certain elementary matrix property [14], [15] for the generalized code construction. Moreover, as in [4], our proposed coding schemes are deterministic and constructive, requiring a symbol alphabet-size of at most (2n \u22122k). Our framework consists of four components: (1) developing a family of codes6 for exact repair of systematic codes based on the common-eigenvector concept; (2) drawing a dual relationship between the systematic and parity node repair; (3) guaranteeing the MDS property of the code; (4) constructing codes with \ufb01nite-\ufb01eld alphabets. The framework covers the case of n \u22652k (and d \u22652k \u22121). It turns out that the (2k, k, 2k \u22121) code case contains the key design ingredients and the case of n \u22652k can be derived from this (see Section VI). Hence, we \ufb01rst focus on the simplest example: (6, 3, 5) E-MSR codes. Later in Section VI, we will generalize this to arbitrary (n, k, d) codes in the class. 6Interestingly, the structure of the code in [4] turns out to work for the exact repair of both systematic and parity nodes provided appropriate repair schemes are developed. October 29, 2018 DRAFT \f11 A. Code Structure for Systematic Node Repair For k \u22653 (more-than-two interfering information units), achieving interference alignment for exact repair turns out to be signi\ufb01cantly more complex than the k = 2 case. Fig. 4 illustrates this dif\ufb01culty through the example of repairing node 1 for a (6, 3, 5) code. By the optimal tradeoff (1), the choice of M = 9 gives \u03b1 = 3 and \u03b3 d = 1. Let a = (a1, a2, a3)t, b = (b1, b2, b3)t and c = (c1, c2, c3)t. We de\ufb01ne 3-by-3 encoding matrices of Ai, Bi and Ci (for i = 1, 2, 3); and 3-dimensional projection vectors v\u03b1i\u2019s. Consider the 5 (= d) equations downloaded from the nodes: \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 0 (A1v\u03b13)t (A2v\u03b14)t (A3v\u03b15)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb a + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 vt \u03b11 0 (B1v\u03b13)t (B2v\u03b14)t (B3v\u03b15)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb b + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 vt \u03b12 (C1v\u03b13)t (C2v\u03b14)t (C3v\u03b15)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb c. In order to successfully recover the desired signal components of \u201ca\u201d, the matrices associated with b and c should have rank 1, respectively, while the matrix associated with a should have full rank of 3. In accordance with the (4, 2) code example in Fig. 3, if one were to set v\u03b13 = B\u22121 1 v\u03b11, v\u03b14 = B\u22121 2 v\u03b12 and v\u03b15 = B\u22121 3 v\u03b11, then it is possible to achieve interference alignment with respect to b. However, this choice also speci\ufb01es the interference space of c. If the Bi\u2019s and Ci\u2019s are not designed judiciously, interference alignment is not guaranteed for c. Hence, it is not evident how to achieve interference alignment at the same time. In order to address the challenge of simultaneous interference alignment, we invoke a common eigenvector concept. The idea consists of two parts: (i) designing the (Ai, Bi, Ci)\u2019s such that v1 is a common eigenvector of the Bi\u2019s and Ci\u2019s, but not of Ai\u2019s7; (ii) repairing by having survivor nodes project their data onto a linear subspace spanned by this common eigenvector v1. We can then achieve interference alignment for b and c at the same time, by setting v\u03b1i = v1, \u2200i. As long as [A1v1, A2v1, A3v1] is invertible, we can also guarantee the decodability of a. See Fig. 5. 7Of course, \ufb01ve additional constraints also need to be satis\ufb01ed for the other \ufb01ve failure con\ufb01gurations for this (6, 3, 5) code example. October 29, 2018 DRAFT \f12 Goal: rank=3 rank=1 rank=1 at bt ct v1 atA1 + btB1 + ctC1 atA2 + btB2 + ctC2 atA3 + btB3 + ctC3 Idea: \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 0 (A1v1)t (A2v1)t (A3v1)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb a + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 vt 1 0 (B1v1)t (B2v1)t (B3v1)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb b + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 vt 1 (C1v1)t (C2v1)t (C3v1)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb c v1 v1 v1 v1 A1v1 v1 v1 A2v1 A3v1 (i) Design Ai\u2019s, Bi\u2019s and Ci\u2019s s.t. v1 is a common eigenvector of the Bi\u2019s and Ci\u2019s, but not of the Ai\u2019s. (ii) Repair by having survivor nodes project their data onto a linear node 1 node 2 node 3 node 4 node 5 node 6 (parity node 1) (parity node 2) (parity node 3) subspace spanned by this common eigenvector v1. Fig. 5. Illustration of exact repair of systematic node 1 for (6, 3, 5) E-MSR codes. The idea consists of two parts: (i) designing (Ai, Bi, Ci)\u2019s such that v1 is a common eigenvector of the Bi\u2019s and Ci\u2019s, but not of Ai\u2019s; (ii) repairing by having survivor nodes project their data onto a linear subspace spanned by this common eigenvector v1. The challenge is now to design encoding matrices to guarantee the existence of a common eigenvector while also satisfying the decodability of desired signals. The dif\ufb01culty comes from the fact that in our (6, 3, 5) code example, these constraints need to be satis\ufb01ed for all six possible failure con\ufb01gurations. The structure of elementary matrices [14], [15] (generalized matrices of Householder and Gauss matrices) gives insights into this. To see this, consider a 3-by-3 elementary matrix A: A = uvt + \u03b1I, (5) where u and v are 3-dimensional vectors. Here is an observation that motivates our proposed structure: the dimension of the null space of v is 2 and the null vector v\u22a5is an eigenvector of October 29, 2018 DRAFT \f13 A, i.e., Av\u22a5= \u03b1v\u22a5. This motivates the following structure: A1 = u1vt 1 + \u03b11I; B1 = u1vt 2 + \u03b21I; C1 = u1vt 3 + \u03b31I A2 = u2vt 1 + \u03b12I; B2 = u2vt 2 + \u03b22I; C2 = u2vt 3 + \u03b32I A3 = u3vt 1 + \u03b13I; B3 = u3vt 2 + \u03b23I; C3 = u3vt 3 + \u03b33I, (6) where vi\u2019s are 3-dimensional linearly independent vectors and so are ui\u2019s. The values of the \u03b1i\u2019s, \u03b2i\u2019s and \u03b3i\u2019s can be arbitrary non-zero values. First consider the simple case where the vi\u2019s are orthonormal. This is for conceptual simplicity. Later we will generalize to the case where the vi\u2019s need not be orthogonal but only linearly independent: namely, bi-orthogonal case. For the orthogonal case, we see that for i = 1, 2, 3, Aiv1 = \u03b1iv1 + ui, Biv1 = \u03b2iv1, Civ1 = \u03b3iv1. (7) Importantly, notice that v1 is a common eigenvector of the Bi\u2019s and Ci\u2019s, while simultaneously ensuring that the vectors of Aiv1 are linearly independent. Hence, setting v\u03b1i = v1 for all i, it is possible to achieve simultaneous interference alignment while also guaranteeing the decodability of the desired signals. See Fig. 5. On the other hand, this structure also guarantees exact repair for b and c. We use v2 for exact repair of b. It is a common eigenvector of the Ci\u2019s and Ai\u2019s, while ensuring [B1v2, B2v2, B3v2] invertible. Similarly, v3 is used for c. We will see that a dual basis property gives insights into the general bi-orthogonal case where {v} := (v1, v2, v3) is not orthogonal but linearly independent. In this case, de\ufb01ning a dual basis {v\u2032} := (v\u2032 1, v\u2032 2, v\u2032 3) gives the solution: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 v\u2032t 1 v\u2032t 2 v\u2032t 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= h v1 v2 v3 i\u22121 . The de\ufb01nition gives the following property: v\u2032t i vj = \u03b4(i \u2212j), \u2200i, j. Using this property, one can see that v\u2032 1 is a common eigenvector of the Bi\u2019s and Ci\u2019s: Aiv\u2032 1 = \u03b1iv\u2032 1 + ui, Biv\u2032 1 = \u03b2iv\u2032 1, Civ\u2032 1 = \u03b3iv\u2032 1. (8) October 29, 2018 DRAFT \f14 So it can be used as a projection vector for exact repair of a. Similarly, we can use v\u2032 2 and v\u2032 3 for exact repair of b and c, respectively. B. Dual Relationship between Systematic and Parity Node Repair We have seen so far how to ensure exact repair of the systematic nodes. We have known that if {v} is linearly independent and so {u} is, then using the code-structure of (6) together with projection direction enables repair, for arbitrary values of (\u03b1i, \u03b2i, \u03b3i)\u2019s. A natural question is now: will this code structure also guarantee exact repair of parity nodes? It turns out that for exact repair of all nodes, we need a special relationship between {v} and {u} through the correct choice of the (\u03b1i, \u03b2i, \u03b3i)\u2019s. We will show that parity nodes can be repaired by drawing a dual relationship with systematic nodes. The procedure has two steps. The \ufb01rst is to remap parity nodes with a\u2032, b\u2032, and c\u2032, respectively: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 a\u2032 b\u2032 c\u2032 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 At 1 Bt 1 Ct 1 At 2 Bt 2 Ct 2 At 3 Bt 3 Ct 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 a b c \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb. Systematic nodes can then be rewritten in terms of the prime notations: at = a\u2032tA\u2032 1 + b\u2032tB\u2032 1 + c\u2032tC\u2032 1, bt = a\u2032tA\u2032 2 + b\u2032tB\u2032 2 + c\u2032tC\u2032 2, ct = a\u2032tA\u2032 3 + b\u2032tB\u2032 3 + c\u2032tC\u2032 3, (9) where the newly mapped encoding matrices (A\u2032 i, B\u2032 i, Ci)\u2019s are de\ufb01ned as: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A\u2032 1 A\u2032 2 A\u2032 3 B\u2032 1 B\u2032 2 B\u2032 3 C\u2032 1 C\u2032 2 C\u2032 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A1 A2 A3 B1 B2 B3 C1 C2 C3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \u22121 . (10) With this remapping, one can dualize the relationship between systematic and parity node repair. Speci\ufb01cally, if all of the A\u2032 i\u2019s, B\u2032 i\u2019s, and C\u2032 i\u2019s are elementary matrices and form a similar codestructure as in (6), exact repair of the parity nodes becomes transparent. The challenge is now how to guarantee the dual structure. In Lemma 1, we show that a special relationship between {u} and {v} through (\u03b1i, \u03b2i, \u03b3i)\u2019s can guarantee this dual relationship of (13). October 29, 2018 DRAFT \f15 a\u2032tA\u2032 1 + b\u2032tB\u2032 1 + c\u2032tC\u2032 1 A\u2032t 3 a\u2032 + B\u2032t 3 b\u2032 + C\u2032t 3 c\u2032 Goal: rank=3 rank=1 rank=1 u1 b\u2032t c\u2032t a\u2032t \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 0 (A\u2032 1u1)t (A\u2032 2u1)t (A\u2032 3u1)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb a\u2032 + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 ut 1 0 (B\u2032 1u1)t (B\u2032 2u1)t (B\u2032 3u1)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb b\u2032 + \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 0 ut 1 (C\u2032 1u1)t (C\u2032 2u1)t (C\u2032 3u1)t \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb c\u2032 a\u2032tA\u2032 2 + b\u2032tB\u2032 2 + c\u2032tC\u2032 2 A\u2032 1u1 u1 u1 u1 u1 u1 u1 A\u2032 2u1 A\u2032 3u1 We provide su\ufb03cient conditions to ensure the dual structure: at bt ct atA1 + btB1 + ctC1 atA2 + btB2 + ctC2 atA3 + btB3 + ctC3 remapping (i) \u03ba[u1, u2, u3] = [v\u2032 1, v\u2032 2, v\u2032 3] \uf8ee \uf8f0 \u03b11 \u03b12 \u03b13 \u03b21 \u03b22 \u03b23 \u03b31 \u03b32 \u03b33 \uf8f9 \uf8fb; (ii) \uf8ee \uf8f0 \u03b11 \u03b12 \u03b13 \u03b21 \u03b22 \u03b23 \u03b31 \u03b32 \u03b33 \uf8f9 \uf8fbis invertible. Fig. 6. Exact repair of a parity node for (6, 3) E-MSR code. The idea is to construct the dual code-structure of (13) by remapping parity nodes and then adding suf\ufb01cient conditions of (11) and (12). Lemma 1: Suppose M := \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 \u03b11 \u03b12 \u03b13 \u03b21 \u03b22 \u03b23 \u03b31 \u03b32 \u03b33 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fbis invertible. (11) Also assume \u03baU = V\u2032M. (12) where U = [u1, u2, u3], V\u2032 = [v\u2032 1, v\u2032 2, v\u2032 3], {v\u2032} := {v\u2032 1, v\u2032 2, v\u2032 3} is the dual basis of {v}, i.e., v\u2032t i vj = \u03b4(i \u2212j) and \u03ba is an arbitrary non-zero value s.t. 1 \u2212\u03ba2 \u0338= 0. Then, we can obtain the October 29, 2018 DRAFT \f16 dual structure of (6) as follows: A\u2032 1 = 1 1 \u2212\u03ba2 \u0000v\u2032 1u\u2032t 1 \u2212\u03ba2\u03b1\u2032 1I \u0001 ; B\u2032 1 = 1 1 \u2212\u03ba2 \u0000v\u2032 1u\u2032t 2 \u2212\u03ba2\u03b1\u2032 2I \u0001 ; C\u2032 1 = 1 1 \u2212\u03ba2 \u0000v\u2032 1u\u2032t 3 \u2212\u03ba2\u03b1\u2032 3I \u0001 A\u2032 2 = 1 1 \u2212\u03ba2 \u0000v\u2032 2u\u2032t 1 \u2212\u03ba2\u03b2\u2032 1I \u0001 ; B\u2032 2 = 1 1 \u2212\u03ba2 \u0000v\u2032 2u\u2032t 2 \u2212\u03ba2\u03b2\u2032 2I \u0001 ; C\u2032 2 = 1 1 \u2212\u03ba2 \u0000v\u2032 2u\u2032t 3 \u2212\u03ba2\u03b2\u2032 3I \u0001 A\u2032 3 = 1 1 \u2212\u03ba2 \u0000v\u2032 3u\u2032t 1 \u2212\u03ba2\u03b3\u2032 1I \u0001 ; B\u2032 3 = 1 1 \u2212\u03ba2 \u0000v\u2032 3u\u2032t 2 \u2212\u03ba2\u03b3\u2032 2I \u0001 ; C\u2032 3 = 1 1 \u2212\u03ba2 \u0000v\u2032 3u\u2032t 3 \u2212\u03ba2\u03b3\u2032 3I \u0001 , (13) where {u\u2032} is the dual basis of {u}, i.e., u\u2032t i uj = \u03b4(i \u2212j) and (\u03b1\u2032 i, \u03b2\u2032 i, \u03b3\u2032 i)\u2019s are the dual basis vectors, i.e., < (\u03b1\u2032 i, \u03b2\u2032 i, \u03b3\u2032 i), (\u03b1j, \u03b2j, \u03b3j) >= \u03b4(i \u2212j): \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 \u03b1\u2032 1 \u03b2\u2032 1 \u03b3\u2032 1 \u03b1\u2032 2 \u03b2\u2032 2 \u03b3\u2032 2 \u03b1\u2032 3 \u03b2\u2032 3 \u03b3\u2032 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 \u03b11 \u03b12 \u03b13 \u03b21 \u03b22 \u03b23 \u03b31 \u03b32 \u03b33 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \u22121 . (14) Proof: See Appendix A. Remark 1: The dual structure of (13) now gives exact-repair solutions for parity nodes. For exact repair of parity node 1, we can use vector u1 (a common eigenvector of the B\u2032 i\u2019s and C\u2032 i\u2019s), since it enables simultaneous interference alignment for b\u2032 and c\u2032, while ensuring the decodability of a\u2032. See Fig. 6. Notice that more conditions of (11) and (12) are added to ensure exact repair of all nodes, while these conditions were unnecessary for exact repair of systematic nodes only. Also note these are only suf\ufb01cient conditions. Remark 2: Note that the dual structure of (13) is quite similar to the primary structure of (6). The only difference is that in the dual structure, {u} and {v} are interchanged to form a transpose-like structure. This reveals insights into how to guarantee exact repair of parity nodes in a transparent manner. C. The MDS-Code Property The third part of the framework is to guarantee the MDS-code property, which allows us to identify speci\ufb01c constraints on the (\u03b1i, \u03b2i, \u03b3i)\u2019s and/or ({v}, {u}). Consider four cases, associated in the Data Collector (DC) who is intended in the source \ufb01le data: (1) 3 systematic nodes; (2) 3 parity nodes; (3) 1 systematic and 2 parity nodes; (4) 1 systematic and 2 parity nodes. The \ufb01rst is a trivial case. The second case has been already veri\ufb01ed in the process of forming the dual code-structure of (13). The invertibility condition of (11) together with (12) suf\ufb01ces to October 29, 2018 DRAFT \f17 ensure the invertibility of the composite matrix. The third case requires the invertibility of all of each encoding matrix. In this case, it is necessary that the \u03b1i\u2019s, \u03b2i\u2019s and \u03b3i\u2019s are non-zero values; otherwise, each encoding matrix has rank 1. Also the non-zero values together with (12) guarantee the invertibility of each encoding matrix. Under these conditions, for example, the inverse of A1 is well de\ufb01ned as: A\u22121 1 = 1 \u03b11 \u0012 \u2212 1 \u03b11 + vt 1u1 u1vt 1 + I \u0013 = 1 \u03b11 \u0012 \u2212 \u03ba \u03b11(\u03ba + 1)u1vt 1 + I \u0013 . where the second equality follows from vt 1u1 = \u03b11 \u03ba due to (12). The last case requires some non-trivial work. Consider a speci\ufb01c example where the DC connects to nodes (3,4,5). In this case, we \ufb01rst recover c from node 3 and subtract the terms associated with c from nodes 4 and 5. We then get: h at bt i \uf8ee \uf8f0A1 A2 B1 B2 \uf8f9 \uf8fb= h at bt i \uf8ee \uf8f0u1vt 1 + \u03b11I u2vt 1 + \u03b12I u1vt 2 + \u03b21I u2vt 2 + \u03b22I \uf8f9 \uf8fb. (15) Using a Gaussian elimination method, we show that the sub-composite matrix is invertible if M2 := \uf8ee \uf8f0\u03b11 \u03b12 \u03b21 \u03b22 \uf8f9 \uf8fbis invertible. (16) Here is the Gaussian elimination method: \uf8ee \uf8f0u1vt 1 + \u03b11I u2vt 1 + \u03b12I u1vt 2 + \u03b21I u2vt 2 + \u03b22I \uf8f9 \uf8fb(a) \u223c \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 vt 1 + \u03b11u\u2032t 1 \u03b12u\u2032t 1 vt 2 + \u03b21u\u2032t 1 \u03b22u\u2032t 1 \u03b11u\u2032t 2 vt 1 + \u03b12u\u2032t 2 \u03b21u\u2032t 2 vt 2 + \u03b22u\u2032t 2 \u03b11u\u2032t 3 \u03b12u\u2032t 3 \u03b21u\u2032t 3 \u03b22u\u2032t 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb (b) \u223c \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 \u03b1\u2032 1vt 1 + \u03b2\u2032 1vt 2 + u\u2032t 1 0t \u03b1\u2032 2vt 1 + \u03b2\u2032 2vt 2 u\u2032t 1 u\u2032t 2 \u03b1\u2032 1vt 1 + \u03b2\u2032 1vt 2 0t \u03b1\u2032 2vt 1 + \u03b2\u2032 2vt 2 + u\u2032t 2 u\u2032t 3 0t 0t u\u2032t 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb . where (a) following from multiplying [u\u2032t 1, 0t; 0t, u\u2032t 1; u\u2032t 2, 0t; 0t, u\u2032t 2; u\u2032t 3, 0t; 0t, u\u2032t 3] to the left; (b) follows from multiplying [M\u22121 2 , 0, 0; 0, M\u22121 2 , 0; 0, 0, M\u22121 2 ] to the left. Here (\u03b1\u2032 i, \u03b2\u2032 i)\u2019s are the dual basis vectors of (\u03b1i, \u03b2i)\u2019s. Note that the resulting matrix is invertible, since {u\u2032} is a dual basis. October 29, 2018 DRAFT \f18 Considering the above 4 cases, the following condition together with (11) and (12) suf\ufb01ces for guaranteeing the MDS-code property: Any submatrix of M of (11) is invertible. (17) D. Code Construction with Finite-Field Alphabets The last part is to design M of (11) and {v} := (v1, v2, v3) in (6) such that {v} is linearly independent and the conditions of (12) and (17) are satis\ufb01ed. First, in order to guarantee (17), we can use a Cauchy matrix, as it was used for the code introduced in [4]. De\ufb01nition 1 (A Cauchy Matrix [16]): A Cauchy matrix M is an m \u00d7 n matrix with entries mij in the form: mij = 1 xi \u2212yj , \u2200i = 1, \u00b7 \u00b7 \u00b7 , m, j = 1, \u00b7 \u00b7 \u00b7 n, xi \u0338= yj, where xi and yj are elements of a \ufb01eld and {xi} and {yj} are injective sequences, i.e., elements of the sequence are distinct. The injective property of {xi} and {yj} requires a \ufb01nite \ufb01eld size of 2s for an s \u00d7 s Cauchy matrix. Therefore, in our (6, 3, 5) code example, the \ufb01nite \ufb01eld size of 6 suf\ufb01ces. The \ufb01eld size condition for guaranteeing linear independence of {v} is more relaxed. E. Summary Using the code structure of (6) and the conditions of (11), (12) and (17), we can now state the following theorem. Theorem 1 ((6, 3, 5) E-MSR Codes): Suppose M of (11) is a Cauchy matrix, i.e., every submatrix of is invertible. Each element of M is in GF(q) and q \u22656. Suppose encoding matrices form the code structure of (6), {v} := (v1, v2, v3) is linearly independent, and {u} satis\ufb01es the condition of (12). Then, the code satis\ufb01es the MDS property and achieves the MSR point under exact repair constraints of all nodes. Remark 3: Note that the code introduced in [4] is a special case of Theorem 1, where V = [v1, v2, v3] = I. October 29, 2018 DRAFT \f19 V. EXAMPLES We provide two numerical examples: (1) an orthogonal code example where V = [v1, v2, v3] is orthogonal, e.g., V = I; (2) an bi-orthogonal code example where V is not orthogonal but invertible. As mentioned earlier, the code in [4] belongs to the case of V = I. We will also discuss the complexity of repair construction schemes for each of these examples. It turns out that the \ufb01rst code has signi\ufb01cantly lower complexity for exact repair of systematic nodes, as compared to that of parity nodes. On the other hand, for the second bi-orthogonal codes, the speci\ufb01c choice of V = \u03ba\u22121Mt gives U = I, thereby providing much simpler parity-node repair schemes instead. Depending on applications of interest, one can choose an appropriate code among our generalized family of codes. A. Orthogonal Case We present an example of (6, 3, 5) E-MSR codes de\ufb01ned over GF(4) where V = I and M = \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 1 1 1 1 2 3 1 3 2 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb, U = \u03ba\u22121V\u2032M = 2 \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 1 1 1 1 2 3 1 3 2 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 2 2 2 2 3 1 2 1 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb, where U is set based on (12) and \u03ba = 2\u22121. We use a generator polynomial of g(x) = x2 +x+1. Notice that we employ a non-Cauchy-type matrix to construct a \ufb01eld-size 4 code (smaller than 6 required when using a Cauchy matrix). Remember that a Cauchy matrix provides only a suf\ufb01cient condition for ensuring the invertibility of any submatrices of M. By (6) and (13), the primary and dual code structures are given by G = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 3 0 0 3 0 0 3 0 0 2 1 0 3 1 0 1 1 0 2 0 1 1 0 1 3 0 1 1 2 0 2 2 0 3 2 0 0 3 0 0 1 0 0 2 0 0 2 1 0 1 2 0 3 3 1 0 2 3 0 2 2 0 2 0 1 2 0 3 3 0 2 1 0 0 3 0 0 2 0 0 1 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb ; G\u22121 = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 2 1 1 3 0 0 3 0 0 0 3 0 1 2 1 0 3 0 0 0 3 0 0 3 1 1 2 2 3 2 2 0 0 1 0 0 0 3 0 1 1 2 0 1 0 0 0 3 0 0 2 1 3 3 2 2 3 1 0 0 2 0 0 0 3 0 1 3 3 0 2 0 0 0 3 0 0 1 1 2 1 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb . (18) October 29, 2018 DRAFT \f20 \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb a1 a2 a3 b1 b2 b3 c1 c2 c3 3a1 + 2a2 + 2a3 + b1 + c1 a2 + 2b1 + 3b2 + 2b3 + c2 a3 + b3 + 2c1 + 2c2 + 3c3 3a1 + 3a2 + a3 + 2b1 + 3c1 a2 + 2b1 + b2 + b3 + 3c2 a3 + 2b3 + 2c1 + 3c2 + 2c3 3a1 + a2 + 3a3 + 3b1 + 2c1 a2 + 2b1 + 2b2 + 3b3 + 2c2 a3 + 3b3 + 2c1 + c2 + 1c3 2a\u2032 1 + 2b\u2032 1 + 2c\u2032 1 a\u2032 1 + 3a\u2032 2 + 3b\u2032 1 + 3b\u2032 2 + 2c\u2032 1 + 3c\u2032 2 a\u2032 1 + 3a\u2032 3 + 2b\u2032 1 + 3b\u2032 3 + 3c\u2032 1 + 3c\u2032 3 3a\u2032 1 + a\u2032 2 + 2b\u2032 1 + b\u2032 2 + c\u2032 1 + c\u2032 2 2a\u2032 2 + b\u2032 2 + 3c\u2032 2 a\u2032 2 + 3a\u2032 3 + 2b\u2032 2 + 2b\u2032 3 + 3c\u2032 2 + c\u2032 3 3a\u2032 1 + a\u2032 3 + b\u2032 1 + b\u2032 3 + 2c\u2032 1 + c\u2032 3 3a\u2032 2 + a\u2032 3 + b\u2032 2 + 3b\u2032 3 + 2c\u2032 2 + 2c\u2032 3 2a\u2032 3 + 3b\u2032 3 + c\u2032 3 a\u2032 1 a\u2032 2 a\u2032 3 b\u2032 1 b\u2032 2 b\u2032 3 c\u2032 1 c\u2032 2 c\u2032 3 c\u2032 1 + c\u2032 2 + c\u2032 3 b\u2032 1 + b\u2032 2 + b\u2032 3 \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb 2a\u2032 1 + 3a\u2032 2 + 3a\u2032 3 +3b\u2032 1 + 3b\u2032 2 + 3b\u2032 3 +3c\u2032 1 + 3c\u2032 2 + 3c\u2032 3 3a\u2032 1 + 2a\u2032 2 + 3a\u2032 3 +2b\u2032 1 + 2b\u2032 2 + 2b\u2032 3 +c\u2032 1 + c\u2032 2 + c\u2032 3 3a\u2032 1 + 3a\u2032 2 + 2a\u2032 3 +b\u2032 1 + b\u2032 2 + b\u2032 3 +2c\u2032 1 + 2c\u2032 2 + 2c\u2032 3 (a) Exact repair of systematic node 1 (b) Exact repair of parity node 1 Fig. 7. Orthogonal case: Illustration of exact pair for a (6, 3, 5) E-MSR code de\ufb01ned over GF(4) where a generator polynomial g(x) = x2 + x + 1. The projection vector solution for systematic node repair is quite simple: v\u03b1i = v1 = (1, 0, 0)t, \u2200i. We download only the \ufb01rst equation from each survivor node; For parity node repair, our new framework provides a simple scheme: setting all of the projection vectors as 2\u22121u1 = (1, 1, 1)t. This enables simultaneous interference alignment, while guaranteeing the decodability of a. where G := \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A1 A2 A3 B1 B2 B3 C1 C2 C3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb; G\u22121 = \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A\u2032 1 A\u2032 2 A\u2032 3 B\u2032 1 B\u2032 2 B\u2032 3 C\u2032 1 C\u2032 2 C\u2032 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb. Fig. 7 shows an example for exact repair of (a) systematic node 1 and (b) parity node 1. Note that the projection vector solution for systematic node repair is quite simple: v\u03b1i = v1 = (1, 0, 0)t, \u2200i. We download only the \ufb01rst equation from each survivor node. Notice that the downloaded \ufb01ve equations contain only \ufb01ve unknown variables of (a1, a2, a3, b1, c1) and three equations associated with a are linearly independent. Hence, we can successfully recover a. On the other hand, exact repair of parity nodes seems non-straightforward. However, our October 29, 2018 DRAFT \f21 framework provides quite a simple repair scheme: setting all of the projection vectors as 2\u22121u1 = (1, 1, 1)t. This enables simultaneous interference alignment, while guaranteeing the decodability of a. Notice that (b\u2032 1, b\u2032 2, b\u2032 3) and (c\u2032 1, c\u2032 2, c\u2032 3) are aligned into b\u2032 1+b\u2032 2+b\u2032 3 and c\u2032 1+c\u2032 2+c\u2032 3, respectively, while three equations associated with a\u2032 are linearly independent. As one can see, the complexity of systematic node repair is a little bit lower than that of parity node repair, although both repair schemes are simple. Hence, one can expect that this orthogonal code is useful for the applications where the complexity of systematic node repair needs to be signi\ufb01cantly low. B. Bi-Orthogonal Case We provide another example of (6, 3, 5) E-MSR codes where V is not orthogonal but invertible. We use the same \ufb01eld size of 4, the same generator polynomial and the same M. Instead we choose non-orthogonal V so that the complexity of parity node repair can be signi\ufb01cantly low. Our framework provides a concrete guideline for designing this type of code. Remember that the projection vector solutions are u1, u2 and u3 for exact repair of each parity node, respectively. For low complexity, we can \ufb01rst set U = I. The condition (12) then gives the following choice: V = Mt\u03ba\u22121 = \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 2 2 2 2 3 1 2 1 3 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb, where we use \u03ba = 2\u22121. By (6) and (13), the primary and dual code structures are given by G = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 3 2 2 1 0 0 1 0 0 0 1 0 2 3 2 0 1 0 0 0 1 0 0 1 2 2 3 3 3 1 2 0 0 3 0 0 0 1 0 2 1 1 0 3 0 0 0 1 0 0 2 2 3 2 3 1 3 3 0 0 2 0 0 0 1 0 2 2 3 0 2 0 0 0 1 0 0 3 2 1 1 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb ; G\u22121 = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 2 0 0 2 0 0 2 0 0 1 3 0 3 3 0 2 3 0 1 0 3 2 0 3 3 0 3 3 1 0 2 1 0 1 1 0 0 2 0 0 1 0 0 3 0 0 1 3 0 2 2 0 3 1 3 0 1 1 0 1 2 0 1 0 3 1 0 1 3 0 2 2 0 0 2 0 0 2 0 0 1 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb . (19) October 29, 2018 DRAFT \f22 \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb a1 a2 a3 b1 b2 b3 c1 c2 c3 3a1 + 3b1 + 3c1 2a1 + a2 + 3b1 + b2 + c1 + c2 2a1 + a3 + b1 + b3 + 3c1 + c3 a1 + 2a2 + 2b1 + 2b2 + 3c1 + 2c2 3a2 + b2 + 2c2 2a2 + a3 + b2 + 2b3 + 3c2 + 3c3 a1 + 2a3 + 3b1 + 2b3 + 2c1 + 2c3 a2 + 2a3 + 3b2 + 3b3 + 2c2 + c3 3a3 + 2b3 + c3 b1 + b2 + b3 c1 + c2 + c3 a\u2032 1 a\u2032 2 a\u2032 3 b\u2032 1 b\u2032 2 b\u2032 3 c\u2032 1 c\u2032 2 c\u2032 3 3a\u2032 3 + b\u2032 3 + c\u2032 1 + 2c\u2032 2 + c\u2032 3 3a\u2032 2 + b\u2032 1 + 3b\u2032 2 + 3b\u2032 3 + 2c\u2032 2 2a\u2032 1 + 2a\u2032 2 + 3a\u2032 3 + b\u2032 1 + 2c\u2032 1 3a\u2032 2 + b\u2032 1 + b\u2032 2 + 2b\u2032 3 + c\u2032 2 3a\u2032 3 + 2b\u2032 3 + c\u2032 1 + 3c\u2032 2 + 2c\u2032 3 2a\u2032 1 + 3a\u2032 2 + 2a\u2032 3 + 2b\u2032 1 + c\u2032 1 3a\u2032 3 + 3b\u2032 3 + c\u2032 1 + c\u2032 2 + 2c\u2032 3 3a\u2032 2 + b\u2032 1 + 2b\u2032 2 + b\u2032 3 + 3c\u2032 2 2a\u2032 1 + a\u2032 2 + a\u2032 3 + 3b\u2032 1 + 3c\u2032 1 \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb \uf8ee \uf8f0 1 1 1 \uf8f9 \uf8fb 3a1 + a2 + a3 +b1 + b2 + b3 +c1 + c2 + c3 a1 + 3a2 + a3 +2b1 + 2b2 + 2b3 +3c1 + 3c2 + 3c3 a1 + a2 + 3a3 +3b1 + 3b2 + 3b3 +2c1 + 2c2 + 2c3 (b) Exact repair of parity node 1 (a) Exact repair of systematic node 1 Fig. 8. Bi-Orthogonal case: Illustration of exact repair for a (6, 3, 5) E-MSR code de\ufb01ned over GF(4) where a generator polynomial g(x) = x2 + x + 1. We use U = I. For parity node repair, the solution for projection vectors is much simpler. We download only the \ufb01rst equation from each survivor node; Systematic node repair is a bit involved: setting all of the projection vectors as 2\u22121v1 = (1, 1, 1)t. Notice that the matrices of (19) have exactly the transpose structure of the matrices of (18). Hence, this code of (19) is a dual solution of (18), thereby providing switched projection vector solutions and lowering the complexity for parity node repair. Fig. 8 shows an example for exact repair of (a) systematic node 1 and (b) parity node 1. Reverse to the previous case, exact repair of parity nodes is now much simpler. In this example, by downloading only the \ufb01rst equation from each survivor node, we can successfully recover a\u2032. On the contrary, systematic node repair is a bit involved: a projection vector solution is 2\u22121v1 = (1, 1, 1)t. Using this vector, we can achieve simultaneous interference alignment, thereby decoding the desired components of a\u2032. October 29, 2018 DRAFT \f23 VI. GENERALIZATION: n \u22652k; d \u22652k \u22121 Theorem 1 gives insights into generalization to (2k, k, k \u22121) E-MSR codes. The key observation is that assuming M = k(d\u2212k+1), storage cost is \u03b1 = M/k = d\u2212k+1 = k and this number is equal to the number of systematic nodes and furthermore matches the number of parity nodes. Notice that the storage size matches the size of encoding matrices, which determines the number of linearly independent vectors of {v} := {v1, \u00b7 \u00b7 \u00b7 }. In this case, therefore, we can generate k linearly independent vectors {v} := {v1, \u00b7 \u00b7 \u00b7 , vk} and corresponding {u} := {u1, \u00b7 \u00b7 \u00b7 , uk} through the appropriate choice of M to design (2k, k, k \u22121) E-MSR codes. A. Case: n = 2k Theorem 2 ((2k, k, k \u22121) E-MSR Codes): Let M be a Cauchy matrix: M = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 m(1) 1 m(2) 1 \u00b7 \u00b7 \u00b7 m(k) 1 m(1) 2 m(2) 2 \u00b7 \u00b7 \u00b7 m(k) 2 . . . . . . ... . . . m(1) k m(2) k \u00b7 \u00b7 \u00b7 m(k) k \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb , where each element m(i) j \u2208GF(q), where q \u22652k. Suppose V = [v1, \u00b7 \u00b7 \u00b7 , vk] is invertible and U = \u03ba\u22121V\u2032M, (20) where V\u2032 = (Vt)\u22121 and \u03ba is an arbitrary non-zero value \u2208Fq such that 1\u2212\u03ba2 \u0338= 0. Also assume that encoding matrices are given by G(1) 1 = u1vt 1 + m(1) 1 I, \u00b7 \u00b7 \u00b7 , G(1) k = u1vt k + m(1) k I, . . . ... . . . G(k) 1 = ukvt 1 + m(k) 1 I, \u00b7 \u00b7 \u00b7 , G(k) k = ukvt k + m(k) k I, (21) where G(i) l indicates an encoding matrix for parity node i, associated with information unit l. Then, the code satis\ufb01es the MDS property and achieves the MSR point under exact repair constraints of all nodes. Proof: See Appendix B. Remark 4: Note that the minimum required alphabet size is 2k. As mentioned earlier, this is because we employ a Cauchy matrix for ensuring the invertibility of any submatrices of M. One may customize codes to \ufb01nd smaller alphabet-size codes. October 29, 2018 DRAFT \f24 B. Case: n \u22652k; d \u22652k \u22121 Now what if k is less than the size (= \u03b1 = d \u2212k + 1) of encoding matrices, i.e., d \u22652k \u22121? Note that this case automatically implies that n \u22652k, since n \u2265d + 1. The key observation in this case is that the encoding matrix size is bigger than k, and therefore we have more degrees of freedom (a larger number of linearly independent vectors) than the number of constraints. Hence, exact repair of systematic nodes becomes transparent. This was observed as well in [4], where it was shown that for this regime, exact repair of systematic nodes only can be guaranteed by judiciously manipulating (2k, k, k \u22121) codes through a puncturing operation. We show that the puncturing technique in [4] (meant for exact repair of systematic nodes and for a special case of our generalized codes) together with our repair construction schemes can also carry over to ensure exact repair of all nodes even for the generalized family of codes. The recipe for this has two parts: 1. Constructing a target code from a larger code through the puncturing technique. 2. Showing that the resulting target code indeed ensures exact repair of all nodes as well as the MDS-code property for our generalized family of codes. The \ufb01rst part contains the following detailed steps: 1(a) Using Theorem 2, construct a larger (2n \u22122k, n \u2212k, 2n \u22122k \u22121) code with a \ufb01nite \ufb01eld size of q \u22652n \u22122k. 1(b) Remove all the elements associated with the (n \u22122k) information units (e.g., from the (k + 1)th to the (n \u2212k)th information unit). The number of nodes is then reduced by (n \u22122k) and so are the number of information units and the number of degrees. Hence, we obtain the (n, k, n \u22121) code. 1(c) Prune the last (n \u22121 \u2212d) equations in each storage node and also the last (n \u22121 \u2212d) symbols of each information unit, while keeping the number of information units and storage nodes. We can then get the (n, k, d) target code. Indeed, based on our framework in Section IV, it can be shown that the resulting punctured code described above guarantees exact repair of all nodes and MDS-code property for our generalized family of codes. Hence, we obtain the following theorem. The proof procedure is tedious and mimics that of Theorem 2. Therefore, details are omitted. Theorem 3 ( k n \u22641 2, d \u22652k \u22121): Under exact repair constraints of all nodes, the optimal tradeOctober 29, 2018 DRAFT \f25 a1 a2 a3 b1 b2 b3 c1 c2 c3 3a2 + b2 + 2c2 3a3 + 2b3 + c3 2a1 + a3 + b1 + b3 + 3c1 + c3 2a1 + a2 + 3b1 + b2 + c1 + c2 3a1 + 3b1 + 3c1 2a2 + a3 + b2 + 2b3 + 3c2 + 3c3 a1 + 2a2 + 2b1 + 2b2 + 3c1 + 2c2 a2 + 2a3 + 3b2 + 3b3 + 2c2 + c3 a1 + 2a3 + 3b1 + 2b3 + 2c1 + 2c3 (6,3,5) E-MSR code (5,2,3) E-MSR code Remove (c1, c2, c3), (a3, b3) and associated elements. 3a2 + b2 2a1 + a2 + 3b1 + b2 3a1 + 3b1 a1 + 2a2 + 2b1 + 2b2 of each storage node. Also remove the third equation a2 + 3b2 a1 + 3b1 a1 a2 b1 b2 Fig. 9. Bi-Orthogonal case: Illustration of the construction of a (5, 2, 3) E-MSR code from a (6, 3, 5) code de\ufb01ned over GF(4). For a larger code, we adopt the (6, 3, 5) code in Fig. 8. First, we remove all the elements associated with the last (n \u22122k) = 1 information unit (\u201cc\u201d). Next, we prune symbols (a3, b3) and associated elements. Also we remove the last equation of each storage node. Finally we obtain the (n, k, d) = (5, 2, 3) target code. off of (1) can be attained with a deterministic scheme requiring a \ufb01eld size of at most 2(n \u2212k). Example 1: Fig. 9 illustrates how to construct an (n, k, d) = (5, 2, 3) target code based on the above recipe. First construct the (2n \u22122k, n \u2212k, 2n \u22122k \u22121) = (6, 3, 5) code, which is larger than the (5, 2, 3) target code, but which belongs to the category of n = 2k. For this code, we adopt the bi-orthogonal case example in Fig. 8. For this code, we now remove all the elements associated with the last (n \u22122k) = 1 information unit, which corresponds to (c1, c2, c3). Next, prune the last symbol (a3, b3) of each information unit and associated elements to shrink the storage size into 2. We can then obtain the (5, 2, 3) target code. Exact repair and the MDS-code property of the resulting code can be veri\ufb01ed based on the proposed framework in Section IV. October 29, 2018 DRAFT \f26 VII. GENERALIZATION: k \u22643 As a side generalization, we consider the case of k \u22643. The interesting special case of the (5, 3) E-MSR code8 will be focused on, since it is not covered by the above case of k n \u22641 2. For this case, we propose another interference alignment technique building on an eigenvector concept. Theorem 4 (k \u22643): The MSR point can be attained with a deterministic scheme requiring a \ufb01nite-\ufb01eld size of at most 2n \u22122k. Proof: The case of k = 1 is trivial. By Theorems 2 and 3, we prove the case of k = 2. However, additional effort is needed to prove the case of k = 3. By Theorems 2 and 3, (n, 3) for n \u22656 can be proved. But (5, 3) codes are not in the class. In Section VII-A, we will address this case to complete the proof. Remark 5: In order to cover general n, we provide a looser bound on the required \ufb01nite-\ufb01eld size: q \u22652n \u22122k. In fact, for the (5, 3) code (that will be shown in Lemma 2), a smaller \ufb01nite\ufb01eld size of q = 3 (< 4 = 2n \u22122k) is enough for construction. We have taken the maximum of the required \ufb01eld sizes of all the cases. A. (5, 3) E-MSR Codes We consider d = 4 and M = 6. The cutset bound (1) then gives the fundamental limits of: storage cost \u03b1 = 2 and repair-bandwidth-per-link=1; hence, the dimension of encoding matrices is 2-by-2. Note that the size is less than the number of systematic nodes. Therefore, our earlier framework does not cover this category. In fact, the (5, 3) code is in the case of n + 1 = 2k, where it was shown in [4] that there exist codes that achieve the cutset bound under exact repair of systematic nodes only (not including parity nodes). We propose an eigenvector-based interference alignment technique to prove the code existence under exact repair of all nodes. Let a = (a1, a2)t, b = (b1, b2)t and c = (c1, c2)t. For exact repair, we connect to 4(= d) nodes to download a one-dimensional scalar value from each node. Fig. 10 illustrates exact repair of node 1. We download four equations from survivor nodes: btv\u03b11; ctv\u03b12; at(A1v\u03b13) + bt(B1v\u03b13) + ct(C1v\u03b13); at(A2v\u03b14) + bt(B2v\u03b14) + ct(C2v\u03b14). The approach is different from that of our earlier proposed framework. Instead an idea here consists 8Independently, the authors in [5] found (5, 3) codes de\ufb01ned over GF(3), based on a search algorithm. October 29, 2018 DRAFT \f27 C1B\u22121 1 v\u03b11 v\u03b12 rank \u0001\u0002 A1B\u22121 1 v\u03b11 A2B\u22121 2 v\u03b11 \u0003\u0004 = 2 v\u03b11 : an eigenvector of B2C\u22121 2 C1B\u22121 1 v\u03b13 = B\u22121 1 v\u03b11 v\u03b12 = C1B\u22121 1 v\u03b11 v\u03b11 A1B\u22121 1 v\u03b11 v\u03b14 = B\u22121 2 v\u03b11 C2B\u22121 2 v\u03b11 A2B\u22121 2 v\u03b11 v\u03b11 at bt atA1 + btB1 + ctC1 atA2 + btB2 + ctC2 v\u03b12 v\u03b13 ct v\u03b14 node 1 node 2 node 3 node 4 (parity node 1) (parity node 2) node 5 \uf8ee \uf8ef \uf8ef \uf8f0 0 0 (A1v\u03b13)t (A2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fba + \uf8ee \uf8ef \uf8ef \uf8f0 vt \u03b11 0 (B1v\u03b13)t (B2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fbb + \uf8ee \uf8ef \uf8ef \uf8f0 0 vt \u03b12 (C1v\u03b13)t (C2v\u03b14)t \uf8f9 \uf8fa \uf8fa \uf8fbc Fig. 10. Eigenvector-based interference alignment for (5, 3) E-MSR codes. First we align interference \u201cb\u201d by setting v\u03b13 = B\u22121 1 v\u03b11 and v\u03b14 = B\u22121 2 v\u03b11. Next, partially align interference of \u201cc\u201d by setting v\u03b12 = C1B\u22121 1 v\u03b11. Finally, choosing v\u03b11 as an eigenvector of B2C\u22121 2 C1B\u22121 1 , we can achieve interference alignment for c. of three steps: (1) choosing projection vectors for achieving interference alignment; (2) gathering all the alignment constraints and the MDS-code constraint; (3) designing the encoding matrices that satisfy all the constraints. Notice the design of encoding matrices is the last part. Here are details. Note that there are 6 unknown variables: 2 desired unknowns (a1, a2) and 4 undesired unknowns (b1, b2, c1, c2). Therefore, it is required to align (b1, b2, c1, c2) onto at least 2-dimensional linear space. We face the challenge that appeared in the (6, 3, 5) code example in Fig. 4. Projection vectors v\u03b13 and v\u03b14 affect interference alignment b and c simultaneously. Therefore, we need simultaneous interference alignment. To solve this problem, we introduce an eigenvector-based interference alignment scheme. First choose v\u03b13 and v\u03b14 such that v\u03b13 = B\u22121 1 v\u03b11 and v\u03b14 = B\u22121 2 v\u03b11, thereby achieving interference alignment for \u201cb\u201d. Observe the interfering vectors associated with \u201cc\u201d: v\u03b12; C1B\u22121 1 v\u03b11; C2B\u22121 2 v\u03b11. The \ufb01rst and second vectors can be aligned by setting v\u03b12 = C1B\u22121 1 v\u03b11. Now what about for the following two vectors: C1B\u22121 1 v\u03b11 and C2B\u22121 2 v\u03b11? Suppose that the associated matrices (C1B\u22121 1 and C2B\u22121 2 ) and the projection vector v\u03b11 are randomly chosen. Then, the these two vectors October 29, 2018 DRAFT \f28 are not guaranteed to be aligned. However, a judicious choice of v\u03b11 makes it possible to align them. The idea is to choose v\u03b11 as an eigenvector of B2C\u22121 2 C1B\u22121 1 . Since v\u03b11 can be chosen arbitrarily, this can be easily done. Lastly consider the condition for ensuring the decodability of desired signals: rank \u0000\u0002 A1B\u22121 1 v\u03b11 A2B\u22121 2 v\u03b11 \u0003\u0001 = 2. We repeat the procedure for exact repair of \u201cb\u201d and \u201cc\u201d. For parity nodes, we employ the remapping technique described earlier: \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 a\u2032 b\u2032 c\u2032 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 At 1 Bt 1 Ct 1 At 2 Bt 2 Ct 2 0 0 I \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 a b c \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb, \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A\u2032 1 A\u2032 2 0 B\u2032 1 B\u2032 2 0 C\u2032 1 C\u2032 2 I \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb:= \uf8ee \uf8ef \uf8ef \uf8ef \uf8f0 A1 A2 0 B1 B2 0 C1 C2 I \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fb \u22121 . (22) We gather all the conditions that need to be guaranteed for exact repair of all nodes: rank \u0000\u0002 A1B\u22121 1 v\u03b11 A2B\u22121 2 v\u03b11 \u0003\u0001 = 2, rank \u0000\u0002 B1C\u22121 1 v\u03b21 B2C\u22121 2 v\u03b21 \u0003\u0001 = 2, rank \u0000\u0002 C1A\u22121 1 v\u03b31 C2A\u22121 2 v\u03b31 \u0003\u0001 = 2, rank \u0000\u0002 A\u2032 1B\u2032\u22121 1 v\u03b1\u20321 A\u2032 2B\u2032\u22121 2 v\u03b1\u20321 \u0003\u0001 = 2, rank \u0000\u0002 B\u2032 1C\u2032\u22121 1 v\u03b2\u20321 B\u2032 2C\u2032\u22121 2 v\u03b2\u20321 \u0003\u0001 = 2, (23) where v\u03b11 : an eigenvector of B2C\u22121 2 C1B\u22121 1 , v\u03b21 : an eigenvector of C2A\u22121 2 A1C\u22121 1 , v\u03b31 : an eigenvector of A2B\u22121 2 B1A\u22121 1 , v\u03b1\u20321 : an eigenvector of B\u2032 2C\u2032\u22121 2 C\u2032 1B\u2032\u22121 1 , v\u03b2\u20321 : an eigenvector of C\u2032 2A\u2032\u22121 2 A\u2032 1C\u2032\u22121 1 . (24) Note that eigenvectors may not exist for the \ufb01nite Galois \ufb01eld. However, the existence is guaranteed by carefully choosing the encoding matrices. We provide an explicit coding scheme in the following lemma. Lemma 2 ((5, 3) E-MSR Codes): Let \u03b1, \u03b2 \u2208GF(3) and be non-zero. Suppose encoding maOctober 29, 2018 DRAFT \f29 a1 a2 b1 c1 c2 b2 2c1 + c2 2a1 + 2a2 + b1 + 2c1 + c2 a2 + 2b1 + 2b2 + 2c2 2a1 + a2 + b1 + c1 + 2c2 b1 2a1 + 2a2 + b1 + (2c1 + c2) 2a2 + 2b1 + b2 + 2c2 2a1 + a2 + b1 + 2(2c1 + c2) \u0001 1 0 \u0002 \u0001 2 1 \u0002 \u0001 1 0 \u0002 \u0001 1 0 \u0002 Fig. 11. Illustration of exact repair of node 1 for a (5, 3) E-MSR code de\ufb01ned over GF(3). The eigenvector-based interference alignment scheme enables to decode 2 desired unknowns (a1, a2) from 4 equations containing 6 unknowns. Notice that interference \u201cb\u201d and \u201cc\u201d are aligned simultaneously although the same projection vectors v\u03b13 and v\u03b14 are used. trices are given by A1 = \uf8ee \uf8f02\u03b1 0 2\u03b2 \u03b2 \uf8f9 \uf8fb, B1 = \uf8ee \uf8f0\u03b1 2\u03b1 0 2\u03b2 \uf8f9 \uf8fb, C1 = \uf8ee \uf8f02\u03b1 0 \u03b2 2\u03b2 \uf8f9 \uf8fb, A2 = \uf8ee \uf8f02\u03b1 0 \u03b2 2\u03b2 \uf8f9 \uf8fb, B2 = \uf8ee \uf8f0\u03b1 2\u03b1 0 \u03b2 \uf8f9 \uf8fb, C2 = \uf8ee \uf8f0\u03b1 0 2\u03b2 2\u03b2 \uf8f9 \uf8fb. (25) Then, the code satis\ufb01es the MDS property and achieves the MSR point (1) under exact repair constraints of all nodes. Proof: See Appendix C. Remark 6: Note that encoding matrices are lower-triangular or upper-triangular. This structure has important properties. Not only does this structure guarantee invertibility, it can in fact guarantee the existence of eigenvectors. It turns out the structure as above satis\ufb01es all of the conditions needed for the MDS property and exact repair. Example 2: Fig. 11 illustrates exact repair of node 1 (a1, a2) for a (5, 3) E-MSR code de\ufb01ned October 29, 2018 DRAFT \f30 over GF(3). Notice that interference \u201cb\u201d and \u201cc\u201d are aligned simultaneously. One can check exact repair of the remaining four nodes based on our proposed method. VIII." + }, + { + "url": "http://arxiv.org/abs/0901.3580v1", + "title": "Feedback Capacity of the Gaussian Interference Channel to Within 1.7075 Bits: the Symmetric Case", + "abstract": "We characterize the symmetric capacity to within 1.7075 bits/s/Hz for the\ntwo-user Gaussian interference channel with feedback. The result makes use of a\ndeterministic model to provide insights into the Gaussian channel. We derive a\nnew outer bound to show that a proposed achievable scheme can achieve the\nsymmetric capacity to within 1.7075 bits for all channel parameters. From this\nresult, we show that feedback provides unbounded gain, i.e., the gain becomes\narbitrarily large for certain channel parameters. It is a surprising result\nbecause feedback has been so far known to provide only power gain (bounded\ngain) in the context of multiple access channels and broadcast channels.", + "authors": "Changho Suh, David Tse", + "published": "2009-01-23", + "updated": "2009-01-23", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION Shannon showed that feedback does not increase capacity in the discrete-memoryless pointto-point channel [1]. However, in the multiple access channel (MAC), Gaarder and Wolf [2] showed that feedback could increase capacity although the channel is memoryless. Inspired by this result, Ozarow [3] found the feedback capacity region for the two-user Gaussian MAC. However, capacity results have been open for more-than-two-user Gaussian MACs and general MACs. Ozarow\u2019s result implies that feedback provides only power gain (bounded gain). The reason of bounded gain is that transmitters cooperation induced by feedback can at most boost signal power (via aligning signal directions) in the MAC. Boosting signal power provides a capacity increase of a constant number of bits. Now a question is \u201cWill feedback help signi\ufb01cantly in other channels where each receiver wants to decode only desired messages in the presence of undesired messages (interferences)?\u201d To answer this question, we focus on the simple two-user Gaussian interference channel where each receiver wants to decode the messages only from its corresponding transmitter. In this channel, we show that feedback can provide unbounded gain for certain channel parameters. For this, we \ufb01rst characterize the symmetric feedback capacity for a linear deterministic model [4] well capturing key properties of the Gaussian channel. Gaining insights from this model, we develop a simple two-staged achievable scheme in the Gaussian channel. We then derive a new outer bound to show that the proposed scheme achieves the symmetric capacity to within 1.7075 bits for all channel parameters. The unbounded gain of feedback can be shown from the generalized degrees-of-freedom (g.d.o.f.) in Fig. 1, de\ufb01ned in [5] as d(\u03b1) \u225c lim SNR,INR\u2192\u221e Csym(SNR, INR) log SNR , (1) where \u03b1 (x-axis) indicates the ratio of INR to SNR in dB scale: \u03b1 \u225clog INR log SNR. Note that in the weak interference regime (0 \u2264\u03b1 \u22642 3) and in the very strong interference regime (\u03b1 \u22652), feedback gain becomes arbitrarily large as SNR and INR go to in\ufb01nity as long as channel parameters keep the certain scale so that \u03b1 remains same. This implies unbounded gain. This is a surprising result because feedback has been so far known to provide only power gain (bounded gain) in the context of multiple access channels and broadcast channels [3], [6]. October 27, 2018 DRAFT \f3 \u0001 \u0002 \u0000\u0003 1 1 2 \u0004 \u0005 Fig. 1 THE GENERALIZED DEGREES-OF-FREEDOM OF THE GAUSSIAN INTERFERENCE CHANNEL WITH FEEDBACK Some work has been done in the interference channel with feedback [7], [8], [9], [10]. In [7], [8], Kramer developed a feedback strategy and derived an outer bound in the Gaussian channel; and later derived a dependence-balance outer bound with Gastpar [9]. However, the gap between those outer bounds and the inner bound is not tight in almost cases, except one speci\ufb01c set of power and channel parameters. For some channel parameters, Kramer\u2019s scheme is worse than the best known non-feedback scheme [11]. Recently, Jiang-Xin-Garg [10] found an achievable region in the discrete memoryless interference channel with feedback, based on the block Markov encoding [12] and the Han-Kobayashi scheme [11]. However, their scheme includes three auxiliary random variables requiring further optimization. Also they did not provide any upper bounds. On the other hand, we propose a simple two-staged achievable scheme which is explicit and has only two stages. Also we derive a tighter outer bound to do tight capacity characterization to within a constant of number of bits. Later we will provide more detailed comparison to Kramer\u2019s scheme [7] in Section IV-D. October 27, 2018 DRAFT \f4 II. MODEL Fig. 2 (a) describes the Gaussian interference channel with feedback. We consider the symmetric interference channel where g11 = g22 = gd, g12 = g21 = gc, and P1 = P2 = P. Without loss of generality, we assume that signal power and noise power are normalized to 1, i.e., Pk = 1, Zk \u223cCN (0, 1), \u2200k = 1, 2. Hence, signal-to-noise ratio and interference-to-noise ratio can be de\ufb01ned to capture channel gains: SNR \u225c|gd|2, INR \u225c|gc|2. (2) There are two independent and uniformly distributed sources, Wk \u2208{1, 2, \u00b7 \u00b7 \u00b7 , Mk} , \u2200k = 1, 2. Due to feedback, the encoded signal Xki of user k at time i is a function of its own message and past output sequences: Xki = f i k \u0000Wk, Yk1, \u00b7 \u00b7 \u00b7 , Yk(i\u22121) \u0001 = f i k \u0000Wk, Y i\u22121 k \u0001 (3) where we use shorthand notation Y i\u22121 k . The symmetric capacity is de\ufb01ned by Csym = sup {R : (R, R) \u2208R} , (4) where R is the capacity region. \u0006\u0007 n \b \t n \u000b n \f \r n Fig. 2 THE GAUSSIAN (AND DETERMINISTIC) INTERFERENCE CHANNELS WITH FEEDBACK October 27, 2018 DRAFT \f5 We \ufb01rst consider the deterministic model as shown in Fig. 2 (b). The symmetric deterministic channel is characterized by two values: n = n11 = n22 and m = n12 = n21, where n and m indicate the number of signal bit levels that we can send through direct link and cross link, respectively. For each level, we assume a modulo-2-addition. This model is useful because in the non-feedback case, the deterministic interference channel approximates the Gaussian channel within a constant gap [13]. In the feedback-case, we expect a similar constant gap as well. In the Gaussian channel, n and m correspond to channel gains in dB scale., i.e., n = \u230alog SNR\u230b, m = \u230alog INR\u230b, and the modulo-2-addition corresponds to a real addition, which causes a fundamental gap between two channels. Our strategy is to \ufb01rst come up with a deterministic scheme, gain insights from it, and then mimic the scheme to the Gaussian channel. III. A DETERMINISTIC INTERFERENCE CHANNEL Theorem 1: The symmetric feedback capacity of a deterministic interference channel is given by Csym = max(n, m) + (n \u2212m)+ 2 . (5) A. Proof of Achievablility Review of a non-feedback scheme [13]: In the non-feedback case, we typically separate into two regimes depending on the strength of interference. In the strong interference channel, the key fact is that all of the feasible rate tuples are decodable at both receivers, i.e., all messages are common. Also since the number of received bit levels is m, an achievable scheme is to send min( m 2 , n). Notice that the number of transmission bits is limited by the number n of direct link bit levels. On the other hand, in the weak interference channel, only part of information is visible to the other receiver. So we spit information into two parts (the Han-Kobayashi scheme): a common part (decodable at both receivers); a private part (decodable only at the desired receiver). An achievable scheme is to send (n \u2212m) bits for private information and to send some number of bits for common information which depends on m n . For the feedback case, we will adopt the above setting. We start with the simpler case: the strong interference regime. October 27, 2018 DRAFT \f6 \u000e \u000f \u0010\u0011\u0012 \u0013\u0014 \u0015 \u0016 \u0017 \u0018\u0019\u001a\u001b \u001c \u001d \u001e\u001f ! \" # $ % & ' ( ) * + , -./0 1 2 3 4 5 6 7 8 9 :;< => ? @ A BCDE F G H I J K L M N OPQ RS T U V WXYZ [ \\ ] ^ _ ` a b cde f g h ijk l m n opq r s t uv w x y z {|} ~ \u007f \u0080 \u0081 \u0082\u0083\u0084 \u0085\u0086 \u0087 \u0088 \u0089 \u008a\u008b\u008c\u008d \u008e \u008f \u0090\u0091 \u0092 \u0093 \u0094 \u0095 \u0096 \u0097\u0098\u0099 \u009a\u009b \u009c \u009d \u009e \u009f \u00a0\u00a1\u00a2 \u00a3 \u00a4 \u00a5\u00a6 \u00a7 \u00a8 \u00a9 \u00aa \u00ab\u00ac\u00ad \u00ae \u00af Fig. 3 AN ACHIEVABLE SCHEME OF THE DETERMINISTIC INTERFERENCE CHANNEL WITH FEEDBACK The strong interference regime (m \u2265n): We will explain a scheme based on a simple example of Fig. 3 (a). Mimicking the non-feedback case, transmitters send only common information. The main point of a scheme is to use two stages. In the \ufb01rst stage, transmitter 1 sends a1, a2, a3 and transmitter 2 sends b1, b2, b3. Note that each transmitter sends the whole m bits instead of min( m 2 , n) (the number of bits sent in the non-feedback case). Due to this, each receiver needs to defer decoding to the second stage. In the second stage, using feedback, each transmitter decodes information of the other user, e.g., transmitter 1 decodes b1, b2, b3 and transmitter 2 decodes a1, a2, a3. Each transmitter then sends information of the other user. Now each receiver can decode its own data by subtracting the received signal in the \ufb01rst stage from the second. Receiver 1 decodes a1, a2, a3 by subtracting b1 from the second received October 27, 2018 DRAFT \f7 signal. Notice that the second stage was used for re\ufb01ning all bits sent previously, without sending additional information. Therefore, the symmetric rate is 3 2. Considering the general case (n, m), we achieve Rsym = m 2 . (6) Note that with feedback, the symmetric rate can exceed n bits, which was the limit for the non-feedback case. This is because the very strong interfering link helps signi\ufb01cantly to relay other messages through feedback. For example, the information \ufb02ow of user 1 is through indirect links (X1 \u2192V1 \u2192feedback \u2192X2 \u2192V2 \u2192Y1) instead of direct link (X1 \u2192Y1). This concept coincides with correlation routing in Kramer\u2019s paper [7]. The weak interference regime (m < n): We will explain a scheme based on an example of Fig. 3 (b). Similar to the non-feedback case, information is split into two parts. But it has two stages. In the \ufb01rst stage, transmitter 1 sends private information a2 on the lower level (invisible to the other receiver) and common information a1 on the upper signal level (visible to the other receiver). Similarly transmitter 2 sends b1 and b2. Similar to the non-feedback case, each transmitter sends (n \u2212m) private bits. However, there is a difference in sending common information. Each transmitter sends m common bits whatever m n is, unlike the non-feedback case where the number of common bits depends on m n . Then, receiver 1 gets the clean signal a1 on the upper level and the interfered signal a2 \u2295b1 on the lower level. In this case (\u03b1 = 1 2), receiver 1 can decode its common information a1 in the \ufb01rst stage. However, for the other case, e.g., \u03b1 = 3 5 (Fig. 3 (c)), receiver 1 cannot fully decode common information in the \ufb01rst stage because a part of it is interfered by common information of the other user. Therefore, each receiver needs to defer decoding to the second stage. In the second stage, with feedback, each transmitter can decode common information of the other user. Transmitter 1 and 2 can decode b1 and a1, respectively. Each transmitter then sends common information of the other user on the upper level. Sending this, receiver 1 can re\ufb01ne the corrupted symbol received in the \ufb01rst stage without causing any interferences to the other receiver. On the lower level, each transmitter sends new private information. Transmitter 1 and 2 send a3 and b3, respectively. Using the \ufb01rst and second received signals, receiver 1 can now decode the corrupted symbol a2 sent in the \ufb01rst stage. At the same time, it can decode new private information a3 by stripping October 27, 2018 DRAFT \f8 a1. During two stages, each receiver can decode three symbols out of two levels. Therefore, the symmetric rate is 3 2\u00b72. This scheme can be easily generalized into the case of (n, m). During two stages, each receiver can decode all of the messages sent in the \ufb01rst stage and a new private message sent in the second stage. Therefore, the symmetric rate is Rsym = n + (n \u2212m) 2 = n \u2212m 2 . (7) Remarks on the achievable scheme: Our two-staged scheme has some similarity with an achievable scheme in [10] in that using feedback each transmitter decodes common information of the other user. However, our scheme is different since it is explicit and has only two stages, while the scheme in [10] employs three auxiliary random variables (requiring further optimization) and the block Markov encoding (requiring a long block length). B. Proof of Converse We have N(R1 + R2) = H(W1) + H(W2) (a) = H(W1|W2) + H(W2) (b) \u2264I(W1; Y N 1 |W2) + I(W2; Y N 2 ) + N\u01ebN (c) = H(Y N 1 |W2) + I(W2; Y N 2 ) + N\u01ebN \u2264H(Y N 1 , V N 1 |W2) + I(W2; Y N 2 ) + N\u01ebN = H(Y N 1 |V N 1 , W2) + H(Y N 2 ) + \u0002 H(V N 1 |W2) \u2212H(Y N 2 |W2) \u0003 + N\u01ebN (d) = H(Y N 1 |V N 1 , W2) + H(Y N 2 ) + N\u01ebN (e) = H(Y N 1 |V N 1 , W2, XN 2 , V N 2 ) + H(Y N 2 ) + N\u01ebN (f) \u2264 N X i=1 [H(Y1i|V1i, V2i) + H(Y2i)] + N\u01ebN where (a) follows from the independence of W1 and W2; (b) follows from Fano\u2019s inequality; (c) follows from the fact that Y N 1 is a function of W1 and W2; (d) follows from H(V N 1 |W2) = H(Y N 2 |W2) (see Claim 1); (e) follows from the fact that XN 2 is a function of (W2, V N\u22121 1 ) (see Claim 2) and V N 2 is a function of XN 2 ; (f) follows from the fact that conditioning reduces entropy. October 27, 2018 DRAFT \f9 Claim 1: H(V N 1 |W2) = H(Y N 2 |W2). Proof: H(Y N 2 |W2) = N X i=1 H(Y2i|Y i\u22121 2 , W2) (a) = N X i=1 H(V1i|Y i\u22121 2 , W2) (b) = N X i=1 H(V1i|Y i\u22121 2 , W2, Xi 2, V i\u22121 1 ) (c) = N X i=1 H(V1i|W2, V i\u22121 1 ) = H(V N 1 |W2), where (a) follows from the fact that Y2i is a function of (X2i, V1i) and X2i is a function of (W2, Y i\u22121 2 ); (b) follows from the fact that Xi 2 is a function of (W2, Y i\u22121 2 ) and V i 2 is a function of Xi 2; (c) follows from the fact that Y i\u22121 2 is a function of (Xi\u22121 2 , V i\u22121 1 ) and Xi 2 is a function of (W2, V i\u22121 1 ) (by Claim 2). Claim 2: For all i \u22651, Xi 1 is a function of (W1, V i\u22121 2 ) and Xi 2 is a function of (W2, V i\u22121 1 ). Proof: By symmetry, it is enough to prove only one. Since the channel is deterministic (noiseless), Xi 1 is a function of W1 and W2. In Fig. 2 (b), we can easily see that information of W2 delivered to the \ufb01rst link must pass through V2i. Also note that X1i depends on the past output sequences until i \u22121 (due to feedback delay). Therefore, Xi 1 is a function of (W1, V i\u22121 2 ). Now let the time index Q be a random variable uniformly distributed over the set {1, 2, \u00b7 \u00b7 \u00b7 , N} and independent of (W1, W2, XN 1 , XN 2 , Y N 1 , Y N 2 ). We de\ufb01ne Xk = XkQ, Vk = VkQ, Yk = YkQ, \u2200k = 1, 2. If (R1, R2) is achievable, then \u01ebN \u21920 as N \u2192\u221e. Hence, we get R1 + R2 \u2264H(Y1|V1, V2) + H(Y2). Since the RHS is maximized when X1 and X2 are uniform and independent, we get Csym \u2264max(n, m) + (n \u2212m)+ 2 . (8) This establishes the converse. October 27, 2018 DRAFT \f10 IV. THE GAUSSIAN INTERFERENCE CHANNEL A. An Achievable Rate Theorem 2: In the strong Gaussian interference channel (INR \u2265SNR), we can achieve Rstrong sym = 1 2 log (1 + INR) . (9) In the weak Gaussian interference channel (INR \u2264SNR), we can achieve Rweak sym = \uf8f1 \uf8f2 \uf8f3 log \u00001 + SNR 2INR \u0001 + 1 2 log (1 + INR) \u22121 2, INR \u22651; log \u00001 + SNR INR+1 \u0001 , INR \u22641. (10) Proof: The strong interference regime (INR \u2265SNR, Fig. 4 (a)): Mimicking the deterministic case, each transmitter sends only common information and employs two stages. In the \ufb01rst stage, each transmitter sends its own signal. In the second stage, each transmitter sends information of the other user after decoding it with the help of feedback. In the Gaussian noisy channel, we need to be careful in how to combine the received signals (during two stages) to decode the message. Alamouti\u2019s scheme [14] gives insights into this. Notice that with feedback both messages are available at transmitters in the second stage. However, in spite of knowing both messages, transmitters cannot control the messages already sent in the \ufb01rst stage. Hence, they can partially collaborate in the second stage. However, the beauty of Alamouti\u2019s scheme is that messages can be designed to be orthogonal (for two time slots), although the messages in the \ufb01rst time slot are sent without any coding. This was well exploited and pointed out in [15]. Therefore, with Alamouti\u2019s scheme, transmitters are able to encode messages so that those are orthogonal. In the interference channel, orthogonality between different messages guarantees to completely remove the other message (interference). This helps improving performance signi\ufb01cantly. In the \ufb01rst stage (block), transmitter 1 and 2 send codewords XN 1 and XN 2 with rates R1 and R2, respectively. In the second stage, using feedback, transmitter 1 decodes XN 2 by stripping its own codeword XN 1 . This can be decoded if R2 \u22641 2 log (1 + INR) bits/s/Hz. (11) Similarly transmitter 2 decodes XN 1 ; hence, we have the same constraint for R1. October 27, 2018 DRAFT \f11 Now we apply Alamouti\u2019s scheme. In the second stage, transmitter 1 sends XN\u2217 2 and transmitter 2 sends \u2212XN\u2217 1 . Then, receiver 1 can gather the signals received during two stages. For 1 \u2264i \u2264N, \uf8ee \uf8f0Y (1) 1i Y (2)\u2217 1i \uf8f9 \uf8fb= \uf8ee \uf8f0 gd gc \u2212g\u2217 c g\u2217 d \uf8f9 \uf8fb \uf8ee \uf8f0X1i X2i \uf8f9 \uf8fb+ \uf8ee \uf8f0Z(1) 1i Z(2)\u2217 1i \uf8f9 \uf8fb (12) To decode X1i, receiver 1 multiplies the row vector orthogonal to the vector corresponding to X2i so we get h g\u2217 d \u2212gc i \uf8ee \uf8f0Y (1) 1i Y (2)\u2217 1i \uf8f9 \uf8fb= (|gd|2 + |gc|2)X1i + g\u2217 dZ(1) 2i \u2212gcZ(2)\u2217 1i . (13) Then, the codeword XN 1 can be decoded if R1 \u22641 2 log (1 + SNR + INR) bits/s/Hz. (14) Similar operations are done at receiver 2. Since (14) is implied by (11), we get the desired result (9). The weak interference regime (INR \u2264SNR, Fig. 4 (b)): Similar to the deterministic case, a scheme has two stages and information is split into common and private parts. Also recall that in the deterministic case, only common information is sent twice during two stages. Therefore, a natural idea is to apply Alamouti\u2019s scheme only for common information. Private information is newly sent for both stages. In the \ufb01rst stage, transmitter 1 independently generates a common codeword XN 1c and a private codeword XN,(1) 1p with rates R1c and R(1) 1p , respectively. For power splitting, we adapt the idea of the simpli\ufb01ed Han-Kobayashi scheme [5] where private power is set such that private information is in the noise level: The scheme is to set \u03bbp = min \u0012 1 INR, 1 \u0013 , \u03bbc = 1 \u2212\u03bbp, (15) Now we assign power \u03bbp and \u03bbc to X(1) 1p,i and X1c,i, \u2200i, respectively; and superpose two signals to form channel input. Similarly transmitter 2 sends XN,(1) 2p + XN 2c. By symmetry, we use the same \u03bbp and \u03bbc. Similar to the deterministic case, each receiver defers decoding to the second stage. In the second stage, Y N,(1) 1 is available at transmitter 1 by feedback. Transmitter 1 subtracts its own codeword XN,(1) 1p and XN 1c from Y N,(1) 1 and then decodes a common codeword XN 2c of October 27, 2018 DRAFT \f12 the other user. We can decode this if R2c \u22641 2 log \u0012 1 + \u03bbcINR \u03bbpINR + 1 \u0013 bits/s/Hz. (16) Similarly transmitter 2 decodes XN 1c. We have the same constraint for R1c. Now we apply Alamouti\u2019s scheme only for common information XN 1c and XN 2c. Transmitter 1 sends XN\u2217 2c and just adds new private information XN,(2) 1p . On the other hand, transmitter 2 sends \u2212XN\u2217 1c and XN,(2) 2p . Then, receiver 1 gets \uf8ee \uf8f0Y (1) 1i Y (2)\u2217 1i \uf8f9 \uf8fb= \uf8ee \uf8f0 gd gc \u2212g\u2217 c g\u2217 d \uf8f9 \uf8fb \uf8ee \uf8f0X1c,i X2c,i \uf8f9 \uf8fb+ \uf8ee \uf8f0gdX(1) 1p,i g\u2217 dX(2)\u2217 1p,i \uf8f9 \uf8fb+ \uf8ee \uf8f0gcX(1) 2p,i g\u2217 cX(2)\u2217 2p,i \uf8f9 \uf8fb+ \uf8ee \uf8f0Z(1) 1i Z(2)\u2217 1i \uf8f9 \uf8fb. (17) To decode X1c,i, we consider h g\u2217 d \u2212gc i \uf8ee \uf8f0Y (1) 1i Y (2)\u2217 1i \uf8f9 \uf8fb= (|gd|2 + |gc|2)X1c,i + |gd|2X(1) 1p,i \u2212gcg\u2217 dX(2)\u2217 1p,i + g\u2217 dgcX(1) 2p,i \u2212|gc|2X(2)\u2217 2p,i + g\u2217 dZ(1) 1i \u2212gcZ(2)\u2217 1i . (18) XN 1c can be decoded if R1c \u22641 2 log \u0012 1 + \u03bbc (SNR + INR) \u03bbp (SNR + INR) + 1 \u0013 bits/s/Hz. (19) This constraint (19) is implied by (16). Similarly receiver 1 can decode XN 2c, so we have the same constraint for R2c. Now receiver 1 subtracts (XN 2c, XN 2c) and then decodes XN,(1) 1p and XN,(2) 1p . This can be decoded if R(1) 1p \u2264log \u0012 1 + \u03bbpSNR \u03bbpINR + 1 \u0013 bits/s/Hz. (20) Similar operations are done at receiver 2. Under the simple power setting (15), we get the desired result (10). Remarks on the achievable scheme: The Alamouti-based phase rotating technique in our scheme looks similar to phase rotating techniques in [3] and [7]. However, it is different because phase rotating in our scheme is to make the desired signal orthogonal to interference, while the purpose of [3], [7] is to align the phase of desired signal to boost power gain. Also our scheme is essentially different from other schemes [3], [7] which are based on Schalkwijk-Kailath scheme. October 27, 2018 DRAFT \f13 B. An Outer Bound Theorem 3: The symmetric capacity of the Gaussian interference channel with feedback is upper-bounded by Csym \u22641 2 sup 0\u2264\u03c1\u22641 \u0014 log \u0012 1 + (1 \u2212\u03c12)SNR 1 + (1 \u2212\u03c12)INR \u0013 + log \u0010 1 + SNR + INR + 2\u03c1 \u221a SNR \u00b7 INR \u0011\u0015 . (21) Proof: For side information, we consider a noisy version of V1: S1 = V1 + Z2 = gcX1 + Z2. (22) Using this, we get N(R1 + R2) = H(W1) + H(W2) = H(W1|W2) + H(W2) \u2264I(W1; Y N 1 |W2) + I(W2; Y N 2 ) + N\u01ebN (a) \u2264I(W1; Y N 1 , SN 1 |W2) + I(W2; Y N 2 ) + N\u01ebN = h(Y N 1 , SN 1 |W2) \u2212h(Y N 1 , SN 1 |W1, W2) + I(W2; Y N 2 ) + N\u01ebN (b) = h(Y N 1 , SN 1 |W2) \u2212 X [h(Z1i) + h(Z2i)] + I(W2; Y N 2 ) + N\u01ebN = h(Y N 1 |SN 1 , W2) \u2212 X h(Z1i) + h(Y N 2 ) \u2212 X h(Z2i) + \u0002 h(SN 1 |W2) \u2212h(Y N 2 |W2) \u0003 + N\u01ebN (c) = h(Y N 1 |SN 1 , W2) \u2212 X h(Z1i) + h(Y N 2 ) \u2212 X h(Z2i) + N\u01ebN (d) = h(Y N 1 |SN 1 , W2, XN 2 ) \u2212 X h(Z1i) + h(Y N 2 ) \u2212 X h(Z2i) + N\u01ebN (e) \u2264 N X i=1 [h(Y1i|S1i, X2i) \u2212h(Z1i) + h(Y2i) \u2212h(Z2i)] + N\u01ebN where (a) follows from the fact that adding information increases mutual information; (b) follows from h(Y N 1 , SN 1 |W1, W2) = P [h(Z1i) + h(Z2i)] (see Claim 3); (c) follows from h(SN 1 |W2) = h(Y N 2 |W2) (see Claim 4); (d) follows from the fact that XN 2 is a function of (W2, SN\u22121 1 ) (see Claim 5); (e) follows from the fact that conditioning reduces entropy. Claim 3: h(Y N 1 , SN 1 |W1, W2) = P [h(Z1i) + h(Z2i)] . Proof: October 27, 2018 DRAFT \f14 h(Y N 1 , SN 1 |W1, W2) = X h(Y1i, S1i|W1, W2, Y i\u22121 1 , Si\u22121 1 ) (a) = X h(Y1i, S1i|W1, W2, Y i\u22121 1 , Si\u22121 1 , X1i, X2i) (b) = X h(Z1i, Z2i|W1, W2, Y i\u22121 1 , Si\u22121 1 , X1i, X2i) (c) = X [h(Z1i) + h(Z2i)] , where (a) follows from the fact that X1i is a function of (W1, Y i\u22121 1 ) and X2i is a function of (W2, Si\u22121 1 ) (by Claim 5); (b) follows from the fact that Y1i = gdX1i + gcX2i + Z1i and S1i = gcX1i+Z2i; (c) follows from the memoryless property of the channel and the independence assumption of Z1i and Z2i. Claim 4: h(SN 1 |W2) = h(Y N 2 |W2). Proof: h(Y N 2 |W2) = X h(Y2i|Y i\u22121 2 , W2) (a) = X h(S1i|Y i\u22121 2 , W2) (b) = X h(S1i|Y i\u22121 2 , W2, Xi 2, Si\u22121 1 ) (c) = X h(S1i|W2, Si\u22121 1 ) = h(SN 1 |W2), where (a) follows from the fact that Y2i is a function of (X2i, S1i) and X2i is a function of (W2, Y i\u22121 2 ); (b) follows from the fact that Xi 2 is a function of (W2, Y i\u22121 2 ) and Si\u22121 1 is a function of (Y i\u22121 2 , Xi\u22121 2 ); (c) follows from the fact that Y i\u22121 2 is a function of (Xi\u22121 2 , Si\u22121 1 ) and Xi 2 is a function of (W2, Si\u22121 1 ) (by Claim 5). Claim 5: For all i \u22651, Xi 1 is a function of (W1, Si\u22121 2 ) and Xi 2 is a function of (W2, Si\u22121 1 ). Proof: By symmetry, it is enough to prove only one. Notice that Xi 2 is a function of (W1, W2, Zi\u22122 1 , Zi\u22121 2 ). In Fig. 2, we can easily see that the information of (W1, Zi\u22122 1 ) delivered to the second link must pass through Si\u22121 1 . Also Si\u22121 1 contains Zi\u22121 2 . Therefore, Xi 2 is a function of (W2, Si\u22121 1 ). Now go back to the main stream of the proof. If (R1, R2) is achievable, then \u01ebN \u21920 as N \u2192\u221e. Hence, we get R1 + R2 \u2264h(Y1|S1, X2) \u2212h(Z1) + h(Y2) \u2212h(Z2). October 27, 2018 DRAFT \f15 Assume that X1 and X2 have covariance \u03c1, i.e., E[X1X\u2217 2] = \u03c1. Then, we get h(Y2) \u2264log 2\u03c0e \u0010 1 + SNR + INR + 2|\u03c1| \u221a SNR \u00b7 INR \u0011 (23) Given (X2, S1), the variance of Y1 is upper bounded by Var [Y1|X2, S1] \u2264KY1 \u2212KY1(X2,S1)K\u22121 (X2,S1)K\u2217 Y1(X2,S1), where KY1 = E \u0002 |Y1|2\u0003 = 1 + SNR + INR + \u03c1g\u2217 dgc + \u03c1\u2217gdg\u2217 c KY1(X2,S1) = E [Y1[X\u2217 2, S\u2217 1]] = [\u03c1gd + gc, g\u2217 cgd + \u03c1\u2217INR] K(X2,S1) = E \uf8ee \uf8f0 \uf8ee \uf8f0|X2|2 X2S\u2217 1 X\u2217 2S1 |S1|2 \uf8f9 \uf8fb \uf8f9 \uf8fb= \uf8ee \uf8f0 1 \u03c1\u2217g\u2217 c \u03c1gc 1 + INR \uf8f9 \uf8fb. (24) By further calculation, h(Y1|X2, S1) \u2264log 2\u03c0e \u0012 1 + (1 \u2212|\u03c1|2)SNR 1 + (1 \u2212|\u03c1|2)INR \u0013 (25) From (23) and (25), we get the desired upper bound. C. Symmetric Capacity to Within 1.7075 Bits Theorem 4: For all channel parameters SNR and INR, we can achieve all rates R up to Csym \u22121.7075. Therefore, the feedback symmetric capacity Csym satis\ufb01es Csym \u22121.7075 \u2264Csym \u2264Csym. (26) Proof: In the weak interference regime, we get October 27, 2018 DRAFT \f16 2(Csym \u2212Rweak sym ) \u2264log \u0012 1 + SNR 1 + INR \u0013 \u2212log \u0012 1 + SNR 2INR \u0013 + log \u0010 1 + SNR + INR + 2 \u221a SNRINR \u0011 \u2212log \u0012 1 + SNR 2INR \u0013 \u2212log (1 + INR) + 1 \u2264log \u00121 + SNR + INR 1 + INR 2INR 2INR + SNR \u0013 + log 1 + SNR + INR + 2 \u221a SNRINR SNR + 2INR 2INR 1 + INR ! + 1 \u2264log 2 + log \u00128 3 \u0013 + 1 \u22483.4150 Therefore, Csym \u2212Rweak sym \u22641.7075. In the strong interference regime, we get 2(Csym \u2212Rstrong sym ) \u2264log \u0012 1 + SNR 1 + INR \u0013 + log 1 + SNR + INR + 2 \u221a SNRINR 1 + INR ! \u2264log 2 + log 4 = 3 Therefore, Csym \u2212Rstrong sym \u22641.5. This completes the proof. D. Comparison to Related Work [7], [8], [9] For the Gaussian channel, Kramer developed a feedback strategy based on Schalkwijk-Kailith scheme and Ozarow\u2019s scheme. However, since the scheme is not expressed as a closed form, we cannot see how his scheme is close to the approximate symmetric capacity we derived. To see this, we \ufb01nd the generalized degrees-of-freedom of his scheme. Lemma 1: The generalized degrees-of-freedom of Kramer\u2019s scheme is given by d(\u03b1) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 \u2212\u03b1, 0 \u2264\u03b1 < 1 3; 3\u2212\u03b1 4 , 1 3 \u2264\u03b1 < 1; 1+\u03b1 4 , \u03b1 \u22651. (27) October 27, 2018 DRAFT \f17 Proof: Let INR = SNR\u03b1. Then, by (29) in [7] and (77*) in [8], we get Rsym = log 1 + SNR + SNR\u03b1 + 2\u03c1\u2217SNR \u03b1+1 2 1 + (1 \u2212\u03c1\u22172)SNR\u03b1 ! , (28) where \u03c1\u2217is the solution between 0 and 1 such that 2SNR 3\u03b1+1 2 \u03c1\u22174 + SNR\u03b1\u03c1\u22173 \u22124(SNR 3\u03b1+1 2 + SNR \u03b1+1 2 )\u03c1\u22172 \u2212(2 + SNR + 2SNR\u03b1)\u03c1\u2217+ 2(SNR 3\u03b1+1 2 + SNR \u03b1+1 2 ) = 0. Notice that for 0 \u2264\u03b1 \u22641 3, SNR is a dominant term for high SNR and \u03c1\u2217is between 0 and 1; hence, we get \u03c1\u2217\u22482SNR 3\u03b1\u22121 2 . From this, we get limSNR\u2192\u221e Rsym log(SNR) = 1 \u2212\u03b1. For 1 3 < \u03b1 < 1, the \ufb01rst and second dominant terms are SNR 3\u03b1+1 2 and SNR. Also for this range, \u03c1\u2217is very close to 1. Hence, we approximately get 1 \u2212\u03c1\u22172 \u2248SNR \u22123\u03b1+1 4 . Therefore, we get the desired result for this range. For \u03b1 \u22651, note that the \ufb01rst and second dominant terms are SNR 3\u03b1+1 2 and SNR; and \u03c1\u2217is very close to 1. So we get 1 \u2212\u03c1\u22172 \u2248SNR\u2212\u03b1+1 4 . From this, we get the desired result. Note that in Fig. 5 his scheme has the unbounded gap with capacity for all values \u03b1 except \u03b1 = 1. To compare the scheme to our results for \u03b1 = 1, we also plot the symmetric rate for \ufb01nite channel parameters as shown in Fig. 6. Notice that Kramer\u2019s scheme is very close to the outer bound when INR is similar to SNR. In fact, we can see capacity theorem in [8], i.e., the Kramer\u2019s scheme achieves the capacity when INR = SNR \u2212 \u221a 2SNR. However, if INR is quite different from SNR, it becomes far away from the outer bound. Also note that our new bound is much better than Gastpar-Kramer\u2019s outer bounds [7], [9]. V. DISCUSSION AND" + } + ], + "Kartik Sreenivasan": [ + { + "url": "http://arxiv.org/abs/2202.12002v2", + "title": "Rare Gems: Finding Lottery Tickets at Initialization", + "abstract": "Large neural networks can be pruned to a small fraction of their original\nsize, with little loss in accuracy, by following a time-consuming \"train,\nprune, re-train\" approach. Frankle & Carbin conjecture that we can avoid this\nby training \"lottery tickets\", i.e., special sparse subnetworks found at\ninitialization, that can be trained to high accuracy. However, a subsequent\nline of work by Frankle et al. and Su et al. presents concrete evidence that\ncurrent algorithms for finding trainable networks at initialization, fail\nsimple baseline comparisons, e.g., against training random sparse subnetworks.\nFinding lottery tickets that train to better accuracy compared to simple\nbaselines remains an open problem. In this work, we resolve this open problem\nby proposing Gem-Miner which finds lottery tickets at initialization that beat\ncurrent baselines. Gem-Miner finds lottery tickets trainable to accuracy\ncompetitive or better than Iterative Magnitude Pruning (IMP), and does so up to\n$19\\times$ faster.", + "authors": "Kartik Sreenivasan, Jy-yong Sohn, Liu Yang, Matthew Grinde, Alliot Nagle, Hongyi Wang, Eric Xing, Kangwook Lee, Dimitris Papailiopoulos", + "published": "2022-02-24", + "updated": "2022-06-02", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG", + "cs.AI", + "cs.CV" + ], + "main_content": "Introduction A large body of research since the 1980s empirically observed that large neural networks can be compressed or sparsi\ufb01ed to a small fraction of their original size while maintaining their predictive accuracy [14\u201316, 20, 23, 28, 41]. Although several pruning methods have been proposed during the past few decades, many of them follow the \u201ctrain, prune, re-train\u201d paradigm. Although the above methods result in very sparse, accurate models, they typically require several rounds of re-training, which is computationally intensive. Frankle & Carbin [9] suggest that this computational burden may be avoidable. They conjecture that given a randomly initialized network, one can \ufb01nd a sparse subnetwork that can be trained to accuracy comparable to that of its fully trained dense counterpart. This trainable subnetwork found at initialization is referred to as a lottery ticket. The study above introduced iterative magnitude pruning (IMP) as a means of \ufb01nding these lottery tickets. Their experimental \ufb01ndings laid the groundwork for what is now known as the Lottery Ticket Hypothesis (LTH). Although Frankle & Carbin [9] establish that the LTH is true for tasks like image classi\ufb01cation on MNIST, they were not able to get satisfactory results for more complex datasets like CIFAR-10 and ImageNet when using deeper networks, such as VGG and ResNets [10]. In fact, subsequent work brought the effectiveness of IMP into question. Su et al. [37] showed that even randomly sampled sparse subnetworks at initialization can beat lottery tickets found by IMP as long as the layerwise sparsities are chosen carefully. Gale et al. [12] showed that methods like IMP which train tickets from initialization cannot compete with the accuracy of a model trained with pruning as part of the optimization process. Frankle et al. [10] explain the failures of IMP using the concept of linear mode connectivity which measures the stability of these subnetworks to SGD noise. Extensive follow-up studies propose \u2217Authors contributed equally to this paper. Preprint. Under review. arXiv:2202.12002v2 [cs.LG] 2 Jun 2022 \fRandomly Initialized Network Partially trained Dense Network IMP Gem-Miner 88.8% accuracy Iteratively Prune and retrain 59.3% accuracy 90.5% accuracy Trained 0.5% Sparse Subnetwork Prune at initialization 0.5% sparse Subnetwork at Initialization Finetune 31.88% accuracy Warmup 0.5% sparse Subnetwork at Warm Initialization Finetune Trained 0.5% Sparse Subnetwork 10% accuracy Figure 1: Conceptual visualization of GEM-MINER vs IMP with warmup. The accuracies listed are on a 0.5% sparse VGG-16 trained on CIFAR-10. Given a randomly initialized network, both methods output a subnetwork which is then \ufb01netuned. IMP requires warmup i.e., few epochs of training before it can \ufb01nd a sparse subnetwork. GEM-MINER \ufb01nds a rare gem, a subnetwork at initialization that achieves high accuracy both before and after weight training. several heuristics for \ufb01nding trainable sparse subnetworks at initialization [24, 38, 39]. However, subsequent work by Frankle et al. [11], Su et al. [37] show experimentally that all of these methods fail simple sanity checks. Most methods seem to merely identify good sparsities at each layer, but given those, random sparse subnetworks can be trained to similar or better accuracy. Frankle et al. [10] show that with a small modi\ufb01cation, IMP can beat these sanity checks; the caveat is that it no longer \ufb01nds these subnetworks at initialization, but after a few epochs of warm-up training. Since these subnetworks are found after initialization, IMP with warmup does not \ufb01nd lottery tickets. As noted in the original work by Frankle & Carbin [9], the importance of \ufb01nding trainable subnetworks at initialization is computational ef\ufb01ciency. It is far preferable to train a sparse model from scratch, rather than having to deal with a large dense model, even if that is for a few epochs (which is what IMP with warmup does). To the best of our knowledge, the empirical validity of the Lottery Ticket Hypothesis, i.e., the hunt for subnetworks at initialization trainable to SOTA accuracy, remains an open problem. Our Contributions. We resolve this open problem by developing GEM-MINER, an algorithm that \ufb01nds sparse subnetworks at initialization, trainable to accuracy comparable or better than IMP with warm-up. GEM-MINER does so by \ufb01rst discovering rare gems. Rare gems are subnetworks at initialization that attain accuracy far above random guessing, even before training. Rare gems can then be re\ufb01ned to achieve near state-of-the-art accuracy. Simply put, rare gems are lottery tickets that also have high accuracy at initialization. High accuracy at initialization is not a requirement for a network to be de\ufb01ned as a lottery ticket. However, if our end goal is high accuracy after training, then having high accuracy at initialization likely helps. Rare gems found by GEM-MINER are the \ufb01rst lottery tickets to beat all baselines in [11, 37]. In Fig. 1 we give a sketch of how our proposed algorithm GEM-MINER compares with IMP with warm start. GEM-MINER \ufb01nds these subnetworks in exactly the same number of epochs that it takes to train them, and is up to 19\u00d7 faster than IMP with warmup. 2 Related Work Lottery ticket hypothesis. Following the pioneering work of Frankle & Carbin [9], the search for lottery tickets has grown across several applications, such as language tasks, graph neural networks and federated learning [3, 4, 13, 25]. While the LTH itself has yet to be proven mathematically, the so-called strong LTH has been derived which shows that any target network can be approximated by pruning a randomly initialized network with minimal overparameterization [27, 29, 30]. Recently, it has been shown that for such approximation results it suf\ufb01ces to prune a random binary network with slightly larger overparameterization [6, 36]. 2 \fTable 1: We compare the different popular pruning methods in the literature on whether they prune at initialization, are \ufb01netunable and pass sanity checks. We also list the amount of computation they need to \ufb01nd a 1.4% sparse subnetwork on ResNet-20, CIFAR-10. For consistency, we do not include the time required to \ufb01netune this subnetwork to full accuracy as it would be equal for all methods. For single-shot pruning method we list it as 1 epoch but this depends on the choice of batch-size. Learning Rate Rewinding which we label Renda et al. [32] is a pruning after training algorithm and just outputs a high accuracy subnetwork and hence the sanity checks do not apply to it. Pruning Method Prunes at initialization Finetunable Passes sanity checks Computation to reach 1.4% sparsity IMP [9] \u0017 \u0013 \u0013 2850 epochs SNIP [24] \u0013 \u0013 \u0017 1 epoch GraSP [39] \u0013 \u0013 \u0017 1 epoch SynFlow [38] \u0013 \u0013 \u0017 1 epoch Edge-popup [31] \u0013 \u0017 \u0017 150 epochs Smart Ratio [37] \u0013 \u0013 \u2013 O(1) Learning Rate Rewinding [32] \u0017 \u2013 \u2013 3000 epochs GEM-MINER \u0013 \u0013 \u0013 150 epochs Pruning at initialization. While network pruning has been studied since the 1980s, \ufb01nding sparse subnetworks at initialization is a more recently explored approach. Lee et al. [24] propose SNIP, which prunes based on a heuristic that approximates the importance of a connection. Tanaka et al. [38] propose SynFlow which prunes the network to a target sparsity without ever looking at the data. Wang et al. [39] propose GraSP which computes the importance of a weight based on the Hessian gradient product. The goal of these algorithms is to \ufb01nd a subnetwork that can be trained to high accuracy. Ramanujan et al. [31] propose Edge-Popup (EP) which \ufb01nds a subnetwork at initialization that has high accuracy to begin with. Unfortunately, they also note that these subnetworks are not conducive to further \ufb01netuning. The above algorithms are all based on the idea that one can assign a \u201cscore\u201d to each weight to measure its importance. Once such a score is assigned, one simply keeps the top fraction of these scores based on the desired target sparsity. This may be done by sorting the scores layer-wise or globally across the network. Additionally, this can be done in one-shot (SNIP, GraSP) or iteratively (SynFlow). Note that IMP can also be \ufb01t into the above framework by de\ufb01ning the \u201cscore\u201d to be the magnitude of the weights and then pruning globally across the network iteratively. More recently, Alizadeh et al. [1] propose ProsPr which utilizes the idea of meta-gradients through the \ufb01rst few steps of optimization to determine which weights to prune. Their intuition is that this will lead to masks at initialization that are more amenable to training to high accuracy within a few steps. While it \ufb01nds high accuracy subnetworks, we show in Section 4.2 that it fails to pass the sanity checks of Frankle et al. [11] and Su et al. [37]. Sanity checks for lottery tickets. A natural question that arises with pruning at initialization is whether these algorithms are truly \ufb01nding interesting and nontrivial subnetworks, or if their performance after \ufb01netuning can be matched by simply training equally sparse, yet random subnetworks. Ma et al. [26] propose more rigorous de\ufb01nitions of winning tickets and study IMP under several settings with careful tuning of hyperparameters. Frankle et al. [11] and Su et al. [37] introduce several sanity checks (i) Random shuf\ufb02ing (ii) Weight reinitialization (iii) Score inversion and (iv) Random Tickets. Even at their best performance, they show that SNIP, GraSP and SynFlow merely \ufb01nd a good sparsity ratio in each layer and fail to surpass, in term of accuracy, fully trained randomly selected subnetworks, whose sparsity per layer is similarly tuned. Frankle et al. [11] show through extensive experiments that none of these methods show accuracy deterioration after random reshuf\ufb02ing. We explain the sanity checks in detail in Section 4 and use them as baselines to test our own algorithm. Pruning during/after training. While the above algorithms prune at/near initialization, there exists a rich literature on algorithms which prune during/after training. Unlike IMP, algorithms in this category do not rewind the weights. They continue training and pruning iteratively. Frankle et al. [11] and Gale et al. [12] show that pruning at initialization cannot hope to compete with these algorithms. While they do not \ufb01nd lottery tickets, they do \ufb01nd high accuracy sparse networks. Zhu & Gupta [41] propose a gradual pruning schedule where the smallest fraction of weights are pruned at a prede\ufb01ned frequency. They show that this results in models up to 95% sparsity with negligible loss in performance on language as well as image processing tasks. Gale et al. [12] and Frankle et al. [11] also study this as a baseline under the name magnitude pruning after training. Renda 3 \fet al. [32] show that rewinding the learning rate as opposed to weights(like in IMP) leads to the best performing sparse networks. However, it is important to remark that these are not Lottery Tickets, merely high accuracy sparse networks. We contrast these different methods in Table 1 in terms of whether they prune at initialization, their \ufb01netunability, whether they pass sanity checks as well as their computational costs. Finally, we note that identifying a good pruning mask can be thought of as training a binary network where the loss is computed over the element-wise product of the original network with the mask. This has been explored in the quantization during training literature [5, 19, 34]. 3 GEM-MINER: Discovering Rare Gems Setting and notation. Let S = {(xi, yi)}n i=1 be a given training dataset for a k-classi\ufb01cation problem, where xi \u2208Rd0 denotes a feature vector and label yi \u2208{1, . . . , k} denotes its label. Typically, we wish to train a neural network classi\ufb01er f(w; x) : Rd0 \u2192{1, . . . , k}, where w \u2208Rd denotes the set of weight parameters of this neural network. The goal of a pruning algorithm is to extract a mask m = {0, 1}d, so that the pruned network is denoted by f(w \u2299m; x), where \u2299denotes the element-wise product. We de\ufb01ne the sparsity of this network to be the fraction of non-zero weights s = \u2225w \u2299m\u22250/d. The loss of a classi\ufb01er on a single sample (x, y) is denoted by \u2113(f(w \u2299m; x), y), which captures a measure of discrepancy between prediction and reality. In what follows, we will denote by w0 \u2208Rd to be the set of random initial weights. The type of randomness will be explicitly mentioned when necessary. On the path to rare gems; \ufb01rst stop: Maximize pre-training accuracy. A rare gem needs to satisfy three conditions: (i) sparsity, (ii) non-trivial pre-training accuracy, and (iii) that it can be \ufb01netuned to achieve accuracy close to that of the fully trained dense network. This is not an easy task as we have two different objectives in terms of accuracy (pre-training and post-training), and it is unclear if a good subnetwork for one objective is also good for the other. However, since pre-training accuracy serves as a lower bound on the \ufb01nal performace, we focus on maximizing that \ufb01rst, and then attempt to further improve it by \ufb01netuning. 100 200 300 400 500 600 Epoch 1 5 50 Sparsity (%) GM: (87.6, 90.67) EP: (91.37, 92.07) GM: (73.38, 89.32) EP: (90.06, 90.93) GM: (87.95, 89.51) EP: (70.69, 84.34) GM: (84.62, 87.37) EP: (10.0, 10.0) GM ( =0) EP (s=50%) GM ( =3e-6) EP (s=20%) GM ( =7e-6) EP (s=5%) GM ( =1.5e-5) EP (s=1.7%) Figure 2: The sparsity of intermediate results, the accuracy of the \ufb01nal output, and the accuracy after \ufb01netuning on MobileNet-V2, CIFAR-10. For GEM-MINER (GM), we also visualize the sparsity upper bounds as dotted lines. As \u03bb increases, note that the sparsity of GEM-MINER\u2019s output decreases. For\u03bb = 3 \u00b7 10\u22126, the iterative freezing algorithm kicks in around epoch 220, regularizing the sparsity thereafter. The gem found by GEM-MINER(\u03bb = 1.5 \u00b7 10\u22125) achieves accuracy of 84.62% before \ufb01netuning and 87.37% after \ufb01netuning, while EP is unable to achieve non-trivial accuracy before or after \ufb01netuning. Our algorithm is inspired by Edge-Popup (EP) [31]. EP successfully \ufb01nds subnetworks with high pre-training accuracy but it has two major limitations: (i) it does not work well in the high sparsity regime (e.g., < 5%), and (ii) most importantly, the subnetworks it \ufb01nds are not conducive to further \ufb01netuning. In the following, we take GEM-MINER apart and describe the components that allow it to surpass these issues. GEM-MINER without sparsity control. Much like EP, GEM-MINER employs a form of backpropagation, and works as follows. Each of the random weights [w0]i in the original network is associated with a normalized score pi \u2208[0, 1]. These normalized scores become our optimization variables and are responsible for computing the supermask m, i.e., the pruning pattern of the network at initialization. For a given set of weights w and scores p, GEM-MINER sets the effective weights as we\ufb00= w\u2299r(p), where r(\u00b7) is an element-wise rounding function, and m = r(p) is the resulting supermask. The rounding function can be changed, e.g., r can perform randomized rounding, in which case pi would be the probability of keeping weight wi in m. In our case, we found that simple deterministic rounding, i.e., r(pi) = 1pi\u22650.5 works well. At every iteration GEM-MINER samples a batch of training data and performs backpropagation on the loss of the effective weights, with respect to the scores p, while projecting back to [0, 1] when needed. During the forward pass, due to the rounding function, the effective network used is indeed 4 \fa subnetwork of the given network. Here, since r(p) is a non-differentiable operation we use the Straight Through Estimator (STE) [2] which backpropagates through the indicator function as though it were the identity function. Note that this vanilla version of GEM-MINER is unable to exercise control over the \ufb01nal sparsity of the model. For reasons that will become evident in below, we will call this version of our algorithm GEM-MINER(0). There is already a stark difference from EP: GEM-MINER(0) will automatically \ufb01nd the optimal sparsity, while EP requires the target sparsity s as an input parameter. However, at the same time, this also signi\ufb01cantly limits the applicability of GEM-MINER(0) as one cannot obtain a highly sparse gem. Shown as a dark blue curve in Fig. 2 is the sparsity of GEM-MINER(0). Here, we run GEM-MINER with a randomly initialized MobileNet-V2 network on CIFAR-10. Note that the sparsity stays around 50% throughout the run, which is consistent with the observation by Ramanujan et al. [31] that accuracy of subnetworks at initialization is maximized at around 50% sparsity. Algorithm 1: GEM-MINER Input: Dataset D = {(xi, yi)}, learning rate \u03b7, rounding function r(\u00b7), number of epochs E, freezing period T, target sparsity s \u2208[0, 1] Output: Mask m = r(p) \u2299q \u2208{0, 1}d 1 c \u2190ln (1/s) E , q \u21901d 2 w, p \u2190random vector in Rd, 3 random vector in [0, 1]d 4 for j in 1, 2, . . . , E do 5 for (xi, yi) \u2208D do 6 we\ufb00\u2190(w \u2299q) \u2299r(p) 7 p \u2190p \u2212\u03b7\u2207p \u2113(f(we\ufb00; xi), yi) 8 /* STE */ 9 p \u2190proj[0,1]d p 10 if mod(j, T) = 0 then 11 I1 \u2190{i : qi = 1} 12 psorted \u2190sort(pi\u2208I1) 13 pbottom \u2190Bottom-(1 \u2212ecT ) fraction 14 of psorted 15 q \u2190q \u22991pi / \u2208pbottom Regularization and Iterative freezing. GEM-MINER(0) is a good baseline algorithm for \ufb01nding accurate subnetworks at initialization, but it cannot be used to \ufb01nd rare gems, which need to be sparse and trainable. To overcome this limitation, we apply a standard trick \u2013 we add a regularization term to encourage sparsity. Thus, in addition to the task loss computed with the effective weights, we also compute the L2 or L1 norm of the score vector p and optimize over the total regularized loss. More formally, we minimize \u2113:= \u2113task + \u03bb\u2113reg, where \u03bb is the hyperparameter and \u2113reg is either L2 or L1 norm of the score vector p. We call this variant GEM-MINER(\u03bb), where \u03bb denotes the regularization weight. This naming convention should explain why we called the initial version GEM-MINER(0). The experimental results in Fig. 2 show that this simple modi\ufb01cation indeed allows us to control the sparsity of the solution. We chose to use the L2 regularizer, however preliminary experiments showed that L1 performs almost identically. By varying \u03bb from \u03bb = 0 to \u03bb = 7 \u00b7 10\u22126 and \u03bb = 1.5 \u00b7 10\u22125, the \ufb01nal sparsity of the gem found by GEM-MINER(\u03bb) becomes 2.5% and 1.4%, respectively. One drawback of this regularization approach is that it only indirectly controls the sparsity. If we have a target sparsity s, then there is no easy way of \ufb01nding the appropriate value of \u03bb such that the resulting subnetwork is s-sparse. If we choose \u03bb to be too large, then it will give us a gem that is way too sparse; too small a \u03bb and we will end up with a denser gem than what is needed. As a simple heuristic, we employ iterative freezing, which is widely used in several existing pruning algorithms, including IMP [9, 12, 41]. More speci\ufb01cally, we can design an exponential function s(j) = e\u2212cj for some c > 0, which will serve as the upper bound on the sparsity. If the total number of epochs is E and the target sparsity is s, we have s(E) = e\u2212cE = s. Thus, we have c = ln (1/s)/E. Once this sparsity upper bound is designed, the iterative freezing mechanism regularly checks the current sparsity to see if the upper bound is violated or not. If the sparsity bound is violated, it \ufb01nds the smallest scores, zeros them out, and freezes their values thereafter. By doing so, we can guarantee the \ufb01nal sparsity even when \u03bb was not suf\ufb01ciently large. To see this freezing mechanism in action, refer the blue curve in Fig. 2. Here, the sparsity upper bounds (decreasing exponential functions) are visualized as dotted lines. Note that for the case of \u03bb = 3 \u00b7 10\u22126, the sparsity of the network does not decay as fast as desired, so it touches the sparsity upper bound around epoch 220. The iterative freezing scheme kicks in here, and the sparsity decay is controlled by the upper bound thereafter, achieving the speci\ufb01ed target sparsity at the end. 5 \fThe full pseudocode of GEM-MINER is provided in Algorithm 1. There are two minor implementation details which differ from the explanation above: (i) we impose the iterative freezing every T epochs, not every epoch and (ii) iterative freezing is imposed even when the sparsity bound is not violated. 50.0 5.0 2.0 Fraction of Remaining Weights (%) 10 20 30 40 50 60 70 80 90 Test Accuracy (%) Weight training GM Renda et al. GM (reinit weight) IMP (with warmup) GM (shuffle mask) EP GM (invert score) SR ResNet-20 50 20 5 2.5 1.4 0.5 Sparsity (%) 50 60 70 80 90 T est Acc. (%) MobileNet-V2 50 20 5 2.5 1.4 Sparsity (%) 50 60 70 80 90 T est Acc. (%) VGG-16 50 20 5 2.5 1.4 0.5 Sparsity (%) 50 60 70 80 90 T est Acc. (%) WideResNet-28-2 50 20 5 2.5 1.4 Sparsity (%) 50 60 70 80 90 T est Acc. (%) 50 20 5 2.5 1.4 0.5 Sparsity (%) 50 60 70 80 90 T est Acc. (%) 50 20 5 2.5 1.4 Sparsity (%) 50 60 70 80 90 T est Acc. (%) 50 20 5 2.5 1.4 0.5 Sparsity (%) 50 60 70 80 90 T est Acc. (%) 50 20 5 2.5 1.4 Sparsity (%) 50 60 70 80 90 T est Acc. (%) Figure 3: Performance of different pruning algorithms on CIFAR-10 for benchmark networks. Top: post-\ufb01netune accuracy; Bottom: sanity check methods suggested in Frankle et al. [11] applied on GEM-MINER (GM). Note that GM achieves similar post-\ufb01netune accuracy as IMP, and typically outperforms it in the sparse regime. GM has higher post-\ufb01netune accuracy than EP and Smart Ratio (SR). GM also passes the sanity checks suggested in Frankle et al. [11]. Finally, GM (which prunes at init) nearly achieves the performance of Renda et al. (which is a pruning after training method) in the sparse regime, e.g., 1.4% sparsity in ResNet-20. 4 Experiments In this section, we present the experimental results2 for the performance of GEM-MINER across various tasks. Tasks. We evaluate our algorithm on (Task 1) CIFAR-10 classi\ufb01cation, on various networks including ResNet-20, MobileNet-V2, VGG-16, and WideResNet-28-2, (Task 2) TinyImageNet classi\ufb01cation on ResNet-18 and ResNet-50, and (Task 3) Finetuning on the Caltech-101 [7] dataset using a ResNet-50 pretrained on ImageNet. The detailed description of the datasets, networks and hyperparameters can be found in Section A of the Appendix. Proposed scheme. We run GEM-MINER with an L2 regularizer. If a network reaches its best accuracy after E epochs of weight training, then we run GEM-MINER for E epochs to get a sparse subnetwork, and then run weight training on the sparse subnetwork for another E epochs. Comparisons. We tested our method against \ufb01ve baselines: dense weight training and four pruning algorithms: (i) IMP [10], (ii) Learning rate rewinding [32], denoted by Renda et al., (iii) Edge-Popup (EP) [31], and (iv) Smart-Ratio (SR) which is the random pruning method proposed by Su et al. [37]. We also ran the following sanity checks, proposed by Frankle et al. [11]: (i) (Random shuf\ufb02ing): To test if the algorithm prunes speci\ufb01c connections, we randomly shuf\ufb02e the mask at every layer. (ii) (Weight reinitialization): To test if the \ufb01nal mask is speci\ufb01c to the weight initialization, we reinitialize the weights from the original distribution. (iii) (Score inversion): Since most pruning algorithms use a heuristic/score function as a proxy to measure the importance of different weights, we invert the scoring function to check whether it is a valid proxy. More precisely, this test involves pruning the weights which have the smallest scores rather than the largest. In all of the above tests, if the accuracy after \ufb01netuning the new subnetwork does not deteriorate signi\ufb01cantly, then the algorithm is merely identifying optimal layerwise sparsities. 4.1 Rare gems obtained by GEM-MINER Task 1. Fig. 3 shows the sparsity-accuracy tradeoff for various pruning methods trained on CIFAR10 using ResNet-20, MobileNet-V2, VGG-16 and WideResNet-28-2. For each column (network), we compare IMP, IMP with learning rate rewinding (Renda et al.), GEM-MINER, EP, and SR in two 2Our codebase can be found at https://anonymous.4open.science/r/pruning_is_enough-F0B0. 6 \f50.0 5.0 2.0 Fraction of Remaining Weights (%) 10 20 30 40 50 60 70 80 90 Test Accuracy (%) Weight training GM Renda et al. GM (reinit weight) IMP (with warmup) GM (shuffle mask) EP GM (invert score) SR CIFAR-100, ResNet-32 50 20 5 2.5 1.4 0.5 Sparsity (%) 20 30 40 50 60 70 80 T est Acc. (%) TinyImageNet, ResNet-18 50 20 5 2.5 1.4 0.5 Sparsity (%) 0 20 40 60 T est Acc. (%) TinyImageNet, ResNet-50 50 20 5 2.5 1.4 0.5 Sparsity (%) 30 40 50 60 Test Acc. (%) Caltech-101, ResNet-50 50 20 5 2 Sparsity (%) 60 80 100 T est Acc. (%) 50 20 5 2.5 1.4 0.5 Sparsity (%) 20 30 40 50 60 70 80 T est Acc. (%) 50 20 5 2.5 1.4 0.5 Sparsity (%) 20 40 60 T est Acc. (%) 50 20 5 2.5 1.4 0.5 Sparsity (%) 0 10 20 30 40 50 60 Test Acc. (%) 50 20 5 2 Sparsity (%) 20 40 60 80 100 T est Acc. (%) Figure 4: Accuracy on image classi\ufb01cation tasks on TinyImageNet, ImageNet and Caltech-101. For Caltech-101, we pruned a pre-trained ImageNet model (ResNet-50). Top: post-\ufb01netune accuracy, bottom: sanity check methods suggested in Frankle et al. [11] applied on GEM-MINER. performance metrics: the top row shows the accuracy of the subnetwork after weight training and bottom row shows the result of the sanity checks on GEM-MINER. As shown in the top row of Fig. 3, GEM-MINER \ufb01nds a lottery ticket at initialization. It reaches accuracy similar to IMP after weight training. Moreover, for in the sparse regime (e.g., below 1.4% for ResNet-20 and MobileNet-V2), GEM-MINER outperforms IMP in terms of post-\ufb01netune accuracy. The bottom row of Fig. 3 shows that GEM-MINER passes the sanity check methods. For all networks, the performance in the sparse regime (1.4% sparsity or below) shows that the suggested GEM-MINER algorithm enjoys 3\u201310% accuracy gap with the best performance among variants. The results in the top row show that GEM-MINER far outperforms the random network with smart ratio (SR). 100 200 300 400 500 600 Epoch 30 50 70 90 Test Accuracy (%) Weight training (at 300 epoch) IMP (at 6000 epoch) GM Figure 5: Convergence plot for CIFAR-10, MobileNet-V2 experiments, where we apply GEM-MINER for 300 epochs and then \ufb01netune the sparse model for another 300 epochs, to reach 1.16% sparse model. We added the accuracy of weight training (dense model) and IMP (1.4% sparse model) as a reference. Note that we compared with 300 epochs of weight training, and compared with IMP using 20 rounds of iterative pruning, i.e., 300 \u00d7 20 = 6000 epochs, to reach 1.4% sparsity. GEM-MINER achieves a higher accuracy than IMP despite its 19\u00d7 shorter runtime to \ufb01nd a sparse subnetwork. Tasks 2\u20134. Fig. 4 shows the sparsity-accuracy tradeoff for Tasks 2\u20134. Similar to Fig. 3, the top row reports the accuracy after weight training, and the bottom row contains the results of the sanity checks. As shown in Fig. 4a and Fig. 4b, the results for Task 2 show that (i) GEM-MINER achieves accuracy comparable to IMP as well as Renda et al. (IMP with learning rate rewinding) even in the sparse regime, (ii) GEM-MINER has non-trivial accuracy before \ufb01netuning (iii) GEM-MINER passes all the sanity checks, and (iv) GEM-MINER outperforms EP and SR. These results show that GEM-MINER successfully \ufb01nds rare gems even in the sparse regime for Task 2. Fig. 4c shows the result for Task 3. Unlike other tasks, GEM-MINER does not reach the post-\ufb01netune accuracy of IMP, but GEM-MINER enjoys over an 8% accuracy gap compared with EP and SR. Moreover, the bottom row shows that GEM-MINER has over 20% higher accuracy than the sanity checks below 5% sparsity showing that the subnetwork found by GEM-MINER is unique in this sparse regime. 4.2 Comparison to ProsPr Alizadeh et al. [1] recently proposed a pruning at init method called ProsPr which utilizes metagradients through the \ufb01rst few steps of optimization to determine which weights to prune, thereby accounting for the \u201ctrainability\u201d of the resulting subnetwork. In Table 2 we compare it against GEMMINER on ResNet-20, CIFAR-10 and also run the (i) Random shuf\ufb02ing and (ii) Weight reinitialization sanity checks from Frankle et al. [11]. We were unable to get ProsPr using their publicly available codebase to generate subnetworks at sparsity below 5% and therefore chose that sparsity. Note 7 \fTable 2: We compare ProsPr [1] vs GEM-MINER on ResNet-20, CIFAR-10 and run the random shuf\ufb02ing as well as the weight reinit sanity checks. Note that GEM-MINER produces a subnetwork that is higher accuracy despite being more sparse. Moreover, ProsPr does not show signi\ufb01cant decay in performance after the sanity checks while GEM-MINER does. Therefore, it is likely that ProsPr is merely identifying good layerwise sparsity ratios. Algorithm Sparsity Accuracy after \ufb01netune Accuracy after Random shuf\ufb02ing Accuracy after Weight reinitialization ProsPr 5% 82.67% 82.15% 81.64% GEM-MINER 3.72% 83.4% 78.73% 78.6% that GEM-MINER produces a subnetwork that is higher accuracy despite being more sparse. After \ufb01netuning for 150 epochs, our subnetwork reaches 83.4% accuracy while the subnetwork found by ProsPr only reaches 82.67% after training for 200 epochs. More importantly, ProsPr does not show signi\ufb01cant decay in performance after the random reshuf\ufb02ing or weight reinitialization sanity checks. Therefore, as Frankle et al. [11] remark, it is likely that it is identifying good layerwise sparsity ratios, rather than a mask speci\ufb01c to the initialized weights. 4.3 Observations on GEM-MINER Convergence of accuracy and sparsity. Fig. 5 shows how the accuracy of GEM-MINER improves as training progresses, for MobileNet-V2 on CIFAR-10 at sparsity 1.4%. This shows that GEMMINER, reaches high accuracy even early in training, and can be \ufb01netuned to accuracy higher than that of IMP (which requires 19\u00d7 the runtime than our algorithm). 50.0 5.0 2.0 Fraction of Remaining Weights (%) 10 20 30 40 50 60 70 80 90 Test Accuracy (%) Weight training GM Renda et al. GM (reinit weight) IMP (with warmup) GM (shuffle mask) EP GM (invert score) SR ResNet-20 50 20 5 2.5 1.4 0.5 Sparsity (%) 10 30 50 70 90 T est Acc. (%) MobileNet-V2 50 20 5 2.5 1.4 Sparsity (%) 10 30 50 70 90 T est Acc. (%) VGG-16 50 20 5 2.5 1.4 0.5 Sparsity (%) 10 30 50 70 90 T est Acc. (%) WideResNet-28-2 50 20 5 2.5 1.4 Sparsity (%) 10 30 50 70 90 T est Acc. (%) Figure 6: Performance of different pruning algorithms before \ufb01netuning on CIFAR-10 for benchmark networks. GEM-MINER \ufb01nds subnetworks that already have reasonably high accuracy even before weight training. Note that, while IMP and SR have scarcely better than random guessing at initialization, subnetworks found by GEM-MINER typically perform even better than EP, especially in the sparse regime. 1 6 11 16 20 Layer ID 50 100 Per Layer Sparsity (%) GM IMP Smart Ratio EP 50% Sparsity 0 20K 1 6 11 16 20 Layer ID 50 100 Sparsity (%) 3.72% Sparsity 0 20K Num Param 1 6 11 16 20 Layer ID 0 20 40 1.4% Sparsity 0 20K 1 6 11 16 20 Layer ID 0 20 40 Sparsity (%) 0.59% Sparsity 0 20K Num Param 1 6 11 16 20 Layer ID 0 20 Figure 7: The layerwise sparsity for ResNet-20 pruned by GEM-MINER, IMP, Smart Ratio, and EP. The dark bar is the layerwise number of parameters. Both GEMMINER and IMP save the most portion of parameter in the \ufb01rst layer and the last layer. High pre-\ufb01netune accuracy. As shown in Fig. 6, GEM-MINER \ufb01nds subnetworks at initialization that have a reasonably high accuracy even before the weight training, e.g., above 90% accuracy for 1.4% sparsity in VGG-16, and 85% accuracy for 1.4% sparsity in MobileNet-V2. Note that, in contrast, IMP and SR have accuracy scarcely better than random guessing at initialization. Clearly, GEM-MINER ful\ufb01lls its objective in maximizing accuracy before \ufb01netuning and therefore \ufb01nds rare gems \u2013 lottery tickets at initialization which already have high accuracy. Limitations of GEM-MINER. We observed that in the dense regime (50% sparsity, 20% sparsity), GEM-MINER sometimes performs worse than IMP. While we believe that this can be resolved by appropriately tuning the hyperparameters, we chose to focus our attention on the sparse regime. We would also like to remark that GEM-MINER is fairly sensitive to the choice of hyperparameters and for some models, we had to choose different hyperparameters for each sparsity 8 \fTable 3: We construct different variants of EP and compare their performance with GEM-MINER, for ResNet-20, CIFAR-10, 0.59% sparsity. We establish that having a global score metric and gradually pruning is key to improved performance. Pruning Method EP Global EP Global EP with Gradual Pruning Global EP with Gradual Pruning and Regularization GEM-MINER Pre-\ufb01netune acc (%) 19.57 22.22 31.56 19.67 45.30 Post-\ufb01netune acc (%) 24.47 34.42 63.54 63.72 66.15 to ensure optimal performance. Though this occurs rarely, we also \ufb01nd that an extremely aggressive choice of \u03bb can lead to layer-collapse where one or more layers gets pruned completely. This happens when all the scores p of that layer drop below 0.5. Layer-wise sparsity. We compare the layer-wise sparsity pattern of different algorithms for ResNet20 trained on CIFAR-10 in Fig. 7. Both GEM-MINER and IMP spend most of their sparsity budget on the \ufb01rst and last layers. By design, SR assigns 30% sparsity to the last layer and the budget decays smoothly across the others. EP maintains the target sparsity ratio at each layer and therefore is always a horizontal line. How does GEM-MINER resolve EP\u2019s failings? An open problem from Ramanujan et al. [31] is why the subnetworks found by EP are not \ufb01ne-tunable. While GEM-MINER is signi\ufb01cantly different from EP, it is reasonable to ask which modi\ufb01cation allowed it to \ufb01nd lottery tickets without forgoing high accuracy at initialization. Table 3 shows how we can modify EP to improve the pre/post-\ufb01netune performance, for ResNet-20, CIFAR-10 at 0.59% sparsity. Here, we compare EP, GEM-MINER, as well as three EP variants that we construct. (i) (Global EP) is a modi\ufb01cation where the bottom-k scores are pruned globally, not layer-wise. This allows the algorithm to trade-off sparsity in one layer for another. (ii) (Gradual pruning) reduces the parameter k gradually as opposed to setting it to the target sparsity from the beginning. (iii) (Regularization): we add an L2 term on the score p of the weights to encourage sparsity. The results indicate that global pruning and gradual pruning signi\ufb01cantly improve both the pre and post-\ufb01netune accuracies of EP. Adding regularization does not improve the performance signi\ufb01cantly. Finally, adding all three features to EP allows it to achieve 63.72% accuracy, while GEM-MINER reaches 66.15% accuracy. It is important to note that even with all three features, EP is inherently different from GEM-MINER in how it computes the supermask based on the scores. But we conjecture that aggressive, layerwise pruning is the key reason for EP\u2019s failings. Table 4: Comparison of GEM-MINER and its longer version, for ResNet-20, CIFAR-10, 1.4% sparsity. LONG GEM-MINER, when given the same number of epochs improves post-\ufb01netune accuracy by 1.5%, rivaling the performance of Renda et al. [32] Method GM (cold) Long GM (cold) IMP (warm) Renda et al. (pruning after training) Number of Epochs 300 3000 3000 3000 Accuracy (%) 77.89 79.50 74.52 80.21 Applying GEM-MINER for longer periods. Recall that GEM-MINER uses 19 \u00d7 fewer training epochs than iterative train-prune-retrain methods like IMP [10] and Learning rate rewinding (Renda et al. [32]), to \ufb01nd a subnetwork at 1.4% sparsity which can then be trained to high accuracy. Here, we consider a long version of GEM-MINER to see if it can bene\ufb01t if it is allowed to run for longer. Table 4 shows the comparison of post-\ufb01netune accuracy for GEM-MINER, LONG GEM-MINER, IMP and Renda et al. [32] tested on ResNet-20, CIFAR-10, sparsity=1.4% setting. Conventional GEM-MINER, applies iterative freezing every 5 epochs to arrive at the target sparsity in 150 epochs. LONG GEM-MINER instead prunes every 150 epochs and therefore reaches the target sparsity in 3000 epochs. We \ufb01nd that applying GEM-MINER for longer periods improves the post-\ufb01netune accuracy in this regime by 1.5%. This shows that given equal number of epochs, GEM-MINER, which prunes at initialization, can close the gap to Learning rate rewinding [32] which is a prune-after-training method. 5" + }, + { + "url": "http://arxiv.org/abs/2110.08996v2", + "title": "Finding Everything within Random Binary Networks", + "abstract": "A recent work by Ramanujan et al. (2020) provides significant empirical\nevidence that sufficiently overparameterized, random neural networks contain\nuntrained subnetworks that achieve state-of-the-art accuracy on several\npredictive tasks. A follow-up line of theoretical work provides justification\nof these findings by proving that slightly overparameterized neural networks,\nwith commonly used continuous-valued random initializations can indeed be\npruned to approximate any target network. In this work, we show that the\namplitude of those random weights does not even matter. We prove that any\ntarget network can be approximated up to arbitrary accuracy by simply pruning a\nrandom network of binary $\\{\\pm1\\}$ weights that is only a polylogarithmic\nfactor wider and deeper than the target network.", + "authors": "Kartik Sreenivasan, Shashank Rajput, Jy-yong Sohn, Dimitris Papailiopoulos", + "published": "2021-10-18", + "updated": "2021-10-22", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG", + "cs.AI" + ], + "main_content": "Introduction As the number of parameters of state-of-the-art networks continues to increase, pruning has become a prime choice for sparsifying and compressing a model. A rich and long body of research, dating back to the 80s, shows that one can prune most networks to a tiny fraction of their size, while maintaining high accuracy (Mozer and Smolensky, 1989; Hassibi and Stork, 1993; Levin et al., 1994; LeCun et al., 1990; Han et al., 2015b;a; Li et al., 2016; Wen et al., 2016; Hubara et al., 2016; 2017; He et al., 2017; Wu et al., 2016; Zhu et al., 2016; He et al., 2018; Zhu and Gupta, 2017; Cheng et al., 2019; Blalock et al., 2020; Deng et al., 2020). A downside of most of the classic pruning approaches is that they sparsify a model once it is trained to full accuracy, followed by signi\ufb01cant \ufb01ne-tuning, resulting in a computationally burdensome procedure. Frankle and Carbin (2018) conjectured the existence of lottery tickets, i.e., sparse subnetworks at (or near) initialization, that can be trained\u2014just once\u2014to reach the accuracy of state-of-the-art dense models. This may help alleviate the computational burden of 1University of Wisconsin-Madison. Correspondence to: Kartik Sreenivasan . prior approaches, as training is predominantly carried on a much sparser model. The conjectured existence of these lucky tickets is referred to as the Lottery Ticket Hypothesis (LTH). Frankle and Carbin (2018) and (Frankle et al., 2020) show that not only do lottery tickets exist, but also the cost of \u201cwinning the lottery\u201d is not very high. Along the LTH literature, a curious phenomenon was observed; even at initialization and in the complete absence of training, one can \ufb01nd sub-networks of the random initial model that have prediction accuracy far beyond random guessing (Zhou et al., 2019; Ramanujan et al., 2020; Wang et al., 2019). Ramanujan et al. (2020) reported this in its most striking form: state-of-the-art accuracy models for CIFAR10 and ImageNet, simply reside within slightly larger, yet completely random networks, and appropriate pruning\u2014and mere pruning\u2014can reveal them! This \u201cpruning is all you need\u201d phenomenon is sometimes referred to as the Strong Lottery Ticket Hypothesis. A recent line of work attempts to establish the theoretical validity of the Strong LTH by studying the following nonalgorithmic question: Can a random network be pruned to approximate a target function f(x)? Here, f represents a bounded range labeling function that acts on inputs x \u2208X, and is itself a neural network of \ufb01nite width and depth. This assumption is not limiting, as neural networks are universal approximators (Stinchombe, 1989; Barron, 1993; Scarselli and Tsoi, 1998; Klusowski and Barron, 2018; Perekrestenko et al., 2018; Hanin, 2019; Kidger and Lyons, 2020). Note that the answer to the above question is trivial if one does not constraint the size of the random initial network, for all interesting cases of \u201crandom\u201d. Indeed, if we start with an exponentially wider random neural network compared to the one representing f, by sheer luck, one can always \ufb01nd weights, for each layer, near-identical to those of any target neural network that is f. Achieving this result with a constrained overparameterization, i.e., the degree by which the random network to be pruned is wider/deeper than f, is precisely why this question is challenging. Malach et al. (2020) were the \ufb01rst to prove that the Strong LTH is true, assuming polynomial-sized overparameteriarXiv:2110.08996v2 [cs.LG] 22 Oct 2021 \fFinding Everything within Random Binary Networks zation. Speci\ufb01cally, under some mild assumptions, they showed that to approximate a target network of width d and depth l to within error \u03b5, it suf\ufb01ces to prune a random network of width e O(d2l2/\u03b52) and depth 2l. Pensia et al. (2020) offered an exponentially tighter bound using a connection to the SubsetSum problem. They showed that to approximate a target network within error \u03b5, it is suf\ufb01cient to prune a randomly initialized network of width O(d log(dl/\u03b5)) and depth 2l. A corresponding lower bound for constant depth networks was also established. Orseau et al. (2020) were also able to reduce the dependence on \u03b5 to logarithmic. They show that in order to approximate a target network within error \u03b5, it suf\ufb01ces to prune a random network of width O(d2 log(dl/\u03b5)) if the weights are initialized with the hyperbolic distribution. However, this bound on overparamaterization is still polynomial in the width d. The above theoretical studies have focused exclusively on continuous distribution for initialization. However, in the experimental work by (Ramanujan et al., 2020), the authors manage to obtain the best performance by pruning networks of scaled, binary weights. Training binary networks has been studied extensively in the past (Courbariaux et al., 2015; Simons and Lee, 2019) as they are compute, memory and hardware ef\ufb01cient, though in many cases they suffer from signi\ufb01cant loss of accuracy. The \ufb01ndings of (Ramanujan et al., 2020) suggest that the accuracy loss may not be fundamental to networks of binary weights, when such networks are learned by pruning. Arguably, since \u201ccarving out\u201d sub-networks of random models is expressive enough to approximate a target function, e.g., according to (Pensia et al., 2020; Malach et al., 2020), one is posed to wonder about the importance of weights altogether. So perhaps, binary weights is all you need. Diffenderfer and Kailkhura (2021) showed that indeed scaled binary networks can be pruned to approximate any target function. The required overparameterization is similar to that of Malach et al. (2020), i.e., polynomial in the width, depth and error of the approximation. Hence in a similar vein to the improvement that Pensia et al. (2020) offered over the bounds of Malach et al. (2020), we explore whether such an improvement is possible on the results of Diffenderfer and Kailkhura (2021). Our Contributions: In this work, we offer an exponential improvement to the theoretical bounds by (Diffenderfer and Kailkhura, 2021), establishing the following. Theorem 1. (informal) Consider a randomly initialized, FC, binary {\u00b11} network of ReLU activations, with depth \u0398 \u0000l log ( dl \u03b5 ) \u0001 and width \u0398 \u0000d log2 ( dl \u03b5\u03b4) \u0001 , with the last layer consisting of scaled binary weights {\u00b1C}. Then, there is always a constant C such that this network can be pruned to approximate any FC ReLU network, up to error \u03b5 > 0 with depth l and width d, with probability at least 1 \u2212\u03b4. Target network Over-parameterized random binary weights High accuracy subnetwork +1 \u22121 \u22121 +1 Overparameterize Prune Figure 1: Approximating a target network with high accuracy by pruning overparameterized random binary network. In this paper, we show that logarithmic overparameterization in both width and depth is suf\ufb01cient. Therefore, we show that in order to approximate any target network, it suf\ufb01ces to just prune a logarithmically overparameterized binary network (Figure 1). In contrast to Diffenderfer and Kailkhura (2021), our construction only requires that the last layer be scaled while the rest of the network is purely binary {\u00b11}. We show a comparison of the known Strong LTH results in Table 1. In light of our theoretical results, one may wonder why in the literature of training, i.e., assigning a sign pattern to \ufb01xed architecture, binary networks, a loss of accuracy is observed, e.g., (Rastegari et al., 2016). Is this an algorithmic artifact, or does pruning random signs offer higher expressivity than assigning the signs? We show that there exist target functions that can be well approximated by pruning binary networks, yet none of all possible, binary, fully connected networks can approximate it. Proposition 1. (informal) There exist a function f that can be represented by pruning a random 2-layer binary network of width d, but not by any 2-layer fully-connected binary network of width d. Note that although \ufb01nding a subnetwork of a random binary network results in a \u201cternary\u201d architecture (e.g., 0 becomes a possible weight), the total number of possible choices of subnetworks is 2N, if N is the total number of weights. This is equal to the total number of sign assignments of the same FC network. Yet, as shown in the proposition above, pruning a random FC network is provably more expressive than \ufb01nding a sign assignment for the same architecture. 2. Preliminaries and Problem Setup Let f(x) : Rd0 \u2192R be the target FC network with l layers and ReLU activations, represented as f(x) = \u03c3(Wl\u03c3(Wl\u22121 . . . \u03c3(W1x))), where x \u2208Rd0 is the input, \u03c3(z) = max{z, 0} is the ReLU activation and Wi \u2208Rdi\u00d7di\u22121 is the weight matrix of layer i \u2208[l]. With slight abuse of terminology, we will refer to f as a network, as opposed to a labeling function. We then consider a binary1 network of depth l\u2032 g(x) = \u03c3((\u03b5\u2032Bl\u2032)\u03c3(Bl\u2032\u22121 . . . \u03c3(B1x))), \fFinding Everything within Random Binary Networks Reference Width Depth Total Params Weights Malach et al. (2020) e O(d2l2/\u03b52) 2l e O(d2l3/\u03b52) Real Orseau et al. (2020) O(d2 log(dl/\u03b5) 2l O(d2l log(dl/\u03b5) Real (Hyperbolic) Pensia et al. (2020) O(d log(dl/ min{\u03b5, \u03b4}) 2l O(dl log(dl/ min{\u03b5, \u03b4}) Real Diffenderfer and Kailkhura (2021) O((ld3/2/\u03b5) + ld log(ld/\u03b4)) 2l O((l2d3/2/\u03b5) + l2d log(ld/\u03b4)) {\u00b1\u03b5} Ours, Theorem 1 O(d log2 (dl/\u03b5\u03b4)) O(l log (dl/\u03b5)) O(dl log3 (dl/\u03b5\u03b4)) Binary-{\u00b11}1 Table 1: Comparing the upper bounds for the overparameterization needed to approximate a target network (of width d and depth l) within error \u03b5 > 0 with probability at least 1 \u2212\u03b4 by pruning a randomly initialized network. where Bi \u2208{\u22121, +1}d\u2032 i\u00d7d\u2032 i\u22121 is a binary weight matrix, with all weights drawn uniformly at random from {\u00b11}, for all layers i \u2208[l\u2032] and the last layer is multiplied by a factor of \u03b5\u2032 > 0. The scaling factor is calculated precisely in Section 3.2.3, where we show that it is unavoidable for function approximation (i.e., regression), rather than classi\ufb01cation. Our goal is to \ufb01nd the smallest network g so that it contains a subnetwork \u02dc g which approximates f well. More precisely, we will bound the overparameterization of the binary network, under which one can \ufb01nd supermask matrices Mi \u2208{0, 1}d\u2032 i\u00d7d\u2032 i\u22121, for each layer i \u2208[l\u2032], such that the pruned network \u02dc g(x) =\u03c3(\u03b5\u2032(Ml\u2032 \u2299Bl\u2032)\u03c3((Ml\u2032\u22121 \u2299Bl\u2032\u22121) . . . . . . \u03c3((M1 \u2299B1)x))) is \u03b5-close to f in the sense of uniform approximation over the unit-ball, i.e., max x\u2208Rd0:||x||\u22641 ||f(x) \u2212\u02dc g(x)|| \u2264\u03b5 for some desired \u03b5 > 0. In this paper, we show g only needs to be polylogarithmically larger than the target network f to have this property. We formalize this and provide a proof in the following sections. Henceforth, we denote [k] = {1, 2, \u00b7 \u00b7 \u00b7 , k} for some positive integer k. Unless otherwise speci\ufb01ed, || \u00b7 || refers to the \u21132 norm. We also use the max norm of a matrix, de\ufb01ned as ||A||max := maxij |Aij|. The element-wise product between two matrices A and B is denoted by A \u2299B. We assume without loss of generality that the weights are speci\ufb01ed in the base-10 system. However, since we don\u2019t specify the base of the logarithm explicitly in our computations, we use the \u0398(\u00b7) notation to hide constant factors that may arise from choosing different bases. 3. Strong Lottery Tickets by Binary Expansion In this section, we formally present our approximation results. We show that in order to approximate any target 1The weights of all the layers are purely binary {\u00b11} except for the last layer which is scaled so that it is {\u00b1\u03b5\u2032} where \u03b5\u2032 = (\u03b5/d2l)l. network f(x) within arbitrary approximation error \u03b5, it suf\ufb01ces to prune a random binary1 network g(x) that is just polylogarithmically deeper and wider than the target network. 3.1. Main Result First, we point out that the scaling factor \u03b5\u2032 in the \ufb01nal layer of g(x) is necessary for achieving arbitrary small approximation error for any target network f(x). In other words, it is impossible to approximate any arbitrary target network with a purely binary {\u00b11} network regardless of the overparameterization. To see this, note that for the simple target function f(x) = \u03b5x, x \u2208[0, 1] and \u03b5 \u2208 [0.5, 1), the best approximation possible by a binary network is g(x) = x and therefore maxx\u2208R:|x|\u22641 |f(x) \u2212g(x)| \u2265 (1 \u2212\u03b5) for any binary network g. We will show that just by allowing the weights of the \ufb01nal layer to be scaled, we can provide a uniform approximation guarantee while the rest of the network remains binary {\u00b11}. Formally, we have the following theorem: Theorem 1. Consider the set of FC ReLU networks F de\ufb01ned as F = {f : f(x) = \u03c3(Wl\u03c3(Wl\u22121 . . . \u03c3(W1x))), \u2200i Wi \u2208Rdi\u00d7di\u22121 ||Wi|| \u22641}, and let d = maxi di. For arbitrary target approximation error \u03b5, let g(x) = \u03c3(\u03b5\u2032Bl\u2032\u03c3(Bl\u2032\u22121 . . . \u03c3(B1x))) (here \u03b5\u2032 = (\u03b5/d2l)l) be a randomly initialized network with depth l\u2032 = \u0398(l log(d2l/\u03b5)) such that every weight is drawn uniformly from {\u22121, +1} and the layer widths are \u0398 \u0010 log d2l/\u03b5 \u00b7 log \u0010 dl log2 (d2l/\u03b5) \u03b4 \u0011\u0011 times wider than f(x). Then, with probability at least 1 \u2212\u03b4, for every f \u2208F, there exist pruning matrices Mi such that max x\u2208Rd0:||x||\u22641 |f(x) \u2212\u02dc g(x)| \u2264\u03b5 holds where \u02dc g(x) :=\u03c3(\u03b5\u2032(Ml\u2032 \u2299Bl\u2032)\u03c3((Ml\u2032\u22121 \u2299Bl\u2032\u22121) . . . . . . \u03c3((M1 \u2299B1)x))). Remark 1. The dimensions of the weight matrices of g(x) in Theorem 1 are speci\ufb01ed more precisely below. Let p = \fFinding Everything within Random Binary Networks (d2l/\u03b5). Since l\u2032 = l log(p), we have \u230alog(p)\u230blayers in g(x) that approximates each layer in f(x). For each i \u2208[l], the dimension of B(i\u22121)\u230alog(p)\u230b+1 is \u0398 \u0012 di\u22121 log(p) log \u0012dl log2 (p) \u03b4 \u0013\u0013 \u00d7 di\u22121, the dimension of Bi\u230alog(p)\u230bis di \u00d7 \u0398 \u0012 di\u22121 log (p) log \u0012dl log2 (p) \u03b4 \u0013\u0013 and the remaining B(i\u22121)\u230alog(p)\u230b+k where 1 < k < \u230alog(p)\u230bhave the dimension \u0398 \u0012 di\u22121 log (p) log \u0012dl log2 (p) \u03b4 \u0013\u0013 \u00d7\u0398 \u0012 di\u22121 log (p) log \u0012dl log2 p \u03b4 \u0013\u0013 . 3.2. Proof of Theorem 1 First, we show in Section 3.2.1 that any target network in f(x) \u2208F can be approximated within \u03b5 > 0, by another network \u02c6 gp(x) having weights of \ufb01nite-precision at most p digits where p is logarithmic in d, l, and \u03b5. Then, in Section 3.2.2, we show that any \ufb01nite precision network can be represented exactly using a binary network where all the weights are binary (\u00b11) except the last layer, and the last layer weights are scaled-binary (\u00b1\u03b5\u2032). The proof sketch is as follows. First, through a simple scaling argument we show that any \ufb01nite-precision network is equivalent to a network with integer weights in every layer except the last. We then present Theorem 2 which shows the deterministic construction of a binary network using diamond-shaped gadgets that can be pruned to approximate any integer network. Lemma 7 extends the result to the case when the network is initialized with random binary weights. Putting these together completes the proof of Theorem 1 as shown in Section 3.2.3. 3.2.1. LOGARITHMIC PRECISION IS SUFFICIENT First, we consider the simplest setting wherein the target network contains a single weight i.e., h(x) = \u03c3(wx), where x, w are scalars, the absolute values of which are bounded by 1. This assumption can be relaxed to any \ufb01nite norm bound. We begin with noting a fact that log(1/\u03b5) digits of precision are suf\ufb01cient to approximate a real number within error \u03b5, as formalized below Fact 1. Let w \u2208R, |w| \u22641 and \u02c6 w be a \ufb01nite-precision truncation of w with \u2308\u0398(log(1/\u03b5))\u2309digits. Then |w \u2212\u02c6 w| \u2264 \u03b5 holds. Now we state the result for the case when the target network contains a single weight w. Lemma 1. Consider a network h(x) = \u03c3(wx) where w \u2208 R, |w| \u22641. For a given \u03b5 > 0, let \u02c6 w be a \ufb01nite-precision truncation of w up to log(1/\u03b5) digits and let \u02c6 glog(1/\u03b5)(x) = \u03c3( \u02c6 wx). Then we have max x\u2208R:|x|\u22641 |h(x) \u2212\u02c6 glog(1/\u03b5)(x)| \u2264\u03b5. Proof. By Fact 1, we know that |w \u2212\u02c6 w| \u2264\u03b5. Applying Cauchy-Schwarz with |x| \u22641 gives us | \u02c6 wx \u2212wx| \u2264\u03b5. Since this holds for any x and ReLU is 1-Lipschitz, the result follows. Lemma 1 can be extended to show that it suf\ufb01ces to consider \ufb01nite-precision truncation up to log(d2l/\u03b5) digits to approximate a network for width d and depth l. This is stated more formally below. Lemma 2. Consider a network h(x) = \u03c3(Wl\u03c3(Wl\u22121 . . . \u03c3(W1x))) where Wi \u2208 Rdi\u00d7di\u22121, ||Wi|| \u2264 1. For a given \u03b5 > 0, de\ufb01ne \u02c6 glog(d2l/\u03b5)(x) = \u03c3(c Wl\u03c3(c Wl\u22121 . . . \u03c3(c W1x))) where c Wi is a \ufb01nite precision truncation of Wi up to log(d2l/\u03b5) digits, where d = maxi di. Then we have max x\u2208Rd0:||x||\u22641 |h(x) \u2212\u02c6 glog(d2l/\u03b5)(x)| \u2264\u03b5. We provide the proof of Lemma 2 as well as approximation results for a single neuron and layer in supplementary materials. 3.2.2. BINARY WEIGHTS ARE SUFFICIENT We begin by showing that any \ufb01nite-precision FC ReLU network can be represented perfectly as a FC ReLU network with integer weights in every layer except the last, using a simple scaling argument. Since ReLU networks are positive homogenous, we have that \u03c3(c \u00b7 z) = c \u00b7 \u03c3(z) for c > 0. Given a network gp where all the weights are of \ufb01niteprecision at most p, we can apply this property layerwise with the scaling factor c = 10p so that, f(x) = \u03c3(Wl\u03c3(Wl\u22121 . . . \u03c3(W1x))) = 1 cl \u03c3(cWl\u03c3(cWl\u22121 . . . \u03c3(cW1x))) = \u03c3(c\u2032 c Wl\u03c3(c Wl\u22121 . . . \u03c3(c W1x))) (1) where c Wi = 10pWi is a matrix of integer weights and c\u2032 = 1 cl . Therefore, the rescaled network has integer weights in every layer except the last layer which has the weight matrix c\u2032 c Wl = (c\u2212l)Wl. \fFinding Everything within Random Binary Networks In the remaining part of this section, we show that any FC ReLU network with integer weights can be represented exactly by pruning a purely binary (\u00b11) FC ReLU network which is just polylogarithmic wider and deeper. More precisely, we prove the following result. Theorem 2. Consider the set of FC ReLU networks with integer weights FW de\ufb01ned as FW = {f : f(x) = \u03c3(Wl\u03c3(Wl\u22121 . . . \u03c3(W1x))), \u2200i Wi \u2208Zdi\u00d7di\u22121 ||Wi||max \u2264W} where W > 0. De\ufb01ne d = maxi di and let g(x) = \u03c3(Bl\u2032\u03c3(Bl\u2032\u22121 . . . \u03c3(B1x))) be a network with depth l\u2032 = \u0398(l log(|W|)) where every weight is uniformrandomly generated from {\u22121, +1} and the layer widths are \u0398 \u0010 log |W| \u00b7 log \u0010 dl log2 |W | \u03b4 \u0011\u0011 times wider than f(x). Then, with probability at least 1 \u2212\u03b4, for every f \u2208F, there exist pruning matrices Mi such that f(x) = \u02dc g(x) holds for any x \u2208 Rd0 where \u02dc g(x) := \u03c3((Ml\u2032 \u2299 Bl\u2032)\u03c3((Ml\u2032\u22121 \u2299Bl\u2032\u22121) . . . \u03c3((M1 \u2299B1)x))). Remark 2. The dimensions of the weight matrices of g(x) in Theorem 2 are speci\ufb01ed more precisely below. Note that we have \u230alog |W|\u230blayers in g(x) that exactly represents each layer in f(x). For each i \u2208[l], the dimension of B(i\u22121)\u230alog |W |\u230b+1 is \u0398 \u0012 di\u22121 log |W| log \u0012dl log2 |W| \u03b4 \u0013\u0013 \u00d7 di\u22121, the dimension of Bi\u230alog |W |\u230bis di \u00d7 \u0398 \u0012 di\u22121 log |W| log \u0012dl log2 |W| \u03b4 \u0013\u0013 and the remaining B(i\u22121)\u230alog |W |\u230b+k where 1 < k < \u230alog |W|\u230bhave the dimension \u0398 \u0012 di\u22121 log |W| log \u0012dl log2 |W| \u03b4 \u0013\u0013 \u00d7\u0398 \u0012 di\u22121 log |W| log \u0012dl log2 |W| \u03b4 \u0013\u0013 Remark 3. Note that \u02dc g(x) is exactly equal to f(x). Furthermore, we provide a uniform guarantee for all networks in F by pruning a single over-parameterized network, like Pensia et al. (2020). Remark 4. Theorem 2 can be made into a deterministic construction for any \ufb01xed target network thereby avoiding the log(1/\u03b4) overparameterization. We extend to the random initialization setting by resampling the construction a suf\ufb01cient number of times. Remark 5. To resolve issues of numerical over\ufb02ow, we can insert scaling neurons after every layer. Remark 6. The integer assumption can easily be converted to a \ufb01nite-precision assumption using a simple scaling argument. Since all networks in practice use \ufb01nite-precision arithmetic, Theorem 2 may be of independent interest to the reader. However, we emphasize here that there is no approximation error in this setting. Practitioners who are interested in small error(10\u2212k) can just apply Theorem 1 and incur an overparameterization factor of O(k). The proof of Theorem 2 will \ufb01rst involve a deterministic construction for a binary network that gives us the desired guarantee. We then extend to the random binary initialization. The construction is based on a diamond-shaped gadget that allows us to approximate a single integer weight by pruning a binary ReLU network with just logarithmic overparameterization. First, consider a target network that contains just a single integer weight i.e., h(x) = \u03c3(wx). We will show that there exists a binary FC ReLU network g(x) which can be pruned to approximate h(x). Lemma 3. Consider a network h(x) = \u03c3(wx) where w \u2208 Z. Then there exists a FC ReLU binary network g(x) of width and depth O(log |w|) that can be pruned to \u02dc g(x) so that \u02dc g(x) = h(x) for all x \u2208R. Proof. Note that since w is an integer, it can be represented using its binary (base-2) expansion w = sign(w) \u230alog2 |w|\u230b X k=0 zk \u00b7 2k, zk \u2208{0, 1}. (2) For ease of notation, we will use log(\u00b7) to represent log2(\u00b7) going forward. Denote n = \u230alog2 |w|\u230b. The construction of gn(x) in Figure 2a shows that 2n can be represented by a binary network with ReLU activations for any n. We will refer to this network as the diamond-shaped \u201cgadget\u201d. Note that the expansion in Equation (2) requires 2k for all 0 \u2264k \u2264n = \u230alog |w|\u230b. Luckily, any of these can be represented by just pruning gn(x) as shown in Figure 2b. However, since our constructions use the ReLU activation, this only works when x \u22650. Using the same trick as (Pensia et al., 2020), we can extend this construction by mirroring it as shown in Figure 3. This gives us f + n\u2212k(x) := 2n\u2212kx and f \u2212 n\u2212k(x) := \u22122n\u2212kx. The correctness of this mirroring trick relies on the simple observation that wx = \u03c3(wx) \u2212\u03c3(\u2212wx). Putting these together, we get gn(x) = \u03c3(\u00b1 Pn k=0 2kx) as shown in Figure 4. By pruning just the weights in the last layer, we can choose which terms to include. Setting n = \u230alog |w|\u230bcompletes the approximation. \fFinding Everything within Random Binary Networks +1 +1 +1 +1 \u22ef +1 +1 +1 +1 times n x (a) gn(x) = 2n max{0, x} times k +1 +1 +1 +1 \u22ef +1 +1 +1 +1 times (n \u2212k) +1 +1 0 0 +1 +1 0 0 \u22ef x (b) gn\u2212k(x) = 2n\u2212k max{0, x} Figure 2: The diamond-shaped binary ReLU networks that compute gn(x) and gn\u2212k(x), respectively. The dashed edges are just weights that have been \u201cpruned\u201d (set to 0). The output neuron is a simple linear activation. To calculate the overparameterization required to approximate h(x) = \u03c3(wx), we simply count the parameters in the above construction. Each gadget gk is a network of width 2 and depth (\u230alog |w|\u230b). To construct f + k , we need two such gadgets. Therefore to construct f + k and f \u2212 k , we need width 4 and depth (\u230alog |w|\u230b). Repeating this for each k \u22081, 2, . . . , \u230alog |w|\u230bshows that our construction is a network of width and depth O(log |w|) which completes the proof of Lemma 3. Remark 7. The network in Fig. 4 used for proving Lemma 3 can be written as g(x) =\u03c3(Mv \u2299v)T [(Mn \u2299Bn)\u03c3((Mn\u22121 \u2299Bn\u22121) . . . . . . \u03c3((M1 \u2299B1)\u03c3((Mu \u2299u)x)))]), where {Mi}i\u2208[n], Mv, Mu are mask matrices and {Bi}i\u2208[n], v, u are binary weight matrices. By pruning elements in u or v, one can obtain h(x) = \u03c3(wx). We will always prune the last layer v as it makes the construction more ef\ufb01cient when we extend it to approximating a layer. Now, we extend the construction to the case where the target function is a neural network with a single neuron i.e., h(x) = \u03c3(wT x). Lemma 4. Consider a network h(x) = \u03c3(wT x) where w \u2208Zd and ||w||\u221e\u2264wmax. Then there exists a FC ReLU binary network g(x) of width O(d log |wmax|) and depth O(log |wmax|) that can be pruned to \u02dc g(x) so that \u02dc g(x) = h(x) for all x \u2208Rd. Proof. A neuron can be written as h(x) = \u03c3(wT x) = \u03c3(Pd i=1 wixi). Therefore, we can just repeat our construction from above for each wi, i \u2208[d]. This results in a network of width O(d log |wmax|) while the depth remains unchanged at O(log |wmax|). Next, we describe how to approximate a single layer target network and avoid a quadratic overparameterization. \u22ef \u22ef gn\u2212k \u22121 gn\u2212k +1 +1 \u22121 \u22ef \u22ef x (a) f + n\u2212k(x) = 2n\u2212kx \u22ef \u22ef gn\u2212k \u22121 gn\u2212k +1 +1 \u22121 \u22ef \u22ef x (b) f + n\u2212k(x) = \u22122n\u2212kx Figure 3: We can use two instances of gn\u2212k to create (a) f + n\u2212k and (b) f \u2212 n\u2212k to approximate both positive weights and negative weights. The output neuron here has a linear activation. Lemma 5. Consider a network h(x) = \u03c3(1T \u03c3(W1x)) where W1 \u2208Zd1\u00d7d0 and ||W1||max \u2264W. Then there exists a FC ReLU binary network g(x) of width O(max{d0, d1} log |W|) and depth O(log |W|) that can be pruned to \u02dc g(x) so that \u02dc g(x) = h(x) for all x \u2208Rd0. Proof. Note that 1 \u2208Rd1 is the vector of 1\u2019s. Naively extending the construction from Lemma 4 would require us to replace each of the d0 neurons in the \ufb01rst layer by a network of width O(d1 log |W|) and depth O(log |W|). This already needs a network of width O(d0d1 log |W|) which is a quadratic overparameterization. Instead, we take advantage of pruning only the last layer in the construction from Lemma 3 to re-use a suf\ufb01cient number of weights and avoid the quadratic overparameterization. An illustration of this idea is shown in Figure 5. The key observation is that by pruning only the last layer of each fn(x) gadget, we leave it available to be re-used to approximate the weights of the remaining (d1 \u22121) neurons. Therefore, the width of the network required to approximate h(x) is just O(max{d0, d1} log |W|) and the depth remains O(log |W|). Now we can tie it all together to show an approximation guarantee for any network in F. Lemma 6. Consider a network f(x) = \u03c3(Wl\u03c3(Wl\u22121 . . . \u03c3(W1x))) where Wi \u2208 Zdi\u00d7di\u22121, ||Wi||max \u2264W. Let d = maxi di. Then, there exists a \fFinding Everything within Random Binary Networks \u22121 +1 \u22ef \u22ef \u22ef \u22ef \u22ef \u22ef \u22ef \u22ef f + n \u22ef (Repeat for ) i \u2208[n \u22121] f \u2212 n f + 0 f \u2212 0 x Figure 4: Illustration of gk(x) = Pn k=0(f + k (x) + f \u2212 k (x)) = \u03c3(\u00b1 Pn k=0 2kx) which can be further pruned to approximate f(x) = \u03c3(wx) for any w : |w| < 2n+1 \u22121 FC ReLU binary network g(x) of width O(d log |W|) and depth O(l log |W|) that can be pruned to \u02dc g(x) so that \u02dc g(x) = f(x) for all x \u2208Rd0. Proof. Note that there is no approximation error in any of the steps above. Therefore, we can just repeat the construction above for each of the l layers in f(x). This results in a network g(x) of width O(d log |W|) and depth O(l log |W|). A more precise description of the dimensions of each layer can be found in the statement of Theorem 1. Finally, below we show that our construction can be extended for networks with randomly initialized weights. The proof can be found in supplementary materials. Lemma 7. Consider a network f(x) = \u03c3(Wl\u03c3(Wl\u22121 . . . \u03c3(W1x))) where Wi \u2208 Zdi\u00d7di\u22121, ||Wi||max \u2264W. De\ufb01ne d = maxi di and let g(x) be a randomly initialized binary network of width \u0398 \u0010 d log \u0010 (dl log2 |W |) \u03b4 \u0011\u0011 and depth \u0398(l log |W|) such that every weight is drawn uniformly from {\u22121, +1}. Then g(x) can be pruned to \u02dc g(x) so that \u02dc g(x) = f(x) for all x \u2208Rd0 with probability at least 1 \u2212\u03b4. Since every network f \u2208FW satis\ufb01es the assumptions of Lemma 7, we can apply it for the entire FW . Note that we do not need to apply the union bound over FW since the randomness is just needed to ensure the existence of a particular deterministic network speci\ufb01cally g(x) from Lemma 6. Therefore, a single random network is suf\ufb01cient to ensure the guarantee for every f \u2208FW . This completes the proof of Theorem 2. \u22ef x \u22ef h(x) = \u03c3(1T\u03c3(W1x)) W1 1 \u22ef x g(x) B1 \u22ef \u22ef \u22ee \u22ef \u22ef \u22ee \u22ee \u22ee \u22ee \u22ee Blog|W| \u22ee 1 Figure 5: Illustration of the construction in Lemma 5: Approximating a 1-hidden layer network h(x) = \u03c3(1T \u03c3(W1x)) by pruning the appropriate binary network g(x). Pruning the last layer allows us to \u201cre-use\u201d weights. 3.2.3. PUTTING EVERYTHING TOGETHER We now put the above results together to complete the proof of Theorem 1. First, note that by Lemma 2, to approximate any f \u2208F within \u03b5 > 0, it suf\ufb01ces to consider \u02c6 g(x) which is a \ufb01nite-precision version of f where the precision of each weight is at most p = log(d2l/\u03b5). Now, applying the scaling trick in Equation (1) we can represent \u02c6 g(x) exactly as a scaled integer network i.e., \u02c6 g(x) = c\u2212l\u03c3(cc Wl\u03c3(cc Wl\u22121 . . . \u03c3(cc W1x))) where c = 10p = (d2l/\u03b5) and all the weight matrices cc Wi are integer. Since \u2225Wi\u2225max \u22641, it is clear that \u2225cc Wi\u2225max \u2264c. Therefore, applying Theorem 2 to the integer network cl\u02c6 g(x) with W = (d2l/\u03b5), we have the following. If h(x) = \u03c3(Bl\u2032\u03c3(Bl\u2032\u22121 . . . \u03c3(B1x))) is a randomly initialized binary network of depth \u0398(l log(d2l/\u03b5)) and width \u0398 \u0010 log(d2l/\u03b5) log \u0010 dl log2(d2l/\u03b5) \u03b4 \u0011\u0011 , then with probability 1 \u2212\u03b4, it can be pruned to \u02dc h(x) so that cl\u02c6 g(x) = \u02dc h(x) for any x in the unit sphere. Therefore, to approximate \u02c6 g(x) we simply push the scaling factor c\u2212l into the last layer Bl\u2032 so that its weights are now scaled binary {\u00b1(\u03b5/d2l)l}. Combining this with the approximation guarantee between \u02c6 g(x) and f(x) completes the proof. 3.3. Binary weights for classi\ufb01cation Theorem 2 can easily be extended for classi\ufb01cation problems using \ufb01nite-precision networks. Since sign(\u00b7) is positive scale-invariant, we no longer even require the \ufb01nal layer to be scaled. Applying the same argument as Sec 3.2.3 and then dropping the c\u2212l factor gives us the following corollary. Corollary 1. Consider the set of binary classi\ufb01cation FC ReLU networks F of width d and depth l, where the weight \fFinding Everything within Random Binary Networks matrices {Wi}l i=1 are of \ufb01nite-precision at most p digits. Let g(x) = sign(Bl\u2032\u03c3(Bl\u2032\u22121 . . . \u03c3(B1x))) be a randomly initialized binary network with depth l\u2032 = \u0398(lp) and width d\u2032 = \u0398(dp log(dlp/\u03b4)) such that every weight is drawn uniformly from {\u22121, +1}. Then, with probability at least 1 \u2212\u03b4, for every f \u2208F, there exist pruning matrices {Mi}l\u2032 i=1 such that f(x) = \u02dc g(x) for any x where \u02dc g(x) := sign((Ml\u2032\u2299Bl\u2032)\u03c3((Ml\u2032\u22121\u2299Bl\u2032\u22121) . . . \u03c3((M1\u2299 B1)x))). 4." + } + ], + "Keon Lee": [ + { + "url": "http://arxiv.org/abs/2207.01063v3", + "title": "DailyTalk: Spoken Dialogue Dataset for Conversational Text-to-Speech", + "abstract": "The majority of current Text-to-Speech (TTS) datasets, which are collections\nof individual utterances, contain few conversational aspects. In this paper, we\nintroduce DailyTalk, a high-quality conversational speech dataset designed for\nconversational TTS. We sampled, modified, and recorded 2,541 dialogues from the\nopen-domain dialogue dataset DailyDialog inheriting its annotated attributes.\nOn top of our dataset, we extend prior work as our baseline, where a\nnon-autoregressive TTS is conditioned on historical information in a dialogue.\nFrom the baseline experiment with both general and our novel metrics, we show\nthat DailyTalk can be used as a general TTS dataset, and more than that, our\nbaseline can represent contextual information from DailyTalk. The DailyTalk\ndataset and baseline code are freely available for academic use with CC-BY-SA\n4.0 license.", + "authors": "Keon Lee, Kyumin Park, Daeyoung Kim", + "published": "2022-07-03", + "updated": "2023-03-13", + "primary_cat": "eess.AS", + "cats": [ + "eess.AS", + "cs.AI", + "cs.CL" + ], + "main_content": "INTRODUCTION Being a part of the conversation process, a TTS system needs to maintain and express the context related to the current dialogue. However, previous TTS models have limitations in context representation since they perceive each utterance separately regardless of the dialogue [1, 2]. Some recent studies proposed context-aware TTS models [3, 4, 5], but they trained the model with an internal dataset, which is not open to the public. Also, some corpora collected from real-world conversations or acts are publicly available as [6], but they have several limitations, including background noise or inconsistent record quality. To resolve this, we introduce DailyTalk, a high-quality dialogue speech dataset for TTS. Deriving dialogues from DailyDialog dataset [7], we construct a new speech dataset by processing and recording selected dialogues. We design DailyTalk to ensure both general and conversational speech synthesis quality: studio-quality audio, simultaneous recording of two participants, and adding \ufb01lling-gap words (uh, umm) in \u2217Equal Contribution \u2020This work was done when Keon Lee was with KAIST as a MS student. 1https://github.com/keonlee9420/DailyTalk the part of the dataset. At the same time, we preserve valuable characteristics of the DailyDialog dataset: academically open license and various dialogue annotations. Together, we show a baseline model for conversational TTS to investigate the effect of contextual information when synthesizing the speech. Based on FastSpeech2 [8] architecture, the baseline consumes dialogue history as an additional input following [3]. Using our baseline, we show training model with our dataset can synthesize natural speech in the scope of both single utterances and whole dialogue. Our major contributions are as below. 1. We provide the \ufb01rst open dataset for conversational TTS with a working baseline. 2. We suggest the entire pipeline to build a conversational TTS system from the text dialogue dataset. 3. We suggest several evaluation criteria to evaluate conversational TTS. In the following sections, we introduce our conversational TTS dataset with recording details, baseline, and experiments. 2. DAILYTALK Construction pipeline of DailyTalk consists of pre-record processing, recording, and post-record processing while taking advantage of details suggested by [3]. To guarantee suf\ufb01cient data size and preserve dialogue characteristics, we assume the conversation is dyadic between one male and one female speaker. Details are introduced in the following sections. 2.1. Pre-record Processing Pre-record processing was the step for selecting and re\ufb01ning dialogues of [7] in two criteria: turn length and participant information. The selected dialogues have more than 5 turns because short dialogues are less likely to have context. Next, we only changed the characters\u2019 names to minimize the gap from the original dataset. In some dialogues with two participants of the same gender, we attempted to change one participant to another gender unless the other annotations (emotion, speech act, arXiv:2207.01063v3 [eess.AS] 13 Mar 2023 \ffeature Male Female Total # clips 11,867 11,906 23,773 # words 124,624 126,721 251,345 total duration (s) 38902 39124 78026 mean duration/clip (s) 3.278 3.286 3.282 mean # phone/clip 29.471 29.884 29.678 # distinct words 7,220 7,329 10,160 # dialogues 2,541 2,541 2,541 mean turns/dialogue 4.670 4.686 9.356 # dialogues w/ fgs 962 689 1,452 Table 1: Detailed stats of DailyTalk. Filling gaps (fgs) indicates uh and umm. topic) were maintained. We excluded the dialogue from our dataset when the gender change harms the other annotations. 2.2. Recording In the recording step, we managed conditions to provide a clear, conversational audio dataset. Two English-\ufb02uent speakers were employed as voice actors, each having lived in the US for at least 3 years. The actors recorded actual conversations rather than just reading the script, and they were directed to follow each utterance\u2019s emotion label. Additionally, we ordered them to add \ufb01lling gaps (i.e. uh, um) in approximately half of the dialogues. This direction enables the TTS model to express \ufb01lling gaps as well. The entire recording is done in a studio. 2.3. Post-record Processing Post-record processing is conducted to correct misalignment between speeches and scripts. Since we directed voice actors not to act exactly to script, recorded speech might differ from the script in aspects of \ufb01lling gaps or some conversational words. For this task, 6 annotators \ufb02uent in English are hired to \ufb01x incorrect transcription for this task. 2.4. Statistics Table 1 shows detailed statistics of DailyTalk, containing 2,541 dialogues which is 20 hours in total. We kept all script-related label distribution (e.g., emotion, dialog act, topic) derived from DailyDialog.2 3. BASELINE As shown in Figure 1, our baseline model for conversational TTS consists of a context encoder added to improved FastSpeech2 [8]. The FastSpeech2 backbone is augmented by replacing Montreal Forced Aligner [9] with a CTC-based aligner 2Detailed statistics are provided in https://github.com/ keonlee9420/DailyTalk Fig. 1: Baseline model architecture. [10] so that duration is learned during training in an end-toend manner. The conversational context encoder is from [3] to condition on historical information. The reason for dropping Tacotron2 [3] as a backbone is dif\ufb01culty aligning when training data becomes expressive [11]. Our implementation is based on two open-sourced projects where the Transformer [12] encoder and decoder are connected by the unsupervised duration modeling 3, and where the conversational context encoder from [3] is applied to FastSpeech2 model 4. 4. EXPERIMENTS We conduct two experiments to verify the ef\ufb01cacy of DailyTalk and provide the baseline. The \ufb01rst experiment compares our dataset with existing datasets, which proves the integrity of ours for neural TTS. Next, we show the performance of the context-aware TTS model trained with DailyTalk to prove the effectiveness in the aspect of context modeling. The models used in experiments are trained with 16 batch sizes using a single NVIDIA RTX 3090. Training hyperparameters such as optimizers are based on each model\u2019s original con\ufb01guration. We stopped the training when the validation loss was no longer dropped. When we trained a model on LJSpeech, we randomly selected 512 sentences as a validation set. For VCTK, we selected two sentences per speaker for validation. In DailyTalk, we set 128 dialogues for the validation set. We randomly sampled 260 utterances (of 15 dialogues) for all the following experiments. We measure all scores with a con\ufb01dence interval of 95%. 3https://github.com/keonlee9420/Comprehensive-Transformer-TTS 4https://github.com/keonlee9420/Expressive-FastSpeech2 \f4.1. Integrity as a TTS dataset Since the primary goal of DailyTalk is training TTS models, TTS models should be successfully trained on our dataset. To show this, we train three neural TTS models: Tacotron2 [13], FastSpeech2 [8] and our baseline. Tacotron2 and FastSpeech2 are selected as representative models of autoregressive TTS and non-autoregressive TTS. Note that our baseline can be trained on DailyTalk only since it requires chat history as an input. We randomly split the training set and test set and used only the training set for model training, where test sets are used for further experiments. To measure the performance of each model trained on each dataset, we conducted a Mean Opinion Score (MOS) test for synthesized and reconstructed speeches. We asked people how natural each synthesized speech is by selecting in a range from 1 (Completely not Human-like) to 5 (Completely Humanlike). Only people from the US participated in the test since English-\ufb02uent people are required for this evaluation. We used data in the test set to synthesize speech when conducting the MOS test. We asked 30 different people to measure the naturalness of each audio, then calculated the average score for each model-dataset pair. Model LJSpeech VCTK DailyTalk GT-Recon 3.90 \u00b1 0.07 3.81 \u00b1 0.07 3.83 \u00b1 0.06 Tacotron2 3.71 \u00b1 0.07 3.72 \u00b1 0.08 3.70 \u00b1 0.07 FastSpeech2 3.85 \u00b1 0.07 3.75 \u00b1 0.07 3.82 \u00b1 0.07 Baseline 3.69 \u00b1 0.08 Table 2: TTS integrity test result. GT-recon refers to the speeches reconstructed from ground truth mel-spectrogram. Baseline is a model suggested in section 3. Table 2 shows the MOS result of each model-dataset pair. Overall, synthesized speeches from models trained on DailyTalk sound as natural as other datasets. In reconstructions, DailyTalk scores between LJSpeech and VCTK, which is coherent considering the number of speakers. Scores of the synthesized speeches from models (Tacotron2, FastSpeech2) show within the con\ufb01dence interval for all three datasets. Although our baseline reports reasonable scores, it is still lower than the others as a result of increased complexity [11]. Note that the lower score does not imply the usefulness of the chat history, which we will explore on in the next section. 4.2. Conversational TTS The major difference between our dataset from the others is the conversational information. Aiming to show how the conversational TTS model differs from general TTS models, we design several evaluation tasks that ask naturalness of dialogue in terms of context preservation. Our assumption throughout the evaluations is that people can be aware of context through Experiment MOS CMOS FS2 Ours Ours-FS2 Dialogue 3.77 \u00b1 0.05 3.79 \u00b1 0.05 -0.29 \u00b1 0.15 Pairwise 3.77 \u00b1 0.04 3.74 \u00b1 0.04 -0.07 \u00b1 0.07 (GT-synth) Pairwise 3.86 \u00b1 0.04 3.90 \u00b1 0.03 -0.07 \u00b1 0.07 (synth pair) User-Sys 3.67 \u00b1 0.07 3.70 \u00b1 0.07 -0.22 \u00b1 0.12 Table 3: Result of experiments in section 4.2. Positive CMOS score means that our baseline scores better than FastSpeech2, and negative score is vice versa. the relation between adjacent utterances. We design the following four novel metrics for our experiments: Varying utterance groups given to evaluators. \u2022 Dialogue-level MOS: A set of audio is an entire dialogue, where the TTS model synthesizes all utterances in the dialogue. \u2022 Pairwise MOS (GT-synthesized): A set of audio is a pair of adjacent utterances, where the prior utterance is ground truth, and the latter is synthesized from the model. \u2022 Pairwise MOS (Synthesized pair): A set of audio is a pair of adjacent utterances, where both utterances are synthesized from the model. \u2022 User-System Simulation MOS: A set of audio is an entire dialogue, where utterances of one speaker are ground truth and of the other speaker are synthesized from the model. In each evaluation, we asked people from the US how natural the spoken dialogue is in terms of context maintenance. We conduct both MOS and Comparative MOS (CMOS) throughout all evaluations. In the MOS test, people evaluate the naturalness in the 1-5 range. In the CMOS test, people listen to two sets of audio synthesized from different models with the same script, then select a preference score from -3 (A is much better) to 3 (B is much better). In all experiments, complete scripts of each dialogue are provided to participants to evoke the context of the given audio set during the test. Table 3 reports the experiment results. Even FastSpeech2 obtains higher naturalness score in section 4.1, our baseline reports higher score than FastSpeech2 in three experiments: dialogue-level, User-System and Pairwise(synth pair) MOS tests. In other words, the audio quality is ranked a bit lower only when the output speech from two models is compared directly. But when it is evaluated under conversational conditions, our baseline beats the competitor for all metrics, showing the effectiveness of chat history, except the GT-synth pair. \fModel Experiment Pairwise(GT-synth) Pairwise(synth pair) MOS FastSpeech2 3.817 \u00b1 0.051 / 3.725 \u00b1 0.049 3.842 \u00b1 0.050 / 3.882 \u00b1 0.047 Ours 3.738 \u00b1 0.051 / 3.737 \u00b1 0.048 3.854 \u00b1 0.048 / 3.937 \u00b1 0.046 CMOS Ours-FastSpeech2 0.003 \u00b1 0.102 / -0.124 \u00b1 0.097 -0.145 \u00b1 0.095 / 0.001 \u00b1 0.089 Table 4: Separated MOS scores of Pariwise(GT-synth), Pairwise(synth pair) experiments. Note that we cannot conduct turn division experiments for dialogue-level and User-System evaluations, because those methods evaluate conversation in dialogue-unit. Since they are not evaluated by each turn, evaluation cannot be divided into turns. Notably, Pairwise (GT-synth) shows a different tendency compared to other metrics, so we discuss it in section 4.3. Pairwise (GT-synth) and pairwise CMOS tests also show decreased gap. Dialogue-level and User-System CMOS results still show a gap that FastSpeech2 synthesizes better than ours. We suspect that listeners are still affected by audio quality even with the guideline. Considering dialogue-level and usersystem CMOS shows a similar gap, the gap tends to increase when people listen to a larger amount of audio. This gap may also come from a failure in a single utterance. Even a little mistake in a single utterance does not affect overall context-awareness much, and it may in\ufb02uence a lot compared to non-failure results. Therefore, we concluded that CMOS results would be said to be affected more by speech quality than MOS results, not context-awareness. Summing up, Dialogue-level criteria can evaluate whether the model synthesizes the entire dialogue correctly. The pairwise criterion would assess the relationship between adjacent utterances, such as context preservation. In pairwise evaluation, the pairwise (GT-synth) metric can be used when synthesized speech needs to resemble ground truth recording. The pairwise (synth pair) test can be used to evaluate whether two synthesized speeches contain a shared context. User-System evaluation can be used at the application level, where the system replies user\u2019s utterance, such as voice secretary. 4.3. Turn Division Usually, contextual information piles up while the dialogue continues. Therefore, it is hard to represent the context in the beginning turns of the dialogue. To show context piling up and the apparent difference in context modeling, we divide the MOS result into half: before/after turn 5. We set the threshold to 5 since the average number of turns per dialogue is 9.3 (see Table 1) and turn 5 is the nearest half-point. Table 4 shows the divided result of MOS test. In both experiments, the MOS result of our baseline remains consistent or increases after turn 5, while the score of FastSpeech2 decreases or increases but is lower than ours. Interestingly, following GT doesn\u2019t guarantee that the model will output more conversationally natural speech. And still, we can assess the performance of the conversational TTS model independently from the similarity to GT. This is proved by several \ufb01ndings from the results as follows. First, the CMOS in GT-synth pairwise reveals that the baseline goes far from the GT after 5 turns, more than FastSpeech2. The CMOS in the synth pair, on the other hand, shows better performance in ours after 5 turns. And this also explains the result of Pairwise(GT-synth) in Table 3. Second, although the performance of ours doesn\u2019t increase after turn 5 in Pairwise (GT-synth) and that of FastSpeech2 increases after turn 5, we would like to highlight that the performance degradation in GT-synth after turn 5 is lower, but the gain in synth pair is huge in ours. 5." + }, + { + "url": "http://arxiv.org/abs/2103.09474v4", + "title": "STYLER: Style Factor Modeling with Rapidity and Robustness via Speech Decomposition for Expressive and Controllable Neural Text to Speech", + "abstract": "Previous works on neural text-to-speech (TTS) have been addressed on limited\nspeed in training and inference time, robustness for difficult synthesis\nconditions, expressiveness, and controllability. Although several approaches\nresolve some limitations, there has been no attempt to solve all weaknesses at\nonce. In this paper, we propose STYLER, an expressive and controllable TTS\nframework with high-speed and robust synthesis. Our novel audio-text aligning\nmethod called Mel Calibrator and excluding autoregressive decoding enable rapid\ntraining and inference and robust synthesis on unseen data. Also, disentangled\nstyle factor modeling under supervision enlarges the controllability in\nsynthesizing process leading to expressive TTS. On top of it, a novel noise\nmodeling pipeline using domain adversarial training and Residual Decoding\nempowers noise-robust style transfer, decomposing the noise without any\nadditional label. Various experiments demonstrate that STYLER is more effective\nin speed and robustness than expressive TTS with autoregressive decoding and\nmore expressive and controllable than reading style non-autoregressive TTS.\nSynthesis samples and experiment results are provided via our demo page, and\ncode is available publicly.", + "authors": "Keon Lee, Kyumin Park, Daeyoung Kim", + "published": "2021-03-17", + "updated": "2021-06-25", + "primary_cat": "eess.AS", + "cats": [ + "eess.AS", + "cs.AI", + "cs.CL", + "cs.SD" + ], + "main_content": "Introduction Despite remarkable improvement in neural text-to-speech (TTS) towards human-level reading performance [1, 2, 3, 4], criticism has been raised for the lack of expressiveness and controllability. Since the style of synthesized speech is determined from the average style of speech in the training dataset [5, 6], the TTS models have been limited to represent expressive voice. In order to resolve such a problem, several approaches [7, 8, 9, 10, 11] has been explored so that they can synthesize and control speech with various style. However, the expressive TTS models have weaknesses in speed and robustness due to autoregressive architecture [12]. Since each frame needs to iterate of all previous time steps to be predicted, decoding consumes signi\ufb01cant training and inference overhead, and collapse in one step may fall into the failure of total synthesis [13, 14]. Also, their unsupervised style modeling reports dif\ufb01cult training and has a fundamental limitation on disentangling features [15]. Meanwhile, speech synthesis frameworks without autoregressive decoding have been proposed recently [12, 16, 17]. Utilizing Transformer [18], some non-autoregressive TTS models use self-attention block for parallel decoding [12, 17]. The substitution of autoregressive decoder into duration prediction 1https://keonlee9420.github.io/STYLER-Demo/ 2https://github.com/keonlee9420/STYLER with supervision shows faster speed and enhances stability. Predicting value only from the single text input, however, these models still have limitations on low expressiveness and weak controllability where the in\ufb02uence of audio decreases. Even with several advances in previous works, none of them covered all criticized problems at once: speed, robustness, expressivity, and controllability. To achieve this, the following conditions should be satis\ufb01ed. First, autoregressive architecture needs to be avoided, which induces speed and robustness weaknesses. Second, style factor modeling from source audio should be introduced to enable more expressive and controllable synthesis, enjoying the opportunity of input variants in addition to a single text. In this paper, we propose STYLER, a fast and robust style modeling TTS framework for expressive and controllable speech synthesis. Upon non-autoregressive decoding, we construct style factor modeling to express and control each style factor. In addition to text input, STYLER takes and splits source audio into \ufb01ve components: duration, pitch, energy, content, and noise. Inspired by speech decomposition via information bottleneck [19, 20] in recent voice conversion task, our model separately encodes each style factor and decodes them into speech. Furthermore, our novel noise modeling decomposes noise from reference speech with no labels while successfully encoding the speech style from noisy input. To the best of our knowledge, STYLER is the \ufb01rst approach of TTS framework equipped with rapidity, robustness, expressivity, and controllability at the same time with high naturalness. The contributions of STYLER are as follows. \u2022 STYLER reveals faster and more robust synthesis in unseen data with a novel audio-text aligning compare to the expressive TTS with the autoregressive decoding. \u2022 STYLER achieves high expressivity and controllability through disentangled style factors with the same level of supervision compare to the non-autoregressive TTS. \u2022 STYLER is also noise-robust, proving that our novel noise modeling pipeline can decompose the noise as other style factors without any additional label. 2. Method In this section, we describe our model \ufb01rst in an abstract level, then in detail. The noise modeling is introduced in the tail. 2.1. Supervised Speech Decomposition There are two primary considerations to achieve our goal, nonautoregressive TTS with style factor modeling. First, unlike the voice conversion task, the output speech\u2019s content is from text, and it may not be matched to the audio content. Second, recent non-autoregressive TTS frameworks bene\ufb01t from supervision, while the speech decomposition is performed in unsupervised arXiv:2103.09474v4 [eess.AS] 25 Jun 2021 \fFigure 1: An architecture of the proposed model. Zt, Zt\u2019, Zd, Zp, Zs, Ze and Zn refer to text, downsampled text, duration, pitch, speaker, energy, and noise encoding, and DAT stands for Domain Adversarial Training. conditions. Solving these issues, we propose Mel Calibrator and a reinterpretation of speech decomposition under supervision. 2.1.1. Mel Calibrator Mismatch of length in text and audio has been solved by attention mechanism [10, 11]. Attention, however, has been claimed to be unstable, especially when synthesizing with unseen input data. In order to mitigate the limitation in robustness, we propose Mel Calibrator, a linear compression or expansion of the audio to the text\u2019s length. Only with the total lengths of both input text and audio, Mel Calibrator averages frames or repeats a frame of audio to assign them to a single phone by the ratio of the size of phoneme sequence over audio frame length. In this physically scaling method, the aligning process becomes a simple frame-wise bottleneck of incoming audio, requiring neither attention nor forced alignment. There are two advantages over the attention mechanism in terms of our objective: it does not bring any robustness issue against the non-autoregression, and it does not being exposed to the text so that audio-related style factors only come from the audio. 2.1.2. Information Bottleneck under Supervision Encoders of prior unsupervised speech decomposition [19, 20] need a tight bottleneck both channel-wise and frame-wise to deliver the necessary information from the source audio. But it makes training dif\ufb01cult and time-consuming. Under the supervision, encoders can utilize pre-obtained features from incoming audio such as pitch contour and energy. Since forced feature selection is not needed, excessive regulation will lead to information shortage and performance degradation [19]. In our model, the bottleneck is adequately mitigated to \ufb01t the information \ufb02ow relishing the supervision during style factor modeling. 2.2. Model Architecture Figure 1 shows an overview architecture of the proposed model. STYLER models total six style factors: text, duration, pitch, speaker, energy, and noise. Text is regarded as a style factor equal to other audio related factors. To achieve rapidity, robustness, expressivity, and controllability at the same time, the elements of non-autoregressive TTS3 [12, 17] and speech decomposition approaches4 [19, 20] are adequately selected and transformed into STYLER. 2.2.1. Encoders Text encoder has two 256 dimensional Feed-Forward Transformer (FFT) blocks [12] with four heads and takes phoneme sequence as input text. Since the text is encoded solely by the text encoder, the content of audio is removed through the bottleneck in other encoders. Duration, pitch, energy, and noise encoder have three 5x1 convolution layers followed by a group normalization [21] of size 16, and channel-wise bottleneck by two layers bidirectional-LSTM of size 64 except the duration encoder where the size is 80. The dimension of the convolution layer is 256 for duration and noise, 320 for pitch and energy. Mel Calibrator is applied after the convolution stack. Encoder outputs are upsampled channel-wise before being sent to each predictor by linear layer with ReLU activation. Length regulator [12, 17] is applied as a frame-wise upsampling to repeat frames for each phone counted by duration encoding. The duration and noise encoder take mel-scaled spectrogram (mel-spectrogram). The pitch encoder takes a speaker normalized pitch contour (mean = 0.5, std = 0.25) so that speaker-independent pitch contour is modeled. Speaker encoding is transferred from pre-trained speaker embedding, as presented in [22]. In this work, Deep Speaker [23] is employed since it is faithful to embed speaker identities without additional features such as noise robustness. The energy encoder takes scaled energy from 0 to 1 to ease the model training. After extracted from audio as in [17], both pitch and energy input are quantized in 256 bins one-hot vector and then processed by encoders. Note that all inputs go through channel-wise bottleneck only. This is because the audio is already processed by the Mel Calibrator, and an additional frame-wise bottleneck acts as the excessive regulation discussed in Section 2.1.2. 2.2.2. Decoders STYLER has three predictors [17], each for the duration, pitch, and energy. The predictors need to consume all necessary information since they predict real values rather than latent codes. The input contains the sum of the corresponding encoding with text encoding. The text encoding is projected to four-dimensional space to balance the dependency. The pitch predictor receives speaker encoding as an additional input to predict the \ufb01nal pitch contour. We empirically \ufb01nd that adding speaker encoding to both downsampled and upsampled pitch encoding improves the decomposition of speaker identity. Decoder, which predicts the \ufb01nal mel-spectrogram from disentangled style factors, has four 256 dimensional FFT blocks with four multi heads. The decoder\u2019s input is a combination of text encoding, pitch embedding (embedding of pitch predictor output), energy embedding (embedding of energy predictor output), and speaker encoding. Speaker encoding is included be3https://github.com/ming024/FastSpeech2 4https://github.com/auspicious3000/SpeechSplit \fcause information between input and target should be matched for the desired mapping. 2.3. Noise modeling On the way of improving the noise robustness of STYLER, the noise can also be decomposed from audio without additional labels. In our model, noise is one of the remaining factors that have been explicitly encoded in model encoders, and it can be de\ufb01ned based on its residual property, excluding other style factors. In order to model noise by the de\ufb01nition, however, other encoders must be constrained to not include noise information even from the noisy input. 2.3.1. Domain Adversarial Training Not including noise information means extracting noiseindependent features. In the TTS domain, this has been tackled by applying Domain Adversarial Training (DAT) [24, 25]. Similar to previous works, the augmentation label and a gradient reversal layer (GRL) are introduced, which is jointly trainable in our model. The label of each predictor acts as a class label. As shown in Figure 1, DAT is applied to every audio-related encoding except noise encoder. After passing through GRL, each encoding is consumed by the augmentation classi\ufb01er to predict the augmentation posterior (original/augmented). The augmentation classi\ufb01er consists of two fully connected layers of 256 hidden size, followed by layer normalization [26] and ReLU activation. Note that each encoder except the noise encoder is now noise-independent, so each predictor\u2019s output is compared to clean labels rather than noisy ones. 2.3.2. Residual Decoding According to the de\ufb01nition of noise and the fact that an explicit label is not given, a novel pipeline called Residual Decoding is designed. It contains two phases: clean decoding and noisy decoding. In clean decoding, all noise-independent encodings are taken to predict a clean mel-spectrogrom, and in noisy decoding, noise encoder output is added to noise-independent encodings to predict a mel-spectrogrom with noise. Since only the noise encoder has to be updated at noisy decoding, the gradient does not \ufb02ow through the other encoders. Residual Decoding can be seen as implicit supervision where the noise is directly compelled to focus on the leftover part of the audio with no explicit label. 2.4. Loss Calculation The total loss of the model without noise modeling is: Lossclean = lmel\u2212clean + lduration + lpitch + lenergy (1) where lmel\u2212clean is mean square error between predicted melspectrogram and target, and predictor losses (lduration, lpitch, and lenergy) are calculated by mean absolute error. Note that each extraction method of duration, pitch, and energy and the loss is the same as [17], which means that the proposed model can disentangle style factors with no additional label. The loss, including noise modeling, then becomes as follows: Losstotal = Lossclean + lmel\u2212noisy + laug (2) where Lossclean refers to Equation 1, lmel\u2212noisy is from noisy decoding calculated by the same way of lmel\u2212clean in Equation 1, and laug is the sum of each augmentation classi\ufb01er loss in Section 2.3.1. 3. Experiment 3.1. Experiment Setup Throughout the experiment, we train and evaluate models on VCTK corpus [27]. For the noisy dataset, we augment the original dataset by mixing each utterance with a randomly selected piece of background noise from WHAM! dataset [28], at a random signal-to-noise ratio (SNR) in a range from 5 to 25. Raw audio is resampled at 22050Hz sampling rate and preprocessed to 1024 \ufb01lter lengths, 1024 window sizes, and 256 hops lengths. We divide the dataset into train, validation, and test set. In the \ufb01rst two sets, 44 speakers of each gender are randomly selected to balance speaker statistics. The remaining speakers are in the test set. The processed dataset is about 26 hours in total, containing 35805, 89, and 8345 utterances for the train, validation, and test set, respectively. For the baseline, we use the following models to compare with STYLER in four aspects of our goal. \u2022 Mellotron [29]: Tacotron2 GST [2, 7] based autoregressive model, with ability to condition pitch and duration explicitly. We use this baseline to compare speed and robustness of expressive and controllable TTS. \u2022 FastSpeech2 [17]: TTS with non-autoregressive decoding and supervision in pitch, duration, energy control. We select this baseline to compare expressiveness and controllability of reading style non-autoregressive TTS. STYLER is optimized by the same optimizer and scheduler in [17] with a batch size of 16. For the fair comparison, we apply pretrained ResCNN Softmax+Triplet model5 for the speaker embedding identically to all models. We use pretrained UNIVERSAL V1 model6 of HiFi-GAN [4] as a vocoder. 3.2. Evaluation Metrics We evaluate our model in four aspects: rapidity, robustness, expressiveness, and controllability. Training and inference time is strictly measured to prove the rapidity. Mean opinion score (MOS) is conducted to show the naturalness and robustness. We further investigate edge cases of weak robustness. Comparative MOS (CMOS) is conducted on a style transfer task to compare the expressiveness. For both MOS and CMOS, we use Amazon Mechanical Turk [30, 31]. Lastly, the controllability is demonstrated through an ablation study. There are two different speaker settings: seen speakers (S) and unseen speakers (US) in training time. And there are two different synthesis environments: parallel (P) indicates audio content and input text are identical, where nonparallel (NP) denotes the opposite. 3.3. Results 3.3.1. Rapidity and Naturalness Table 1: Time Measurement & MOS Results Model Time (sec) S US Training Inference MOS-P MOS-NP MOS-P MOS-NP FastSpeech2 0.102 0.019 3.753 3.750 Mellotron 0.914 0.215 3.600 3.256 3.572 3.165 STYLER 0.262 0.024 3.594 3.573 3.632 3.650 Table 1 shows experiment results of model speed and audio naturalness. For the model speed, we measure average training 5https://github.com/philipperemy/deep-speaker 6https://github.com/jik876/hi\ufb01-gan \ftime per step during the \ufb01rst 1k steps and average inference time per single text from a set of speakers without vocoding. Even with increased times compared to FastSpeech2 due to the style factor modeling, STYLER shows faster training and inference speed about 3.49\u00d7 and 8.96\u00d7 than Mellotron. Comparing MOS score against FastSpeech2 con\ufb01rms that STYLER can synthesize natural speech even with increased model complexity. STYLER is ranked higher than Mellotron for all conditions except S & P where only a trivial gap exists. While both baselines show performance degradation under US or NP conditions, STYLER shows better naturalness for both cases. In our auxiliary experiment, STYLER begins to tilt on average style at a particular point of uncertainty, bringing more naturalness than expressive synthesis. It leads to higher MOS score in US than S with an increasing MOS score in US. In edge case investigation, Mellotron shows repeated and ignored words due to the broken alignment map, which is not the case in STYLER. The naturalness tendency and edge cases indicate that our model synthesizes more natural speech than Mellotron especially when unseen data is consumed as input, proving unseen data robustness. 3.3.2. Style Transfer Table 2: CMOS Results Model S US CMOS-P CMOS-NP CMOS-P CMOS-NP STYLER 0 0 0 0 FastSpeech2 -0.303 -0.478 -0.388 -0.266 Mellotron -0.334 -0.422 -0.25 -0.359 Table 2 reports CMOS of style transfer performance where STYLER score is \ufb01xed to 0. Comparison with FastSpeech2 and Mellotron denotes that our method represents better expressiveness than both baselines. Also, the style transfer performance decreases in all conditions of Mellotron, showing our model\u2019s robustness on unseen data over the baseline. In another auxiliary experiment, there exists a trade-off between the naturalness and style transfer performance according to the dependency of each model input. Naturalness score, in detail, is proportional to the in\ufb02uence of text rather than audio, while style transfer performance is inversely proportional. FastSpeech2 is an extreme case where the model has text dependency only, and hence the best naturalness yet in average style. The phenomenon is relatively severe in duration so that the duration modeling could be less precise than other style factors. Note that even with such a trade-off, STYLER outperforms Mellotron in all conditions. 3.3.3. Style Factor Modeling Figure 2 shows our ablation study results as mel-spectrogram. Validating the functionality of style factor modeling, we synthesize speech while excluding each encoding one-by-one. Synthesized from noisy reference audio (top-left), the resulting speech contains background noise (top-right) or not (middle-left) depending on whether the noise decoding is activated. Only the noise is synthesized when the noise encoding is solely activated (middle-right). From these results, it is proven that STYLER can produce clean audio from the noisy reference audio, con\ufb01rming our model\u2019s noise-robustness. We also \ufb01nd that noise modeling predicts the different noise levels of input audio. Summing up, these experiments conclude that our novel noise modeling successfully decomposes noise from reference Figure 2: Results of style factor decomposition. The orange line represents pitch contour, and the purple line represents energy. The content is \u201dThe party has never fully recovered.\u201d and spoken by a male speaker. Both input text and audio are not shown in the training time. The \ufb01gure is best shown in color. audio like other style factors. When the noise-independent style factors are ignored, the corresponding encodings become \ufb01xed style (bottom-left). Excluding pitch encoding, for instance, synthesizes speech has \ufb02at pitch contour. Also, given another speaker, the speaker\u2019s identity is changed while the other style factors are unchanged. When the text encoder is deactivated, audio-related style factors are modeled correctly, while synthesized speech is unintelligible (bottom-right). From all ablation study results, we conclude that encoders successfully capture expected style factors. In FastSpeech2, the duration, pitch, and energy are already controllable. However, the values are predicted from the same text and can be controlled only after being synthesized. In contrast, STYLER distributes the dependency equally among text and style factors and controls them at the input levels. Therefore, with disentangled style factors, STYLER can control output speech by input variants in addition to a single text, where all inputs can be of all different sources. 4." + } + ], + "Soheun Yi": [ + { + "url": "http://arxiv.org/abs/2211.15604v1", + "title": "Convergence Analyses of Davis-Yin Splitting via Scaled Relative Graphs II: Convex Optimization Problems", + "abstract": "The prior work of [arXiv:2207.04015, 2022] used scaled relative graphs (SRG)\nto analyze the convergence of Davis-Yin splitting (DYS) iterations on monotone\ninclusion problems. In this work, we use this machinery to analyze DYS\niterations on convex optimization problems and obtain state-of-the-art linear\nconvergence rates.", + "authors": "Soheun Yi, Ernest K. Ryu", + "published": "2022-11-28", + "updated": "2022-11-28", + "primary_cat": "math.OC", + "cats": [ + "math.OC" + ], + "main_content": "Introduction Consider the problem minimize x\u2208H f(x) + g(x) + h(x), (1) where H is a Hilbert space, f, g, and h are convex, closed, and proper functions, and h is di\ufb00erentiable with L-Lipschitz continuous gradients. The Davis\u2013Yin splitting (DYS) [1] solves this problem by performing the \ufb01xedpoint iteration with \ud54b= \ud540\u2212Prox\u03b1g + Prox\u03b1f(2Prox\u03b1g \u2212\ud540\u2212\u03b1\u2207hProx\u03b1g), (2) where \u03b1 > 0, Prox\u03b1f and Prox\u03b1g are the proximal operators with respect to \u03b1f and \u03b1g, and \ud540is the identity mapping. DYS has been used as a building \u2217Corresponding author Email addresses: lsdluis@snu.ac.kr (Soheun Yi), ernestryu@snu.ac.kr (Ernest K. Ryu) 1 arXiv:2211.15604v1 [math.OC] 28 Nov 2022 \fblock for various algorithms for a diverse range of optimization problems [2, 3, 4, 5, 6, 7]. Much prior work has been dedicated to analyzing the convergence rate of DYS iterations [1, 8, 9, 10, 11, 12, 13]. Recently, Lee, Yi, and Ryu [14] leveraged the recently introduced scaled relative graphs (SRG) [15] to obtain tighter analyses. However, the focus of [14] was on DYS applied to general class monotone operators, rather than the narrower class of subdi\ufb00erential operators. In this paper, we use the SRG theory of [14] to analyze the linear convergence rates of DYS applied to convex optimization problems and obtain state-of-the-art rates. 1.1. Prior works For explaining and inducing many convex optimization algorithms, splitting methods for monotone inclusion problems has been a potent tool[16, 17, 18]. Renowned examples of this methodology include forward-backward splitting (FBS) [19, 20], Douglas\u2013Rachford splitting (DRS) [21, 22, 23], and alternating directions method of multipliers (ADMM) [24], which have been widely used in application regimes. While FBS and DRS are concerned with the sum of two monotone operators, Davis and Yin proposed a new splitting method for the sum of three monotone operators [1], thereby uniting the aforementioned methods. This has come up with a variety of applications [2, 3, 4, 5, 6, 7] and many variants, including stochastic DYS [25, 26, 27, 28, 29], inexact DYS [30], adaptive DYS [31], inertial DYS [32], and primal-dual DYS[33]. Compared to the substantial amount of these studies on DYS, there is not much literature on linear convergence analysis of the DYS iteration. One approach is to formulate SDPs that numerically \ufb01nd the tight contraction factors of DYS: Ryu, Taylor, Bergeling, and Giselsson [11] and Wang, Fazlyab, Chen, and Preciado [12] carried out this approach using the performance estimation problem (PEP) and integral quadratic constraint (IQC), respectively. However, this approach does not give an analytical expression of the contraction factors. There is far less literature that gives an analytical expression of the contraction factors. Two of them are respectively done by Davis and Yin [1], and Condat and Richtarik [13] via inequality analysis. On the other hand, Lee, Yi, and Ryu [14] made a di\ufb00erent approach utilizing scaled relative graphs. 2 \fThis novel tool, the scaled relative graphs (SRG) [15], renders a new approach to analyzing the behavior of multi-valued operators (in particular, including nonlinear operators) by mapping them to the extended complex plane. This theory was further studied by Huang, Ryu, and Yin [34], where they identi\ufb01ed the SRG of normal matrices. Furthermore, Pates leveraged the Toeplitz\u2013Hausdor\ufb00theorem to identify SRGs of linear operators [35]. This approach has also been used for analyzing nonlinear operators. To prove the tightness of the averaged coe\ufb03cient of the composition of averaged operators [36] and the DYS operator [1], Huang, Ryu, and Yin performed analyses of SRGs of those operators [37]. Moreover, there is certain literature on applying SRG to control theory. SRGs have been leveraged by Cha\ufb00ey, Forni, and Rodolphe to examine input-output properties of feedback systems [38, 39]. Cha\ufb00ey and Sepulchre have further found its application to characterize behaviors of a given model by leveraging it as an experimental tool [40, 41, 42]. 1.2. Preliminaries Multi-valued operators.. In general, we follow notations regarding multivalued operators as in [16, 18]. Write H for a real Hilbert space with inner product \u27e8\u00b7, \u00b7\u27e9and norm \u2225\u00b7\u2225. To represent that \ud538is a multi-valued operator de\ufb01ned on H, write \ud538: H \u21d2H, and de\ufb01ne its domain as dom \ud538= {x \u2208H | \ud538x \u0338= \u2205}. We say \ud538is single-valued if all outputs of \ud538are singletons or the empty set, and identify \ud538with the function from dom \ud538to H. De\ufb01ne the graph of an operator \ud538as graph(\ud538) = {(x, u) \u2208H \u00d7 H | u \u2208\ud538x}. We do not distinguish \ud538and graph(\ud538) for the sake of notational simplicity. For instance, it is valid to write (x, u) \u2208\ud538to mean u \u2208\ud538x. De\ufb01ne the inverse of \ud538as \ud538\u22121 = {(u, x) | (x, u) \u2208\ud538}, scalar multiplication with an operator as \u03b1\ud538= {(x, \u03b1u) | (x, u) \u2208\ud538}, the identity operator as \ud540= {(x, x) | x \u2208H}, 3 \fand \ud540+ \u03b1\ud538= {(x, x + \u03b1u) | (x, u) \u2208\ud538} for any \u03b1 \u2208R. De\ufb01ne the resolvent of \ud538with stepsize \u03b1 > 0 as \ud541\u03b1\ud538= (\ud540+ \u03b1\ud538)\u22121. Note that \ud541\u03b1\ud538is a single-valued operator if \ud538is monotone, or equivalently if \u27e8x \u2212y, u \u2212v\u27e9\u22650 for all (x, u), (y, v) \u2208\ud538. De\ufb01ne addition and composition of operators \ud538: H \u21d2H and \ud539: H \u21d2H as \ud538+ \ud539= {(x, u + v) | (x, u) \u2208\ud538, (x, v) \u2208\ud539} , \ud538\ud539= {(x, s) | \u2203u such that (x, u) \u2208\ud539, (u, s) \u2208\ud538} . We call A a class of operators if it is a set of operators. For any real scalar \u03b1 \u2208R, de\ufb01ne \u03b1A = {\u03b1\ud538| \ud538\u2208A} and \ud540+ \u03b1A = {\ud540+ \u03b1\ud538| \ud538\u2208A}. De\ufb01ne A\u22121 = {\ud538\u22121 | \ud538\u2208A} and \ud541\u03b1A = (\ud540+ \u03b1A)\u22121 for \u03b1 > 0. Subdi\ufb00erential operators.. Unless otherwise stated, functions de\ufb01ned on H are extended real-valued, which means f : H \u2192R \u222a{\u00b1\u221e} . For a function f, we de\ufb01ne the subdi\ufb00erential operator \u2202f via \u2202f(x) = {g \u2208H | f(y) \u2265f(x) + \u27e8g, y \u2212x\u27e9, \u2200y \u2208H} (we allow \u221e\u2265\u221eand \u2212\u221e\u2265\u2212\u221e). In some cases, the subdi\ufb00erential operator \u2202f is a single-valued operator. Then, we write \u2207f = \u2202f. Proximal operators.. We call f a CCP function if it is a convex, closed, and proper [18, 16]. For a CCP function f : H \u2192R\u222a{\u00b1\u221e} and \u03b1 > 0, we de\ufb01ne the proximal operator with respect to \u03b1f as Prox\u03b1f(x) = argmin y\u2208H \u001a \u03b1f(y) + 1 2\u2225x \u2212y\u22252 \u001b . Then, \ud541\u03b1\u2202f = Prox\u03b1f. 4 \fClass of functions and subdi\ufb00erential operators.. De\ufb01ne f : H \u2192R \u222a{\u00b1\u221e} being \u00b5-strongly convex (for \u00b5 \u2208(0, \u221e)) and L-smooth (for L \u2208(0, \u221e)) as they are de\ufb01ned in [43]. Write F\u00b5,L = {f | f is \u00b5-strongly convex, L-smooth, and CCP.} . for collection of functions that are \u00b5-strongly convex and L-smooth at the same time. For notational simplicity, we extend F\u00b5,L to allow \u00b5 = 0 or L = \u221eby de\ufb01ning F0,L = {f | f is L-smooth and CCP.} , F\u00b5,\u221e= {f | f is \u00b5-strongly convex and CCP.} , F0,\u221e= {f | f is CCP.} . for \u00b5, L \u2208(0, \u221e). Subdi\ufb00erential operators of any functions in F\u00b5,L are denoted \u2202F\u00b5,L = {\u2202f | f \u2208F\u00b5,L} . Complex set notations.. Denote C = C \u222a{\u221e}, and de\ufb01ne 0\u22121 = \u221eand \u221e\u22121 = 0 in C. For A \u2282C and \u03b1 \u2208C, de\ufb01ne \u03b1A = {\u03b1z | z \u2208A} , \u03b1 + A = {\u03b1 + z | z \u2208A} , A\u22121 = \b z\u22121 | z \u2208A \t . For A \u2286C, de\ufb01ne the boundary of A \u2202A = A \\ intA. We clarify that the usage of \u2202operator is di\ufb00erent when it is applied to a function or a complex set; the former is the subdi\ufb00erential operator, and the latter is the boundary operator. For circles and disks on the complex plane, write Circ(z, r) = {w \u2208C | |w \u2212z| = r}, Disk(z, r) = {w \u2208C | |w \u2212z| \u2264r} for z \u2208C and r \u2208(0, \u221e). Note, Circ(z, r) = \u2202Disk(z, r). 5 \fScaled relative graphs [15].. De\ufb01ne the SRG of an operator \ud538: H \u21d2H as G(\ud538) = \u001a\u2225u \u2212v\u2225 \u2225x \u2212y\u2225exp [\u00b1i\u2220(u \u2212v, x \u2212y)] \f \f \f u \u2208\ud538x, v \u2208\ud538y, x \u0338= y \u001b \u0012 \u222a{\u221e} if \ud538is not single-valued \u0013 . where the angle between x \u2208H and y \u2208H is de\ufb01ned as \u2220(x, y) = ( arccos \u0010 \u27e8x,y\u27e9 \u2225x\u2225\u2225y\u2225 \u0011 if x \u0338= 0, y \u0338= 0 0 otherwise. Note, SRG is a subset of C. De\ufb01ne the SRG of a class of operators A as G(A) = [ \ud538\u2208A G(\ud538). Say A is SRG-full if \ud538\u2208A \u21d4 G(\ud538) \u2286G(A). Fact 1 (Theorem 4, 5 [15]). If A is a class of operators, then G(\u03b1A) = \u03b1G(A), G(\ud540+ A) = 1 + G(A), G(A\u22121) = G(A)\u22121. where \u03b1 is a nonzero real number. If A is furthermore SRG-full, then \u03b1A, \ud540+ A, and A\u22121 are SRG-full. Fact 2 (Proposition 2 [15]). Let 0 < \u00b5 < L < \u221e. Then G(\u2202F0,\u221e) = \u222a{\u221e} {z | Re z \u22650} \u222a{\u221e} G(\u2202F\u00b5,\u221e) = \u222a{\u221e} \u00b5 {z | Re z \u2265\u00b5} \u222a{\u221e} G(\u2202F0,L) = L G(\u2202F\u00b5,L) = L \u00b5 6 \fDYS operators.. Let \ud54b\ud538,\ud539,\u2102,\u03b1,\u03bb = \ud540\u2212\u03bb\ud541\u03b1\ud539+ \u03bb\ud541\u03b1\ud538(2\ud541\u03b1\ud539\u2212\ud540\u2212\u03b1\u2102\ud541\u03b1\ud539) be the DYS operator for operators \ud538: H \u21d2H, \ud539: H \u21d2H, and \u2102: H \u21d2H with stepsize \u03b1 \u2208(0, \u221e) and averaging parameter \u03bb \u2208(0, \u221e). In this paper, we usually take \ud538= \u2202f, \ud539= \u2202g, and \u2102= \u2207h for some CCP functions f, g, and h de\ufb01ned on H, to obtain \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb = \ud540\u2212\u03bbProx\u03b1g + \u03bbProx\u03b1f(2Prox\u03b1g \u2212\ud540\u2212\u03b1\u2207hProx\u03b1g) what we call the subdi\ufb00erential DYS operator. Let \ud54bA,B,C,\u03b1,\u03bb = {\ud54b\ud538,\ud539,\u2102,\u03b1,\u03bb | \ud538\u2208A, \ud539\u2208B, \u2102\u2208C} be the class of DYS operators for operator classes A, B, and C with \u03b1, \u03bb \u2208 (0, \u221e). De\ufb01ne \u03b6DYS(zA, zB, zC; \u03b1, \u03bb) = 1 \u2212\u03bbzB + \u03bbzA(2zB \u22121 \u2212\u03b1zCzB) = 1 \u2212\u03bbzA \u2212\u03bbzB + \u03bb(2 \u2212\u03b1zC)zAzB, which exhibits symmetry with respect to zA and zB, and ZDYS A,B,C,\u03b1,\u03bb = {\u03b6DYS(zA, zB, zC; \u03b1, \u03bb) | zA \u2208G (\ud541\u03b1A) , zB \u2208G (\ud541\u03b1B) , zC \u2208G (C)} for operator classes A, B, and C with \u03b1, \u03bb \u2208(0, \u221e). Identifying the tight Lipschitz coe\ufb03cient via SRG.. Say a subset of C is a generalized disk if it is a disk, {z | Re z \u2265a} \u222a{\u221e}, or {z | Re z \u2264a} \u222a{\u221e} for a real number a. The following is the key fact for calculating the Lipschitz coe\ufb03cients of the DYS operators via SRG. Fact 3 (Corollary 1 of [14]). Let \u03b1, \u03bb > 0. Let A and B be SRG-full classes of monotone operators where G (\ud540+ \u03b1A) forms a generalized disk. Let C be an SRG-full class of single-valued operators. Assume G (A), G (B), and G (C) are nonempty. Then, sup \ud54b\u2208\ud54bA,B,C,\u03b1,\u03bb x,y\u2208dom \ud54b,x\u0338=y \u2225\ud54bx \u2212\ud54by\u2225 \u2225x \u2212y\u2225 = sup z\u2208ZDYS A,B,C,\u03b1,\u03bb |z|. 7 \fIndeed, the original version of Fact 3 is consistent with G (\ud540+ \u03b1A) having a more general property, namely an arc property. We can calculate bounds for the right-hand-side of the equality in Fact 3 e\ufb03ciently by using the following fact. Fact 4 (Lemma 1 of [14]). Let f : C3 \u2192C be a polynomial of three complex variables. Let A, B, and C be compact subsets of C. Then, max zA\u2208A,zB\u2208B, zC\u2208C |f(zA, zB, zC)| = max zA\u2208\u2202A,zB\u2208\u2202B, zC\u2208\u2202C |f(zA, zB, zC)|. 2. Lipschitz factors of subdi\ufb00erential DYS We present Lipschitz factors of subdi\ufb00erential DYS operators. To the best of our knowledge, the following results are the best linear convergence rates of the DYS iteration compared to Theorem 9 of [13] and Theorems 3, 4, and 5 of [14]. To clarify, our rates are not slower than the prior rates in all cases and faster in most cases. Theorem 1. Let f \u2208F\u00b5f,Lf, g \u2208F\u00b5g,Lg, and h \u2208F\u00b5h,Lh, where 0 \u2264\u00b5f < Lf \u2264\u221e, 0 \u2264\u00b5g < Lg \u2264\u221e, 0 \u2264\u00b5h < Lh < \u221e. Let \u03bb > 0 be an averaging parameter and \u03b1 > 0 be a step size. Throughout this theorem, de\ufb01ne r/\u221e= 0 for any real number r. Write Cf = 1 2 \u0012 1 1 + \u03b1\u00b5f + 1 1 + \u03b1Lf \u0013 , Cg = 1 2 \u0012 1 1 + \u03b1\u00b5g + 1 1 + \u03b1Lg \u0013 , Rf = 1 2 \u0012 1 1 + \u03b1\u00b5f \u2212 1 1 + \u03b1Lf \u0013 , Rg = 1 2 \u0012 1 1 + \u03b1\u00b5g \u2212 1 1 + \u03b1Lg \u0013 , d = max{|2 \u2212\u03bb \u2212\u03b1\u00b5h|, |2 \u2212\u03bb \u2212\u03b1Lh|}. If \u03bb < 1/Cf, then \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1f-Lipschitz, where \u03c12 f = \u0012 1 \u2212\u03bbC2 f \u2212R2 f Cf \u0013 max \u001a \u0012 1 \u2212 \u03bb 1 + \u03b1\u00b5g \u00132 + \u03bbd2 1/Cf \u2212\u03bb \u0012 1 1 + \u03b1\u00b5g \u00132 , \u0012 1 \u2212 \u03bb 1 + \u03b1Lg \u00132 + \u03bbd2 1/Cf \u2212\u03bb \u0012 1 1 + \u03b1Lg \u00132 \u001b . 8 \fSymmetrically, if \u03bb < 1/Cg, then \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1g-Lipschitz, where \u03c12 g = \u0012 1 \u2212\u03bbC2 g \u2212R2 g Cg \u0013 max \u001a \u0012 1 \u2212 \u03bb 1 + \u03b1\u00b5f \u00132 + \u03bbd2 1/Cg \u2212\u03bb \u0012 1 1 + \u03b1\u00b5f \u00132 , \u0012 1 \u2212 \u03bb 1 + \u03b1Lf \u00132 + \u03bbd2 1/Cg \u2212\u03bb \u0012 1 1 + \u03b1Lf \u00132 \u001b . Theorem 2. Let f, g, h, \u00b5f, Lf, \u00b5g, Lg, \u00b5h, Lh, \u03bb, and \u03b1 be the same as in Theorem 1. Additionally, assume \u03bb < 2 \u2212\u03b1(\u00b5h+Lh) 2 . De\ufb01ne \u221e/\u221e2 = 0. Write \u03bdf = min \u001a 2\u00b5f + \u00b5h (1 + \u03b1\u00b5f)2, 2Lf + \u00b5h (1 + \u03b1Lf)2 \u001b , \u03bdg = min \u001a 2\u00b5g + \u00b5h (1 + \u03b1\u00b5g)2, 2Lg + \u00b5h (1 + \u03b1Lg)2 \u001b , \u03b8 = 2 4 \u2212\u03b1(\u00b5h + Lh). Then, \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1-contractive, where \u03c12 = 1 \u2212\u03bb\u03b8 + \u03bb q (\u03b8 \u2212\u03b1\u03bdf) (\u03b8 \u2212\u03b1\u03bdg). 2.1. Proofs of Theorems 1 and 2 To leverage Fact 3, we need to set adequate SRG-full operator classes A, B, and C to apply it. The most natural choice is to set A = \u2202F\u00b5f,Lf, B = \u2202F\u00b5g,Lg, C = \u2202F\u00b5h,Lh. However, this choice is not appropriate since these are not SRG-full classes. To overcome this issue, we introduce the following operator classes: Df = {\ud538: H \u21d2H | G(\ud538) \u2286G(\u2202F\u00b5f,Lf)}, Dg = {\ud539: H \u21d2H | G(\ud539) \u2286G(\u2202F\u00b5g,Lg)}, Dh = {\u2102: H \u21d2H | G(\u2102) \u2286G(\u2202F\u00b5h,Lh)}. To elaborate, we gather all operators that have its SRG within G(\u2202F\u00b5f,Lf) to form Df, and so on. Then, Df, Dg, and Dh are SRG-full classes by de\ufb01nition. We now consider A = Df, B = Dg, and C = Dh in the following proof. We mention two elementary facts. 9 \fFact 5. For a, b, c, d \u2208[0, \u221e), \u0010\u221a ab + \u221a cd \u00112 \u2264(a + c)(b + d). Proof. This inequality is an instance of Cauchy\u2013Schwartz. Fact 6. Let k, l, and r be positive real numbers, and b, c be a real number. For z \u2208Circ(c, r), k|z \u2212b|2 + l|z|2 is maximized at z = c \u2212r or z = c + r. Proof to Fact 6. Observe that k|z \u2212b|2 + l|z|2 = (k + l) \f \f \f \fz \u2212 kb k + l \f \f \f \f 2 + klb2 k + l. and distance from kb k+l to z \u2208Circ(c, r) is maximized at z = c \u2212r if kb k+l > c and z = c + r otherwise. We now prove Theorem 1, 2. We use the similar proof technique introduced in the previous work [14]. Proof to Theorem 1. We \ufb01rst prove the \ufb01rst statement and show the other by the same reasoning. Invoking Fact 3 and Fact 4, it su\ufb03ces to show that |\u03b6DYS(zf, zg, zh; \u03b1, \u03bb)|2 \u2264\u03c12 f for zf \u2208\u2202G \u0000\ud541\u03b1Df \u0001 = Circ (Cf, Rf) , zg \u2208\u2202G \u0000\ud541\u03b1Dg \u0001 = Circ (Cg, Rg) , zh \u2208\u2202G (Dh) = Circ \u0012Lh + \u00b5h 2 , Lh \u2212\u00b5h 2 \u0013 (refer to Fact 2 and Fact 1 to see why \u2202G \u0000\ud541\u03b1Df \u0001 , \u2202G \u0000\ud541\u03b1Dg \u0001 , G (Dh) are formed correspondingly) when \u03bb < 1/Cf holds. De\ufb01ne r = d 1/Cf \u2212\u03bb. 10 \fThen, |\u03b6DYS(zf, zg, zh; \u03b1, \u03bb)|2 = |1 \u2212\u03bbzf \u2212\u03bbzg + \u03bb(2 \u2212\u03b1zh)zfzg|2 = |(1 \u2212\u03bbzf)(1 \u2212\u03bbzg) + \u03bb(2 \u2212\u03bb \u2212\u03b1zh)zfzg|2 \u2264(|(1 \u2212\u03bbzf)(1 \u2212\u03bbzg)| + |\u03bb(2 \u2212\u03bb \u2212\u03b1zh)zfzg|)2 (i) \u2264(|(1 \u2212\u03bbzf)(1 \u2212\u03bbzg)| + \u03bbd|zfzg|)2 (ii) \u2264 \u0000|1 \u2212\u03bbzf|2 + \u03bbdr\u22121|zf|2\u0001 \u0000|1 \u2212\u03bbzg|2 + \u03bbdr|zg|2\u0001 . (3) We have used |2 \u2212\u03bb \u2212\u03b1zh| \u2264max{|2 \u2212\u03bb \u2212\u03b1\u00b5h|, |2 \u2212\u03bb \u2212\u03b1Lh|} = d in (i) and Fact 5 in (ii). Recall that \u2202G \u0000\ud541\u03b1Df \u0001 = Circ(Cf, Rf). This renders |1 \u2212\u03bbzf|2 + \u03bbdr\u22121|zf|2 = \u03bb Cf |zf \u2212Cf|2 + 1 \u2212\u03bbCf = 1 \u2212\u03bbC2 f \u2212R2 f Cf . (4) For the latter term, zg = 1 1+\u03b1\u00b5g or zg = 1 1+\u03b1Lg give the maximum, invoking Fact 6. Therefore, |1 \u2212\u03bbzg|2 + \u03bbdr|zg|2 \u2264max \u001a \u0012 1 \u2212 \u03bb 1 + \u03b1\u00b5g \u00132 + \u03bbd2 1/Cf \u2212\u03bb \u0012 1 1 + \u03b1\u00b5g \u00132 , \u0012 1 \u2212 \u03bb 1 + \u03b1Lg \u00132 + \u03bbd2 1/Cf \u2212\u03bb \u0012 1 1 + \u03b1Lg \u00132 \u001b . (5) (4) and (5) together give |\u03b6DYS(zf, zg, zh; \u03b1, \u03bb)|2 \u2264\u03c12 f which concludes the proof for the \ufb01rst statement. The second statement can be proven in the same reasoning. Proof to Theorem 2. By the same reasoning in the proof of Theorem 1, it su\ufb03ces to show that |\u03b6DYS(zf, zg, zh; \u03b1, \u03bb)| \u2264\u03c1 for zf \u2208\u2202G \u0000\ud541\u03b1Df \u0001 , zg \u2208\u2202G \u0000\ud541\u03b1Dg \u0001 , zh \u2208\u2202G (Dh) . 11 \fRecall that \u03b8 = 2 4\u2212\u03b1(\u00b5h+Lh). Note that \u2202G (Dh) = Circ \u0012Lh + \u00b5h 2 , Lh \u2212\u00b5h 2 \u0013 thus |2 \u2212\u03b8\u22121 \u2212\u03b1zh| = \u03b1 \f \f \f \fzh \u2212Lh + \u00b5h 2 \f \f \f \f = \u03b1Lh \u2212\u00b5h 2 = 2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121. (6) Now, observe |\u03b6DYS(zf, zg, zh; \u03b1, \u03bb) \u2212(1 \u2212\u03bb\u03b8)|2 = \u03bb2|\u03b8 \u2212zf \u2212zg + (2 \u2212\u03b1zh)zfzg|2 = \u03bb2|\u03b8\u22121(zf \u2212\u03b8)(zg \u2212\u03b8) + (2 \u2212\u03b8\u22121 \u2212\u03b1zh)zfzg|2 \u2264\u03bb2 \u0000\u03b8\u22121|zf \u2212\u03b8||zg \u2212\u03b8| + |2 \u2212\u03b8\u22121 \u2212\u03b1zh||zf||zg| \u00012 (i) = \u03bb2 \u0000\u03b8\u22121|zf \u2212\u03b8||zg \u2212\u03b8| + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zf||zg| \u00012 (ii) \u2264\u03bb2 \u0000\u03b8\u22121|zf \u2212\u03b8|2 + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zf|2\u0001 \u0000\u03b8\u22121|zg \u2212\u03b8|2 + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zg|2\u0001 . (7) Here, (i) follows from (6) and (ii) follows from Fact 5. Invoking Fact 6, \u03b8\u22121|zf \u2212\u03b8|2 + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zf|2 is maximized at either zf = 1 1+\u03b1Lf or zf = 1 1+\u03b1\u00b5f . The \ufb01rst term evaluates to \u03b8\u22121|zf \u2212\u03b8|2 + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zf|2 = \u03b8 \u2212\u03b1 2Lf + \u00b5h (1 + \u03b1Lf)2 when zf = 1 1+\u03b1Lf and similarly to \u03b8\u22121|zf \u2212\u03b8|2 + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zf|2 = \u03b8 \u2212\u03b1 2\u00b5f + \u00b5h (1 + \u03b1\u00b5f)2 when zf = 1 1+\u03b1\u00b5f . Hence, \u03b8\u22121|zf \u2212\u03b8|2 + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zf|2 \u2264\u03b8 \u2212\u03b1 min \u001a 2Lf + \u00b5h (1 + \u03b1Lf)2, 2\u00b5f + \u00b5h (1 + \u03b1\u00b5f)2 \u001b = \u03b8 \u2212\u03b1\u03bdf. (8) 12 \fSimilarly, we have \u03b8\u22121|zg \u2212\u03b8|2 + (2 \u2212\u03b1\u00b5h \u2212\u03b8\u22121)|zg|2 \u2264\u03b8 \u2212\u03b1\u03bdg. (9) Combining (7), (8), and (9), we obtain |\u03b6DYS(zf, zg, zh; \u03b1, \u03bb)| \u22641 \u2212\u03bb\u03b8 + \u03bb q (\u03b8 \u2212\u03b1\u03bdf) (\u03b8 \u2212\u03b1\u03bdg). which is the desired bound. 2.2. Comparison with previous results We now compare our linear convergence rates with prior results and show that our results are better in general. 2.2.1. Comparison with the result of [13]. Condat and Richtarik [13] showed the following rate. Fact 7 (Setting \u03c9 = 0 in Theorem 9 of [13]). Assume \u00b5g > 0 or \u00b5h > 0, and that Lf, Lh \u2208(0, \u221e). Furthermore, suppose that \u03b1 \u2208(0, 2/Lh). Consider problem (1) where f \u2208F0,Lf, g \u2208F\u00b5g,\u221e, and h \u2208F\u00b5h,Lh. Let x\u22c6be a solution for (1). The DYS iteration converges with the geometric rate of, \u03c12 old = max \u001a(1 \u2212\u03b1\u00b5h)2 1 + \u03b1\u00b5g , (1 \u2212\u03b1Lh)2 1 + \u03b1\u00b5g , \u03b1Lf \u03b1Lf + 2 \u001b . Assuming the same setting of Fact 7, we get the following as a direct result of Theorem 1. Corollary 1. Assume the same conditions as in Fact 7. Then, \ud54b\u2202f,\u2202g,\u2207h,\u03b1,1 is \u03c1new-contractive, where d = max{|1 \u2212\u03b1\u00b5h|, |1 \u2212\u03b1Lh|} and \u03c12 new = max \u001a d2 1 + 2\u03b1\u00b5g , 1 (1 + \u03b1Lf)2 \u0012 \u03b12L2 f + d2 1 + 2\u03b1\u00b5g \u0013\u001b . (10) The following proposition claims that the contraction factor introduced in Corollary 1 is strictly better than that of Fact 7 in general cases. 13 \fProposition 1. Let \u00b5g, \u00b5h, Lf, Lh, \u03b1 be as in Fact 7. Let \u03c1old and \u03c1new be as in Fact 7 and Corollary 1, respectively. Then, \u03c1new < \u03c1old if \u00b5g > 0, and \u03c1new = \u03c1old if \u00b5g = 0. Proof to Proposition 1. Denote d = max{|1\u2212\u03b1\u00b5h|, |1\u2212\u03b1Lh|} and recall that \u03c12 old = max \u001a(1 \u2212\u03b1\u00b5h)2 1 + \u03b1\u00b5g , (1 \u2212\u03b1Lh)2 1 + \u03b1\u00b5g , \u03b1Lf \u03b1Lf + 2 \u001b = max \u001a d2 1 + \u03b1\u00b5g , \u03b1Lf \u03b1Lf + 2 \u001b \u03c12 new = max \u001a d2 1 + 2\u03b1\u00b5g , 1 (1 + \u03b1Lf)2 \u0012 \u03b12L2 f + d2 1 + 2\u03b1\u00b5g \u0013\u001b . First, consider the case where d2 1 + 2\u03b1\u00b5g \u2265 \u03b1Lf \u03b1Lf + 2. In this case, d2 1 + 2\u03b1\u00b5g \u2265 1 (1 + \u03b1Lf)2 \u0012 \u03b12L2 f + d2 1 + 2\u03b1\u00b5g \u0013 holds, and together with the assumption that \u00b5g > 0, \u03c12 new = d2 1 + 2\u03b1\u00b5g < d2 1 + \u03b1\u00b5g \u2264\u03c12 old. Otherwise if d2 1 + 2\u03b1\u00b5g < \u03b1Lf \u03b1Lf + 2, observe that \u03c12 new = 1 (1 + \u03b1Lf)2 \u0012 \u03b12L2 f + d2 1 + 2\u03b1\u00b5g \u0013 < 1 (1 + \u03b1Lf)2 \u0012 \u03b12L2 f + \u03b1Lf \u03b1Lf + 2 \u0013 = \u03b1Lf \u03b1Lf + 2 \u2264\u03c12 old. 14 \f2.2.2. Comparison with [14]. Lee, Yi, and Ryu introduced contraction factors in Theorems 3, 4, and 5 of [14]. The following proposition exhibits those contraction factors and claims ours are strictly better in most cases due to stronger assumptions on the operators. Proposition 2. Let \u00b5f, \u00b5g, Lf, Lg, Lh, \u03b1, and \u03bb be as in Theorem 1. Additionally, assume \u03b1Lh < 4 and \u03bb < 2 \u2212\u03b1Lh 2 . Then, 1. Assume f \u2208F\u00b5f,Lf, g \u2208F0,\u221e, and h \u2208F0,Lh. Then, Theorem 3 of [14] implies \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1old-contractive, where \u03c1old = 1 \u2212 2\u03bb 4 \u2212\u03b1Lh + \u03bb v u u t 2 4 \u2212\u03b1Lh 2 4 \u2212\u03b1Lh \u2212 2\u03b1\u00b5f \u03b12L2 f + 2\u03b1\u00b5f + 1 ! . Meanwhile, Theorem 2 implies \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1new-contractive, where \u03c1new = 1\u2212 2\u03bb 4 \u2212\u03b1Lh +\u03bb s 2 4 \u2212\u03b1Lh \u0012 2 4 \u2212\u03b1Lh \u2212\u03b1 min \u001a 2\u00b5f (1 + \u03b1\u00b5f)2, 2Lf (1 + \u03b1Lf)2 \u001b\u0013 . and \u03c1new < \u03c1old if \u00b5f > 0, and \u03c1new = \u03c1old if \u00b5f = 0. 2. Assume f \u2208F0,Lf, g \u2208F\u00b5g,\u221e, and h \u2208F0,Lh. Then, \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1old-contractive where \u03c1old = v u u t1 \u22122\u03bb\u03b1 min ( (2 \u2212\u03bb)\u00b5g (1 + \u03b12L2 f)(2 \u2212\u03bb + 2\u03b1\u00b5g), (2 \u2212\u03bb)(\u00b5g + Lf) + 2\u03b1\u00b5gLf (1 + \u03b1Lf)2(2 \u2212\u03bb + 2\u03b1\u00b5g) ) . Meanwhile, Theorem 1 implies \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1new-contractive where \u03c1new = s 1 \u22122\u03bb\u03b1 min \u001a (2 \u2212\u03bb)\u00b5g 2 \u2212\u03bb + 2\u03b1\u00b5g , (2 \u2212\u03bb)(\u00b5g + Lf) + 2\u03b1\u00b5gLf (1 + \u03b1Lf)2(2 \u2212\u03bb + 2\u03b1\u00b5g) \u001b . and \u03c1new \u2264\u03c1old, and \u03c1new < \u03c1old if (2\u2212\u03bb)(1\u22122\u03b1\u00b5g +\u03b12L2 f)+2\u03b1\u00b5g(1+ \u03b12L2 f) > 0 and \u00b5g > 0. 3. Assume f \u2208F0,Lf, g \u2208F0,\u221e, and h \u2208F\u00b5h,Lh. Then, \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1old-contractive where \u03c1old = v u u u t1 \u22122\u03bb\u03b1 min \uf8f1 \uf8f2 \uf8f3 \u00b5h \u0010 1 \u2212 \u03b1Lh 2(2\u2212\u03bb) \u0011 1 + \u03b12L2 f , Lf + \u00b5h \u0010 1 \u2212 \u03b1Lh 2(2\u2212\u03bb) \u0011 (1 + \u03b1Lf)2 \uf8fc \uf8fd \uf8fe. 15 \fMeanwhile, Theorem 1 implies \ud54b\u2202f,\u2202g,\u2207h,\u03b1,\u03bb is \u03c1new-contractive where \u03c1new = s 1 \u22122\u03bb\u03b1 min \u001a \u03be, Lf + \u03be (1 + \u03b1Lf)2 \u001b . where \u03be = min \u001a \u00b5h \u0012 1 \u2212 \u03b1\u00b5h 2(2 \u2212\u03bb) \u0013 , Lh \u0012 1 \u2212 \u03b1Lh 2(2 \u2212\u03bb) \u0013\u001b . Furthermore, \u03c1new < \u03c1old if \u00b5h > 0, and \u03c1new = \u03c1old if \u00b5h = 0. Proof to Proposition 2-1. To show \u03c1new < \u03c1old given \u00b5f > 0, it su\ufb03ces to show min \u001a 2\u00b5f (1 + \u03b1\u00b5f)2, 2Lf (1 + \u03b1Lf)2 \u001b > 2\u00b5f \u03b12L2 f + 2\u03b1\u00b5f + 1, or equivalently 2\u00b5f (1 + \u03b1\u00b5f)2 > 2\u00b5f \u03b12L2 f + 2\u03b1\u00b5f + 1 2Lf (1 + \u03b1Lf)2 > 2\u00b5f \u03b12L2 f + 2\u03b1\u00b5f + 1 which are straightforward from Lf > \u00b5f > 0. Proof to Proposition 2-2. \u03c1new \u2264\u03c1old is obvious. By straightforward calculations, the mentioned conditions imply (2 \u2212\u03bb)\u00b5g (1 + \u03b12L2 f)(2 \u2212\u03bb + 2\u03b1\u00b5g) < (2 \u2212\u03bb)(\u00b5g + Lf) + 2\u03b1\u00b5gLf (1 + \u03b1Lf)2(2 \u2212\u03bb + 2\u03b1\u00b5g) . (11) Thus, \u03c1old = s 1 \u22122\u03bb\u03b1 (2 \u2212\u03bb)\u00b5g (1 + \u03b12L2 f)(2 \u2212\u03bb + 2\u03b1\u00b5g) \u03c1new = s 1 \u22122\u03bb\u03b1 min \u001a (2 \u2212\u03bb)\u00b5g 2 \u2212\u03bb + 2\u03b1\u00b5g , (2 \u2212\u03bb)(\u00b5g + Lf) + 2\u03b1\u00b5gLf (1 + \u03b1Lf)2(2 \u2212\u03bb + 2\u03b1\u00b5g) \u001b holds. Then, (2 \u2212\u03bb)\u00b5g (1 + \u03b12L2 f)(2 \u2212\u03bb + 2\u03b1\u00b5g) < (2 \u2212\u03bb)\u00b5g 2 \u2212\u03bb + 2\u03b1\u00b5g together with (11) gives \u03c1new < \u03c1old. 16 \fProof to Proposition 2-3. First, observe that 0 < \u00b5h < Lh renders \u03be > \u00b5h \u0012 1 \u2212 \u03b1Lh 2(2 \u2212\u03bb) \u0013 . Therefore, \u03be > \u00b5h \u0010 1 \u2212 \u03b1Lh 2(2\u2212\u03bb) \u0011 1 + \u03b12L2 f , Lf + \u03be (1 + \u03b1Lf)2 > Lf + \u00b5h \u0010 1 \u2212 \u03b1Lh 2(2\u2212\u03bb) \u0011 (1 + \u03b1Lf)2 which give \u03c1new < \u03c1old. 3. Discussion and conclusion The reduction of Fact 3 allows us to obtain the Lipschitz coe\ufb03cients Theorems 1 and 2 by characterizing the maximum modulus of ZDYS A,B,C,\u03b1,\u03bb = \b \u03b6DYS(zf, zg, zh; \u03b1, \u03bb) \f \f zf \u2208G \u0000\ud541\u03b1Df \u0001 , zg \u2208G \u0000\ud541\u03b1Dg \u0001 , zh \u2208G (Dh) \t , where \u03b6DYS = 1\u2212\u03bbzB+\u03bbzA(2zB\u22121\u2212\u03b1zCzB) is a relatively simple polynomial. This only requires elementary mathematics, and it is considerably easier than directly analyzing \u001a\u2225\ud54bx \u2212\ud54by\u2225 \u2225x \u2212y\u2225 \f \f \f \f \ud54b\u2208\ud54bDf,Dg,Dh,\u03b1,\u03bb, x, y \u2208dom \ud54b, x \u0338= y \u001b . Furthermore, by obtaining tighter bounds on the set ZDYS A,B,C,\u03b1,\u03bb, one can improve upon the contraction factors presented in this work. We leave the tighter analysis to future work. We point out that the explicit and simple description of ZDYS A,B,C,\u03b1,\u03bb allows one to investigate it in a numerical and computer-assisted manner. Sampling points from ZDYS Df,Dg,Dh,\u03b1,\u03bb is straightforward, and doing so provides a numerical estimate of the maximum modulus. For example, Figure 1 depicts ZDYS Df,Dg,Dh,\u03b1,\u03bb with a speci\ufb01c choice of \u00b5f, \u00b5g, \u00b5h, Lf, Lg, Lh, \u03b1, and \u03bb. It shows that \u03c1g, the contraction factor of Theorem 1, is valid but not tight; the gap between ZDYS Df,Dg,Dh,\u03b1,\u03bb and Circ(0, \u03c1g) indicates the contraction factor has 17 \froom for improvement. Interestingly, if we modify the proof of Theorem 1 to choose r in (3) more carefully, we seem to obtain a tight contraction factor in the instance of Figure 1. Speci\ufb01cally, when we numerically minimize \u03c1 as a function of r, we observe that Circ(0, \u03c1) touches ZDYS Df,Dg,Dh,\u03b1,\u03bb in Figure 1 and the contact indicates tightness. Figure 1: ZDYS Df ,Dg,Dh,\u03b1,\u03bb with Circ(0, \u03c1f), Circ(0, \u03c1g), and Circ(0, \u03c1), where \u00b5f = 0.7, \u00b5g = 2, \u00b5h = 0.8, Lf = 1.5, Lg = 3, Lh = 1.3, \u03b1 = 0.9, and \u03bb = 1." + } + ], + "Tsachy Weissman": [ + { + "url": "http://arxiv.org/abs/1109.0351v2", + "title": "Directed Information, Causal Estimation, and Communication in Continuous Time", + "abstract": "A notion of directed information between two continuous-time processes is\nproposed. A key component in the definition is taking an infimum over all\npossible partitions of the time interval, which plays a role no less\nsignificant than the supremum over \"space\" partitions inherent in the\ndefinition of mutual information. Properties and operational interpretations in\nestimation and communication are then established for the proposed notion of\ndirected information. For the continuous-time additive white Gaussian noise\nchannel, it is shown that Duncan's classical relationship between causal\nestimation and information continues to hold in the presence of feedback upon\nreplacing mutual information by directed information. A parallel result is\nestablished for the Poisson channel. The utility of this relationship is then\ndemonstrated in computing the directed information rate between the input and\noutput processes of a continuous-time Poisson channel with feedback, where the\nchannel input process is constrained to be constant between events at the\nchannel output. Finally, the capacity of a wide class of continuous-time\nchannels with feedback is established via directed information, characterizing\nthe fundamental limit on reliable communication.", + "authors": "Tsachy Weissman, Young-Han Kim, Haim H. Permuter", + "published": "2011-09-02", + "updated": "2012-10-31", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION The directed information I(Xn \u2192Y n) between two random n-sequences Xn = (X1, . . . , Xn) and Y n = (Y1, . . . , Yn) is a natural generalization of Shannon\u2019s mutual information to random objects obeying causal relations. Introduced by Massey [1], this notion has been shown to arise as the canonical answer to a variety of problems with causally dependent components. For example, it plays a pivotal role in characterizing the capacity CFB of a communication channel with feedback. Massey [1] showed that the feedback capacity is upper bounded as CFB \u2264lim n\u2192\u221e max p(xn||yn\u22121) 1 nI(Xn \u2192Y n), (1) where I(Xn \u2192Y n) = Pn i=1 I(Xi; Yi|Y i\u22121) and p(xn||yn\u22121) = Qn i=1 p(xi|xi\u22121, yi\u22121); see also Kramer [2] that streamlines the notion of directed information by causal conditioning. The upper bound in (1) is tight for certain This work is partially supported by the NSF grant CCF-0729195, BSF grant 2008402, and the Center for Science of Information (CSoI), an NSF Science and Technology Center, under grant agreement CCF-0939370. H. H. Permuter has been partially supported by the Marie Curie Reintegration fellowship. Author\u2019s emails: tsachy@stanford.edu, yhk@ucsd.edu, haimp@bgu.ac.il \f2 classes of ergodic channels, such as general nonanticipatory channels satisfying certain regularity conditions [3], channels with \ufb01nite input memory and ergodic noise [4], and indecomposable \ufb01nite-state channels [5], paving the road to a computable characterization of feedback capacity; see [6]\u2013[8] for examples. Directed information and its variants also characterize (via multiletter expressions) the capacity for two-way channels [2], multiple access channels with feedback [2], [9], broadcast channels with feedback [10], and compound channels with feedback [11], as well as the rate\u2013distortion function with feedforward [12], [13]. In another context, directed information captures the difference in growth rates of wealth in horse race gambling due to causal side information [14]. This provides a natural interpretation of I(Xn \u2192Y n) as the amount of information about Y n causally provided by Xn on the \ufb02y. Similar interpretations for directed information can be drawn for other problems in science and engineering [15]. This paper is dedicated to extending the mathematical notion of directed information to continuous-time random processes and to establishing results that demonstrate the operational signi\ufb01cance of this notion in estimation and communication. Our contributions include the following: \u2022 We introduce the notion of directed information in continuous time. Given a pair of continuous-time processes in a time interval and its partition consisting of n subintervals, we \ufb01rst consider the (discrete-time) directed information for the two sequences of length n whose components are the sample paths on the respective subintervals. The resulting quantity depends on the speci\ufb01c partition of the time interval. We de\ufb01ne directed information in continuous time by taking the in\ufb01mum over all \ufb01nite time partitions. Thus, in contrast to mutual information in continuous time which can be de\ufb01ned as a supremum of mutual information over \ufb01nite \u201cspace\u201d partitions [16, Ch. 2.5], [17, Ch. 3.5], inherent to our notion of directed information is a similar supremum followed by an in\ufb01mum over time partitions. We explain why this de\ufb01nition is natural by showing that the continuous-time directed information inherits key properties of its discrete-time origin and by establishing new properties that are meaningful in continuous time. \u2022 We show that this notion of directed information arises in extending classical relationships between information and estimation in continuous time\u2014Duncan\u2019s theorem [18] that relates the minimum mean squared error (MMSE) in causal estimation of a target signal based on an observation through an additive white Gaussian noise channel to the mutual information between the target signal and the observation, and its counterpart for the Poisson channel\u2014to the scenarios in which the channel input process can causally depend on the channel output process, whereby corresponding relationships now hold between directed information and estimation. \u2022 We illustrate these relationships between directed information and estimation by characterizing the directed information rate and the feedback capacity of a continuous-time Poisson channel with inputs constrained to be constant between events at the channel output. \u2022 We establish the fundamental role of continuous-time directed information in characterizing the feedback capacity of a large class of continuous-time channels. In particular, we show that for channels where the output is a function of the input and some stationary ergodic \u201cnoise\u201d process, the continuous-time directed information characterizes the feedback capacity of the channel. \f3 The remainder of the paper is organized as follows. Section II is devoted to the de\ufb01nition of directed information and related quantities in continuous time, which is followed by a presentation of key properties of continuous-time directed information in Section III. In Section IV, we establish the generalizations of Duncan\u2019s theorem and its Poisson counterpart that accommodate the presence of feedback. In Section V, we apply the relationship between the causal estimation error and directed information for the Poisson channel to compute the directed information rate between the input and the output of this channel in a scenario that involves feedback. In Section VI, we study a general feedback communication problem in which our notion of directed information in continuous time emerges naturally in the characterization of the feedback capacity. Section VII concludes the paper with a few remarks. II. DEFINITION AND REPRESENTATION OF DIRECTED INFORMATION IN CONTINUOUS TIME Let P and Q be two probability measures on the same space and dP dQ be the Radon\u2013Nikodym derivative of P with respect to Q. The relative entropy between P and Q is de\ufb01ned as D(P\u2225Q) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 R \u0000log dP dQ \u0001 dP if dP dQ exists, \u221e otherwise. (2) For jointly distributed random objects U and V , the mutual information between them is de\ufb01ned as I(U; V ) := D(PU,V \u2225PU \u00d7 PV ), (3) where PU \u00d7PV denotes the product distribution under which U and V are independent but maintain their respective marginal distributions. As an alternative, the mutual information is de\ufb01ned [16, Ch. 2.5] as I(U; V ) := sup I([U]; [V ]), (4) where the supremum is over all \ufb01nite quantizations of U and V . That the two notions coincide has been established in, e.g., [19], [17, Ch. 3.5]. We write I(PU,V ) instead of I(U; V ) when we wish to emphasize the dependence on the joint distribution PU,V . For a jointly distributed random triple (U, V, W) with components in arbitrary measurable spaces, we de\ufb01ne the conditional mutual information between U and V given W as I(U; V |W) := sup I([U]; [V ]|W), (5) where the supremum is over all \ufb01nite quantizations of U and V . This quantity, due to Wyner [20], is always well de\ufb01ned and satis\ufb01es all the basic properties of conditional mutual information for discrete and continuous random variables, in particular: 1) Nonnegativity: I(U; V |W) \u22650 with equality iff U \u2192W \u2192V form a Markov chain (that is, U and V are conditionally independent given W). 2) Chain rule: I(U; V, X|W) = I(U; V |W) + I(U; X|V, W). 3) Data processing inequality: If U \u2192(W, X) \u2192V form a Markov chain, then I(U; X|W) \u2265I(U; V |W) with equality iff I(U; V |W, X) = 0. \f4 The de\ufb01nition in (5) coincides with Dobrushin\u2019s more restrictive de\ufb01nition [17, p. 29] Z I(PU,V |W=w) dPW (w), (6) where PU,V |W=w is a regular version of the conditional probability law of (U, V ) given {W = w} (cf. [21, Ch. 6]) if it exists. Let (Xn, Y n) be a pair of random n-sequences. The directed information from Xn to Y n is de\ufb01ned as I(Xn \u2192Y n) := n X i=1 I(Xi; Yi|Y i\u22121). (7) Note that, unlike mutual information, directed information is asymmetric in its arguments, i.e., I(Xn \u2192Y n) \u0338= I(Y n \u2192Xn) in general. Let us now develop the notion of directed information between two continuous-time stochastic processes on the time interval [0, T ). For a continuous-time process {Xt}, let Xb a = {Xs : a \u2264s < b} denote the process in the time interval [a, b). Let t = (t0, t1, . . . , tn) denote a vector with components satisfying 0 = t0 < t1 < \u00b7 \u00b7 \u00b7 < tn = T. (8) Let XT,t 0 denote the sequence of length n resulting from \u201cchopping up\u201d the continuous-time signal XT 0 into consecutive segments as XT,t 0 = (Xt1 0 , Xt2 t1 , . . . , XT tn\u22121). (9) Note that each component of the sequence is a continuous-time stochastic process. For a pair of jointly distributed stochastic processes (XT 0 , Y T 0 ), de\ufb01ne It(XT 0 \u2192Y T 0 ) := I(XT,t 0 \u2192Y T,t 0 ) (10) = n X i=1 I(Y ti ti\u22121; Xti 0 \f \fY ti\u22121 0 ), (11) where on the right side of (12) is the directed information between two sequences of length n de\ufb01ned in (7); and in (13) we note that the conditional mutual information terms, de\ufb01ned as in (5), are between two continuous-time processes, conditioned on a third. We extend this de\ufb01nition to It(XT 0 \u2192Y T 0 |V ), where V is a random object jointly distributed with (XT 0 , Y T 0 ), in the obvious way, namely It(XT 0 \u2192Y T 0 |V ) := I(XT,t 0 \u2192Y T,t 0 |V ) (12) := n X i=1 I(Y ti ti\u22121; Xti 0 \f \fY ti\u22121 0 , V ). (13) We de\ufb01ne T (a, b) to be the set of all \ufb01nite partitions of the time interval [a, b). The quantity It(XT 0 \u2192Y T 0 ) is monotone in t in the following sense: Proposition 1. Let t and t\u2032 be partitions in T (0, T ). If t\u2032 is a re\ufb01nement of t, i.e., {ti} \u2282{t\u2032 i}, then It\u2032(XT 0 \u2192 Y T 0 ) \u2264It(XT 0 \u2192Y T 0 ). \f5 Proof: It suf\ufb01ces to prove the claim assuming t as in (8) and that t\u2032 is the (n + 2)-dimensional vector with components 0 = t0 < t1 < \u00b7 \u00b7 \u00b7 < ti\u22121 < t\u2032 < ti < \u00b7 \u00b7 \u00b7 < tn = T. (14) For such t and t\u2032, we have from (13) It(XT 0 \u2192Y T 0 ) \u2212It\u2032(XT 0 \u2192Y T 0 ) (15) = I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) \u2212 \u0002 I(Y t\u2032 ti\u22121; Xt\u2032 0 |Y ti\u22121 0 ) + I(Y ti t\u2032 ; Xti 0 |Y t\u2032 0 ) \u0003 (16) = I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) \u2212 \u0002 I(Y t\u2032 ti\u22121; Xt\u2032 0 |Y ti\u22121 0 ) + I(Y ti t\u2032 ; Xti 0 |Y ti\u22121 0 , Y t\u2032 ti\u22121) \u0003 (17) = I(Xt\u2032 0 , Xti t\u2032 ; Y t\u2032 ti\u22121, Y ti t\u2032 |Y ti\u22121 0 ) \u2212 \u0002 I(Y t\u2032 ti\u22121; Xt\u2032 0 |Y ti\u22121 0 ) + I(Y ti t\u2032 ; Xt\u2032 0 , Xti t\u2032 |Y ti\u22121 0 , Y t\u2032 ti\u22121) \u0003 (18) = I(Xt\u2032 0 , Xti t\u2032 ; Y t\u2032 ti\u22121, Y ti t\u2032 |Y ti\u22121 0 ) \u2212I(Xt\u2032 0 Xti t\u2032 \u2192Y t\u2032 ti\u22121, Y ti t\u2032 |Y ti\u22121 0 ) (19) \u22650, (20) where the last inequality follows since directed information (between two sequences of length 2 in this case) is upper bounded by the mutual information [1, Th. 2]. The following de\ufb01nition is now natural: De\ufb01nition 1. Let (XT 0 , Y T 0 ) be a pair of stochastic processes. The directed information from XT 0 to Y T 0 is de\ufb01ned as I(XT 0 \u2192Y T 0 ) := inf t\u2208T (0,T ) It(XT 0 \u2192Y T 0 ). (21) If V is another random object jointly distributed with (XT 0 , Y T 0 ) we de\ufb01ne the conditional directed information I(XT 0 \u2192Y T 0 |V ) as I(XT 0 \u2192Y T 0 |V ) := inf t\u2208T (0,T ) It(XT 0 \u2192Y T 0 |V ). (22) Note that the de\ufb01nitions and conventions preceding De\ufb01nition 1 imply that the directed information I(XT 0 \u2192Y T 0 ) is a nonnegative extended real number (i.e., as an element of [0, \u221e]). It is also worth noting, by recalling (4), that each of the conditional mutual information terms in (13), and hence the sum, is a supremum over \u201cspace\u201d partitions of the stochastic process in the corresponding time intervals. Thus the directed information in (21) is an in\ufb01mum over time partitions of a supremum over space partitions. Also note that I(XT 0 \u2192Y T 0 ) = lim \u03b5\u21920+ inf t:ti\u2212ti\u22121\u2264\u03b5,\u2200i It(XT 0 \u2192Y T 0 ), (23) where the in\ufb01mum is over all partitions in T (0, T ) with subinterval lengths uniformly bounded by \u01eb > 0. Indeed, for any \u01eb > 0 and any partition t \u2208T (0, T ), have inft\u2032:t\u2032 i\u2212t\u2032 i\u22121\u2264\u03b5,\u2200i It\u2032(XT 0 \u2192Y T 0 ) \u2264It(XT 0 \u2192Y T 0 ), since a re\ufb01nement of the time interval does not increase the directed information as seen in Proposition 1. By the arbitrariness of t \u2208T (0, T ), this implies inf t\u2032:t\u2032 i\u2212t\u2032 i\u22121\u2264\u03b5,\u2200i It\u2032(XT 0 \u2192Y T 0 ) \u2264 inf t\u2208T (0,T ) It(XT 0 \u2192Y T 0 ) = I(XT 0 \u2192Y T 0 ), (24) \f6 which in turn implies I(XT 0 \u2192Y T 0 ) \u2265lim\u03b5\u21920+ inft:ti\u2212ti\u22121\u2264\u03b5,\u2200i It(XT 0 \u2192Y T 0 ) by the arbitrariness of \u03b5 > 0. Since the reverse inequality I(XT 0 \u2192Y T 0 ) \u2264lim\u03b5\u21920+ inft:ti\u2212ti\u22121\u2264\u03b5,\u2200i It(XT 0 \u2192Y T 0 ) is immediate from the de\ufb01nition of I(XT 0 \u2192Y T 0 ), we have (23). As is clear from its de\ufb01nition in (7), the discrete-time directed information satis\ufb01es I(Xn \u2192Y n) \u2212I(Xn\u22121 \u2192Y n\u22121) = I(Yn; Xn|Y n\u22121). (25) A continuous-time analogue would be that, for small \u03b4 > 0, I(Xt+\u03b4 0 \u2192Y t+\u03b4 0 ) \u2212I(Xt 0 \u2192Y t 0 ) \u2248I(Y t+\u03b4 t ; Xt+\u03b4 0 |Y t 0 ). (26) Thus, if our proposed notion of directed information in continuous time is to be a natural extension of that in discrete time, one might expect the approximate relation (26) to hold in some sense. Toward a precise statement, denote it := lim \u03b4\u21920+ 1 \u03b4 I(Y t+\u03b4 t ; Xt+\u03b4 0 |Y t 0 ) for t \u2208(0, T ) (27) whenever the limit exists. Assuming it exists, let \u03b7(t, \u03b4) := 1 \u03b4 I(Y t+\u03b4 t ; Xt+\u03b4 0 |Y t 0 ) \u2212it (28) and note that (27) is equivalent to lim \u03b4\u21920+ \u03b7(t, \u03b4) = 0. (29) Proposition 2. Fix 0 < t < T . Suppose that it is continuous at t and that the convergence in (29) is uniform in the interval [t, t + \u03b3) for some \u03b3 > 0. Then d+ dt I(Xt 0 \u2192Y t 0 ) = it. (30) Note that Proposition 2 formalizes (26) by implying that the left and right hand sides of (26), when normalized by \u03b4, coincide in the limit of small \u03b4. Proof of Proposition 2: Note \ufb01rst that the stipulated uniform convergence in (29) implies the existence of \u03b3 > 0 and a monotone function f(\u03b4) such that |\u03b7(t\u2032, \u03b4)| \u2264f(\u03b4) for all t\u2032 \u2208[t, t + \u03b3) (31) and lim \u03b4\u21920+ f(\u03b4) = 0. (32) Fix now 0 < \u03b5 \u2264\u03b3 and consider I(Xt+\u03b5 0 \u2192Y t+\u03b5 0 ) = inf t\u2208T (0,t+\u03b5) It(Xt+\u03b5 0 \u2192Y t+\u03b5 0 ) (33) = inf t\u2208T (0,t+\u03b5) n X i=1 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) (34) \f7 = inf t\u2208(T (0,t) S T (t,t+\u03b5)) n X i=1 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) (35) = inf t\u2208T (0,t) n X i=1 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) + inf t\u2208T (t,t+\u03b5) n X i=1 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) (36) = I(Xt 0 \u2192Y t 0 ) + inf t\u2208T (t,t+\u03b5) n X i=1 (ti \u2212ti\u22121) 1 ti \u2212ti\u22121 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) (37) = I(Xt 0 \u2192Y t 0 ) + inf t\u2208T (t,t+\u03b5) n X i=1 (ti \u2212ti\u22121) \u00b7 [iti\u22121 + \u03b7(ti\u22121, ti \u2212ti\u22121)], (38) where the equality in (35) follows since the in\ufb01mum over all partitions does not change by restricting to partitions that have an interval up to time t and from time t and the last equality follows by the de\ufb01nition of the function \u03b7 in (28). Now, inf t\u2208T (t,t+\u03b5) n X i=1 (ti \u2212ti\u22121) \u00b7 [iti\u22121 + \u03b7(ti\u22121, ti \u2212ti\u22121)] \u2264 inf t\u2208T (t,t+\u03b5) n X i=1 (ti \u2212ti\u22121) \u00b7 \u0014 sup t\u2032\u2208[t,t+\u03b5) it\u2032 + f(\u03b5) \u0015 (39) = \u03b5 \u0014 sup t\u2032\u2208[t,t+\u03b5) it\u2032 + f(\u03b5) \u0015 , (40) where the inequality in (39) is due to (31) and the monotonicity of f, which implies f(ti \u2212ti\u22121) \u2264f(\u03b5), as ti \u2212ti\u22121 is the length of a subinterval in [t, t + \u03b5). Bounding the \u03b7 terms in (39) from the other direction, we similarly obtain inf t\u2208T (t,t+\u03b5) n X i=1 (ti \u2212ti\u22121) \u00b7 [iti\u22121 + \u03b7(ti\u22121, ti \u2212ti\u22121)] \u2265\u03b5 \u0014 inf t\u2032\u2208[t,t+\u03b5) it\u2032 \u2212f(\u03b5) \u0015 . (41) Combining (38), (40), and (41) yields inf t\u2032\u2208[t,t+\u03b5) it\u2032 \u2212f(\u03b5) \u2264I(Xt+\u03b5 0 \u2192Y t+\u03b5 0 ) \u2212I(Xt 0 \u2192Y t 0 ) \u03b5 \u2264 sup t\u2032\u2208[t,t+\u03b5) it\u2032 + f(\u03b5) for all \u03b5 > 0. (42) The continuity of it at t implies lim\u03b5\u21920+ inft\u2032\u2208[t,t+\u03b5) it\u2032 = lim\u03b5\u21920+ supt\u2032\u2208[t,t+\u03b5) it\u2032 = it and thus, taking the limit \u03b5 \u21920+ in (42) and applying (32) \ufb01nally yields lim \u03b5\u21920+ I(Xt+\u03b5 0 \u2192Y t+\u03b5 0 ) \u2212I(Xt 0 \u2192Y t 0 ) \u03b5 = it, (43) which completes the proof of Proposition 2. Beyond the intuitive appeal of Proposition 2 in formalizing (26), it also provides a useful formula for computing directed information. Indeed, the integral version of (30) is I(XT 0 \u2192Y T 0 ) = Z T 0 it dt. (44) As the following example illustrates, evaluating the right hand side of (44) (via the de\ufb01nition of it in (27)) can be simpler than tackling the left hand side directly via De\ufb01nition 1. Example 1. Let {Bt} be a standard Brownian motion and A \u223cN(0, 1) be independent of {Bt}. Let Xt \u2261A for all t and dYt = Xtdt + dBt. Letting J(P, N) = (1/2) ln((P + N)/N) denote the mutual information between a \f8 Gaussian random variable of variance P and its corrupted version by an independent Gaussian noise of variance N, we have for every t \u2208[0, T ) I(Y t+\u03b4 t ; Xt+\u03b4 0 |Y t 0 ) = J \u0012 1/t 1 + 1/t, 1 \u03b4 \u0013 = 1 2 ln \u0012 1 + \u03b4 t + 1 \u0013 . With such an explicit expression for I(Y t+\u03b4 t ; Xt+\u03b4 0 |Y t 0 ), it can be obtained directly from its de\ufb01nition: it = lim \u03b4\u21920+ 1 2\u03b4 ln \u0012 1 + \u03b4 t + 1 \u0013 = 1 2(t + 1). (45) We can now compute the directed information by applying Proposition 2: I(XT 0 \u2192Y T 0 ) = Z T 0 itdt = Z T 0 1 2(t + 1)dt = 1 2 ln(1 + T ). (46) Note that in this example I(XT 0 ; Y T 0 ) = J(1, 1/T ) = 1 2 ln(1 + T ) and thus, by (46), we have I(XT 0 \u2192Y T 0 ) = I(XT 0 ; Y T 0 ). This equality between mutual information and directed information holds in more general situations, as elaborated in the next section. The directed information we have just de\ufb01ned is between two processes on [0, T ). We extend this de\ufb01nition to processes of different durations by zero-padding at the beginning of the shorter process. For instance, I(XT \u2212\u03b4 0 \u2192Y T 0 ) := I((0\u03b4 0XT \u2212\u03b4 0 ) \u2192Y T 0 ), (47) where (0\u03b4 0XT \u2212\u03b4 0 ) denotes a process on [0, T ) formed by concatenating a process that is equal to the constant 0 for the time interval [0, \u03b4) and then the process XT \u2212\u03b4 0 . De\ufb01ne now I(XT \u2212 0 \u2192Y T 0 ) := lim sup \u03b4\u21920+ I(XT \u2212\u03b4 0 \u2192Y T 0 ) (48) and I(XT \u2212 0 \u2192Y T 0 ) := lim inf \u03b4\u21920+ I(XT \u2212\u03b4 0 \u2192Y T 0 ). (49) Finally, de\ufb01ne the directed information I(XT \u2212 0 \u2192Y T 0 ) by I(XT \u2212 0 \u2192Y T 0 ) := lim \u03b4\u21920+ I(XT \u2212\u03b4 0 \u2192Y T 0 ) (50) when the limit exists, or equivalently, when I(XT \u2212 0 \u2192Y T 0 ) = I(XT \u2212 0 \u2192Y T 0 ). As we shall see below (in the last part of Proposition 3), I(XT \u2212 0 \u2192Y T 0 ) is guaranteed to exist whenever I(XT 0 ; Y T 0 ) < \u221e. III. PROPERTIES OF THE DIRECTED INFORMATION IN CONTINUOUS TIME The following proposition collects some properties of directed information in continuous time: Proposition 3. Let (XT 0 , Y T 0 ) be a pair of jointly distributed stochastic processes. Then: 1) Monotonicity: I(Xt 0 \u2192Y t 0 ) is monotone nondecreasing in 0 \u2264t \u2264T . 2) Invariance to time dilation: For \u03b1 > 0, if \u02dc Xt = Xt\u03b1 and \u02dc Yt = Yt\u03b1, then I( \u02dc XT/\u03b1 0 \u2192\u02dc Y T/\u03b1 0 ) = I(XT 0 \u2192Y T 0 ). More generally, if \u03c6 is monotone strictly increasing and continuous, and ( \u02dc X\u03c6(t), \u02dc Y\u03c6(t)) = (Xt, Yt), then I(XT 0 \u2192Y T 0 ) = I( \u02dc X\u03c6(T ) \u03c6(0) \u2192\u02dc Y \u03c6(T ) \u03c6(0) ). (51) \f9 3) Coincidence of directed information and mutual information: If the Markov relation Y t 0 \u2192Xt 0 \u2192XT t holds for all 0 \u2264t < T , then I(XT 0 \u2192Y T 0 ) = I(XT 0 ; Y T 0 ). (52) 4) Equivalence between discrete time and piecewise constancy in continuous time: Let (U n, V n) be a pair of jointly distributed n-tuples and suppose (t0, t1, . . . , tn) satisfy (8). Let the pair (XT 0 , Y T 0 ) be de\ufb01ned as the piecewise-constant process satisfying (Xt, Yt) = (Ui, Vi) if ti\u22121 \u2264t < ti (53) for i = 1, . . . , n. Then I(XT 0 \u2192Y T 0 ) = I(U n \u2192V n). (54) 5) Conservation law: For any 0 < \u03b4 \u2264T we have I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) + I(Y T \u2212\u03b4 0 \u2192XT 0 ) = I(XT 0 ; Y T 0 ). (55) Further, if I(XT 0 ; Y T 0 ) < \u221ethen I(Y T \u2212 0 \u2192XT 0 ) exists and I(XT 0 \u2192Y T 0 ) + I(Y T \u2212 0 \u2192XT 0 ) = I(XT 0 ; Y T 0 ). (56) Remarks. 1) The \ufb01rst, second, and fourth parts in the proposition present properties that are known to hold for mutual information (when all the directed information expressions in those items are replaced by the corresponding mutual information), which follow immediately from the data processing inequality and the invariance of mutual information to one-to-one transformations of its arguments. That these properties hold also for directed information is not as obvious in view of the fact that directed information is, in general, not invariant to oneto-one transformations nor does it satisfy the data processing inequality in its second argument. 2) The third part of the proposition is a natural analogue of the fact that I(Xn; Y n) = I(Xn \u2192Y n) whenever Y i \u2192Xi \u2192Xn i+1 form a Markov chain for all 1 \u2264i \u2264n. It covers, in particular, any scenario where XT 0 and Y T 0 are the input and output of any channel of the form Yt = gt(Xt 0, W T 0 ), where the process W T 0 (which can be thought of as the internal channel noise) is independent of the channel input process XT 0 . To see this, note that in this case we have (Xt 0, W T 0 ) \u2192Xt 0 \u2192XT t for all 0 \u2264t \u2264T , implying Y t 0 \u2192Xt 0 \u2192XT t since Y t 0 is determined by the pair (Xt 0, W T 0 ). 3) Particularizing even further, we obtain I(XT 0 \u2192Y T 0 ) = I(XT 0 ; Y T 0 ) whenever Y T 0 is the outcome of corrupting XT 0 with additive noise, i.e., Yt = Xt + Wt, where XT 0 and W T 0 are independent. 4) The \ufb01fth part of the proposition can be considered the continuous-time analogue of the discrete-time conservation law [22] I(U n \u2192V n) + I(V n\u22121 \u2192U n) = I(U n; V n). (57) It is consistent with, and in fact generalizes, the third part. Indeed, if the Markov relation Y t 0 \u2192Xt 0 \u2192XT t holds for all 0 \u2264t \u2264T then our de\ufb01nition of directed information is readily seen to imply that I(Y T \u2212\u03b4 0 \u2192 \f10 XT 0 ) = 0 for all \u03b4 > 0 and therefore that I(Y T \u2212 0 \u2192XT 0 ) exists and equals zero. Thus (56) in this case reduces to (52). Proof of Proposition 3: The \ufb01rst part of the proposition follows immediately from the de\ufb01nition of directed information in continuous time (De\ufb01nition 1) and from the fact that, in discrete time, I(U m \u2192V m) \u2264I(U n \u2192V n) for m \u2264n. The second part follows from De\ufb01nition 1 upon noting that, under a dilation \u03c6 as stipulated, due to the invariance of mutual information to one-to-one transformations of its arguments, for any partition t of [0, T ), It(XT 0 \u2192Y T 0 ) = I\u03c6(t)( \u02dc X\u03c6(T ) \u03c6(0) \u2192\u02dc Y \u03c6(T ) \u03c6(0) ), (58) where \u03c6(t) is shorthand for (\u03c6(t0, \u03c6(t1), . . . , \u03c6(tn)). Thus I(XT 0 \u2192Y T 0 ) = inf t\u2208T (0,T ) It(XT 0 \u2192Y T 0 ) (59) = inf t\u2208T (0,T ) I\u03c6(t)( \u02dc X\u03c6(T ) \u03c6(0) \u2192\u02dc Y \u03c6(T ) \u03c6(0) ) (60) = inf t\u2208T (\u03c6(0),\u03c6(T )) It( \u02dc X\u03c6(T ) \u03c6(0) \u2192\u02dc Y \u03c6(T ) \u03c6(0) ) (61) = I( \u02dc X\u03c6(T ) \u03c6(0) \u2192\u02dc Y \u03c6(T ) \u03c6(0) ), (62) where (59) and (62) follow from De\ufb01nition 1, (60) follows from (58), and (61) is due to the strict monotonicity and continuity of \u03c6 which implies that {\u03c6(t) : t is a partition of [0, T )} = {t : t is a partition of [\u03c6(0), \u03c6(T ))}. (63) Moving to the proof of the third part, assume that the Markov relation Y t 0 \u2192Xt 0 \u2192XT t holds for all 0 \u2264t \u2264T and \ufb01x t = (t0, t1, . . . , tn) as in (8). Then It(XT 0 \u2192Y T 0 ) = I(XT,t 0 \u2192Y T,t 0 ) (64) = N X i=1 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) (65) = N X i=1 I(Y ti ti\u22121; XT 0 |Y ti\u22121 0 ) (66) = I(XT 0 ; Y T 0 ), (67) where (66) follows since Y ti 0 \u2192Xti 0 \u2192XT ti for each 1 \u2264i \u2264N, and (67) is due to the chain rule for mutual information. The proof of the third part of the proposition now follows from the arbitrariness of t. To prove the fourth part, consider \ufb01rst the case n = 1. In this case Xt \u2261U1 and Yt \u2261V1 for all t \u2208[0, T ). It is an immediate consequence of the de\ufb01nition of directed information that I((U, U, . . . , U) \u2192(V, V, . . . , V )) = I(U; V ) and therefore that It(XT 0 \u2192Y T 0 ) = I(U1; V1) = I(U1 \u2192V1) for all t. Consequently I(XT 0 \u2192Y T 0 ) = I(U1 \u2192V1), which establishes the case n = 1. For the general case n \u22651, note \ufb01rst that it is immediate from the de\ufb01nition of It(XT 0 \u2192Y T 0 ) and from the construction of (XT 0 , Y T 0 ) based on (Xn, Y n) in (53) that for t = (t0, t1, . . . , tn) \f11 consisting of the time epochs in (53) we have It(XT 0 \u2192Y T 0 ) = I(U n \u2192V n). Thus I(XT 0 \u2192Y T 0 ) \u2264It(XT 0 \u2192 Y T 0 ) = I(U n \u2192V n). We now argue that Is(XT 0 \u2192Y T 0 ) \u2265I(U n \u2192V n) (68) for any partition s. By Proposition 1, it suf\ufb01ces to establish (68) with equality assuming s is a re\ufb01nement of the particular t just discussed, that is, s is of the form 0 = t0 = s0,0 < s0,1 < \u00b7 \u00b7 \u00b7 < s0,J0 < t1 = s1,0 < s1,1 < \u00b7 \u00b7 \u00b7 < s1,J1 < t2 = s2,0 < \u00b7 \u00b7 \u00b7 < sn\u22121,Jn\u22121 < tn = T. (69) Then, Is(XT 0 \u2192Y T 0 ) = I(XT,s 0 \u2192Y T,s 0 ) (70) = n X i=1 Ji\u22121 X j=1 I(Y si\u22121,j si\u22121,j\u22121; Xsi\u22121,j 0 |Y si\u22121,j\u22121 0 ) (71) = n X i=1 I(Ui; V i|U i\u22121) (72) = I(U n \u2192V n), (73) where (72) follows by applying a similar argument as in the case n = 1. Moving to the proof of the \ufb01fth part of the proposition, \ufb01x t = (t0, t1, . . . , tn) as in (8) with t1 = \u03b4 > 0. Applying the discrete-time conservation law (57), we have It(XT 0 \u2192Y T 0 ) + It(Y T \u2212\u03b4 0 \u2192XT 0 ) = I(XT 0 ; Y T 0 ) (74) and consequently, for any \u03b5 > 0, inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} It(XT 0 \u2192Y T 0 ) + inf {t:maxi ti\u2212ti\u22121\u2264\u03b5} It(Y T \u2212\u03b4 0 \u2192XT 0 ) (75) = inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} It(XT 0 \u2192Y T 0 ) + inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} It(Y T \u2212\u03b4 0 \u2192XT 0 ) (76) = inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} \u0002 It(XT 0 \u2192Y T 0 ) + It(Y T \u2212\u03b4 0 \u2192XT 0 ) \u0003 (77) = I(XT 0 ; Y T 0 ), (78) where the equality in (76) follows since due to its de\ufb01nition in (47), It(Y T \u2212\u03b4 0 \u2192XT 0 ) does not decrease by re\ufb01ning the time interval t in the [0, \u03b4) interval; the equality in (77) follows from the re\ufb01nement property in Proposition 1, which implies that for arbitrary processes XT 0 , Y T 0 , ZT 0 , W T 0 and partitions t and t\u2032 there exists a third partition t\u2032\u2032 (which will be a re\ufb01nement of both) such that It(XT 0 \u2192Y T 0 ) + It\u2032(ZT 0 \u2192W T 0 ) \u2265It\u2032\u2032(XT 0 \u2192Y T 0 ) + It\u2032\u2032(ZT 0 \u2192W T 0 ); (79) and the equality in (78) follows since (74) holds for any t = (t0, t1, . . . , tn) with t1 = \u03b4. Hence, I(XT 0 ; Y T 0 ) = lim \u03b5\u21920+ \u0014 inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} It(XT 0 \u2192Y T 0 ) + inf {t:maxi ti\u2212ti\u22121\u2264\u03b5} It(Y T \u2212\u03b4 0 \u2192XT 0 ) \u0015 (80) \f12 = lim \u03b5\u21920+ inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} It(XT 0 \u2192Y T 0 ) + lim \u03b5\u21920+ inf {t:maxi ti\u2212ti\u22121\u2264\u03b5} It(Y T \u2212\u03b4 0 \u2192XT 0 ) (81) = lim \u03b5\u21920+ inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} \" I(X\u03b4 0; Y \u03b4 0 ) + n X i=2 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) # + I(Y T \u2212\u03b4 0 \u2192XT 0 ) (82) = I(X\u03b4 0; Y \u03b4 0 ) + lim \u03b5\u21920+ inf {t:t1=\u03b4,maxi\u22652 ti\u2212ti\u22121\u2264\u03b5} n X i=2 I(Y ti ti\u22121; Xti 0 |Y ti\u22121 0 ) + I(Y T \u2212\u03b4 0 \u2192XT 0 ) (83) = I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) + I(Y T \u2212\u03b4 0 \u2192XT 0 ), (84) where the equality in (80) follows by taking the limit \u03b5 \u21920 from both sides of (78); the equality in (82) follows by writing out It(XT 0 \u2192Y T 0 ) explicitly for t with t1 = \u03b4 and using (23) to equate the second limit in (81) with I(Y T \u2212\u03b4 0 \u2192XT 0 ); and the equality in (84) follows by applying (23) on the conditional distribution of the pair (XT 0 , (0\u03b4 0Y T \u03b4 )) given Y \u03b4 0 . We have thus proven (55) or, equivalently, the identity I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) = I(XT 0 ; Y T 0 ) \u2212I(Y T \u2212\u03b4 0 \u2192XT 0 ). (85) Toward the proof of (56), for t \u2208T (0, T ) and \u03b4 < t1 let t\u03b4 denote the re\ufb01nement of t obtained by adding an additional point at \u03b4. Then It(XT 0 \u2192Y T 0 ) \u2265 It\u03b4(XT 0 \u2192Y T 0 ) (86) = I(X\u03b4 0; Y \u03b4 0 ) + It\u03b4(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) (87) \u2265 I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ), (88) where the \ufb01rst inequality follows since t\u03b4 is a re\ufb01nement of t, the equality by writing out the sum that de\ufb01nes It\u03b4(XT 0 \u2192Y T 0 ) and isolating its \ufb01rst term, and the second inequality by the in\ufb01mum over partitions inherent in the de\ufb01nition of I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ). The arbitrariness of \u03b4 < t1 in (88) implies lim sup \u03b4\u21920+ I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) \u2264It(XT 0 \u2192Y T 0 ) (89) which, by the arbitrariness of t \u2208T (0, T ), implies lim sup \u03b4\u21920+ I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) \u2264I(XT 0 \u2192Y T 0 ). (90) On the other hand, for any \u03b4 > 0, we clearly have I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) \u2265I(XT 0 \u2192Y T 0 ), (91) as the right hand side, by its de\ufb01nition, is an in\ufb01mum over all partitions in T (0, T ), while the left hand side corresponds to an in\ufb01mum over the subset consisting only of those partitions with t1 = \u03b4. By the arbitrariness of \u03b4 in (91) we obtain lim inf \u03b4\u21920+ I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) \u2265I(XT 0 \u2192Y T 0 ) (92) which, when combined with (90), \ufb01nally implies lim \u03b4\u21920+ I(X\u03b4 0; Y \u03b4 0 ) + I(XT 0 \u2192Y T \u03b4 |Y \u03b4 0 ) = I(XT 0 \u2192Y T 0 ). (93) \f13 Existence of the limit in (93), when combined with (55) and the added assumption I(XT 0 ; Y T 0 ) < \u221e, implies existence of the limit lim\u03b4\u21920+ I(Y T \u2212\u03b4 0 \u2192XT 0 ) = I(Y T \u2212 0 \u2192XT 0 ) and that I(XT 0 \u2192Y T 0 ) + I(Y T \u2212 0 \u2192XT 0 ) = I(XT 0 ; Y T 0 ), (94) thus completing the proof. IV. DIRECTED INFORMATION, FEEDBACK, AND CAUSAL ESTIMATION A. The Gaussian Channel In [18], Duncan discovered the following fundamental relationship between the minimum mean squared error (MMSE) in causal estimation of a target signal corrupted by an additive white Gaussian noise (AWGN) in continuous time and the mutual information between the clean and noise-corrupted signals: Theorem 1 (Duncan [18]). Let XT 0 be a signal of \ufb01nite average power R T 0 E[X2 t ]dt < \u221e, independent of a standard Brownian motion {Bt}. Let Y T 0 satisfy dYt = Xtdt + dBt. Then 1 2 Z T 0 E \u0002 (Xt \u2212E[Xt|Y t 0 ])2\u0003 dt = I(XT 0 ; Y T 0 ). (95) A remarkable aspect of Duncan\u2019s theorem is that the relationship (95) holds regardless of the distribution of XT 0 . Among its rami\ufb01cations is the invariance of the causal MMSE to the \ufb02ow of time, or more generally, to any reordering of time [23], [24]. It should also be mentioned that, although this exact relationship holds in continuoustime, approximate versions that hold in discrete-time can be derived from it, as is done in [24, Theorem 9]. A key stipulation in Duncan\u2019s theorem is the independence between the noise-free signal XT 0 and the channel noise {Bt}, which excludes scenarios in which the evolution of Xt is affected by the channel noise, as is often the case in signal processing (e.g., target tracking) and communication (e.g., in the presence of feedback). Indeed, the identity (95) does not hold in the absence of such a stipulation. As an extreme example, consider the case where the channel input is simply the channel output with some delay, i.e., Xt+\u03b5 = Yt (96) for some \u03b5 > 0 (and Xt \u22610 for t \u2208[0, \u03b5)). In this case the causal MMSE on the left side of (95) is clearly 0, while the mutual information on its right side is in\ufb01nite. On the other hand, in this case the directed information I(XT 0 \u2192Y T 0 ) = 0, as can be seen by noting that It(XT 0 \u2192Y T 0 ) = 0 for all t satisfying maxi(ti \u2212ti\u22121) \u2264\u03b5 (since for such t, Xti 0 is determined by Y ti\u22121 0 for all i). The third remark following Proposition 3 implies that Theorem 1 could be equivalently stated with I(XT 0 ; Y T 0 ) on the right side of (95) replaced by I(XT 0 \u2192Y T 0 ). Furthermore, such a modi\ufb01ed identity would be valid in the extreme example in (96). This is no coincidence and is a consequence of the result that follows, which generalizes Duncan\u2019s theorem. To state it formally we assume a probability space (\u2126, F, P) with an associated \ufb01ltration {Ft} satisfying the \u201cusual conditions\u201d (right-continuous and F0 contains all the P-negligible events in F, cf., e.g., [25, \f14 De\ufb01nition 2.25]). Recall also that when the standard Brownian motion is adapted to {Ft} then, by de\ufb01nition, it is implied that, for any s < t, Bt \u2212Bs is independent of Fs (rather than merely of Bs 0, cf., e.g., [25, De\ufb01nition 1.1]). Theorem 2. Let {(Xt, Bt)}T t=0 be adapted to the \ufb01ltration {Ft}T t=0, where XT 0 is a signal of \ufb01nite average power R T 0 E[X2 t ]dt < \u221eand BT 0 is a standard Brownian motion. Let Y T 0 be the output of the AWGN channel whose input is XT 0 and whose noise is driven by BT 0 , i.e., dYt = Xtdt + dBt. (97) Suppose that the regularity assumptions of Proposition 2 are satis\ufb01ed for all 0 < t < T . Then 1 2 Z T 0 E \u0002 (Xt \u2212E[Xt|Y t 0 ])2\u0003 dt = I(XT 0 \u2192Y T 0 ). (98) Note that unlike in Theorem 1, where the channel input process is independent of the channel noise process, in Theorem 2 no such stipulation exists and thus the setting in the latter accommodates the presence of feedback. Furthermore, since I(XT 0 \u2192Y T 0 ) is not invariant to the direction of the \ufb02ow of time in general, Theorem 2 implies, as should be expected, that neither is the causal MMSE for processes evolving in the generality afforded by the theorem. That Theorem 1 can be extended to accommodate the presence of feedback has been established for a communication theoretic framework by Kadota, Zakai, and Ziv [26]. Indeed, in communication over the AWGN channel where XT 0 = XT 0 (M) is the waveform associated with message M, in the absence of feedback the Markov relation M \u2192XT 0 \u2192Y T 0 implies that I(XT 0 ; Y T 0 ) on the right hand side of (95), when applying Theorem 1 in this restricted communication framework, can be equivalently written as I(M; Y T 0 ). The main result of [26] is that this relationship between the causal estimation error and I(M; Y T 0 ) persists in the presence of feedback, i.e., that 1 2 Z T 0 E \u0002 (Xt \u2212E[Xt|Y t 0 ])2\u0003 dt = I(M; Y T 0 ) (99) with or without feedback, even though, in the presence of feedback, one no longer has I(M; Y T 0 ) = I(XT 0 ; Y T 0 ) and therefore (95) is no longer true. The combination of Theorem 2 with the main result of [26] (namely, with (99)) thus implies that in communication over the AWGN channel, with or without feedback, we have I(M; Y T 0 ) = I(XT 0 \u2192Y T 0 ). This equality holds well beyond the Gaussian channel, as is elaborated in Section VI. Evidently, Theorem 2 can be considered an extension of the Kadota\u2013Zakai\u2013Ziv result as it holds in settings more general than communication, where there is no message but merely a signal observed through additive white Gaussian noise, adapted to a general \ufb01ltration. Theorem 2 is a direct consequence of Proposition 2 and the following lemma. Lemma 1 ( [27]). Let P and Q be two probability laws governing (XT 0 , Y T 0 ), under which (97) and the stipulations of Theorem 2 are satis\ufb01ed. Then D(PY T 0 \u2225QY T 0 ) = 1 2EP \u0014Z T 0 (Xt \u2212EQ[Xt|Y t 0 ])2 \u2212(Xt \u2212EP [Xt|Y t 0 ])2dt \u0015 . (100) \f15 Lemma 1 was implicit in [27]. It follows from the second part of [27, Theorem 2], put together with the exposition in [27, Subsection IV-D] (cf., in particular, equations (148) through (161) therein). Proof of Theorem 2: Consider I(Y t+\u03b4 t ; Xt+\u03b4 0 |Y t 0 ) = D(PY t+\u03b4 t |Xt+\u03b4 t ,Y t 0 \u2225PY t+\u03b4 t |Y t 0 |PY t 0 ,Xt+\u03b4 t ) (101) = Z D(PY t+\u03b4 t |Xt+\u03b4 t =xt+\u03b4 t ,Y t 0 =yt 0\u2225PY t+\u03b4 t |Y t 0 =yt 0)dPY t 0 ,Xt+\u03b4 t (yt 0, xt+\u03b4 t ) (102) = 1 2 Z E \u0014Z t+\u03b4 t (xs \u2212E[Xs|Y s 0 ])2 \u2212(xs \u2212xs)2ds \f \f \f \f yt 0, xt+\u03b4 t \u0015 dPY t 0 ,Xt+\u03b4 t (yt 0, xt+\u03b4 t ) (103) = 1 2 Z t+\u03b4 t E \u0002 (Xs \u2212E[Xs|Y s 0 ])2\u0003 ds, (104) where the equality in (103) follows by applying (100) to the integrand in (102) as follows: replacing the time interval [0, T ) by [t, t + \u03b4), substituting P by the law of (Xt+\u03b4 t , Y t+\u03b4 t ) conditioned on (yt 0, xt+\u03b4 t ) (note that Xt+\u03b4 t is deterministic at xt+\u03b4 t under this law), and substituting Q by the law of (Xt+\u03b4 t , Y t+\u03b4 t ) conditioned on yt 0. The last step is obtained by switching between the integral R t+\u03b4 t and R E and then using the de\ufb01nition of conditional expectation. The switch between the integrals is possible due to Fubini\u2019s theorem and the fact that the signal has \ufb01nite average power R T 0 E[X2 t ]dt < \u221e. It follows that it de\ufb01ned in (27) exists and is given by it = 1 2E \u0002 (Xt \u2212E[Xt|Y t 0 ])2\u0003 , (105) which completes the proof by an appeal to Proposition 2. B. The Poisson Channel Consider the function \u2113: [0, \u221e) \u00d7 [0, \u221e) \u2192[0, \u221e] given by \u2113(x, \u02c6 x) = x log(x/\u02c6 x) \u2212x + \u02c6 x. (106) That this function is natural for quantifying the loss when estimating nonnegative quantities is implied in [28, Section 2], where some of its basic properties are exposed. Among them is that conditional expectation is the optimal estimator not only under the squared error loss but also under \u2113, i.e., for any nonnegative random variable X jointly distributed with Y , min \u02c6 X(\u00b7) E h \u2113(X, \u02c6 X(Y )) i = E [\u2113(X, E(X|Y ))] , (107) where the minimum is over all (measurable) maps from the domain of Y into [0, \u221e). With this loss function, the analogue of Duncan\u2019s theorem for the case of doubly stochastic Poisson process (i.e., the intensity is a random process) can be stated as: Theorem 3 ( [28], [29]). Let Y T 0 be a doubly stochastic Poisson process and XT 0 be its intensity process (i.e., conditioned on XT 0 , Y T 0 is a nonhomogenous Poisson process with rate function XT 0 ) satisfying E R T 0 |Xt log Xt|dt < \u221e. Then Z T 0 E[\u2113(Xt, E[Xt|Y t 0 ])]dt = I(XT 0 ; Y T 0 ). (108) \f16 We remark that for \u03c6(\u03b1) = \u03b1 log \u03b1, one has E \u0002 \u03c6(Xt) \u2212\u03c6(E[Xt|Y t 0 ]) \u0003 = E \u0002 \u2113(Xt, E[Xt|Y t 0 ]) \u0003 , (109) and thus (108) can equivalently be expressed as Z T 0 E \u0002 \u03c6(Xt) \u2212\u03c6(E[Xt|Y t 0 ]) \u0003 dt = I(XT 0 ; Y T 0 ), (110) as was done in [29] and other classical references. But it was not until [28] that the left hand side was established as the minimum mean causal estimation error under an explicitly identi\ufb01ed loss function, thus completing the analogy with Duncan\u2019s theorem. The condition stipulated in the third item of Proposition 3 is readily seen to hold when Y T 0 is a doubly stochastic Poisson process and XT 0 is its intensity process. Thus, the above theorem could equivalently be stated with directed information rather than mutual information on the right hand side of (108). Indeed, with continuous-time directed information replacing mutual information, this relationship remains true in much wider generality, as the next theorem shows. In the statement of the theorem, we use the notions of a point process and its predictable intensity, as developed in detail in, e.g., [30, Chapter II]. Theorem 4. Let Yt be a point process and Xt be its FY t -predictable intensity, where FY t is the \u03c3-\ufb01eld \u03c3(Y t 0 ) generated by Y t 0 . Suppose that E R T 0 |Xt log Xt|dt < \u221e, and that the assumptions of Proposition 2 are satis\ufb01ed for all 0 < t < T . Then Z T 0 E[\u2113(Xt, E[Xt|Y t 0 ])]dt = I(XT 0 \u2192Y T 0 ). (111) Paralleling the proof of Theorem 2, the proof of Theorem 4 is a direct application of Proposition 2 and the following: Lemma 2 ( [28]). Let P and Q be two probability laws governing (XT 0 , Y T 0 ) under the setting and stipulations of Theorem 4. Then D(PY T 0 \u2225QY T 0 ) = EP \"Z T 0 \u2113(Xt, EQ[Xt|Y t 0 ]) \u2212\u2113(Xt, EP [Xt|Y t 0 ])dt # . (112) Lemma 2 is implicit in [28], following directly from [28, Theorem 4.4] and the discussion in [28, Subsection 7.5]. Equipped with it, the proof of Theorem 4 follows similarly as that of Theorem 2, the role of (100) being played here by (112). V. EXAMPLE: POISSON CHANNEL WITH FEEDBACK The Poisson channel (e.g., [31]\u2013[38]) is a channel where the input at time t, Xt, determines the intensity of the doubly stochastic Poisson process Yt occurring at the output of the channel. A Poisson channel with feedback refers to the case where the input signal Xt may depend on the previous observation of the output Y t. In this section we consider a special case of Poisson channel with feedback. Let X = {Xt} and Y = {Yt} be the input and output processes of the continuous-time Poisson channel with feedback, where each time an event \f17 occurs at the channel output, the channel input changes to a new value, drawn according to the distribution of a positive random variable X, independently of the channel input and output up to that point in time. The channel input remains \ufb01xed at that value until the occurrence of the next event at the channel output, and so on. Throughout this section, the shorthand \u201cPoisson channel with feedback\u201d will refer to this scenario, with its implied channel input process. The Poisson channel we use here is similar to the well-known Poisson channel model (e.g., [31]\u2013[38]) with one difference that the intensity of the Poisson channel changes according to the input X only when there is an event at the output of the channel. Note that the channel description given here uniquely determines the joint distribution of the channel input and output processes. In the \ufb01rst part of this section, we derive, using Theorem 4, a formula for the directed information rate of this Poisson channel with feedback. In the second part, we demonstrate the use of this formula by computing and plotting the directed information rate for a special case in which the intensity alphabet is of size 2. A. Characterization of the Directed Information Rate For jointly distributed processes (X, Y) de\ufb01ne the directed information rate I(X \u2192Y) by I(X \u2192Y) = lim T \u2192\u221e 1 T I(XT 0 \u2192Y T 0 ), (113) when the limit exists. Proposition 4. Assume that X is \ufb01nite-valued with probability mass function (pmf) pX(x). The directed information rate between the input and output processes of the Poisson channel with feedback I(X \u2192Y) exists and is given by I(X \u2192Y) = I(X; Y ) E[1/X] , (114) where, in I(X; Y ) on the right hand side, Y |{X = x} \u223cExp(x), i.e., the conditional density of Y given {X = x} is f(y|x) = xe\u2212yx \u00b7 1{y\u22650}. The key component in the proof of the proposition is the use of Theorem 4 for directed information in continuous time as a causal mean estimation error. An intuition for the expression in (114) can be obtained by considering rate per unit cost [39], i.e., R = I(X; Y )/E[b(X)], where b(x) is the cost of the input. In our case, the \u201ccost\u201d of X is proportional to the average duration of time until the channel can be used again, i.e., b(x) = 1/x. Finally, we remark that the assumption of discreteness of X in Proposition 4 is made for simplicity of the proof, though the result carries over to more generally distributed X. To prove Proposition 4, let us \ufb01rst collect the following observations: Lemma 3. Let X \u223cpX(x) and Y |{X = x} \u223cExp(x). De\ufb01ne g(t) := E[X|Y \u2265t] = P x xe\u2212txpX(x) P x e\u2212txpX(x) , t \u22650. (115) \f18 Then the following statement holds. 1) The marginal distribution of Xt is P{Xt = x} = (1/x)pX(x) P x\u2032(1/x\u2032)pX(x\u2032) (116) and consequently E[Xt log Xt] = E[log X] E[1/X] . (117) 2) Let \u2113= \u2113(Y 0 \u2212\u221e) denote the time of occurrence of the last (most recent) event at the channel output prior to time 0 and de\ufb01ne \u03c4 := \u2212\u2113. The density of \u03c4 is f\u03c4(t) = P x e\u2212txpX(x) E[1/X] , t \u22650. (118) 3) For \u03c4 distributed as in (118), E[g(\u03c4) log g(\u03c4)] = 1 \u2212h(Y ) E[1/X] . (119) Proof: For the \ufb01rst part of the lemma, note that Xt is an ergodic continuous-time Markov chain and thus P{Xt = x} is equal to the fraction of time that Xt spends in state x which is proportional to (1/x)pX(x), accounting for (116), which, in turn, yields E[Xt log Xt] = X x (1/x)pX(x) P x\u2032(1/x\u2032)pX(x\u2032)x log x = P x pX(x) log x P x\u2032(1/x\u2032)pX(x\u2032) = E[log X] E[1/X] , (120) accounting for (117). To prove the second part of the lemma, observe that (a) the interarrival times of the process Y are independent and identically distributed (i.i.d.) copies of a random variable Y ; (b) Y has a density fY (y) = X x pX(x)xe\u2212xy, y \u22650, (121) (c) the probability density of the length of the interarrival interval of the Y process around 0 is proportional to fY (y) \u00b7 y; and (d) given the length of the interarrival interval around 0 is y, its left point is uniformly distributed on [\u2212y, 0]. Letting Unif[0, y](\u00b7) denote the density of a random variable uniformly distributed on [0, y], it follows that the density of \u03c4 is f\u03c4(t) = Z \u221e 0 fY (y) \u00b7 y R \u221e 0 fY (y\u2032) \u00b7 y\u2032dy\u2032 Unif[0, y](t)dy (122) = Z \u221e t fY (y) \u00b7 y R \u221e 0 fY (y\u2032) \u00b7 y\u2032dy\u2032 1 y dy (123) = P x pX(x)x R \u221e t e\u2212xydy P x pX(x)x R \u221e 0 e\u2212xy\u2032 \u00b7 y\u2032dy\u2032 (124) = P x pX(x)x e\u2212tx x P x pX(x)x 1 x2 (125) \f19 = P x pX(x)e\u2212tx E[1/X] , (126) where (122) follows by combining observations (c) and (d), and (124) follows by substituting from (121). We have thus proven the second part of the lemma. To establish the third part, let FY (t) denote the cumulative distribution function of Y and consider E[g(\u03c4) log g(\u03c4)] = Z \u221e 0 f\u03c4(t)g(t) log g(t) (127) = Z \u221e 0 P x pX(x)e\u2212tx E[1/X] P x xe\u2212txpX(x) P x e\u2212txpX(x) log P x xe\u2212txpX(x) P x e\u2212txpX(x) dt (128) = 1 E[1/X] Z \u221e 0 X x xe\u2212txpX(x) log P x xe\u2212txpX(x) P x e\u2212txpX(x) dt (129) = 1 E[1/X] Z \u221e 0 fY (t) log fY (t) 1 \u2212FY (t)dt (130) = 1 E[1/X] \u0012Z \u221e 0 fY (t) log 1 1 \u2212FY (t)dt \u2212h(Y ) \u0013 (131) = 1 E[1/X] \u0012Z 1 0 log 1 1 \u2212udu \u2212h(Y ) \u0013 (132) = 1 E[1/X](1 \u2212h(Y )), (133) where (128) follows by substituting from the second part of the lemma and (130) follows by substituting from (121) and noting that X x e\u2212txpX(x) = X x pX(x)xe\u2212tx x = X x pX(x)x Z \u221e t e\u2212xydy = Z \u221e t X x pX(x)xe\u2212xydy = Z \u221e t fY (y)dy = 1 \u2212FY (t). (134) We have thus established the third and last part of the lemma. Proof of Proposition 4: We have I(X \u2192Y) = lim T \u2192\u221e 1 T I(XT 0 \u2192Y T 0 ) (135) = lim T \u2192\u221e 1 T Z T 0 E \u0002 Xt log Xt \u2212E[Xt|Y t 0 ] log E[Xt|Y t 0 ] \u0003 dt (136) = E \u0002 X0 log X0 \u2212E[X0|Y 0 \u2212\u221e] log E[X0|Y 0 \u2212\u221e] \u0003 (137) = E[log X] E[1/X] \u2212E \u0002 E[X0|Y 0 \u2212\u221e] log E[X0|Y 0 \u2212\u221e] \u0003 , (138) where (136) follows from the relation between directed information and causal estimation in (111); (137) follows from the stationarity and martingale convergence. Speci\ufb01cally, by martingale convergence E[X0|Y 0 \u2212t] \u2192 E[X0|Y 0 \u2212\u221e] as t \u2192\u221ea.s. and thus E \u0002 Xt log Xt \u2212E[Xt|Y t 0 ] log E[Xt|Y t 0 ] \u0003 , which by stationarity is equal to E \u0002 X0 log X0 \u2212E[X0|Y 0 \u2212t] log E[X0|Y 0 \u2212t] \u0003 , converges to E \u0002 X0 log X0 \u2212E[X0|Y 0 \u2212\u221e] log E[X0|Y 0 \u2212\u221e] \u0003 by the bounded convergence theorem (recall that X0 is \ufb01nite-valued); and (138) follows from the \ufb01rst part of Lemma 3. \f20 Now, recalling the de\ufb01nition of the function g in (115) we note that E[X0|\u2113(Y 0 \u2212\u221e)] = g(\u2212\u2113(Y 0 \u2212\u221e)). (139) Thus E \u0002 E[X0|Y 0 \u2212\u221e] log E[X0|Y 0 \u2212\u221e] \u0003 = E \u0002 E[X0|\u2113(Y 0 \u2212\u221e)] log E[X0|\u2113(Y 0 \u2212\u221e)] \u0003 (140) = E \u0002 g(\u2212\u2113(Y 0 \u2212\u221e)) log g(\u2212\u2113(Y 0 \u2212\u221e)) \u0003 (141) = E[g(\u03c4) log g(\u03c4)] (142) = 1 \u2212h(Y ) E[1/X] , (143) where (140) follows from the Markov relation Y 0 \u2212\u221e\u2192\u2113(Y 0 \u2212\u221e) \u2192X0, (141) follows from (139), and (143) from the last part of Lemma 3. Thus I(X \u2192Y) = h(Y ) \u22121 + E[log X] E[1/X] (144) = h(Y ) \u2212h(Y |X) E[1/X] (145) = I(X; Y ) E[1/X] , (146) where (144) follows by combining (138) with (143), and (145) follows by noting that h(Y |X) = X x h(Y |X = x)pX(x) = X x (1 \u2212log x)pX(x) = 1 \u2212E[log X]. (147) This completes the proof of Proposition 4. B. Evaluation of the Directed Information Rate Fig. 1 depicts the directed information rate I(X \u2192Y) for the case where X takes only two values \u03bb1 and \u03bb2. We have used numerical evaluation of I(X; Y ) in the right hand side of (114) to compute the directed information rate. The \ufb01gure shows the in\ufb02uence of p = P{X = \u03bb1} on the directed information rate where \u03bb1 = 1 and \u03bb2 = 2. As expected, the maximum is achieved when there is higher probability that the encoder output will be the higher rate \u03bb2, which would imply more channel uses per unit time, but not much higher as otherwise the input value will be close to deterministic. Fig. 2 depicts the maximal value (optimized w.r.t. P{X = \u03bb1}) of the directed information rate when \u03bb1 is \ufb01xed and is equal to 1 and \u03bb2 varies. This value is the capacity of the Poisson channel with feedback, when the inputs are restricted to one of the two values \u03bb1 or \u03bb2. When \u03bb2 = 0 the capacity is obviously zero since any use of X = \u03bb2 as input will cause the channel not to change any further. It is also obviously zero at \u03bb2 = 1 since in this case \u03bb1 = \u03bb2, so there is only one possible input to the channel. As \u03bb2 increases, the capacity of the channel increases without bound since, for \u03bb2 \u226b\u03bb1, the channel effectively operates as a noise-free binary channel, where one symbol \u201ccosts\u201d an average duration of 1 while the other a vanishing average duration. Thus the limiting capacity with increasing \u03bb2 is equal to limp\u21930 H(p)/p = \u221e. \f21 0.2 0.6 1 0.02 0.06 0.1 p := P{X = \u03bb1} I(X \u2192Y) Fig. 1. The directed information rate between the input and output processes for the continuous-time Poisson channel with feedback, as a function of P (x), the pmf of the input to the channel. The input to the channel is one of two possible values \u03bb1 = 1 and \u03bb2 = 2, and it is the intensity of the Poisson process at the output of the channel until the next event. 10 \u22124 10 \u22122 10 0 10 2 0 1 2 3 4 5 \u03bb2 max I(X \u2192Y) Fig. 2. Capacity of the Poisson channel with feedback, in case where channel input is constrained to the binary set {\u03bb1, \u03bb2}, when \u03bb1 is \ufb01xed and is equal to 1 and \u03bb2 varies. One can consider a discrete-time memoryless channel, where the input X is discrete (\u03bb1 or \u03bb2) and the output Y is distributed according to Exp(X). Consider now a random cost b(X) = Y , where Y is the output of the channel. Using the result from [39] we obtain that the capacity per unit cost of the discreet memoryless channel is max P (x) I(X; Y ) E[Y ] = max P (x) I(X; Y ) E[1/X] , (148) where the equality follows since E[Y ] = E[E[Y |X]] = E[1/X]. Finally, we note that the capacity of the Poisson channel in the example above is the capacity per unit cost of the discrete memoryless channel. Thus, by Proposition 4 we can conclude that the continuous-time directed information rate characterizes the capacity of the Poisson channel with feedback. In the next section we will see that the continuous-time directed information rate characterizes the capacity of a large family of continuous-time channels. \f22 VI. COMMUNICATION OVER CONTINUOUS-TIME CHANNELS WITH FEEDBACK We \ufb01rst review the de\ufb01nition of a block-ergodic process as given by Berger [40]. Let (X, X, \u00b5) denote a continuous-time process {Xt}t\u22650 drawn from a space X according to the probability measure \u00b5. For t > 0, let T t be a t-shift transformation, i.e., (T tx)s = xs+t. A measurable set A is t-invariant if it does not change under the t-shift transformation, i.e., T tA = A. A continuous-time process (X, X, \u00b5) is \u03c4-ergodic if every measurable \u03c4-invariant set of processes has either probability 1 or 0, i.e., for any \u03c4-invariant set A, in other words, \u00b5(A) = (\u00b5(A))2. The de\ufb01nition of \u03c4-ergodicity means that if we take the process {Xt}t\u22650 and slice it into time-blocks of length \u03c4, then the new discrete-time process (X\u03c4 0 , X2\u03c4 \u03c4 , X3\u03c4 2\u03c4 , . . .) is ergodic. A continuous-time process (X, X, \u00b5) is block-ergodic if it is \u03c4-ergodic for every \u03c4 > 0. Berger [40] showed that weak mixing (therefore also strong mixing) implies block ergodicity. Message Message estimate M \u2208{1, . . . , 2nT} Xt Delay \u2206 Yt\u2212\u2206 Yt \u02c6 M xt(m, yt\u2212\u2206 0 ) Encoder g(Xt, Zt) Channel \u02c6 m(yT 0 ) Decoder Fig. 3. Continuous-time communication with delay \u2206and channel of the form Yt = g(Xt, Zt), where Zt is a block ergodic process. Now let us describe the communication model of our interest (see Fig. 3) and show that the continuous-time directed information characterizes the capacity. Consider a continuous-time channel that is speci\ufb01ed by \u2022 the channel input and output alphabets X and Y, respectively, that are not necessarily \ufb01nite, and \u2022 the channel output at time t Yt = g(Xt, Zt) (149) corresponding to the channel input Xt at time t, where {Zt} is a stationary ergodic noise process on an alphabet Z and g : X \u00d7 Z \u2192Y is a given measurable function. A (2T R, T ) code with delay \u2206> 0 for the channel consists of \u2022 a message set {1, 2, . . ., 2\u230aT R\u230b}, \u2022 an encoder that assigns a symbol xt(m, yt\u2212\u2206 0 ) (150) to each message m \u2208{1, 2, . . ., 2\u230aT R\u230b} and past received output signal yt\u2212\u2206 0 \u2208Y[0,t\u2212\u2206) for t \u2208[0, T ), where xt : {1, 2, . . ., 2\u230aT R\u230b} \u00d7 Y[0,t\u2212\u2206) \u2192X is measurable, and \u2022 a decoder that assigns a message estimate \u02c6 m(yT 0 ) \u2208{1, 2, . . ., 2\u230aT R\u230b} to each received output signal yT 0 \u2208 Y[0,T ), where \u02c6 m : Y[0,T ) \u2192{1, 2, . . ., 2\u230aT R\u230b} is measurable. \f23 We assume that the message M is uniformly distributed on {1, 2, . . ., \u230a2T R\u230b} and independent of the noise process {Zt}. By the de\ufb01nition of the channel in (149), the de\ufb01nition of the encoding function in (150), and the independence of M and {Zt}, it follows that for any \u03b4 > 0 and any t \u22650, M \u2192(Xt+\u03b4 0 , Y t 0 ) \u2192Y t+\u03b4 t (151) form a Markov chain. This is analogous to the assumption in the discrete case that p(yn+1|xn+1, yn, m) = p(yn+1|xn+1, yn); the analogy is exact when we convert a discrete time channel to a continuous time channel with constant piecewise process between the time samples. Furthermore, for any t \u22650, \u03b4 > 0, and \u2206\u2265\u03b4, Xt+\u03b4 t \u2192(Xt 0, Y t+\u03b4\u2212\u2206 0 ) \u2192Y t+\u03b4 t+\u03b4\u2212\u2206 (152) form a Markov chian. This is analogous to the assumption in the discrete case that whenever there is feedback of delay d \u22651, p(xn+1|xn, yn) = p(xn+1|xn, yn+1\u2212d). Similar communication settings with feedback in continuous time were studied by Kadota, Zakai, and Ziv [41] for continuous-time memoryless channels, where it is shown that feedback does not increase the capacity, and by Ihara [42], [43] for the Gaussian case. Our main result in this section is showing that the operational capacity, de\ufb01ned below, can be characterized by the information capacity, which is the maximum of directed information from the channel input process to the output process. Next we de\ufb01ne an achievable rate, the operational feedback capacity, and the information feedback capacity for our setting. De\ufb01nition 2. A rate R is said to be achievable with feedback delay \u2206if for each T there exists a family of (2RT , T ) codes such that lim T \u2192\u221eP{M \u0338= \u02c6 M(Y T 0 )} = 0. (153) De\ufb01nition 3. Let C(\u2206) = sup{R : R is achievable with feedback delay \u2206} (154) be the (operational) feedback capacity with delay \u2206, and let the (operational) feedback capacity be C \u225csup \u2206>0 C(\u2206). (155) From the monotonicity of C(\u2206) in \u2206we have sup\u2206>0 C(\u2206) = lim\u2206\u21920 C(\u2206). This de\ufb01nition coincides with the feedback capacity de\ufb01nition of continuous time channels given in [41], where there also was assumed a positive but arbitrary small delay in the feedback capacity. De\ufb01nition 4. Let CI(\u2206) be the information feedback capacity de\ufb01ned as CI(\u2206) = lim T \u2192\u221e 1 T sup S\u2206 I(XT 0 \u2192Y T 0 ), (156) \f24 where the supremum in (156) is over S\u2206, which is the set of all channel input processes of the form Xt = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 gt(Ut, Y t\u2212\u2206 0 ) t \u2265\u2206, gt(Ut) t < \u2206, (157) some family of measurable functions {gt}T t=0, and some process U T 0 which is independent of the channel noise process ZT 0 (appearing in (149)) and has a \ufb01nite cardinality that may depend on T . The limit in (156) is shown to exist in Lemma 4 using the superadditivity property. We now characterize C(\u2206) in terms of CI(\u2206) for the class of channels de\ufb01ned in (149). Theorem 5. For the channel de\ufb01ned in (149), C(\u2206) \u2264CI(\u2206), (158) C(\u2206) \u2265CI(\u2206\u2032) for all \u2206\u2032 > \u2206. (159) Since CI(\u2206) is a decreasing function in \u2206, (159) may be written as C(\u2206) \u2265lim\u03b4\u2192\u2206+ CI(\u03b4), and the limit exists because of the monotonicity. Since the function is monotonic then CI(\u2206) = lim\u03b4\u2192\u2206+ CI(\u03b4) with a possible exception of the points of \u2206of a set of measure zero [44, p. 5]. Therefore C(\u2206) = CI(\u2206) for any \u2206\u22650 except of a set of points of measure zero. Furthermore (158) and (159) imply that sup\u2206>0 C(\u2206) = sup\u2206>0 CI(\u2206), hence we also have C = sup\u2206>0 CI(\u2206) = lim\u2206\u21920 CI(\u2206). Before proving the theorem we show that the limits in (156) exist. Lemma 4. The term supS\u2206I(XT 0 \u2192Y T 0 ) is superadditive, namely, sup S\u2206 I(XT1+T2 0 \u2192Y T1+T2 0 ) \u2265sup S\u2206 I(XT1 0 \u2192Y T1 0 ) + sup S\u2206 I(XT2 0 \u2192Y T2 0 ), (160) and therefore the limit in (156) exists and is equal to lim T \u2192\u221e 1 T sup S\u2206 I(XT 0 \u2192Y T 0 ) = sup T 1 T sup S\u2206 I(XT 0 \u2192Y T 0 ) (161) To prove Lemma 4 we use the following result: Lemma 5. Let {(Xi, Yi)}n+m i=1 be a pair of discrete-time processes such that Markov relation Xi \u2192 (Xi\u22121, Y i\u22121) \u2192(Xi\u22121 n+1, Y i\u22121 n+1) holds for i \u2208{n + 1, n + 2, . . . , n + m}. Then I(Xn+m \u2192Y n+m) \u2265I(Xn \u2192Y n) + I(Xn+m n+1 \u2192Y n+m n+1 ), (162) Proof: The result is a consequence of the identity [4, Eq. (11)] I(Xn \u2192Y n) = n X i=1 I(Xi; Y n i |Xi\u22121, Y i\u22121). (163) Consider I(Xn+m \u2192Y n+m) = n+m X i=1 I(Xi; Y n+m i |Xi\u22121, Y i\u22121) (164) \f25 = n X i=1 I(Xi; Y n+m i |Xi\u22121, Y i\u22121) + n+m X i=n+1 I(Xi; Y n+m i |Xi\u22121, Y i\u22121) (165) \u2265 n X i=1 I(Xi; Y n i |Xi\u22121, Y i\u22121) + n+m X i=n+1 I(Xi; Y n+m i |Xi\u22121 n+1, Y i\u22121 n+1) (166) = I(Xn \u2192Y n) + I(Xn+m n+1 \u2192Y n+m n+1 ), (167) where (164) follows from the identity given in (163), and (166) follows from the Markov chain assumption in the lemma. Proof of Lemma 4: First note that we do not increase the term inft It(XT1+T2 0 \u2192Y T1+T2 0 ) by restricting the time-partition t to have an interval starting at point T1. Now \ufb01x three time-partitions: t1 in [0, T1), t2 in [T1, T1 + T2), and t in [0, T1 + T2) such that t is a concatenation t1 and t2. For XT1 0 and XT1+T2 T1 , \ufb01x the input functions of the form of (157) and \ufb01x the arguments U T1 and U T1+T2 T1 which corresponds to XT1 0 and XT1+T2 T1 , respectively. The construction is such that the random processes U T1 and U T1+T2 T1 are independent of each other. Let XT1+T2 0 be a concatenation of XT1 0 and XT1+T2 T1 . Applying Lemma 5 on the discrete-time process {(Xi, Yi)}n+m i=1 , where (Xi, Yi) = (Xti+1 ti , Y ti+1 ti ) for i = 1, 2, . . . , n + m we obtain that for any \ufb01xed t1, t2, XT1 0 , XT1+T2 T1 , U T1, and U T1+T2 T1 as described above, we have It(XT1+T2 0 \u2192Y T1+T2 0 ) \u2265It1(XT1 0 \u2192Y T1 0 ) + It2(XT1+T2 T1 \u2192Y T1+T2 T1 ). (168) Note that the Markov condition Xi \u2192(Xi\u22121 0 , Y i\u22121) \u2192(Xi\u22121 n+1, Y i\u22121 n+1) indeed holds because of the construction of XT1+T2 0 . Furthermore, because of the stationarity of the noise (168) implies (160). Finally, using Fekete\u2019s lemma [45, Ch. 2.6] and the superadditivity in (160) implies the existence of the limit in (161). The proof of Theorem 5 consists of two parts: the proof of the converse, i.e., (158), and the proof of achievability, i.e., (159). Proof of the converse for Theorem 5: Fix an encoding scheme {ft}T t=0 with rate R and probability of decoding error, P (T ) e = P{M \u0338= \u02c6 M(Y T 0 )}. In addition, \ufb01x a partition t of length n such that ti \u2212ti\u22121 < \u2206for any i \u2208[1, 2, . . ., n] and let tn = T . Consider RT = H(M) (169) = H(M) + H(M|Y T 0 ) \u2212H(M|Y T 0 ) (170) \u2264I(M; Y T 0 ) + T \u01ebT (171) = I(M; Y t1 0 , Y t2 t1 , . . . , Y tn tn\u22121) + T \u01ebT (172) = n X i=1 I(M; Y ti ti\u22121|Y ti\u22121 0 ) + T \u01ebT (173) = n X i=1 I(M, Xti\u22121+\u2206 0 ; Y ti ti\u22121|Y ti\u22121 0 ) + T \u01ebT (174) = n X i=1 I(M, Xti 0 , Xti\u22121+\u2206 ti ; Y ti ti\u22121|Y ti\u22121 0 ) + T \u01ebT (175) \f26 = n X i=1 I(M, Xti 0 ; Y ti ti\u22121|Y ti\u22121 0 ) + I(Xti\u22121+\u2206 ti ; Y ti ti\u22121|Y ti\u22121 0 , M, Xti 0 ) + T \u01ebT (176) = n X i=1 I(Xti 0 ; Y ti ti\u22121|Y ti\u22121 0 ) + I(Xti\u22121+\u2206 ti ; Y ti ti\u22121|Y ti\u22121 0 , M, Xti 0 ) + T \u01ebT (177) = n X i=1 I(Xti 0 ; Y ti ti\u22121|Y ti\u22121 0 ) + T \u01ebT (178) = It(XT 0 \u2192Y T 0 ) + T \u01ebT, (179) where the equality in (169) follows since the message is distributed uniformly, the inequality in (171) follows from Fano\u2019s inequality, where \u01ebT = 1 T + P (T ) e R, the equality in (174) follows from the fact that Xti\u22121+\u2206 0 is a deterministic function of M and Y ti\u22121 0 , the equality in (175) follows from the assumption that ti \u2212ti\u22121 < \u2206, the equality in (177) follows from (151), and the equality in (178) follows from (152). Hence, we obtained that for every t R \u22641 T It(XT 0 \u2192Y T 0 ) + \u01ebT . (180) Since the number of codewords is \ufb01nite, we may consider the input signal of the form xT,t 0 with xti ti\u22121 = f(uT 0 , yti\u2212\u2206 0 ), where the cardinality of uT 0 is bounded, i.e., |UT 0 | < \u221efor any given T (the bound may depend on T ), independently of the partition t. Furthermore, R \u2264inf t 1 T It(XT 0 \u2192Y T 0 ) + \u01ebT , = 1 T I(XT 0 \u2192Y T 0 ) + \u01ebT . (181) Finally, for any R that is achievable there exists a sequence of codes such that limT \u2192\u221eP (T ) e = 0, hence \u01ebT \u21920 and we have established (159). Note that as a byproduct of the sequence of equalities (171)\u2013(179), we conclude that for the communication system depicted in Fig. 3, I(M; Y T 0 ) = inf t:ti\u2212ti\u22121\u2264\u03b4 It(XT 0 \u2192Y T 0 ) = I(XT 0 \u2192Y T 0 ). (182) The only assumptions that we used to prove (171)\u2013(179) is that the encoders uses a strictly causal feedback of the form given in (157) and that the channel satis\ufb01es the benign assumption given in (151). This might be a valuable result by itself that provides a good intuition why directed information characterizes the capacity of a continuoustime channel. Furthermore, the interpretations of the measure I(M; Y T 0 ), for instance, as given in [26], should also hold for directed information and vice versa. For the proof of achievability we will use the following result for discrete-time channels. Lemma 6. Consider the discrete-time channel, where the input Ui at time i has a \ufb01nite alphabet, i.e., |U| < \u221e, and the output Yi at time i has an arbitrary alphabet Y. We assume that the relation between the input and the output is given by Yi = g(Ui, Zi), (183) \f27 where the noise process {Zi}i\u22651 is stationary and ergodic with an arbitrary alphabet Z. Then, any rate R is achievable for this channel if R < max p(u) I(U; Y ), (184) where the joint distribution of (U, Y ) is induced by the input distribution p(u), the stationary distribution of Z, and (183). Proof: Fix the pmf p(u) that attains the maximum in (184). Since I(U; Y ) can be approximated arbitrarily close by a \ufb01nite partition of Y [16], assume without loss of generality that Y is \ufb01nite. The proof uses the random codebook generation and joint typicality decoding in [46, Ch. 3]. Randomly and independently generate 2nR codewords un(m), m = 1, 2, . . ., 2nR, each according to Qn i=1 pU(ui). The decoder \ufb01nds the unique \u02c6 m such that (un(m), yn) is jointly typical. (For the de\ufb01nition and properties of joint typicality, refer to [47], [46, Ch. 2].) Now, assuming that M = 1 is sent, the decoder makes an error only if (U n(1), Y n) is not typical or (U n(m), Y n) is typical for some m \u0338= 1. By the packing lemma ( [46, Ch. 3]), the probability of the second event tends to zero as n \u2192\u221eif R < I(U; Y ). To bound the probability of the \ufb01rst event, recall from [48, Th. 10.3.1] that if {Ui} is i.i.d. and {Zi} is stationary ergodic, independent of {Ui}, then the pair {(Ui, Zi)} is jointly stationary ergodic. Consequently, from the de\ufb01nition of the channel in (183), {(Ui, Yi)} is jointly stationary ergodic. Thus, by Birkhoff\u2019s ergodic theorem, the probability that (U n(1), Y n) is not typical tends to zero as n \u2192\u221e. Therefore, any rate R < I(U; Y ) is achievable. The proof of achievability is based on the lemma above and the de\ufb01nition of directed information for continuous time. It is essential to divide into small time-interval as well as increasing the feedback delay by a small but positive value \u03b4 > 0. Proof of achivability for Theorem 5: Let \u2206\u2032 = \u2206+\u03b4, where \u03b4 > 0. In addition, let t = (0 = t0, t1, . . . , tn = T ) be such that ti \u2212ti\u22121 \u2264\u03b4 for all i = 1, 2, . . ., n. Let XT,t 0 be of the form Xti ti\u22121 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f(U T 0 , Y ti\u2212\u2206\u2032 0 ) ti \u2265\u2206\u2032, f(U T 0 ) ti < \u2206\u2032, (185) where the cardinality of U T 0 is bounded. Then we show that any rate R < 1 T It(XT,t 0 \u2192Y T 0 ), (186) is achievable. Assume that the communication is over the time interval [0, nT ], where T is \ufb01xed and n may be chosen to be as large as needed. Partition the time interval [0, nT ] into n subintervals of length T and in each subinterval [jT, jT + T ), which we index by j, \ufb01x the relation XjT +ti jT +ti\u22121 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f(U jT +T jT , Y jT +ti\u2212\u2206\u2032 jT ) ti \u2265\u2206\u2032, f(U jT +T jT ) ti < \u2206\u2032. (187) \f28 Note that this coding scheme is possible with feedback delay \u2206since ti\u22121 \u2212\u2206\u2265ti \u2212\u2206\u2032. This follows from the assumption that ti \u2212ti\u22121 \u2264\u03b4 and \u2206\u2032 \u2212\u2206\u2265\u03b4. Now, let us de\ufb01ne a discrete-time channel where the input at time j + 1 is \u02dc Uj+1 = U jT +T jT (which has an alphabet [1, . . . , 2nT ]), the output at time j + 1 is the vector \u02dc Yj+1 = (Y jT +t1 jT , . . . , Y jT +ti jT +ti\u22121, . . . , Y jT +T jT +tn\u22121) and the noise at time j + 1 is \u02dc Zj+1 = ZjT +T jT . Note that since ZjT +T jT is a stationary and block-ergodic the noise process { \u02dc Zj+1}j\u22650 is stationary and ergodic. Furthermore the relation \u02dc Yj+1 = \u02dc f( \u02dc Uj+1, \u02dc Zj+1) holds and the alphabet of \u02dc Uj+1 is \ufb01nite. Hence by Lemma 6, any rate R < max p(\u02dc u) I( \u02dc U; \u02dc Y ), (188) is achievable. Now using the de\ufb01nition of the discrete-time channel and the properties of directed information, we obtain I( \u02dc U; \u02dc Y ) = I(U T 0 ; Y T 0 ) (189) = I(U T 0 ; Y t1 0 , Y t2 t1 , . . . , Y tn tn\u22121) (190) = It(XT,t 0 \u2192Y T,t 0 ), (191) where the equality in (189) follows from the de\ufb01nition of the discrete-time channel and the equality in (191) follows from the same sequence of equalities as in (171)\u2013(179). Since (191) holds for any t such that ti \u2212ti\u22121 \u2264\u03b4 we conclude that C(\u2206) \u2265inf t It(XT 0 \u2192Y T 0 ). (192) Finally, by the de\ufb01nition of directed information and by the fact that (192) holds for any T we have established (159). VII. CONCLUDING REMARKS We have introduced and developed a notion of directed information between continuous-time stochastic processes. It emerges naturally in the characterization of the fundamental limit on reliable communication for a wide class of continuous-time channels with feedback, quite analogously to the discrete-time setting. It also arises in estimation theoretic relations as the replacement for mutual information when extending the scope to the presence of feedback. In particular, with continuous-time directed information replacing mutual information, Duncan\u2019s theorem generalizes to estimation problems in which the evolution of the target signal is affected by the past channel noise. An analogous relationship based on the directed information holds for the Poisson channel. We have illustrated the use of the latter in an explicit computation of the directed information rate between the input and output of a Poisson channel where the input intensity changes only when there is an event at the channel output. One important direction for future exploration is to use the \u201cmultiletter\u201d characterization of capacity developed here to compute or approximate the feedback capacity of interesting continuous-time channels. ACKNOWLEDGMENTS The authors thank the Associate Editor and the anonymous reviewers for their careful reading of the original manuscript and many valuable comments that helped improve the presentation. \f29" + }, + { + "url": "http://arxiv.org/abs/0904.2311v2", + "title": "Source Coding with a Side Information \"Vending Machine\"", + "abstract": "We study source coding in the presence of side information, when the system\ncan take actions that affect the availability, quality, or nature of the side\ninformation. We begin by extending the Wyner-Ziv problem of source coding with\ndecoder side information to the case where the decoder is allowed to choose\nactions affecting the side information. We then consider the setting where\nactions are taken by the encoder, based on its observation of the source.\nActions may have costs that are commensurate with the quality of the side\ninformation they yield, and an overall per-symbol cost constraint may be\nimposed. We characterize the achievable tradeoffs between rate, distortion, and\ncost in some of these problem settings. Among our findings is the fact that\neven in the absence of a cost constraint, greedily choosing the action\nassociated with the `best' side information is, in general, sub-optimal. A few\nexamples are worked out.", + "authors": "Tsachy Weissman, Haim H. Permuter", + "published": "2009-04-15", + "updated": "2009-04-30", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION The role and potential bene\ufb01t of Side Information (S.I.) in lossless and lossy data compression is a central theme in information theory. In ways that are well understood for various source coding systems, S.I. can be a valuable resource, resulting in signi\ufb01cant performance boosts relative to the case where it is absent. In the problems studied thus far, the lack or availability of the S.I., and its quality, are a given. But what if the system can take actions that affect the availability, quality, or nature of the S.I.? For example, consider a source coding system where the S.I. is a sequence of noisy measurements of the source sequence to be compressed, each S.I. symbol acquired via a sensor. The quality of each S.I. symbol may be commensurate with resources, such as power or time expended by the sensor for obtaining it, which are limited. Alternatively, or in addition, a sensor may have freedom to choose, for each source symbol, how many independent noisy measurements to observe, with a constraint on the overall number of measurements. It is then natural to wonder how these resources, which may or may not be limited, should best be used, and what would the corresponding optimum performance be. We abstract this problem by assuming a memoryless source PX, a conditional distribution of the side information given the source and an action PY |X,A, a function assigning costs to the possible actions, and a distortion measure. The \ufb01rst scenario we focus on is that depicted in Figure 1, where the actions are taken at the decoder: Based on its observation of the source sequence Xn, which is i.i.d.\u223cPX, the encoder gives an index to the decoder. Having received the index, the decoder chooses the action sequence An. Nature then generates the side information sequence Y n as the output of the memoryless Haim Permuter is with the Department of Electrical and Computer Engineering, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel. (Email: haimp@bgu.ac.il). Tsachy Weissman is with the Department of Electrical Engineering, Technion, Haifa 32000, Israel and on leave from Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA. (Email: tsachy@ee.technion.ac.il). \f2 PSfrag replacements Xn Encoder Decoder Vender T (Xn) \u22082nR \u02c6 Xn(Y n, T ) An(T ) Y n Fig. 1. Rate distortion with a side information vender at the decoder. The source Xn is i.i.d. \u223cPX, and Y n is the output of the side information channel PY |X,A in response to the pair of sequences Xn and An, where An is the action sequence chosen by the decoder. channel PY |X,A whose input is the pair (Xn, An). The reconstruction sequence \u02c6 Xn is then based on the index and on the side information sequence. The setting of Figure 1 can be considered the source coding dual of coding for channels with action-dependent states, where the transmitter chooses an action sequence that affects the formation of the channel states, and then creates the channel input sequence based on the state sequence, as considered in [11]. We characterize the achievable tradeoff between rate, distortion, and cost in Section II. We demonstrate, by a few examples, that greedily choosing the action associated with the \u2018best\u2019 side information may be sub-optimal even in the absence of a cost constraint. Further, in the presence of a cost constraint, time-sharing between schemes that are optimal for different cost values is, in general, sub-optimal. We also characterize the fundamental limits for the case where the reconstruction is con\ufb01ned to causal dependence on the side information sequence, and the case where the encoder observes a noisy observation of the source rather than the source itself. The second scenario we consider is that depicted in Figure 2, where actions are taken at the encoder: Based on its observation of the source sequence Xn, the encoder chooses a sequence of actions An. Nature then generates the side information sequence Y n as the output of the memoryless channel PY |X,A whose input is the pair (Xn, An). The encoder now chooses the index to be given to the decoder on the basis of the source and possibly the side information sequence (according to whether or not the switch is closed). The reconstruction sequence \u02c6 Xn is then based on the index and on the side information sequence. Though we leave the general case open, in Section III we characterize the achievable tradeoff between rate, distortion, and cost for three important special cases: the (near) lossless case, the Gaussian case (where Y = A+X +N, with X and N being independent Gaussian random variables), and the case of the Markov relation Y \u2212A \u2212X (i.e., when PY |X,A is of the form PY |A). We end that section with Subsection III-D, giving lower and upper bounds on the achievable rates for the general case. We summarize the paper and related open directions in Section IV. The family of problems we consider in this work includes scenarios arising naturally in the coding or compression of sources for which the S.I. arises from noisy measurements of the source components. The acquisition, handling, processing and storage of these measurements may require system resources that come at a cost. This premise, that the acquisition of source measurements may be costly and is to be done sparingly, is in fact central in the emerging Compressed Sensing \f3 PSfrag replacements Xn Encoder Decoder Vender T \u22082nR \u02c6 Xn(Y n, T ) An(Xn) Y n Y n Fig. 2. Rate distortion with side information vender at the encoder, where the side information is known at the decoder and may or may not be known to the encoder. The source Xn is i.i.d.\u223cPX and side information is generated as the output of the memoryless channel PY |X,A in response to the input (Xn, An), where the action sequence An is chosen by the encoder. paradigm [1], [2], [5], arising naturally in the study of an increasing array of sensing problems. In many such problems, the system has the freedom to choose how many sensors to deploy in each region of the phenomenon it is trying to gauge, subject to an overall budget of sensors. Assuming each sensor provides an independent measurement of the source region in which it was deployed, this setting corresponds to our model, with Ai \u2208{0, 1, 2, . . .} representing the number of sensors, PY |X,A = QA j=1 PZj|X representing A independent measurements from the \u2018sensor channel\u2019 PZ|X, and \u039b(Ai) = Ai assuming all sensors are equally costly. The cost constraint C then corresponds to the budget of sensors to deploy, in number of sensors per source region. We are not aware of previous work on source coding for systems allowed to take S.I.-affecting actions from a Shannon theoretic perspective. We refer to [7] and some references therein for other recent Shannon theoretic studies of new problems involving source coding in the presence of S.I. II. SIDE INFORMATION VENDING MACHINE AT THE DECODER Throughout the paper we let upper case, lower case, and calligraphic letters denote, respectively, random variables, speci\ufb01c or deterministic values they may assume, and their alphabets. For two jointly distributed random objects X and Y , let PX, PX,Y , and PX|Y respectively denote the distribution of X, the joint distribution of X, Y , and the conditional distribution of X given Y . In particular, when X and Y are discrete, PX|Y represents the stochastic matrix whose elements are PX|Y (x|y) = P(X = x|Y = y). The term Xn m denotes the n \u2212m + 1-tuple (Xm, . . . , Xn) when m \u2264n and the empty set otherwise. The term Xn is shorthand for Xn 1 , and Xn\\i stands for the n \u22121-tuple consisting of all the components of Xn but Xi. A. The Setup A source with action dependent decoder side information is characterized by the source distribution PX and by the conditional distribution of the side information given the source and an action PY |X,A. The difference between this and previously studied scenarios is that here, after receiving the index from the encoder, the decoder may choose actions that will affect the nature of the side information it will get to observe. Speci\ufb01cally, a scheme in this setting for blocklength n and rate R is characterized by an encoding function T : X n \u2192{1, 2, . . ., 2nR}, an action strategy f : {1, 2, . . ., 2nR} \u2192An, and a decoding function g : {1, 2, . . ., 2nR} \u00d7 Yn \u2192\u02c6 X n that operate as follows: \f4 \u2022 The source n-tuple Xn is i.i.d. \u223cPX \u2022 Encoding: based on Xn give index T = T (Xn) to the decoder \u2022 Decoding: \u2013 given the index, choose an action sequence An = f(T ) \u2013 the side information Y n will be the output of the memoryless channel PY |X,A whose input is (Xn, An) \u2013 let \u02c6 Xn = g(T, Y n) A triple (R, D, C) is said to be achievable if for all \u03b5 > 0 and suf\ufb01ciently large n there exists a scheme as above for blocklength n and rate R + \u03b5 satisfying both E \" n X i=1 \u03c1(Xi, \u02c6 Xi) # \u2264n(D + \u03b5) (1) and E \" n X i=1 \u039b(Ai) # \u2264n(C + \u03b5), (2) where \u03c1 and \u039b are, respectively, given distortion and cost functions. The rate distortion (and cost) function R(D, C) is de\ufb01ned as R(D, C) = inf{R\u2032 : the triple (R\u2032, D, C) is achievable}. (3) B. The Rate Distortion Cost Tradeoff De\ufb01ne R(I)(D, C) = min [I(X; A) + I(X; U|Y, A)] , (4) where the joint distribution of X, A, Y, U in (4) is of the form PX,A,U,Y (x, a, u, y) = PX(x)PA,U|X(a, u|x)PY |X,A(y|x, a), (5) and the minimization is over all PA,U|X under which E h \u03c1 \u0010 X, \u02c6 Xopt(U, Y ) \u0011i \u2264D, E [\u039b(A)] \u2264C, (6) where \u02c6 Xopt(U, Y ) denotes the best estimate of X based on U, Y , U is an auxiliary random variable. We show below that the cardinality of U may be restricted to |U| \u2264|X||A| + 1. Our main result pertaining to R(I)(D, C) is the following: Theorem 1: The rate distortion cost function, as de\ufb01ned in (3), is given by R(I)(D, C) in (4), i.e., R(D, C) = R(I)(D, C). (7) Remark: Write RWZ(PX, PY |X, D) for the explicit dependence of the Wyner-Ziv rate distortion function [15] on the distribution of the source and the conditional distribution of the source given the side information. It is clear that R(D, C) \u2264min (X a PA(a)RWZ(PX, PY |X,A=a, Da) : X a PA(a)Da \u2264D, X a PA(a)\u039b(a) \u2264C ) , (8) \f5 since the right hand side can be achieved by letting the decoder take actions according to a pre-speci\ufb01ed sequence with the symbol a fraction PA(a) of the time, and performing Wyner-Ziv coding at distortion level Da separately on each subsequence associated with each action symbol. It is natural to wonder whether the inequality in (8) can be strict. We will see through some examples below that, in general, it may very well be strict. Indeed, even in the absence of a cost constraint, we give examples showing that greedily selecting the action associated with the side information which is best in the Wyner-Ziv sense, that is the action a minimizing RWZ(PX, PY |X,A=a, D), may be suboptimal. The following lemma will be useful in proving Theorem 1. Lemma 1: Properties of the expressions de\ufb01ning R(I)(D, C): 1) For any \ufb01xed PX and PY |A,X, the set of distributions of the form given in (5) is a convex set in PA,U|X. 2) For any \ufb01xed PX and PY |A,X, the expression I(X; A) + I(X; U|Y, A) is convex in PA,U|X (assuming the joint distribution given in (5)). 3) To exhaust R(I)(D, C), it is enough to restrict the alphabet of U to satisfy |U| \u2264|X||A| + 2. (9) 4) It suf\ufb01ces to restrict the minimization in (4) to joint distributions where A is a deterministic function of U, i.e., of the form PX(x)PU|X(u|x)1{a=f(u)}PY |X,A(y|x, a). (10) Proof: 1) Since the set of conditional distributions PA,U|X is a convex set, and since PX and PY |X,A are \ufb01xed, the set of distributions PX,A,U,Y of the form given in (5) is a convex set. 2) Using the de\ufb01nition of mutual information we have the identity, I(X; A) + I(X; U|Y, A) = I(X; U, Y, A) + H(Y |A, X) \u2212H(Y |A). (11) We show now that the right-hand part of (11) is convex in PU,A|X for a \ufb01xed PX and PY |A,X. The expression I(X; U, Y, A) is convex in PU,Y,A|X, hence it is also convex in PU,A|X. For \ufb01xed PX and PY |A,X, the expression H(Y |A, X) is linear in PA|X. Finally, we show that \u2212H(Y |A) is convex using the the log sum inequality that states that for non negative number, a1, a2 and b1, b2 a1 log a1 b1 + a2 log a2 b2 \u2265(a1 + a2) log a1 + a1 b1 + b2 . (12) Now let P 3 A|X = \u03b1P 1 A|X + \u03b1P 2 A|X, where 0 \u2264\u03b1 \u22641 and \u03b1 = 1 \u2212\u03b1. Let us denote P i Y,A,X and Hi(A|X), the joint distribution and the conditional entropy induced by P i A|X and the \ufb01xed pmfs PX and PY |A,X for i = 1, 2, 3. Consider, P 3 Y,A(y, a) log P 3 Y,A(y, a) P 3 A(a) (a) = \u0000\u03b1P 1 Y,A(y, a) + \u03b1P 2 Y,A(y, a) \u0001 log \u03b1P 1 Y,A(y, a) + \u03b1P 2 Y,A(y, a) \u03b1P 1 A(a) + \u03b1P 2 A(a) (b) \u2264 \u03b1P 1 Y,A(y, a) log P 1 Y,A(y, a) P 1 A(a) + \u03b1P 2 Y,A(y, a) log P 2 Y,A(y, a) P 2 A(a) , (13) \f6 where (a) follows from the de\ufb01nition of P i Y,A,X and (b) follows from the log sum inequality. Since (13) holds for any a \u2208A and any y \u2208Y, we obtain that \u2212H(Y |A) is convex, i.e., \u2212H3(Y |A) \u2264\u2212\u03b1H1(Y |A) \u2212\u03b1H2(Y |A) (14) 3) We invoke the support lemma [4]. The external random variable U must have |X||A| \u22121 letters to preserve PX,A, plus two more to preserve the distortion constraint, the cost constraint and I(A; X) + I(X; U|Y, A). This results in alphabet of size |X||A| + 2. 4) Note that it suf\ufb01ces to restrict the minimization in (4) to joint distributions where A is a deterministic function of U, i.e., of the form PX(x)PU|X(u|x)1{a=f(u)}PY |X,A(y|x, a), (15) in lieu of (5). To see the equivalence note that a distribution of the form in (5) assumes the form in (10) by taking (U, A) as the auxiliary variable. Proof of Theorem 1: Achievability: We brie\ufb02y and informally outline the achievability part, which is based on standard arguments: A codebook of size 2n(I(X;A)+\u03b5) is generated with codewords that are i.i.d.\u223cPA. For each such codeword, generate 2n(I(X;U|A)+\u03b5) codewords according to PU|A. Distribute these codewords uniformly at random into 2n(I(X;U|Y,A)+2\u03b5) bins. Given the source realization, n(I(X; A) + \u03b5) bits are used by the encoder to communicate the identity of a codeword from the \ufb01rst codebook jointly typical with it (with high probability there is at least one such codeword). The decoder now performs the actions according to the action sequence conveyed to it. The encoder now uses an additional n(I(X; U|Y, A) + 2\u03b5) number of bits to describe the bin index of the codeword from the second code-book which is jointly typical with the source and the \ufb01rst codeword. With high probability there is at least one such codeword (since more than 2nI(X;U|A) such were generated), and it is the only codeword in its bin which is jointly typical with the \ufb01rst codeword (which the decoder already knows) and the side information sequence that it has generated and is observed at the decoder, since the size of each bin is no larger than \u22482n(I(X;U|A)\u2212I(X;U|Y,A)\u2212\u03b5) = 2n(I(Y ;U|A)\u2212\u03b5). For the reconstruction, the decoder now employs the mapping \u02c6 Xopt in a symbol-by-symbol fashion on the components of the pair consisting of the second codeword and the side information sequence. Converse: For the converse part, \ufb01x a scheme of rate \u2264R for a block of length n and consider: nR \u2265 H(T ) = H(T, An) = H(An) + H(T |An) \u2265 H(An) \u2212H(An|Xn) + H(T |An, Y n) \u2212H(T |Y n, An, Xn) = I(Xn; An) + I(Xn; T |An, Y n) \f7 = I(Xn; An) + H(Xn|An, Y n) \u2212H(Xn|An, Y n, T ). (16) Now I(Xn; An) + H(Xn|An, Y n) \u2265 H(Xn) \u2212H(Xn|An) + H(Xn, Y n|An) \u2212H(Y n|An) = H(Xn) \u2212H(Xn|An) + H(Xn|An) + H(Y n|An, Xn) \u2212H(Y n|An) = H(Xn) + H(Y n|An, Xn) \u2212H(Y n|An) = n X i=1 H(Xi) + H(Yi|Ai, Xi) \u2212H(Yi|Y i\u22121, An) (17) \u2265 n X i=1 H(Xi) + H(Yi|Ai, Xi) \u2212H(Yi|Ai) = n X i=1 H(Xi) \u2212I(Yi; Xi|Ai) = n X i=1 I(Xi; Ai) + H(Xi|Yi, Ai). (18) Combining (16) and (18) yields nR \u2265 n X i=1 I(Xi; Ai) + H(Xi|Yi, Ai) \u2212H(Xi|Xi\u22121, An, Y n, T ) (a) = n X i=1 I(Xi; Ai) + H(Xi|Yi, Ai) \u2212H(Xi|Yi, Ai, Ui) = n X i=1 I(Xi; Ai) + I(Xi; Ui|Yi, Ai), (19) where (a) follows by taking Ui = (An\\i, Y n\\i, Xi\u22121, T ). Noting that \u02c6 Xi = \u02c6 Xi(T, Y n) is a function of the pair (Ui, Yi), and the Markov relation Ui \u2212(Ai, Xi) \u2212Yi, the proof is now completed in the standard way upon considering the joint distribution of (X\u2032, A\u2032, U \u2032, Y \u2032, \u02c6 X\u2032) \u25b3 = (XJ, AJ, UJ, YJ, \u02c6 XJ), where J is randomly generated uniformly at random from the set {1, . . . , n}, independent of (Xn, An, U n, Y n, \u02c6 Xn), and noting that: PX\u2032 = PX, U \u2032 \u2212(A\u2032, X\u2032) \u2212Y \u2032, PY \u2032|X\u2032,A\u2032 = PY |X,A, (20) \u02c6 X\u2032 = \u02c6 X\u2032(U \u2032, Y \u2032), (21) E \" n X i=1 \u03c1(Xi, \u02c6 Xi) # = nE\u03c1(X\u2032, \u02c6 X\u2032), E \" n X i=1 \u039b(Ai) # = nE\u039b(A\u2032) (22) and 1 n n X i=1 I(Xi; Ai) + I(Xi; Ui|Yi, Ai) \u2265I(X\u2032; A\u2032) + I(X\u2032; U \u2032|Y \u2032, A\u2032), (23) where last inequality follows from item 2 in Lemma 1, which states that I(X; A) + I(X; U|Y, A) is convex over the set of distributions that satis\ufb01es (20). \f8 It is natural to wonder whether the characterization above remains valid when the choice of the actions is allowed to depend on the side information symbols generated thus far, that is, for the ith action to be of the form Ai = Ai(T, Y i\u22121). The converse in the proof above does not carry over to this case since the inequality H(Y n|An, Xn) \u2265Pn i=1 H(Yi|Ai, Xi), used in (17), may no longer hold. Whether the best achievable rate could, in general, be better (less) when allowing such schemes remains open. C. Actions taken by the decoder before the index is seen Consider the setting as in Figure 1, where the actions An are taken by the decoder before the index T is seen. In such a case An is independent of Xn. For this case, the rate distortion cost function is similar to R(I)(D, C) de\ufb01ned in the previous section, but with an additional constraint that A is independent of X. De\ufb01ne R(I) A\u22a5X(D, C) = min I(X; U|Y, A), (24) where the joint distribution of X, A, Y, U is of the form PX,A,U,Y (x, a, u, y) = PX(x)PA(a)PU|X,A(u|x, a)PY |X,A(y|x, a), (25) and the minimization is over all PA and PU|X,A under which E h \u03c1 \u0010 X, \u02c6 Xopt(U, Y ) \u0011i \u2264D, E [\u039b(A)] \u2264C, (26) where \u02c6 Xopt(U, Y ) denotes the best estimate of X based on U, Y , where U is an auxiliary random variable with a cardinality |U| \u2264|X||A| + 2. Theorem 2: The rate distortion cost function for the setting where actions taken by the decoder before the index is seen, is given by R(I) A\u22a5X(D, C). Proof: The proof is similar to the proof of Theorem 1, but taking into account that An is independent of Xn, and therefore Ai is independent of Xi. If the cost is unlimited, then the greedy policy is optimal, namely the decoder blindly chooses the action a minimizing RWZ(PX, PY |X,A=a, D), (27) and an optimal Wyner-Ziv code for the source PX and channel PY |X,A=a is employed. For the more general case, in the presence of a cost constraint, as can be expected and is straightforward to check, R(I) A\u22a5X(D, C) in (24) coincides with the minimum on the right hand side of (8). D. Examples 1) The Lossless Case: As a very special case of Theorem 1 we get that, in the absence of a cost constraint on the actions, the minimum rate needed for a near lossless reconstruction at the decoder is given by min I(X; A) + H(X|Y, A), (28) \f9 where the joint distribution of X, A, Y in (28) is of the form PX,A,Y (x, a, y) = PX(x)PA|X(a|x)PY |X,A(y|x, a), (29) and the minimization is over all PA|X. Letting RSW (PX, PY |X) denote the conditional entropy H(X|Y ) induced by the pair (PX, PY |X) (the subscript SW standing for \u2018Slepian-Wolf\u2019 [9]), it is natural to wonder whether the above minimum rate can be strictly better (smaller) than mina RSW (PX, PY |X,A=a), which is what would be achieved if the decoder greedily takes the one action leading to S.I. which is best in the sense of inducing lowest H(X|Y ), irrespective of any information from the encoder, and then proceeding as in Slepian-Wolf coding. The following is an example showing that this greedy strategy may be suboptimal. PSfrag replacements X \u223c \u001a 0 w.p. 1/2 1 w.p. 1/2 \u001b Encoder Decoder Vender R \u02c6 Xn A Y X 1 1 1 1 \u22121 0 0 0 0 1 \u2212\u03b4 1 \u2212\u03b4 \u03b4 \u03b4 Z channel S channel Fig. 3. An example of vending side information, where the action chooses between Z-channel and S-channel with parameter \u03b4. Consider the case X = A = Y = {0, 1} where X is a fair coin \ufb02ip, PY |X,A=0 is the Z-channel with crossover probability \u03b4 from 1 to 0, and PY |X,A=1 is the S-channel with crossover probability \u03b4 from 0 to 1. The setting is depicted in Figure 3. Symmetry implies that the PA|X minimizing I(X; A) + H(X|Y, A) satis\ufb01es PA|X(0|1) = PA|X(1|0), in other words, there is a BSC connecting X to A (or A to X). Assuming this BSC has crossover probability \u03b1, an elementary calculation yields I(X; A) + H(X|Y, A) = 1 \u2212h(\u03b1) + h \u0012 \u03b1\u03b4 1 \u2212\u03b1 + \u03b1\u03b4 \u0013 (1 \u2212\u03b1 + \u03b1\u03b4). (30) Thus, letting Rmin(\u03b4) denote the minimum in (28) for this scenario, Rmin(\u03b4) = min \u03b1\u2208[0,1] \u0014 1 \u2212h(\u03b1) + h \u0012 \u03b1\u03b4 1 \u2212\u03b1 + \u03b1\u03b4 \u0013 (1 \u2212\u03b1 + \u03b1\u03b4) \u0015 . (31) In contrast, the minimum rate achieved by a \u2018greedy\u2019 strategy which chooses actions without regard to the information from the encoder is given by the conditional entropy of the input given the output of the Z-channel(\u03b4) whose input is a fair coin \ufb02ip, namely Rgreedy(\u03b4) = h \u0012 \u03b4 1 + \u03b4 \u0013 1 + \u03b4 2 . (32) For example, elementary calculus shows that Rmin(1/2) is achieved by \u03b1\u2217= 2/5, assuming the value \u22480.678072, which is about a 1.5% improvement over Rgreedy(1/2) \u22480.688722. Figure 4 plots the difference between Rgreedy(\u03b4) and Rmin(\u03b4). In the presence of a cost constraint, Theorem 1 implies that the minimum rate needed for a near lossless reconstruction \f10 0.4 \u03b4 0.2 0.6 0.8 1 Rgreedy(\u03b4) \u2212Rmin(\u03b4) 0.006 0.008 0.01 0.004 0.002 Encoder Decoder Unknown Source Unknown Source General channel Denoiser Systematic Encoder Known Denoiser DMC Decoder Fig. 4. Plot of Rgreedy(\u03b4) \u2212Rmin(\u03b4) is given by the minimum in (28), with the additional constraint E\u039b(A) \u2264C. Let Rmin(\u03b4, C) denote this minimum for our present example, assuming cost 0 for using say the \ufb01rst Z-channel and 1 for using the second channel. Clearly Rmin(\u03b4, 0) = Rgreedy(\u03b4), Rmin(\u03b4, 1/2) = Rmin(\u03b4) and consequently, by a time-sharing argument, Rmin(\u03b4, C) \u22642CRmin(\u03b4) + (1 \u22122C)Rgreedy(\u03b4) 0 \u2264C \u22641/2. (33) As it turns out, the inequality in (33) is strict, i.e., in our example one can do better than time-sharing between the respective optimum schemes for the different costs (to the level allowed by the cost constraint). Figure 5 contains a plot of Rmin(1/2, C), which is seen to be better (lower) than the straight line represented by the right side of (33). 0.68 0.682 PSfrag replacements 0.2 C 0.4 0.1 0.3 0.5 Rmin(1/2, C) 0.684 0.686 0.01 0.004 0.002 Rgreedy(1/2) \u2192 Rmin(1/2) zn \u02c6 xn DMC Encoder Decoder Unknown Source Unknown Source General channel Denoiser Systematic Encoder Known Denoiser DMC Decoder Fig. 5. Plot of Rmin(1/2, C) as a function of the cost C. 2) The Lossy Case: Ternary Source and Binary Side Information of Unit Cost: Consider a ternary X taking values in {\u22121, 0, 1}, distributed according to X = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 w.p. 1/4 0 w.p. 1/2 \u22121 w.p. 1/4. (34) The actions are binary, taking values in {0, 1}, where action 0 corresponds to no S.I. while action 1 corresponds to obtaining a binary noisy measurement of X, taking values in {\u22121, 1}, which is the output of the following channel: \f11 X \u223c \uf8f1 \uf8f2 \uf8f3 1 w.p. 1/4 0 w.p. 1/2 \u22121 w.p. 1/4 \uf8fc \uf8fd \uf8fe Encoder Decoder Vender R \u02c6 Xn A Y X 1 1 \u22121 \u22121 0 0.5 0.5 Fig. 6. Ternary example. PY |X(1|1) = PY |X(\u22121|\u22121) = 1 and PY |X(1|0) = PY |X(\u22121|0) = 1/2. Suppose that there is a unit cost for obtaining such a noisy measurement of the source, i.e.: \u039b(a) = a, a \u2208{0, 1}. The conditional entropy of X given Y is 1 bit. Thus, lossless compression of X is achievable at a rate of 1 bit per source symbol at a cost of 1 per source symbol with a greedy decoder who chooses to observe the noisy source measurement of all symbols. Can one do better than this greedy policy? This rate is achievable at half the cost via the following scheme: the encoder uses one bit per source symbol to describe whether or not the symbol is 0. The decoder then needs to use the noisy measurement of the source only for those symbols that are not 0 (in which case the measurement will completely determine the source symbol). This corresponds to rate I(X; A) + H(X|Y, A) under PA|X(1|1) = PA|X(1| \u22121) = PA|X(0|0) = 1, which is readily veri\ufb01ed to be the minimum of achievable rates under a cost constraint of 1/2. In the lossy case, under Hamming distortion, we note that: \u2022 When the S.I. is available to both encoder and decoder (at no cost) the problem is reduced to one of lossy compression for the binary symmetric source, thus RX|Y (D) = 1 \u2212h(D). \u2022 This rate is achievable even when the S.I. is absent at the encoder, as can be seen by letting W be the output of a BSC(D) whose input Q(X) is the quantized version of X, de\ufb01ned by Q(0) = 0 and Q(1) = Q(\u22121) = 1, where W \u2212X \u2212Y . It is readily seen that the optimal estimate of X based on (W, Y ) satis\ufb01es P(X \u0338= \u02c6 X(W, Y )) = D and that I(X; W|Y ) = 1 \u2212h(D). Thus RWZ X|Y (D) = RX|Y (D) = 1 \u2212h(D). \u2022 RWZ X|Y (D) in the above item corresponds to a decoder that observes all of the S.I. symbols. Can the same performance be achieved with fewer observations? In other words, assuming unit cost per observation, can the same performance be achieved at a cost less than 1? We now argue that the same performance can be achieved at half the cost: letting, as before, A = 0 correspond to no observation and A = 1 correspond to an observation, consider a conditional distribution PA|X given by PA|X(1|1) = PA|X(1| \u22121) = 1 \u2212PA|X(1|0) = D and where U = A. Then I(X; A) + I(X; U|Y, A) = I(X; A) = H(A) \u2212H(A|X) = 1 \u2212h(D) and the optimal estimate of X based on (U, Y ) has P(X \u0338= \u02c6 X(U, Y )) = D. The cost here is P(A = 1) = 1/2. Evidently, the rate-distortion-cost function R(D, C) in (4) satis\ufb01es R(D, 1/2) \u22641 \u2212h(D) and in fact R(D, 1/2) = 1 \u2212h(D) since obviously R(D, 1/2) \u2265RWZ X|Y (D). Thus the rate 1 \u2212h(D) is achievable even if the decoder is allowed to access only half of the observations. \f12 3) Binary Action: To Observe or Not to Observe the S.I.: Consider a given source and side information distribution PX,Y . The action is to either observe the side information symbol or not, where an observation has unit cost. Thus 0 \u2264C \u22641 is a constraint on the fraction of side information symbols the decoder will be allowed to observe. Let us arbitrarily take A = {0, 1}, with A = 1 corresponding to observation of the side-information symbol and A = 0 to lack of it. Noting that the second mutual information term in (4) corresponds to Wyner-Ziv coding conditional on A, the specialization of Theorem 1 for this case gives R(D, C) = min PA|X:A\u2212X\u2212Y,P (A=1)=C,(1\u2212C)D0+CD1=D I(X; A) + R(PX|A=0, D0) \u00b7 P(A = 0) + RWZ(PX,Y |A=1, D1) \u00b7 P(A = 1) = min PA|X:A\u2212X\u2212Y,P (A=1)=C,(1\u2212C)D0+CD1=D I(X; A) + R(PX|A=0, D0) \u00b7 (1 \u2212C) + RWZ(PX,Y |A=1, D1) \u00b7 C, (35) where R(PX, D) denotes the rate distortion function of the source PX and RWZ(PX,Y , D) denotes the Wyner-Ziv rate distortion function when source and side information are distributed according to PX,Y . A very special case is when Y = X. Thus the action is either to observe the source symbol or not. Assuming a non-negative distortion measure satisfying min\u02c6 x \u03c1(x, \u02c6 x) = 0 for all x, (35) becomes R(D, C) = min PA|X:P (A=1)=C I(X; A) + R \u0012 PX|A=0, D 1 \u2212C \u0013 \u00b7 (1 \u2212C). (36) When X is a fair coin \ufb02ip and distortion is Hamming, (36) becomes (for D, C in the non-trivial region) R(D, C) = min PA|X:P (A=1)=C I(X; A) + Rb \u0012 PX=1|A=0, D 1 \u2212C \u0013 \u00b7 (1 \u2212C) = min PA|X:P (A=1)=C 1 \u2212[hb(PX=1|A=1)C + hb(PX=1|A=0)(1 \u2212C)] + hb \u0000PX=1|A=0 \u0001 \u2212hb \u0012 D 1 \u2212C \u0013 \u00b7 (1 \u2212C), = min PA|X:P (A=1)=C 1 \u2212hb(PX=1|A=1)C \u2212hb \u0012 D 1 \u2212C \u0013 \u00b7 (1 \u2212C), (a) = 1 \u2212C \u2212hb \u0012 D 1 \u2212C \u0013 \u00b7 (1 \u2212C), = Rb \u00121 2, D 1 \u2212C \u0013 \u00b7 (1 \u2212C) = RA\u22a5X(D, C) (37) where Rb(p, D) = [hb(p) \u2212hb(D)]+ is the rate distortion function of the Bernoulli(p) source and step (a) is due to the fact that \u2212hb(PX=1|A=1) is minimized (at the value \u22121) by taking A independent of X. To see that R(D, C) can be strictly smaller than RA\u22a5X(D, C) in the observe/not-observe binary action scenario, consider the case where X is a fair coin \ufb02ip and Y is the output of an erasure channel with erasure probability e (whose input is X). Recalling that RWZ(PX,Y , D) = eR(PX, D/e) when Y is the erased version of X (cf. [8], [10]), we specialize the \f13 right hand side of (35) for this case to obtain R(D, C) = min PA|X:A\u2212X\u2212Y,P (A=1)=C,(1\u2212C)D0+CD1=D 1 \u2212H(X|A) + Rb(PX|A=0(1), D0) \u00b7 (1 \u2212C) + eRb(PX|A=1(1), D1/e) \u00b7 C = min 1 \u2212 \u0014 hb \u0012 \u03b2 2(1 \u2212C) \u0013 (1 \u2212C) + hb \u00121 \u2212\u03b2 2C \u0013 C \u0015 + Rb \u0012 \u03b2 2(1 \u2212C), D \u2212CD1 1 \u2212C \u0013 (1 \u2212C) + eRb \u00121 \u2212\u03b2 2C , D1/e \u0013 C, (38) where the last minimum is over max{0, 1\u22122C} \u2264\u03b2 \u2264min{1, 2\u22122C} and 0 \u2264D1 \u2264min{D/C, e}. For the extreme points we get, as expected: R(D, 0) = Rb \u0000 1 2, D \u0001 and R(D, 1) = eRb \u0000 1 2, D/e \u0001 . Figure 7 plots the curve in (38) for D = 1/4, e = 1/2 and 0 \u2264C \u22641. PSfrag replacements 0.4 C 0.2 0.6 0.8 1 R \u0000D = 1 4, C \u0001 0.1 0.125 0.075 0.025 0.05 0.15 0.175 Encoder Decoder Vender T (Xn) \u22082nR \u02c6 Xi(Y i, T ) Ai(T ) Yi \u223cPYi|Ai,Xi Fig. 7. Rate distortion cost function R(D, C), 0 \u2264C \u22641, for the case where X is a fair coin \ufb02ip, Y its erased version where e = 1/2, C is fraction of places where decoder is allowed to observe S.I., and D = 1/4. In this case R(D, 0) = Rb ` 1 2, 1 4 \u00b4 \u22480.188722, R(D, 1) = eRb ` 1 2, D/e\u00b4 = 1 2Rb ` 1 2, 1 2 \u00b4 = 0. The strict concavity implies sub-optimality of time-sharing optimal schemes according to the available observation budget. E. Causal Decoder Side Information Consider the setting presented in Figure 8, which is similar to that described in Section II-A, the only difference being that the reconstruction is allowed causal dependence on the side information, i.e., to be of the form \u02c6 Xi = \u02c6 Xi(T, Y i) (motivation for why this might be interesting can be found in [12]). De\ufb01ne R(I) causal(D, C) = min I(X; U, A), (39) where the joint distributions of X, A, Y, U is of the form PX,A,U,Y (x, a, u, y) = PX(x)PA,U|X(a, u|x)PY |X,A(y|x, a), (40) \f14 PSfrag replacements Xn Encoder Decoder Vender T (Xn) \u22082nR \u02c6 Xi(Y i, T ) Ai(T ) Yi \u223cPYi|Ai,Xi Fig. 8. Rate distortion with causal side information vender at the decoder. and the minimization is over all PA,U|X under which E h \u03c1 \u0010 X, \u02c6 Xopt(U, Y ) \u0011i \u2264D, E [\u039b(A)] \u2264C, (41) where \u02c6 Xopt(U, Y ) denotes the best estimate of X based on U, Y , where U is an auxiliary random variable. The cardinality of U may be restricted to |U| \u2264|X||A|+2 as shown in item 3 Lemma 1. One can also denote U, A as \u02dc U, and an equivalent representation would be R(I) causal(D, C) = min I(X; \u02dc U), (42) where PX,A, \u02dc U,Y (x, a, \u02dc u, y) = PX(x)P \u02dc U|X(\u02dc u|x)1{a=f(\u02dc u)}PY |X,A(y|x, a). Theorem 3: The rate distortion cost function for the setting where actions taken by the decoder before the index is seen, is given by R(I) causal(D, C). Proof: Achievability: The achievability proof is based on the fact that the encoder and decoder generate a joint type PX,A using a rate that is I(X; A) + \u01eb, and since both the encoder and decoder know the sequence of actions an, they can time-share between |A| causal schemes such that if the action is a a rate I(X; U|a) + \u01eb would achieve the distortion constraint [12]. Hence, the total rate is I(X; A) + \u01eb + P PA(a)(I(X; U|a) + \u01eb) = I(X; A, U) + 2\u01eb. Converse: for the converse part, \ufb01x a scheme of rate R for a block of length n and consider: nR \u2265 H(T ) \u2265 I(Xn; T ) = n X i=1 H(Xi) \u2212H(Xi|Xi\u22121, T ) (a) = n X i=1 H(Xi) \u2212H(Xi|Xi\u22121, T, Y i\u22121) \u2265 n X i=1 H(Xi) \u2212H(Xi|T, Y i\u22121) (43) where step (a) is due to the Markov chain Xi \u2212(Xi\u22121, T ) \u2212Y i\u22121. Now let us denote \u02dc Ui := (T, Y i\u22121), and we obtain that R \u22651 nI(Xi; \u02dc Ui). (44) \f15 The proof is now completed in the standard way upon considering the joint distribution of (X\u2032, A\u2032, \u02dc U \u2032, Y \u2032, \u02c6 X\u2032) \u25b3 = (XJ, AJ, ( \u02dc UJ, J), YJ, \u02c6 XJ), where J is randomly generated uniformly at random from the set {1, . . ., n}, independent of (Xn, An, U n, Y n, \u02c6 Xn), and noting that: PX\u2032 = PX, \u02dc U \u2032 \u2212(A\u2032, X\u2032) \u2212Y \u2032, PY \u2032|X\u2032,A\u2032 = PY |X,A, (45) \u02c6 X\u2032 = \u02c6 X\u2032( \u02dc U \u2032, Y \u2032), A\u2032 = f( \u02dc U \u2032), (46) E \" n X i=1 \u03c1(Xi, \u02c6 Xi) # = nE\u03c1(X\u2032, \u02c6 X\u2032), E \" n X i=1 \u039b(Ai) # = nE\u039b(A\u2032) (47) and 1 n n X i=1 I(Xi; \u02dc Ui) = I(X\u2032; \u02dc U \u2032). (48) F. Indirect Rate Distortion with Action-Dependent Side Information PSfrag replacements Xn Zn Encoder Decoder Vender T (Zn) \u22082nR \u02c6 Xn(Y n, T ) An(T ) Y n PZ|X Fig. 9. Indirect rate distortion with a side information vender at the decoder. The source Xn is i.i.d. PX and the encoder observes a noisy version of the source, Zn, where the pairs (Xi, Zi) are i.i.d.\u223cPX,Z. Side information is generated as the output of the channel PY |X,Z,A in response to the noise-free, noisy, and action sequences (Xn, Zn, An), where the action sequence An is generated on the basis of the index from the encoder. Consider the case shown in Figure 9 where, rather than the source X, the encoder observes a noisy version of it, Z. The decoder, based on the index conveyed to it from the encoder, will then select an action sequence that will result in the side information Y , as output from the channel PY |X,Z,A. The reconstruction, as before, will be a function of the index and the side information. Speci\ufb01cally, a scheme in this setting for blocklength n and rate R is characterized by an encoding function T : Zn \u2192{1, 2, . . ., 2nR}, an action strategy f : {1, 2, . . ., 2nR} \u2192An, and a decoding function g : {1, 2, . . ., 2nR} \u00d7 Yn \u2192\u02c6 X n that operate as follows: \u2022 The source n-tuple Xn is i.i.d.\u223cPX goes through a DMC PZ|X to yield its noisy observation sequence Zn. Thus, overall the clean and noisy source are characterized by a given joint distribution PX,Z \u2022 Encoding: based on Zn give index T = T (Zn) to the decoder \u2022 Decoding: \u2013 given the index, choose an action sequence An = f(T ) \f16 \u2013 the side information Y n will be the output of the memoryless channel PY |X,Z,A whose input is (Xn, Zn, An) \u2013 let \u02c6 Xn = g(T, Y n) The rate-distortion-cost for this case is now de\ufb01ned similarly as in subsection II-A. Let us denote it by RID(D, C), the subscript standing for \u2018indirect\u2019. Theorem 1 is generalized to this case as follows: Theorem 4: RID(D, C) is given by RID(D, C) = min [I(Z; A) + I(Z; U|Y, A)] , (49) where the joint distribution of X, Z, A, Y, U is of the form PX,Z,A,U,Y (x, z, a, u, y) = PX,Z(x, z)PA,U|Z(a, u|z)PY |X,Z,A(y|x, z, a), (50) and the minimization is over all PA,U|Z under which E h \u03c1 \u0010 X, \u02c6 Xopt(U, Y ) \u0011i \u2264D, E [\u039b(A)] \u2264C, (51) where \u02c6 Xopt(U, Y ) denotes the best estimate of X based on U, Y , and U is an auxiliary random variable whose cardinality is bounded as |U| \u2264|Z||A| + 2. Proof outline: The achievability part is very similar to the original. The random generation of the scheme is performed in the same way, with the noisy source replacing the original noise-free source. This guarantees that (Zn, An, U n, Y n) are, with high probability, jointly typical. The joint typicality also with Xn, namely the joint typicality of (Xn, Zn, An, U n, Y n), then follows from an application of the Markov lemma. The converse part also follows similarly to the one from the noise-free case: that nR \u2265I(Zn; An) + H(Zn|An, Y n) \u2212H(Zn|An, Y n, T ) (52) follows identically as in (16) by replacing Xn by Zn. That I(Zn; An) + H(Zn|An, Y n) \u2265 n X i=1 I(Zi; Ai) + H(Zi|Yi, Ai) (53) follows similarly as (18) by replacing Xn with Zn, upon noting that H(Y n|An, Zn) = Pn i=1 H(Yi|Ai, Zi), which follows from the Markov relation (Xi, Yi)\u2212(Ai, Zi)\u2212(An\\i, Zn\\i, Y i\u22121) (which a fortiori implies Yi\u2212(Ai, Zi)\u2212(An\\i, Zn\\i, Y i\u22121)). Combining (52) and (53) now yields nR \u2265 n X i=1 I(Zi; Ai) + I(Xi; Ui|Yi, Ai) (54) similarly as in Step (a) in (19) upon de\ufb01ning Ui = (An\\i, Y n\\i, Zi\u22121, T ). The proof of the converse is concluded by verifying that: \u2022 \u02c6 Xi = \u02c6 Xi(T, Y n) is a function of the pair (Ui, Yi) \u2022 the Markov relation Xi \u2212Zi \u2212(Ai, Ui) holds (which follows from Xi \u2212Zi \u2212(Zn, Y n\\i)) \u2022 the Markov relation Ui \u2212(Xi, Zi, Ai) \u2212Yi holds (which follows from (Zn, Y n\\i) \u2212(Xi, Zi, Ai) \u2212Yi) and invoking the convexity of the informational rate distortion function de\ufb01ned on the right hand side of (49), which is \f17 established similarly as in Lemma 1. III. SIDE INFORMATION VENDING MACHINE AT THE ENCODER In this section we consider the setting where the action sequence An is chosen at the encoder and the side information is available at the decoder and possibly at the encoder too. The setting is depicted in Figure 2. Speci\ufb01cally, a communication scheme in this setting for blocklength n and rate R is characterized by an action strategy f : X n \u2192An, (55) an encoding function T : X n \u00d7 Yn \u2192{1, 2, . . ., 2nR} (when side information is available at the encoder), T : X n \u2192{1, 2, . . ., 2nR} (when side information is not available at the encoder), and a decoding function g : {1, 2, . . ., 2nR} \u00d7 Yn \u2192\u02c6 X n. (56) As in the case where the actions were chosen by the decoder, the side information Y n will be the output of the memoryless channel PY |X,A whose input is (Xn, An). Furthermore, a triple (R, D, C) is said to be achievable if for all \u03b5 > 0 and suf\ufb01ciently large n there exists a scheme as above for blocklength n and rate R + \u03b5 satisfying both E \" n X i=1 \u03c1(Xi, \u02c6 Xi) # \u2264n(D + \u03b5) (57) and E \" n X i=1 \u039b(Ai) # \u2264n(C + \u03b5). (58) The rate distortion (and cost) function Re(D, C) (The letter e stands for encoder) is de\ufb01ned as Re(D, C) = inf{R\u2032 : the triple (R\u2032, D, C) is achievable}. (59) The general case remains open, however we present here a characterization of three important cases: lossless case (where Pr(Xn = \u02c6 Xn) \u21921), Gaussian case (where Y = A + X + N and X and N are independent Gaussian random variables), and a case where the Markov form Y \u2212A \u2212X holds. In all three cases, Re(D, C) is independent of whether or not the S.I. is available at the encoder. A. Lossless case Here we consider the lossless case, namely, for any \u01eb > 0 there exists an n such that Pr(Xn = \u02c6 Xn) > 1 \u2212\u01eb. De\ufb01ne R(I) e (C) = min [H(X|A, Y ) + I(X; A) \u2212I(Y ; A)] , (60) where PX and PY |A,X are determined by the problem setting and the minimization is over PA|X such that E [\u039b(A)] \u2264C. The term [H(X|A, Y ) + I(X; A) \u2212I(Y ; A)] is convex in PA|X since the term \u2212I(Y ; A, X) is convex in PA|X and the \f18 following identity holds H(X|A, Y ) + I(X; A) \u2212I(Y ; A) = H(X|A, Y ) + H(X) \u2212H(X|A) \u2212H(Y ) + H(Y |A) = H(X) \u2212I(X; Y |A) \u2212H(Y ) + H(Y |A) = H(X) \u2212H(Y |A) + H(Y |A, X) \u2212H(Y ) + H(Y |A) = H(X) \u2212I(Y ; A, X) (61) Let us denote the minimum (operational) rate that is needed to reconstruct the source at the encoder losslessy where with a cost of the action less than C as Re(C). Theorem 5: For the setting in Figure 2 where the actions are chosen by the encoder and the side information Y is known to the decoder and may or may not be known to the encoder the minimum rate that is needed to reconstruct the source under a cost constraint C is given by Re(C) = R(I) e (C). (62) Proof: Achievability : The achievability proof is divided into two cases according to the sign of the term I(X; A) \u2212I(Y ; A). In the \ufb01rst case we assume I(X; A) \u2212I(Y ; A) > 0 and we use a coding scheme that is based on Wyner-Ziv coding [15] for rate distortion theory where side information known at the decoder. In the second case, we assume I(Y ; A) \u2212I(X; A) > 0 and we use a coding scheme that is based on Gel\u2019fand-Pinsker coding [6] for channel with states where the state is known to the encoder. First case I(X; A) \u2212I(Y ; A) > 0 : We \ufb01rst generate a codebook of sequences of actions An that covers Xn; hence, the size of the codebook needs to be 2n(I(A;X)+\u01eb), where \u01eb > 0. Then, similarly to Wyner-Ziv coding scheme [15], we bin the codebook into 2n(I(A;X)\u2212I(Y ;A)+2\u01eb) bins such that into each bin we have 2n(I(Y ;A)\u2212\u01eb) codebooks. Similarly to Wyner-Ziv scheme, we look in the codebook for a sequence An that is jointly typical with Xn and transmit the number of the bin that contains the sequence. The decoder receives the bin number and looks which of the sequences of An in the bin that its number is received are jointly typical with the side information Y n. Similar to the analysis in Wyner-ziv setting, with high probability there will be only one codeword that is jointly typical with Y n (The Markov form that is needed in the analysis of Wyner-ziv setting is not needed here, since the side information Y n is generated according to PY |A,X and therefore if (An, Xn) are jointly typical then with high probability the triple (An, Xn, Y n) would also be jointly typical). In the \ufb01nal step the encoder uses a Slepian-Wolf scheme for transmitting Xn losslessy to the decoder that has side information (Y n, An); hence additional rate of H(X|Y, A) is needed. Second case I(Y ; A) \u2212I(X; A) > 0 : First we notice that the expression in (62) can be written as H(X|A, Y ) \u2212 (I(Y ; A) \u2212I(X; A)). The actions can be considered as input to a channel with states where the output of the channels is Y and the state is X and the conditional probability of the channel is PY |X,A. The capacity of this channel is achieved by Gel\u2019fand-Pinsker coding scheme [6] and is given as I(Y ; A) \u2212I(X; A). In addition the Gel\u2019fand-Pinsker coding scheme induces a triple (Xn, Y n, An) that is jointly typical. Hence, we can use the message in order to reduce the needed rate \f19 H(X|Y, A) as in Slepian-Wolf scheme to H(X|A, Y ) \u2212(I(Y ; A) \u2212I(X; A)). Converse: for the converse part, \ufb01x a scheme of rate R for a block of length n with a probability of error Pr(Xn \u0338= Xn) = P (n) e and consider: nR \u2265 H(T ) \u2265 H(T |Y n) = H(Xn, T |Y n) \u2212H(Xn|T, Y n) (a) \u2265 H(Xn, T |Y n) \u2212n\u01ebn (b) = H(Xn, An|Y n) \u2212n\u01ebn = H(Xn, An, Y n) \u2212H(Y n) \u2212n\u01ebn = H(Xn) + H(An|Xn) + H(Y n|An, Xn) \u2212H(Y n) \u2212n\u01ebn (c) \u2265 n X i=1 H(Xi) + H(Yi|Ai, Xi) \u2212H(Yi) \u2212n\u01ebn \u2265 min [n (H(X) + H(Y |A, X) \u2212H(Y ))] \u2212n\u01ebn (63) where \u01ebn = P (n) e log |X|+ 1 n and step (a) follows Fano\u2019s inequality. Step (b) follows the fact that An and T are deterministic functions of Xn. Step (c) follows the following four relations: P(xn) = Qn i=1 P(xi), P(yn|an, xn) = Qn i=1 P(yi|ai, xi) H(An|Xn) = 0, and H(Y n) \u2264Pn i=1 H(Yi). The minimization in the last step is over all conditional distribution PA|X that satisfy the cost constrain, namely E [\u039b(A)] \u2264C, and the inequality follows from the fact that the expression H(X) + H(Y |A, X) \u2212H(Y ) is convex in PA|X for \ufb01xed PX and PY |A,X. The converse proof is completed by invoking the fact that since R is an achievable rate there exists a sequence of codes at rate R such that \u01ebn \u21920. We have seen that, in the absence of a cost constraint on the actions, the minimum rate needed for a near lossless reconstruction at the decoder is given by min I(X; A) + H(X|Y, A) \u2212I(Y ; A) (64) (regardless of whether or not the side information is present at the encoder). Thus, I(Y ; A) represents the saving in rate relative to the case where the actions are taken by the decoder (recall (28) for the minimum rate at that case). To see that this can be signi\ufb01cant, recall the example X = A = Y = {0, 1}, where X is a fair coin \ufb02ip, PY |X,A=0 is the Z-channel with crossover probability \u03b4 from 1 to 0, and PY |X,A=1 is the S-channel with crossover probability \u03b4 from 0 to 1. It is easily seen that in this case I(X; A) + H(X|Y, A) \u2212I(Y ; A) = 0 and so, a fortiori, the minimum in (64) is zero. That the source can be reconstructed losslessly with zero rate in this case is equally easy to see from an operational standpoint, since taking actions Ai = Xi ensures that Yi = Xi with probability one. B. Gaussian Case Here we consider the case where \f20 \u2022 the source has a Gaussian distribution with zero mean and variance \u03c32 X, i.e., X \u223cN(0, \u03c32 X), (65) \u2022 the relation between Y, X, A is given by Y = X + A + N, (66) where N is a random variable independent of (A, X) and has a Gaussian distribution with zero mean and variance \u03c32 N, i.e., W \u223cN(0, \u03c32 N), (67) \u2022 the distortion is a mean square error distortion, i,e, E hPn i=1(Xi \u2212\u02c6 Xi)2i and it has to be less than D \u2022 the cost of the actions is E \u0002Pn i=1 A2 i \u0003 and has to be less than C. Without loss of generality, we assume that C = \u03b12\u03c32 X where \u03b1 > 0. Theorem 6: For the Gaussian setting of Figure 2, as described above, Re(D, C) = ( 1 2 log h \u03c32 N (1+ \u221a C/\u03c3X)2\u03c32 X+\u03c32 N \u00b7 \u03c32 X D i if [(1 + \u221a C/\u03c3X)2\u03c32 X + \u03c32 N] \u00b7 D \u2264\u03c32 X\u03c32 N 0 otherwise. (68) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.5 1 1.5 2 2.5 3 PSfrag replacements R(D, C) D R(D, C = 0) R(D, C = 0.3) R(D, C = 0.6) R(D, C = 1) Fig. 10. Re(D, C) in the Gaussian case, for \u03c32 X = \u03c32 N = 1. The boundary of the region where Re(D, C) = 0 is the curve D = 1 (1+ \u221a C)2+1. Indeed, this distortion level can be achieved with zero rate by estimating X on the basis of Y = X + N/(1 + \u221a C). Before proving Theorem 6 we would like to point out that the state ampli\ufb01cation problem [16], [17] is tangent to the vending side information problem described here. In the state ampli\ufb01cation problem, the goal is to design a communication scheme for a channel with i.i.d. states sequence, Sn, which is known to the encoder. The purpose of the scheme is to send a message through the channel, and at the same time to describe to the decoder the state sequence Sn. The case where there is no message to send, namely, the input to the channel is used only to describe the state sequence, is equivalent to the problem presented here when Re(D, C) = 0, namely, we when are using only the actions to describe the source and no additional message is sent. If Re(D, C) = 0, we obtain from (68) that for the Gaussian source coding problem, the \f21 minimum mean square error satis\ufb01es D \u2265 \u03c32 X\u03c32 N (\u03c3X + \u221a C)2 + \u03c32 N , (69) a result that was also obtained in [17, Theorem 2], where the channel is the Gaussian channel and the goal is to describe the state sequence with minimum mean square error distortion. Proof of Theorem 6: Achievability: The encoder chooses the actions to be A = \u03b1X and then it uses a coding for the Gaussian Wyner-Ziv with side information at the decoder [13]. The side information satis\ufb01es Y = X + A + N = (1 + \u03b1)X + N, which is equivalent to having a side information Y = X + N (1+\u03b1). Denote by N \u2032 = N 1+\u03b1. Using the Gaussian Wyner-Ziv result, a rate R = 1 2 log \u03c32 N \u2032 \u03c32 X + \u03c32 N \u2032 \u03c32 X D = 1 2 log \u03c32 N (1 + \u03b1)2\u03c32 X + \u03c32 N \u03c32 X D , (70) is achievable. Converse: We prove the converse in two steps. First we derive the lower bound Re(D, C) \u2265 min PA|X,P \u02c6 X|X,A,Y I(X; \u02c6 X) \u2212I(Y ; X, A), (71) which holds for any PX and PY |A,X (not necessarily Gaussian), and then we evaluate it for the Gaussian case. Fix a scheme at rate R for a block of length n and consider nR \u2265 H(T ) \u2265 H(T |Y n) \u2265 I(Xn; T |Y n) (a) = I(Xn; Y n, T ) \u2212I(Xn; Y n) (b) \u2265 I(Xn; \u02c6 Xn) \u2212I(Xn, An; Y n) (c) \u2265 n X i=1 I(Xi; \u02c6 Xi) \u2212I(Xi, Ai; Yi), (72) where (a) follows from [14, Lemma3.2], which asserts that for arbitrary random variables I(X, Z; Y ) = I(Z; Y )+I(X; Y |Z). Step (b) follows from the facts that \u02c6 Xn is a determinstic function of the pair (T, Y n), and An is a deterministic function of Xn. Step (c) follows from the facts that H(Xn) = Pn i=1 H(Xi), H(Y n|An, Xn) = Pn i=1 H(Yi|Ai, Xi) and conditioning reduces entropy. Since the expression in (72) is convex in PA|X, P \u02c6 X|X for \ufb01xed PX and PY |A,X, we obtain the lower bound Re(D, C) \u2265min I(X; \u02c6 X) \u2212I(Y ; X, A), (73) and the minimization is over conditional distributions PA|X, P \u02c6 X|X that satisfy the distortion and cost constraints. Now we evaluate the lower bound for the Gaussian case. I(X; \u02c6 X) \u2212I(Y ; X, A) = H(X) \u2212H(X| \u02c6 X) \u2212H(Y ) + H(Y |A, X) = H(X) \u2212H(X| \u02c6 X) \u2212H(Y ) + H(N) \f22 (a) \u2265 1 2 log 2\u03c0e\u03c32 X \u22121 2 log 2\u03c0eD \u22121 2 log 2\u03c0e((1 + \u03b1)2\u03c32 X + \u03c32 N)) + 1 2 log 2\u03c0e\u03c32 N = 1 2 log \u03c32 N (1 + \u03b1)2\u03c32 X + \u03c32 N \u03c32 X D , (74) where inequality (a) follows from the fact that H(X| \u02c6 X) \u2264H(X \u2212\u02c6 X) \u2264 1 2 log 2\u03c0eD (because of the constraint that E(X \u2212\u02c6 X)2 \u2264D) and H(Y ) \u22641 2 log 2\u03c0e\u03c32 Y , where \u03c32 Y = E[X2] + 2E[AX] + E[A2] + E[N 2] \u2264 E[X2] + 2 p E[X2]E[A2] + E[A2] + E[N 2] \u2264 \u03c32 X + 2\u03b1\u03c32 X + \u03b12\u03c32 X + \u03c32 N = \u03c32 X(1 + \u03b1)2 + \u03c32 N. (75) C. Markov Form Y-A-X Here we consider the case where the Markov form X \u2212A \u2212Y holds. Theorem 7: The rate distortion (and cost) function Re(D, C) for the setting in Figure 2 when PY |A,X(y|a, x) = PY |A(y|a) satis\ufb01es Re(D, C) = h min(I(X; \u02c6 X) \u2212I(A; Y )) i+ , (76) where [\u00b7]+ denotes min{\u00b7, 0} and the minimization is over joint distributions of the form PA,X,Y, \u02c6 X(a, x, y, \u02c6 x) = PX(x)PA|X(a|x)PY |A(y|a)P \u02c6 X|X(\u02c6 x|x) satisfying E\u03c1(X, \u02c6 X) \u2264D and E\u039b(A) \u2264C. It is interesting to note that the solution is the difference between a rate-distortion expression minP \u02c6 X|X I(X; \u02c6 X) and channel capacity expression maxPA I(A; Y ) . I.e., Re(D, C) = [R(PX, D) \u2212Cap(PY |A, C)]+, (77) where Cap(PY |A, C) denotes the capacity of the channel PY |A under a cost-constraint C. proof of Theorem 7: Achievability: Design a regular rate distortion code, which needs a rate larger than I(X; \u02c6 X), and then transmits part of the rate through the channel which has an input A and output Y . Therefore the total rate that is needed to be transmitted through the index T (Xn) is the difference I(X; \u02c6 X) \u2212I(A; Y ). Converse: We invoke the lower bound given in (73) and obtain Re(D, C) \u2265 I(X; \u02c6 X) \u2212I(Y ; X, A) = I(X; \u02c6 X) \u2212I(Y ; A), (78) where the last equality is due to the Markov form X \u2212A \u2212Y . \f23 D. Upper and Lower Bounds for the General Case 1) Achievable Rates: \u2022 Absence of S.I. at Encoder: For the setting of Figure 2 with an open switch, i.e., when the encoder has no access to the S.I., the following is an achievable rate: I(U; X|A, Y ) + I(X; A) \u2212I(Y ; A) (79) under any joint distribution of the form PX(x)PA|X(a|x)PY |X,A(y|x, a)PU|X,A(u|x, a) such that E\u03c1(X, \u02c6 Xopt(A, Y, U)) \u2264D and E\u039b(A) \u2264C. The argument for why this rate is achievable is similar to that given in Subsection III-A for why the right side of (62) is achievable, the difference being that the H(X|A, Y ) term in (62), corresponding to Slepian-Wolf coding of Xn conditioned on An, is replaced by I(U; X|A, Y ), corresponding to Wyner-Ziv coding conditioned on An. \u2022 S.I. Available at Encoder: For the setting of Figure 2 with a closed switch, i.e., when the encoder has access to the S.I., the following is an achievable rate: I( \u02c6 X; X|A, Y ) + I(X; A) \u2212I(Y ; A) (80) under any joint distribution of the form PX(x)PA|X(a|x)PY |X,A(y|x, a)P \u02c6 X|X,A,Y (\u02c6 x|x, a, y) such that E\u03c1(X, \u02c6 X) \u2264D and E\u039b(A) \u2264C. The argument for why this rate is achievable is similar to that for why the right side of (80) is achievable, the difference being that the I(U; X|A, Y ) terms, corresponding to Wyner-Ziv coding conditioned on An, is replaced by I( \u02c6 X; X|A, Y ), corresponding to standard rate distortion coding conditioned on An, Y n. 2) Lower Bound on Achievable Rate: As pointed out in Subsection III-B, the proof of the converse part of Theorem 6 is valid for the general case (i.e., beyond the Gaussian scenario), and shows that the rate needed to achieve distortion D at cost C, regardless of whether or not S.I. is available at the encoder, is at least as large as I(X; \u02c6 X) \u2212I(Y ; X, A) (81) for some joint distribution of the form PX(x)PA|X(a|x)PY |X,A(y|x, a)P \u02c6 X|X,A,Y (\u02c6 x|x, a, y) satisfying the distortion and cost constraints. It is worthwhile to note that this rate was shown to be achievable for the three special cases considered in the previous three subsections. Indeed, this fact was shown explicitly for the cases of Subsection III-B and Subsection III-C, and in the lossless case (81) becomes H(X) \u2212I(Y ; X, A) = H(X|A, Y ) + I(A; X) \u2212I(A; Y ), \f24 which coincides with the expression on the right hand side of (62). To see that the lower bound in (81) may not be tight in general, even when the S.I. is available at the encoder, consider the standard case of rate distortion coding with S.I. available to both encoder and decoder. In this case A is degenerate, so the right hand side of (81) reduces to I(X; \u02c6 X) \u2212I(Y ; X, A) = H(X|Y ) \u2212H(X| \u02c6 X) (82) while the tight lower bound on the achievable rate for this scenario is well-known to be given by I(X; \u02c6 X|Y ) = H(X|Y ) \u2212H(X| \u02c6 X, Y ), (83) which may be strictly larger than the expression in (82). IV. SUMMARY AND OPEN QUESTIONS We have studied source coding in the presence of side information, when the system can take actions that affect the availability, quality, or nature of the side information. We have given a full characterization of the rate-distortion-cost tradeoff when the actions are taken by the decoder. For the case where the actions are taken by the encoder, we have characterized this tradeoff in a few important special cases, while providing upper and lower bounds on the achievable rate for the general case. The most signi\ufb01cant question left open by our work is a full characterization of the rate-distortion-cost tradeoff for the setting of actions taken at the encoder (beyond the special cases considered here), with S.I. that may or may not be available at the encoder (Figure 2). Another question left open, for the setting of actions taken by the decoder, is whether the rate distortion cost tradeoff can be improved when each action is allowed to depend on the side information symbols generated thus far, that is, when the ith action is allowed to be of the form Ai = Ai(T, Y i\u22121) (rather than Ai(T )). ACKNOWLEDGEMENT We are grateful to Paul Cuff for suggesting the title of this paper and for helpful discussions." + } + ], + "Taeho Yoon": [ + { + "url": "http://arxiv.org/abs/2205.11093v1", + "title": "Accelerated Minimax Algorithms Flock Together", + "abstract": "Several new accelerated methods in minimax optimization and fixed-point\niterations have recently been discovered, and, interestingly, they rely on a\nmechanism distinct from Nesterov's momentum-based acceleration. In this work,\nwe show that these accelerated algorithms exhibit what we call the merging path\n(MP) property; the trajectories of these algorithms merge quickly. Using this\nnovel MP property, we establish point convergence of existing accelerated\nminimax algorithms and derive new state-of-the-art algorithms for the\nstrongly-convex-strongly-concave setup and for the prox-grad setup.", + "authors": "TaeHo Yoon, Ernest K. Ryu", + "published": "2022-05-23", + "updated": "2022-05-23", + "primary_cat": "math.OC", + "cats": [ + "math.OC" + ], + "main_content": "Introduction Minimax optimization problems of the form minimize x\u2208X maximize y\u2208Y L(x, y) (1) have recently received increased attention in many machine learning applications including robust optimization [86], adversarial training [55], fair machine learning [29], and GANs [32]. Consequently, there has been a large body of research on ef\ufb01cient algorithms for solving minimax optimization. For the deterministic smooth convex-concave setup, the classical extragradient (EG) [47] and the optimistic gradient descent (OG) [78, 80, 21] methods converge with O(1/k) rate on the squared gradient norm [85, 33], and accelerated algorithms [100, 49, 92] with O(1/k2) rate, based on an anchoring mechanism retracting iterates toward the initial point, were recently discovered. In a different line of work on the \ufb01xed-point problem \ufb01nd z\u2208Rd z = \ud54b(z), (2) the Halpern iteration [35], which averages the iterates with the initial point, was shown to achieve an accelerated O(1/k2) rate on the squared \ufb01xed-point residual norm when executed with optimal parameters [51, 43]. Interestingly, both the Halpern iteration and the accelerated minimax algorithms rely on the anchoring mechanism, but a precise theoretical explanation of such apparent similarity was, to the best of our knowledge, yet missing. Contribution. In this work, we identify the merging path (MP) property among anchoring-based algorithms, which state that the trajectories of these algorithms quickly merge, and use this novel property to develop new accelerated algorithms. First, we show that known accelerated minimax algorithms [100, 49, 92] have O(1/k2)-merging paths with the optimized Halpern iteration [35, 51] and thereby identify their point convergence. Second, we present a new accelerated minimax algorithm for smooth strongly-convex-strongly-concave setup designed to approximate, in the MP sense, the OC-Halpern method, which has an optimal accelerated rate. Third, we present a nearoptimal accelerated proximal gradient type algorithm, designed to approximate, in the MP sense, the Halpern-accelerated Douglas-Rachford splitting [27, 53]. Preprint. Under review. arXiv:2205.11093v1 [math.OC] 23 May 2022 \f\u22122.0 \u22121.5 \u22121.0 \u22120.5 0.0 3.0 3.1 3.2 3.3 3.4 3.5 initial point AGM, a=2 AGM, a=4 AGM, a=6 AGM, a=8 (a) Divergent paths of AGM (\u03b1 = 0.025) \u22122.0 \u22121.5 \u22121.0 \u22120.5 0.0 2.95 3.00 3.05 3.10 3.15 3.20 3.25 initial point iteration 10 iteration 20 iteration 50 iteration 100 iteration 1000 FEG OHM APS EAG (b) Merging paths of anchored methods (\u03b1 = 0.1) Figure 1: Trajectories of (left) Nesterov\u2019s accelerated algorithms (AGM [69, 8, 11]; described in Appendix A.2) with distinct momentum parameters and (right) anchoring-based algorithms. On the right, paths quickly merge and become indistinguishable in 50 iterations. All algorithms are executed on the convex minimzation problem with f(x1, x2) = 4x2 1 x2 , starting at (x1, x2) = (\u22122, 3). 1.1 The merging path (MP) property Let A be a deterministic algorithm. Write A(x0; P) = (x0, x1, x2, . . . ) to denote that A applied to problem P with starting point x0 produces iterates x1, x2, . . . . We say two algorithms A1 and A2 have O(r(k))-merging paths if for any problem P and x0 \u2208Rd, the iterates A\u2113(x0; P) = \u0000x0, x(\u2113) 1 , x(\u2113) 2 , . . . \u0001 for \u2113= 1, 2 satisfy \r \rx(1) k \u2212x(2) k \r \r2 = O(r(k)). More concisely, we say A1 and A2 are O(r(k))-MP if they have O(r(k))-merging paths. The MP property precisely formalizes a notion of near-equivalence of algorithms. In particular, it is a stronger notion of approximation compared to the conceptual ones of prior work [63, 1]. Classical \ufb01rst-order algorithms for smooth convex minimization are not O(r(k))-MP with r(k) \u21920. Figure 1 shows that accelerated gradient methods with different momentum coef\ufb01cients have divergent paths, even though their function values (of course) converge to the same optimal value. In this paper, we show that the accelerated minimax algorithms for smooth convex-concave minimax optimization are O(1/k2)-MP (which is faster than the o(1)-point convergence). Figure 1b shows that the algorithms\u2019 paths indeed merge far before they converge to the limit. 1.2 Related work An overview of convex-concave minimax optimization. The extragradient (EG) [47] and Popov\u2019s algorithm [78], also known as optimistic gradient (OG) descent [80, 21], have initiated the long stream of research in the optimization literature on the closely interconnected topics of minimax optimization [68, 38, 46, 61, 63, 64, 5, 36], variational inequalities [74, 71, 73, 42, 9, 10, 59, 56, 16, 40, 57], and monotone inclusion problems [87, 93, 65, 66, 54]. A number of advances in minimax optimization have been made in the context of practical machine learning applications, including the multi-player game dynamics [20, 88, 31, 39] and GANs [97, 30, 14, 50, 61, 85, 77]. A recent trend of the \ufb01eld focuses on achieving near-optimal ef\ufb01ciency with respect to primal-dual gap or distance to the optimum for convex-concave minimax problems with distinct strong convexity, strong concavity and smoothness parameters for each variable [91, 52, 95, 2, 99, 41]. Proximal point methods and proximal approximation approaches. The proximal point method (PPM) [67, 60, 83] is an implicit algorithm for solving convex minimization, minimax optimization, and, more generally, monotone inclusion problems. Although PPM is not always directly applicable, it has been used as the conceptual basis for developing many algorithms with approximate proximal evaluations [82, 34, 87, 68, 37, 13, 28, 23, 79]. Several prior works [63, 1] interpreted prominent forward algorithms such as Nesterov\u2019s accelerated gradient method (AGM), EG, and OG as conceptual approximations of PPM. Acceleration with respect to gradient norm. Since the seminal work of Nesterov [69], acceleration of \ufb01rst-order algorithms has been extensively studied for convex minimization problems 2 \f[70, 8, 44, 6, 75, 89]. Lately, there has been a rapid progress in the discovery of accelerated algorithms with respect to the gradient magnitude of the objective for smooth convex problems [45, 72, 26, 48, 101] and smooth convex-concave minimax problems [24, 100, 49, 92]. Acceleration of PPM has been achieved for convex minimization [34] and more recently for monotone inclusions in terms of the \ufb01xed-point residual [43, 76]. Notably, the latter has been shown [76, 84] to be an instance of the Halpern iteration [35] with optimal parameters [51]. While the accelerated minimax algorithms bear an apparent resemblance to the Halpern iteration [24, 92], the latest results with optimal rates [100, 49, 92] were not formally understood through this connection. Composite minimax problems and prox-grad-type algorithms. The particular case of bilinearlycoupled minimax problems, L(x, y) = f(x) + \u27e8x, Ay\u27e9\u2212g(y) for some convex functions f and g and a linear operator A, have been studied by many researchers. A line of works [70, 96, 90] considered the setups where f and g are smooth and one can only access their gradients. Another line of works, more closely connected to the setup we study in Section 5, considered the cases when f or g is proximable, and used primal-dual splitting algorithms to solve those problems [12, 18, 94, 15, 38, 46, 98, 84]. Primal-dual problem setups can often be reformulated into a more \ufb02exible format of composite monotone inclusion problems and addressed via forward-backward splitting algorithms [93, 17, 18, 56, 19, 58], which also extend the prox-grad-type methods including the iterative shrinkage-thresholding algorithms (ISTA) [22, 8]. Our result of Section 5 can be understood as acceleration of forward-backward algorithms with respect to the forward-backward residual, which generalizes the gradient magnitude. For the special case of the projected-gradient setup, the prior work [24] has also achieved a near-optimal acceleration. 2 Background and preliminaries Convex-concave functions and minimax optimization. A function L(\u00b7, \u00b7) : X \u00d7 Y \u2192R is convex-concave if X and Y are convex sets, L(x, y) is convex as a function of x for all y, and L(x, y) is concave as a function of y for all x. A point (x\u22c6, y\u22c6) \u2208X \u00d7 Y is a saddle point, or (minimax) solution of (1) if L(x\u22c6, y) \u2264L(x\u22c6, y\u22c6) \u2264L(x, y\u22c6) for all (x, y) \u2208X \u00d7 Y. Operators and monotonicity. An operator \ud538on Rd, denoted \ud538: Rd \u21d2Rd, is a set-valued function, i.e., \ud538(z) \u2286Rd for all z \u2208Rd. For simplicity, we often write \ud538z = \ud538(z). If \ud538z contains exactly one element for all z \u2208Rd, then we say \ud538is single-valued and view it as a function. An operator \ud538: Rd \u21d2Rd is monotone if \u27e8\ud538z \u2212\ud538z\u2032, z \u2212z\u2032\u27e9\u22650, \u2200z, z\u2032 \u2208Rd, where the notation means that \u27e8u \u2212v, z \u2212z\u2032\u27e9\u22650 for any u \u2208\ud538z and v \u2208\ud538z\u2032. For \u00b5 > 0, an operator \ud538is \u00b5-strongly monotone if \u27e8\ud538z \u2212\ud538z\u2032, z \u2212z\u2032\u27e9\u2265\u00b5\u2225z \u2212z\u2032\u22252, \u2200z, z\u2032 \u2208Rd. The graph of \ud538is denoted Gra \ud538= {(z, u) | u \u2208\ud538z}. A monotone operator \ud538is maximally monotone if there is no monotone operator \ud538\u2032 such that Gra \ud538\u2282Gra \ud538\u2032 strictly. In particular, if \ud538is monotone, single-valued, and continuous as a function, then it is maximally monotone [7, Corollary 20.28]. De\ufb01ne (\ud538+ \ud539)(z) = {u + v \u2208Rd | u \u2208\ud538z, v \u2208\ud539z}, (\u03b1\ud538)(z) = {\u03b1u \u2208Rd | u \u2208\ud538z} for operators \ud538and \ud539and scalar \u03b1 \u2208R. The inverse of an operator \ud538is the operator \ud538\u22121 : Rd \u21d2Rd de\ufb01ned by \ud538\u22121(u) = {z \u2208Rd | u \u2208\ud538z}. Monotone inclusion and \ufb01xed-point problems. In a monotone inclusion problem, \ufb01nd z\u2208Rd 0 \u2208\ud538(z), (3) where \ud538: Rd \u21d2Rd is monotone, one \ufb01nds zeros of an operator \ud538. For any \u03b1 > 0, we have 0 \u2208\ud538(z) \u21d0 \u21d2z \u2208(\ud540+ \u03b1\ud538)(z) \u21d0 \u21d2z \u2208(\ud540+ \u03b1\ud538)\u22121(z). Therefore, (3) is equivalent to the \ufb01xed-point problem (2) with \ud54b= (\ud540+ \u03b1\ud538)\u22121 = \ud541\u03b1\ud538. We call \ud541\u03b1\ud538the resolvent of \u03b1\ud538. We say \ud54b: Rd \u2192Rd is nonexpansive if it is 1-Lipschitz continuous and contractive if it is \u03c1-Lipschitz with \u03c1 < 1. If \ud538is maximally monotone, then \ud541\u03b1\ud538has full domain [62] and is nonexpansive [83] for any \u03b1 > 0. Denoting Zer \ud538= {z \u2208Rd | 0 \u2208\ud538(z)} and Fix \ud54b= {z \u2208Rd | z = \ud54b(z)}, we have Zer \ud538= Fix \ud541\u03b1\ud538. We call the algorithm zk+1 = \ud541\u03b1\ud538(zk) the proximal point method (PPM) for \ud538. 3 \fMinimax optimization as monotone inclusion problems. For differentiable and convex-concave L: Rn \u00d7 Rm \u2192R, de\ufb01ne \u2207\u00b1L(x, y) = (\u2207xL(x, y), \u2212\u2207yL(x, y)) and call it the saddle operator of L. Then \u2207\u00b1L is monotone [82], and z\u22c6= (x\u22c6, y\u22c6) is a solution of (1) if and only if \u2207\u00b1L(z\u22c6) = 0. Therefore, a convex-concave minimax optimization problem (1) can be reformulated as a monotone inclusion problem (3) with \ud538= \u2207\u00b1L and as a \ufb01xed-point problem (2) with \ud54b= \ud541\u2207\u00b1L. The \ufb01xed-point residual norm \u2225z \u2212\ud54bz\u2225quanti\ufb01es the rate of convergence of an algorithm for \ufb01nding \ufb01xed points. When \ud54b= \ud541\u03b1\ud539, we have z \u2212\ud54bz = z \u2212\ud541\u03b1\ud539(z) \u2208\u03b1\ud539(\ud541\u03b1\ud539(z)), because u = \ud541\u03b1\ud539(z) = (\ud540+ \u03b1\ud539)\u22121(z) \u21d0 \u21d2z \u2208(\ud540+ \u03b1\ud539)(u) = u + \u03b1\ud539u. Thus, when \ud54b= \ud541\u03b1\ud539and \ud539is single-valued, a convergence rate on \u2225z \u2212\ud54bz\u2225is equivalent to a convergence rate on \u2225\ud539(\u00b7)\u2225. 3 Accelerated algorithms for smooth minimax optimization are O(1/k2)-MP In this section, we prove the O(1/k2)-MP property among accelerated minimax algorithms and the optimized Halpern iteration. The problem setup is: minimax problems with convex-concave L such that \u2207\u00b1L is L-Lipschitz. For notational convenience, write \ud539= \u2207\u00b1L. Our MP result provides new insight into these acceleration mechanisms and allows us to establish their point convergence. Preliminaries: Accelerated minimax and proximal algorithms. Recently, several accelerated minimax algorithms with rate \u2225\ud539zk\u22252 = O(1/k2) have been proposed. The list includes the extra anchored gradient (EAG) algorithm [100] zk+1/2 = \u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b1\ud539zk zk+1 = \u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b1\ud539zk+1/2, the fast extragradient (FEG) algorithm [49] zk+1/2 = \u03b2kz0 + (1 \u2212\u03b2k) (zk \u2212\u03b1\ud539zk) zk+1 = \u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b1\ud539zk+1/2, and the anchored Popov\u2019s scheme (APS) [92], with v0 = z0, vk+1 = \u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b1\ud539vk zk+1 = \u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b1\ud539vk+1. Prior to the development of these minimax algorithms, the optimized Halpern\u2019s method (OHM) [51, 43] was presented for solving the \ufb01xed-point problem (2) with nonexpansive operator \ud54b: Rd \u2192 Rd: wk+1/2 = \u03b2kw0 + (1 \u2212\u03b2k)wk wk+1 = \ud54b(wk+1/2), (4) where \u03b2k = 1 k+1. OHM converges with rate \u2225wk+1/2 \u2212\ud54bwk+1/2\u22252 \u22644\u2225w0 \u2212w\u22c6\u22252/(k + 1)2, if a \ufb01xed point w\u22c6exists. To clarify, the prior works [100, 92] study further general forms of EAG and APS with iterationdependent step-sizes. For the sake of simplicity, we consider the case of constant step-sizes and defer the discussion on varying step-sizes to Appendix A and F. MP property among accelerated algorithms. The accelerated forward algorithms, EAG, FEG, and APS, do not merely resemble OHM in their form. The algorithms are, in fact, near-equivalent in the sense that they have quickly merging paths. Theorem 1 (EAG\u2248FEG\u2248APS\u2248OHM). Let \ud539: Rd \u2192Rd be monotone and L-Lipschitz and assume z\u22c6\u2208Zer \ud539= Fix \ud541\u03b1\ud539exists. Let \ud54b= \ud541\u03b1\ud539and \u03b2k = 1 k+1. Then FEG and OHM are O(\u2225z0 \u2212z\u22c6\u22252/k2)-MP for \u03b1 \u2208 \u00000, 1 L \u0001 , and there exists \u03b7 > 0 such that EAG, APS, and OHM are O(\u2225z0 \u2212z\u22c6\u22252/k2)-MP for \u03b1 \u2208 \u00000, \u03b7 L \u0001 . 4 \fProof. We provide the proof for the FEG case and defer the proofs for EAG and APS to Appendix B.2. Let {wk}k=0,1,... be the OHM iterates and {zk}k=0,1,... the FEG iterates, and assume z0 = w0. Then zk+1 \u2212wk+1 = \u0000\u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b1\ud539zk+1/2 \u0001 \u2212(\u03b2kw0 + (1 \u2212\u03b2k)wk \u2212\u03b1\ud539wk+1) = (1 \u2212\u03b2k)(zk \u2212wk) + \u03b1 \u0000\ud539wk+1 \u2212\ud539zk+1/2 \u0001 where we used (\ud540+ \u03b1\ud539)(wk+1) = wk+1/2 in the \ufb01rst equality and z0 = w0 in the second. Thus, \u2225zk+1 \u2212wk+1\u22252 = (1 \u2212\u03b2k)2 \u2225zk \u2212wk\u22252 + 2 \u03b1(1 \u2212\u03b2k)(zk \u2212wk), \ud539wk+1 \u2212\ud539zk+1/2 \u000b + \u03b12 \r \r\ud539wk+1 \u2212\ud539zk+1/2 \r \r2 . (5) Now use the similar identity zk+1/2 \u2212wk+1 = (1 \u2212\u03b2k)(zk \u2212wk) \u2212\u03b1(1 \u2212\u03b2k)\ud539zk + \u03b1\ud539wk+1 to rewrite the inner product term in (5) as 2 \u03b1(1 \u2212\u03b2k)(zk \u2212wk), \ud539wk+1 \u2212\ud539zk+1/2 \u000b = 2 \u03b1(zk+1/2 \u2212wk+1) + \u03b12(1 \u2212\u03b2k)\ud539zk \u2212\u03b12\ud539wk+1, \ud539wk+1 \u2212\ud539zk+1/2 \u000b \u22642\u03b12 (1 \u2212\u03b2k)\ud539zk \u2212\ud539wk+1, \ud539wk+1 \u2212\ud539zk+1/2 \u000b , where the last inequality follows from monotonicity of \ud539. Combining this with (5), we obtain \u2225zk+1 \u2212wk+1\u22252 \u2264(1 \u2212\u03b2k)2 \u2225zk \u2212wk\u22252 + 2\u03b12 (1 \u2212\u03b2k)\ud539zk \u2212\ud539wk+1, \ud539wk+1 \u2212\ud539zk+1/2 \u000b + \u03b12 \r \r\ud539wk+1 \u2212\ud539zk+1/2 \r \r2 = (1 \u2212\u03b2k)2 \u2225zk \u2212wk\u22252 \u2212\u03b12 \u2225\ud539wk+1\u22252 + 2\u03b12(1 \u2212\u03b2k)\u27e8\ud539zk, \ud539wk+1\u27e9 | {z } \u2264\u03b12(1\u2212\u03b2k)2\u2225\ud539zk\u22252 \u22122\u03b12(1 \u2212\u03b2k)\u27e8\ud539zk, \ud539zk+1/2\u27e9+ \u03b12\u2225\ud539zk+1/2\u22252. Plugging in \u03b2k = 1 k+1 and multiplying both sides by (k + 1)2, we obtain (k + 1)2 \u2225zk+1 \u2212wk+1\u22252 \u2264k2 \u2225zk \u2212wk\u22252 + \u03b12\u2225k\ud539zk \u2212(k + 1)\ud539zk+1/2\u22252. The conclusion then follows from the following Lemma 2. Lemma 2. Let \ud539: Rd \u2192Rd be monotone and L-Lipschitz and assume z\u22c6\u2208Zer \ud539exists. Let {zk}k=0,1,... be the iterates of FEG with \u03b2k = 1 k+1. For \u03b1 \u2208 \u00000, 1 L \u0001 , \u221e X k=0 \u2225k\ud539zk \u2212(k + 1)\ud539zk+1/2\u22252 \u2264 1 \u03b12(1 \u2212\u03b12L2)\u2225z0 \u2212z\u22c6\u22252 < \u221e. Proof outline of Lemma 2. We show Vk = \u03b1k2 2 \u2225\ud539zk\u22252 + k\u27e8\ud539zk, zk \u2212z0\u27e9+ 1 2\u03b1\u2225z0 \u2212z\u22c6\u22252 satisfy Vk \u2212Vk+1 \u2265\u03b1(1 \u2212\u03b12L2) 2 \u2225k\ud539zk \u2212(k + 1)\ud539zk+1/2\u22252 and Vk \u22650 for k = 0, 1, . . . (details presented in Appendix B.1). The conclusion follows by taking N \u2192\u221ein \u03b1(1\u2212\u03b12L2) 2 PN k=0 \u2225k\ud539zk \u2212(k+1)\ud539zk+1/2\u22252 \u2264V0 \u2212VN+1 \u2264V0 = 1 2\u03b1\u2225z0 \u2212z\u22c6\u22252. As it is well known [7, Theorem 30.1] that OHM converges to \u03a0Fix\ud54b(z0) = argmin z\u2208Fix \ud54b \u2225z \u2212z0\u2225, Theorem 1 immediately implies point convergence of EAG, FEG, and APS. Corollary 3. In the setup of Theorem 1, iterates of EAG, FEG, and APS converge to \u03a0Zer \ud539(z0). 5 \f4 Fastest algorithm for smooth strongly-convex-strongly-concave minimax optimization with respect to gradient norm In Section 3, we identi\ufb01ed that prior accelerated minimax algorithms approximate, in the MP sense, the optimal proximal algorithm OHM. In this section, we design a novel algorithm to approximate, in the MP sense, the optimal proximal algorithm OC-Halpern [76] and thereby achieve the state-ofthe-art rate for minimax problems with L-smooth \u00b5-strongly-convex-strongly-concave L (so \u2207\u00b1L is L-Lipschitz and \u00b5-strongly monotone). Later in Section 5, we repeat the strategy of designing an algorithm to approximate, in the MP sense, a known accelerated proximal algorithm. We design the novel algorithm Strongly Monotone Extra Anchored Gradient+1 zk+1/2 = \u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b7k\u03b1\ud539zk zk+1 = \u03b2kz0 + (1 \u2212\u03b2k)zk \u2212\u03b1\ud539zk+1/2, (SM-EAG+) where \ud539= \u2207\u00b1L, \u03b2k = 1 Pk j=0(1+2\u03b1\u00b5)j , \u03b7k = 1\u2212\u03b2k 1+2\u03b1\u00b5, and 0 < \u03b1 \u2264 \u221a L2+\u00b52+\u00b5 L2 , to approximate OC-Halpern in the MP sense. SM-EAG+ inherits the accelerated rate of OC-Halpern and achieves the fastest known rate in the setup with respect to the gradient norm, improving upon the prior rates of EG and OG by factors of 8 and 4, respectively. (OC-Halpern is not a forward algorithm as it uses the proximal operator \ud541\u03b1\ud539, while SM-EAG+ is a forward algorithm using evaluations of \ud539.) Preliminaries: Proximal algorithm with exact optimal complexity. For the \ufb01xed-point problem (2) with a \u03b3\u22121-contractive \ud54b, the recently presented OC-Halpern [76, Corollary 3.3, Theorem 4.1], which has the same form (4) as OHM but with \u03b2k = \u0010Pk j=0 \u03b32j\u0011\u22121 , achieves the exact optimal accelerated rate of \u2225wk+1/2 \u2212\ud54bwk+1/2\u22252 \u2264 \u00001 + \u03b3\u22121\u00012 \u0010Pk j=0 \u03b3j\u0011\u22122 \u2225w0 \u2212w\u22c6\u22252, where w\u22c6\u2208 Fix \ud539. We consider OC-Halpern with \ud54b= \ud541\u03b1\ud539, which is contractive if \ud539is maximally monotone and \u00b5-strongly monotone [7, Proposition 23.8, 23.13]. MP property and convergence analysis of SM-EAG+. SM-EAG+ approximates OC-Halpern in the MP sense. Lemma 4 (SM-EAG+\u2248OC-Halpern). Let \ud539: Rd \u2192Rd be \u00b5-strongly monotone and L-Lipschitz with 0 < \u00b5 \u2264L. Let z\u22c6be the zero of \ud539. If \u03b1 \u2208 \u00000, \u221a L2+\u00b52+\u00b5 L2 \u0001 , then for any \u03f5 \u2208(0, 1), SM-EAG+ and OC-Halpern with \ud54b= \ud541\u03b1\ud539and \u03b2k = 1 Pk j=0(1+2\u03b1\u00b5)j are O \u0010 \u2225z0\u2212z\u22c6\u22252 (1+2\u03b1\u00b5(1\u2212\u03f5))k \u0011 -MP. To clarify, in Lemma 4, we view \ud541\u03b1\ud539as a \u03b3\u22121-contractive operator with \u03b3\u22121 = 1 \u221a1+2\u03b1\u00b5 > 1 1+\u03b1\u00b5, where 1 1+\u03b1\u00b5 is the tight contraction factor for \ud541\u03b1\ud539[7, Proposition 23.13], and then apply OC-Halpern. This slack, which is negligible in the regime L/\u00b5 \u2192\u221e, is necessary for proving the MP result. SM-EAG+ inherits the convergence rate of OC-Halpern, since the paths of OC-Halpern and SMEAG+ merge at rate O \u0010 \u2225z0\u2212z\u22c6\u22252 (1+2\u03b1\u00b5(1\u2212\u03f5))k \u0011 for any \u03f5 > 0, arbitrarily close to the order of convergence \u2225\u03b1\ud539wk+1\u22252 = O \u0010 \u2225w0\u2212w\u22c6\u22252 (1+2\u03b1\u00b5)k \u0011 of OC-Halpern [76]. On the other hand, EG and OG can be viewed as approximations of the proximal point method (PPM) [63], which is slower than OC-Halpern. Additionally, it is unclear whether EG and OG exhibit an MP property to PPM fast enough to preserve PPM\u2019s convergence rate since EG and OG have no anchoring mechanism inducing the MP property. Theorem 5 (Fast rate of SM-EAG+). Let \ud539: Rd \u2192Rd be \u00b5-strongly monotone and L-Lipschitz with 0 < \u00b5 \u2264L. Let z\u22c6be the zero of \ud539. For \u03b1 \u2208 \u00000, \u221a L2+\u00b52+\u00b5 L2 \u0003 , then SM-EAG+ exhibits the rate \u2225\ud539zk\u22252 \u2264 \u0000\u221a1 + 2\u03b1\u00b5 + 1 \u00012 \u03b12 \u0010Pk\u22121 j=0(1 + 2\u03b1\u00b5) j 2 \u00112 \u2225z0 \u2212z\u22c6\u22252, for k = 1, 2, . . . . 1The \u201c+\u201d follows the nomenclature of [25], where the two-time-scale EG was named EG+. 6 \fWith the choice \u03b1 = \u221a L2+\u00b52+\u00b5 L2 , Theorem 5 implies \u2225\ud539zk\u22252 = O \u0010 e\u22122k \u00b5 L L2\u2225z0 \u2212z\u22c6\u22252\u0011 as L/\u00b5 \u2192\u221e. Therefore, SM-EAG+ requires roughly L \u00b5 log L\u2225z0\u2212z\u22c6\u2225 \u03f5 iterations to achieve \u2225\ud539zk\u2225\u2264 \u03f5. On the other hand, simultaneous gradient descent-ascent has the rate [7, Proposition 26.16] \u2225\ud539zk\u22252 \u2264L2\u2225zk \u2212z\u22c6\u22252 = O \u0012 e\u2212k \u00b52 L2 L2\u2225z0 \u2212z\u22c6\u22252 \u0013 as L/\u00b5 \u2192\u221e, and therefore requires 2L2 \u00b52 log L\u2225z0\u2212z\u22c6\u2225 \u03f5 \u226b L \u00b5 log L\u2225z0\u2212z\u22c6\u2225 \u03f5 iterations to achieve \u2225\ud539zk\u2225\u2264\u03f5. EG and OG has the rate [63, 5]: \u2225\ud539zk\u22252 \u2264L2\u2225zk \u2212z\u22c6\u22252 = O \u0010 e\u2212k \u00b5 4L L2\u2225z0 \u2212z\u22c6\u22252\u0011 , as L/\u00b5 \u2192\u221e, and therefore requires 8L \u00b5 log L\u2225z0\u2212z\u22c6\u2225 \u03f5 iterations to achieve \u2225\ud539zk\u2225\u2264\u03f5. Thus, SM-EAG+ is 8\u00d7 faster than EG and 4\u00d7 faster than OG (since OG makes a single call to the gradient oracle at each iteration). As a \ufb01nal remark, when \u00b5 = 0, SM-EAG+ coincides with FEG [49], which has the fastest known rate of \u2225\ud539zk\u22252 \u22644L2\u2225z0\u2212z\u22c6\u22252 k2 when \ud539is (non-strongly) monotone and L-Lipschitz. This rate is precisely recovered by Theorem 5 when \u00b5 = 0, with \u03b1 = 1 L. 5 Near-optimal prox-grad-type accelerated minimax algorithm Consider the setup with minimax objective of the form L(x, y) = Lp(x, y) + Ls(x, y), where Ls is L-smooth. Informally assume that Lp is nonsmooth and proximable (we clarify this notion below). We reformulate the minimax optimization problem into the monotone inclusion problem \ufb01nd z\u2208Rd 0 \u2208(\u2202\u00b1Lp + \u2207\u00b1Ls)(z), (6) where \u2202\u00b1 is the analog of \u2207\u00b1 for nonsmooth convex-concave functions (precisely de\ufb01ned in Section 5.1). We design the Anchored Proximal Gradient* (APG*): zk = SM-EAG+ (\ud540+ \u03b1\ud539\u2212\u03bek, \u03bek, \u03f5k) \u03bek+1 = \u03b2kz0 + (1 \u2212\u03b2k) (\ud541\u03b1\ud538(zk \u2212\u03b1\ud539zk) + \u03b1\ud539zk) , (APG*) where \ud538= \u2202\u00b1Lp, \ud539= \u2207\u00b1Ls, \u03be0 \u2208Rd is the initial point, and SM-EAG+ (\ud540+ \u03b1\ud539\u2212\u03bek, \u03bek, \u03f5k) denotes the execution of SM-EAG+ with the 1-strongly monotone (1 + \u03b1L)-smooth operator z 7\u2192 (\ud540+ \u03b1\ud539)(z) \u2212\u03bek with \u03bek as the initial point using the step-size \u221a (1+\u03b1L)2+1+1 (1+\u03b1L)2 until reaching the tolerance \u2225(\ud540+ \u03b1\ud539)(\u00b7) \u2212\u03bek\u2225\u2264\u03f5k. APG* is a proximal-gradient-type algorithm; each iteration, APG* uses a single proximal operation \ud541\u03b1\u2202\u00b1Lp and multiple (logarithmically many) gradient operations \u2207\u00b1Ls. (The superscript * in the name APG* indicates the multiple gradient evaluations per iteration.) Thus, APG* is useful when Lp is proximable in the sense that \ud541\u03b1\u2202\u00b1Lp can be computed ef\ufb01ciently. APG* is designed to approximate the accelerated proximal algorithm OHM-DRS, described soon, in the MP sense and thereby achieves a near-optimal accelerated rate. 5.1 Preliminaries: Subdifferential operators and forward-backward residual If Lp : Rn \u00d7 Rm \u2192R is a \ufb01nite-valued, not necessarily differentiable convex-concave function, then its (saddle) subdifferential operator \u2202\u00b1Lp : Rn \u00d7 Rm \u21d2Rn \u00d7 Rm de\ufb01ned by \u2202\u00b1Lp(x, y) = {(v, w) \u2208Rn \u00d7 Rm | v \u2208\u2202xLp(x, y), w \u2208\u2202y(\u2212Lp)(x, y)} is maximal monotone [7, Theorem 35.1, Corollary 37.5.2], where \u2202x and \u2202y respectively denote the convex subdifferential with respect to the xand y-variables. However, in order to encompass a broader range of problems, such as constrained minimax optimization problems, it is necessary 7 \fto consider Lp taking values in R \u222a{\u00b1\u221e}. In this case, \u2202Lp is maximal monotone if Lp is closed and proper [81], which we precisely de\ufb01ne and detail in Appendix G. For now, we simply state that most convex-concave saddle functions considered in practice are closed and proper, and if that holds, z\u22c6is a solution of (1) if and only if 0 \u2208\u2202\u00b1L(z\u22c6). Therefore, in our setup with L = Lp + Ls, we rewrite (1) into (6), using the fact that \u2202\u00b1 = \u2207\u00b1 for differentiable convex-concave functions. Of particular interest are problems with L(x, y) = f(x) + \u27e8Ax, y\u27e9\u2212g(y), with some A \u2208Rm\u00d7n, or, more generally, L(x, y) = f(x) + Ls(x, y) \u2212g(y), with some L-smooth convex-concave Ls. In this case, \u2202\u00b1Lp(x, y) = (\u2202f(x), \u2202g(y)), and \ud541\u03b1\u2202\u00b1Lp(x, y) = (Prox\u03b1f(x), Prox\u03b1g(y)), where Prox\u03b1f(x) = argmin x\u2032\u2208Rn \u001a \u03b1f(x\u2032) + 1 2\u2225x\u2032 \u2212x\u22252 \u001b , Prox\u03b1g(y) = argmin y\u2032\u2208Rm \u001a \u03b1g(y\u2032) + 1 2\u2225y\u2032 \u2212y\u22252 \u001b are the proximal operators. When f(x) = \u03b4X (x) = \u001a0 x \u2208X +\u221e x / \u2208X , the proximal operator becomes the projection operator: Prox\u03b1\u03b4X (x) = \u03a0X (x) for any \u03b1 > 0. Therefore, the constrained problem (1) with X \u2282Rn, Y \u2282Rm is recovered with Lp(x, y) = \u03b4X (x) \u2212\u03b4Y(y). Note that z\u22c6is a solution of (6) if and only if z\u22c6= \ud541\u03b1\ud538(z\u22c6\u2212\u03b1\ud539z\u22c6). Therefore, the norm of the forward-backward residual \ud53e\u03b1(z) = 1 \u03b1(z \u2212\ud541\u03b1\ud538(z \u2212\u03b1\ud539z)) serves as a measure for the convergence of algorithms solving (6), and it generalizes the operator norm and the \ufb01xed-point residual norm. 5.2 Preliminaries: Accelerated proximal splitting algorithm Consider wk = \ud541\u03b1\ud539(uk) uk+1 = \u03b2ku0 + (1 \u2212\u03b2k)(\ud541\u03b1\ud538(wk \u2212\u03b1\ud539wk) + \u03b1\ud539wk) (OHM-DRS) with \u03b2k = 1 k+2, which is the application of OHM to the Douglas\u2013Rachford splitting (DRS) operator [27, 53] de\ufb01ned by \ud54bDRS = \ud540\u2212\ud541\u03b1\ud539+\ud541\u03b1\ud538\u25e6(2\ud541\u03b1\ud539\u2212\ud540). Since \ud54bDRS is nonexpansive [53], OHM-DRS exhibits the accelerated rate \u2225\u03b1\ud53e\u03b1(wk)\u22252 = \u2225uk \u2212\ud54bDRS(uk)\u22252 = O(1/k2). (Here we use the alternative representation of OHM uk+1 = \u03b2ku0 + (1 \u2212\u03b2k)\ud54buk with \u03b2k = 1 k+2, which is equivalent to (4) under the identi\ufb01cation uk = wk+1/2. We clarify some details in Appendix A.6). OHM-DRS will serve as a conceptual algorithm (not implemented) that our APG* (implementable) is designed to O(1/k2)-MP approximate, so that APG* inherits the accelerated rate of OHM-DRS. (OHM-DRS is not prox-grad-type, while APG* is.) 5.3 MP property and convergence of APG* We now describe the design of APG*. The forms of APG* and OHM-DRS bear a clear resemblance. The second lines are identical. In the \ufb01rst lines, the resolvent computation \ud541\u03b1\ud539of OHM-DRS is replaced in APG* by an inner loop with SM-EAG+. Note, z = \ud541\u03b1\ud539(\u03bek) if and only if it solves \ufb01nd z\u2208Rd 0 = z + \u03b1\ud539z \u2212\u03bek = (\ud540+ \u03b1\ud539\u2212\u03bek)(z), (7) and \ud540+ \u03b1\ud539\u2212\u03bek is a 1-strongly monotone, (1 + \u03b1L)-Lipschitz operator. Therefore, (7) is an instance of the problem class of Section 4, and SM-EAG+ can ef\ufb01ciently solve it to accuracy \u2225(\ud540+\u03b1\ud539)(z)\u2212\u03bek\u2225\u2264\u03f5k in O(log(1/\u03f5k)) iterations. Therefore the zk-iterate of APG* approximates the wk = \ud541\u03b1\ud539(uk)-iterate of OHM-DRS. Theorem 6 (APG*\u2248OHM-DRS). With \u03b1 \u2208 \u00000, 1 L \u0001 , \u03b2k = 1 k+2, and \u03f5k = 1+L\u22121\u2225\ud539\u03be0\u2225 (k+1)2(k+2), APG* is O(1/k2)-MP to OHM-DRS; speci\ufb01cally, if \u03bek, zk are the APG* iterates and uk, wk are the iterates of OHM-DRS, and if \u03be0 = u0, then max \b \u2225\u03bek \u2212uk\u22252, \u2225zk \u2212wk\u22252\t \u2264 C(\u03be0)2 L2(k + 1)2 , for k = 0, 1, . . . , where C(\u03be0) = L(\u2225\u03be0 \u2212\u03be\u22c6\u2225+ 1) + \u2225\ud539\u03be\u22c6\u2225and \u03be\u22c6= \u03a0Fix (\ud54bDRS)(\u03be0). 8 \fCorollary 7 (Fast rate of APG*). With the conditions and notations of Theorem 6, \u03bek \u2192\u03be\u22c6and zk \u2192\ud541\u03b1\ud539(\u03be\u22c6) \u2208Zer (\ud538+ \ud539) as k \u2192\u221e. Additionally, \u2225\ud53e\u03b1(zk)\u22252 \u2264(3 + \u03b1L)2C(\u03be0)2 \u03b12L2(k + 1)2 , for k = 0, 1, . . . . Corollary 8 (Oracle complexity of APG*). With the conditions and notations of Theorem 6, the kth inner loop of SM-EAG+ in APG* requires O (log (k (\u2225\u03be0 \u2212\u03be\u22c6\u2225+ 1))) oracle calls of \ud539for k = 0, 1, . . . . This implies that the total computational cost needed to achieve \u2225\ud53e\u03b1(zk)\u2225\u2264\u03f5 is O \u0012C(\u03be0) \u03f5 \u0012 C\ud538+ C\ud539log C(\u03be0) \u03f5 \u0013\u0013 , where C\ud538, C\ud539are upper bounds on the respective costs of evaluating \ud541\u03b1\ud538and \ud539. Discussion. We clarify some key differences between APG* and the prior work of Diakonikolas [24, Algorithm 3]. First, APG* is a more general proximal-gradient-type algorithm, while the prior algorithm of Diakonikolas is a projected-gradient-type algorithm. Second, APG* is an anytime algorithm, as it requires no a priori speci\ufb01cation of the terminal accuracy or the total iteration count, while the prior algorithm is not an anytime algorithm, as it requires the terminal accuracy \u03f5 to be pre-speci\ufb01ed. Third, APG* has O \u0000 C\ud538 \u03f5 + C\ud539 \u03f5 log 1 \u03f5 \u0001 total cost (hiding dependency on \u03be0, \u03be\u22c6and L), while the prior algorithm has O \u0000 C\ud538+C\ud539 \u03f5 log 1 \u03f5 \u0001 total cost, as its inner loops require projections (corresponding to \ud541\u03b1\ud538). Therefore, APG* offers a speedup for constrained minimax problems when the projection represents the dominant cost, i.e., when C\ud538\u226bC\ud539. The initial point \u03bek for the inner loop SM-EAG+ (\ud540+ \u03b1\ud539\u2212\u03bek, \u03bek, \u03f5k) of APG* is a crude choice. However, while a more careful warm starting may improve the iteration complexity by a constant factor, we do not believe it alone can eliminate the O(log(1/\u03f5))-factor. In our analysis of APG* presented in Appendix D, we induce the MP property by solving the kth inner loop up to accuracy \u03f5k satisfying P\u221e k=0 k\u03f5k < \u221e. Therefore, unless a mechanism for \ufb01nding a warm-starting point z satisfying \u2225(\ud540+ \u03b1\ud539)(z) \u2212\u03bek\u2225= O(\u03f5k) = o(1/k) is proposed (which seems nontrivial), the O(log(1/\u03f5))-factor will persist. Eliminating the logarithmic factor probably requires a new additional insight, and we leave it as a future work. 6" + }, + { + "url": "http://arxiv.org/abs/2202.11910v1", + "title": "Robust Probabilistic Time Series Forecasting", + "abstract": "Probabilistic time series forecasting has played critical role in\ndecision-making processes due to its capability to quantify uncertainties. Deep\nforecasting models, however, could be prone to input perturbations, and the\nnotion of such perturbations, together with that of robustness, has not even\nbeen completely established in the regime of probabilistic forecasting. In this\nwork, we propose a framework for robust probabilistic time series forecasting.\nFirst, we generalize the concept of adversarial input perturbations, based on\nwhich we formulate the concept of robustness in terms of bounded Wasserstein\ndeviation. Then we extend the randomized smoothing technique to attain robust\nprobabilistic forecasters with theoretical robustness certificates against\ncertain classes of adversarial perturbations. Lastly, extensive experiments\ndemonstrate that our methods are empirically effective in enhancing the\nforecast quality under additive adversarial attacks and forecast consistency\nunder supplement of noisy observations.", + "authors": "TaeHo Yoon, Youngsuk Park, Ernest K. Ryu, Yuyang Wang", + "published": "2022-02-24", + "updated": "2022-02-24", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG" + ], + "main_content": "INTRODUCTION Time series forecasting is among the most important tasks in the automation and optimization of business processes. In retail, for example, determining how many units of each item to purchase and where to store them depends on forecasts of future demand over di\ufb00erent regions. In cloud computing, the estimated Proceedings of the 25th International Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2022, Valencia, Spain. PMLR: Volume 151. Copyright 2022 by the author(s). Perturbed series Original series Figure 1: Predictions from vanilla DeepAR model (top) and its version with randomized smoothing we propose (bottom). Forecasts are separated from input series via the black vertical line. The smoothed model is more robust in the sense that its probabilistic outputs are less a\ufb00ected by adversarial input perturbation (which perturbs the blue series to red). future usage of services and infrastructure components guides capacity planning (Park et al., 2019, 2020). The real-time forecasting is essential as a subroutine for vehicle control and planning (Kim et al., 2020) and other numerous applications (Petropoulos et al., 2022). Due to its crucial role in downstream decision making, there are two desirable properties of a forecaster: 1) the ability to generate probabilistic forecasts that allows for uncertainty estimation; 2) reliability, in the sense of being robust to (potentially adversarial) input perturbations. In the present work, we investigate robust probabilistic forecasting models which aim to satisfy the both requirements. In the classical time series literature where statistical methods were predominant, studies on robust forecasting were mainly focused on model stability against outliers (Connor et al., 1994; Gelper et al., 2010). More *Work done as an intern at Amazon Research. Correspondence to: TaeHo Yoon , Youngsuk Park . arXiv:2202.11910v1 [cs.LG] 24 Feb 2022 \fRobust Probabilistic Time Series Forecasting recently, deep learning models increasingly gained popularity and has gradually become the main workhorse of scalable forecasting (Li et al., 2019b; Oreshkin et al., 2019; Sen et al., 2019; Fan et al., 2019; Chen et al., 2020; Arik et al., 2020; Zhou et al., 2021). A distinct concept of adversarial robustness has emerged as an issue because deep neural networks are notoriously prone to small adversarial input perturbations (Szegedy et al., 2013; Goodfellow et al., 2015). Particularly in the context of forecasting, in Dang-Nhu et al. (2020), the authors showed that deep autoregressive time series forecasting models with probabilistic outputs could su\ufb00er from such attacks. Despite its importance, to the best of our knowledge, there is no prior work which has developed the formal concept of adversarial robustness for probabilistic forecasting models. This leaves the forecasters under the threat of adversarial attacks, endangering the decision making of mission-critical tasks. Furthermore, the time series data possess a unique characteristic (i.e. the time dimension), which allows for robustness notions that are fundamentally di\ufb00erent from lp-adversarial robustness, e.g., forecasts\u2019 stability under temporal window shift (translation) or the presence of statistical outliers. Therefore, it is necessary to establish a notion of robustness for probabilistic time series forecasting models that is general enough to encompass diverse classes of input changes and the corresponding practical requirements. Contributions. In this paper, we develop a framework of robust probabilistic time series forecasting, handling both theory and practice. To this aim, we \ufb01rst generalize the concept of adversarial perturbations in terms of abstract input and output transformations and provide a formal mathematical notion of robustness in terms of distributional stability of outputs when inputs are perturbed. With these foundations established, we propose randomized smoothing for probabilistic forecasters, which enjoys theoretical robustness guarantees for distinct classes of input perturbations and potentially improves a baseline model\u2019s robustness without requiring separate learning procedures. To establish even more pragmatic robust methods, we combine the smoothing techniques with randomized training, i.e., augmentation of training data with random noises, which is generally known to be e\ufb00ective in generating robust base learners. Finally, we empirically verify that the randomizing procedures are indeed e\ufb00ective in rendering robust probabilistic forecasting models through extensive experiments on multiple real datasets. 1.1 Related Work Classically robust forecasting via statistical methods. Earlier ideas on robust forecasting have mostly focused on adapting the classical techniques to deal with outliers, missing data, or change points. A plethora of works have been developed in multiple directions, including robust versions of exponential and Holt\u2013Winters smoothing (Cipra, 1992; Cipra et al., 1995; Gelper et al., 2010), outlier-robust Kalman \ufb01lters (Cipra and Romera, 1997; Ting et al., 2007; Agamennoni et al., 2011; Chang, 2014), and statistical frameworks based on detection or \ufb01ltering of anomalies (Connor et al., 1994; Garnett et al., 2009; Ristanoski et al., 2013; Anava et al., 2015; Guo et al., 2016). Deep learning for time series forecasting. Neural network has been applied to time series forecasting for more than half a century, and the earliest work dates back to 1960s (Hu and Root, 1964). Despite the early start, neural networks found little success in the time series forecasting literature until recently. With the explosive production of time series data and advances in neural architectures, deep learning has become increasingly popular. The strong performance of the deep forecasting models is especially prominent in the \ufb01elds where a large collection of time series is available, such as demand forecasting in large retailers. Among the deep forecasting models, more relevant to the present work are the approaches that generate probabilistic forecasts. This is typically achieved by two avenues. The \ufb01rst approach, including Salinas et al. (2020, 2019); de B\u00e9zenac et al. (2020); Rangapuram et al. (2018); Wang et al. (2019), uses neural networks as backbone sequential model and the last layer is customized via a likelihood function. An alternative approach directly generates the desired quantile forecasts; see for example Wen et al. (2017); Lim et al. (2021); Park et al. (2021); Eisenach et al. (2022). Other classes of works include variance reduced training (Lu et al., 2021) or domain adaptaton based techniques (Jin et al., 2022). For comprehensive study of deep forecasting models, we refer interested readers to Benidis et al. (2020); Hewamalage et al. (2021); Alexandrov et al. (2020). Adversarial attacks and time series. The seminal work of Szegedy et al. (2013) demonstrated that image classi\ufb01cation models based on deep neural networks, in spite of their high test accuracy, tend to be susceptible to hardly human-perceptible changes, which cause them to completely misclassify the inputs. This inspired a number of works to further study e\ufb00ective attack schemes (Goodfellow et al., 2015; Madry et al., 2018; Papernot et al., 2017; Athalye et al., 2018). In the time series domain, earlier works (Fawaz et al., 2019; Karim et al., 2021) mainly focused on attacking time series classi\ufb01cation models, and attack against probabilistic forecasting models was \ufb01rst devised by Dang-Nhu et al. (2020) using reparametrization tricks. \fTaeHo Yoon, Youngsuk Park, Ernest K. Ryu, Yuyang Wang Certi\ufb01ed adversarial defenses. While the adversarial training (Kurakin et al., 2017; Madry et al., 2018) has been arguably the most successful defense scheme against adversarial attacks on the empirical side, its theoretical performance guarantee over perturbed data has not been established. Towards developing defense scheme with certi\ufb01cates, a series of works (Dvijotham et al., 2018; Wong and Kolter, 2018; Wong et al., 2018; Raghunathan et al., 2018b,a; Gowal et al., 2018) suggested to directly control the local Lipschitz constant of feedforward neural networks, but such approaches were constrained to moderate-sized models. Another line of works (Lecuyer et al., 2019; Cohen et al., 2019; Salman et al., 2019; Li et al., 2019a) studied randomized smoothing as a more scalable and model-agnostic approach, and successfully provided practical accuracy guarantees on classi\ufb01cation problems up to the ImageNet scale under adversarial perturbations. Randomized smoothing have also had applications in regression problems in the context of certi\ufb01ably robust object detection (Chiang et al., 2020). However, we are not aware of any prior works along this direction which have considered adversarial defenses for models with probabilistic outputs, which is the standard framework for time series forecasting. Exposure bias and translation robustness. An autoregressive (or conditional) sequence generation model may behave signi\ufb01cantly di\ufb00erently in training and inference stages because in the test time, it generates outputs based on its own previous outputs, whose distribution may deviate from the ground-truth and the resulting error can be propagated (Bengio et al., 2015; Bowman et al., 2016; Ranzato et al., 2016). This phenomenon, often referred to as exposure bias, has been studied and empirically addressed by a number of prior works on language models (Norouzi et al., 2016; Schmidt, 2019) and more recently on time series forecasting (Sangiorgio and Dercole, 2020). These works are partially related to our translation robustness framework where we control propagation of errors caused by appending noisy or adversarial observations, but our approach is fundamentally distinct in that we focus on worst-case perturbations (rather than regarding the data distributions) and theoretically guaranteed solutions (rather than empirical remedies). 2 PRELIMINARIES 2.1 Probabilistic Time Series Forecasting Suppose we are given a dataset of N time series, where the i-th time series consist of observation xi,t P R with (optional) input covariates zi,t P Rd at time t. We drop the time series index whenever the context is clear. Examples of the input covariates include price and promotion at a certain time with the observations being the sales. For each time series, we observe T past targets x \u201c x1:T P X \u201c \u01648 T \u201c1 RT and all covariates z1:T `\u03c4 P Z to predict \u03c4 future targets xT `1:T `\u03c4 P Y \u201c R\u03c4. We denote a global1 probabilistic forecaster f : X \u02c6Z \u00d1 PpYq where PpYq denotes the probability distribution on the prediction space Y. With a slight abuse of notation, we describe the forecaster as pY1, . . . , Y\u03c4q \u201c fpx1, . . . , xT , z1 . . . , zT `\u03c4q, where pY1, . . . , Y\u03c4q are random variables associated with future targets pxT `1, . . . , xT `\u03c4q and their full distributions are speci\ufb01ed in terms of quantiles (Wen et al., 2017; Park et al., 2021) or parametric forms (Salinas et al., 2020), e.g., Gaussian, Student\u2019s t, or negative binomial. For notational simplicity, we omit the covariates z1:T `\u03c4 and concisely write Y \u201c pY1, . . . , Y\u03c4q \u201c fpxq. (1) We respectively denote as xT `1, . . . , xT `\u03c4 the groundtruth future targets and y \u201c py1, . . . , y\u03c4q \u201c p\u02c6 xT `1, . . . , \u02c6 xT `\u03c4q the sampled prediction (or realization) of future targets from probabilistic forecaster f. Since probabilistic forecast fpxq essentially carries distributional information, we abuse notation to allow sampling from it: p\u02c6 xT `1, . . . , \u02c6 xT `\u03c4q \u201e fpxq. 2.2 Adversarial Attacks on Probabilistic Autoregressive Forecasting Models In the probabilistic forecasting setting, the adversarial perturbation (or attack) \u03b4 on the input x with given adversarial target values tadv P Rm and a statistic2 \u03c7 : R\u03c4 \u00d1 Rm can be found by minimizing argmin \u03b4:}\u03b4}\u010f\u03b7 \u203a \u203aEfpy|x`\u03b4qr\u03c7pY1, . . . , Y\u03c4qs \u00b4 tadv \u203a \u203a2 2 (2) where \u03b7 \u011b 0 is the attack threshold, and the norm }\u03b4} is chosen depending on the context. The expectation Efpy|x`\u03b4q is taken over the randomness in pY1, . . . , Y\u03c4q \u201c fpx ` \u03b4q, the output of the probabilistic forecaster (1) on the input x ` \u03b4. The target value tadv is chosen to be signi\ufb01cantly di\ufb00erent from Efpy|xqr\u03c7pY1, . . . , Y\u03c4qs. For the case of stock price predictions, the choice of \u03c7 may be varied to express \ufb01nancial quantities such as buyor sell-option prices; see Dang-Nhu et al. (2020) for details. For practical experiments, we focus on attacking subsets of prediction outputs. In other words, we consider 1Forecast models are the same across all N time series. 2Dang-Nhu et al. (2020) limits \u03c7 and tadv to scalar ones (m \u201c 1). \fRobust Probabilistic Time Series Forecasting statistics of the form \u03c7HpY1, . . . , Y\u03c4q \u201c pYh1, . . . , Yhmq (3) in (2), where H is a subset of prediction indices with size m, i.e., H :\u201c th1, . . . , hmu \u0102 t1, . . . , \u03c4u. In this case, the adversary searches for a minimal norm perturbation x1 \u201c x ` \u03b4 for which the subset of perturbed forecasts is signi\ufb01cantly di\ufb00erent from the original forecasts at corresponding indices, and potentially from the ground-truth values as well. Constrained optimization (2) can be relaxed into a regularized optimization problem as follows: min \u03b4 Lp\u03b4q :\u201c }\u03b4}2 ` \u03bb \u00a8 \u203a \u203aEfpy|x`\u03b4qrYHs \u00b4 tadv \u203a \u203a2 2 (4) where \u03bb \u0105 0 is a hyperparameter. The derivative of the relaxed objective (4) with respect to \u03b4 could be computed via the reparametrization trick3 as in Dang-Nhu et al. (2020), which allows us to solve the regularized problem using any \ufb01rst-order optimizer. 3 DEFINING ROBUSTNESS FOR PROBABILISTIC TIME SERIES FORECASTING The adversarial attack in time series forecasting has been proposed only in terms of additive input perturbation with respect to lp norm as in (2). However, the unique properties of time series data including the existence of time dimension, periodicity, or seasonality potentially allow for a number of distinct types of perturbation. In this section, we generalize the notion of adversarial input perturbations in probabilistic forecasting, in order to incorporate distinct classes of input changes. Then we de\ufb01ne the corresponding notion of robustness which properly quanti\ufb01es model sensitivity to those input perturbations. 3.1 Generalized Input Perturbations in Time Series Forecasting We consider abstract input perturbation TX : X \u00d1 X and output transformation TY : Y \u00d1 Y. Given a probabilistic forecaster f and an input x P X, we describe the forecast output from f on x under the input perturbation TX as fpTX pxqq, and the original forecast output with the output transformation TY applied as pTYq# fpxq. Ultimately, we want to have f \u02dd TX \u00ab TY \u02dd f (5) 3The forecaster should support sampling from a distribution with location and/or scale parameters, which is the case for autoregressive forecasting models. in probabilistic sense, toward achieving robust forecasters. Before formally providing the detailed concept of robustness (in Section 3.2), we \ufb01rst walk through two example classes of perturbations: additive adversarial attack and time shift with new noisy observations, and demonstrate how the transformations TX and TY can be speci\ufb01ed. 3.1.1 Additive Adversarial Perturbation Consider the additive advsersarial perturbation which deceives the forecaster to deviate from its original forecasts on the subset H of prediction indices. We model the corresponding TX as TX pxq \u201c x ` \u03b4\u2039pxq, (6) where \u03b4\u2039pxq \u201c argmax }\u03b4}\u010f\u03b7 \u203a \u203aEfpy|x`\u03b4qrYHs \u00b4 Efpy|xqrYHs \u203a \u203a2 . (7) Simply taking the output transformation as the identity map, i.e., TY \u201c Id, the requirement f \u02dd TX \u00ab TY \u02dd f reduces to fpx ` \u03b4\u2039pxqq \u00ab fpxq. That is, we want our forecaster to be insensitive to adversarially constructed additive noise. If f is both robust in this sense and also has good prediction performance over the clean (unattacked) test dataset, its forecast quality will be retained even when it is given with adversarially perturbed time series data. 3.1.2 Time Shift with New Noisy Observations Consider the scenario where one initially has an input series x \u201c px1, . . . , xT q, and later a set of k ! \u03c4 new observations t\u02dc xT `1, . . . , \u02dc xT `ku arrives and the entire prediction task is shifted by k time steps. Suppose that we want the initial forecasts fpxq \u201c pY1, Y2, . . . , Yk`1, Yk`2, . . . q to be consistent with the forecasts fpx; \u02dc xT `1, . . . , \u02dc xT `kq \u201c pY 1 k`1, Y 1 k`2, . . . q produced after given the new observations. In this case, we let TX pxq \u201c px; \u02dc xT `1, . . . , \u02dc xT `kq be the augmentation by new observations, and TYpy1, y2, . . . , yk`1, yk`2, . . . q \u201c pyk`1, yk`2, . . . q be the left translation operator by k time steps. Then f \u02dd TX \u00ab TY \u02dd f requires that Yk`1 \u00ab Y 1 k`1, Yk`2 \u00ab Y 1 k`2, . . . , \fTaeHo Yoon, Youngsuk Park, Ernest K. Ryu, Yuyang Wang i.e., the two sets of forecasts should be consistent. Note that \u02dc x\u2019s are inherited in TX , and when some of \u02dc x\u2019s are highly anomalous or have been manipulated by an adversary, TX may largely impact the original forecasts. In our experiments, we consider the speci\ufb01c case of appending a single adversarial observation \u02dc xT `1 :\u201c p1 ` \u03c1qxT `1, (8) which is (de-)ampli\ufb01ed relative to the ground truth according to the adversarial parameter \u03c1 \u0105 \u00b41. 3.2 Formal Mathematical De\ufb01nition of Robustness Given a probabilistic forecaster f and a transformation pair TX , TY suitable for time series setting, both fpTX pxqq and pTYq# fpxq are random variables in R\u03c4 whose distributions are speci\ufb01ed. Let us denote, for each prediction time point t \u201c 1, . . . , \u03c4, the associated marginal distributions by \u00b5t and \u00b51 t respectively, i.e., pfpTX pxqqqt \u201e \u00b5t and ppTYq# fpxqqt \u201e \u00b51 t. We aim to formally quantify the informal notion (5) in terms of certain metric between these distributions. As a \ufb01nal ingredient for establishing the precise de\ufb01nition, we de\ufb01ne dpx; TX q, which is a measure of how signi\ufb01cant the change due to the transformation TX is, or in other words, a dissimilarity measure between x and TX pxq. For the case of additive adversarial perturbation in Section 3.1.1, a natural choice for d would be dpx; TX q \u201c }\u03b4\u2039pxq}. For the time-shift setup in Section 3.1.2, we take d of the form dpx; TX q \u201c D p\u02dc xT `1:T `k; \u02c6 xT `1:T `kq where \u02c6 xT `1:T `k are the model\u2019s initial point forecasts on the \ufb01rst k future time points, and D \u011b 0 satis\ufb01es Dpy1; yq \u201c 0 i\ufb00y1 \u201c y. We take Dpy1; yq \u201c }y1 \u00b4 y}2 in this paper for the sake of establishing theoretical guarantees, but D can also be chosen in ways that involve likelihood to less penalize large deviations, e.g., Dpy1; yq \u201c \u00b4 log ` qpy1q L qpyq \u02d8 , where q is the joint density function for the model\u2019s k-step ahead predictions. De\ufb01nition 1. Let f be a probabilistic forecaster, x P X, and TX , TY the input, output transformations with marginal distributions \u00b5t, \u00b51 t, respectively. Then, f is \u03b5\u2013\u03b7 robust at x with respect to the transformation pair pTX , TYq if, provided that dpx; TX q \u0103 \u03b7, for any t \u201c 1, . . . , \u03c4, we have W1 ` \u00b5t, \u00b51 t \u02d8 \u0103 \u03b5. (9) Connection to adversarial attacks. In the additive adversarial attack we detailed in Section 2.2, it is assumed that \u03c7 is chosen by the adversary. Therefore, a defense scheme against the attack (2) would naturally involve minimizing an objective of the form sup }\u03b4}\u010f\u03b7 sup \u03c7PF pEr\u03c7pfpx ` \u03b4qqs \u00b4 Er\u03c7pfpxqqsq , (10) where F denotes the collection of \u03c7\u2019s which the adversary could choose from. Note that if } \u00a8 } \u201c } \u00a8 }2, and if F consists of L-Lipschitz continuous functions for some L \u0105 0, the inner maximization in (10) results in a constant multiple of 1-Wasserstein (or W1) distance between the distributions of fpx ` \u03b4q and fpxq, due to Kantorovich-Rubinstein duality. This interpretation motivates our choice of 1-Wasserstein distance as the measure of local distributional change in the robustness de\ufb01nition (9). Connection to quantile forecasts. Here we provide another, more general perspective on the reason for formulating De\ufb01nition 1 in terms of W1 distance. Probabilistic forecasts are often characterized via quantiles; MQ-RNN (Wen et al., 2017) directly performs quantile regression, and for sampling-based forecasters such as DeepAR (Salinas et al., 2020), sample quantiles are used to compute the prediction intervals. This practice of using quantiles as important quantities is fortuitously aligned with the following interpretation of W1 distance as the average quantile di\ufb00erence; if F, G are respectively the cumulative distribution function of a real-valued random variable and \u00b5, \u03bd are the corresponding probability distributions, then W1p\u00b5, \u03bdq \u201c \u017c 1 0 |F \u00b41puq \u00b4 G\u00b41puq| du. (11) That is, our robustness de\ufb01nition requires that a model\u2019s quantile estimates, in the average sense over probability levels, should not be signi\ufb01cantly a\ufb00ected by small input perturbations with respect to dX . 4 THEORY AND FRAMEWORKS In this section, we develop methodologies for robust forecasting based on randomized smoothing, covering the two classes of adversarial perturbations we considered in Section 3.2. We establish and discuss the theoretical robustness guarantees of these smoothingbased methods. Additionally, we revisit the randomized training (data augmentation with noises) widely adopted by practitioners as a strategy for enhancing base forecasters for randomized smoothing techniques. 4.1 Randomized Smoothing for Guaranteed Robustness Against l2 Perturbations We \ufb01rst develop a robustness framework using randomized smoothing for additive adversarial perturbations (covered in Section 3.1.1) with dpx; TX q \u201c }\u03b4}2. \fRobust Probabilistic Time Series Forecasting Algorithm 1: Randomized smoothing for probabilistic forecasters Input: Multi-horizon sample-based forecaster f, Input series x \u201c px1, . . . , xT q, \u03c4, n, \u03c32 Output: n sample paths \u02c6 xpjq T `1:T `\u03c4 from g\u03c3 (j \u201c 1, . . . , n) for j \u201c 1, . . . , n do \u03b61, . . . , \u03b6T \u201e Np0, \u03c32q i.i.d. \u02dc x \u00d0 px1 ` \u03b61, . . . , xT ` \u03b6T q \u02c6 xpjq T `1:T `\u03c4 \u201e fp\u02dc xq end One-step ahead predictors. Consider the simple setting where f is a random function from RT to R, which we will extend to multivariate cases later. Let us denote by Fx the cumulative distribution function (cdf) of the random variable fpxq, i.e., Fxprq :\u201c Pr rfpxq \u010f rs for r P R. Given the smoothing paramter \u03c3 \u0105 0, we de\ufb01ne the smoothed version g\u03c3 of f as the random function from RT to R with cdf Gx,\u03c3prq \u201c Pr rg\u03c3pxq \u010f rs (12) \u201c E z\u201eN p0,\u03c32IqrFx`zprqs \u201c \u017c Fx`zprqp\u03c3pzq dz, where p\u03c3pzq is the density function of the multivariate Gaussian distribution Np0, \u03c32Iq. Below, we provide a theoretical guarantee on the robustness of the smoothed probabilistic predictor g\u03c3, analogous to prior results (Cohen et al., 2019; Salman et al., 2019; Chiang et al., 2020) for deterministic setups. Theorem 1. Let f be a random function from RT to R, and let g\u03c3 be as in (12). Given x P RT , we have the inequality Ropx; \u03c3q :\u201c lim sup }\u03b4}2\u00d10 W1pGx,\u03c3, Gx`\u03b4,\u03c3q }\u03b4}2 \u010f 1 \u03c3 \u017c 8 \u00b48 \u03c6 ` \u03a6\u00b41pGx,\u03c3prqq \u02d8 dr (13) provided that the integral on the right hand side is locally bounded at x, where \u03c6, \u03a6 denote the pdf and cdf of the standard normal distribution. Note that if Ropx; \u03c3q \u0103 8, then f is Op\u03b7q\u2013\u03b7 robust for \u03b7 small enough, in the sense of De\ufb01nition 1 with respect to adversarial perturbations, in the one-step ahead prediction case. Finiteness of Ropx; \u03c3q. Provided that the cdf Gx,\u03c3prq of the smoothed random variable g\u03c3pxq has nonzero derivative at all r P R, we can make the change of variable u \u201c Gx,\u03c3prq to rewrite (13) as Ropx; \u03c3q \u010f 1 \u03c3 \u017c 1 0 \u03c6p\u03a6\u00b41puqq ` G\u00b41 x,\u03c3 \u02d81 puq du. Note that the quantity ` G\u00b41 x,\u03c3 \u02d81 puq is the quantile density function of g\u03c3pxq, while \u03c6 ` \u03a6\u00b41puq \u02d8 is the inverse quantile density of the standard Gaussian. Therefore, intuitively speaking, we expect Ropx; \u03c3q to be \ufb01nite unless the quantiles of g\u03c3pxq blow up too quickly compared to those of the normal distribution as u \u00d1 0, 1. Indeed, the following Lemmas show that Ropx; \u03c3q \u0103 8 holds under mild, practical assumptions on f. Lemma 2. Let f and g\u03c3 be as in Theorem 1. Suppose that there exists a function \u03d5 : R` \u00d1 R` such that Pr r|fpxq| \u011b rs \u010f \u03d5prq for any x P RT and r \u0105 0, and \u017c 8 0 \u03d5prq dr \u0103 8. (14) Then there exists C \u0105 0, depending only on \u03d5 and \u03c3, such that Ropx; \u03c3q \u0103 C for all x P RT . Lemma 3. Suppose that f is parametrized with globally bounded mean and variance. Then f satis\ufb01es the assumptions of Lemma 2. In particular, the most commonly used distributions including Gaussian, generalized Student\u2019s t (\u03bd \u0105 2), or negative binomial are all allowed if the mean and scaling parameters are bounded. Multi-horizon predictors. In the case when f is a random function from RT to R\u03c4 as in the multi-horizon forecasting setup, we can apply Theorem 1 and Lemmas 2, 3 to each component function of f. This directly implies that f, under suitable assumption, is Op\u03b7q\u2013\u03b7 robust with respect to the l2 adversarial perturbation in the sense of De\ufb01nition 1, which we formally restate in the following as Corollary 4. Therefore, we take the (component-wisely) smoothed version g\u03c3 of the baseline predictor f as our predictor with robustness guarantees, which is presented in Algorithm 1. Note that we implement g\u03c3 by directly sampling its output paths, which is done by sampling independently over randomness in z and f. Corollary 4. Let f be a probabilistic multi-horizon forecaster, denoted fpxq \u201c pf1pxq, . . . , f\u03c4pxqq. Suppose that \u03d5 : R` \u00d1 R` satis\ufb01es Pr r|fjpxq| \u011b rs \u010f \u03d5prq for any x P RT , r \u0105 0 and j \u201c 1, . . . , \u03c4, and (14) holds. Then there exists a constant C \u0105 0 depending only on \u03d5 and \u03c3 such that for any \u03b7 \u0105 0, the smoothed forecaster g\u03c3 de\ufb01ned by applying the smoothing (12) to each component fjpxq is C\u03b7\u2013\u03b7 robust at all x with respect to TX as in (6) with } \u00a8 } \u201c } \u00a8 }2, TY \u201c Id and dpx; TX q \u201c }\u03b4\u2039pxq}. \fTaeHo Yoon, Youngsuk Park, Ernest K. Ryu, Yuyang Wang Algorithm 2: Future smoothing for probabilistic autoregressive forecasters Input: Probabilistic autoregressive forecaster f, Input series x \u201c px1, . . . , xT q, \u03c4, n, \u03c32, (Optionally) Noisy observations py1, . . . , ykq \u201c p\u02dc xT `1, . . . , \u02dc xT `kq Output: Sample forecasts \u02c6 xpjq T `i (i \u201c k ` 1, . . . , k ` \u03c4, j \u201c 1, . . . , n) for i \u201c 1, . . . , k ` \u03c4 do if i \u010f k then yi\u00b41 \u00d0 \u02dc xT `i\u00b41 else yi\u00b41 \u00d0 \u02c6 xT `i\u00b41 for j \u201c 1, . . . , n do \u03b61, . . . , \u03b6i\u00b41 \u00d0 Np0, \u03c32q i.i.d. \u02c6 xpjq T `i \u00d0 f px; y1 ` \u03b61, . . . , yi\u00b41 ` \u03b6i\u00b41q end \u02c6 xT `i \u00d0 1 n \u0159n j\u201c1 \u02c6 xpjq T `i end end Possible generalization of the theorems. We point out that the domain of f could be replaced with some representation space W without altering the essence of the theorems. If \u03a8 : X \u00d1 W is an invertible representation map such that dX px; TX q \u201c }\u03a8pTX pxqq \u00b4 \u03a8pxq}2, then one can simply apply Theorem 1 and Corollary 4 to f \u02dd \u03a8\u00b41, which is a random function from W to R. Thus, albeit the theorems seem to only provide guarantees for local l2 perturbations, one can extend the results to fundamentally distinct types of perturbations by choosing an appropriate W, e.g., the frequency domain. 4.2 Noisy Autoregressive Inference for Robustness Under Time Shift In this section, we propose a strategy, Algorithm 2, for achieving robustness against the time-shift setup of Section 3.1.2. As randomized smoothing bu\ufb00ers the e\ufb00ect of input perturbation by averaging over noised inputs, we exploit random noises to bu\ufb00er the e\ufb00ect of appending the noisy observation \u02dc xT `1. Note that because the uncertainty is now within future times, we perform smoothing over future time indices. An autoregressive forecaster f can be generally described in the form fpxq \u201c pY1, Y2, . . . q, where x P RT is an input series and Yh \u201c f phq px; Y1, . . . , Yh\u00b41q (15) for some random functions f phq from RT `h\u00b41 to R, for h \u201c 1, 2, . . . . For \ufb01xed \u03c3 \u0105 0 and x P RT , consider the smoothed version gphq \u03c3 of f phq, de\ufb01ned as gphq \u03c3 px; y1, . . . , yh\u00b41q \u201c E \u03b6\u201eN p0,\u03c32Ih\u00b41q \u201d f phqpx; y1 ` \u03b61, \u00a8 \u00a8 \u00a8 , yh\u00b41 ` \u03b6h\u00b41q \u0131 where we noise only the variables y1, . . . , yh\u00b41 but not x. Now we de\ufb01ne g\u03c3, the smoothed version of f, by g\u03c3pxq \u201c pY1, Y2, . . . q, where Yh \u201c gphq \u03c3 px; y1, . . . , yh\u00b41q (16) for h \u201c 1, 2, . . . , and yj P R (j \u201c 1, . . . , h \u00b4 1) denotes either 1) the mean forecast from g\u03c3 at that time point (i.e. expected value of Yj) if the true value is unobserved, or 2) the given ground-truth value if a new observation at that time has arrived. Algorithm 2 presents the procedural implementation of g\u03c3 based on sampling. The theoretical justi\ufb01cation for performing smoothing in this way is Corollary 5, which states the robustness property of the smoothed forecaster g\u03c3 with respect to the notions provided in Sections 3.1.2 and 3.2, which is done by applying the smoothing framework developed in Section 4.1 under a slightly di\ufb00erent setting. Corollary 5. Let f be an autoregressive forecaster de\ufb01ned as (15) and suppose that \u03d5 : R` \u00d1 R` satis\ufb01es (14) and Pr \u201d\u02c7 \u02c7 \u02c7f phqpx; y1, . . . , yh\u00b41q \u02c7 \u02c7 \u02c7 \u011b r \u0131 \u010f \u03d5prq holds for each h \u201c 1, 2, . . . . Then, for all \u03b7 \u0105 0, the forecaster g\u03c3 de\ufb01ned as (16) is Op\u03b7q\u2013\u03b7 robust at all x P RT with respect to TX , TY of Section 3.1.2 and dpx; TX q \u201c }\u02dc xT `1:T `k \u00b4 \u02c6 xT `1:T `k}2 , where \u02dc xT `1:T `k are the values appended by TX and \u02c6 xT `1:T `k are the k-step mean predictions from the initial forecasts g\u03c3pxq. 4.3 Training the Baseline Model via Random Data Augmentation Having a baseline model f with solid prediction performance over noised inputs is the key for achieving success in randomized smoothing, which is expected to improve the theoretical certi\ufb01cate (13) as well as practical performance (Cohen et al., 2019). This encourages us to additionally apply randomized training, which augments the training dataset with random noises to adapt the baseline forecaster to random perturbations. Note that randomized training and smoothing are di\ufb00erent schemes; in particular, the former requires training from scratch while the latter does not. In the end, we take advantage of both techniques to achieve e\ufb00ective robust forecasters. \fRobust Probabilistic Time Series Forecasting 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Attack threshold 0.05 0.10 0.15 0.20 0.25 Normalized deviation Vanilla RT RS RT+RS (a) Exchange Rate, \u03c4 \u201c 30, tHu \u201c t\u03c4u 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Attack threshold 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Normalized deviation Vanilla RT RS RT+RS (b) M4-Daily, \u03c4 \u201c 14, tHu \u201c t\u03c4u 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Attack threshold 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 Normalized deviation Vanilla RT RS RT+RS (c) Tra\ufb03c, \u03c4 \u201c 24, tHu \u201c t\u03c4u Figure 2: NDH from DeepAR models on di\ufb00erent datasets. Randomized smoothing (RS) of baseline models uses \u03c3 \u201c 0.5. Randomized training (RT) uses \u03c3tr \u201c 0.1. 5 EXPERIMENTS In this section, we demonstrate the e\ufb00ects of our frameworks for the DeepAR (Salinas et al., 2020) implementation within GluonTS (Alexandrov et al., 2020) on real datasets, including the M4-Daily, Exchange Rate, Tra\ufb03c and UCI Electricity datasets preprocessed as in Salinas et al. (2019). We use DeepAR as it is the standard model with properties of being samplingbased and autoregressive, which Algorithms 1 and 2 respectively require. We speci\ufb01cally focus on how point forecasts from the model change under input transformations of Sections 3.1.1 and 3.1.2, and quantitatively assess them using the normalized deviation (ND) NDH \u201c \u0159N k\u201c1 \u0159 hPH |\u02c6 xk,T `h \u00b4 xref k,T `h| \u0159N k\u201c1 \u0159 hPH |xref k,T `h| . (17) Here H \u0102 t1, . . . , \u03c4u is the set of prediction indices of interest, N is the size of the test dataset, \u02c6 xk,T `h is the model\u2019s prediction for pT ` hq-th time step for the kth (possibly transformed) test series, and xref k,T `h is the corresponding reference value (which may either be the ground-truth value or the model output before transformation, depending on the setup). 5.1 Prediction Performance Under Additive Adversarial Attacks In this section, we consider the setup of Section 3.1.1, where the input transformation corresponds to the adversarial attack of Dang-Nhu et al. (2020). Experimental setup. Following Dang-Nhu et al. (2020), we choose the relative l2 norm }\u03b4}x :\u201c \u02dc t \u00ff i\u201c1 p\u03b4i{xiq2 \u00b81{2 as the measure of perturbation magnitude. Given the set H \u0102 t1, . . . , \u03c4u of attack indices and attack threshold \u03b7 \u0105 0, we solve the problem (4) with di\ufb00erent choices of tadv and \u03bb, and among the resulting approximate solutions \u03b4 meeting the norm constraint }\u03b4}x \u010f \u03b7, we measure the largest point forecast error in terms of normalized deviation (17). Here we take the ground-truth future values as reference values, i.e., xref k,T `h \u201c xk,T `h. As the attack is performed under norm constraint with respect to } \u00a8 }x, we accordingly use relative noises in both randomized smoothing and randomized training. That is, if the given input is x \u201c px1, . . . , xT q and the randomizing variance is \u03c32, then we use px1p1 ` \u03b61q, . . . , xtp1 ` \u03b6tqq as noised inputs, where \u03b61, . . . , \u03b6t are i.i.d. samples from Np0, \u03c32q. Note that in this case, our theoretical results can be applied with locally scaled version of f with respect to each input x. To distinguish the two randomizing procedures, we respectively denote by \u03c32 and \u03c32 tr the variance values of noises used in smoothing and data augmentation for training. Results and discussion. Figure 2 respectively compares the performance of vanilla DeepAR models (trained without data augmentation; solid green line) with their corresponding smoothed versions (labeled RS; dashed blue line), and the random-trained models (labeled RT; solid red line) with their smoothed versions (labeled RS`RT; dashed purple line). We observe three important points. First, for the majority of cases, randomized smoothing provides statistically signi\ufb01cant improvement to vanilla models\u2019 performance under attacks with moderate to high threshold values \u03b7 (see Table 2 in the appendix), at the cost of possibly slightly worsening the performance for small \u03b7. Second, RT models tend to strongly outperform the corresponding vanilla models, uniformly over the levels of attack threshold \u03b7. Third, further smoothing the RT model (RS`RT) can improve upon RT and in these cases, RS`RT achieves the best empirical performance \fTaeHo Yoon, Youngsuk Park, Ernest K. Ryu, Yuyang Wang \u22121.0 \u22120.5 0.0 0.5 1.0 log10(1 + \u03c1) 0.0 0.1 0.2 0.3 0.4 Normalized deviation Vanilla RT FS RT+FS (a) Exchange Rate \u22121.0 \u22120.5 0.0 0.5 1.0 log10(1 + \u03c1) 0.0 0.2 0.4 0.6 Normalized deviation Vanilla RT FS RT+FS (b) M4-daily \u22121.0 \u22120.5 0.0 0.5 1.0 log10(1 + \u03c1) 0.05 0.10 0.15 Normalized deviation Vanilla RT FS RT+FS (c) Tra\ufb03c Figure 3: NDrel from DeepAR models on di\ufb00erent datasets. Randomized training (RT) uses \u03c3tr \u201c 0.1. Future smoothing (FS) uses \u03c3 \u201c 1.0 for the \ufb01rst two datasets, and \u03c3 \u201c 0.1 for the Tra\ufb03c dataset. against attacks with high values of \u03b7. Table 2, provided in the appendix, displays full experiment results for all datasets and distinct attack indices. We emphasize that only the smoothed models are the ones that come with robustness certi\ufb01cates, and in particular, we \ufb01nd that RS`RT is a promising methodology, supported from both theoretical and empirical sides. Additionally, randomized smoothing is readily applicable to any pre-trained baseline model as a postprocessing step without the cost of model retraining, and still o\ufb00ers a potential direction for improving the model\u2019s robustness. We believe that these points constitute important practical values of the smoothing technique in general. Randomized training and prediction performance. As an aside, we unexpectedly \ufb01nd that RT tends to improve the usual prediction performance of forecasting models (see Table 1 in the appendix). That is, randomizing the training data may positively impact a model\u2019s generalization in the time series domain, which we believe, is an interesting phenomenon in its own right. In the appendix, we provide further discussion on this point, connecting the observation to prior works on generalization and training with noise. 5.2 Forecast Consistency Under Time Shift with Noisy Observation In this section, we examine the setup of Section 3.1.2 with k \u201c 1, where we append an adversarial observation to input series. Experimental setup. Given a series x P RT , we append \u02dc xT `1 as in (8) with various values of \u03c1 within the range \u00b41 \u010f log10p1 ` \u03c1q \u010f 1. As a metric, we compute the ND (17) with xref k,T `h \u201c \u02c6 xk,T `h as reference values, for h P H \u201c t2, . . . , \u03c4u. This measures the relative discrepancy between the forecasts p\u02c6 xT `1, . . . , \u02c6 xT `\u03c4q and p\u02c6 x1 T `2, . . . , \u02c6 x1 T `\u03c4`1q, respectively based on x and px; \u02dc xT `1q, at common indices. The lower this value is, the more consistent the forecasts are, before and after the arrival of adversarial observation \u02dc xT `1. We scale each input series x before applying noises; i.e., x is replaced by x{Sx for some Sx \u0105 0 computed by the model to process each series within a consistent scale. Results and discussion. Figure 3 compares the metrics from vanilla DeepAR model, its smoothed version using Algorithm 2 (labeled FS), the randomtrained (RT) model as in the previous section, and its smoothed version (RT`FS), on each dataset. The horizontal axes represent the adversarial parameter in a logarithmic scale log10p1 ` \u03c1q, and the vertical axes represent the relative ND. The vanilla models already have a desirable behavior around \u03c1 \u201c 0, but as |\u03c1| grows, their forecast consistency is progressively broken. On the other hand, RT models tend to be more resilient compared to vanilla models. FS provides statistically signi\ufb01cant improvement in forecast consistency to vanilla and RT models for large values of \u03c1 in many cases (see Table 3 in the appendix). We elicit a message similar to that of Section 5.1; RT`FS is theoretically well-supported, and often achieves superior empirical performance as well. We provide Table 3 containing all experiment results in the appendix. 6" + }, + { + "url": "http://arxiv.org/abs/2102.07922v2", + "title": "Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with $\\mathcal{O}(1/k^2)$ Rate on Squared Gradient Norm", + "abstract": "In this work, we study the computational complexity of reducing the squared\ngradient magnitude for smooth minimax optimization problems. First, we present\nalgorithms with accelerated $\\mathcal{O}(1/k^2)$ last-iterate rates, faster\nthan the existing $\\mathcal{O}(1/k)$ or slower rates for extragradient, Popov,\nand gradient descent with anchoring. The acceleration mechanism combines\nextragradient steps with anchoring and is distinct from Nesterov's\nacceleration. We then establish optimality of the $ \\mathcal{O}(1/k^2)$ rate\nthrough a matching lower bound.", + "authors": "TaeHo Yoon, Ernest K. Ryu", + "published": "2021-02-16", + "updated": "2021-06-10", + "primary_cat": "math.OC", + "cats": [ + "math.OC" + ], + "main_content": "Introduction Minimax optimization problems, or minimax games, of the form minimize x\u2208Rn maximize y\u2208Rm L(x, y) (1) have recently gained signi\ufb01cant interest in the optimization and machine learning communities due to their application in adversarial training (Goodfellow et al., 2015; Madry et al., 2018) and generative adversarial networks (GANs) (Goodfellow et al., 2014). Prior works on minimax optimization often consider compact domains X, Y for x, y and use the duality gap Errgap(x, y) := sup \u02dc y\u2208Y L(x, \u02dc y) \u2212inf \u02dc x\u2208X L(\u02dc x, y) to quantify suboptimality of algorithms\u2019 iterates in solving (1). However, while it is a natural analog of minimization error for minimax problems, the duality gap can be dif\ufb01cult to measure directly in practice, and it is unclear how to generalize the notion to non-convex-concave problems. In contrast, the squared gradient magnitude \u2225\u2207L(x, y)\u22252, when L is differentiable, is a more directly observable 1Department of Mathematical Sciences, Seoul National University, Seoul, Korea. Correspondence to: Ernest K. Ryu . Proceedings of the 38 th International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s). value for quantifying suboptimality. Moreover, the notion is meaningful for differentiable non-convex-concave minimax games. Interestingly, very few prior works have analyzed convergence rates on the gradient norm for minimax problems, and the optimal convergence rate or corresponding algorithms were hitherto unknown. Contributions. In this work, we introduce the extra anchored gradient (EAG) algorithms for smooth convexconcave minimax problems and establish an accelerated \u2225\u2207L(zk)\u22252 \u2264O(R2/k2) rate, where R is the Lipschitz constant of \u2207L. The rate improves upon the O(R2/k) rates of prior algorithms and is the \ufb01rst O(R2/k2) rate in this setup. We then provide a matching \u2126(R2/k2) complexity lower bound for gradient-based algorithms and thereby establish optimality of EAG. Beyond establishing the optimal complexity, our results provide the following observations. First, different suboptimality measures lead to materially different acceleration mechanisms, since reducing the duality gap is done optimally by the extragradient algorithm (Nemirovski, 2004; Nemirovsky, 1992). Also, since our optimal accelerated convergence rate is on the non-ergodic last iterate, neither averaging nor keeping track of the best iterate is necessary for optimally reducing the gradient magnitude in the deterministic setup. 1.1. Preliminaries and notation We say a saddle function L: Rn \u00d7 Rm \u2192R is convexconcave if L(x, y) is convex in x \u2208Rn for all \ufb01xed y \u2208Rm and L(x, y) is concave in y \u2208Rm for all \ufb01xed x \u2208Rn. We say (x\u22c6, y\u22c6) is a saddle point of L if L(x\u22c6, y) \u2264L(x\u22c6, y\u22c6) \u2264L(x, y\u22c6) for all x \u2208Rn and y \u2208Rm. Solutions to the minimax problem (1) are de\ufb01ned to be saddle points of L. For notational conciseness, write z = (x, y). When L is differentiable, de\ufb01ne the saddle operator of L at z = (x, y) by GL(z) = \u0014 \u2207xL(x, y) \u2212\u2207yL(x, y) \u0015 . (2) (When clear from the context, we drop the subscript L.) The saddle operator is monotone (Rockafellar, 1970), i.e., arXiv:2102.07922v2 [math.OC] 10 Jun 2021 \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm \u27e8G(z1)\u2212G(z2), z1\u2212z2\u27e9\u22650 for all z1, z2 \u2208Rn\u00d7Rm. We say L is R-smooth if GL is R-Lipschitz continuous. Note that \u2207L \u0338= GL due to the sign change in the y gradient, but \u2225\u2207L\u2225= \u2225GL\u2225, and we use the two forms interchangeably. Because z\u22c6= (x\u22c6, y\u22c6) is a saddle point of L if and only if 0 = GL(z\u22c6), the squared gradient magnitude is a natural measure of suboptimality at a given point for smooth convexconcave problems. 1.2. Prior work Extragradient-type algorithms. The \ufb01rst main component of our proposed algorithm is the extragradient (EG) algorithm of Korpelevich (1977). EG and its variants, including the algorithm of Popov (1980), have been studied in the context of saddle point and variational inequality problems and have appeared in the mathematical programming literature (Solodov & Svaiter, 1999; Tseng, 2000; Noor, 2003; Censor et al., 2011; Lyashko et al., 2011; Malitsky & Semenov, 2014; Malitsky, 2015; 2020). More recently in the machine learning literature, similar ideas such as optimism (Chiang et al., 2012; Rakhlin & Sridharan, 2013a), prediction (Yadav et al., 2018), and negative momentum (Gidel et al., 2019; Zhang et al., 2020) have been presented and used in the context of multi-player games (Daskalakis et al., 2011; Rakhlin & Sridharan, 2013b; Syrgkanis et al., 2015; Antonakopoulos et al., 2021) and GANs (Gidel et al., 2018; Mertikopoulos et al., 2019; Liang & Stokes, 2019; Peng et al., 2020). O(R/k) rates on duality gap. For minimax problems with an R-smooth L and bounded domains for x and y, Nemirovski (2004) presented the mirror-prox algorithm generalizing EG and established ergodic O(R/k) convergence rates on Errgap. Nesterov (2007); Monteiro & Svaiter (2010; 2011) extended the O(R/k) complexity analysis to the case of unbounded domains. Mokhtari et al. (2020b) showed that the optimistic descent converges at O(R/k) rate with respect to Errgap. Since there exists \u2126(R/k) complexity lower bound on Errgap for black-box gradient-based minimax optimization algorithms (Nemirovsky, 1992; Nemirovski, 2004), in terms of duality gap, these algorithms are order-optimal. Convergence rates on squared gradient norm. Using standard arguments (e.g. (Solodov & Svaiter, 1999, Lemma 2.3)), one can show min i=0,...,k \u2225G(zi)\u22252 \u2264O(R2/k) convergence rate of EG, provided that L is R-smooth. Ryu et al. (2019) showed that optimistic descent algorithms also attain O(R2/k) convergence in terms of the best iterate and proposed simultaneous gradient descent with anchoring, which pulls iterates toward the initial point z0, and established O(R2/k2\u22122p) convergence rates in terms of squared gradient norm of the last iterate (where p > 1 2 is an algorithm parameter; see Section A). Notably, anchoring resembles the Halpern iteration (Halpern, 1967; Lieder, 2020), which was used in Diakonikolas (2020) to develop a regularization-based algorithm with near-optimal (optimal up to logarithmic factors) complexity with respect to the gradient norm of the last iterate. Anchoring turns out to be the second main component of the acceleration; combining EG steps with anchoring, we obtain the optimal last-iterate convergence rate of O(R2/k2). Structured minimax problems. For structured minimax problems of the form L(x, y) = f(x) + \u27e8Ax, y\u27e9\u2212g(y), where f, g are convex and A is a linear operator, primaldual splitting algorithms (Chambolle & Pock, 2011; Condat, 2013; V\u02dc u, 2013; Yan, 2018; Ryu & Yin, 2021) and Nesterov\u2019s smoothing technique (Nesterov, 2005a;b) have also been extensively studied (Chen et al., 2014; He & Monteiro, 2016). Notably, when g is of \u201csimple\u201d form, Neterov\u2019s smoothing framework achieves an accelerated rate O \u0010 \u2225A\u2225 k + Lf k2 \u0011 on duality gap. Additionally, Chambolle & Pock (2016) have shown that splitting algorithms can achieve O(1/k2) or linear convergence rates under appropriate strong convexity and smoothness assumptions on f and g, although they rely on proximal operations. Kolossoski & Monteiro (2017); Hamedani & Aybat (2018); Zhao (2019); Alkousa et al. (2020) generalized these accelerated algorithms to the setting where the coupling term \u27e8Ax, y\u27e9is replaced by non-bilinear convex-concave function \u03a6(x, y). Complexity lower bounds. Ouyang & Xu (2021) presented a \u2126 \u0010 \u2225A\u2225 k + Lf k2 \u0011 complexity lower bound on duality gap for gradient-based algorithms solving bilinear minimax problems with proximable g, establishing optimality of Nesterov\u2019s smoothing. Zhang et al. (2019) presented lower bounds for strongly-convex-strongly-concave problems. Golowich et al. (2020) proved that with the narrower class of 1-SCLI algorithms, which includes EG but not EAG, the squared gradient norm of the last iterate cannot be reduced beyond O(R2/k) in R-smooth minimax problems. These approaches are aligned with the information-based complexity analysis, introduced in (Nemirovsky & Yudin, 1983) and thoroughly studied in (Nemirovsky, 1991; 1992) for the special case of linear equations. Other problem setups. Nesterov (2009) and Nedi\u00b4 c & Ozdaglar (2009) proposed subgradient algorithms for nonsmooth minimax problems. Stochastic minimax and variational inequality problems were studied in (Nemirovski et al., 2009; Juditsky et al., 2011; Lan, 2012; Ghadimi & Lan, 2012; 2013; Chen et al., 2014; 2017; Hsieh et al., 2019). Strongly monotone variational inequality problems \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm or strongly-convex-strongly-concave minimax problems were studied in (Tseng, 1995; Nesterov & Scrimali, 2011; Gidel et al., 2018; Mokhtari et al., 2020a; Lin et al., 2020b; Wang & Li, 2020; Zhang et al., 2020; Azizian et al., 2020). Recently, minimax problems with objectives that are either strongly convex or nonconvex in one variable were studied in (Ra\ufb01que et al., 2018; Thekumparampil et al., 2019; Jin et al., 2019; Nouiehed et al., 2019; Ostrovskii et al., 2020; Lin et al., 2020a;b; Lu et al., 2020; Wang & Li, 2020; Yang et al., 2020; Chen et al., 2021). Minimax optimization of composite objectives with smooth and nonsmoothbut-proximable convex-concave functions were studied in (Tseng, 2000; Csetnek et al., 2019; Malitsky & Tam, 2020; B` ui & Combettes, 2021). 2. Accelerated algorithms: Extra anchored gradient We now present two accelerated EAG algorithms that are qualitatively very similar but differ in the choice of stepsizes. The two algorithms present a tradeoff between the simplicity of the step-size and the simplicity of the convergence proof; one algorithm has a varying step-size but a simpler convergence proof, while the other algorithm has a simpler constant step-size but has a more complicated proof. 2.1. Description of the algorithms The proposed extra anchored gradient (EAG) algorithms have the following general form: zk+1/2 = zk + \u03b2k(z0 \u2212zk) \u2212\u03b1kG(zk) zk+1 = zk + \u03b2k(z0 \u2212zk) \u2212\u03b1kG(zk+1/2) (3) for k \u22650, where z0 \u2208Rn\u00d7Rm is the starting point. We use G de\ufb01ned in (2) rather than describing the xand yupdates separately to keep the notation concise. We call \u03b1k > 0 step-sizes and \u03b2k \u2208[0, 1) anchoring coef\ufb01cients. Note that when \u03b2k = 0, EAG coincides with the unconstrained extragradient algorithm. The simplest choice of {\u03b1k}k\u22650 is the constant one. Together with the choice \u03b2k = 1 k+2 (which we clarify later), we get the following simpler algorithm. EAG with constant step-size (EAG-C) zk+1/2 = zk + 1 k + 2(z0 \u2212zk) \u2212\u03b1G(zk) zk+1 = zk + 1 k + 2(z0 \u2212zk) \u2212\u03b1G(zk+1/2) where \u03b1 > 0 is \ufb01xed. Theorem 1. Assume L: Rn \u00d7 Rm \u2192R is an R-smooth convex-concave function with a saddle point z\u22c6. Assume \u03b1 > 0 satis\ufb01es 1 \u22123\u03b1R \u2212\u03b12R2 \u2212\u03b13R3 \u22650 1 \u22128\u03b1R + \u03b12R2 \u22122\u03b13R3 \u22650. (4) Then EAG-C converges with rate \u2225\u2207L(zk)\u22252 \u22644(1 + \u03b1R + \u03b12R2) \u03b12(1 + \u03b1R) \u2225z0 \u2212z\u22c6\u22252 (k + 1)2 for k \u22650. Corollary 1. In the setup of Theorem 1, \u03b1 \u2208 \u00000, 1 8R \u0003 satis\ufb01es (4), and the particular choice \u03b1 = 1 8R yields \u2225\u2207L(zk)\u22252 \u2264260R2\u2225z0 \u2212z\u22c6\u22252 (k + 1)2 for k \u22650. While EAG-C is simple in its form, its convergence proof (presented in the appendix) is complicated. Furthermore, the constant 260 in Corollary 1 seems large and raises the question of whether it could be reduced. These issues, to some extent, are addressed by the following alternative version of EAG. EAG with varying step-size (EAG-V) zk+1/2 = zk + 1 k + 2(z0 \u2212zk) \u2212\u03b1kG(zk) zk+1 = zk + 1 k + 2(z0 \u2212zk) \u2212\u03b1kG(zk+1/2), where \u03b10 \u2208 \u00000, 1 R \u0001 and \u03b1k+1 = \u03b1k 1 \u2212\u03b12 kR2 \u0012 1 \u2212 (k + 2)2 (k + 1)(k + 3)\u03b12 kR2 \u0013 = \u03b1k \u0012 1 \u2212 1 (k + 1)(k + 3) \u03b12 kR2 1 \u2212\u03b12 kR2 \u0013 (5) for k \u22650. As the recurrence relation (5) may seem unfamiliar, we provide the following lemma describing the behavior of the resulting sequence. Lemma 1. If \u03b10 \u2208 \u00000, 3 4R \u0001 , then the sequence {\u03b1k}k\u22650 of (5) monotonically decreases to a positive limit. In particular, when \u03b10 = 0.618 R , we have limk\u2192\u221e\u03b1k \u22480.437 R . We now state the convergence results for EAG-V. Theorem 2. Assume L: Rn \u00d7 Rm \u2192R is an R-smooth convex-concave function with a saddle point z\u22c6. Assume \u03b10 \u2208 \u00000, 3 4R \u0001 , and de\ufb01ne \u03b1\u221e= limk\u2192\u221e\u03b1k. Then EAG-V converges with rate \u2225\u2207L(zk)\u22252 \u22644 \u00001 + \u03b10\u03b1\u221eR2\u0001 \u03b12 \u221e \u2225z0 \u2212z\u22c6\u22252 (k + 1)(k + 2) for k \u22650. \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm Corollary 2. EAG-V with \u03b10 = 0.618 R satis\ufb01es \u2225\u2207L(zk)\u22252 \u226427R2\u2225z0 \u2212z\u22c6\u22252 (k + 1)(k + 2) for k \u22650. 2.2. Proof outline We now outline the convergence analysis for EAG-V, whose proof is simpler than that of EAG-C. The key ingredient of the proof is a Lyapunov analysis with a nonincreasing Lyapunov function, the Vk of the following lemma. Lemma 2. Let {\u03b2k}k\u22650 \u2286(0, 1) and \u03b10 \u2208 \u00000, 1 R \u0001 be given. De\ufb01ne the sequences {Ak}k\u22650, {Bk}k\u22650 and {\u03b1k}\u22650 by the recurrence relations Ak = \u03b1k 2\u03b2k Bk (6) Bk+1 = Bk 1 \u2212\u03b2k (7) \u03b1k+1 = \u03b1k\u03b2k+1(1 \u2212\u03b12 kR2 \u2212\u03b22 k) \u03b2k(1 \u2212\u03b2k)(1 \u2212\u03b12 kR2) (8) for k \u22650, where B0 = 1. Suppose that \u03b1k \u2208(0, 1 R) holds for all k \u22650. Assume L is R-smooth and convex-concave. Then the sequence {Vk}k\u22650 de\ufb01ned as Vk := Ak\u2225G(zk)\u22252 + Bk\u27e8G(zk), zk \u2212z0\u27e9 (9) for EAG iterations in (3) is nonincreasing. In Lemma 2, the choice of \u03b2k = 1 k+2 leads to Bk = k + 1, Ak = \u03b1k(k+2)(k+1) 2 , and (5). Why the Lyapunov function of Lemma 2 leads to the convergence guarantee of Theorem 2 may not be immediately obvious. The following proof provides the analysis. Proof of Theorem 2. Let \u03b2k = 1 k+2 as speci\ufb01ed by the de\ufb01nition of EAG-V. By Lemma 2, the quantity Vk de\ufb01ned by (9) is nonincreasing in k. Therefore, Vk \u2264\u00b7 \u00b7 \u00b7 \u2264V0 = \u03b10\u2225G(z0)\u22252 \u2264\u03b10R2\u2225z0 \u2212z\u22c6\u22252. Next, we have Vk = Ak\u2225G(zk)\u22252 + Bk\u27e8G(zk), zk \u2212z0\u27e9 (a) \u2265Ak\u2225G(zk)\u22252 + Bk\u27e8G(zk), z\u22c6\u2212z0\u27e9 (b) \u2265Ak\u2225G(zk)\u22252 \u2212Ak 2 \u2225G(zk)\u22252 \u2212B2 k 2Ak \u2225z0 \u2212z\u22c6\u22252 (c) = \u03b1k 4 (k + 1)(k + 2)\u2225G(zk)\u22252 \u2212 k + 1 \u03b1k(k + 2)\u2225z0 \u2212z\u22c6\u22252 (d) \u2265\u03b1\u221e 4 (k + 1)(k + 2)\u2225G(zk)\u22252 \u2212 1 \u03b1\u221e \u2225z0 \u2212z\u22c6\u22252, where (a) follows from the monotonicity inequality \u27e8G(zk), zk \u2212z\u22c6\u27e9\u22650, (b) follows from Young\u2019s inequality, (c) follows from plugging in Ak = \u03b1k(k+1)(k+2) 2 and Bk = k + 1, and (d) follows from Lemma 1 (\u03b1k \u2193\u03b1\u221e). Reorganize to get \u03b1\u221e 4 (k + 1)(k + 2)\u2225G(zk)\u22252 \u2264Vk + 1 \u03b1\u221e \u2225z0 \u2212z\u22c6\u22252 \u2264 \u0012 \u03b10R2 + 1 \u03b1\u221e \u0013 \u2225z0 \u2212z\u22c6\u22252, and divide both sides by \u03b1\u221e 4 (k + 1)(k + 2). 2.3. Discussion of further generalizations The algorithms and results of Sections 2.1 and 2.2 remain valid when we replace G with an R-Lipschitz continuous monotone operator; neither the de\ufb01nition of the EAG algorithms nor any part of the proofs of Theorems 1 and 2 utilize properties of saddle functions beyond the monotonicity of their subdifferentials. For EAG-C, the step-size conditions (4) in Theorem 1 can be relaxed to accommodate larger values of \u03b1. However, we do not pursue such generalizations to keep the already complicated and arduous analysis of EAG-C manageable. Also, larger step-sizes are more naturally allowed in EAG-V and Theorem 2. Finally, although (4) holds for values of \u03b1 up to 0.1265 R , we present a slightly smaller range \u00000, 1 8R \u0003 in Corollary 1 for simplicity. For EAG-V, the choice \u03b2k = 1 k+2 was obtained by roughly, but not fully, optimizing the bound on EAG-V originating from Lemma 2. If one chooses \u03b2k = 1 k+\u03b4 with \u03b4 > 1, then (6) and (7) become Ak = \u03b1k(k + \u03b4)(k + \u03b4 \u22121) 2(\u03b4 \u22121) , Bk = k + \u03b4 \u22121 \u03b4 \u22121 . As the proof of Theorem 2 illustrates, linear growth of Bk and quadratic growth of Ak leads to O(1/k2) convergence of \u2225G(zk)\u22252. The value \u03b10 = 0.618 R in Lemma 1 and Corollary 2 was obtained by numerically minimizing the constant 4 \u03b12 \u221e \u00001 + \u03b10\u03b1\u221eR2\u0001 in Theorem 2 in the case of \u03b4 = 2. The choice \u03b4 = 2, however, is not optimal. Indeed, the constant 27 of Corollary 2 can be reduced to 24.44 with (\u03b4\u22c6, \u03b1\u22c6 0) \u2248(2.697, 0.690/R), which was obtained by numerically optimizing over \u03b4 and \u03b10. Finally, there is a possibility that a choice of \u03b2k not in the form of \u03b2k = 1 k+\u03b4 leads to an improved constant. In the end, we choose to present EAG-C and EAG-V with the simple choice \u03b2k = 1 k+2. As we establish in Section 3, the EAG algorithms are optimal up to a constant. \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm 3. Optimality of EAG via a matching complexity lower bound Upon seeing an accelerated algorithm, it is natural to ask whether the algorithm is optimal. In this section, we present a \u2126(R2/k2) complexity lower bound for the class of deterministic gradient-based algorithms for smooth convexconcave minimax problems. This result establishes that EAG is indeed optimal. For the class of smooth minimax optimization problems, a deterministic algorithm A produces iterates (xk, yk) = zk for k \u22651 given a starting point (x0, y0) = z0 and a saddle function L, and we write zk = A(z0, . . . , zk\u22121; L) for k \u22651. De\ufb01ne Asim as the class of algorithms satisfying zk \u2208z0 + span{GL(z0), . . . , GL(zk\u22121)}, (10) and Asep as the class of algorithms satisfying xk \u2208x0 + span \b \u2207xL(x0, y0), . . . , \u2207xL(xk\u22121, yk\u22121) \t yk \u2208y0 + span \b \u2207yL(x0, y0), . . . , \u2207yL(xk\u22121, yk\u22121) \t . (11) To clarify, algorithms in Asim access and utilize the xand ysubgradients simultaneously. So Asim contains simultaneous gradient descent, extragradient, Popov, and EAG (if we also count intermediate sequences zk+1/2 as algorithms\u2019 iterates). On the other hand, algorithms in Asep can access and utilize the xand y-subgradients separately. So Asim \u2282 Asep, and alternating gradient descent-ascent belongs to Asep but not to Asim. In this section, we present a complexity lower bound that applies to all algorithms in Asep, not just the algorithms in Asim. Although EAG-C and EAG-V are in Asim, we consider the broader class Asep to rule out the possibility that separately updating the xand y-variables provides an improvement beyond a constant factor. We say L(x, y) is biaf\ufb01ne if it is an af\ufb01ne function of x for any \ufb01xed y and an af\ufb01ne function of y for any \ufb01xed x. Biaf\ufb01ne functions are, of course, convex-concave. We \ufb01rst establish a complexity lower bound on minimiax optimization problems with biaf\ufb01ne loss functions. Theorem 3. Let k \u22650 be \ufb01xed. For any n \u2265k + 2, there exists an R-smooth biaf\ufb01ne function L on Rn \u00d7 Rn for which \u2225\u2207L(zk)\u22252 \u2265R2\u2225z0 \u2212z\u22c6\u22252 (2\u230ak/2\u230b+ 1)2 (12) holds for any algorithm in Asep, where \u230a\u00b7\u230bis the \ufb02oor function and z\u22c6is the saddle point of L closest to z0. Moreover, this lower bound is optimal in the sense that it cannot be improved with biaf\ufb01ne functions. Since smooth biaf\ufb01ne functions are special cases of smooth convex-concave functions, Theorem 3 implies the optimality of EAG applied to smooth convex-concave mimimax optimization problems. Corollary 3. For R-smooth convex-concave minimax problems, an algorithm in Asep cannot attain a worst-case convergence rate better than R2\u2225z0 \u2212z\u22c6\u22252 (2\u230ak/2\u230b+ 1)2 with respect to \u2225\u2207L(zk)\u22252. Since EAG-C and EAG-V have rates O(R2\u2225z0 \u2212z\u22c6\u22252/k2), they are optimal, up to a constant factor, in Asep. 3.1. Outline of the worst-case biaf\ufb01ne construction Consider biaf\ufb01ne functions of the form L(x, y) = \u27e8Ax \u2212b, y \u2212c\u27e9, where A \u2208Rn\u00d7n and b, c \u2208Rn. Then, \u2207xL(x, y) = A\u22ba(y \u2212c), \u2207yL(x, y) = Ax \u2212b, G is \u2225A\u2225-Lipschitz, and solutions to minimize x\u2208X maximize y\u2208Y \u27e8Ax \u2212b, y \u2212c\u27e9 are characterized by Ax \u2212b = 0 and A\u22ba(y \u2212c) = 0. Through translation, we may assume without loss of generality that x0 = 0, y0 = 0. In this case, (11) becomes xk \u2208span{A\u22bac, A\u22ba(AA\u22ba)c, . . . , A\u22ba(AA\u22ba)\u230ak\u22121 2 \u230bc} + span{A\u22bab, A\u22ba(AA\u22ba)b, . . . , A\u22ba(AA\u22ba)\u230ak 2 \u230b\u22121b} yk \u2208span{b, (AA\u22ba)b, . . . , (AA\u22ba)\u230ak\u22121 2 \u230bb} + span{AA\u22bac, . . . , (AA\u22ba)\u230ak 2 \u230bc} (13) for k \u22652. (We detail these arguments in the appendix.) Furthermore let A = A\u22baand b = A\u22bac = Ac. Then the characterization of Asep further simpli\ufb01es to xk, yk \u2208Kk\u22121(A; b) := span{b, Ab, A2b, . . . , Ak\u22121b}. Note that Kk\u22121(A; b) is the order-(k \u22121) Krylov subspace. Consider the following lemma. Its proof, deferred to the appendix, combines arguments from Nemirovsky (1991; 1992). Lemma 3. Let R > 0, k \u22650, and n \u2265k + 2. Then there exists A = A\u22ba\u2208Rn\u00d7n such that \u2225A\u2225\u2264R and b \u2208R(A), satisfying \u2225Ax \u2212b\u22252 \u2265 R2\u2225x\u22c6\u22252 (2\u230ak/2\u230b+ 1)2 (14) for any x \u2208Kk\u22121(A; b), where x\u22c6is the minimum norm solution to the equation Ax = b. \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm Take A and b as in Lemma 3 and c = x\u22c6. Then z\u22c6= (x\u22c6, x\u22c6) is the saddle point of L(x, y) = \u27e8Ax \u2212b, y \u2212c\u27e9 with minimum norm. Finally, \u2225\u2207L(xk, yk)\u22252 = \u2225A\u22ba(yk \u2212c)\u22252 + \u2225Axk \u2212b\u22252 = \u2225Ayk \u2212b\u22252 + \u2225Axk \u2212b\u22252 \u2265 R2\u2225x\u22c6\u22252 (2\u230ak/2\u230b+ 1)2 + R2\u2225x\u22c6\u22252 (2\u230ak/2\u230b+ 1)2 = R2\u2225z\u22c6\u2212z0\u22252 (2\u230ak/2\u230b+ 1)2 , for any xk, yk \u2208Kk\u22121(A; b). This completes the construction of the biaf\ufb01ne L of Theorem 3. 3.2. Optimal complexity lower bound We now formalize the notion of complexity lower bounds. This formulation will allow us to precisely state and prove the second statement of Theorem 3 regarding the optimality of the lower bound. Let F be a function class, PF = {Pf}f\u2208F a class of optimization problems (with some common form), and E(\u00b7; Pf) a suboptimality measure for the problem Pf. De\ufb01ne the worst-case complexity of an algorithm A for PF at the k-th iteration given the initial condition \u2225z0 \u2212z\u22c6\u2225\u2264D, as C (A; PF, D, k) := sup z0\u2208B(z\u22c6;D) f\u2208F E \u0000zk; Pf \u0001 , where zj = A(z0, . . . , zj\u22121; f) for j = 1, . . . , k and B(z; D) denotes the closed ball of radius D centered at z. The optimal complexity lower bound with respect to an algorithm class A is C (A; PF, D, k) : = inf A\u2208A C (A; PF, D, k) = inf A\u2208A sup z0\u2208B(z\u22c6;D) f\u2208F E \u0000zk; Pf \u0001 . A complexity lower bound is a lower bound on the optimal complexity lower bound. Let LR(Rn\u00d7Rm) be the class of R-smooth convex-concave functions on Rn \u00d7 Rm, PL the minimax problem (1), and E(z; PL) = \u2225\u2207L(z)\u22252. With this notation, the results of Section 2 can be expressed as C \u0000EAG; PLR(Rn\u00d7Rm), D, k \u0001 = O \u0012R2D2 k2 \u0013 . Let Lbiaff R (Rn \u00d7Rm) be the class of R-smooth biaf\ufb01ne functions on Rn \u00d7 Rm. Then the \ufb01rst statement of Theorem 3, the existence of L, can be expressed as C \u0010 Asep; PLbiaff R (Rn\u00d7Rn), D, k \u0011 \u2265 R2D2 (2\u230ak/2\u230b+ 1)2 (15) for n \u2265k + 2. As an aside, the argument of Corollary 3 can be expressed as: for any A \u2208Asep, we have C \u0000A; PLR(Rn\u00d7Rn), D, k \u0001 \u2265C \u0000Asep; PLR(Rn\u00d7Rn), D, k \u0001 \u2265C \u0010 Asep; PLbiaff R (Rn\u00d7Rn), D, k \u0011 \u2265 R2D2 (2\u230ak/2\u230b+ 1)2 . The \ufb01rst inequality follows from A \u2208Asep, the second from Lbiaff R \u2282LR, and the third from Theorem 3. Optimality of lower bound of Theorem 3. Using above notations, our goal is to prove that for n \u2265k + 2, C \u0010 Asep; PLbiaff R (Rn\u00d7Rn), D, k \u0011 = R2D2 (2\u230ak/2\u230b+ 1)2 . (16) We establish this claim with the chain of inequalities: R2D2 (2\u230ak/2\u230b+ 1)2 \u2264C \u0010 Asep; PLbiaff R (Rn\u00d7Rn), D, k \u0011 (17) \u2264C \u0010 Asim; PLbiaff R (Rn\u00d7Rn), D, k \u0011 (18) \u2264C \u0010 Alin; P2n,skew R,D , k \u0011 (19) \u2264C \u0000Alin; P2n R,D, k \u0001 (20) \u2264 R2D2 (2\u230ak/2\u230b+ 1)2 . (21) Inequality (17) is what we established in Section 3.1. Inequality (18) follows from Asim \u2282Asep and the fact that the in\ufb01mum over a larger class is smaller. Roughly speaking, the quantities in lines (19) and (20) are the complexity lower bounds for solving linear equations using only matrix-vector products, which were studied thoroughly in (Nemirovsky, 1991; 1992). We will show inequalities (19), (20), and (21) by establishing the connection of Nemirovsky\u2019s work with our setup of biaf\ufb01ne saddle problems. Once this is done, equality holds throughout and (16) is proved. We \ufb01rst provide the de\ufb01nitions. Let P2n R,D be the collection of linear equations with 2n \u00d7 2n matrices B satisfying \u2225B\u2225\u2264R and v = Bz\u22c6for some z\u22c6\u2208B(0; D). Let P2n,skew R,D \u2282P2n R,D be the subclass of equations with skewsymmetric B. Let Alin be the class of iterative algorithms solving linear equations Bz = v using only matrix multiplication by B and B\u22bain the sense that zk \u2208span{v0, . . . , vk}, (22) where v0 = 0, v1 = v, and for k \u22652, vk = Bvj or B\u22bavj for some j = 0, . . . , k \u22121. \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm The optimal complexity lower bound for a class of linear equation instances is de\ufb01ned as C \u0000Alin; P2n R,D, k \u0001 := inf A\u2208Alin sup \u2225B\u2225\u2264R v=Bz\u22c6,\u2225z\u22c6\u2225\u2264D \r \rBzk \u2212v \r \r2 . De\ufb01ne C \u0010 Alin; P2n,skew R,D , k \u0011 analogously. Now we relate the optimal complexity lower bounds for biaf\ufb01ne minimax problems to those for linear equations. For L(x, y) = b\u22bax + x\u22baAy \u2212c\u22bay, we have GL(x, y) = \u0014 O A \u2212A\u22ba O \u0015 \u0014x y \u0015 + \u0014b c \u0015 . Therefore, the minimax problem PL for L \u2208Lbiaff R (Rn \u00d7 Rn) is equivalent to solving the linear equation Bz = v with B = \u0014 O \u2212A A\u22ba O \u0015 and v = \u0014b c \u0015 \u2208R2n, which belongs to P2n,skew R,D with D = \u2225z\u22c6\u2225. For both algorithm classes Asim and Alin, we may assume without loss of generality that z0 = 0 through translation. Then, the span condition (10) for Asim becomes zk \u2208Kk\u22121(B; v). (23) Note that (22) reduces to (23) as B is skew-symmetric, so Asim and Alin are effectively the same class of algorithms under the identi\ufb01cation PLbiaff R (Rn\u00d7Rn) \u2282P2n,skew R,D . Since the supremum over a larger class of problems is larger, inequality (19) holds. Similarly, inequality (20) follows from P2n,skew R,D \u2282P2n R,D. Finally, (21) follows from the following lemma, using arguments based on Chebyshev-type matrix polynomials from Nemirovsky (1992). Its proof is deferred to the appendix. Lemma 4. Let R > 0 and k \u22650. Then there exists A \u2208 Alin such that for any m \u22651, B \u2208Rm\u00d7m, and v = Bz\u22c6 satisfying \u2225B\u2225\u2264R and \u2225z\u22c6\u2225\u2264D, the zk-iterate produced by A satis\ufb01es \r \rBzk \u2212v \r \r2 \u2264 R2D2 (2\u230ak/2\u230b+ 1)2 . 3.3. Broader algorithm classes via resisting oracles In (10) and (11), we assumed the subgradient queries are made within the span of the gradients at the previous iterates. This requirement (the linear span assumption) can be removed, i.e., a similar analysis can be done on general deterministic black-box gradient-based algorithms (formally de\ufb01ned in the appendix, Section C.5), using the resisting oracle technique (Nemirovsky & Yudin, 1983) at the cost of slightly enlarging the required problem dimension. We informally state the generalized result below and provide details in the appendix. Theorem 4 (Informal). Let n \u22653k + 2. For any gradientbased deterministic algorithm, there exists an R-smooth biaf\ufb01ne function L on Rn \u00d7 Rn such that (12) holds. Although we do not formally pursue this, the requirement that the algorithm is not randomized can also be removed using the techniques of Woodworth & Srebro (2016), which exploit near-orthogonality of random vectors in high dimensions. 3.4. Discussion We established that one cannot improve the lower bound of Theorem 3 using biaf\ufb01ne functions, arguably the simplest family of convex-concave functions. Furthermore, this optimality statement holds for both algorithm classes Asep and Asim as established through the chain of inequalities in Section 3.2. However, as demonstrated by Drori (2017), who introduced a non-quadratic lower bound for smooth convex minimization that improves upon the classical quadratic lower bounds of Nemirovsky (1992) and Nesterov (2013), a non-biaf\ufb01ne construction may improve the constant. In our setup, there is a factor-near-100 difference between the upper and lower bounds. (Note that each EAG iteration requires 2 evaluations of the saddle subdifferential oracle.) We suspect that both the algorithm and the lower bound can be improved upon, but we leave this to future work. Golowich et al. (2020) establishes that for the class of 1SCLI algorithms (S is for stationary), a subclass of Asim for biaf\ufb01ne objectives, one cannot achieve a rate faster than \u2225\u2207L(zk)\u22252 \u2264O(1/k). This lower bound applies to EG but not EAG; EAG is not 1-SCLI, as its anchoring coef\ufb01cients 1 k+2 vary over iterations, and its convergence rate breaks the 1-SCLI lower bound. On the other hand, we can view EAG as a non-stationary CLI algorithm (Arjevani & Shamir, 2016, De\ufb01nition 2). We further discuss these connections in the appendix, Section E. 4. Experiments We now present experiments illustrating the accelerated rate of EAG. We compare EAG-C and EAG-V against the prior algorithms with convergence guarantees: EG, Popov\u2019s algorithm (or optimistic descent) and simultaneous gradient descent with anchoring (SimGD-A). The precise forms of the algorithms are restated in the appendix. Figure 1(a) presents experiments on our \ufb01rst example, constructed as follows. For \u03f5 > 0, de\ufb01ne f\u03f5(u) = ( \u03f5|u| \u22121 2\u03f52 if |u| \u2265\u03f5, 1 2u2 if |u| < \u03f5. Next, for 0 < \u03f5 \u226a\u03b4 \u226a1, de\ufb01ne L\u03b4,\u03f5(x, y) = (1 \u2212\u03b4)f\u03f5(x) + \u03b4xy \u2212(1 \u2212\u03b4)f\u03f5(y), (24) \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm 100 101 102 103 104 105 Iteration count 10\u22128 10\u22126 10\u22124 10\u22122 100 ||G(zk)||2 EG Popov EAG-V EAG-C SimGD-A (a) Two-dimensional example L\u03b4,\u03f5 of (24) 100 101 102 103 104 105 106 Iteration count 10\u22124 10\u22122 100 102 104 106 ||G(zk)||2 EG Popov EAG-V EAG-C SimGD-A (b) Lagrangian of linearly constrained QP of (25) Figure 1. Plots of \u2225G(zk)\u22252 versus iteration count. Dashed lines indicate corresponding theoretical upper bounds. \u22121.0 \u22120.5 0.0 0.5 1.0 \u22121.0 \u22120.5 0.0 0.5 1.0 EG EAG-C SimGD-A (a) Discrete trajectories with L\u03b4,\u03f5 \u22121.0 \u22120.5 0.0 0.5 1.0 \u22121.0 \u22120.5 0.0 0.5 1.0 Regularized subdifferential flow Anchored subdifferential flow (b) Moreau\u2013Yosida regularized \ufb02ow with \u03bb = 0.01 and the anchored \ufb02ow with L(x, y) = xy Figure 2. Comparison of the discrete trajectories and their corresponding continuous-time \ufb02ow. Trajectories from EAG-C and SimGD-A virtually coincide and resemble the anchored \ufb02ow. However, SimGD-A progresses slower due to its diminishing step-sizes. where x, y \u2208R. Since f\u03f5 is a 1-smooth convex function, L\u03b4,\u03f5 has smoothness parameter 1, which is almost tight due to the quadratic behavior of L\u03b4,\u03f5 within the region |x|, |y| \u2264\u03f5. This construction was inspired by Drori & Teboulle (2014), who presented f\u03f5 as the worst-case instance for gradient descent. We choose the step-size \u03b1 = 0.1 as this value is comfortably within the theoretical range of convergent parameters for EG, EAG-C, and Popov. For EAG-V, we set \u03b10 = 0.1. We use N = 105, \u03b4 = 10\u22122, and \u03f5 = 5 \u00d7 10\u22125, and the initial point z0 has norm 1. Figure 1(b) presents experiments on our second example L(x, y) = 1 2x\u22baHx \u2212h\u22bax \u2212\u27e8Ax \u2212b, y\u27e9, (25) where x, y \u2208Rn, A \u2208Rn\u00d7n, b \u2208Rn, H \u2208Rn\u00d7n is positive semide\ufb01nite, and h \u2208Rn. Note that this is the Lagrangian of a linearly constrained quadratic minimization problem. We adopted this saddle function from Ouyang & Xu (2021), where the authors constructed H, h, A and b to provide a lower bound on duality gap. The exact forms of H, h, A, and b are restated in the appendix. We use n = 200, N = 106, \u03b1 = 0.5 for EG and Popov, \u03b1 = 0.1265 for EAG-C and \u03b10 = 0.618 for EAG-V. Finally, we use the initial point z0 = 0. ODE Interpretation Figure 2(a) illustrates the algorithms applied to (24). For |x|, |y| \u226b\u03f5, GL\u03b4,\u03f5(x, y) = \u0014(1 \u2212\u03b4)\u03f5 + \u03b4y (1 \u2212\u03b4)\u03f5 \u2212\u03b4x \u0015 \u2248\u03b4 \u0014 y \u2212x \u0015 , so the algorithms roughly behave as if the objective is the bilinear function \u03b4xy. When \u03b4 is suf\ufb01ciently small, trajectories of the algorithms closely resemble the corresponding continuous-time \ufb02ows with L(x, y) = xy. Csetnek et al. (2019) demonstrated that Popov\u2019s algorithm can be viewed as discretization of the Moreau\u2013Yosida regularized \ufb02ow \u02d9 z(t) = \u2212G\u2212(Id+\u03bbG)\u22121 \u03bb (z(t)) for some \u03bb > 0, \fAccelerated O(1/k2) Rate for Smooth Convex-Concave Minimax Problems on Squared Gradient Norm and a similar analysis can be performed with EG. This connection explains why EG\u2019s trajectory in Figure 2(a) and the regularized \ufb02ow depicted in Figure 2(b) are similar. On the other hand, EAG and SimGD-A can be viewed as a discretization of the anchored \ufb02ow ODE \u02d9 z(t) = \u2212G(z(t)) + 1 t (z0 \u2212z(t)). The anchored \ufb02ow depicted in Figure 2(b) approaches the solution much more quickly due to the anchoring term dampening the cycling behavior. The trajectories of EAG and SimGD-A iterates in Figure 2(a) are very similar to the anchored \ufb02ow. However, SimGD-A requires diminishing step-sizes 1\u2212p (k+1)p (both theoretically and experimentally) and therefore progresses much slower. 5." + } + ], + "Sehyun Kwon": [ + { + "url": "http://arxiv.org/abs/2310.18297v4", + "title": "Image Clustering Conditioned on Text Criteria", + "abstract": "Classical clustering methods do not provide users with direct control of the\nclustering results, and the clustering results may not be consistent with the\nrelevant criterion that a user has in mind. In this work, we present a new\nmethodology for performing image clustering based on user-specified text\ncriteria by leveraging modern vision-language models and large language models.\nWe call our method Image Clustering Conditioned on Text Criteria (IC|TC), and\nit represents a different paradigm of image clustering. IC|TC requires a\nminimal and practical degree of human intervention and grants the user\nsignificant control over the clustering results in return. Our experiments show\nthat IC|TC can effectively cluster images with various criteria, such as human\naction, physical location, or the person's mood, while significantly\noutperforming baselines.", + "authors": "Sehyun Kwon, Jaeseung Park, Minkyu Kim, Jaewoong Cho, Ernest K. Ryu, Kangwook Lee", + "published": "2023-10-27", + "updated": "2024-02-22", + "primary_cat": "cs.CV", + "cats": [ + "cs.CV", + "cs.AI" + ], + "main_content": "INTRODUCTION Image clustering has been studied as a prototypical unsupervised learning task, and it has been used to organize large volumes of visual data (Platt et al., 2003), to reduce the cost of labeling an unlabeled image dataset (Russell et al., 2008; Schmarje et al., 2022), and to enhance image retrieval systems (Wu et al., 2000; J\u00b4 egou and Chum, 2012). Modern deep image clustering methods are often evaluated against pre-defined class labels of datasets viewed as the ground truth. In practice, however, a user may have a criterion in mind for how to cluster or organize a set of images. The user may even want multiple clustering results of the same dataset based on different criteria. (See Figure 1.) But, classical clustering methods offer no direct mechanism for the user to control the clustering criterion; the clustering criteria for existing methods are likely determined by the inductive biases of the neural networks and the loss function, data augmentations, and feature extractors used within the method. This necessitates a new paradigm in image clustering, enabling diverse outcomes from a single dataset based on user-specified criteria and revolutionizing the conventional, implicitly dictated clustering processes. Recently, foundation models have received significant recent interest due to their ability to understand and follow human instructions at an unprecedented level. Large language models (LLMs) (Brown et al., 2020; Chowdhery et al., 2022; Touvron et al., 2023a;b; Chiang et al., 2023; OpenAI, 2023; Adams et al., 2023) perform remarkably well on a wide range of natural language tasks such as understanding, summarizing, and reasoning in zeroor few-shot settings. Vision-language models (VLMs) (Alayrac et al., 2022; Liu et al., 2023; Awadalla et al., 2023; Dai et al., 2023; Li et al., 2023a; Zhu et al., 2023; Gong et al., 2023) interpret natural language instructions in visual contexts and produce responses that seemingly exhibit in-depth image analyses and complex reasoning. In this work, we present a new methodology based on foundation models for performing image clustering based on user-specified criteria provided in natural language text. We call our method Image Clustering Conditioned on Text Criteria (IC|TC), and it represents a different paradigm of image clustering: the user directs the method with the relevant clustering criterion, the same dataset can be clustered with multiple different criteria, and if the clustering results are not satisfactory, the user can edit the text criterion to iteratively refine the clustering results. IC|TC requires a minimal and practical degree of human intervention and grants the user significant control over the clustering results in return, and we argue that this makes IC|TC more practical and powerful compared to the classical purely unsupervised clustering methods. 1 Work done at KRAFTON. 2 Our code is available at https://github.com/sehyunkwon/ICTC. 1 arXiv:2310.18297v4 [cs.CV] 22 Feb 2024 \fPublished as a conference paper at ICLR 2024 Blowing bubbles Applauding Jumping Shooting an arrow \u2026 Criterion Action Restaurant Educational institute Store Sports facility \u2026 Criterion Location Criterion Mood Joyful Adventurous Focused Relaxed (a) Sample images from the clustering results on the Stanford 40 Action dataset. Each result is obtained using a different text criterion: Action, Location, and Mood. Criterion Instrument (K=2) Brass instruments String instruments Trumpet Flute Cello Guitar Criterion Instrument (K=7) \u2026 (b) Sample images from the clustering results on the PPMI dataset using the text criterion Instrument with different cluster numbers K = 2 and 7. Figure 1: Sample images from clustering results of IC|TC. The method finds clusters consistent with the user-specified text criterion. Furthermore, IC|TC provides cluster names (texts above each image cluster) along with the clusters, enhancing the interpretability of clustering results. 2 \fPublished as a conference paper at ICLR 2024 1.1 CONTRIBUTION Our main contributions are the proposal of the novel task of image clustering conditioned on text criteria and our method IC|TC for solving this task. The task is interesting because the setup where the user is willing and able to provide a textual description of the clustering criterion is practical, arguably more practical than the classical purely unsupervised clustering setup. The method IC|TC is interesting because it leverages modern multi-modal foundation models and solves the task well; our experiments demonstrate that IC|TC can indeed produce satisfactory clustering results consistent with the user-specified criteria. 2 TASK DEFINITION: IMAGE CLUSTERING CONDITIONED ON ITERATIVELY REFINED TEXT CRITERIA The main task we consider in this work is defined as follows: Given a set of images, a number of clusters K, and a user-specified criterion expressed in natural language, partition the set of images into K clusters such that the semantic meanings of the clusters are distinguished in a manner that is consistent with the specified user criterion. Recent image clustering methods (Van Gansbeke et al., 2020; Park et al., 2021; Niu and Wang, 2021) find clusters that agree with pre-defined class labels for datasets such as CIFAR-10 (\u223c90% accuracy). The semantic meanings of the clusters tend to correspond to the category of the foreground object, and the inductive biases of the neural networks and the loss function, data augmentations, and feature extractors used within the method are likely the cause of the clusters being chosen in this manner. In a given setup, however, the clusters returned by such classical clustering methods may not be consistent with the relevant criterion that a user has in mind. Iterative refinement of text criteria. Under our main task, the text criterion is chosen through a process of iterative refinement: The user specifies a text criterion, performs clustering, examines the clustering results, and, if not satisfied, edits the text criterion to iteratively refine the clustering results. Sometimes, a user-defined text criterion immediately leads to a clustering result that is sufficiently consistent with what the user has in mind, but if not, this iterative prompt engineering procedure provides a practical means for converging to desired results. In practice, hyperparameters of classical clustering algorithms are chosen through an iterative process where the user inspects the clustering output and adjusts the parameters accordingly. In this work, we explicitly acknowledge the process of iteratively determining the text criterion and consider it to be part of the main task. Comparison with classical clustering. Our task differs from classical clustering in that the user provides information characterizing the relevant criterion by which the images should be clustered. In contrast, classical clustering methods are purely unsupervised and use no such information. Deep clustering methods are often evaluated against a pre-defined set of labels of a dataset, and such labels tend to focus on the type of object in the foreground. However, the question of how clustering algorithms could (or cannot) perform clustering with arbitrary criteria has been raised and studied in several prior works (Wolpert and Macready, 1997; Kleinberg, 2002; Caruana et al., 2006; Cui et al., 2007; von Luxburg et al., 2012; Caruana, 2013; McCarthy et al., 2020; Viswanathan et al., 2023). The use of user-defined text criteria makes our task not an instance of (classical) unsupervised clustering, but providing a text criterion is a necessary and practical intervention from the user if the goal is to perform clustering with arbitrary criteria. Comparison with zero-shot classification. Our task differs from zero-shot classification in that zero-shot classification requires a pre-defined set of classes, and the goal is merely to assign images to these classes. In contrast, our task requires both finding the clusters and assigning images to the clusters. In fact, zero-shot classification can be considered an instance of our task when the user explicitly and precisely describes all K clusters in the clustering criterion. 3 \fPublished as a conference paper at ICLR 2024 Figure 2: The IC|TC method. (Step 1) Vision-language model (VLM) extracts detailed relevant textual descriptions of images. (Step 2) Large language model (LLM) identifies the names of the clusters. (Step 3) LLM conducts clustering by assigning each description to the appropriate cluster. The entire procedure is guided by a user-specified text criterion (TC). (Optional TC Refinement). The user can update the text criterion if the clustering results are unsatisfactory. See Appendix B.4 for an unabridged sample output. 3 IC|TC: IMAGE CLUSTERING CONDITIONED ON TEXT CRITERIA Our main method consists of 3 stages with an optional iterative outer loop. The user-specified text criterion TC is incorporated into 3 stages via text prompts roughly of the following form. Pstep1(TC) = \"Characterize the image using a well-detailed description\" + TC Pstep2a(TC) = \"Given a description of an image, label the image\" + TC Pstep2b(TC, N, K) = \"Given a list of {N} labels, cluster them into {K} words\" + TC Pstep3(TC) = \"Based on the image description, determine the most appropriate cluster\" + TC The precise prompt for each experimental setup in this work is specified in Appendix B.3.1. 3.1 STEP 1: EXTRACT SALIENT FEATURES FROM THE IMAGE In Step 1, the vision-language model (VLM) extracts salient features from the image in the form of text descriptions. Step 1 Vision-language model (VLM) extracts salient features Input: Image Dataset Dimg, Text Criteria TC, Descriptions Ddes \u2190[] Output: Ddes 1: for img in Dimg do 2: Ddes.append( VLM(img, Pstep1(TC) ) //append image description to Ddes 3: end for The user\u2019s criterion TC determines the relevant features the VLM should focus on. For example, the user may wish to cluster with respect to the mood of a person in the image or the overall mood (atmosphere) of the scene. In such cases, the TC may slightly vary: Criterion 1: Focus on the mood of the person in the center. Criterion 2: Describe the general mood by inspecting the background. 4 \fPublished as a conference paper at ICLR 2024 3.2 STEP 2: OBTAINING CLUSTER NAMES In Step 2, the large language model (LLM) discovers the cluster names in two sub-steps. In Step 2a, the LLM outputs raw initial labels of the images. Since the number of distinct initial labels is usually larger than K, in Step 2b, the LLM aggregates the raw initial labels into appropriate names of K clusters. (Combining Steps 2a and 2b and asking the LLM to discover K cluster names from N image descriptions is infeasible due to the limited token lengths of the LLMs.) Step 2 Large Language Model (LLM) obtains K cluster names Input: Descriptions Ddes, Text Criteria TC, Dataset size N, Number of clusters K, Lraw \u2190[] Output: List of cluster names Cname 1: for description in Ddes do 2: Lraw.append( LLM(description + Pstep2a(TC)) ) //append raw label to Lraw 3: end for 4: Cname = LLM(Lraw+Pstep2b(TC, N, K)) //Step 2b can be further optimized The simplest instance of Step 2b, described above, directly provides Lraw, the full list of raw labels. However, we find that it is more efficient to convert Lraw to a dictionary with labels being the keys and numbers of occurrences of the labels being the values. When the same raw label occurs many times, this optimization significantly reduces the token length of the input to the LLM of Step 2b. Careful prompt engineering of Pstep2b(TC, N, K) allows the user to refine the clusters to be consistent with the user\u2019s criteria. For example, the user may append additional text prompts such as: When categorizing the classes, consider the following criteria: 1. Merge similar clusters. For example, [sparrow, eagle, falcon, owl, hawk] should be combined into \u2019birds of prey.\u2019 2. Clusters should be differentiated based on the animal\u2019s habitat. 3.3 STEP 3: CLUSTERING BY ASSIGNING IMAGES In Step 3, images are assigned to one of the final K clusters. The text criterion TC, text description of the images from Step 1, and the K cluster names from Step 2 are provided to the LLM. Step 3 Large Language Model (LLM) assigns clusters to images Input: Descriptions Ddes, Text Criteria TC, List of cluster names Cname, RESULT\u2190[] Output: RESULT 1: for description in Ddes do 2: RESULT.append( LLM(description+Pstep3(TC)) ) //append assigned cluster 3: end for 3.4 ITERATIVELY EDITING THE ALGORITHM THROUGH TEXT PROMPT ENGINEERING Main method IC|TC Input: Dataset Dimg, Text Criteria TC, ADJUST \u2190True 1: while ADJUST do 2: RESULT \u2190do Steps 1\u20133 conditioned on TC 3: if User determines RESULT satisfactory then 4: ADJUST \u2190False 5: else 6: TC \u2190Update TC //user writes updated TC 7: end if 8: end while Our main method IC|TC is described above. Upon performing the clustering once, if the clusters are not sufficiently consistent with the specified text criterion TC or if the TC turns out to not precisely specify what the user had in mind, the user can update the TC. This iterative process may continue until the clustering result is satisfactory, as judged by the user. 5 \fPublished as a conference paper at ICLR 2024 Table 1: Clustering with varying text criteria. Accuracies labeled with * are evaluated by having a human provide ground truth labels for 1000 randomly sampled images. In this experiment, we used LLaVA for VLM and GPT-4 for LLM. Dataset Criterion SCAN Ours Stanford 40 Action Action 0.397 0.774 Location 0.359* 0.822* Mood 0.250* 0.793* PPMI M.I. (K=7) 0.632 0.964 M.I. (K=2) 0.850 0.977 Location (K=2) 0.512 0.914 CIFAR-10-Gen Object 0.989 0.987 10 11 10 12 Number of Parameters 20 40 60 80 100 Accuracy (%) CIFAR-10 STL-10 CIFAR-100 LLAVA only Llama 2 (7B) Llama 2 (13B) Llama 2 (70B) GPT-3.5 GPT-4 Figure 3: Effect of LLM selection. 3.5 PRODUCING CLUSTER LABELS Classically, the unsupervised clustering task does not require the method to produce labels or descriptions of the output clusters. Notably, however, IC|TC produces names describing the clusters. This is a significant advantage of IC|TC as it makes the clustering results more directly and immediately interpretable. 4 EXPERIMENTS We now present experimental results demonstrating the effectiveness of IC|TC. In this section, we partially describe the settings and results while deferring much of the details to the appendix. In particular, the precise text prompts used can be found in Appendix B.3.1. IC|TC crucially relies on the use of foundation models, specifically a vision-language model (VLM) and a large language model (LLM) that have undergone instruction tuning. In our experiments, we mainly use LLaVA (Liu et al., 2023) for the VLM and GPT-4 (OpenAI, 2023) for the LLM, but Section 4.5 and Appendix B.2 presents ablation studies investigating how the performance is affected when other foundation models are used. 4.1 CLUSTERING WITH VARYING TEXT CRITERIA In this experiment, we show that varying the text criterion TC indeed leads to varying clustering results of a single image dataset. The results demonstrate that IC|TC is highly flexible and can accommodate a variety of text criteria. We use the Stanford 40 Action Dataset (Yao et al., 2011), which contains 9,532 images of humans performing various actions. The dataset comes with image labels describing a subject\u2019s action among 40 classes, such as reading, phoning, blowing bubbles, playing violin, etc. We additionally define two different collections of labels. The first collection contains 10 classes describing the location, such as restaurant, store, sports facility, etc. The second collection contains 4 classes describing the mood of the scene, specifically joyful, adventurous, relaxed, and focused. We utilize three text criteria, Action, Location, and Mood, to obtain three distinct clustering results. We evaluate the results based on how accurately the methods recover the three collections of labels described previously. This degree of control would be difficult or impossible for classical deep clustering methods. We compare our results against the prior deep clustering method SCAN (Van Gansbeke et al., 2020) and present the results in Table 1. Image samples are in Figure 1a. (Note that we do not have the 9,532 ground truth labels for the Location and Mood criteria. Therefore, we evaluate accuracy by having a human provide ground truth labels on 1000 randomly sampled images.) 6 \fPublished as a conference paper at ICLR 2024 Figure 4: Sample images from the clustering results on the PPMI dataset using text criterion Location and cluster number K = 2. 4.2 CLUSTERING WITH VARYING GRANULARITY In this experiment, we show that IC|TC can automatically control the granularity of clustering results by adjusting K, the number of clusters. We find that the cluster descriptions returned by IC|TC are highly interpretable and that the images are assigned to the clusters well for various values of K. We use the People Playing Musical Instrument (PPMI) dataset (Wang et al., 2010; Yao and Fei-Fei, 2010), which contains 1,200 images of humans interacting with 12 different musical instruments. We select 700 images across 7 classes from the original dataset to reduce the size and difficulty of the task. We use the text criterion Musical Instrument with number of clusters K = 2 and K = 7. With K = 7, images are indeed grouped into clusters such as violin, guitar, and other specific instruments, and 96.4% accuracy against the ground truth label of PPMI is achieved. With K = 2, images are divided into 2 clusters of brass instrument and string instrument and achieve a 97.7% accuracy. To clarify, we did not specifically instruct IC|TC to group the 7 instruments into brass and string instruments; the hierarchical grouping was discovered by IC|TC. As an additional experiment, we also cluster the same set of images with the text criterion Location and K = 2. In this case, the images are divided into 2 clusters of indoor and outdoor, and achieve a 91.4% accuracy. We again compare our results against SCAN (Van Gansbeke et al., 2020) and present the results in Table 1. Image samples are provided in Figure 4. 4.3 COMPARISON WITH CLASSICAL CLUSTERING METHODS In this experiment, we compare IC|TC against several classical clustering algorithms on CIFAR10, STL-10, and CIFAR-100. The three datasets have 10, 10, and 20 classes and 10,000, 8,000, and 10,000 images, respectively. We use the text criterion Object with the number of clusters equal to the number of classes in the dataset. The results in Table 2 show that IC|TC significantly outperforms classical clustering methods on CIFAR-10, STL-10 and CIFAR-100. Clustered sample images are provided in Appendix B.6. This comparison is arguably unfair against the classical clustering methods as they do not utilize foundation models or any pre-trained weights. Nevertheless, our results demonstrate that IC|TC is competitive when the goal is to cluster images based on the foreground object type. 4.4 FAIR CLUSTERING THROUGH TEXT CRITERION REFINEMENT Existing clustering methods sometimes exhibit biased results, and measures to mitigate such biases have been studied (Li et al., 2020; Zeng et al., 2023). Since foundation models are known to learn biases in their training data (Bommasani et al., 2022), IC|TC has the risk of propagating such biases into the clustering results. In this experiment, we show that by simply adding a prompt along the line of \"Do not consider gender\" to the text criterion, we can effectively mitigate biases in the clustering results. FACET (Gustafson et al., 2023) is a benchmark dataset for evaluating the robustness and algorithmic fairness of AI and machine-learning vision models. It comprises 32,000 diverse images labeled with 7 \fPublished as a conference paper at ICLR 2024 Table 2: Comparison with classical clustering methods using criterion Object. IC|TC outperforms state-of-the-art methods on CIFAR-10, STL-10 and CIFAR-100. Method CIFAR-10 STL-10 CIFAR-100 ACC \u2191 NMI \u2191 ARI \u2191 ACC \u2191 NMI \u2191 ARI \u2191 ACC \u2191 NMI \u2191 ARI \u2191 IIC (Ji et al. (2019)) 0.617 0.511 0.411 0.596 N/A N/A 0.257 N/A N/A SCAN (Van Gansbeke et al. (2020)) 0.883 0.797 0.772 0.809 0.698 0.646 0.507 0.468 0.301 SPICE (Niu and Wang (2021)) 0.926 0.865 0.852 0.938 0.872 0.870 0.584 0.583 0.422 RUC (Park et al. (2021)) 0.903 N/A N/A 0.867 N/A N/A 0.543 N/A N/A TCL (Yunfan et al. (2022)) 0.887 0.819 0.780 0.868 0.799 0.757 0.531 0.529 0.357 LLaVA only 0.647 0.455 0.442 0.774 0.587 0.589 0.097 0.022 0.014 Ours (LLaVA + Llama 2) 0.884 0.789 0.759 0.974 0.939 0.944 0.526 0.554 0.374 Ours (BLIP-2 + GPT-4) 0.975 0.941 0.947 0.993 0.982 0.985 0.584 0.690 0.429 Ours (LLaVA + GPT-4) 0.910 0.823 0.815 0.986 0.966 0.970 0.589 0.642 0.422 Actual Craftsman Prediction Laborer Samples (a) Biased results Craftsman Laborer Dancer Gardener Craftsman Laborer Dancer Gardener 0 50 Base prompt Fair prompt Male Ratio (%) Female Ratio (%) (b) Gender ratio Figure 5: (a) Biased results showing that male \u2018Craftsman\u2019 tend to be misclassified as \u2018Laborer\u2019. (b) Gender ratio of each cluster. When the ratio between males and females differs by more than 10%, the bar is colored red. Bias is mitigated by refining the text criterion into a \u2018Fair prompt\u2019. several attributes, including 52 occupation classes. For this experiment, we sampled 20 images each for men and women from the craftsman, laborer, dancer, and gardener occupation classes, 160 images in total. For this experiment, we define fairness to be achieved when each cluster maintains an equal proportion of genders. When we use the text criterion Occupation, IC|TC exhibited a gender bias. To mitigate this bias, we introduced a simple negative prompt, instructing IC|TC to not take gender into consideration and instead to focus on the activity. When the clustering was repeated, the results were promising: the gender ratio disparities in the craftsman and laborer clusters improved by 27.2% \u21924.4% and 11.6% \u21923.2%, respectively. Furthermore, the Dancer and Gardener clusters also experienced marginal reductions in disparities by 2.8% \u21922.6% and 10.6% \u21929.0%, respectively. The results are shown in Figure 5. 4.5 FURTHER ANALYSES Ablation studies of LLMs and VLMs. We conduct an ablation study to evaluate whether LLMs actually serve a significant role in our methodology since one may wonder whether the visionlanguage model (VLMs) alone is sufficient. When we perform a \u2018LLaVA only\u2019 experiment that does not utilize an LLM, the performance is considerably lower. However, when we use LLMs of varying sizes, the performance is not affected significantly. The results and details are provided in Figure 3 and Appendix A.2. The results lead us to conclude that the LLM serves a crucial role (the VLM by itself is not sufficient), but the size of the LLM does not seem to be very important. We also fix the LLM to GPT-4 and perform an ablation study on the choice of vision-language model (VLM). As an image captioning model, ClipCap (Mokady et al., 2021) cannot perform text conditioning, and this leads to poor performance. Blip-2 (Li et al., 2023b) and LLaVA (Liu et al., 2023) can extract information relevant to the text criteria, and they exhibit strong strong performance. The results and details are provided in Appendix A.1. 8 \fPublished as a conference paper at ICLR 2024 Data Contamination. When evaluating research using foundation models, the potential of data contamination is a significant concern (Wei et al., 2022; Du et al., 2022). The datasets we use to measure accuracy, namely CIFAR10, STL-10, CIFAR-100, and Stanford 40 Action, may have been used in the training of LLaVA. If so, the validity of the accuracy measurements comes into question. To address this concern, we conducted an experiment with synthetically generated images. Specifically, we use Stable Diffusion XL (Rombach et al., 2022) and the CIFAR-10 labels to generate 1000 CIFAR-10-like images, and we call this dataset CIFAR-10-Gen. See Appendix B for further details. On this synthetic data, IC|TC achieves 98.7% accuracy. The fact that the accuracy on CIFAR-10Gen is no worse than the accuracy on the actual CIFAR-10 dataset gives us confidence that the strong performance of IC|TC is likely not due to data contamination. (Strictly speaking, the training data for Stable Diffusion may contain the CIFAR-10 images, and if so, we are not completely free from the risk of data contamination. However, the CIFAR-10-Gen dataset does not seem to contain exact copies of CIFAR-10 images, and we argue that the synthetic generation significantly mitigates the risk of data contamination.) 5 RELATED WORK Image clustering. Modern deep clustering methods (Van Gansbeke et al., 2020; Park et al., 2021; Niu and Wang, 2021; Yunfan et al., 2022) adopt a multi-stage training approach. They begin with representation learning, which finds a representation that maps similar images to similar features, and then perform unsupervised clustering based on these feature representations. Additionally, to obtain more meaningful semantics, Zhong et al. (2021); Shen et al. (2021) proposed contrastive learning at not only the instance level but also at the cluster level. Misra and Maaten (2020); Cho et al. (2021); Kwon et al. (2023); Long et al. (2023); Metaxas et al. (2023) proposed specially designed representation learning for certain clustering criteria. The concurrent work Li et al. (2023c) is particularly relevant to our work as it presents Text-Aided Clustering (TAC), which leverages text as external knowledge to enhance image clustering performance. Specifically, Li et al. (2023c) enhanced feature discriminability by selecting specific WordNet nouns of images and mutually distilled the neighborhood information between the text and image modalities. Foundation models. In recent years, foundation models have been improving at a remarkable pace, and combined with instruction tuning (Sanh et al., 2022; Ouyang et al., 2022; Wei et al., 2022), these foundation models can be applied more flexibly to downstream tasks. Vision-Language Models (VLMs) (Alayrac et al., 2022; Liu et al., 2023; Awadalla et al., 2023; Dai et al., 2023; Li et al., 2023a; Zhu et al., 2023; Gong et al., 2023) can provide users with appropriate descriptions of given images according to the requirements of the input prompt. Large language models (LLMs) (Chowdhery et al., 2022; Touvron et al., 2023a;b; OpenAI, 2023) exhibit remarkable abilities in a wide range of natural language processing tasks such as text summarization. Recently, Radford et al. (2021); Jia et al. (2021); Li et al. (2022); Dinh et al. (2022); Geng et al. (2023); Menon and Vondrick (2023); Zhang et al. (2022); Cai et al. (2023); Ren et al. (2023) have shown computer vision problems with no direct connection to language can be successfully addressed using large language models. Image retrieval. Image retrieval aims to find images from a database that are relevant to a given query. This crucially differs from clustering in that clustering requires both finding the clusters and assigning the images to them; image retrieval techniques are very relevant to the sub-task of cluster assignment but not to the sub-task of finding the clusters. The fundamental approach in image retrieval is to assess the similarity among image features. Current approaches focus on two kinds of image representations: global features and local features. For global representations, Babenko et al. (2014); Tolias et al. (2015); Gordo et al. (2016); Cao et al. (2020); Lee et al. (2023) extracts activations from deep CNNs and aggregates them for obtaining global features. For local representations, Yi et al. (2016); Noh et al. (2017); Vassileios Balntas and Mikolajczyk (2016); DeTone et al. (2018); He et al. (2018); Dusmanu et al. (2019); Revaud et al. (2019) proposed well-embedded representations for all regions of interest. Recent state-of-the-art methods (Noh et al., 2017; Simeoni et al., 2019; Cao et al., 2020; Zhang et al., 2023; Wu et al., 2023) typically followed a two-stage paradigm: initially, candidates are retrieved using global features, and then they are re-ranked with local features. Recently, Vo et al. (2019); Liu et al. (2021); Baldrati et al. (2022); Tian et al. (2023) proposed to condition retrieval on user-specified language. 9 \fPublished as a conference paper at ICLR 2024 ACKNOWLEDGMENTS EKR was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) [NRF-2022R1C1C1010010] and the Creative-Pioneering Researchers Program through Seoul National University. SK and EKR were partly supported by the Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) [NO.2021-0-01343-004, Artificial Intelligence Graduate School Program (Seoul National University)]. We thank Dimitris Papailiopoulos and Yong Jae Lee for providing insightful discussion. We thank Byeong-Uk Lee for providing valuable feedback on the manuscript. ETHICS STATEMENT Our methodology provides users with direct control over the clustering results, but this agency could be used maliciously to produce unfair and discriminatory results. However, it is unlikely that our work will be responsible for new unfair results that could not already be produced with a malicious user\u2019s direct and overt intervention. On the other hand, it is possible for biases already in foundation models to propagate into our clustering methodology. Section 4.4 explicitly discusses this possibility and offers measures to mitigate such biases, and a well-intentioned user following the guidance of Section 4.4 is unlikely to amplify biases in the foundation models through the use of our method. REPRODUCIBILITY STATEMENT In this work, we use publically available datasets, describe the methodology in precise detail, and make our code available at https://github.com/sehyunkwon/ICTC. Of the two main foundation models we use, the vision-language model LLaVA (Liu et al., 2023) is fully open-source. However, the large language model GPT-4 (OpenAI, 2023) is a proprietary model, and we accessed it through the API offered by OpenAI. The API cost to conduct the experiments presented in this work was less than $3,000 (USD), so we argue that the proprietary API cost does not pose a significant barrier in terms of reproducibility. However, if OpenAI were to discontinue access to the GPT-4 version that we used, namely api-version=2023-03-15-preview, or if OpenAI discontinues access to GPT-4 altogether, then our experiments will no longer be exactly reproducible. To address this concern, we carry out an ablation study that uses the open-source large language model Llama 2 (Touvron et al., 2023b) and observe that a similar, albeit slight worse, performance is attained. See Figure 3 and Appendix A.2. Therefore, even if GPT-4 becomes unavailable in the future, the results of this work will be similarly reproducible by using Llama 2 or any other large language model of power comparable to or stronger than Llama 2 and GPT-4." + }, + { + "url": "http://arxiv.org/abs/2304.13995v2", + "title": "Rotation and Translation Invariant Representation Learning with Implicit Neural Representations", + "abstract": "In many computer vision applications, images are acquired with arbitrary or\nrandom rotations and translations, and in such setups, it is desirable to\nobtain semantic representations disentangled from the image orientation.\nExamples of such applications include semiconductor wafer defect inspection,\nplankton microscope images, and inference on single-particle cryo-electron\nmicroscopy (cryo-EM) micro-graphs. In this work, we propose Invariant\nRepresentation Learning with Implicit Neural Representation (IRL-INR), which\nuses an implicit neural representation (INR) with a hypernetwork to obtain\nsemantic representations disentangled from the orientation of the image. We\nshow that IRL-INR can effectively learn disentangled semantic representations\non more complex images compared to those considered in prior works and show\nthat these semantic representations synergize well with SCAN to produce\nstate-of-the-art unsupervised clustering results.", + "authors": "Sehyun Kwon, Joo Young Choi, Ernest K. Ryu", + "published": "2023-04-27", + "updated": "2023-06-12", + "primary_cat": "cs.CV", + "cats": [ + "cs.CV", + "cs.AI" + ], + "main_content": "Introduction In many computer vision applications, images are acquired with arbitrary or random rotations and translations. Examples of such applications include semiconductor wafer defect inspection (Wang, 2008; Wang & Chen, 2019; 2020), plankton microscope images (Zhao et al., 2009), and inference on single-particle cryo-electron microscopy (cryo-EM) micrographs (Zhong et al., 2021). In such applications, the rotation and translation of images serve as nuisance parameters (Cox & Hinkley, 1979, \u00a77.3) that may interfere with the 1Interdisciplinary Program in Articifial Intelligence, Seoul National University 2Department of Mathematical Sciences, Seoul National University. Correspondence to: Ernest K. Ryu . Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s). inference of the semantic meaning of the image. Therefore, it is desirable to obtain semantic representations that are not dependent on such nuisance parameters. Obtaining low-dimensional \u201cdisentangled\u201d representations is an active area of research in the area of representation learning. Prior works such as \u03b2-VAE (Higgins et al., 2017) and Info-GAN (Chen et al., 2016) propose general methods for disentangling latent representations so that components correspond to semantically independent factors. However, such fully general approaches are limited in the extent of disentanglement that they can accomplish. Alternatively, Spatial-VAE (Bepler et al., 2019) and TARGET-VAE (Nasiri & Bepler, 2022) explicitly, and therefore much more effectively, disentangle nuisance parameters from the semantic representation using an encoder with a so-called spatial generator. However, we find that these prior methods are difficult to train on more complex datasets such as semiconductor wafer maps or plankton microscope images, as we demonstrate in Section 3.4. We also find that the learned representations do not synergize well with modern deep-learning-based unsupervised clustering methods, as we demonstrate in Section 4.3. In this work, we propose Invariant Representation Learning with Implicit Neural Representation (IRL-INR), which uses an implicit neural representation (INR) with a hypernetwork to obtain semantic representations disentangled from the orientation of the image. Through our experiments, we show that IRL-INR can learn disentangled semantic representations on more complex images. We also show that these semantic representations synergize well with SCAN (Van Gansbeke et al., 2020) to produce state-of-the-art clustering results. Finally, we show a scaling phenomenon in which the clustering performance improves as the dimension of the semantic representation increases. 2. Related Works Disentangled representation learning. Finding disentangled latent representations corresponding to semantically independent factors is a classical problem in machine learning (Comon, 1994; Hyv\u00a8 arinen & Oja, 2000; Shakunaga & Shigenari, 2001). Recently, generative models have been 1 arXiv:2304.13995v2 [cs.CV] 12 Jun 2023 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations used extensively for this task. DR-GAN (Tran et al., 2017), TC-\u03b2-VAE (Chen et al., 2018), DIP-VAE (Kumar et al., 2018), Deformation Autoencoder (Shu et al., 2018), \u03b2-VAE (Higgins et al., 2017), StyleGAN (Karras et al., 2019), and Locatello et al. (2020) are prominent prior work finding disentangled representations of images. However these methods are post-hoc approaches that do not explicitly structure the latent space to separate the semantic representations from the known factors to be disentangled. In contrast, Spatial-VAE (Bepler et al., 2019) attempts to explicitly separate latent space into semantic representation of a image and its rotation and translation information, but only the generative part of spatial-VAE ends up being equivariant to rotation and translation. TARGET-VAE (Nasiri & Bepler, 2022) is the first method to successfully disentangle rotation and translation information from the semantic representation in an explicit manner. However, we find that TARGET-VAE fails to obtain meaningful semantic representation of complex data such as semiconductor wafer maps and plankton image considered in Figure 2. Invariant representation learning. Recently, contrastive learning methods have been widely used to learn invariant representations (Wang & Gupta, 2015; Sermanet et al., 2018; Wu et al., 2018; Dwibedi et al., 2019; Hjelm et al., 2019; He et al., 2020; Misra & Maaten, 2020; Chen et al., 2020; Yeh et al., 2022). Contrastive learning maximizes the similarity of positive samples generated by data augmentation and maximizes dissimilarity to negative samples. Since positive samples are defined by data augmentation such as rotation, translation, crop, color jitter and etc., contrastive learning forces data representations to be invariant under the designated data augmentation. Siamese networks is another approach for learning invariant representation (Bromley et al., 1993). The approach is to maximize the similarity between an image and its augmented image. Since only maximizing similarity may lead to a bad trivial solution, having an additional constraint is essential. For example, momentum encoder (Grill et al., 2020), stop gradient method (Chen & He, 2021), and reconstruction loss (Chen & Salman, 2011; Giancola et al., 2019; Zhou et al., 2020; Liu et al., 2020) were used to avoid the trivial solution. Our IRL-INR methodology can be interpreted as an instance of the Siamese network that uses reconstruction loss as a constraint. Implicit neural representations. It is natural to view an image as a discrete and finite set of measurements of an underlying continuous signal or image. To model this view, Stanley (2007) proposed using a neural network to represent a function f that can be evaluated at any input position (x, y) as a substitute for the more conventional approach having a neural network to output a 2D array representing an image. The modern literature now refers to this approach as an Implicit Neural Representation (INR). For example, Dupont et al. (2022); Sitzmann et al. (2019; 2020) uses deep neural networks to parameterize images and uses hypernetworks to obtain the parameters of such neural networks representing a continuous image (Ha et al., 2017). Taking the coordinate as an input makes INR, by definition, symmetric or equivariant under rotation and translation. Leveraging the equivariant structure of INR, Bepler et al. (2019); Mildenhall et al. (2020); Anokhin et al. (2020); Zhong et al. (2021); Karras et al. (2021); Deng et al. (2021); Chen et al. (2021); Nasiri & Bepler (2022) proposed the generative networks that are equivariant under rotation or translation, and our method uses the equivariance property to learn invariant representations. Deep clustering. Representation learning plays an essential role in modern deep clustering. Many deep-learningbased clustering methods utilize a pretext task to extract a clustering-friendly representation. Early methods such as Tian et al. (2014); Xie et al. (2016) used the auto-encoder to learn low-dimensional representation space and directly clustered on this obtained representation space. Later, Ji et al. (2017); Zhou et al. (2018); Zhang et al. (2021) proposed a subspace representation learning as a pretext task, where images are well separated by mapping into a suitable low-dimensional subspace. More recently, Van Gansbeke et al. (2020); Dang et al. (2021); Li et al. (2021); Shen et al. (2021) established state-of-the-art performance on many clustering benchmarks by utilizing contrastive learningbased pretext tasks such as SimCLR (Chen et al., 2020) or MOCO (He et al., 2020). However, none of the pretext tasks considered in prior work explicitly take into account rotation and translation invariant clustering. 3. Method Our method Invariant Representation Learning with Implicit Neural Representation (IRL-INR) obtains a representation that disentangles the semantic representation from the rotation and translation of the image, using an implicit neural representation (INR) with a hypernetwork. Our main framework is illustrated in Figure 1, and we describe the details below. 3.1. Data and its measurement model Our data J(1), . . . , J(N) are images with resolution P (number of pixels) and C color channels. In the applications we consider, C = 1 or C = 3. We index the images with the spatial indices reshaped into a single dimension, so that J(i) \u2208RC\u00d7P and J(i) p \u2208RC, p = 1, . . . , P 2 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations I J Encoder E\u03d5 \u02c6 \u03b8 \u02c6 \u03c4 z S\u02c6 \u03b8,\u02c6 \u03c4(xp, yp) (xp, yp) Hypernetwork H\u03c8 INR network I(\u00b7, \u00b7; \u03b7) \u2243 \u03b7 M T \u25e6R Figure 1. The IRL-INR framework. Encoder E\u03d5 takes an image J as input and outputs rotation representation \u02c6 \u03b8, translation representation \u02c6 \u03c4 and semantic representation z. Hypernetwork H\u03c8 takes z as an input and then outputs the weights and biases of INR network. INR network I outputs the pixel (image) value corresponding to the input (x, y) coordinate. for i = 1, . . . , N. We assume J(i) represents measurements of a true underlying continuous image I(i) that has been randomly rotated and translated for i = 1, . . . , N. We further detail our measurement model below. We assume there exist continuous 2-dimensional images I(1), . . . , I(N) (so I(i)(x, y) \u2208RC for any x, y \u2208R) . We observe/measure a randomly rotated and translated version of I(1), . . . , I(N) on a discretized finite grid, to obtain J(1), . . . , J(N). Mathematically, we write J(i) = M[T\u03c4 (i)[R\u03b8(i)[I(i)]]], i = 1, . . . , N, where R\u03b8(i) denotes rotation by angle \u03b8(i) \u2208[0, 2\u03c0), T\u03c4 (i) denotes translation by direction \u03c4 (i) \u2208R2, and M is a measurement operator that measures a continuous image on a finite grid. More specifically, given a continuous image \u02dc I, the measurement M[\u02dc I] is a finite image (M[\u02dc I])p = \u02dc I(xp, yp) \u2208RC, p = 1, . . . , P with a pre-specified set of gridpoints {(xp, yp)}P p=1, which we take to be a uniform grid on [\u22121, 1]2. Throughout this work, we assume that \u03b8(1), . . . , \u03b8(N) IID \u223cUniform([0, 2\u03c0]), i.e., that the rotations sampled uniformly at random, and that translations \u03c4 (1), . . . , \u03c4 (N) are sampled IID from some distribution. To clarify, we do not have access to the true underlying continuous images I(1), . . . , I(N), so we do not use them on our framework. Also, the rotation \u03b8(i) and translation \u03c4 (i) of I(i) that produced the observed image J(i) for i = 1, . . . , N are impossible to learn without additional supervision, so we do not attempt to learn it. 3.2. Implicit neural representation with a hypernetwork Our framework takes in, as input, a discrete image J, which we assume originates from a true underlying continuous image I. The framework, as illustrated in Figure 1, uses the rotation and translation operators R\u03b8 and T\u03c4 and three neural networks E\u03d5, H\u03c8, and I. Define the rotation operation R\u03b8 and translation operation T\u03c4 on points and images as follows. For notational convenience, define S\u03b8,\u03c4 = R\u03b8 \u25e6T\u03c4. When translating and rotating a point in R2, define S\u03b8,\u03c4 as S\u03b8,\u03c4(x, y) = \u0014 cos \u03b8 \u2212sin \u03b8 sin \u03b8 cos \u03b8 \u0015 \u0012\u0014 x y \u0015 + \u03c4 \u0013 \u2208R2. For rotating and translating a continuous image I, define S\u22121 \u03b8,\u03c4[I](x, y) = I (S\u03b8,\u03c4(x, y)) , where S\u22121 \u03b8,\u03c4 = T \u22121 \u03c4 \u25e6R\u22121 \u03b8 = T\u2212\u03c4 \u25e6R\u2212\u03b8. For rotating and translating a discrete image J, we use an analogous formula with nearest neighbor interpolation. The encoder network E\u03d5(J) = (z, \u02c6 \u03b8, \u02c6 \u03c4) \u2208Rd \u00d7 R \u00d7 R2, where J is an input image and \u03d5 is a trainable parameter, is trained such that the semantic representation z \u2208Rd captures a representation of I disentangled from the arbitrary orientation J is presented in. The rotation representation \u02c6 \u03b8 \u2208[0, 2\u03c0) and translation representation \u02c6 \u03c4 \u2208R2 are trained to be estimates of the rotation and translation with respect to a certain canonical orientation. Specifically, given an image J and its canonical 3 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations orientation J(can), we define (\u02c6 \u03b8, \u02c6 \u03c4) such that J(can) = S\u02c6 \u03b8,\u02c6 \u03c4[J], and the equivariance property (1) that we soon discuss implies that E\u03d5(J(can)) = (z, 0, 0). This canonical orientation J(can) is not (cannot be) the orientation of I. Rather, it is an orientation that we designate through the symmetry braking technique that we soon describe in Section 3.4. The hypernetwork has the form H\u03c8(z) = \u03b7, where the semantic representation z \u2208Rd is the input and \u03c8 is a trainable parameter. (Notably, \u02c6 \u03b8 and \u02c6 \u03c4 are not inputs.) The output H\u03c8(z) = \u03b7 = (w1, b1, w2, b2, . . . , wk, bk) will be used as the weights and biases of the k layers of the INR network, to be defined soon. We train the hypernetwork so that the INR network produces a continuous image representation approximating I. The implicit neural representation (INR) network has the form I(x, y; \u03b7) \u2208RC, where x, y \u2208R and \u03b7 is the output of the hypernetwork. The IRL-INR framework is trained so that I(\u00b7, \u00b7; \u03b7(i)) \u2248I(i)(\u00b7, \u00b7) in some sense, where \u03b7(i) is produced by H\u03c8 and E\u03d5 with J(i) provided as input. More specifically, we view I(x, y; \u03b7) as a continuous 2-dimensional image with inputs (x, y) and fixed parameter \u03b7, and we want I(x, y; \u03b7(i)) and I(i)(x, y) to be the same image in a different orientation. The INR network is a deep neural network (specifically, we use an MLP), but it has no trainable parameters as its weights and biases \u03b7 are generated by the hypernetwork H\u03c8(z). 3.3. Reconstruction and consistency losses We train IRL-INR with the loss L(\u03d5, \u03c8) = \u03bbrecon Lrecon + \u03bbconsis Lconsis + \u03bbsymm Lsymm, where \u03bbrecon > 0, \u03bbconsis > 0, and \u03bbsymm > 0. We define Lrecon and Lconsis in this section and define Lsymm in Section 3.4. 3.3.1. RECONSTRUCTION LOSS We use the reconstruction loss Lrecon(\u03d5, \u03c8) = EJ[ \u02c6 Lrecon(J; \u03d5, \u03c8)], with the per-image loss \u02c6 Lrecon(J; \u03d5, \u03c8) defined as (z, \u02c6 \u03b8, \u02c6 \u03c4) = E\u03d5(J) \u03b7 = H\u03c8(z) (\u02dc xp, \u02dc yp) = S\u02c6 \u03b8,\u02c6 \u03c4(xp, yp), p = 1, . . . , P \u02c6 Lrecon(J; \u03d5, \u03c8) = 1 P P X p=1 h \u2225Jp \u2212I(\u02dc xp, \u02dc yp; \u03b7)\u22252i . Given an image J and its canonical orientation J(can), minimizing the reconstruction loss induces Jp \u2248I(\u02dc xp, \u02dc yp; \u03b7), which is roughly equivalent to J(can) p \u2248I(xp, yp; \u03b7) for p = 1, . . . , P. This requires the latent representation (z, \u02c6 \u03b8, \u02c6 \u03c4) = E\u03d5(J) to contain sufficient information about J so that H\u03c8 and I are capable of reconstructing J. This is a similar role as those served by the reconstruction losses of autoencoders and VAEs. We believe that the INR structure already carries a significant inductive bias that promotes disentanglement between the semantic representation and the orientation information (\u02c6 \u03b8, \u02c6 \u03c4). However, it is still possible that the same image in different orientations produces different semantic representations z while still producing the same reconstruction. (Two different latent vectors can produce the same reconstructed image in autoencoders and INRs.) Therefore, we use an additional consistency loss to further enforce disentanglement between the semantic representation and the orientation of the image. 3.3.2. CONSISTENCY LOSS We use the consistency loss Lconsis(\u03d5) = EJ[ \u02c6 Lconsis(J; \u03d5)] with the per-image loss \u02c6 Lconsis(J; \u03d5) is defined as \u03c41, \u03c42 \u223cN(0, \u03c32I2) \u03b81, \u03b82 \u223cUniform([0, 2\u03c0]) (zi, \u02c6 \u03b8i, \u02c6 \u03c4i) = E\u03d5(S\u03b8i,\u03c4i[J]), i = 1, 2 \u02c6 Lconsis(J; \u03d5) = 1 \u2212 z1 \u00b7 z2 \u2225z1\u2225\u2225z2\u2225. Note that this is the cosine similarity between z1 and z2. Since S\u03b81,\u03c41[J] and S\u03b82,\u03c42[J] are also measurements of the same underlying continuous image I, minimizing this consistency loss enforces E\u03d5 to produce the same semantic representation z regardless of the orientation in which J is provided. (Of course, E\u03d5 produces different \u02c6 \u03b8 and \u02c6 \u03c4 depending on the orientation of J.) It is possible to use other distance measures, such as the MSE loss, instead of the cosine similarity in measuring the discrepancy between z1 and z2. However, we found that the cosine similarity distance synergized well with the SCAN-based clustering of Section 4.3. 4 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations Equivariance of encoder. Minimizing the reconstruction and consistency losses induces the following equivariance property. If E\u03d5(J) = (z, \u02c6 \u03c4, \u02c6 \u03b8), then E\u03d5(S\u03b8,\u03c4[J]) \u2248(z, R\u02c6 \u03b8\u2212\u03b8[\u02c6 \u03c4] \u2212\u03c4, \u02c6 \u03b8 \u2212\u03b8) (1) for all \u03c4 \u2208R2 and \u03b8 \u2208[0, 2\u03c0), where \u02c6 \u03b8\u2212\u03b8 \u2208[0, 2\u03c0) should be understood in the sense of modulo 2\u03c0. In other words, rotating J by \u03b8 will subtract \u03b8 to the rotation predicted by E\u03d5. For translation, the rotation effect must be taken into account. To see why, note that minimizing the consistency loss enforces E\u03d5(J) and E\u03d5(S\u03b8,\u03c4[J]) to produce an (approximately) equal semantic representation z, and therefore, the corresponding \u03b7 = H\u03c8(z) will be (approximately) equal. Minimizing the reconstruction loss implies 0 (a) \u2248\u02c6 Lrecon(J; \u03d5, \u03c8) = 1 P P X p=1 \u0014\r \r \rJp \u2212I(S\u02c6 \u03b8,\u02c6 \u03c4[(xp, yp)]; \u03b7) \r \r \r 2\u0015 (b) \u22481 P P X p=1 \u0014\r \r \r(S\u03b8,\u03c4[J])p \u2212I(S\u02c6 \u03b8\u2212\u03b8,R\u02c6 \u03b8\u2212\u03b8[\u02c6 \u03c4]\u2212\u03c4[(xp, yp)]; \u03b7) \r \r \r 2\u0015 , (c) \u2248\u02c6 Lrecon(S\u03b8,\u03c4[J]; \u03d5, \u03c8) (a) \u22480. Steps (a) holds since the recontruction loss is minimized. Step (b) holds since if two images are similar, then their rotated and translated versions are also similar. More precisely, let J\u2032 be the discrete image defined as J\u2032 p = I(S\u02c6 \u03b8,\u02c6 \u03c4[(xp, yp)]; \u03b7) for p = 1, . . . P. If J \u2248J\u2032, then S\u03b8,\u03c4[J] \u2248S\u03b8,\u03c4[J\u2032]. Furthermore, (S\u03b8,\u03c4[J\u2032])p (d) \u2248I(S\u22121 \u03b8,\u03c4S\u02c6 \u03b8,\u02c6 \u03c4[(xp, yp)]; \u03b7) (d) \u2248I(S\u02c6 \u03b8\u2212\u03b8,R\u02c6 \u03b8\u2212\u03b8[\u02c6 \u03c4]\u2212\u03c4[(xp, yp)]; \u03b7), where we the approximation of (d) captures interpolation artifacts. Step (c) holds since the left-hand-side and the righthand-side are both approximately 0. Finally, the fact that (c) holds implies that the equivariance property (1) holds. 3.4. Symmetry-breaking loss Our assumed data measurement model is symmetric/invariant with respect to the group of rotations and translations. More specifically, let G be the group generated by rotations and translations, then for any image J and g \u2208G, the images J, \u02dc J = g[J] are equally likely observations and carry exactly the same information about the true underlying continuous image I.1 1We point out two technicalities in this statement. First, strictly speaking, J and \u02dc J do not have exactly the same pixel values due Ground Truth TARGET-VAE IRL-INR w/o s.b. IRL-INR w/ s.b. Figure 2. Methods without symmetry fail to reconstruct WM811k and WHOI-Plankton images. However, our framework inevitably decides on a canonical orientation as it disentangles the semantic representation z from the orientation information (\u02c6 \u03b8, \u02c6 \u03c4), such that input image J and its canonical orientation J(can) satisfy J(can) = S\u02c6 \u03b8,\u02c6 \u03c4[J], E\u03d5(J(can)) \u2248(z, 0, 0). This canonical orientation is not orientation of the true underlying continuous image I. The prior work of Bepler et al. (2019); Nasiri & Bepler (2022) allows the canonical orientation to be determined by the neural networks and their training process. Some of the datasets used in Nasiri & Bepler (2022) are fully rotationally symmetric (such as \u201cMNIST(U)\u201d) and for those setups, the symmetry makes the determination of the canonical orientation an arbitrary choice. We find that if we break the symmetry by manually prescribing a rule for the canonical orientation, the trainability of the framework significantly improves as we soon demonstrate. We propose a symmetry breaking based on the center of mass of the image. Given a continuous image I, we define its center of mass as (mx, my) = 1 \u2225I\u22251 Z \u221e \u2212\u221e Z \u221e \u2212\u221e (x, y)\u2225I(x, y)\u22251 dxdy \u2208R2 where \u2225I\u22251 = R \u221e \u2212\u221e R \u221e \u2212\u221e|I(x, y)| dxdy is L1 norm. For a discrete image J, we use an analogous discretized formula. Given an image J with center of mass m = (mx, my), let \u03c4 = \u2212m and let \u03b8 \u2208[0, 2\u03c0) such that m = \u2225\u03c4\u2225(cos \u03b8, \u2212sin \u03b8). to interpolation artifacts, except when the translation and rotation exactly aligns with the image grid. Second, the invariance with respect to translation holds only if \u03c4 has a uniform prior on R2, which is an improper prior. On the other hand, the rotation group is compact and we do assume the rotation is uniformly distributed on [0, 2\u03c0), which is a proper prior. 5 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations (a) MNIST (U) (b) WM811K (c) 5HDB (d) dSprites (e) WHOI-Plankton (f) Galaxy Zoo Figure 3. To validate the disentanglement of semantic representations, we verify that the reconstructions are indeed invariant under rotation and translation. The first row of (a)\u2013(f) are rotated by 2\u03c0 7 degrees. The second row of (a)\u2013(f) are reconstructions using only the semantic representation z, without any rotation or translation. We see that the reconstructions are invariant with respect to the rotations and translations. The setup is further detailed in Appendix F and more images are provided in Figure 10. Then J(can) = S\u02c6 \u03b8,\u02c6 \u03c4[J] and J(can) has its center of mass at (0, 0). We use the symmetry-breaking loss Lsymm(\u03d5) = EJ[ \u02c6 Lsymm(J; \u03d5)] with the per-image loss \u02c6 Lsymm(J; \u03d5) is defined as (z, \u02c6 \u03c4, \u02c6 \u03b8) = E\u03d5(J) m = CoM(J) \u02c6 Lsymm(J; \u03d5) = \r \r \rm \u2212\u2225\u02c6 \u03c4\u2225(cos \u02c6 \u03b8, \u2212sin \u02c6 \u03b8) \r \r \r 2 , where CoM(J) denotes the center of mass of J. The use of an INR with a hypernetwork is essential in directly enforcing the representation to disentangled while allowing the network to be sufficiently expressive to be able to learn sufficiently complex tasks. Specifically, we show in Figure 2 that we could not train TARGET-VAE and IRLINR to reconstruct the WM811k dataset without using the symmetry breaking technique. 4. Experiments 4.1. Experimental setup The encoder network E\u03d5(J) uses the ResNet18 architecture (He et al., 2016) with an MLP as the head. The hypernetwork H\u03c8(z) is an MLP with input dimension d. The INR network I(x, y; \u03b7) uses a random Fourier feature (RFF) encoding in the style of (Rahimi & Recht, 2007; Tancik et al., 2020) followed by an MLP with output dimension 1 (for grayscale images) or 3 (for rgb images). The architectures for H\u03c8(z) and I(x, y; \u03b7) are inspired by Dupont et al. (2022). Further details of the architecture can be found in Appendix A or the code provided as supplementary materials. We use the Adam optimizer with learning rate 1 \u00d7 10\u22124, weight decay 5 \u00d7 10\u22124, and batch size 128. For the loss function scaling coefficients, we use \u03bbrecon = \u03bbconsis = 1 and \u03bbsymm = 15. We use the MSE and cosine similarity distances for the consistency loss for our results of Section 4.2 and Section 4.3, respectively. We evaluate the performance of IRL-INR against the recent prior work TARGET-VAE (Nasiri & Bepler, 2022). For TARGET-VAE experiments, we mostly use the code and settings provided by the authors. We use the TARGET-VAE 6 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations with P16 and d = 32, which Nasiri & Bepler (2022) report to perform the best for the clustering. For more complex datasets, such as WM811K or WHOI-PLANKTON, we increase the number of layers from 2 to 6 in their \u201cspatial generator\u201d network, as the authors did for the cryo-EM dataset. For the clustering experiments of Sections Section 4.3 and Section 4.4, we separate the training set and test set and evaluate the accuracy on the test set. 4.1.1. DATASETS MNIST(U) is derived from MNIST with random rotations and translations respectively sampled from Uniform([0, 2\u03c0)) and N(0, 52). To accommodate the translations, we embed the images into 50 \u00d7 50 pixels as was done in (Nasiri & Bepler, 2022). WM811k is a dataset of silicon wafer maps classified into 9 defect patterns (Wu et al., 2015). The wafer maps are circular, and the semiconductor fabrication process makes the data rotationally invariant. Because the original full dataset has a severe class imbalance, we distill the dataset into 7350 training set and 3557 test set images with reasonably balanced 9 defect classes and resize the images to 32 \u00d7 32 pixels. 5HDB consists of 20,000 simulated projections of integrin \u03b1IIb in complex with integrin \u03b23 (Lin et al., 2015; Bepler et al., 2019) with varying orientations. There are 16,000 training set and 4000 test set images of 40 \u00d7 40 pixels. dSprites consists of 2D shapes procedurally generated from 6 ground truth independent latent factors (Higgins et al., 2017). All possible combinations of these latents are present exactly once, resulting in 737,280 total images. WHOI-Plankton is an expert-labeled image dataset for plankton (Orenstein et al., 2015). The orientation of the plankton with respect to the microscope is random, so the dataset exhibits rotation and translation invariance. However, the original dataset has a severe class imbalance, so we distill the dataset into 10 balanced classes with 1000 training set and 200 test set images. We also perform a circular crop and resize the images to 32 \u00d7 32 pixels. Galaxy Zoo consists of 61,578 RGB color images of galaxies from the Sloan Digital Sky Survey (Lintott et al., 2008). Each image is cropped and downsampled to 64 \u00d7 64 pixels following common practice (Dieleman et al., 2015). We divide into 50,000 training set and 11,578 test set images. 4.2. Validating disentanglement In this section, we validate whether the encoder network E\u03d5(J) = (z, \u02c6 \u03b8, \u02c6 \u03c4) is indeed successfully trained to produce a semantic representation z disentangled from the orientation of the input image J. Translation Rotation Spatial-VAE 0.982, 0.983 0.005 TARGET-VAE P4 0.975, 0.976 0.80 TARGET-VAE P8 0.972, 0.971 0.859 TARGET-VAE P16 0.974, 0.971 0.93 IRL-INR 0.999, 0.999 0.9891 Table 1. Correlation between true rotation and predicted rotation and true translation and predcited translation from MNIST(U). Figure 3 shows images and their reconstructions with MNIST(U), WM811k, 5HDB, dSprites, WHOI-Plankton, and Galaxy Zoo datasets. The first row of each subfigure shows images that have been rotated or translated from a given image, and we compute E\u03d5(J) = (z, \u02c6 \u03b8, \u02c6 \u03c4). The second row of each figure is the reconstruction of these images by the disentangled semantic representation z. More specifically, the reconstructions correspond to I(x, y; H\u03c8(z)) with (x, y) not rotated or translated. (So the (\u02c6 \u03b8, \u02c6 \u03c4) output by E\u03d5(J) is not used.) We can see that the reconstruction is indeed (approximately) invariant regardless of the orientation of the input image J. For comparison, TARGET-VAE was unable to learn representations from the WM811k and WHOI-Plankton datasets, as discussed in Section 3.4. Table 1 and Figure 4 shows how well the predicted rotation \u02c6 \u03b8 and predicted translation \u02c6 \u03c4 matched the true rotation \u03b8 and true translation \u03c4. Table 1 shows the Pearson correlation between the predicted rotation \u02c6 \u03b8 and the true rotation \u03b8, predicted translation \u02c6 \u03c4 and true translation \u03c4. We confirmed that our method has the highest correlation value. Also, in Figure 4 we plotted values of \u03b8 and \u02c6 \u03b8. We can observe that most of predicted rotation degree are exactly same with true rotation. Interestingly, in the case of the WM811k, there were many cases where predicted degree and true degree are differed by 2\u03c0, which is acceptable because the rotation degree is equivalent under mod 2\u03c0. 0 /2 3 /2 2 true rotation 0 /2 3 /2 2 predicted rotation MNIST(U) 0 /2 3 /2 2 true rotation 0 /2 3 /2 2 predicted rotation WM811k Figure 4. Difference between predicted rotation values and true rotation values on MNIST(U) and WM811. 7 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations 4.3. Clustering with semantic representations and SCAN As the semantic representations are disentangled from the orientation of the image, they should be more amenable to be used for clustering, provided that the semantic meaning of the underlying dataset is invariant under different orientations of the image. In this section, we use the semantic representations to perform clustering based on two approaches: directly with z and using SCAN. Table 2 shows the results of applying k-means and agglomerative clustering on the semantics representation z. For TARGET-VAE, we used the original authors\u2019 code and hyperparameters to best reproduce their results. We see that the semantic representation produced by our framework IRLINR has better clustering accuracies and has significantly less variability. MNIST(U) WM811k Spatial-VAE (K-means) 31.87 \u00b1 3.72 27.4 \u00b1 1.16 Spatial-VAE (Agg) 35.62 \u00b1 2.08 28.73 \u00b1 1.39 Target-VAE (K-means) 64.63 \u00b1 4.4 39.6 \u00b1 1.29 Target-VAE (Agg) 68.8 \u00b1 4.39 40.11 \u00b1 2.7 IRL-INR (K-means) 59.6 \u00b1 1.12 55.06 \u00b1 1.83 IRL-INR (Agg) 71.53 \u00b1 1.01 56.74 \u00b1 1.15 Table 2. Clustering on semantics representation z. The confidence interval is a single standard deviation measured over 5 runs. To further improve the clustering accuracy, we combine our framework with one of the state-of-the-art deep-learningbased method SCAN (Van Gansbeke et al., 2020). The original SCAN framework adopted SimCLR (Chen et al., 2020) as its pretext task. We instead use the training of IRL-INR as a pretext task and then use the trained encoder network E\u03d5 with only the z output for the SCAN framework. (The (\u02c6 \u03b8, \u02c6 \u03c4) are not used with SCAN.) Since SimCLR is based on InfoNCE loss (van den Oord et al., 2018), which uses cosine similarity, we also use cosine similarity distance in training IRL-INR. Table 3 shows that IRL-INL synergizes well with SCAN to produce state-of-the-art performance. Specifically, the clustering accuracy significantly outperforms vanilla SCAN (with InfoNCE loss) and combining Target-VAE with SCAN yields little or no improvement compared to directly clustering the semantic representations of Target-VAE. The confusion matrix Figure 5 shows that there is significant misclassification between 6 and 9. In Appendix D, we present our clustering results with the class 9 removed, and IRL-INR + SCAN achieves a 98% accuracy. MNIST(U) WM811k TARGET-VAE + SCAN 63.09 \u00b1 1.7 43.39 \u00b1 4.55 SimCLR + SCAN 85.4 \u00b1 1.46 57.1 \u00b1 2.81 IRL-INR + SCAN 90.4 \u00b1 1.74 64.6 \u00b1 1.01 Table 3. Using IRL-INR as pretext task for SCAN outperformed other combinations using TARGET-VAE and SimCLR. Here, d is the dimension of the semantic representation z. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 99 98 69 98 94 95 82 84 98 82 MNIST(U) 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 81 64 65 85 24 50 64 51 36 WM811k Figure 5. Confusion matrices of clustering of IRL-INR + SCAN. 4.4. Scaling latent dimension d Table 4 shows that clustering accuracy of IRL-INR scales (improves) as the latent dimension d, the size of the semantic representation z, becomes larger. This phenomenon may seem counterintuitive as one might think that a smaller semantic representation is a more compressed and, therefore, better representation. We also tried scaling the output dimension of the SimCLR + SCAN\u2019s backbone model, but we did not find any noticeable performance gain or trend. To clarify, SimCLR + SCAN and IRL-INR + SCAN used the same backbone model, ResNet18, but only IRL-INR + SCAN exhibited the scaling improvement. We also conducted a similar scaling experiment with TARGET-VAE, but we did not find any performance gain or trend with or without SCAN. IRL-INR + SCAN MNIST(U) WM811k d = 32 84.18 \u00b1 2.11 53.78 \u00b1 3.41 d = 64 86 \u00b1 1.78 55.4 \u00b1 1.35 d = 128 85.8 \u00b1 1.46 56.2 \u00b1 1.16 d = 256 87 \u00b1 0.89 58.6 \u00b1 1.62 d = 512 90.4 \u00b1 1.74 64.6 \u00b1 1.01 Table 4. Increasing latent dimension d of IRL-INR leads to better clustering performance. 8 \fRotation and Translation Invariant Representation Learning with Implicit Neural Representations 4.5. Ablation studies IRL-INR uses a hypernetwork-INR architecture, but one can alternatively consider: (1) using a simple MLP generator or (2) using a standard autoencoder while directly rotating the generated output image through pixel interpolation. We find that both alternatives fail in the following sense. For the more complex images, such as the plankton microscope or silicon wafer maps, if we use the losses for requiring the semantic representation to be invariant, then the models fail the image reconstruction pretext task in the sense that the reconstruction comes out to be a blur with no discernable content. When we nevertheless proceeded to cluster the latents, the accuracy was poor. Using the hypernetwork was the only option that allowed us to do clustering successfully for the silicon wafer maps. Also, the loss function of IRL-INR consists of three components: (1) reconstruction loss (2) consistency loss (3) symmetry breaking loss. We conducted ablation study on the different loss terms and found that all components are essential to reconstruction and clustering. For example, removing the consistency loss does not affect the reconstruction quality but does significantly reduce the clustering accuracy. Also, as we see in Figure 2, removing the symmetry-breaking loss significantly degrades the reconstruction quality, thereby worsening the clustering results. 5." + } + ], + "Amir Ingber": [ + { + "url": "http://arxiv.org/abs/1312.2063v1", + "title": "The Minimal Compression Rate for Similarity Identification", + "abstract": "Traditionally, data compression deals with the problem of concisely\nrepresenting a data source, e.g. a sequence of letters, for the purpose of\neventual reproduction (either exact or approximate). In this work we are\ninterested in the case where the goal is to answer similarity queries about the\ncompressed sequence, i.e. to identify whether or not the original sequence is\nsimilar to a given query sequence. We study the fundamental tradeoff between\nthe compression rate and the reliability of the queries performed on compressed\ndata. For i.i.d. sequences, we characterize the minimal compression rate that\nallows query answers, that are reliable in the sense of having a vanishing\nfalse-positive probability, when false negatives are not allowed. The result is\npartially based on a previous work by Ahlswede et al., and the inherently\ntypical subset lemma plays a key role in the converse proof. We then\ncharacterize the compression rate achievable by schemes that use lossy source\ncodes as a building block, and show that such schemes are, in general,\nsuboptimal. Finally, we tackle the problem of evaluating the minimal\ncompression rate, by converting the problem to a sequence of convex programs\nthat can be solved efficiently.", + "authors": "Amir Ingber, Tsachy Weissman", + "published": "2013-12-07", + "updated": "2013-12-07", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "cs.DB", + "cs.IR", + "math.IT" + ], + "main_content": "INTRODUCTION Traditionally, data compression deals with concisely representing data, e.g., a sequence of letters, for the purpose of eventual reproduction (either exact or approximate). More generally, one wishes to know something about the source from its compressed representation. In this work, we are interested in compression when the goal is to identify whether the original source sequence is similar to a given query sequence. A typical scenario where this problem arises is in database queries. Here, a large database containing many sequences {x1, ..., xM} is required to answer queries of the sort \u201cwhat are the sequences in the database that are close to the sequence y?\u201d. Such a scenario (see Fig. 1) appears, for example, in computation biology (where the sequences can be, e.g., DNA sequences), forensics (where the sequences represent \ufb01ngerprints) and internet search. Speci\ufb01cally, our interest is in the case where for each sequence x in the database we keep a short signature T(x), and the similarity queries are performed based on T(x) and y. Our setting differs from classical compression in that we do not require that the original data be reproducible from the signatures, so the signatures are not meant to replace the original database. There are many instances where such compression is desirable. For example, the set of signatures can be thought of as a cached version of the original database which, due to its smaller size, can be stored on a faster media (e.g. RAM), or even hosted on The authors are with the Dept. of Electrical Engineering, Stanford University, Stanford, CA 94305. Email: {ingber, tsachy}@stanford.edu. This work is supported in part by the NSF Center for Science of Information under grant agreement CCF-0939370, and by a Google research award \f2 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Fig. 1. Similarity queries in a database. Fig. 2. Answering similarity queries using a compressed database: \ufb01rst the user receives a set of potential matches, and then asks the original database for the actual sequences. In the example, x23 is a false positive (FP). many location in order to reduce the burden on the main database. Typically, the user will eventually request the relevant sequences (and only them) from the original database \u2013 see Fig. 2. Naturally, when the queries are answered from the compressed data, one cannot expect to get accurate answers all the time. There are two error events that may occur: the \ufb01rst is a false positive (FP), where the query returns a positive answer (\u201cthe sequences are similar\u201d) but the answer is wrong (the sequences are in fact dissimilar). The second is a false negative (FN), when the query returns a negative answer (\u201cthe sequences are not similar\u201d) but the sequences are actually similar1. Therefore the interesting tradeoff to consider is between the compression rate (the amount of space required to represent T(x)) and the reliability of the 1In the statistics literature, the FP and FN events are also known as type 1 and type 2 errors, respectively, while in the engineering literature they are also known as false alarm and misdetection events, respectively. \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 3 query answers (measured by the FP and FN probabilities, under an appropriately de\ufb01ned probabilistic model). The problem was \ufb01rst studied from an information-theoretic viewpoint in the seminal work by Ahlswede et al. [1]. In this work, the source and query sequences are assumed to be drawn i.i.d. from a known distribution, and both false negatives and false positives are allowed. In [1], the authors \ufb01rst considered the case where the probability of false positives and the probability of false negatives are only required to vanish with the dimension n (in the same spirit of the de\ufb01nition of an achievable rate for communication as a rate for which the error probability can be made to vanish). However, it was shown in [1] that this de\ufb01nition leads to degeneracy: there exist schemes for compression at rates that are arbitrarily close to zero while the two error probabilities vanish. Then, the authors in [1] moved on to consider the case where the FP and FN probabilities are required to vanish exponentially, with prescribed exponents \u03b1 and \u03b2, respectively, and were able to \ufb01nd the optimal compression rate in that setting2. We note that this case is atypical in information theory: in the channel coding setting, the highest achievable rate (the channel capacity) is the same, regardless of whether an achievable rate is de\ufb01ned by an error probability vanishing exponentially or just vanishing. The same holds for the lowest compression rate for lossy reproduction (the rate-distortion function). In this paper, we consider the case where no false negatives are allowed. The main motivation is that false negatives cause an undetected error in the system where, in contrast, false positives can be easily detected (after retrieving the sequences \u2018\ufb02agged\u2019 as potential matches, it is easy to \ufb01lter out the false positives). This is important in several applications, where one cannot compromise on the accuracy of the results (e.g. in a forensic database), but still would like to enjoy the bene\ufb01ts of compression. While it is natural to ask what can be gained when the FN probability is nonzero but \u2018very very small\u2019 \u2013 it is important to recall that this probability is based on a probabilistic model of the data, which may not be accurate (in fact, it is rarely the case, especially in source coding settings, where the probabilistic model matches the actual data very closely). The contributions of the current paper are as follows. 1) We \ufb01nd the minimal compression rate for reliable similarity identi\ufb01cation with no false negatives. This rate, called the identi\ufb01cation rate and denoted RID(D), turns out to be the in\ufb01mal rate at which the \u201cfalse-positive\u201d exponent of [1] is positive. Our result holds for both \ufb01xed and variable length compression schemes. 2) In the case where x and y have the same alphabet, and the similarity measure satis\ufb01es the triangle inequality, we analyze two schemes for compression that are based on the notion of lossy compression. In those schemes the signature is used for producing a reconstruction \u02c6 x of the source, and the decision whether x and y is done according to the distance between \u02c6 x and y. We show that those schemes, although simpler for analysis and implementation, attain rates that are generally suboptimal, i.e. strictly greater than RID(D). 3) The identi\ufb01cation rate RID(D) is stated as a non-convex optimization program with an auxiliary random variable. We provide two results that facilitate the computation of RID(D). First, we improve a bound on the cardinality of the auxiliary RV. Then, we propose a method of transforming the said non-convex optimization program into a 2This optimal rate, however, is uncomputable, since the expression depends on an auxiliary random variable with unbounded cardinality. \f4 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY sequence of convex optimization programs, and by that allowing ef\ufb01cient computation of RID(D) for small alphabet sizes. We demonstrate the effectiveness of this approach by calculating RID(D) for several sources. The paper is organized as follows. In the next section we provide an extended literature survey, which compares the setting discussed in the paper with other ideas, including different hashing schemes. In Sec. III we formulate the problem, and in Sec. IV we state and discuss our main results. In Sec. V we prove the result for the identi\ufb01cation rate, Sec. VI contains the analysis of the schemes based on lossy compression, and in Sec. VII we describe the results enabling the computation of the identi\ufb01cation rate. Sec. VIII delivers concluding remarks. II. RELATED LITERATURE A. Directly related work In the current paper we focus on discrete alphabets only, following [1]. A parallel result, with complete characterization of the identi\ufb01cation rate (and exponent) for the Gaussian case and quadratic distortion, appears in [2],[3]. The identi\ufb01cation exponent problem was originally studied in [1] for the variable length case, where the resulting exponent depends on an auxiliary random variable with unbounded cardinality. A bound on the cardinality has been obtained recently in [4], where the exponent for \ufb01xed-length schemes is also found (and is different than that of the variable length schemes, unlike the identi\ufb01cation rate \u2013 see Prop. 1 below). In the special case of exact match queries (i.e. identi\ufb01cation with D = 0 for Hamming distance), the exponent was studied in [5]. B. Other work in information theory Another closely related work is the one by Tuncel et al. [6], where a similar setting of searching in a database was considered. In that work the search accuracy was addressed by a reconstruction requirement with a single-letter distortion measure that is side-information dependent (and the tradeoff between compression and accuracy is that of a Wyner-Ziv [7] type). In contrast, in the current paper the search accuracy is measured directly by the probability of false positives. A different line of work geared at identifying the fundamental performance limits of database retrieval includes [8], [9], which characterize the maximum rate of entries that can be reliably identi\ufb01ed in a database. These papers were extended in [10], [11] allowing compression of the database, and in [12] to the case where sequence reconstruction is also required. In all of these papers, the underlying assumption is that the original sequences are corrupted by noise before enrolled in the database, the query sequence is one of those original sequences, and the objective is to identify which one. There are two fundamental differences between this line of work and the one in the current paper. First, in our case the query sequence is random (i.e. generated by nature) and does not need to be a sequence that has already been enrolled in the database. Second, in our problem we are searching for sequences that are similar to the query sequence (rather than an exact match). \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 5 C. Hashing and related concepts The term hashing (see, e.g. [13]) generally refers to the process of representing a complex data entity with a short signature, or hash. Classically, hashing is used for quickly identifying exact matches, by simply comparing the hash of the source sequence and that of the query sequence. Hashing has been extended to detect membership in sets, a method known as the Bloom Filter [14] (with many subsequent improvements, e.g. [15]). Here, however, we are interested in similarities, or \u201capproximate\u201d matches. The extension of the hashing concept to similarity search is called Locality Sensitive Hashing (LSH), which is a framework for data structures and algorithms for \ufb01nding similar items in a given set (see [16] for a survey). LSH trades off accuracy with computational complexity and space, and false negatives are allowed. Several fundamental points are different in our setting. First, we study the information-theoretic aspect of the problem, i.e. concentrate on space only (compression rate) and ignore computational complexity in an attempt to understand the amount of information relevant to querying that can be stored in the short signatures. Second, we do not allow false negatives, which, as discussed above, are inherent for LSH. Third, in the general framework of LSH (and also in hashing), the general assumption is that the data is \ufb01xed and that the hashing functions are random. This means that the performance guarantees are given as low failure probabilities, where the probability space is that of the random functions. However, for a given database, the hashing function is eventually \ufb01xed, which means that there always exist source and/or query sequences for which the scheme will always fail. In our case, the scheme is deterministic, false negatives never occur (by design), and the probability of false positive depends on the probabilistic assumptions on the data. Another related idea is that of dimensionality reduction techniques that preserve distances, namely based on Johnson-Lindenstrauss type embeddings [17]. Such embeddings take a set of points in space, and transform each point to a point in a lower-dimensional space, with a guarantee that the distance between these points is approximately preserved. However, note that such mappings generally depend on the elements in the database \u2013 so that the distance preservation property cannot apply to any query element outside the database, making the guarantee for zero false negative impossible without further assumptions. In fact, the original proof of the lemma in [17] results in a guarantee for any two points in space, but this guarantee is probabilistic, and therefore cannot match our setting (similarly to LSH). The process of compressing a sequence to produce a short signature can be also thought of as a type of sketching (see, e.g. [18]), which is a computational framework for succinct data representation that still allows performing different operations with the data. D. Practical examples of compression for similarity identi\ufb01cation The idea of using compression for accelerating similarity search in databases is not new. Earlier practical examples include the VA-\ufb01le scheme [19], which uses scalar quantization of each coordinate of the source sequence in order to form the signature. The VA-\ufb01le approach demonstrates that compression-based similarity search systems can outperform tree-based systems for similarity search, providing further motivation to study the fundamental tradeoff between compression and search accuracy. The VA-\ufb01le scheme has been generalized to vector quantization in [20], showing further improvements in both computational time and number of disk I/O operations. \f6 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY In the machine learning literature, the term \u2018semantic hashing\u2019 [21] refers to a transformation that maps similar elements to bit strings that have low Hamming distance. Extensions of this concept include [22], [23]. We comment that in neither of these papers there is a guarantee for zero false negatives, as in the setting considered in the current paper. We emphasize again that the results in the current paper are concerned with the amount of compression only, and ignore the computational complexity (as is typical for information theoretical results). Nevertheless, the fundamental limits, such as RID(D), characterize the playing \ufb01eld at which practical schemes should be evaluated. III. PROBLEM FORMULATION A. Notation Throughout this paper, boldface notation x denotes a column vector of elements [x1, ...xn]T . Capital letters denote random variables (e.g. X, Y ), and X, Y denote random vectors. We use calligraphic fonts (e.g. X , Y) to represent the \ufb01nite alphabets. log(\u00b7) denotes the base-2 logarithm, while ln(\u00b7) is used for the usual natural logarithm. We measure the similarity between symbols with an arbitrary per-letter distortion measure \u03c1 : X \u00d7 Y \u2192R+. For length n vectors, the distortion is given by d(x, y) \u225c1 n n X i=1 \u03c1(xi, yi). (1) We say that x and y are D-similar when d(x, y) \u2264D, or simply similar when D is clear from the context. B. Identi\ufb01cation Systems A rate-R identi\ufb01cation system (T, g) consists of a signature assignment T : X n \u2192{1, 2, . . . , 2nR} (2) and a decision function g : {1, 2, . . . , 2nR} \u00d7 Yn \u2192{no, maybe}. (3) A system (T, g) is said to be D-admissible, if for any x, y satisfying d(x, y) \u2264D, we have g(T(x), y) = maybe. (4) This notion of D-admissibility motivates the use of \u201cno\u201d and \u201cmaybe\u201d in describing the output of g: \u2022 If g(T(x), y) = no, then x and y can not be D-similar. \u2022 If g(T(x), y) = maybe, then x and y are possibly D-similar. Stated another way, a D-admissible system (T, g) does not produce false negatives. Thus, a natural \ufb01gure of merit for a D-admissible system (T, g) is the frequency at which false positives occur (i.e., where g(T(x), y) = maybe and d(x, y) > D). To this end, let PX and PY be probability distributions on X , Y respectively, and assume that the vectors X and Y are independent of each other and drawn i.i.d. according to PX and PY respectively. De\ufb01ne the false positive event E = {g(T(X), Y) = maybe, d(X, Y) > D}, (5) \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 7 and note that, for any D-admissible system (T, g), we have Pr{g(T(X), Y) = maybe} = Pr{g(T(X), Y) = maybe|d(X, Y) \u2264D} Pr{d(X, Y) \u2264D} + Pr{g(T(X), Y) = maybe, d(X, Y) > D} (6) = Pr{d(X, Y) \u2264D} + Pr{E}, (7) where (7) follows since Pr{g(T(X), Y) = maybe|d(X, Y) \u2264D} = 1 by D-admissibility of (T, g). Since Pr{d(X, Y) \u2264D} does not depend on what scheme is employed, minimizing the false positive probability Pr{E} over all D-admissible schemes (T, g) is equivalent to minimizing Pr{g(T(X), Y) = maybe}. Also note, that the only interesting case is when Pr{d(X, Y) \u2264D} \u21920 as n grows, since otherwise almost all the sequences in the database will be similar to the query sequence, making the problem degenerate (since almost all the database needs to be retrieved, regardless of the compression). In this case, it is easy to see from (6) that Pr{E} vanishes if and only if the conditional probability Pr{g(T(X), Y) = maybe|d(X, Y) > D} (8) vanishes as well. In view of the above, we henceforth restrict our attention to the behavior of Pr{g(T(X), Y) = maybe}. In particular, we study the tradeoff between the rate R and Pr{g(T(X), Y) = maybe}. This motivates the following de\ufb01nitions: De\ufb01nition 1: For given distributions PX, PY and a similarity threshold D, a rate R is said to be D-achievable if there exists a sequence of admissible schemes (T (n), g(n)) with rates at most R, satisfying lim n\u2192\u221ePr \b g(n) \u0000T (n)(X), Y \u0001 = maybe \t = 0. (9) De\ufb01nition 2: For given distributions PX, PY and a similarity threshold D, the identi\ufb01cation rate RID(D, PX, PY ) is the in\ufb01mum of D-achievable rates. That is, RID(D) \u225cinf{R : R is D-achievable}, (10) where an in\ufb01mum over the empty set is equal to \u221e. It is not hard to see that RID(D) must be nondecreasing. To see this, note that any sequence of schemes at rate R that achieve vanishing probability of maybe for similarity threshold D, is also admissible for any threshold D\u2032 \u2264D, so if R is D-achievable, then it is also D\u2032-achievable. In other words, a higher similarity threshold is a more dif\ufb01cult task (i.e. requires higher compression rate). Therefore, analogously to the de\ufb01nition of RID(D), we de\ufb01ne DID(R) as the maximal achievable similarity threshold for \ufb01xed rate schemes. The de\ufb01nitions of an achievable rate and the identi\ufb01cation rate are in the same spirit of the rate distortion function (the rate above which a vanishing probability for excess distortion is achievable), and also in the spirit of the channel capacity (the rate below which a vanishing probability of error can be obtained). See, for example, Gallager [24]. C. Variable Length Identi\ufb01cation Systems In [1], the authors study a similar setting, where the compression is of variable length. In that spirit, we de\ufb01ne the corresponding variable-length quantities: A variable length identi\ufb01cation system (Tvl, gvl) consists of a signature assignment Tvl : X n \u2192B, (11) \f8 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY where B \u2286{0, 1}\u2217is a pre\ufb01x-free set, and a decision function gvl : B \u00d7 Yn \u2192{no, maybe}. (12) The rate of the system is given by R = 1 nE [length (Tvl(X))] . (13) As before, a scheme is said to be admissible, if for any x, y satisfying d(x, y) \u2264D, we have gvl(Tvl(x), y) = maybe. (14) Analogously to De\ufb01nitions 1 and 2, we de\ufb01ne the variable-length identi\ufb01cation rate, denoted Rvl ID as the in\ufb01mum of achievable rates for variable length identi\ufb01cation systems. Clearly, any rate R that is achievable with \ufb01xed-length schemes is also achievable with variable length schemes, and therefore RID \u2265Rvl ID. It turns out that both quantities are actually equal: Proposition 1: The identi\ufb01cation rate for variable rate is the same as that for \ufb01xed rate, i.e. RID(D) = Rvl ID(D) (15) The proof of the proposition, given in detail in Appendix A, is based on a simple metaargument, that essentially says that any variable length scheme can be used as a building block to construct a \ufb01xed-length scheme. The argument is based on the concatenation of several input sequences into a larger one, and then applying a variable length scheme to each of the sequences. This will result in a variable length scheme, but with high probability, most of the signatures will have overall length bounded by some \ufb01xed length, enabling the conversion to a \ufb01xed-length scheme. There are two direct consequences of Prop. 1 that will enable the evaluation of RID(D): In order to show that a given rate is achievable with \ufb01xed-rate schemes, it is possible to consider variable length schemes, as in [1]. On the other hand, in order to prove a converse, it suf\ufb01ces to consider \ufb01xed-length schemes, slightly simplifying the proof. This is the path we take in the paper. IV. MAIN RESULTS A. The Identi\ufb01cation Rate De\ufb01ne the following distance between distributions PX, PY : \u00af \u03c1(PX, PY ) \u225cmin E[\u03c1(X, Y )], (16) where the minimization is w.r.t. all random variables X, Y with marginal distributions PX and PY , respectively. This distance goes by many names, such as the Wasserstein (or Vasershtein) distance, the Kantorovich distance and also the Transport distance (see [25], and also [26] for a survey). De\ufb01ne the (informational) identi\ufb01cation rate as \u00af RID(D) = min PU|X:P u\u2208U PU(u)\u00af \u03c1(PX|U(\u00b7|u),PY )\u2265D I(X; U), (17) \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 9 where U is any random variable with \ufb01nite3 alphabet U, that is independent of Y . It follows from [1, Theorem 2] that when R > \u00af RID(D), there exist (variable-length) identi\ufb01cation schemes with FP probability that vanishes exponentially with n (the explicit connection to the limiting rate \u00af RID(D) is made in [1, Eq. (2.21)]). This fact, combined with Prop. 1 above implies that RID(D) \u2264\u00af RID(D). However, it remains open whether RID(D) is even strictly positive. In the related case studied in [1], where the probability of both FN and FP events are required to vanish, it was shown [1, Thm. 1] that the achievable rate in this sense is equal to zero, so according to [1], \u201cthe only problem left to investigate is the case tradeoff between the rate R and the two error exponents [of the FP and FN events]\u201d. Our \ufb01rst result below shows that the restriction to the case of no FN (also called a \u2018one-sided error\u2019) is, in fact, very interesting. Theorem 2 (The Identi\ufb01cation Rate Theorem): RID(D) = \u00af RID(D), (18) i.e. the identi\ufb01cation rate is given in (17). Moreover, if R < RID(D), then the probability of maybe converges to 1 exponentially fast. A few comments are in order regarding the theorem, whose proof is given in detail in Sec. V: \u2022 Theorem 2 states that the case where the probability of FN events is equal to zero is inherently different from the case \ufb01rst discussed in [1, Thm. 1], where the FN probability is only required to vanish with n: here the rate problem is not degenerate (since the minimal achievable rate question does not always give zero), and is in fact, more along the lines of the classical results in information theory, such as channel capacity and classical rate distortion compression. \u2022 As discussed above, the direct part of the theorem follows from [1] and Prop. 1. Nevertheless, in Sec. V we shall also outline how to prove the achievability part directly, based on a version of the type covering lemma. \u2022 The techniques used in [1] for proving a converse result on the error exponents are not strong enough for proving the converse for Thm. 2. In the proof here, we utilize some of the tools developed in [1], namely the inherently typical subset lemma, and augment them with the blowing-up lemma ([27], see also [28, Lemma 12]). The purpose of the blowing up lemma in this context is to take an event whose probability is exponentially small, but with a very small exponent, and transform it to a related event, whose probability is very close to 1. See Sec. V for details. B. Special Case: Distortion Measures Satisfying the Triangle Inequality Consider the special case where X = Y, and for all x, y, z \u2208X , we have \u03c1(x, y) + \u03c1(y, z) \u2265\u03c1(x, y). (19) In other words, the distortion measure satis\ufb01es the triangle inequality, which is a common property of distance / similarity measures. For simplicity, also assume that the measure is symmetric, i.e. that for all x, y \u2208X , \u03c1(x, y) = \u03c1(y, x) (later on we discuss the generalization 3The cardinality of U can be taken as |X | + 2, according to [1, Lemma 3]. However, in the sequel we improve the cardinality bound, see Subsection IV-C. \f10 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Fig. 3. Illustration of the triangle-inequality decision rule. If d(\u02c6 x, y) > D + d(x, \u02c6 x), then it is certain that d(x, y) > D, and we can therefore safely declare that x and y are not similar. to the non-symmetric case). In this case there are intuitive compression schemes that naturally come to mind. The main idea is to use the compressed representation for reconstructing an approximation \u02c6 x of the source, and then to use this reconstruction to decide whether x and y are similar or not. For example, suppose that the source was indeed reconstructed as \u02c6 x, and also assume that we know the value of d(x, \u02c6 x) (adding this value to the compressed representation of \u02c6 x is a negligible addition to the rate). Consider the following decision rule, based on the pair (\u02c6 x, d(x, \u02c6 x)): g((\u02c6 x, d(x, \u02c6 x)), y) = \u001a no, d(\u02c6 x, y) > d(x, \u02c6 x) + D; maybe, otherwise. (20) This rule is admissible because in cases where x and y are D-similar, it follows by the triangle inequality that d(\u02c6 x, y) \u2264d(\u02c6 x, x) + d(x, y) (21) \u2264d(\u02c6 x, x) + D, (22) resulting in the decision function in (20) returning a maybe. The process is illustrated in Fig. 3. The remaining question is, then, how to choose the lossy representation \u02c6 x, and whether this results in an optimal scheme (i.e. whether the optimal RID(D) is achieved). We survey two schemes based on the triangle inequality principle, and discuss the compression rate achievable with each scheme. In general, neither can be shown to achieve RID(D). The naive scheme: LC-\u25b3(Lossy Compression signatures and triangle ineq. decision rule) In this scheme, we use standard lossy compression in order to represent \u02c6 x with \ufb01xed rate R, i.e. we optimize for a reconstruction that minimizes d(x, \u02c6 x). When the compression rate is R, it is known that the attained distortion d(x, \u02c6 x) is close, with high probability and for long enough sequences, to the distortion-rate function D(R). The full details of the LC-\u25b3scheme are given in Section VI, along with the proof of the following theorem, which characterizes the similarity threshold supported by any given compression rate: Theorem 3: Any similarity threshold below DLC-\u25b3 ID (R) can be attained with a LC-\u25b3 compression scheme of rate R, where DLC-\u25b3 ID (R) \u225cE[\u03c1( \u02c6 X, Y )] \u2212D(R), (23) \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 11 where D(R) is the classical distortion-rate function of the source X, and \u02c6 X, which is independent of Y , is distributed according to the marginal distribution of the D(R) achieving distribution (if there are several D(R)-achieving distributions, take one that maximizes E[\u03c1( \u02c6 X, Y )]). We denote by RLC-\u25b3 ID (D) the inverse function of DLC-\u25b3 ID (R), i.e. the compression rate that is achieved for a similarity threshold D. This scheme is suboptimal in general, i.e. there are cases for which RLC-\u25b3 I D(D) > RID(D), but in some cases they are equal. For example, the symmetric binary source with Hamming distortion is one case in which this naive scheme is optimal. This case is discussed in detail in [29], where an actual scheme is implemented based on lossy compression algorithms. An improved scheme: TC-\u25b3(\u201cType Covering\u201d signatures and triangle ineq. decision rule) The expression in (23) gives rise to the following intuitive idea: in the distortion rate case, we wish to minimize the distortion, with a constraint on the mutual information (that controls the compression rate). The free variable in the optimization is the transition probability P \u02c6 X|X(\u02c6 x|x). So far this is in agreement with (23), as we wish that the similarity threshold will be maximized. However, the expectation term in (23) also depends on the transition probability P \u02c6 X|X(\u02c6 x|x), raising the question whether both terms should be optimized together. The answer to this question is positive. The key step is to use a general type covering lemma for generating \u02c6 x (using a distribution that does not necessarily minimize the distortion between X and \u02c6 X). This idea is made concrete in the following theorem (whose proof is given in section VI): Theorem 4: Any similarity threshold below DTC-\u25b3 ID (R) can be attained by a TC-\u25b3compression scheme of rate R, where DTC-\u25b3 ID (R) \u225c max P \u02c6 X|X:I(X; \u02c6 X)\u2264R E[\u03c1( \u02c6 X, Y )] \u2212E[\u03c1(X, \u02c6 X)], (24) where on the RHS, \u02c6 X and Y are independent. We denote by RTC-\u25b3 ID (D) the inverse function of DTC-\u25b3 ID (R), i.e. the compression rate that is achieved by a TC-\u25b3scheme for a similarity threshold D. It can be also written as RTC-\u25b3 ID (D) = min P \u02c6 X|X:E[\u03c1( \u02c6 X,Y )]\u2212E[\u03c1(X, \u02c6 X)]\u2265D I(X; \u02c6 X). (25) We also note that the TC-\u25b3scheme is a natural extension of the scheme given in [3, Theorem 3], which applies for continuous sources and quadratic distortion. It is not hard to see that RTC-\u25b3 ID (D) \u2264RLC-\u25b3 ID (D), since the distortion-rate achieving distribution in (23) is a feasible transition probability for the expression in (24). However, the TC-\u25b3scheme is not optimal in general, as we shall see later on. So far we have, in general, that RID(D) \u2264RTC-\u25b3 ID (D) \u2264RLC-\u25b3 ID (D). (26) There are special cases, however, where some of the above inequalities are actually equalities (proved in Sec. VI): \u2022 For the binary-Hamming case, we have RID(D) = RTC-\u25b3 ID (D). (27) \f12 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY \u2022 If P y\u2208Y PY (y)\u03c1(\u02c6 x, y) is constant for all \u02c6 x \u2208X , then RTC-\u25b3 ID (D) = RLC-\u25b3 ID (D) = R(D0 \u2212D), (28) where R(\u00b7) is the standard rate-distortion function and D0 \u225cP y\u2208Y PY (y)\u03c1(\u02c6 x, y). \u2022 As a consequence, in the binary-Hamming case where Y is symmetric, both (27) and (28) hold, and we have RID(D) = RTC-\u25b3 ID (D) = RLC-\u25b3 ID (D) = R( 1 2 \u2212D), (29) where R(\u00b7) is the rate distortion function for the source X and Hamming distortion. Next, we provide an easily computable lower bound on the identi\ufb01cation rate, that holds for the case of Hamming distortion: Theorem 5: For Hamming distortion, the identi\ufb01cation rate is always lower bounded by RID(D) \u2265 \u0002 2D2 log e \u2212D(PX||PY ) \u0003+ . (30) In particular, when PX = PY , we have RID(D) \u22652D2 log e. (31) Proof: Appendix B In Fig. 4 we plot the three rates: RLC-\u25b3 ID (D), RTC-\u25b3 ID (D) and RID(D) for the case of X = Y = {0, 1, 2}, PX = PY = [.8.1.1] and Hamming distortion. As seen in the \ufb01gure, the three rates are different, indicating that neither of the LC-\u25b3and TC-\u25b3schemes is optimal. We also plot the lower bound from Theorem 5. C. Computing the Identi\ufb01cation Rate The identi\ufb01cation rate RID(D) is given in (17) as an optimization program of single-letter information-theoretic quantities, with an auxiliary random variable with alphabet U. In order to actually compute the value of RID(D), one has to (a) obtain a bound on the cardinality of U, and (b) be able to ef\ufb01ciently solve the optimization program, which is non-convex. In order to tackle the cardinality issue, let us de\ufb01ne Rk ID(D) \u225c min PU|X:P u\u2208U PU(u)\u00af \u03c1(PX|U(\u00b7|u),PY )\u2265D I(X; U), (32) where the alphabet U is given by k. Clearly, Rk ID(D) is a decreasing function of k (since a lower value of k can be considered a special case of the higher k). In [1, Lemma 3], the standard support lemma (see, e.g. [30, Lemma 15.4]) is used to show that RID(D) = R|X|+2 ID (D). In other words, taking |U| = |X | + 2 suf\ufb01ces in order to obtain the true value of RID(D). We improve this result as follows: Theorem 6: For any D, we have RID(D) = R|X|+1 ID (D). (33) Furthermore, the entire curve RID(D) can be obtained by calculating the curve R|X| ID (D) for all D, and then taking the lower convex envelope. Remarks: \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 13 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 H(X) RLC\u2212\u25b3 ID (D) RTC\u2212\u25b3 ID (D) RID(D) 2D2 loge E[\u03c1(X, Y )] Compression rate (bits) Similarity threshold D (Hamming) Fig. 4. Three compression rates for the ternary source PX = [.8, .1.1]. The distribution PY of the query sequence is the same as PX. The distortion measure is Hamming. \u2022 The proof, given in detail in Sec. VII, is based on the support lemma, but uses it in a more re\ufb01ned way than in [1, Lemma 3], in a manner similar in spirit to [31]. \u2022 The second part of the result follows from the fact that whenever (RID(D), D) is an exposed point of the convex region of achievable pairs, RID(D) = R|X| ID (D). (34) See Sec. VII for details. \u2022 Taking the lower convex envelope is also necessary. In other words, we sometimes have a strict inequality of the form R|X|+1 ID (D) < R|X| ID (D). For example, in the case of ternary alphabet and Hamming distortion as in Fig. 4 above, we have such a strict inequality for D = 0.32. The harder problem is the non-convexity of the optimization in (17) (not to be confused with the fact that the function RID(D) itself is a convex function of D, see [1, Lemma 3]). While the target function (the mutual information) is convex, the feasibility region is not, which makes the optimization hard. In order to tackle this issue, we show that this region is the complementary of a convex polytope. Then, we propose a method for reducing the optimization program (17) to a sequence of convex optimization programs that can be solved easily (e.g. via cvx [32]), and the minimum among the solutions of those programs is equal to RID(D). To illustrate this idea, consider the optimization of a convex function f : Rn \u2192R over \f14 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY the set Rn \\ \u039e, where \u039e is a convex polytope: inf z\u2208Rn\\\u039e f(z). (35) Any polytope can be written as \u039e = {z \u2208Rn : aT i z \u2264bi for all 1 \u2264i \u2264m}, (36) where {ai} are m length-n vectors and {bi} are m scalars, where m corresponds to the number of facets of the polytope \u039e. Rewriting the optimization program gives inf z\u2208Rn\\\u039e f(z) = min z\u2208Rn:\u2203i:aT i z\u2265bi f(z) (37) = min 1\u2264i\u2264m min z\u2208Rn:aT i z\u2265bi f(z). (38) Each of the new optimization programs is a minimization of a the original convex function, but now with linear constraints, and therefore can be calculated easily. While this method does not scale well with the alphabet size (since the number of facets of the polytope grows very quickly with |X |), it still provides a method for calculating RID(D) for small values of |X |. For example, the RID(D) curve in Fig. 4 was obtained with this method. The full details of the reduction method, along with simpli\ufb01cations for the Hamming distortion case, are given in Sec. VII. V. PROOF OF THE IDENTIFICATION RATE THEOREM In this section we prove Theorem 2. After giving a high-level overview of the proof, we introduce additional notation and review the basic tools that are used in the proof. The proof itself is given in Subsection V-F. A. Proof roadmap We start with an informal overview of the proof. First, note that it suf\ufb01ces to consider only typical sequences x, i.e. sequences whose (\ufb01rst-order) empirical distribution is close to the true one PX (denoted TPX, see a formal de\ufb01nition below). The same holds for the query sequences y. The achievability scheme is based on constructing a code, which is a set of sequences from another alphabet Un, that \u201ccovers\u201d the typical set TPX. The covering is in the sense that for each x \u2208TPX, there exists a word u in the code, s.t. x, u will have a \ufb01rst-order empirical distribution that is close to some given joint distribution (which is given as a design parameter PU|X, along with the size of the alphabet U). Such a covering is guaranteed to exist by a version of the type covering lemma (stated below), and the code rate is given by the mutual information between X and U, which de\ufb01ne the joint distribution. The signature of a sequence x, T(x), is de\ufb01ned as the index to the sequence u in the code that covers x. The decision process g(\u00b7, \u00b7) simply declares maybe (given u and y), if there exists a typical sequence x that is mapped to u, that is also similar to y. The scheme is admissible, and all that remains is to evaluate the probability of maybe, i.e. the probability that Y falls in the D-\u2018expansion\u2019 of the set of sequences x that are mapped to u. See Fig. 5 for an illustration. It can be shown that if the similarity threshold D satis\ufb01es D < X u PU(x)\u00af \u03c1(PX|U(\u00b7|u), PY ), (39) \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 15 Fig. 5. Illustration of the achievability scheme. Each sequence in the typical set TPX is mapped to (the index of) a single code point u. A maybe is declared if y falls in the D-expansion of the set of x-sequences that are mapped to u. the fraction of sequences for which a maybe is declared vanishes with n. This leads to the achievablity of the rate \u00af RID(D). For the converse part, we follow the steps of [1] and essentially show that any compression scheme performs as well as a scheme that was considered in the achievability part. In particular, we show that there exists a distribution PU|X for which each of the sets of xsequences mapped to the same i, contains a set which is called \u2018inherently typical\u2019 w.r.t. PUX (see below). While in the achievability part we claimed that if (39) holds then the probability of maybe vanishes, here we need to claim the opposite, i.e. that if (39) does not hold, then the probability of maybe cannot go to zero. We note that in [1] the argument can only lead to a claim of the sort \u201cif (39) does not hold, then the probability of maybe cannot vanish exponentially with n\u201d. Here, however, we require a stronger result. In order to proceed, we use the blowing-up lemma (see below), and show that if (39) does not hold, then the probability of maybe converges to 1, and this convergence is exponential in n. This can be regarded as a \u2018exponentially strong converse\u2019 (see, e.g. [33] for an overview of converse types). B. Additional Notation We shall use the method of types [30]. Let P(X ) denote the set of all probability distributions on the alphabet X . We denote by P(X \u2192Y) the set of all stochastic matrices (or, equivalently, conditional distributions, or channels) from alphabet X to Y. On occasions, we deal with vectors of different lengths. For this purpose, we use the notation xk as short hand for the vector [x1, ..., xk], so, for example, xn and x shall denote the same thing. For a sequence x \u2208X n and a \u2208X , let N(a|x) denote the number of occurrences of a in x. The type of the sequence x, denoted Px, is de\ufb01ned as the vector \f16 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 1 n[N(1|x), N(2|x), ..., N(|X | |x)]. For any sequence length n, let Pn(X ) denote the set of all possible types of sequences of length n, (also called n-types): Pn(X ) \u225c{P \u2208P(X )|\u2200x \u2208X , nP(x) \u2208Z+} . (40) For a type P \u2208Pn(X ), the type class TP is de\ufb01ned as the set of all sequences x \u2208X n with type P (or, equivalently, that Px = P). More generally, for a distribution P \u2208P(X ) (not necessarily an n-type), and a constant \u03b3 > 0, the set of typical sequences TP,\u03b3 is de\ufb01ned as the set of all sequences x \u2208X n for which: 1) |P(x) \u2212Px(x)| \u2264\u03b3 \u2200x \u2208X , 2) Px(x) = 0 whenever P(x) = 0. If X is a random variable distributed according to P, we shall sometimes write TX,\u03b3 for TP,\u03b3. Similarly, for V \u2208P(X \u2192Y) and x \u2208X n, we denote by TV,\u03b3(x) the set of conditionally typical sequences, i.e. that 1) |N(x\u2032, y\u2032|x, y) \u2212N(x\u2032|x)V (y\u2032|x\u2032)| \u2264n \u00b7 \u03b3 \u2200x\u2032 \u2208X , y\u2032 \u2208Y, 2) N(x\u2032, y\u2032|x, y) = 0 whenever V (y|x) = 0. For random variables X, Y where Y is the output of the channel V with input X, we\u2019ll sometimes use the notation TY |X,\u03b3(x) for TV,\u03b3(x). Let \u03b4n be de\ufb01ned according to the delta convention [30]; i.e. that \u03b4n \u21920, but also that n\u03b42 n \u2192\u221e(e.g. \u03b4n \u225cn\u2212\u03b1, with some \u03b1 \u2208(0, 0.5)). With this convention, we have (see [30, Lemma 2.12]): \u2022 If X is distributed i.i.d. according to a distribution P, then Pr{X \u2208TP,\u03b4n} \u22651 \u2212|X | 4n\u03b42 n . (41) \u2022 If Y is the output of the DMC V with input x \u2208X n, then Pr{Y \u2208TV,\u03b4n(x)} \u22651 \u2212|X ||Y| 4n\u03b42 n . (42) Recall that we have de\ufb01ned an arbitrary distortion measure \u03c1 : X \u00d7 Y \u2192R+. The maximal possible distortion is denoted by \u03c1max: \u03c1max \u225c max x\u2208X,y\u2208Y \u03c1(x, y). (43) For a set A \u2286X n, we denote its D-expansion by \u0393D(A) \u225c{y \u2208Yn : \u2203x \u2208A s.t. d(x, y) \u2264D} . (44) When we use the Hamming distance as the distortion measure, we denote the expansion by \u0393D H(A). \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 17 C. A covering lemma The main building block for the achievability proofs in the paper is the following version of the covering lemma (see e.g. [34, Prop. 1]): Lemma 1 (A covering lemma): For any distribution PX \u2208P(X ) and any channel V \u2208 P(X \u2192U), there exists a mapping \u03c9 : TPX,\u03b4n \u2192Un s.t. \u03c9(x) \u2208TV,\u03b4n(x) for all x \u2208TPX,\u03b4n, (45) and |{\u03c9(x) : x \u2208TPX,\u03b4n}| \u22642n(I(PX,V )+\u03b5n), (46) where \u03b5n = O (\u03b4n log(1/\u03b4n)), with constants that depend only on |X | and |U|. Essentially, this lemma says that there exists a code C \u2286Un of rate I(PX, W) that covers the typical set TPX,\u03b4n, in the sense that for every x \u2208TPX,\u03b4n, there exists a \u2018codeword\u2019 u \u2208C such that the sequences x, u have a joint type that is very close to (PX, W). D. The inherently typical subset lemma The proof of the converse of Theorem 2 relies heavily on the inherently typical subset lemma, due to Ahlswede et al.[1]. An inherently typical set is a generalization of the conditional type class concept, as detailed below. Loosely speaking, the lemma says that every set contains an inherently typical subset of essentially the same size. Before stating the lemma we require several de\ufb01nitions. De\ufb01nition 3 (Pre\ufb01x set): For A \u2286X n and 1 \u2264k \u2264n, let Ak denote the set of pre\ufb01xes of sequences of A, i.e. Ak \u225c{xk \u2208X k : \u2203\u02dc xn \u2208A s.t. xk = \u02dc xk}. (47) We denote A0 = {\u039b}, where \u039b denotes the empty string. Note that, by our convention, for any x \u2208A, x0 = \u039b. De\ufb01nition 4 (Causal mapping): For A \u2286X n and an arbitrary alphabet U, a mapping \u03c6 : A \u2192Un is said to be causal, if there exist mappings {\u03c6i : Ai \u2192U}n\u22121 i=0 , s.t. for all x \u2208A we have \u03c6(x) = [\u03c61(x0), \u03c61(x1), ..., \u03c6n(xn\u22121)]. (48) Let Um denote a discrete alphabet of size |Um| = |Pm(X )| = \u0012 |X | + m \u22121 m \u0013 . (49) The alphabet Um is exactly the set of m-types over the alphabet X . For example, if X = {0, 1}, we have |Um| = m + 1. De\ufb01nition 5 (Inherently typical set [1]): A set A \u2286X n is said to be m-inherently typical, if there exist a causal mapping \u03c6 : A \u2192Un m and an n-type Q \u2208Pn(X \u00d7 Um), s.t. 1) For every x \u2208A, the sequences x, \u03c6(x) have the joint type Q, i.e. Px,\u03c6(x) = Q. (50) 2) If X0, U0 are jointly distributed according to Q, then 1 n log |A| \u2264H(X0|U0) \u22641 n log |A| + log2 m m . (51) \f18 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Lemma 2 (The inherently typical subset lemma [1]): Let m > 216|X|2, and let n be large enough s.t. (m + 1)5|X|+4 ln(n + 1)/n \u22641. Then for any set A \u2286X n there exists a subset \u02dc A \u2286A that is m-inherently typical, whose size satis\ufb01es 1 n log |A| | \u02dc A| \u2264|X |(m + 1)|X|log(n + 1) n . (52) Note that the lemma holds starting at m = 216|X|2, which is extremely large, even for X being binary. It is therefore expected that \ufb01nite blocklength versions of results based on the lemma will be very loose. Nevertheless, it is suf\ufb01cient for proving a converse for the error exponent (as in [1]), and also for proving the converse for the rate (as we show next). E. The Blowing Up Lemma The blowing up lemma (see, e.g., [28, Lemma 12]) will play a key role in proving the converse part of Theorem 2. Lemma 3 (Blowing-up Lemma): Let Z be distributed i.i.d. according to PZ and let B \u2286 Zn be a given set. Then: Pr{Z / \u2208\u0393\u03b4 H(B)} \u2264exp \uf8ee \uf8f0\u22122n \u03b4 \u2212 s 1 2n ln \u0012 1 Pr{Z \u2208B} \u0013!2\uf8f9 \uf8fb, (53) \u2200\u03b4 > s 1 2n ln \u0012 1 Pr{Z \u2208B} \u0013 . F. Proof of Theorem 2 As discussed above, the direct part of the theorem follows from the combination of [1, Thm. 2] and Prop. 1 above. Nevertheless, for completeness, we give a direct proof here. It will be simpler than the proof of [1, Thm. 2], since it only involves the achievable rate (and not error exponents). After reading through the direct part, it should be easier to the reader to follow the converse proof. Proof of Theorem 2, direct part: Let W : P(X \u2192U) be an arbitrary channel into an arbitrary (discrete) alphabet U. Following Lemma 1, let \u03c9 : TPX,\u03b4n \u2192Un be the mapping that covers the typical set TPX,\u03b4n. Let C = {\u03c9(x) : x \u2208TPX,\u03b4n} denote the set of codewords. Let u = \u03c9(x). The mapping T(\u00b7) is de\ufb01ned as follows: T(x) = \u001a [u, Pxu], x \u2208TPX,\u03b4n; e, otherwise. (54) In other words, the signature T(x) consists of the (index to the) codeword u, and also the joint type of x and u. The special symbol e denotes \u2018erasure\u2019. The decision function g(T(x), y) shall return maybe only if one of the following occurs: \u2022 The erasure symbol was received, i.e. T(x) = e, or \u2022 y was not typical, i.e. y / \u2208TPY ,\u03b4n, or \u2022 y is typical, T(x) \u0338= e, and there exists a joint type Q \u2208P(X \u00d7 U \u00d7 Y) s.t.: 1) The marginal of Q w.r.t. U, Y is Puy (i.e. the empirical joint distribution of the sequences u, y), \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 19 2) the marginal of Q w.r.t. X, U is Pxu and 3) the marginal of Q w.r.t. X, Y satis\ufb01es E[\u03c1(X, Y )] \u2264D. First, let us see why this scheme is admissible, i.e. there are no false negatives. Note that the only case where the scheme may return no is when the signature consists of [u, Pxu]. If d(x, y) \u2264D, then the joint type Q = Pxuy satis\ufb01es the conditions 1) 3) above, and therefore the scheme would return maybe, as required. Next, note that the rate of the scheme is arbitrarily close to I(PX, W), since the joint n-type Px,u adds O(log(n)/n) to the rate, and the signature e adds a negligible amount to the rate. The last point that needs to be proved is that the probability of maybe vanishes. Consider the three error events: \u2022 T(x) = e. This happens if and only of x / \u2208TPX,\u03b4n, which, following (41) and the de\ufb01nition of \u03b4n, vanishes with n. \u2022 For a similar reason, the probability that y / \u2208TPY ,\u03b4n vanishes with n. \u2022 The remaining event is when x and y are both typical (i.e. have empirical distributions near PX and PY , respectively), and there exists a joint type Q that satis\ufb01es 1)-3) above. Fix a sequence x \u2208TPX,\u03b4n and let P \u2032 Y \u2208Pn(Y) be a given n-type. Conditioned on Y \u2208TP \u2032 Y , we know that Y is distributed uniformly on TP \u2032 Y . Among those sequences, the fraction of these sequences that trigger a maybe are the ones residing in the QY |U(u)-shell of some distribution Q that satis\ufb01es 1)-3). Note that the requirement 3) is on the distribution Q, and not on the sequence y. Therefore the probability of maybe, conditioned on X = x and y \u2208TP \u2032 Y , is upper bounded, for any \u03b5 > 0 and large enough n, by X Q:QXU=Pxu,EQ[\u03c1(X\u2032,Y \u2032)]\u2264D,QY =P \u2032 Y 2\u2212n(I(Y \u2032;U\u2032)\u2212\u03b5), (55) where X\u2032, U\u2032, Y \u2032 are jointly distributed according to Q. Since P \u2032 Y is arbitrarily close to PY and Pxu is arbitrarily close to (PX, W), we can sum over the (polynomial number of) types P \u2032 Y , and bound the probability for maybe, now conditioned on x \u2208TPX,\u03b4n and y \u2208TPY ,\u03b4n, is upper bounded, for any \u03b5 > 0 and large enough n, by max Q:QXU=(PX,W ),EQ[\u03c1(X\u2032,Y \u2032)]\u2264D,QY =PY 2\u2212n(I(Y \u2032;U\u2032)\u2212\u03b5). (56) It therefore remains to show that if W is chosen according to (17), then there are no types Q that satisfy 1)-3), for which U\u2032 and Y \u2032 are independent. This fact will make the exponent in (56) to be strictly positive, completing the proof. In order to show this, let \u2206R, \u2206D > 0 be arbitrarily small constants, s.t. R = \u2206R + \u00af RID(D \u2212\u2206D). (57) In other words, we know that (PX, W) satis\ufb01es X u\u2208U PU(u)\u00af \u03c1(PX|U(\u00b7|u), PY ) \u2265D + \u2206D. (58) and we need to prove that for all joint distributions Q that satisfy QXU = (PX, W), QY = PY , and EQ[\u03c1(X, Y )] \u2264D, we must have I(Y ; U) > 0. To prove this, assume, \f20 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY for contradiction, that Q is such a distribution, where Y and U are independent. Then we can write the following: D \u2265EQ[\u03c1(X, Y )] (59) D \u2265 X u\u2208U PU(u) X x,y QXY |U(x, y|u)\u03c1(x, y) (60) D \u2265 X u\u2208U PU(u) X x,y QY |U(y|u)QX|Y U(x, y|u)\u03c1(x, y) (61) D \u2265 X u\u2208U PU(u) X x,y PY (y)QX|Y U(x, y|u)\u03c1(x, y) (62) Note that for all u, the term PY (y)QX|Y U(x, y|u) de\ufb01nes a joint distribution on X, Y , with marginals PY and QX|U(\u00b7|u) = PX|U(\u00b7|u). Therefore the inner summation is an upper bound, by de\ufb01nition, for the term \u00af \u03c1(PX|U(\u00b7|u), PY ), and we get that D \u2265 X u\u2208U PU(u)\u00af \u03c1(PX|U(\u00b7|u), PY ), (63) which contradicts (58). Therefore U and Y can never be independent, and the exponent in (56) is strictly positive. This makes sure that the third error event also has a vanishing probability, and the proof is concluded by using the union bound on the error events. To prove the converse part, we start by following the steps similar to that of the converse of [1, Thm. 2]. In our case these steps are actually slightly simpler than those of [1, Thm. 2] because of the restriction to \ufb01xed-length compression schemes. As mentioned before, the key step that is missing from [1, Thm. 2] is the ability to transform an event with probability that vanishes with an exponent that is arbitrarily small, to an event whose probability goes to 1. The key to achieving this is the blowing-up lemma, as detailed in what follows. Proof of Theorem 2, converse part: Let \u2206R, \u2206D > 0, and let T, g be a sequence of schemes with rate R < \u00af RID(D+\u2206D)\u2212\u2206R. Our goal is to show that the probability for maybe cannot vanish with n, for arbitrarily small \u2206R and \u2206D. Let i \u2208[1 : 2nR], and let T \u22121(i) \u225c{x \u2208X n : T(x) = i}. (64) Since the given scheme is admissible, we must have Pr{maybe} = 2nR X i=1 Pr{T(X) = i} Pr{maybe|T(X) = i} (65) \u2265 2nR X i=1 Pr {T(X) = i} Pr \b Y \u2208\u0393D \u0000T \u22121(i) \u0001\t . (66) For some \u03b3 > 0, de\ufb01ne the set Ai \u225cT \u22121(i) \u2229TPX,\u03b3, (67) \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 21 Since by de\ufb01nition Ai \u2286T \u22121(i), we also have Pr{maybe} \u2265 2nR X i=1 Pr {T(X) = i} Pr \b Y \u2208\u0393D(Ai) \t . (68) It appears that it is enough to consider only Ai\u2019s that are \u2018large\u2019, by the following lemma: Lemma 4 (Most Ai\u2019s are large): There exists n0 = n0(\u03b3) > 0, s.t. for all n > n0, X i:|Ai|\u22642n[H(PX)\u2212R\u22122\u03b3\u2032] Pr{X \u2208Ai} \u22642\u2212\u03b3\u2032n, (69) where \u03b3\u2032 \u225c2\u03b3|X | log(1/\u03b3). Proof: Appendix C. Next, consider a speci\ufb01c A = Ai, and suppose that |A| \u22652n(H(X)\u2212R\u22122\u03b3\u2032) (by the previous lemma we know that this occurs with high probability). We invoke the inherently typical subset lemma and conclude that for any m > 216|X|2 and large enough n, there exists a subset \u02dc A \u2286A, for which: 1) The size of the set \u02dc A is essentially the same as A: 1 n log |A| | \u02dc A| \u2264|X |(m + 1)|X|log(n + 1) n . (70) 2) There exists an n-type Q \u2208Pn(X \u00d7 Um), and a causal mapping \u03c6 : X n \u2192Un m, s.t. for every x \u2208\u02dc A, Px,\u03c6(x) = Q. (71) 3) If X0, U0 are jointly distributed according to Q, then 1 n log | \u02dc A| \u2264H(X0|U0) \u22641 n log | \u02dc A| + log2 m m . (72) Since A \u2286TPX,\u03b3, we must have that the marginal of Q w.r.t. X has a type P that satis\ufb01es \u2225P \u2212PX\u2225\u221e\u2264\u03b3. Since \u02dc A \u2286A, we have Pr{Y \u2208\u0393DA} \u2265Pr{Y \u2208\u0393D \u02dc A}. (73) Let \u03b5 > 0 and let V : X \u00d7 Um \u2192Y be a stochastic matrix, s.t. X x,y,u Q(x, u)V (y|x, u)\u03c1(x, y) \u2264D \u22122\u03b5. (74) In other words, if X0, U0, Y0 are distributed according to (Q, V ), then E[\u03c1(X0, Y0)] \u2264D\u22122\u03b5. Following (74), for all x \u2208\u02dc A and large enough n, y \u2208TY0|X0U0,\u03b4n(x, \u03c6(x)) implies d(x, y) \u2264D \u2212\u03b5. Next, de\ufb01ne the set F to be the union of all such conditional type classes as follows: F \u225c [ x\u2208\u02dc A TY0|X0U0,\u03b4n(x, \u03c6(x)). (75) Clearly, from the above it follows that F \u2286\u0393D\u2212\u03b5 \u0010 \u02dc A \u0011 . \f22 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Let \u03b5\u2032 \u225c\u03b5/\u03c1max. Note the following fact: \u0393\u03b5\u2032 H \u0010 \u0393D\u2212\u03b5 \u0010 \u02dc A \u0011\u0011 \u2286\u0393D \u0010 \u02dc A \u0011 . (76) To see this, suppose y \u2208\u0393\u03b5\u2032 H \u0010 \u0393D\u2212\u03b5 \u0010 \u02dc A \u0011\u0011 . Then there exists y\u2032 \u2208\u0393D\u2212\u03b5( \u02dc A) s.t. dH(y, y\u2032) \u2264 \u03b5\u2032. Also, there exists x \u2208\u02dc A s.t. d(x, y\u2032) \u2264D \u2212\u03b5. Now write n \u00b7 d(x, y) = n X i=1 d(xi, yi) (77) = X i:yi=y\u2032 i d(xi, y\u2032 i) + X i:yi\u0338=y\u2032 i d(xi, y\u2032 i) (78) \u2264 n X |i=1 d(xi, y\u2032 i) + X i:yi\u0338=y\u2032 i \u03c1max (79) \u2264n(D \u2212\u03b5) + n\u03b5\u2032\u03c1max (80) = nD. (81) With (76) above we have Pr \b y \u2208\u0393D(A) \t \u2265Pr n y \u2208\u0393\u03b5\u2032 H (F) o . (82) Next, we apply the blowing up lemma to the set F, and have: Pr n y \u2208\u0393\u03b5\u2032 H (F) o \u22651 \u2212exp \uf8ee \uf8f0\u22122n \u03b5\u2032 \u2212 s 1 2n ln \u0012 1 Pr{Y \u2208F} \u0013!2\uf8f9 \uf8fb, (83) whenever \u03b5\u2032 > s 1 2n ln \u0012 1 Pr{Y \u2208F} \u0013 . (84) Since \u03b5\u2032 > 0 is a constant, all that is left is to prove that Pr{Y \u2208F} does not vanish exponentially. Note that we only need to show the existence of a single channel V for which Pr{Y \u2208F} does not vanish exponentially. To show this, we follow the steps of [1]: First, since F is a union of V -shells, we deduce that all the sequences in it are typical: F \u2286TY0,\u03b4n\u00b7|X||Um|, (85) where Y0 is an RV with the marginal distribution of (Q, V ). It follows that Pr{Y \u2208F} \u2265 |F| 2nH(Y0)2\u2212n(D(Y0||Y )+\u03ben), (86) where \u03ben \u21920 with n. We see that the key to evaluating Pr{Y \u2208F} is evaluating |F|. Let \u02dc X be uniformly distributed on \u02dc A, and let \u02dc U \u225c\u03c6( \u02dc X). Note that \u02dc X, \u02dc U are random vectors which are not i.i.d., but have the joint type Q with probability 1. Next, de\ufb01ne \u02dc Y to be the output of the memoryless channel V with inputs \u02dc X, \u02dc U. Here are several facts regarding the random vector \u02dc Y, expressed by the random variables X0, U0, Y0 that are distributed according to (Q, V ). \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 23 \u2022 Since \u02dc Y is conditionally typical w.h.p. (given \u02dc X = x \u2208\u02dc A), Pr{ \u02dc Y \u2208F| \u02dc X = x} \u2265Pr{ \u02dc Y \u2208TY0|X0U0,\u03b4n(x, \u03c6(x))| \u02dc X = x} (87) \u22651 \u2212|X||Y||Um| 4n\u03b4n . (88) \u2022 Since the above holds for any x \u2208\u02dc A, we also have Pr{ \u02dc Y \u2208F} = X x\u2208\u02dc A Pr{ \u02dc X = x} Pr{ \u02dc Y \u2208F| \u02dc X = x} \u22651 \u2212|X||Y||Um| 4n\u03b4n . (89) For convenience, let \u03b3n \u225c|X||Y| 4n\u03b42 n . Note that by the delta convention, we have \u03b3n \u21920. De\ufb01ne the indicator RV \u03c7F as \u03c7F( \u02dc Y) \u225c1{ \u02dc Y \u2208F} (90) It follows that H( \u02dc Y) = H \u0010 \u02dc Y, \u03c7F( \u02dc Y) \u0011 (91) = H \u0010 \u02dc Y|\u03c7F( \u02dc Y) \u0011 + H(\u03c7F( \u02dc Y)) (92) = H \u0010 \u02dc Y|\u03c7F( \u02dc Y) \u0011 + h(Pr{ \u02dc Y \u2208F}) (93) (a) \u2264H \u0010 \u02dc Y|\u03c7F( \u02dc Y) \u0011 + h(\u03b3n) (94) = H \u0010 \u02dc Y| \u02dc Y \u2208F \u0011 Pr{ \u02dc Y \u2208F} + H \u0010 \u02dc Y| \u02dc Y / \u2208F \u0011 Pr{ \u02dc Y / \u2208F} + h(\u03b3n) (95) \u2264H \u0010 \u02dc Y| \u02dc Y \u2208F \u0011 + n log |Y| Pr{ \u02dc Y / \u2208F} + h(\u03b3n) (96) \u2264log |F| + n log |Y| Pr{ \u02dc Y / \u2208F} + h(\u03b3n) (97) \u2264log |F| + \u03b3nn log |Y| + h(\u03b3n). (98) where (a) follows from (89) for n large enough s.t. \u03b3n < 1/2. The last derivation reveals that we can bound H( \u02dc Y) to get a lower bound on |F| (and by that a lower bound on Pr{Y \u2208\u0393D(A)}). The next step follows [1, (4.26)-(4.30)] almost verbatim. For completeness, we pack the argument into a lemma, and prove it in the appendix (with simpler notation than in [1]). Lemma 5: With \u02dc Y de\ufb01ned above, we have 1 nH( \u02dc Y) \u2265H(Y0|U0) \u2212log2 m m . (99) Proof: Appendix D. We conclude that 1 n log |F| \u2265H(Y0|U0) \u2212log2 m m \u22121 nh(\u03b3n) \u2212\u03b3n log |Y|. (100) \f24 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Substituting back into (86) gives Pr{Y \u2208F} \u2265 |F| 2nH(Y0)2\u2212n(D(Y0||Y )+\u03ben) (101) \u22652 \u2212n \u0014 I(Y0;U0)+D(Y0||Y )+\u03ben+ log2 m m + 1 n h(\u03b3n)+\u03b3n log |Y| \u0015 . (102) Our next step is to show that the exponent in (102) can be made arbitrarily small, by a proper selection of the channel V . For that purpose, write: I(X0; U0) \u2264H(X0) \u2212H(X0|U0) (103) (a) \u2264H(X) \u2212H(X0|U0) + \u03b3\u2032 (104) (b) \u2264H(X) \u22121 n log | \u02dc A| + \u03b3\u2032 (105) (c) \u2264H(X) \u22121 n log |A| + \u03b3\u2032 + \u03b6n (106) (d) \u2264R + 3\u03b3\u2032 + \u03b6n (107) (e) \u2264\u00af RID(D \u2212\u2206D) \u2212\u2206R + 3\u03b3\u2032 + \u03b6n. (108) In the above, (a) follows by [30, Lemma 2.7] (cf. also the proof of Lemma 4). (b) and (c) follow from (72) and (70) respectively, where \u03b6n \u225c|X |(m + 1)|X| log(n+1) n . (d) follows from the assumption that |A| \u22652n(H(X)\u2212R\u22122\u03b3\u2032), and (e) follows from the assumption at the beginning of the proof. Next, we use the fact that X and X0 have distributions that are very close (with closeness quanti\ufb01ed by \u03b3), and write: \u00af RID(D \u2212\u2206D) = min PU|X:P u\u2208U PU(u)\u00af \u03c1(PX|U(\u00b7|u),PY )\u2265D\u2212\u2206D I(X; U), (109) \u2264 min PU|X0:P u\u2208U PU(u)\u00af \u03c1(PX0|U(\u00b7|u),PY )\u2265D\u2212\u2206D+\u03b3\u2032\u2032 I(X0; U) + \u03b3\u2032\u2032, (110) for some \u03b3\u2032\u2032 > 0 that vanishes with \u03b3. Next, let \u03b5, de\ufb01ned above to be arbitrarily small but positive, take the value of 1 3\u2206D. Also let \u03b3 be small enough s.t. \u03b3\u2032\u2032 < \u03b5, and that 3\u03b3\u2032 + \u03b3\u2032\u2032 \u22641 2\u2206R. This way, whenever n is large enough so that \u03b6n \u22641 2\u2206R, we have I(X0; U0) \u2264 min PU|X0:P u\u2208U PU(u)\u00af \u03c1(PX0|U(\u00b7|u),PY )\u2265D\u22122\u03b5 I(X0; U). (111) From the equation above, we deduce that X u\u2208Um PU0(u)\u00af \u03c1(PX0|U0(\u00b7|u), PY ) < D \u22122\u03b5. (112) By the de\ufb01nition of \u00af \u03c1(\u00b7, \u00b7), there exist distributions \u03a8u(x, y), for each u \u2208Um, s.t. X u\u2208Um PU0(u)E\u03a8u[\u03c1(X, Y )] \u2264D \u22122\u03b5. (113) Furthermore, the marginals of \u03a8u are PX0|U0(\u00b7|u) and PY . With PU0, \u03a8u de\ufb01nes a joint distribution with the following properties: \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 25 \u2022 The marginal of the distribution w.r.t. X0, U0 is exactly Q. Therefore the conditional distribution w.r.t. Y is a feasible choice for V according to (74). Denote the RV at the output of this channel by Y0. \u2022 The marginal distribution w.r.t. U0, Y0 is given by PU0 \u00d7 PY , i.e. U0 and Y0 are independent, and we also have that PY0 = PY . Indeed, we choose V to be de\ufb01ned according to PU0, \u03a8u, and conclude that I(Y0; U0) + D(Y0||Y ) = 0. (114) Substituting back into (102), we see that the desired exponent is arbitrarily close to log2 m m . We then set m to be large enough s.t. the condition (84) holds. To summarize, so far we have shown that if |A| \u22652n(H(X)\u2212R\u22122\u03b3\u2032), then Pr{Y \u2208\u0393D(A)} \u2265\u03b7n, (115) where \u03b7n approaches 1 (exponentially fast), as a result of the blowing up lemma. Finally, we repeat this for each of the sets Ai, and write: Pr{maybe} \u2265 2nR X i=1 Pr {T(X) = i} Pr \b Y \u2208\u0393D(Ai) \t (116) \u2265 X i:|Ai|\u22652n(H(X)\u2212R\u22122\u03b3\u2032) Pr {T(X) = i} Pr \b Y \u2208\u0393D(Ai) \t (117) (a) \u2265 X i:|Ai|\u22652n(H(X)\u2212R\u22122\u03b3\u2032) Pr {T(X) = i} \u03b7n (118) = \u03b7n X i:|Ai|\u22652n(H(X)\u2212R\u22122\u03b3\u2032) Pr {T(X) = i} (119) (b) \u2265\u03b7n \uf8ee \uf8f01 \u2212 X i:|Ai|\u22642n(H(X)\u2212R\u22122\u03b3\u2032) Pr {T(X) = i} \uf8f9 \uf8fb (120) \u2265\u03b7n h 1 \u22122\u2212n\u03b3\u2032i . (121) In the above, (a) follows from (115) and (b) follows from Lemma 4. Since \u03b7n approaches 1 exponentially fast, we conclude that the probability for maybe approaches 1, also exponentially fast. This concludes the proof of the converse. VI. SCHEMES BASED ON THE TRIANGLE INEQUALITY In this section we discuss the triangle-inequality based schemes: the lossy compression triangle inequality (LC-\u25b3) and the type covering triangle inequality (TC-\u25b3). A. Lossy compression with triangle inequality (LC-\u25b3) Here we prove that any compression rate above RLC-\u25b3 ID (D) [de\ufb01ned as the inverse function of DLC-\u25b3 ID (R), see (23)] can be attained via a scheme that employs standard lossy compression for the signature assignment and the triangle inequality for the decision rule. \f26 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Proof of Theorem 3: We will show that any pair (R, D) s.t. D > DLC-\u25b3 ID (R) is achievable, where DLC-\u25b3 ID (R) = E[\u03c1( \u02c6 X, Y )] \u2212D(R), (122) where D(R) is the classical distortion-rate function of the source X, and \u02c6 X, which is independent of Y , is distributed according to any marginal distribution of the D(R) achieving distribution. Let PX| \u02c6 X be an achieving distribution for the standard distortion rate function at rate R, and let \u02c6 X be the corresponding marginal distribution. Next, use the covering lemma (Lemma 1) to show the existence of a code C that covers the typical set4 TPX,\u03b4n with the distribution PX \u02c6 X. In other words, for each sequence in the typical set x \u2208TPX,\u03b4n, there exists a sequence \u03c9(x) = \u02c6 x \u2208C s.t. x, \u02c6 x are strongly jointly typical according to the distribution PX \u02c6 X (formally, \u02c6 x \u2208TV,\u03b4n(x)). We also know by the covering lemma that the code rate is upper bounded by I(X; \u02c6 X) + \u03b5n where \u03b5n vanishes as \u03b4n \u21920. Since PX, \u02c6 X is the distortion-rate achieving distribution, we know that d(x, \u02c6 x) \u2264D(R) + \u03b5\u2032 n, with some \u03b5\u2032 n that vanishes as \u03b4n \u21920. So far, we have constructed a standard code for lossy compression: for an input x, if it is typical, then its compressed representation is the index to \u02c6 x. If x is not typical, then declare an \u2018erasure\u2019 e. Therefore with probability approaching one, we have a guarantee that the distortion between the source and the reconstruction is at most D(R). Next, we use this code in order to construct a compression scheme for identi\ufb01cation. We only need to specify the decision process g(\u00b7, \u00b7), which proceeds as follows. If T(x) = e, we set g(T(x), y) = maybe. Otherwise, we reconstruct \u02c6 x, and have g(T(x), y) = \u001a maybe, if d(\u02c6 x, y) \u2264D + D(R) + \u03b5\u2032 n ; no, otherwise. (123) It follows from the triangle inequality that whenever d(x, y) \u2264D and when d(x, \u02c6 x) \u2264 D(R) + \u03b5\u2032 n, then d(\u02c6 x, y) \u2264D + D(R) + \u03b5\u2032 n, triggering a maybe. Therefore the scheme is admissible. Since, by construction, the rate of the scheme is arbitrarily close to R, we only need to verify that the probability of maybe vanishes. Next, assume that the similarity threshold D satis\ufb01es D = DLC-\u25b3 ID (R) = \u2206D for some \u2206D > 0. We analyze the probability of maybe as follows: Pr{maybe} \u2264Pr{X / \u2208TPX,\u03b4n} + Pr{Y / \u2208TPY ,\u03b4n} + Pr{maybe|X \u2208TPX,\u03b4n, Y \u2208TPY ,\u03b4n}. (124) The \ufb01rst two terms in the summation vanish with n. To bound the third term, we need to evaluate the probability that a sequence Y will be in a \u2018ball\u2019 of radius D + D(R) + \u03b5\u2032 n centered at \u02c6 x. Let \u02c6 x be the reconstruction sequence with type P \u2032 \u02c6 X. We know that it has a type close to P \u02c6 X, the marginal of the D(R)-achieving distribution. Also, let P \u2032 Y \u2208Pn(Y) be a given n-type. Conditioned on Y \u2208TP \u2032 Y , Y is now distributed uniformly on TP \u2032 Y . Among those sequences, the fraction of sequences that trigger a maybe is given (up to sup-exponential factors) by X 2nH(Y | \u02c6 X) 2nH(Y ) , (125) 4Recall that \u03b4n is de\ufb01ned according to the delta-convention [30]. See also Sec. V. \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 27 where the summation is over all joint n-types for Y, \u02c6 X with marginals P \u2032 Y , P \u2032 \u02c6 X s.t. E[\u03c1( \u02c6 X, Y )] \u2264D + D(R) + \u03b5n. The exponent of this expression is given by min I(Y ; \u02c6 X), (126) where the minimization is the same as in (125). We can see that this exponent can be made strictly positive: if it was zero, this would imply that there exist independent \u02c6 X, Y s.t. E[\u03c1( \u02c6 X, Y )] > D + D(R) + \u03b5n, which contradicts the assumption that D > DLC-\u25b3 ID (R). The next step follows by standard type arguments showing that P \u2032 Y and P \u2032 \u02c6 X are arbitrarily close to the real PY , P \u02c6 X (see also the direct part of the proof of Theorem 2). Finally, we see that all three terms in the expression for the probability of maybe in (124) vanish with n, as required. B. Type covering with triangle inequality (TC-\u25b3) Here we prove that for any similarity threshold D > DTC-\u25b3 ID (R), there exists a sequence of rate-R schemes that are D-admissible. Proof of Theorem 4: The proof follows the steps of the proof of Theorem 3 above, with the following key exception. The conditional distribution determining the code that describes X remains a design parameter and is optimized at the end (rather than at the beginning of the proof as in Theorem 3, where it is set to be the one that minimizes the expected distortion between X and \u02c6 X). More formally, let P \u02c6 X|X be a conditional distribution s.t. I(X; \u02c6 X) > R. Again, we use the covering lemma (Lemma 1) to show the existence of a code C that covers the typical set TPX,\u03b4n with the distribution PX \u02c6 X. In other words, for each x \u2208TPX,\u03b4n, there exists a sequence \u03c9(x) = \u02c6 x \u2208C s.t. x, \u02c6 x are strongly jointly typical according to the distribution PX \u02c6 X (formally, \u02c6 x \u2208TV,\u03b4n(x)). Again, we know by the covering lemma that the code rate is upper bounded by I(X; \u02c6 X) + \u03b5n where \u03b5n vanishes as \u03b4n \u21920. Since x, \u02c6 x are strongly jointly typical, we know that d(x, \u02c6 x) \u2264E[\u03c1(X, \u02c6 X)] + \u03b5\u2032 n, with some \u03b5\u2032 n that vanishes as \u03b4n \u21920. The rest of the proof is identical to that of Theorem 3, where D(R) is replaced by E[\u03c1(X, \u02c6 X)]. If we choose P \u02c6 X|X s.t. E[\u03c1( \u02c6 X, Y )] \u2212E[\u03c1(X, \u02c6 X)] \u2265D, we can verify that the exponent of the third term in the equivalent of (124) is positive, proving that the overall probability of maybe vanishes, as required. Remarks: \u2022 It is obvious that RTC-\u25b3 ID (D) \u2264RLC-\u25b3 ID (D), since the distortion-rate achieving distribution in (23) is a feasible transition probability for the expression in (24). Therefore the TC-\u25b3 scheme can be regarded as a generalization of LC-\u25b3. \u2022 In order to simplify the discussion, we have assumed that the distortion measure \u03c1(\u00b7, \u00b7) is symmetric. Similar results can be obtained for the non-symmetric case, where the only difference is that the triangle inequality is applied in the following form: d(\u02c6 x, y) \u2264d(\u02c6 x, x) + d(x, y). (127) \f28 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Therefore the compression of x needs to be done (in the LC-\u25b3scheme) for the distortion measure \u03c1\u2032(x, \u02c6 x) \u225c\u03c1(\u02c6 x, x). The decision rule (in the typical case) is given by g(T(x), y) = \u001a maybe, if d(\u02c6 x, y) \u2264D + E[\u03c1( \u02c6 X, X)] + \u03b5\u2032 n ; no, otherwise. (128) \u2022 If, in addition to the symmetry condition, we would require that \u03c1(x, y) = 0 if and only if x = y, this would make the measure a metric. However, there is no need for this third condition in order for the results to hold. \u2022 In principle, one could add another condition that rules out sequences that are not similar, using the modi\ufb01ed decision rule: g(T(x), y) = \u001a maybe, if D(R) \u2212D \u2264d(\u02c6 x, y) \u2264D + D(R) ; no, otherwise. (129) This condition retains the admissibility of the scheme by another usage of the triangle inequality. In essence, it allows us to rule out sequences y that are too close to \u02c6 x, since we know that x is at distance approximately D(R) from it. A similar argument holds for the LC-\u25b3scheme as well. However, this condition does not improve the achievable rate, and even in practice, the performance gain is generally negligible (see [29]). C. Special cases In general, RID(D) \u2264RTC-\u25b3 ID (D) \u2264RLC-\u25b3 ID (D), (130) where the inequalities may be strict (see Fig. 4). There are cases, however, where some of the inequalities in (130) are equalities. We review some of those cases here. Theorem 7: If there exists a constant D0 (that may depend on PY ), s.t. for all \u02c6 x \u2208X X y\u2208X PY (y)\u03c1(\u02c6 x, y) = D0, (131) then RTC-\u25b3 ID (D) = RLC-\u25b3 ID (D) = R(D0 \u2212D), (132) where R(\u00b7) is the rate-distortion function under distortion measure \u03c1(\u00b7, \u00b7). Proof: Under the stipulation in the theorem, we have that for any RV \u02c6 X that is independent of Y , E[\u03c1( \u02c6 X, Y )] = X \u02c6 x,y PY (y)P \u02c6 X(\u02c6 x)\u03c1(\u02c6 x, y) (133) = X \u02c6 x P \u02c6 X(\u02c6 x) X y PY (y)\u03c1(\u02c6 x, y) (134) = D0. (135) It follows that DTC-\u25b3 ID (R) = max P \u02c6 X|X:I(X; \u02c6 X)\u2264R D0 \u2212E[\u03c1(X, \u02c6 X)] (136) = D0 \u2212 min P \u02c6 X|X:I(X; \u02c6 X)\u2264R E[\u03c1(X, \u02c6 X)] (137) = D0 \u2212D(R). (138) \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 29 The proof follows by noting that (137) is equal to DLC-\u25b3 ID (R). The conditions for the above theorem holds, for example, in the following cases: \u2022 If Y is equiprobable on Y = X , and the columns of \u03c1(\u00b7, \u00b7) are permutations of each other (e.g. if \u03c1(\u00b7, \u00b7) is a \u2018difference\u2019 distortion measure), then D0 is given by D0 = X y 1 |X |\u03c1(y, 1). (139) \u2022 A special case of the above is the Hamming distortion. In this case, (where PY is still equiprobable), D0 = |X | \u22121 |X | . (140) Theorem 7 implies that in simple cases the LC-\u25b3scheme is equivalent (in the rate sense) to the TC-\u25b3scheme. If X and Y are binary and equiprobable, and the distortion measure is Hamming, it follows from [29, Theorem 1] that RTC-\u25b3 ID (D) = RID(D), i.e. the TC-\u25b3 and the LC-\u25b3schemes are optimal. However, if X and Y are not equiprobable (and the distortion measure is still Hamming), the LC-\u25b3scheme differs from the TC-\u25b3scheme (see [35, Fig. 2]). Is this TC-\u25b3scheme optimal for the binary-Hamming case? The following theorem answers this question in the af\ufb01rmative. Theorem 8: For the binary-Hamming case, i.e. X \u223cBer(p) and Y \u223cBer(q), RID(D) = RTC-\u25b3 ID (D). (141) Proof: We \ufb01rst show that it is suf\ufb01cient to take the cardinality of U in (17) to be equal to 2. To that end, note that for the binary-Hamming case we have \u00af \u03c1(p, q) = |p \u2212q|. Let U be an arbitrary (but \ufb01nite) alphabet, and let PU|X be a given channel from X to U, that attains the minimum in RID(D). It has to satisfy X u\u2208U PU(u)\u00af \u03c1(PX|U(\u00b7|u), PY ) \u2265D, (142) i.e. it is feasible for the optimization in RID(D). Suppose there exist u1 and u2 for which both PX|U(1|u1) \u2265q and PX|U(1|u2) \u2265q hold. Next, de\ufb01ne a new channel P \u2032 U|X that is the result of the channel PU|X, followed by a merge of u1 and u2 into a new symbol u\u2217. By the data processing inequality, the mutual information does not increase (in fact, it decreases unless PX|U(1|u1) = PX|U(1|u2)). The new reverse channel given u\u2217is easily calculated as P \u2032 X|U(1|u\u2217) = PU(u1)PX|U(1|u1) + PU(u2)PX|U(1|u2) PU(u1) + PU(u2) , (143) and the new prior is P \u2032 U(u\u2217) = PU(u1) + PU(u2). (144) Next, observe that PU(u1)\u00af \u03c1(PX|U(\u00b7|u1), PY ) + PU(u2)\u00af \u03c1(PX|U(\u00b7|u2), PY ) = PU(u1) \f \fPX|U(\u00b7|u1) \u2212q \f \f + PU(u2) \f \fPX|U(\u00b7|u2) \u2212q \f \f = PU(u1) \u0000PX|U(\u00b7|u1) \u2212q \u0001 + PU(u2) \u0000PX|U(\u00b7|u2) \u2212q \u0001 = P \u2032 U(u\u2217) \u0000P \u2032 X|U(\u00b7|u\u2217) \u2212q \u0001 . \f30 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Since for all the other values of u \u2208U, PX|U(x|u) = P \u2032 X|U(x|u), we conclude that the new channel is also feasible, and attains lower mutual information. Therefore the cardinality of U can be safely reduced by 1, assuring that the value of the optimization of RID(D) will not be higher because of this reduction. The same holds for the case where both PX|U(1|u1) \u2264q and PX|U(1|u2) \u2264q hold. This process can be applied for any channel W with more than one value of u for which PX|U(1|u) at the same side of q. Therefore for the optimal channel, it suf\ufb01ces to check only channels with |U| = 2. Next, we aim to show that min PU|X:P u\u2208U PU(u)\u00af \u03c1(PX|U(\u00b7|u),PY )\u2265D I(X; U) = min PU|X:E[\u03c1(U,Y )]\u2212E[\u03c1(X,U)]\u2265D I(X; U). (145) Since it suf\ufb01ces to look at binary U, the feasibility condition on the LHS of (145) can be rewritten as PU(0) \f \fPX|U(1|0) \u2212q \f \f + PU(1) \f \fPX|U(1|1) \u2212q \f \f \u2265D. (146) Let P \u2217 X|U be a minimizing channel for the LSH of (145). Denote the reverse channel by P \u2217 U|X. Next, assume that P \u2217 X|U(1|0), P \u2217 X|U(1, 1) are on different sides of q. Then, there is no loss of generality when assuming that P \u2217 X|U(1|0) \u2264q and that P \u2217 X|U(1|1) \u2265q (if this is not the case, then it can be achieved by reversing the roles of u = 1 and u = 0). In this case, the feasibility condition can be rewritten as PU(0) \u0000q \u2212P \u2217 X|U(1|0) \u0001 + PU(1) \u0000P \u2217 X|U(1|1) \u2212q \u0001 \u2265D. (147) On the other hand, the feasibility condition on W for R\u25b3(D) is E[\u03c1(U, Y )] \u2212E[\u03c1(U, X)] \u2265D, (148) which is the same as PU(0) \u0000PY (1) \u2212PX|U(1|0) \u0001 + PU(1) \u0000PY (0) \u2212PX|U(0|1) \u0001 \u2265D, (149) or, equivalently, PU(0) \u0000q \u2212PX|U(1|0) \u0001 + PU(1) \u0000PX|U(1|1) \u2212q \u0001 \u2265D. (150) We conclude that the channel P \u2217 U|X is also feasible for the RHS of (145), thereby proving (145), i.e. that RID(D) = RTC-\u25b3 ID (D). If P \u2217 X|U(1|0), P \u2217 X|U(1|1) are on the same side of q, then u = 0 and u = 1 can be merged, following the steps of the merging process at the beginning of the proof. If they are merged, this means that U is no longer a random variable, but a constant, and that X and U are therefore independent. This implies that RID(D) = 0, and also that |p \u2212q| \u2265D. In this special case, we show that RTC-\u25b3 ID (D) = 0 directly. Our goal is to \ufb01nd a channel W that will make X, U independent, and at the same time be feasible for the minimization of RTC-\u25b3 ID (D) according to (147). For this purpose, choose the channel PU|X to be PU|X(0|x) = \u03b1; PU|X(1|x) = 1 \u2212\u03b1, for any x \u2208{0, 1}. (151) It is easy to see that U and X are independent, i.e. I(X; U) = 0, and that PU(0) = \u03b1. Next, in order to satisfy (147), choose either \u03b1 = 0 or \u03b1 = 1, according to whether p < q or q < p (if p = q, this implies that D = 0 and then any choice of \u03b1 will work). \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 31 Theorems 7 and 8, when combined, result in the following corollary. Corollary 1: If X \u223cBer(p), Y \u223cBer( 1 2), and the distortion measure is Hamming, then RID(D) = RTC-\u25b3 ID (D) = RLC-\u25b3 ID (D) = R( 1 2 \u2212D), (152) where R(\u00b7) is the rate distortion function of the source X and Hamming distortion. Note that this result is slightly more general than [29, Theorem 1], since here X is not restricted to be symmetric. VII. COMPUTING THE IDENTIFICATION RATE Calculating the value of the achievable rates RLC-\u25b3 ID (D) and RTC-\u25b3 ID (D) is relatively easy. The term DLC-\u25b3 ID (R), shown in (23), can be calculated from the distortion rate function (and the achieving reconstruction distribution), which can be calculated with, e.g. the well known Blahut-Arimoto algorithm, or simply as a minimization problem of a linear function over a convex set. The term DTC-\u25b3 ID (R), shown in (24), is also given as a minimization problem of a linear function with convex constraints, and therefore can be solved easily. The general term RID(D), however, is posed as a minimization problem with nonconvex constraints, making its computation a challenge. In this section we give two results that facilitate the computation of this quantity. In Subsection VII-A we improve the bound on the cardinality of U, which reduces the dimensions of the optimization problem. In Subsection VII-B we describe the process of transforming the (non-convex) problem into a sequence of convex problems that can be solved ef\ufb01ciently. A. Cardinality of the auxiliary RV U For the evaluation of RID(D), it was already shown in [1, Lemma 3] that it suf\ufb01ces to consider only |U| = |C|+2. Here we prove an improvement of the cardinality bound, stated in Theorem 6. Proof of Theorem 6: We start by proving that taking |U| = |X | + 1 suf\ufb01ces to calculate RID(D). The proof follows the idea from [36], i.e. using the strengthened version of Carath\u00b4 eodory\u2019s theorem due to Fenchel and Eggleston. De\ufb01ne |X | + 1 functions \u03a8i : P(U \u2192X ) \u2192R. In other words, the functions \u03a8i take a conditional distribution from U to X , and return a real number. The functions are given by: \u03a8x(Q) = Q(x), , for x = 1, ..., |X | \u22121; (153) \u03a8|X|(Q) = \u00af \u03c1 (Q, PY ) ; (154) \u03a8|X|+1(Q) = H(PX) \u2212H(Q). (155) Note that in the optimization function RID(D), the objective function can be written as I(X; U) = X u\u2208U PU(u)\u03a8|X|(PX|U(\u00b7|u)), (156) and the constraint can be written as X u\u2208U PU(u)\u03a8|X|+1(PX|U(\u00b7|u)) \u2265D. (157) De\ufb01ne the set A to be the set of tuples (\u03a81(Q), ..., \u03a8|X+1|(Q)) for all Q \u2208P(X ). Note that A is a closed and connected set, and therefore any point in the convex hull of A can \f32 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY be represented as a convex combination of at most |X | + 1 elements of A (this is due to the Fenchel-Eggleston-Carath\u00b4 eodory theorem, see, e.g. [37, Theorem 18]). De\ufb01ne B to the convex hull of A. Further, de\ufb01ne BPX as BPX \u225c \b (\u03c81, ..., \u03c8|X|+1 : \u03c8i = PX(i), 1 \u2264i \u2264|X | \u22121 \t . (158) In other words, the set BPX contains only convex combinations of A that correspond to combinations of distributions on X that, when averaged with the convex combination (which represents the distribution on U), result in the distribution PX. Finally, let P \u2217 U|X be an achieving distribution for RID(D) and let P \u2217 U and P \u2217 X|U be the induced marginal on U and the reverse conditional distribution, respectively. P \u2217 U and P \u2217 X|U can be associated with a point in the set BPX, where the attained (R, D) pair is given in the last two coordinates of the vector in BPX. As claimed before, any point in B can be represented as a convex combination of at most |X |+1 points of A, and in other words, the same pair (R, D) can be achieved with a distribution that averages only |X |+1 distributions of X , i.e. the cardinality of U can be limited to |X | + 1. For the second part, recall the following facts: \u2022 RID(D) is a convex function of D [1, Lemma 3]. Denote the region of achievable pairs (R, D) as R \u225c{(R, D) : R \u2265RID(D)} , (159) where5 D \u2208[0, \u03c1max] and R \u2264log |X |. Therefore the set is closed, bounded and convex. \u2022 For a convex set, an extreme point is a point in the set that cannot be represented as a nontrivial6 convex combination of other points in the set. It is a well-known theorem that any convex set is equal to the convex hull of its extreme points (e.g. [38, Corr. 18.5.1]). For our proof we require the more delicate notion of exposed points. \u2022 An exposed point p of a convex set is a point on the boundary of the set, s.t. there exists a supporting hyperplane of the set at p (a hyperplane that touches the set at p, but the set is at one side of the hyperplane), whose intersection with the set itself is equal to {p}. Any exposed point is also an extreme point. The most useful fact about exposed points is the fact that any closed and bounded convex set is equal to the closure of a convex hull of its exposed points (a special case of [38, Theorem 18.7]). We shall use this fact directly. To begin the proof, let (R0, D0) (where R0 = RID(D0)) be an exposed point of the achievable region. Our goal is to show that this point D0, R0 is achievable with a conditional distribution PU|X s.t. the distribution of U is supported on at most |X | elements. Next, let (c, \u03bb) be constants s.t. R = c+\u03bbD is a supporting line (hyperplane in 2D) of the achievable region at (D0, R0), s.t. the intersection of R and the line contains this point only. Such a line is guaranteed to exist by the assumption that (R0, D0) is an exposed point. A typical image is shown in Fig. 6. Recall the RID(D) is given by the minimization expression (17). Let P \u2217 U|X be an achieving conditional distribution at D0, i.e. that minimizes (17). If RID(D0) = 0, this implies that U and X are independent, and therefore \u00af \u03c1(PX, PY ) \u2265D. This means that 5The restriction on the values of D and R is due to the fact that any rate above log |X | is trivially achievable by using the sequence x itself as the signature, and for a distortion threshold above \u03c1max renders all sequences similar to each other, making the problem degenerate. 6A convex combination is considered trivial if all the coef\ufb01cients are zero except for one of them (which is equal to one). \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 33 Fig. 6. The achievable region R, an exposed point (D, R) and a supporting line. RID(D0) can be attained by a trivial distribution of U (where U is a constant). In the general case where RID(D0) > 0, we conclude that the constraint in (17) is active, and therefore R0 = I(X; U); and D0 = E[\u00af \u03c1(P \u2217 X|U(\u00b7, U), PY )], (160) where X, U are distributed according to PX, P \u2217 U|X. Next, note that P \u2217 U|X also minimizes the expression I(X; U) \u2212\u03bbE[\u00af \u03c1(PX|U(\u00b7, U), PY )], (161) where the minimization is without constraints (other than the fact that PU|X is a conditional distribution). This fact follows since an existence of a better minimum would imply a distribution P \u2032 U|X for which (I(X; U), E[\u00af \u03c1(P \u2032 X|U(\u00b7, U), PY )]) falls outside the achievable region (due to the supporting hyperplane property), leading to a contradiction. Next, claim that from P \u2217 U|X, we can construct a distribution P \u2217\u2217 U|X that attains the same minimum in (161), for which the distribution of PU has at most |X | elements. This can be shown by following the same steps as in the \ufb01rst part of the proof (where the cardinality was shown to be bounded by |X | + 1), but now we replace the two functions \u03a8|X| and \u03a8|X|+1 by a single function \u03a8|X| that is equal to (161). Since the new distribution P \u2217\u2217 U|X attains the same minimum in (161) as P \u2217 U|X does, we conclude that the point (D1, R1), given by R1 = I(X; U); and D1 = E[\u00af \u03c1(P \u2217 X|U(\u00b7, U), PY )], (162) \f34 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY where X, U are distributed according to PX, P \u2217\u2217 U|X, satis\ufb01es the same line equation R1 = c + \u03bbD1. However, since we assumed that (D0, R0) is an exposed point of the achievable region, then by de\ufb01nition (D1, R1) = (D0, R0). (163) In other words, the distribution P \u2217\u2217 U|X attains the minimum of the original minimization problem (17). The proof is concluded since, as noted before, a bounded, closed and convex set is equal to the closure of the convex hull of its exposed points. As a result, the achievable region R can be calculated by calculating Rk ID(D), and then taking the lower convex envelope (the closure operation has no practical effect). B. Conversion to a set of convex functions Consider \ufb01rst the case where the distortion measure is Hamming, and PX, PY are arbitrary distributions on X . In this case, it is not hard to verify that \u00af \u03c1(PX, PY ) = 1 2\u2225PX \u2212PY \u22251 (164) = 1 2 X x\u2208X |PX(x) \u2212PY (x)|. (165) With this fact, we can rewrite the identi\ufb01cation rate as min PU|X I(X; U), (166) where the minimization is w.r.t. all conditional distributions PU|X s.t. X u\u2208U PU(u) X x\u2208X \f \fPX|U(x|u) \u2212PY (x) \f \f \u22652D. (167) De\ufb01ne F to be the set of all functions that take a pair in X \u00d7 U and return a binary value. With this, the condition (167) is equivalent to max f\u2208F \"X u\u2208U PU(u) X x\u2208X (\u22121)f(x,u) \u0000PX|U(x|u) \u2212PY (x) \u0001 # \u22652D. (168) Alternatively, we may require PU|X to satisfy X u\u2208U PU(u) X x\u2208X (\u22121)f(x,u) \u0000PX|U(x|u) \u2212PY (x) \u0001 \u22652D, (169) for some function f \u2208F. De\ufb01ne the LHS of (169) as Lf(PU|X), and rewrite it as Lf(PU|X) \u225c X u\u2208U PU(u) X x\u2208X (\u22121)f(x,u) \u0000PX|U(x|u) \u2212PY (x) \u0001 (170) = X u\u2208U X x\u2208X (\u22121)f(x,u) \u0000PU|X(u|x)PX(x) \u2212PU(u)PY (x) \u0001 (171) = X u\u2208U X x\u2208X (\u22121)f(x,u) \u0000PU|X(u|x)PX(x) \u2212PU(u)PY (x) \u0001 , (172) \fINGBER AND WEISSMAN: THE MINIMAL COMPRESSION RATE FOR SIMILARITY IDENTIFICATION 35 |X | |F| |F\u2032| 2 16 1 3 512 20 4 65536 1001 5 34 \u00b7 106 142 \u00b7 103 10 1.3 \u00b7 1030 3.3 \u00b7 1023 TABLE I NUMBER OF CONVEX OPTIMIZATION PROBLEMS TO BE SOLVED IN ORDER TO CALCULATE RID(D). which shows that Lf(PU|X) is a linear function of the optimization variable PU|X. Finally, we can rewrite the optimization problem as RID(D) = min f\u2208F min Lf (PU|X)\u22652D I(X; U). (173) The expression (173) gives rise to the following scheme for computing RID(D): since Lf(\u00b7) is a linear function, the inner optimization in (173) is that of a convex target function with linear constraints, and can be solved ef\ufb01ciently (e.g. via cvx [32]). To get the value of RID(D), simply repeat the inner optimization for all f \u2208F, and take the overall minimal value. The main problem with this approach is that the number of optimization problems can be very high. Assume that U = X , following the previous subsection. The size of F is 2|X|2. For |X | = 5, for example, one would need to solve 225 \u223c = 33.5 \u00d7 106 optimization problems. We can slightly improve the situation by utilizing symmetries in the expression (169). Theorem 9: De\ufb01ne the set F \u2032 \u2286F as follows. The set F \u2032 shall contain only functions f(\u00b7, \u00b7) where: \u2022 For all u, the function f(\u00b7, u) cannot be constant (in x). In other words, for the inner summation in (169), some of the summands must be \ufb02ipped and some not. \u2022 There are no u1 \u0338= u2 \u2208U s.t. \u2200x\u2208Xf(x, u1) = f(x, u2). Then in the double optimization of the form (173), it suf\ufb01ces to consider functions f \u2208F \u2032 as de\ufb01ned above. In addition, the number of such functions is given by |F \u2032| = \u0012 2|X| \u22122 |U| \u0013 . (174) Proof: Appendix E. For quick reference, we show in Table I the improvement in the number of optimization problems that is suf\ufb01cient to solve as a result of Theorem 9. For example, the identi\ufb01cation rate in Fig. 4 above, for ternary alphabet, was calculated using the method above. At each point, we have solved 20 convex optimization programs and took the minimum value. As seen in Table I, the proposed method for the computation of RID(D), although it improves on the naive (173), is only effective for small values of |X |. It is therefore an open problem how to calculate RID(D) effectively for larger alphabets. Finally, we brie\ufb02y note how to extend the process described here for arbitrary distortion measures. The key fact in the computation of RID(D) in the Hamming case is the fact that the function f(P) = \u00af \u03c1(P, PY ) can be represented as a maximum of linear functions of \f36 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY P. This fact holds in the general case as well, following the fact that the epigraph of the function f, de\ufb01ned as epif \u225c{(P, D) \u2208P(X ) \u00d7 R : f(P) \u2264D} , (175) is always a polyhedron. For the proof of this fact, see [4]. Once the \u00af \u03c1-distance has been represented as a maximization of linear functions, the process described in Equations (164)(173) can be followed. Note that as in the Hamming case, it is expected that this approach will only allow easy computation of RID(D) in cases where the alphabet X is small. VIII." + }, + { + "url": "http://arxiv.org/abs/1307.6609v1", + "title": "Compression for Quadratic Similarity Queries", + "abstract": "The problem of performing similarity queries on compressed data is\nconsidered. We focus on the quadratic similarity measure, and study the\nfundamental tradeoff between compression rate, sequence length, and reliability\nof queries performed on compressed data. For a Gaussian source, we show that\nqueries can be answered reliably if and only if the compression rate exceeds a\ngiven threshold - the identification rate - which we explicitly characterize.\nMoreover, when compression is performed at a rate greater than the\nidentification rate, responses to queries on the compressed data can be made\nexponentially reliable. We give a complete characterization of this exponent,\nwhich is analogous to the error and excess-distortion exponents in channel and\nsource coding, respectively.\n For a general source we prove that, as with classical compression, the\nGaussian source requires the largest compression rate among sources with a\ngiven variance. Moreover, a robust scheme is described that attains this\nmaximal rate for any source distribution.", + "authors": "Amir Ingber, Thomas Courtade, Tsachy Weissman", + "published": "2013-07-24", + "updated": "2013-07-24", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION For a database consisting of many long sequences, it is natural to perform queries of the form: which sequences in the database are similar to a given sequence y? In this paper, we study the problem of compressing this database so that queries about the original data can be answered reliably given only the compressed version. This goal stands in contrast to the traditional compression paradigm, where data is compressed so that it can be reconstructed \u2013 either exactly or approximately \u2013 from its compressed form. Speci\ufb01cally, for each sequence x in the database we only keep a short signature, denoted T(x), where T(\u00b7) is a signature assignment function. Queries are performed using only y and T(x) as input, rather than the original (uncompressed) sequence x. This setting is illustrated in Fig. 1. As alluded to above, we generally do not require that the original data be reproducible from the signatures. Therefore the set of signatures is not meant to replace the database itself. Nevertheless, there are many instances where such compression is desirable. For example, the set of signatures can The material in this paper was presented in part at the 2013 Data Compression Conference (DCC), Snowbird, UT. The authors are with the Dept. of Electrical Engineering, Stanford University, Stanford, CA 94305. Email: {ingber, courtade, tsachy}@stanford.edu. This work is supported in part by the NSF Center for Science of Information under grant agreement CCF-0939370. \f2 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY PSfrag replacements x1 x2 xM. . . T(\u00b7) t1 t2 tM y yes/no is x1 \u223c = y ? is x2 \u223c = y ? is xM \u223c = y ? Fig. 1. Answering a query from compressed data be thought of as a cached version of the original database (possibly hosted at many locations due to its relatively small size). By performing queries only on the cached (i.e., compressed) database, query latency can be reduced and the computational burden on the server hosting the uncompressed database can be lessened. In many scenarios (e.g,. querying a criminal forensic database), query responses which are false negatives are not acceptable. A false negative occurs if a query performed on T(x) and y indicates that x and y are not similar, but they are in truth. Therefore, we impose the restriction in our model that false negatives are not permitted. With this in mind, we regard the query responses from the compressed data as either \u201cno\u201d or \u201cmaybe\u201d. Since minimizing the probability that a query returns maybe is equivalent to minimizing the probability of returning a false positive1, any good compression scheme will have a corresponding query function which returns maybe with small probability. We note brie\ufb02y that a false positive does not cause an error per se. Rather, it only introduces a computational burden due to the need for further veri\ufb01cation. In our setting we assume that the query and database sequences are independent from one another, and all entries are drawn i.i.d. according to a given distribution. The setting is closely related to the problem considered by Ahlswede et al. [1], where the focus was only on discrete sources. In [1], the authors attempt to attack the more general problem where both false positives and false negatives are allowed. In this general case, it was demonstrated in [1] that the question of \u2018achievable rate\u2019 is uninteresting and only the error exponent is studied (in the current paper, where false negatives are not allowed, we show that the rate question becomes interesting again). We should also note that the error exponent results in [1] are parameterized by an auxiliary random variable with unbounded alphabet cardinality, rendering those quantities incomputable, and therefore of limited practical interest. Another closely related work is the one by Tuncel et al. [2], where the search accuracy was addressed by a reconstruction requirement with a single-letter distortion measure that is side-information dependent 1Complementary to false negatives, a false positive occurs if a query performed on T (x) and y indicates that x and y are similar (i.e., returns maybe), but they are not in truth. \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 3 (and the tradeoff between compression and accuracy is that of a Wyner-Ziv [3] type). In contrast, in the current paper the search accuracy is measured directly by the accuracy of the query answers. A different line of work attempting to identify the fundamental performance limits of database retrieval includes [4], [5], which characterized the maximum rate of entries that can be reliably identi\ufb01ed in a database. This line of work was extended independently in [6], [7] allowing compression of the database, and in [8] to the case where sequence reconstruction is also required. In each of these works, the underlying assumption is that the original sequences are corrupted by noise before their enrollment in the database, the query sequence is one of those original sequences, and the objective is to identify which one. There are two fundamental differences between this line of work and ours. First, in our case the query sequence is random (i.e. generated by nature) and does not need to be a sequence that has already been enrolled in the database. Second, in our problem we attempt to identify sequences that are similar to the query sequence, rather than an exact match. Other related ideas in the literature include Bloom Filters [9] (with many subsequent improvements, e.g. [10]), which are ef\ufb01cient data structures enabling queries without false negatives. The Bloom Filter only applies for exact matches (where here we are interested in similarity queries) so it is not applicable to our problem. Nevertheless, as surveyed in [11], Bloom \ufb01lters demonstrate the potential of answering queries from compressed data. Another related notion is that of Locality Sensitive Hashing (LSH), which is a framework for data structures and algorithms for \ufb01nding similar items in a given set (see [12] for a survey). LSH trades off accuracy with computational complexity and space, and false negatives are allowed. Two fundamental points are different in our approach. First, we study the information-theoretic aspect of the problem, i.e., we concentrate on space only (compression rate) and ignore computational complexity in an attempt to understand the amount of information relevant to querying that can be stored in the short signatures. Second, we do not allow false negatives, which, as discussed above, are inherent for LSH. Other approaches for similarity search from compressed data involve dimensionality reduction techniques that preserve distances, namely those based on Johnson-Lindenstrauss-type embeddings [13] (see also sketching, e.g. [14]). A recent interesting application of this approach involves image retrieval for an augmented reality setting [15]. However, note that such mappings generally depend on the elements in the database; the distance preservation property cannot apply to any query element outside the database, making the guarantee for zero false negatives impossible without further assumptions. This paper is organized as follows. In the next section we formally de\ufb01ne the problem and the quantities we study (i.e., the identi\ufb01cation rate and the identi\ufb01cation exponent). In Section III we state and discuss our main results. Section IV provides the proofs of these results, and Section V delivers concluding remarks. \f4 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY II. PROBLEM FORMULATION Throughout this paper, boldface notation x denotes a column vector of elements [x1, ...xn]T. Capital letters denote random variables (e.g. X, Y ), and X, Y denote random vectors. Throughout the paper log(\u00b7) denotes the base-2 logarithm, while ln(\u00b7) is used for the usual natural logarithm. We focus on the basic notion of quadratic similarity (sometimes called mean square error, or MSE). To this end, for any length-n real sequences x and y de\ufb01ne d(x, y) \u225c1 n n X i=1 (xi \u2212yi)2 = 1 n\u2225x \u2212y\u22252, (1) where \u2225\u00b7\u2225denotes the standard Euclidean norm. We say that x and y are D-similar when d(x, y) \u2264D, or simply similar when D is clear from context. A rate-R identi\ufb01cation system (T, g) consists of a signature assignment T : Rn \u2192{1, 2, . . . , 2nR} (2) and a query function g : {1, 2, . . . , 2nR} \u00d7 Rn \u2192{no, maybe}. (3) A system (T, g) is said to be D-admissible, if for any x, y satisfying d(x, y) \u2264D, we have g(T(x), y) = maybe. (4) This notion of D-admissibility motivates the use of \u201cno\u201d and \u201cmaybe\u201d in describing the output of g: \u2022 If g(T(x), y) = no, then x and y can not be D-similar. \u2022 If g(T(x), y) = maybe, then x and y are possibly D-similar. Stated another way, a D-admissible system (T, g) does not produce false negatives, i.e., indicate that x and y are not similar, when they are in truth. Thus, a natural \ufb01gure of merit for a D-admissible system (T, g) is the frequency at which false positives occur (i.e., where g(T(x), y) = maybe and d(x, y) > D). To this end, let PX and PY be probability distributions on R, and assume (X, Y) \u223cQn i=1 PX(xi)PY (yi). That is, the vectors X and Y are independent of each other and drawn i.i.d. according to PX and PY respectively. De\ufb01ne the false positive event E = {g(T(X), Y) = maybe, d(X, Y) > D}, (5) and note that, for any D-admissible system (T, g), we have Pr{g(T(X), Y) = maybe} = Pr{g(T(X), Y) = maybe|d(X, Y) \u2264D} Pr{d(X, Y) \u2264D} + Pr{g(T(X), Y) = maybe, d(X, Y) > D} (6) = Pr{d(X, Y) \u2264D} + Pr{E}, (7) \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 5 where (7) follows since Pr{g(T(X), Y) = maybe|d(X, Y) \u2264D} = 1 by D-admissibility of (T, g). Since Pr{d(X, Y) \u2264D} does not depend on what scheme is employed, minimizing the false positive probability Pr{E} over all D-admissible schemes (T, g) is equivalent to minimizing Pr{g(T(X), Y) = maybe}. Also note, that the only interesting case is when Pr{d(X, Y) \u2264D} \u21920 as n grows, since otherwise almost all the sequences in the database will be similar to the query sequence, making the problem degenerate (since almost all the database needs to be retrieved, regardless of the compression). In this case, it is easy to see that Pr{E} vanishes if and only if the conditional probability Pr{g(T(X), Y) = maybe|d(X, Y) > D} (8) vanishes as well. In view of the above, we henceforth restrict our attention to the behavior of Pr{g(T(X), Y) = maybe}. In particular, we study the tradeoff between the rate R and Pr{g(T(X), Y) = maybe}. This motivates the following de\ufb01nitions: De\ufb01nition 1: For given distributions PX, PY and a similarity threshold D, a rate R is said to be D-achievable if there exists a sequence of rate-R admissible schemes (T (n), g(n)) satisfying lim n\u2192\u221ePr \b g(n) \u0000T (n)(X), Y \u0001 = maybe \t = 0. (9) De\ufb01nition 2: For given distributions PX, PY and a similarity threshold D, the identi\ufb01cation rate RID(D, PX, PY ) is the in\ufb01mum of D-achievable rates. That is, RID(D, PX, PY ) \u225cinf{R : R is D-achievable}, (10) where an in\ufb01mum over the empty set is equal to \u221e. The above de\ufb01nitions are in the same spirit of the rate distortion function (the rate above which a vanishing probability for excess distortion is achievable), and also in the spirit of the channel capacity (the rate below which a vanishing probability of error can be obtained). See, for example, Gallager [16].2 Having de\ufb01ned RID(D, PX, PY ), the rate at which Pr{g(T(X), Y) = maybe} vanishes is also of signi\ufb01cant interest. We expect the vanishing rate to be exponential as in the traditional source coding setting, motivating the following de\ufb01nition: De\ufb01nition 3: Fix R \u2265RID(D, PX, PY ). The identi\ufb01cation exponent is de\ufb01ned as EID(R, D, PX, PY ) \u225clim sup n\u2192\u221e\u22121 n log inf g(n),T (n) Pr \b g(n) \u0000T (n)(X), Y \u0001 = maybe \t , (11) where the in\ufb01mum is over all D-admissible systems (g(n), T (n)) of rate R and blocklength n. The analogous quantity in source coding is the excess distortion exponent, \ufb01rst studied by Marton [18] for discrete sources and by Ihara and Kubo [19] for the Gaussian source (see also [20] and [21] for other sources). 2See, for example, Cover and Thomas [17] for the alternative approach based on average distortion rather than excess distortion probability. \f6 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY We pause to make a few additional remarks on the connection between Pr{g(T(X), Y) = maybe} and Pr{E}, where E is the false positive event de\ufb01ned in (5). If PX and PY have identical means and \ufb01nite variances \u03c32 X and \u03c32 Y , respectively, then the weak law of large numbers implies lim n\u2192\u221ePr{d(X, Y) \u2264D} = 0 (12) when D < \u03c32 X + \u03c32 Y . Thus, the relation (7) implies that vanishing Pr{E} is attainable if and only if R > RID(D, PX, PY ) when D < \u03c32 X + \u03c32 Y . Finally, observe that (7) implies the relationship EID(R, D, PX, PY ) = lim sup n\u2192\u221e\u22121 n log max \u0014 Pr{d(X, Y) \u2264D}, inf g(n),T (n) Pr \b E(n)\t\u0015 , (13) where E(n) is the false positive event de\ufb01ned via (5) for the system (g(n), T (n)), and the in\ufb01mum is taken over all D-admissible systems (g(n), T (n)) of rate R and blocklength n. III. MAIN RESULTS This section delivers our main results; all proofs are given in Section IV. The Gaussian distribution plays a prominent role in this section, therefore we use the shorthand notation PX = N(\u00b5, \u03c32) to denote that PX is the Gaussian distribution on R with mean \u00b5 and variance \u03c32. A. The Identi\ufb01cation Rate for Gaussian Sources Theorem 1: If PX = N(\u00b5, \u03c32 X) and PY = N(\u00b5, \u03c32 Y ), then RID(D, PX, PY ) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 for 0 \u2264D < (\u03c3X \u2212\u03c3Y )2 log 2\u03c3X\u03c3Y \u03c32 X+\u03c32 Y \u2212D for (\u03c3X \u2212\u03c3Y )2 \u2264D < \u03c32 X + \u03c32 Y \u221e for D \u2265\u03c32 X + \u03c32 Y . (14) Before proceeding, we make a few observations about the behavior of RID(D, PX, PY ) under the assumptions of Theorem 1. First, the fact that RID(D, PX, PY ) = \u221efor D \u2265\u03c32 X +\u03c32 Y is not surprising. Indeed, if D \u2265\u03c32 X +\u03c32 Y , then X and Y are inherently D-similar. That is, Pr{d(X, Y) \u2264D} is bounded away from zero (it actually converges to 1), and therefore (13) reveals that Pr{g(T(X), Y) = maybe} can never vanish, regardless of what scheme is used. Second, (14) is symmetric with respect to \u03c32 X and \u03c32 Y . Though this might be expected, it is not obviously true from the outset. Finally, for \ufb01xed \u03c32 X and D < \u03c32 X, the function RID(D, PX, PY ) given by (14) is maximized when \u03c32 Y = \u03c32 X \u2212D. In Fig. 2 we plot (14) for different values of \u03c32 Y in order to illustrate some of its properties. As an immediate corollary to Theorem 1, we obtain the following concise result for the symmetric case of PX = PY = N(\u00b5, \u03c32). \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 7 0 0.5 1 1.5 2 2.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 \u03c32 Y Rate [bits] RID(D, \u03c32 X, \u03c32 Y ) RID(D, \u03c32 X, \u03c32 X) RID(D, \u03c32 X, \u03c32 Y ) maximized RID(D, \u03c32 X, \u03c32 Y ) = RID(D, \u03c32 X) Fig. 2. The identi\ufb01cation rate RID(D, \u03c32 X, \u03c32 Y ) := RID(D, N(\u00b5, \u03c32 X), N(\u00b5, \u03c32 Y )) for different values of \u03c32 Y . Here \u03c32 X = 1 and D = 0.4. Corollary 1: If PX = PY = N(\u00b5, \u03c32), then RID(D, PX, PY ) = ( log \u0010 2\u03c32 2\u03c32\u2212D \u0011 for 0 \u2264D < 2\u03c32 \u221e for D \u22652\u03c32. (15) We remark that (15) is reminiscent of the Gaussian rate distortion function R(D) = h 1 2 log \u03c32 D i+ (cf. [17]). The identi\ufb01cation rate RID(D, N(\u00b5, \u03c32), N(\u00b5, \u03c32)) and rate distortion function R(D) for a Gaussian source are plotted in Fig. 3, and as seen in the \ufb01gure, R(D) is monotonically decreasing in D, while (15) is monotone increasing. This can be intuitively explained by thinking of the compression scheme as a quantizer, where all the x sequences mapped to the same i \u2208{1, 2, . . . , 2nR} de\ufb01ne a quantization cell. Since the scheme must answer maybe for all sequences y similar to x, it therefore has to answer maybe for all y in the D-expansion of the quantization cell (all sequences that are at distance D from any point in the cell). The probability of maybe is, therefore, the probability that Y falls in the expanded cell, and this probability increases as either D grows, or as the size of the quantization cell itself grows (i.e. the rate decreases). B. The Identi\ufb01cation Exponent for Gaussian Sources Having established the identi\ufb01cation rate for Gaussian sources, we now turn our attention to the identi\ufb01cation exponent. In order to simplify the notation for the identi\ufb01cation exponents, we de\ufb01ne the \f8 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 D/\u03c32 Rate [bits] RID(D) R(D) Fig. 3. The identi\ufb01cation rate RID(D) := RID(D, N(\u00b5, \u03c32), N(\u00b5, \u03c32)) and the rate distortion function R(D) for a Gaussian source with variance \u03c32. following functions EZ(\u03c1) \u225c 1 2 ln 2(\u03c1 \u22121 \u2212ln \u03c1) (16) \u2118(R, D, z1, z2) \u225c\u2212log sin min \u0014 \u03c0 2, \u0012 arcsin \u00002\u2212R\u0001 + arccos z1 + z2 \u2212D 2\u221az1z2 \u0013\u0015 . (17) Theorem 2: Let PX = N(\u00b5, \u03c32 X) and PY = N(\u00b5, \u03c32 Y ). For any \ufb01xed rate R > RID(D, PX, PY ), EID(R, D, PX, PY ) = min \u03c1X,\u03c1Y EZ(\u03c1X) + EZ(\u03c1Y ) + \u2118(R, D, \u03c1X\u03c32 X, \u03c1Y \u03c32 Y ), (18) where the minimization is over all \u03c1X, \u03c1Y > 0 satisfying \f \f \f \f q \u03c1X\u03c32 X \u2212 q \u03c1Y \u03c32 Y \f \f \f \f < \u221a D, \u03c1X\u03c32 X + \u03c1Y \u03c32 Y \u2265D. (19) Remark 1: We note that, for PX = N(\u00b5, \u03c32 X) and PY = N(\u00b5, \u03c32 Y ), the exponent EID(R, D, PX, PY ) is strictly positive for R > RID(D, PX, PY ), and is equal to zero at R = RID(D, PX, PY ). Therefore, the direct part of Theorem 1 is implied by Theorem 2. However, the converse part of Theorem 1 is not implied by Theorem 2, as the latter does not exclude the possibility that the probability of maybe can be made to vanish with a sub-exponential decay rate when the exponent is equal to zero. In light of Theorem 2, it is instructive to revisit the relationship between false-positive and maybe probabilities speci\ufb01ed in (13). To this end, consider the setting where PX = N(\u00b5, \u03c32 X), PY = N(\u00b5, \u03c32 Y ), and D \u2264\u03c32 X + \u03c32 Y . In this case, the random variable 1 n(\u03c32 X+\u03c32 Y )\u2225X \u2212Y\u22252 has a chi-squared distribution \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 9 with n degrees of freedom. Therefore, it follows by Cramer\u2019s Theorem (cf. [22, Theorem 2.2.3]) that lim n\u2192\u221e\u22121 n log Pr {d(X, Y) \u2264D} = EZ \u0012 D \u03c32 X + \u03c32 Y \u0013 . (20) In this setting, it is a straightforward algebraic exercise to see that EID(R, D, PX, PY ) < EZ \u0012 D \u03c32 X + \u03c32 Y \u0013 (21) for R < \u221eby putting \u03c1X = \u03c32 XD + \u03c32 Y (\u03c32 X + \u03c32 Y ) (\u03c32 X + \u03c32 Y )2 , \u03c1Y = \u03c32 Y D + \u03c32 X(\u03c32 X + \u03c32 Y ) (\u03c32 X + \u03c32 Y )2 (22) in (18). Therefore, EID(R, D, PX, PY ) also precisely characterizes the best-possible exponent corresponding to the probability of a false positive event in this setting due to the relation (13). In the case where PX = PY = N(\u00b5, \u03c32), the symmetry in (18) can be exploited to yield the following corollary. Corollary 2: Let PX = PY = N(\u00b5, \u03c32). For any \ufb01xed rate R > RID(D, PX, PY ), EID(R, D, PX, PY ) = min \u03c1 2EZ(\u03c1) + \u2118(R, D, \u03c1\u03c32, \u03c1\u03c32), (23) where the minimization is over all \u03c1 satisfying 2\u03c32 \u22652\u03c1\u03c32 \u2265D. (24) A formal proof is given in Section IV. The identi\ufb01cation exponent (23) for the case of D/\u03c32 = 1.5 is illustrated in Fig. 4. Before proceeding, we brie\ufb02y note that the identi\ufb01cation exponent EID(R, D, PX, PY ) can sometimes be strictly positive at R \u21920 3. For instance, if \f \f \f \f 1 \u221anE\u2225X\u2225\u22121 \u221anE\u2225Y\u2225 \f \f \f \f > \u221a D + \u01eb (25) for some \u01eb > 0, then the signature T(X) can simply indicate whether \f \f \f 1 \u221an\u2225X\u2225\u2212 1 \u221anE\u2225X\u2225 \f \f \f > \u01eb/2, requiring rate R = 1/n. Then, the query function g returns maybe only if \f \f \f \f 1 \u221an\u2225X\u2225\u22121 \u221anE\u2225X\u2225 \f \f \f \f > \u01eb/2, or (26) \f \f \f \f 1 \u221an\u2225Y\u2225\u22121 \u221anE\u2225Y\u2225 \f \f \f \f > \u01eb/2. (27) If neither (26) nor (27) occur, then it is readily veri\ufb01ed that d(X, Y) > D using the triangle inequality. Whenever the random variables X2 and Y 2 satisfy a large deviations principle (as in the Gaussian case, 3Note that whenever R is equal to zero, the probability of maybe is equal to 1 (unless the supports of PX and PY are disjoint in a way making any two sequences x and y dissimilar, making the problem degenerate). \f10 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 0 1 2 3 4 5 6 7 8 9 10 0 0.005 0.01 0.015 0.02 0.025 0.03 RID(D) R [bits] EID(R) EID(R) EZ \u00a1 D 2\u03c32 \u00a2 Fig. 4. Plot of EID(R) := EID(R, D, N(\u00b5, \u03c32), N(\u00b5, \u03c32)) for D/\u03c32 = 1.5. In this case, RID(D, N(\u00b5, \u03c32), N(\u00b5, \u03c32)) = 2 bits per symbol. and for many other distributions, cf. [22]), we see that g returns maybe with probability exponentially decaying in n, and we can conclude that limR\u21920+ EID(R, D, PX, PY ) > 0. If this is indeed the case, then it also follows that RID(D, PX, PY ) = 0 by de\ufb01nition. Though this discussion applies for arbitrary distributions PX, PY , this latter point is concretely re\ufb02ected in Theorems 1 and 2 for the case where D \u2264(\u03c3X \u2212\u03c3Y )2. C. Upper Bounds on the Identi\ufb01cation Rate In the previous two subsections, we focused our attention primarily to the case where PX and PY were Gaussian distributions. In the sequel, we consider more general distributions and show that Gaussian PX, PY constitute an extremal case in terms of the identi\ufb01cation rate. Theorem 3: Suppose PX and PY are distributions with \ufb01nite second moments \u03c32 X and \u03c32 Y , respectively. Then RID(D, PX, PY ) \u2264RID(D, PX, PY ) \u225cinf P \u02c6 X|X I(X; \u02c6 X), (28) where the in\ufb01mum is taken over all conditional distributions P \u02c6 X|X satisfying v u u tE \"\u0012r\u03c3X \u03c3Y Y \u2212\u02c6 X \u00132# \u2265 v u u tE \"\u0012r\u03c3Y \u03c3X X \u2212\u02c6 X \u00132# + p D \u2212(\u03c3X \u2212\u03c3Y )2 (29) \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 11 for (Y, X, \u02c6 X) \u223cPY (y)PX(x)P \u02c6 X|X(\u02c6 x|x). Moreover, EID(R, D, PX, PY ) > 0 (30) for any R > RID(D, PX, PY ). Remark 2: Note that Theorem 3 does not require PX and PY to have identical means. Remark 3: Also note, that the achievability result and the proof technique carry over to general distortion criteria satisfying the triangle inequality. We omit the details as the focus of this paper is on the quadratic similarity criterion. For general source distributions PX, PY , we lack a matching lower bound on RID(D, PX, PY ). However, such a converse was proved in the Gaussian setting (see Theorem 1). The key ingredient in the proof of Theorem 1 is the isoperimetric inequality on the surface of a hypersphere \u2013 the set on which the probability of a high dimensional Gaussian random vector concentrates (see Section IV for details). In general, precise isoperimetric inequalities are unknown and therefore establishing a general converse appears to be extremely dif\ufb01cult. In spite of this, an application of Theorem 3 reveals the interesting fact that Gaussian PX and PY correspond to sources which are \u201cmost dif\ufb01cult\u201d to compress for queries. This is analogous to the setting of classical lossy compression, where the Gaussian source requires the maximum rate for compression subject to a quadratic distortion constraint. Formally, Theorem 4: Suppose PX and PY have identical means and \ufb01nite variances \u03c32 X and \u03c32 Y , respectively. Then RID(D, PX, PY ) \u2264RID(D, N(0, \u03c32 X), N(0, \u03c32 Y )). (31) In particular, Gaussian PX and PY demand the largest identi\ufb01cation rate for given variances. D. Robust Identi\ufb01cation Schemes In addition to the extremal property of Gaussian sources described in Theorem 4, there exists a sequence of rate-R identi\ufb01cation schemes {T (n), g(n)}n\u2192\u221e, where (T (n), g(n)) denotes a blocklength-n identi\ufb01cation scheme, designed for Gaussian sources which are robust in the following sense. Using the construction described in the achievability proof of Theorem 1, we can construct a sequence of D-admissible, rate-R schemes {T (n), g(n)}n\u2192\u221ewhich satisfy lim n\u2192\u221ePr \b g(n) \u0000T (n)(X), Y \u0001 = maybe \t = 0 (32) when X, Y \u223cQn i=1 PX(xi)PY (yi), PX = N(0, \u03c32 X), PY = N(0, \u03c32 Y ) and R > RID \u0000D, N(0, \u03c32 X), N(0, \u03c32 Y ) \u0001 . (33) It turns out that this particular sequence {T (n), g(n)}n\u2192\u221eis robust to the source distributions in the \f12 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY sense that we also have lim n\u2192\u221ePr n g(n) \u0010 T (n)( \u02dc X), \u02dc Y \u0011 = maybe o = 0 (34) when \u02dc X, \u02dc Y \u223cQn i=1 P \u02dc X(\u02dc xi)P \u02dc Y (\u02dc yi), and P \u02dc X, P \u02dc Y are zero-mean distributions with variances \u03c32 X and \u03c32 Y , respectively. Moreover, the sequence {T (n), g(n)}n\u2192\u221econtinues to be D-admissible for the sources \u02dc X, \u02dc Y. Thus, roughly speaking, a scheme (T, g) which is \u201cgood\u201d for Gaussian sources X, Y can be expected to perform well for arbitrary sources \u02dc X, \u02dc Y, provided the respective variances match their Gaussian counterparts and the blocklength n is large. The proof of this robustness property is given in Section IV-F. IV. PROOFS In this section, we prove each of the main results. Proofs are organized by subsection. We begin with a primer on the key geometric ideas that are used throughout the proofs. A. Geometric Preliminaries For the proofs we require the following de\ufb01nitions related to n-dimensional Euclidean geometry. For r > 0, u \u2208Rn, let BALLr(u) \u2286Rn denote the ball with radius r centered at u: BALLr(u) \u225c{x \u2208Rn : \u2225x \u2212u\u2225\u2264r} . (35) BALLr(0) will be denoted BALLr. Denote by Sr \u2286Rn the spherical shell with radius r centered at the origin: Sr \u225c{x \u2208Rn : \u2225x\u2225= r} . (36) For any two vectors x1, x2 \u2208Rn \\ {0}, the angle between them shall be denoted by \u2220(x1, x2) \u225carccos \u0012 xT 1 x2 \u2225x1\u2225\u2225x2\u2225 \u0013 \u2208[0, \u03c0]. (37) For \u03b8 \u2208[0, \u03c0] and a point u \u2208Rn \\ {0}, de\ufb01ne the cone with half angle \u03b8 and axis going through u: CONE(u, \u03b8) \u225c{x \u2208Rn : \u2220(u, x) \u2264\u03b8} . (38) Note that CONE(u, 0) is the half-in\ufb01nite line {\u03b1u : \u03b1 > 0}, that CONE(u, \u03c0/2) is the half-space containing u that is bordered by the hyperplane orthogonal to u which passes through the origin, and that CONE(u, \u03c0) is the entire space Rn. Also, note that CONE(u1, \u03b8) = CONE(u2, \u03b8) for any u1 = \u03bbu2, \u03bb > 0. For r > 0, u \u2208Rn \\ {0} and \u03b8 \u2208[0, \u03c0], denote by CAPr(u, \u03b8) the spherical cap: CAPr(u, \u03b8) \u225cSr \u2229CONE(u, \u03b8). (39) \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 13 Let \u2126(\u03b8) denote the fraction of the (hyper-)surface area of Sr that is occupied by CAPr(u, \u03b8): \u2126(\u03b8) \u225c|CAPr(u, \u03b8)| |Sr| . (40) Note that the value of \u2126(\u03b8) depends neither on r nor on u. The following bounds on \u2126(\u03b8) will be useful: Lemma 1: [23, Corrolary 3.2] For 0 < \u03b8 < arccos(1/\u221an) < \u03c0 2, we have \u2126(\u03b8) < 1 p 2\u03c0(n \u22121) \u00b7 1 cos \u03b8 \u00b7 sinn\u22121 \u03b8, (41) \u2126(\u03b8) > 1 3 \u221a 2\u03c0n \u00b7 1 cos \u03b8 \u00b7 sinn\u22121 \u03b8. (42) For positive r1 \u2264r2 \u2208R, let Sr1,r2 \u2286Rn be a spherical shell of inner radius r1 and outer radius r2: Sr1,r2 \u225c{x \u2208Rn : r1 \u2264\u2225x\u2225\u2264r2} . (43) For a given half-angle \u03b8 \u2208[0, \u03c0], de\ufb01ne the (r1, r2)-spherical cap with half-angle \u03b8 and axis going through u as CAPr1,r2(u, \u03b8) \u225cCONE(u, \u03b8) \u2229Sr1,r2. (44) For a set A \u2286Rn and D > 0, the D-expansion of A, denoted \u0393D(A) is de\ufb01ned as \u0393D(A) \u225c{y \u2208Rn : \u2203x\u2208Ad(x, y) \u2264D} (45) = A + BALL\u221a nD, (46) where we have used + to denote the Minkowski sum. B. Codes that cover a spherical shell De\ufb01nition 4: Let Sr \u2286Rn be the spherical shell with radius r. We say that a set of points C = {u1, . . . , um : ui \u2208Rn} is a code that D-covers Sr if Sr \u2286 [ u\u2208C BALL\u221a nD(u). (47) The rate of C is de\ufb01ned as 1 n log m. When not explicitly stated, the ambient dimension n of the code C will be clear from context. Lemma 2 (Following [24]): Fix \u03c32 > 0 and the dimension n. For any 0 < D0 < \u03c32, there exists a code C that D0-covers S\u221a n\u03c32 with rate R0 = 1 n log |C| \u22641 2 log \u03c32 D0 + O \u0012log n n \u0013 . (48) \f14 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Moreover, for all u \u2208C, we have \u2225u\u2225= p n(\u03c32 \u2212D0), and CAP\u221a n\u03c32(u, \u03b80) = S\u221a n\u03c32 \u2229BALL\u221anD0(u), (49) where \u03b80 \u225carcsin( p D0/\u03c32) < \u03c0 2 . (50) Proof: Appendix A. It is no surprise that the term 1 2 log \u03c32 D0 appearing in (48) is identical to the rate-distortion function for the Gaussian source with variance \u03c32 evaluated at distortion-level D0. We could have therefore used any standard (random code-like) construction. However, using Lemma 2 will be more convenient for our purposes since each point in Sr is guaranteed to be covered, and hence we do not need to account for another error event. This fact will make the subsequent proofs more straightforward. C. Identi\ufb01cation Rate The proof of Theorem 1 is somewhat lengthy, so we \ufb01rst give the key ideas here before moving onto the formal details. The proof of the theorem relies on the fact that a high-dimensional Gaussian random vector \u2013 with independent entries having zero mean and variance \u03c32 X \u2013 concentrates near a thin hyper-spherical shell of radius r0 \u225c p n\u03c32 X, which we call the typical sphere. The signature assignment constructed in the direct part of the proof quantizes the surface of the typical sphere into regions roughly described by spherical caps. The query function g, knowing which cap X lies in from the received signature, returns maybe only if Y lies within Euclidean distance \u221a D of the cap in which X lies. Thus, the goal in the direct part is to show that, for suf\ufb01ciently large rate R, the probability Y falls into the \u0393D-expansion of any given cap is vanishing. The key ingredient in proving the converse is the isoperimetric inequality on the surface of the hypersphere, known as Levy\u2019s lemma (see e.g. [25, Theorem 1.1]). In a nutshell, we apply Levy\u2019s lemma to prove that any given identi\ufb01cation system (T, g) requires a rate that is essentially as large as an identi\ufb01cation system that uniquely assigns caps on the typical sphere to signatures (as is done by the achievability scheme). The apparent need for a re\ufb01ned isoperimetric inequality to prove the converse distinguishes our problem from the class of standard rate-distortion problems. Proof of Theorem 1: Before beginning the proof, we \ufb01rst note that it is suf\ufb01cient to consider D in the interval (\u03c3X \u2212\u03c3Y )2 < D < \u03c32 X + \u03c32 Y . The claims that RID(D, PX, PY ) = 0 for D \u2264(\u03c3X \u2212\u03c3Y )2, and RID(D, PX, PY ) = \u221efor D \u2265\u03c32 X + \u03c32 Y then follow from monotonicity of RID(D, PX, PY ) in D. Direct Part: Fix a small \u01eb > 0, and de\ufb01ne rX \u225c p n\u03c32 X (i.e., the radius of the typical sphere). Let D be a desired similarity threshold in the interval (\u03c3X \u2212\u03c3Y )2 < D < \u03c32 X + \u03c32 Y , and let \u03b7 > 0 be \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 15 suf\ufb01ciently small so that (1 \u2212\u01eb) \u0014\u03c32 X + \u03c32 Y \u2212D 2\u03c3X\u03c3Y \u00152 < \" \u03c32 X + \u03c32 Y \u22122\u03b7 \u2212D 2 p (\u03c32 X + \u03b7)(\u03c32 Y + \u03b7) #2 . (51) Next, de\ufb01ne a constant D0 satisfying (1 \u2212\u01eb)\u03c32 X \" \u03c32 X + \u03c32 Y \u22122\u03b7 \u2212D 2 p (\u03c32 X + \u03b7)(\u03c32 Y + \u03b7) #2 < D0 < \u03c32 X \" \u03c32 X + \u03c32 Y \u22122\u03b7 \u2212D 2 p (\u03c32 X + \u03b7)(\u03c32 Y + \u03b7) #2 . (52) The motivation behind the choices of \u03b7 and D0 satisfying (51) and (52) will become clear as the proof proceeds. By our assumption that D > (\u03c3X \u2212\u03c3Y )2, it follows that 0 < D0 < \u03c32 X. By Lemma 2, there exists a code C which D0-covers SrX with rate R0 bounded by R0 \u22641 2 log \u03c32 X D0 + O \u0012log n n \u0013 . (53) Let T0 : SrX \u2192C be the quantization operation de\ufb01ned by T0(x) = arg min u\u2208C \u2225x \u2212u\u2225for x \u2208SrX. (54) That is, the function T0(x) maps x \u2208SrX to the closest reconstruction point u \u2208C. Since C is a code that D0-covers SrX, it follows that \u2225T0(x) \u2212x\u2225\u2264\u03c10 \u225c p nD0 for all x \u2208SrX. (55) Denote the points in SrX that are mapped to u by T \u22121 0 (u). With this notation, it follows by construction that T \u22121 0 (u) \u2286CAPrX(u, \u03b80), (56) where \u03b80 \u225carcsin( p D0/\u03c32 X) courtesy of Lemma 2. The set CAPrX(u, \u03b80) is illustrated in Fig. 5. De\ufb01ne Styp X to be the set of all vectors x \u2208Rn s.t. \u03c32 X \u2212\u03b7 \u22641 n\u2225x\u22252 \u2264\u03c32 X + \u03b7. In other words, Styp X \u225cSr\u2212,r+, (57) where r\u00b1 \u225c p n(\u03c32 X \u00b1 \u03b7). Note that Pr{X / \u2208Styp X } vanishes with n (in fact, it vanishes exponentially), which motivates the notation Styp X . Next, we construct a mapping T : Styp X \u2192C de\ufb01ned as follows: T(x) = T0 x \u00b7 p n\u03c32 X \u2225x\u2225 ! . (58) \f16 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY rX \u03b80 u \u03c10 Fig. 5. Illustration of a single cap CAPrX(u, \u03b80) (denoted in grey). Since T \u22121 0 (u) is contained in CAPrX(u, \u03b80), we similarly have that the inverse map T \u22121 satis\ufb01es T \u22121(u) \u2286CAPr\u2212,r+(u, \u03b80). (59) The signature assignment for our identi\ufb01cation scheme for x \u2208Styp X shall be given by the function T(\u00b7) de\ufb01ned above. For x / \u2208Styp X we de\ufb01ne T(x) = e, where e is an additional \u201cerasure\u201d symbol, denoting the fact that the signature does not convey any information about x in this case (and the decision function g(\u00b7, \u00b7) must output maybe). Note that the additional rate incurred by the erasure symbol is negligible and we still have that the signature assignment\u2019s rate R is bounded by R = 1 n log (|C| + 1) (60) \u22641 2 log \u03c32 X D0 + O \u0012log n n \u0013 (61) \u2264log 2\u03c3X\u03c3Y \u03c32 X + \u03c32 Y \u2212D + log 1 1 \u2212\u01eb + O \u0012log n n \u0013 , (62) where the \ufb01nal inequality follows from (51) and (52). The query function g(\u00b7, \u00b7) is de\ufb01ned to be the optimal one given the signature mapping T(\u00b7): g(t, y) = ( maybe If t = e or if \u2203x\u2032 \u2208T \u22121(t) s.t. d(x\u2032, y) \u2264D no otherwise. (63) Using the shorthand notation Pr{maybe} \u225cPr{g(T(X), Y) = maybe}, (64) \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 17 we analyze Pr{maybe} as follows. First, de\ufb01ne a typical set for the Y-sequences: Styp Y \u225cSr\u2212 Y ,r+ Y , (65) where r\u00b1 Y \u225c p n(\u03c32 Y \u00b1 \u03b7), and write Pr{maybe} \u2264Pr{maybe|X \u2208Styp X , Y \u2208Styp Y } + Pr{X / \u2208Styp X } + Pr{Y / \u2208Styp Y }. (66) Note that the latter two terms in (66) vanish as n grows large, thus we focus on bounding the \ufb01rst term. To this end, we require the following lemma. Lemma 3: Let C and \u03b7 be as de\ufb01ned above. For any u \u2208C, we have \u0393D \u0000T \u22121(u) \u0001 \u2229Styp Y \u2286CONE(u, \u03b8\u2032), (67) where \u03b8\u2032 \u225c\u03b80 + \u03b81 < \u03c0 2 , (68) and the angles \u03b80 and \u03b81 are given by \u03b80 \u225carcsin s D0 \u03c32 X ! (69) \u03b81 \u225carccos \u03c32 X + \u03c32 Y \u22122\u03b7 \u2212D 2 p (\u03c32 X + \u03b7)(\u03c32 Y + \u03b7) ! . (70) Proof: Appendix B. Fig. 6 illustrates the claim in the lemma. Let \u03b8\u2032 be as de\ufb01ned in Lemma 3 above. We continue with Pr{maybe|X \u2208Styp X , Y \u2208Styp Y } (a) = Pr \b Y \u2208\u0393D \u0000T \u22121(T(X)) \u0001 |X \u2208Styp X , Y \u2208Styp Y \t (b) \u2264Pr \b Y \u2208CONE(T(X), \u03b8\u2032)|X \u2208Styp X , Y \u2208Styp Y \t (c) = \u2126(\u03b8\u2032) (d) \u2264 1 p 2\u03c0(n \u22121) \u00b7 1 cos \u03b8\u2032 \u00b7 sinn\u22121 \u03b8\u2032. (71) Identity (a) follows by de\ufb01nition of the query function g(\u00b7, \u00b7). Inequality (b) follows from Lemma 3. Equality (c) follows since Y is uniformly distributed within each shell Sr of radius r > 0 (due to the spherical symmetry of the Gaussian distribution), and the probability of falling in a cap of a given half-angle \u03b8\u2032 is precisely the fraction of the surface that is occupied by the cap, \u2126(\u03b8\u2032). Inequality (d) follows since \u03b8\u2032 \u2264arccos(1/\u221an) for suf\ufb01ciently large n, and therefore (41) applies. Since \u03b8\u2032 < \u03c0/2, we have sin \u03b8\u2032 < 1, and it therefore follows from (71) that the probability Pr{maybe|X \u2208Styp, Y \u2208Styp Y } vanishes with n. Thus, since \u01eb was arbitrary, recalling (62) completes \f18 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY rX \u03b80 rY \u03b81 \u221a nD \u221a nD Styp Y u Styp X Fig. 6. Illustration for Lemma 3. The black region marks CAPr\u2212,r+(u, \u03b80). The grey area denotes \u0393D \u0000CAPr\u2212,r+(u, \u03b80) \u0001 , and the dark grey region marks the intersection \u0393D \u0000CAPr\u2212,r+(u, \u03b80) \u0001 \u2229Styp Y . the direct part of the proof. Remark 4: The alert reader will observe that the direct part also follows from the direct part of Theorem 2. However, we have chosen to include an explicit proof here to introduce the notations and ideas crucial for proving Theorem 2. Converse Part: Let \u03b7 > 0 and de\ufb01ne Styp X as in (57). Let T : Rn \u2192{1, ..., 2nR} be a given signature function corresponding to a D-admissible system (T, g), and assume that Pr{g(T(X), Y) = maybe} \u22641 4 (72) since we are only interested in D-achievable rates R. As before, we will use the shorthand notation Pr{maybe} \u225cPr{g(T(X), Y) = maybe} to simplify the presentation. We shall restrict our attention to the typical sphere. To this end, de\ufb01ne the mapping \u02dc T : Styp X \u2192 {1, ..., 2nR}, where \u02dc T(x) = T(x) for x \u2208Styp X . Let \u02dc T \u22121(\u00b7) denote the inverse mapping of \u02dc T(\u00b7), i.e. \u02dc T \u22121(i) \u225c{x \u2208Styp X : T(x) = i} (73) = T \u22121(i) \u2229Styp X . (74) \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 19 Let pi \u225cPr{X \u2208\u02dc T \u22121(i)|X \u2208Styp X }. Clearly, we have P2nR i=1 pi = 1. De\ufb01ne the set Ai \u2286SrX to be projection of \u02dc T \u22121(i) onto the sphere SrX: Ai = \u001a rX x \u2225x\u2225: x \u2208\u02dc T \u22121(i) \u001b (75) Let \u03b1i denote the fraction of the surface area of SrX that is occupied by Ai. By the spherical symmetry of the pdf of X, \u03b1i is also equal to the probability that the projection of X onto Sr0 lies in Ai. Therefore \u03b1i \u2265pi, with equality if and only if \u02dc T \u22121(i) is a thick cap with inner and outer radii r\u00b1 \u225c p n(\u03c32 X \u00b1 \u03b7). Let D\u2032 \u225c( \u221a D + p \u03c32 X \u2212\u03b7 \u2212 p \u03c32 X)2 < D. It can easily be veri\ufb01ed that \u0393D\u2032 (Ai) \u2286\u0393D \u0010 \u02dc T \u22121(i) \u0011 . (76) Now let D\u2032\u2032 \u225c( \u221a D\u2032+ p \u03c32 Y \u2212\u03b7\u2212 p \u03c32 Y )2, and let the set Bi denote the D\u2032\u2032-expansion of Ai, restricted to the sphere SrY , i.e. Bi \u225c\u0393D\u2032\u2032 (Ai) \u2229SrY . (77) The set Bi can also be thought of an expansion of a set \u02dc Ai \u225c\u03c3Y \u03c3X \u00b7 Ai, with the alternative distance measure \u02d8 d(\u00b7, \u00b7) de\ufb01ned over the sphere SrY that measures the arc-length between the two points (i.e., the geodesic distance). Also note that \u03b1i = |Ai| |SrX | = | \u02dc Ai| |SrY | where |\u00b7| is used to denote the (hyper-) surface area. Let \u03b2i = |Bi| |Sr0| denote the fraction of Sr0 that is occupied by Bi. Let the set Ci denote the r\u2212 Y , r+ Y thickening of Bi as follows: Ci = \u001a y \u2208Styp Y : rY y \u2225y\u2225\u2208Bi \u001b . (78) Next, it can also be veri\ufb01ed that Ci \u2286\u0393D\u2032 (Ai) . (79) Suppose that x \u2208Styp X and that T(x) = i. Then we have: Pr{maybe|X = x \u2208Styp X } \u2265Pr \b Y \u2208\u0393D \u0000T \u22121(i) \u0001\t (a) \u2265Pr n Y \u2208\u0393D \u0010 \u02dc T \u22121(i) \u0011o (b) \u2265Pr n Y \u2208\u0393D\u2032 (Ai) o (c) \u2265Pr {Y \u2208Ci} , where (a) follows since \u02dc T \u22121(i) \u2286T \u22121(i), and (b) and (c) follow from (76) and (79) respectively. \f20 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Let fY be the density of Y. Then, we continue with Pr {Y \u2208Ci} = Z Ci fY(y)dy = \u03b2i \u00b7 Pr{Y \u2208Styp Y }, where the second equality follows from the spherical symmetry of fY(y). We now arrive at the main step in proving the converse. The key ingredient we require is the wellknown isoperimetric inequality on the hypersphere (cf. [25, Theorem 1.1]) which states that, among all subsets of the hypersphere with a given surface area, spherical caps have minimum D-expansion measured under geodesic distance. As noted before, the set Bi \u2286SrY is an expansion of the set \u02dc Ai \u2286 SrY with the arclength (i.e., geodesic) distance measure. Therefore, it follows from the isoperimetric inequality that |Bi| = \f \f \f\u0393D\u2032\u2032 (Ai) \u2229SrY \f \f \f \u2265 \f \f \f\u0393D\u2032\u2032 (CAPrX(u, \u03b8i)) \u2229SrY \f \f \f (80) = |CAPrY (u, \u03b8i + \u03b8D\u2032\u2032)| , (81) where u is an arbitrary point and \u03b8i \u225c\u2126\u22121(\u03b1i) (82) \u03b8D\u2032\u2032 \u225carccos \u0012\u03c32 X + \u03c32 Y \u2212D\u2032\u2032 2\u03c3X\u03c3Y \u0013 . (83) Therefore, we can conclude that if x \u2208Styp X and T(x) = i, then Pr{maybe|X = x} \u2265Pr{Y \u2208Styp Y } \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(\u03b1i) \u0001 . (84) Now, the average quantity Pr{maybe|X \u2208Styp X } is bounded as follows Pr{maybe|X \u2208Styp X } = 2nR X i=1 Pr{T(X) = i|X \u2208Styp X } Pr{maybe|T(X) = i, X \u2208Styp X } \u2265 2nR X i=1 pi \u00b7 Pr{Y \u2208Styp Y } \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(\u03b1i) \u0001 (85) \u2265Pr{Y \u2208Styp Y } \u00b7 2nR X i=1 pi \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(pi) \u0001 , (86) where the last inequality follows since \u03b1i \u2265pi and the function \u2126(\u03b8D\u2032\u2032 +\u2126\u22121(\u00b7)) is monotone increasing. If the scheme at hand were to satisfy pi = 2\u2212nR for all i, then we could simply continue with analyzing \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(2\u2212nR) \u0001 . However, in general this might not be the case. We therefore require the following lemma: Lemma 4: Let 0 < \u2126\u2217< 1 and 0 < c < 1 be given constants. De\ufb01ne p\u2217to be the solution to \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 21 \u2126(\u03b8D\u2032\u2032 + \u2126\u22121(p)) = \u2126\u2217. Then if 2nR X i=1 pi \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(pi) \u0001 \u2264c \u00b7 \u2126\u2217, (87) then R \u22651 n log 1 \u2212c p\u2217. (88) Proof: Appendix C. For our purposes4 we set \u2126\u2217= 1 2 so that \u2126(\u03b8D\u2032\u2032 + \u2126\u22121(p\u2217)) = 1 2. Now use (41) to upper bound \u2126(\u00b7) and evaluate p\u2217: p\u2217= \u2126 \u0010\u03c0 2 \u2212\u03b8D\u2032\u2032 \u0011 \u2264 1 p 2\u03c0(n \u22121) \u00b7 1 cos \u0000\u03c0 2 \u2212\u03b8D\u2032\u2032\u0001 \u00b7 sinn\u22121 \u0010\u03c0 2 \u2212\u03b8D\u2032\u2032 \u0011 \u2264 1 p 2\u03c0(n \u22121) \u00b7 cosn\u22121 (\u03b8D\u2032\u2032) . Recalling the de\ufb01nition of \u03b8D\u2032\u2032, we have cos (\u03b8D\u2032\u2032) = \u03c32 X + \u03c32 Y \u2212D\u2032\u2032 2\u03c3X\u03c3Y , therefore 1 n log 1 p\u2217= log 2\u03c3X\u03c3Y \u03c32 X + \u03c32 Y \u2212D\u2032\u2032 + O \u0012log n n \u0013 . (89) Our goal, now, is to show that the rate R must be lower bounded by the identi\ufb01cation rate from (14). Recalling (72), it follows that 1 4 \u2265Pr{maybe} (90) = Pr{X \u2208Styp X } \u00b7 Pr{maybe|X \u2208Styp X } (91) + Pr{X / \u2208Styp X } \u00b7 Pr{maybe|X / \u2208Styp X } (92) \u2265Pr{X \u2208Styp X } \u00b7 Pr{maybe|X \u2208Styp X } (93) \u2265Pr{X \u2208Styp X } \u00b7 Pr{Y \u2208Styp Y } \u00b7 2nR X i=1 pi \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(pi) \u0001 , (94) where the \ufb01nal inequality is simply (86). Since Pr{Y \u2208Styp Y } and Pr{X \u2208Styp X } both approach 1 as n grows, we may assume that both probabilities are above 3 4 (for large enough n). Then, we can now invoke Lemma 4 with c = 8/9 and 4We shall use Lemma 4 again for proving the identi\ufb01cation exponent results, but with a different \u2126\u2217. \f22 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY \u2126\u2217= 1/2, combined with (89), to conclude that R \u2265log 2\u03c3X\u03c3Y \u03c32 X + \u03c32 Y \u2212D\u2032\u2032 + O \u0012log n n \u0013 . (95) As \u03b7 can be taken to be arbitrarily small, D\u2032\u2032 can be arbitrarily close to D, completing the proof of the converse. D. Identi\ufb01cation Exponent As with Theorem 1, the proof of Theorem 2 is rather involved, so we \ufb01rst sketch the main ideas before moving on to the formal proof. Characterizing the optimal exponent requires a slightly more sophisticated scheme than characterizing the identi\ufb01cation rate, but the proofs are very similar in spirit. The achievability proof builds upon that of Theorem 1 in the sense that we re\ufb01ne the signature assignment to quantize x/\u2225x\u2225and \u2225x\u2225separately. Intuitively, we can think of our scheme as quantizing the direction and amplitude of the vector x (similarly to \u2018shape-gain\u2019 quantizers [26, Ch. 12]). Similar to the achievability proof of Theorem 1, the set of vectors x/\u2225x\u2225are quantized by covering the unit sphere with regions roughly described by caps. It will turn out that the achievable identi\ufb01cation exponent emerges through the analysis of quantizing the amplitudes x. For the converse proof, we take the \u03c1\u2217 X, \u03c1\u2217 Y to minimize (18), and focus on the case where X lies in a spherical shell with radius p n\u03c1\u2217 X\u03c32 X and small, nonzero thickness. Then, the converse proceeds similar to that of Theorem 1, in the sense that the \u201ctypical shell\u201d is replaced by the new shell that depends on \u03c1\u2217 X. Proof of Theorem 2: Direct Part: We will rely on the code construction given in the achievability proof of Theorem 1, and hence we adopt the notation previously de\ufb01ned there. To this end, let (T, g) be the rate-R, D-admissible identi\ufb01cation system de\ufb01ned in the achievability proof of Theorem 1. Recall that \u2220 \u0012 T \u0012 rX x \u2225x\u2225 \u0013 , x \u2225x\u2225 \u0013 \u2264\u03b80, (96) where \u03b80 was de\ufb01ned as (69). In a variation on the scheme used previously, we describe the amplitude \u2225x\u2225by quantization as follows. Let \u03c32 max(n) \u225cn \u00b7 \u03c32 X, and recall that \u03b7 was chosen to be a small positive constant. De\ufb01ne the spherical shells S(i) as follows: S(i) \u225cSr(i),r(i+1), (97) where r(i) \u225c\u221an \u00b7 i \u00b7 \u03b7. The modi\ufb01ed signature assignment T \u2032 then describes the \u201cdirection\u201d and \u201camplitude\u201d of x as follows: \u2022 If 1 n\u2225x\u22252 \u2264\u03c32 max(n), then T \u2032(x) = \u0010 T \u0010 rX x \u2225x\u2225 \u0011 , i \u0011 , where i is chosen to satisfy x \u2208S(i). \u2022 If 1 n\u2225x\u22252 > \u03c32 max(n), then the signature T \u2032(x) is de\ufb01ned to be the erasure symbol e. \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 23 The overall rate of the modi\ufb01ed signature assignment T \u2032 described above is R (i.e., the rate of T(\u00b7)), plus an additional 1 n log \u03c32 max(n) \u03b7 = O( log n n ) (required for the quantization of \u2225x\u2225), and therefore remains essentially unchanged. Therefore, the upper bound (62) also upper bounds the rate of the modi\ufb01ed signature assignment function. Let g\u2032 be the optimal query function corresponding to T \u2032 (de\ufb01ned in an analogous manner to (63)). Thus, we only need to analyze the exponent attained by the proposed scheme. To this end, let Z be a Chi-square random variable with n degrees of freedom. The pdf of Z is given by fZ(z) = z n 2 \u22121e\u2212z 2 2n/2\u0393( n 2), (98) where \u0393 in (98) is the usual Gamma function, and should not be confused with the set-expansion operator \u0393D de\ufb01ned previously. Now, de\ufb01ne the random variables ZX \u225c 1 \u03c32 X \u2225X\u22252 and ZY \u225c 1 \u03c32 Y \u2225Y\u22252. Note that both ZX and ZY are distributed according to (98). In order to proceed, we require the following lemma. Lemma 5: The probability Pr \b 1 n\u2225X\u22252 > \u03c32 max(n) \t vanishes super-exponentially with n. Proof: Appendix D. Now, we are in a position to analyze Pr {maybe}, where we again employ the shorthand notation Pr{maybe} \u225cPr{g\u2032(T \u2032(X), Y) = maybe} to simplify the presentation. Pr{maybe} \u2264Pr \u001a maybe, 1 n\u2225X\u22252 \u2264\u03c32 max(n), 1 n\u2225Y\u22252 \u2264\u03c32 max(n) \u001b + Pr \u001a 1 n\u2225X\u22252 > \u03c32 max(n) \u001b + Pr \u001a1 n\u2225Y\u22252 > \u03c32 max(n) \u001b . (99) By Lemma 5, and a similar argument for Pr \b 1 n\u2225Y\u22252 > \u03c32 max(n) \t , the last two terms of the above expression vanish super-exponentially and do not affect the exponent of Pr{maybe}. We therefore concentrate on the \ufb01rst term. We can now write Pr \b maybe, 1 n\u2225X\u22252 \u2264\u03c32 max(n), 1 n\u2225Y\u22252 \u2264\u03c32 max(n) \t (100) = Pr n maybe, ZX \u2264n2, ZY \u2264 \u03c32 Y \u03c32 X n2o (101) = Z n2 0 Z \u03c32 Y \u03c32 X n2 0 Pr {maybe | ZX = zX, ZY = zY } fZ(zX)fZ(zY )dzY dzX (102) \u2264\u03c32 Y \u03c32 X n4 max 0\u2264zX\u2264n2 0\u2264zY \u2264n2\u03c32 Y /\u03c32 X Pr {maybe | ZX = zX, ZY = zY } fZ(zX)fZ(zY ) (103) \u2264\u03c32 Y \u03c32 X n4 max 0\u2264\u03c1X,\u03c1Y Pr {maybe | ZX = n\u03c1X, ZY = n\u03c1Y } fZ(n\u03c1X)fZ(n\u03c1Y ), (104) where \u03c1X \u225czX/n and \u03c1Y \u225czY /n. The event {maybe} coincides with the event {Y \u2208\u0393D(T \u2032\u22121(T \u2032(X)))}. Let U = T(rXX/\u2225X\u2225), and \f24 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY observe that if 1 n\u2225X\u22252 \u2264\u03c32 max(n), then \u0393D(T \u2032\u22121(T \u2032(X))) \u2286\u0393D(CAPr(i),r(i+1)(U, \u03b80)) (105) \u2286\u0393D\u2032(CAP\u2225X\u2225(U, \u03b80))), (106) where (105) follows from similar arguments leading to (59), and (106) follows with D\u2032 \u225c( \u221a D +\u221a\u03b7)2. We therefore continue with Pr {maybe | ZX = n\u03c1X, ZY = n\u03c1Y } \u2264Pr \u001a Y \u2208\u0393D\u2032(CAP\u2225X\u2225(U, \u03b80))) | 1 n\u2225X\u22252 = \u03c1X\u03c32 X, 1 n\u2225Y\u22252 = \u03c1Y \u03c32 Y \u001b = Pr \u001a Y \u2208\u0393D\u2032(CAP\u221a n\u03c1X\u03c32 X(U, \u03b80))) | 1 n\u2225Y\u22252 = \u03c1Y \u03c32 Y \u001b (107) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 if | p \u03c1X\u03c32 X \u2212 p \u03c1Y \u03c32 Y | \u2265 \u221a D\u2032 1 if \u03c1X\u03c32 X + \u03c1Y \u03c32 Y \u2264D\u2032 \u2126(\u03b80 + \u03b8\u2032 1) otherwise. (108) where (107) follows by spherical symmetry of the Gaussian distribution, and \u03b8\u2032 1 \u225carccos \u03c1x\u03c32 X + \u03c1Y \u03c32 Y \u2212D\u2032 2 p \u03c1X\u03c32 X \u00b7 \u03c1Y \u03c32 Y . (109) The identity (109) follows from the law of cosines. The geometric image now is similar to that depicted in Fig. 6, where here rX \u225c p n\u03c32\u03c1X and rY \u225c p n\u03c32\u03c1Y denote the actual radii of the vectors X and Y (as opposed to their average value in the proof of Theorem 1). Next, using the bound (41) we have 1 n log 1 \u2126(\u03b8) \u2265 ( \u2212log sin \u03b8 + c n log n, 0 < \u03b8 < arccos(1/\u221an); 0, otherwise. (110) where c is a universal constant. Combined with (110), we compactly write the exponent corresponding to expression (108) as E\u2126(\u03b80, D\u2032, \u03c32 X, \u03c32 Y , \u03c1X, \u03c1Y ) \u225c = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 \u221e, if | p \u03c1X\u03c32 X \u2212 p \u03c1Y \u03c32 Y | \u2265 \u221a D\u2032 0, if \u03c1X\u03c32 X + \u03c1Y \u03c32 Y \u2264D\u2032 \u2212log sin min \u0002\u03c0 2, \u03b80 + \u03b8\u2032 1 \u0003 , otherwise. (111) with \u03b8\u2032 1 given in (109). Before we plug the above result into (104), we note that by Stirling\u2019s approximation we may write, \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 25 for any \ufb01xed \u03c1 > 0: fZ(n\u03c1) = 1 n\u03c1 \u0010n\u03c1 2 \u0011n/2 exp(\u2212n\u03c1/2) 1 \u0393(n/2) = 1 n\u03c1 \u0010n\u03c1 2 \u0011n/2 exp(\u2212n\u03c1/2) 1 q 4\u03c0 n \u0000 n 2e \u0001n/2 \u00001 + O \u0000 1 n \u0001\u0001 = exp \u0014 \u2212n \u0012\u03c1 2 \u22121 2 \u22121 2 log \u03c1 \u0013\u0015 1 \u03c1 \u221a 4\u03c0n \u00001 + O \u0000 1 n \u0001\u0001 \u22642\u2212nEZ(\u03c1) \u00b7 nc, (112) where EZ(\u00b7) was de\ufb01ned in (16) and c is a universal constant. Finally, we plug (111) and (112) into the upper bound (104) on the (conditional) probability for maybe and conclude that the following exponent is achievable: min \u03c1X,\u03c1Y \u22650 EZ(\u03c1X) + EZ(\u03c1Y ) + E\u2126(\u03b80, D\u2032, \u03c32 X, \u03c32 Y , \u03c1X, \u03c1Y ). (113) Since \u03b7 is arbitrarily small we may replace D\u2032 with D in the above. We may therefore rewrite the achievable exponent as min \u03c1X,\u03c1Y \u22650 EZ(\u03c1X) + EZ(\u03c1Y ) + E\u2126 \u0000arcsin(2\u2212R), D, \u03c32, \u03c1X, \u03c1Y \u0001 . (114) In order to simplify matters further, note that in (114), the minimizing (\u03c1X, \u03c1Y ) must satisfy: \f \f \f \f q \u03c1X\u03c32 X \u2212 q \u03c1Y \u03c32 Y \f \f \f \f < \u221a D (115) \u03c1X\u03c32 X + \u03c1Y \u03c32 Y \u2265D. (116) The condition (115) must hold because otherwise the term E\u2126is in\ufb01nite [see (111)]. To prove that (116) must hold, assume, for contradiction, that (114) is minimized for (\u03c1\u2217 X, \u03c1\u2217 Y ) that satisfy \u03c1\u2217 X\u03c32 X + \u03c1\u2217 Y \u03c32 Y < D. (117) In this case, the value of (114) at the minimizing point is EZ(\u03c1\u2217 X) + EZ(\u03c1\u2217 Y ). If, say \u03c1\u2217 X > 1, then we may replace it with another value 0 < \u03c1\u2217\u2217 X < 1 that satis\ufb01es EZ(\u03c1\u2217\u2217 X ) = EZ(\u03c1\u2217 X) that is guaranteed to exist (see the de\ufb01nition of EZ(\u00b7)). The same argument holds for \u03c1\u2217 Y , and therefore we may assume that in this case both \u03c1\u2217 X, \u03c1\u2217 Y \u2208(0, 1]. Next, since EZ(\u03c1) is monotone decreasing for \u03c1 \u2208(0, 1), we may increase \u03c1\u2217 X and \u03c1\u2217 Y , while still in (0, 1]2, until (116) is met with an equality. Since the value of the objective function decreases, we arrive at a contradiction, meaning that (116) must hold for any minimizing \u03c1X, \u03c1Y . Therefore the achievable exponent can be simpli\ufb01ed to the expression (18) and the proof of the direct part is concluded. \f26 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Converse Part: Let \u03c1\u2217 X, \u03c1\u2217 Y denote the minimizers of (18) (in light of the discussion above, we can assume without loss of generality that \u03c1\u2217 X, \u03c1\u2217 Y satisfy (19)). The proof of the converse proceeds by focusing on values of X and Y that satisfy 1 n\u2225X\u22252 \u223c = \u03c1\u2217 X\u03c32 X and 1 n\u2225Y\u22252 \u223c = \u03c1\u2217 Y \u03c32 Y . The details are as follows: Let 0 < \u03b7 < min(\u03c1\u2217 X, \u03c1\u2217 Y ) be a small but \ufb01xed value. De\ufb01ne the following spherical caps: S\u2217 X \u225cSr\u2212 X,r+ X, S\u2217 Y \u225cSr\u2212 Y ,r+ Y , (118) where r\u00b1 X \u225c p n\u03c32 X(\u03c1\u2217 X \u00b1 \u03b7) and r\u00b1 Y \u225c p n\u03c32 Y (\u03c1\u2217 Y \u00b1 \u03b7). We then write the following: Pr{maybe} \u2265Pr {maybe, X \u2208S\u2217 X, Y \u2208S\u2217 Y } = Pr {maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y } \u00b7 Pr{X \u2208S\u2217 X} \u00b7 Pr{Y \u2208S\u2217 Y }. (119) Consider the term Pr{X \u2208S\u2217 X}: Pr {X \u2208S\u2217 X} = Pr \u001a 1 n\u03c32 X \u2225X\u22252 \u2208(\u03c1\u2217 X \u2212\u03b7, \u03c1\u2217 X + \u03b7) \u001b (120) = Z n(\u03c1\u2217 X+\u03b7) n(\u03c1\u2217 X\u2212\u03b7) fZ(z)dz (121) \u22652n\u03b7 min z\u2208[n(\u03c1\u2217 X\u2212\u03b7),n(\u03c1\u2217 X+\u03b7)] fZ(z) (122) \u22652n\u03b7 \u00b7 nc \u00b7 2 \u2212n max\u03c1X \u2208[\u03c1\u2217 X\u2212\u03b7,\u03c1\u2217 X +\u03b7] EZ(\u03c1X), (123) where (123) follows from Stirling\u2019s approximation similar to (112). A similar derivation applies for Pr{Y \u2208S\u2217 Y }. Thus, it follows from (119) and continuity of EZ(\u00b7) that \u22121 n log Pr{maybe} \u2264\u22121 n log [Pr {maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y }] + EZ(\u03c1\u2217 X) + EZ(\u03c1\u2217 Y ) + \u03b7\u2032 + O \u0012log n n \u0013 , (124) where \u03b7\u2032 is a quantity tending to zero as \u03b7 \u21920. We now concentrate on the term Pr {maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y }, and proceed in a manner similar to the converse proof of Theorem 1. To this end, let T : Rn \u2192{1, ..., 2nR} denote the signature assignment for the scheme at hand. De\ufb01ne the mapping \u02dc T : S\u2217 X \u2192{1, ..., 2nR} as \u02dc T(x) = T(x) for all x \u2208S\u2217 X. That is, \u02dc T(\u00b7) is the restriction of T(\u00b7) to S\u2217 X. Let \u02dc T \u22121(\u00b7) denote the inverse mapping of \u02dc T(\u00b7): \u02dc T \u22121(i) \u225c{x \u2208S\u2217 X : T(x) = i} (125) = T \u22121(i) \u2229S\u2217 X. (126) Let pi \u225cPr{X \u2208\u02dc T \u22121(i)|X \u2208S\u2217 X}, so P2nR i=1 pi = 1. De\ufb01ne rX \u225c p n\u03c32 X\u03c1\u2217 X, and let the set Ai \u2286SrX \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 27 denote the projection of \u02dc T \u22121(i) onto the sphere SrX. In other words, Ai = \u001a rX x \u2225x\u2225: x \u2208\u02dc T \u22121(i) \u001b . (127) Let \u03b1i denote the fraction of the surface area of SrX that is occupied by Ai. By the spherical symmetry of the distribution of X, \u03b1i is also equal to the probability that the projection of X onto SrX lies in Ai. Therefore \u03b1i \u2265pi, with equality if and only if \u02dc T \u22121(i) is a thick cap with inner and outer radii r\u2212 X and r+ X respectively. Let D\u2032 \u225c( \u221a D + p \u03c32 X(\u03c1\u2217 X \u2212\u03b7) \u2212 p \u03c32 X\u03c1\u2217 X)2. As in (76) we have that \u0393D\u2032 (Ai) \u2286\u0393D \u0010 \u02dc T \u22121(i) \u0011 . (128) Now let D\u2032\u2032 \u225c( \u221a D\u2032 + p \u03c32 Y (\u03c1\u2217 Y \u2212\u03b7)\u2212 p \u03c32 Y \u03c1\u2217 Y )2, and let the set Bi \u2286SrY denote the D\u2032\u2032-expansion of Ai, restricted to the sphere SrY , where rY \u225c p n\u03c32 Y \u03c1\u2217 Y , i.e. Bi \u225c\u0393D\u2032\u2032 (Ai) \u2229SrY . (129) Let \u03b2i denote the fraction of SrY that is occupied by Bi. Let the set Ci denote the r\u2212 Y , r+ Y thickening of Bi as follows: Ci = \u001a y \u2208S\u2217 Y : rY y \u2225y\u2225\u2208Bi \u001b . (130) As in (79) we have that Ci \u2286\u0393D\u2032 (Ai) . (131) Suppose that X = x \u2208S\u2217 X and that T(x) = i. Then we have, with the aid of (128) and (131): Pr{maybe|X = x \u2208S\u2217 X, Y \u2208S\u2217 Y } \u2265Pr \b Y \u2208\u0393D \u0000T \u22121(i) \u0001 |Y \u2208S\u2217 Y \t (132) \u2265Pr n Y \u2208\u0393D \u0010 \u02dc T \u22121(i) \u0011 |Y \u2208S\u2217 Y o (133) \u2265Pr n Y \u2208\u0393D\u2032 (Ai) |Y \u2208S\u2217 Y o (134) \u2265Pr {Y \u2208Ci|Y \u2208S\u2217 Y } (135) = \u03b2i, (136) where the last equality follows from the spherical symmetry of the pdf of Y. As in the proof of the converse of Theorem 1, we apply the isoperimetric inequality on the sphere for the sets Ai and Bi. We get that the set A\u2217 i that minimizes \u03b2i for given \u03b1i is the set CAPrX(u, \u03b8i), where u is an arbitrary point, \u03b8i \u225c\u2126\u22121(\u03b1i), and B\u2217 i is the set CAPrY (u, \u03b8\u2032 i), de\ufb01ned by \u03b8\u2032 i \u225c\u03b8i + \u03b8D\u2032\u2032 (137) \f28 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY where \u03b8D\u2032\u2032 \u225carccos \u03c1\u2217 X\u03c32 X + \u03c1\u2217 Y \u03c32 Y \u2212D\u2032\u2032 2 p \u03c1\u2217 X\u03c32 X \u00b7 \u03c1\u2217 Y \u03c32 Y . (138) Therefore the (normalized) surface area of B\u2217 i is given by \u03b2\u2217 i = \u2126(\u03b8\u2032 i). It follows that Pr{maybe|X = x \u2208S\u2217 X, Y \u2208S\u2217 Y } \u2265\u2126(\u03b8D\u2032\u2032 + \u2126\u22121(pi)), (139) and the average (conditional) probability Pr{maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y } is bounded by Pr{maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y } (140) = 2nR X i=1 Pr{T(X) = i|X \u2208S\u2217 X} Pr{maybe|T(X) = i, X \u2208S\u2217 X, Y \u2208S\u2217 Y } (141) \u2265 2nR X i=1 pi \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(pi) \u0001 . (142) Now, let 0 < c < 1, and invoke Lemma 4 to conclude that R \u22651 n log 1 \u2212c p\u2217, (143) where p\u2217is the solution to \u2126(\u03b8D\u2032\u2032 + \u2126\u22121(p)) = c\u22121 Pr{maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y }. (144) Since \u2126(\u00b7) is monotone increasing, so is \u2126\u22121(\u00b7). Therefore, (143) and (144) imply the inequality Pr{maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y } \u2265c \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121 \u0000(1 \u2212c)2\u2212nR\u0001\u0001 . (145) It is a straightforward exercise to verify (e.g., by Taylor series expansion) that \u2126\u22121 \u0000(1 \u2212c)2\u2212nR\u0001 = arcsin \u00002\u2212R\u0001 + O \u0000 log n n \u0001 . (146) If \u03b8D\u2032\u2032 + arcsin \u00002\u2212R\u0001 \u2265\u03c0/2, then (145) and the de\ufb01nition of \u2126(\u00b7) yield Pr{maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y } \u2265c/2, (147) which, combined with (124), yields the desired upper bound \u22121 n log Pr{maybe} \u2264\u2212log sin \u0010\u03c0 2 \u0011 + EZ(\u03c1\u2217 X) + EZ(\u03c1\u2217 Y ) + \u03b7\u2032 + O \u0012log n n \u0013 . (148) On the other hand, if \u03b8D\u2032\u2032 + arcsin \u00002\u2212R\u0001 < \u03c0/2, then the hypothesis of Lemma 1 is satis\ufb01ed for n suf\ufb01ciently large, and the estimate (42) gives \u22121 n log Pr{maybe|X \u2208S\u2217 X, Y \u2208S\u2217 Y } \u2264\u2212log sin \u0000\u03b8D\u2032\u2032 + \u2126\u22121 \u0000(1 \u2212c)2\u2212nR\u0001\u0001 + O \u0012log n n \u0013 . (149) \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 29 By letting \u03b7 be arbitrarily small we can infer from (148) and (149) that any sequence of identi\ufb01cation schemes {g(n), T (n)}n\u2192\u221emust satisfy lim sup n\u2192\u221e\u22121 n log Pr{g(n)(T (n)(X), Y) = maybe} \u2264EZ(\u03c1\u2217 X) + EZ(\u03c1\u2217 Y ) \u2212log sin min \" \u03c0 2, arcsin \u00002\u2212R\u0001 + arccos \u03c1\u2217 X\u03c32 X + \u03c1\u2217 Y \u03c32 Y \u2212D 2 p \u03c1\u2217 X\u03c32 X \u00b7 \u03c1\u2217 Y \u03c32 Y # , as desired. Proof of Corollary 2: Let \u03c1X, \u03c1Y satisfy (19). We claim that the quantity EZ(\u03c1X) + EZ(\u03c1Y ) + \u2118(R, D, \u03c1X\u03c32, \u03c1Y \u03c32) (150) can not increase if \u03c1X and \u03c1Y are both replaced by their average \u03c1 := (\u03c1X + \u03c1Y )/2, which continues to satisfy (19). To see that this is indeed the case, note that EZ(\u00b7) is convex, and therefore Jensen\u2019s inequality implies EZ(\u03c1X) + EZ(\u03c1Y ) \u22652EZ(\u03c1). (151) Next, the inequality of arithmetic and geometric means implies \u03c1X\u03c32 + \u03c1Y \u03c32 Y \u2212D 2\u03c32\u221a\u03c1X\u03c1Y \u22652\u03c1\u03c32 \u2212D 2\u03c1\u03c32 , (152) and therefore, since arccos(x) is monotone decreasing on x \u2208[0, 1], arccos \u03c1X\u03c32 + \u03c1Y \u03c32 Y \u2212D 2\u03c32\u221a\u03c1X\u03c1Y \u2264arccos 2\u03c1\u03c32 \u2212D 2\u03c1\u03c32 . (153) Since \u2212log sin(x) is decreasing on x \u2208[0, \u03c0/2], (153) implies \u2118(R, D, \u03c1X\u03c32, \u03c1Y \u03c32) \u2265\u2118(R, D, \u03c1\u03c32, \u03c1\u03c32), (154) which proves that (150) can not increase if \u03c1X and \u03c1Y are both replaced by their average \u03c1. The observation that 2EZ(\u03c1) + \u2118(R, D, \u03c1\u03c32, \u03c1\u03c32) (155) is monotone increasing for \u03c1 > 1 completes the proof. E. General Sources and the Extremal Property of the Gaussian The proof of Theorem 3 can be accomplished by restricting our attention to the setting where X and Y are discrete random variables. Therefore, the usual typicality machinery will be useful to us, and we review a few facts before beginning the proof of Theorem 3. We should also note that the method of types is used in the proofs in [1], but the proof here, which is similar in spirit, is signi\ufb01cantly simpler \f30 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY and shorter, partially because we are only interested in the achievable rate (and not in the exponent). To this end, let T (n) \u01eb denote the usual \u01eb-typical set (cf. [27, Chapter 2]). That is, we de\ufb01ne the empirical pmf of w \u2208Wn as \u03c0(w|w) = |i : wi = w| n for w \u2208W, (156) and, for W \u223cPW, the set of \u01eb-typical n-sequences is de\ufb01ned by T (n) \u01eb (W) = {w : |\u03c0(w|w) \u2212PW(w)| \u2264\u01ebPW(w) for all w \u2208W} . (157) Observe that if W \u223cQn i=1 PW(wi), then the union of events bound and Hoeffding\u2019s inequality imply Pr \b W / \u2208T (n) \u01eb (W) \t \u2264 X w\u2208W Pr {|\u03c0(w|W) \u2212PW(w)| > \u01ebPW(w)} (158) \u2264 X w\u2208W: PW (w)>0 2 exp \u0000\u2212n (\u01ebPW(w))2\u0001 . (159) Therefore, if |W| < \u221e, Pr \b W / \u2208T (n) \u01eb (W) \t \u2264exp (\u2212n\u03b4(\u01eb)) , (160) where \u03b4(\u01eb) denotes a positive quantity satisfying lim\u01eb\u21920 \u03b4(\u01eb) = 0. One useful fact is the so-called Typical Average Lemma [27, Section 2.4]: Lemma 6 (Typical Average Lemma): If w \u2208T (n) \u01eb (W), then (1 \u2212\u01eb)E[f(W)] \u22641 n n X i=1 g(wi) \u2264(1 + \u01eb)E[f(W)] for any nonnegative function f(w) on W. Now, we state a simple variant of the Covering Lemma [27, Lemma 3.3]: Lemma 7: Let PW V be a joint probability distribution on the \ufb01nite alphabet W\u00d7V, with corresponding marginals PW and PV . Let W \u223cQn i=1 PW(wi) and let V(m), m \u2208{1, 2, . . . , 2nR}, be random sequences, independent of each other and of W, each distributed according to Qn i=1 PV (vi). Then, for n suf\ufb01ciently large, there exists positive functions \u03b4(\u01eb), \u02dc \u03b4(\u01eb) satisfying lim\u01eb\u21920 \u03b4(\u01eb) = lim\u01eb\u21920 \u02dc \u03b4(\u01eb) = 0 and Pr \b (W, V(m)) / \u2208T (n) \u01eb (W, V ) for all m \t \u2264exp (\u2212n\u03b4(\u01eb)) + exp \u0010 \u22122n(R\u2212I(W ;V )\u2212\u02dc \u03b4(\u01eb))\u0011 . Proof: The proof follows that of [27, Lemma 3.3] verbatim, invoking (160) where appropriate. We require one more result before moving on to the proof of Theorem 3. Lemma 8: Let PW and PV be probability distributions with \ufb01nite second moments \u03c32 W and \u03c32 V , \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 31 respectively. If w \u2208T (n) \u01eb (W), v \u2208T (n) \u01eb (V ), and 1 n\u2225w \u2212v\u22252 \u2264D, then 1 n \r \r \r \r r \u03c3V \u03c3W w \u2212 r\u03c3W \u03c3V v \r \r \r \r 2 \u2264D \u2212(\u03c3W \u2212\u03c3V )2 + \u01eb|\u03c32 W \u2212\u03c32 V |. (161) Proof: Without loss of generality, assume \u03c3V \u2265\u03c3W. Note that the assumption 1 n\u2225w \u2212v\u22252 \u2264D implies \u22122 nwTv \u2264D \u22121 n\u2225w\u22252 \u22121 n\u2225v\u22252. (162) Moreover, Lemma 6 implies the following inequalities 1 n\u2225w\u22252 \u2264(1 + \u01eb)\u03c32 W (163) 1 n\u2225v\u22252 \u2265(1 \u2212\u01eb)\u03c32 V . (164) Therefore, it follows that 1 n \r \r \r \r r \u03c3V \u03c3W w \u2212 r\u03c3W \u03c3V v \r \r \r \r 2 = 1 n \u0012 \u03c3V \u03c3W \u2225w\u22252 + \u03c3W \u03c3V \u2225v\u22252 \u22122wTv \u0013 (165) \u2264D + 1 n \u0012\u0012 \u03c3V \u03c3W \u22121 \u0013 \u2225w\u22252 + \u0012\u03c3W \u03c3V \u22121 \u0013 \u2225v\u22252 \u0013 (166) \u2264D + (1 + \u01eb)\u03c32 W \u0012 \u03c3V \u03c3W \u22121 \u0013 + (1 \u2212\u01eb)\u03c32 V \u0012\u03c3W \u03c3V \u22121 \u0013 (167) = D \u2212(\u03c3W \u2212\u03c3V )2 + \u01eb(\u03c32 V \u2212\u03c32 W). (168) Considering the symmetric case where \u03c3V \u2264\u03c3W gives 1 n \r \r \r \r r \u03c3V \u03c3W w \u2212 r\u03c3W \u03c3V v \r \r \r \r 2 \u2264D \u2212(\u03c3W \u2212\u03c3V )2 + \u01eb(\u03c32 W \u2212\u03c32 V ), (169) completing the proof. Proof of Theorem 3: We can assume that X and Y are discrete random variables with \ufb01nite alphabet X \u2282R. The extension to continuous distributions with \ufb01nite second moments follows by the usual quantization arguments and continuity of \u2225\u00b7 \u2225. Fix \u01eb > 0 and a conditional pmf P \u02c6 X|X(\u02c6 x|x), where the alphabet \u02c6 X is an arbitrary subset of R with \ufb01nite support. Throughout, the random variables (Y, X, \u02c6 X) are drawn according to the joint distribution PY X \u02c6 X(y, x, \u02c6 x) = PY (y)PX \u02c6 X(x, \u02c6 x) = PY (y)PX(x)P \u02c6 X|X(\u02c6 x|x). (170) Random signature assignment. Randomly and independently generate 2nR sequences \u02c6 x(t), t \u2208 {1, 2, . . . , 2nR}, each according to Qn i=1 P \u02c6 X(\u02c6 xi). Given a sequence x, \ufb01nd an index t such that (x, \u02c6 x(t)) \u2208 T (n) \u01eb (X, \u02c6 X) and put T(x) = t. If there is more than one such index, break ties arbitrarily. If there is no such index, put T(x) = e. Observe that the rate R is negligibly affected by the addition of the \f32 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY additional \u201cerasure\u201d signature e (as in the proofs of Theorems 1 and 2). De\ufb01nition of the query function. In order to simplify notation, de\ufb01ne the quantity \u03a8 \u225c v u u t(1 + \u01eb)E \"\u0012r\u03c3Y \u03c3X X \u2212\u02c6 X \u00132# + q D \u2212(\u03c3X \u2212\u03c3Y )2 + \u01eb|\u03c32 X \u2212\u03c32 Y |. (171) For a signature t \u2208{1, 2, . . ., 2nR} \u222a{e} and a sequence y, de\ufb01ne g(t, y) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 maybe if \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 y / \u2208T (n) \u01eb (Y ), or t = e, or 1 \u221an \r \r \r q \u03c3X \u03c3Y y \u2212\u02c6 x(t) \r \r \r \u2264\u03a8 and t \u0338= e no otherwise. Scheme analysis. First, we check to ensure that g(\u00b7, \u00b7) does not produce any false negatives; that is, we need to verify that (T, g) is D-admissible. Note that g(T(x), y) returns maybe if y / \u2208T (n) \u01eb (Y ) or T(x) = e. Therefore, we only need to show that g(T(x), y) returns maybe if y \u2208T (n) \u01eb (Y ), (x, \u02c6 x(T(x))) \u2208 T (n) \u01eb (X, \u02c6 X), and 1 n\u2225x \u2212y\u22252 \u2264D. Under these assumptions, note that Lemma 6 implies 1 n \r \r \r \r r\u03c3Y \u03c3X x \u2212\u02c6 x(t) \r \r \r \r 2 \u2264(1 + \u01eb)E \"\u0012r\u03c3Y \u03c3X X \u2212\u02c6 X \u00132# . (172) Next, recall that (x, \u02c6 x(T(x))) \u2208T (n) \u01eb (X, \u02c6 X) implies x \u2208T (n) \u01eb (X). Hence, under the assumption that 1 n\u2225x \u2212y\u22252 \u2264D, Lemma 8 implies 1 n \r \r \r \r r\u03c3Y \u03c3X x \u2212 r\u03c3X \u03c3Y y \r \r \r \r 2 \u2264D \u2212(\u03c3X \u2212\u03c3Y )2 + \u01eb|\u03c32 X \u2212\u03c32 Y |. (173) Combining the triangle inequality, (172), (173), and (171), we have 1 \u221an \r \r \r \r r\u03c3X \u03c3Y y \u2212\u02c6 x(t) \r \r \r \r \u2264 1 \u221an \r \r \r \r r\u03c3Y \u03c3X x \u2212\u02c6 x(t) \r \r \r \r + 1 \u221an \r \r \r \r r\u03c3Y \u03c3X x \u2212 r\u03c3X \u03c3Y y \r \r \r \r \u2264\u03a8. (174) Hence, g(T(x), y) returns maybe if y \u2208T (n) \u01eb (Y ), (x, \u02c6 x(T(x))) \u2208T (n) \u01eb (X, \u02c6 X), and 1 n\u2225x \u2212y\u22252 \u2264D. Therefore, (T, g) is D-admissible as desired. Next, we check to ensure that Pr{g(T(X), Y) = maybe} is small. To this end, consider the events E0 = \b y / \u2208T (n) \u01eb (Y ) \t , E1 = {T(X) = e} , E2 = \u001a 1 \u221an \r \r \r \r r\u03c3X \u03c3Y Y \u2212\u02c6 X(T(X)) \r \r \r \r \u2264\u03a8 \u001b , and observe that Pr{g(T(X), Y) = maybe} \u2264Pr{E0}+Pr{E1}+Pr{E2} by the union of events bound. \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 33 We have already seen in (160) that Pr{E0} \u2264exp (\u2212n\u03b4(\u01eb)) (175) for some positive \u03b4(\u01eb) satisfying lim\u01eb\u21920 \u03b4(\u01eb) = 0. Next, Lemma 7 implies that, for n suf\ufb01ciently large, ET [Pr{E1}] \u2264exp (\u2212n\u03b4(\u01eb)) + exp \u0010 \u22122n(R\u2212I(X; \u02c6 X)\u2212\u02dc \u03b4(\u01eb))\u0011 , (176) where ET [Pr{E1}] denotes the value of Pr(E1) averaged over the random choice of the signature assignment T(\u00b7). Let \u02c6 X be distributed according to Qn i=1 P \u02c6 X(\u02c6 xi), independent of Y \u223cQn i=1 PY (yi). An application of Hoeffding\u2019s inequality implies Pr \uf8eb \uf8ed1 \u221an \r \r \r \r r\u03c3X \u03c3Y Y \u2212\u02c6 X \r \r \r \r \u2264 v u u tE \"\u0012r\u03c3X \u03c3Y Y \u2212\u02c6 X \u00132# \u2212\u01eb \uf8f6 \uf8f8\u2264exp(\u2212n\u03b4(\u01eb)). (177) Since the sequence Y is independent of X, and is therefore also independent of \u02c6 X(T(X)), (177) implies that ET [Pr{E2}] \u2264exp (\u2212n\u03b4(\u01eb)) (178) if \u03a8 \u2264 v u u tE \"\u0012r\u03c3X \u03c3Y Y \u2212\u02c6 X \u00132# \u2212\u01eb. (179) Therefore, if (179) holds, we have ET [Pr{g(T(X), Y) = maybe}] \u22643 exp (\u2212n\u03b4(\u01eb)) + exp \u0010 \u22122n(R\u2212I(X; \u02c6 X)\u2212\u02dc \u03b4(\u01eb))\u0011 , (180) implying the existence of a sequence of D-admissible, rate R > I(X; \u02c6 X) schemes for which Pr{g(T(X), Y) = maybe} is exponentially small in n. Since \u01eb was arbitrary, the proof is complete. Proof of Theorem 4: Since d(\u00b7, \u00b7) is translation invariant, we can assume without loss of generality that PX and PY have mean zero. Also, note that it is suf\ufb01cient to consider D in the interval (\u03c3X \u2212\u03c3Y )2 \u2264 D \u2264\u03c32 X + \u03c32 Y . Indeed, if D > \u03c32 X + \u03c32 Y , then the theorem asserts that RID(D, PX, PY ) \u2264\u221e, which is trivially true. On the other hand, we can argue that RID(D, PX, PY ) = 0 for D < (\u03c3X \u2212\u03c3Y )2 by monotonicity of RID(D, PX, PY ) in D and the fact that RID((\u03c3X \u2212\u03c3Y )2, PX, PY ) = 0. Therefore, assume (\u03c3X \u2212\u03c3Y )2 < D \u2264\u03c32 X +\u03c32 Y and consider the conditional distribution P \u02c6 X|X de\ufb01ned \f34 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY by \u02c6 X = \u03c1 q \u03c3Y \u03c3X X + Z, where Z \u223cN(0, \u03c32 Z) is independent of X and \u03c1, \u03c32 Z are given by \u03c1 = (\u03c3X + \u03c3Y )2 \u2212D (2\u03c3X\u03c3Y ) \u03c32 Z = ((\u03c3X + \u03c3Y )2 \u2212D)(\u03c32 X + \u03c32 Y \u2212D)2 4\u03c3X\u03c3Y (D \u2212(\u03c3X \u2212\u03c3Y )2) . (181) With P \u02c6 X|X de\ufb01ned in this way, the following identities are readily veri\ufb01ed v u u tE \"\u0012r\u03c3X \u03c3Y Y \u2212\u02c6 X \u00132# = q \u03c3X\u03c3Y (1 + \u03c12) + \u03c32 Z = 2\u03c3X\u03c3Y p D \u2212(\u03c3X \u2212\u03c3Y )2 (182) v u u tE \"\u0012r\u03c3Y \u03c3X X \u2212\u02c6 X \u00132# = q \u03c3X\u03c3Y (1 \u2212\u03c1)2 + \u03c32 Z = \u03c32 X + \u03c32 Y \u2212D p D \u2212(\u03c3X \u2212\u03c3Y )2. (183) Therefore, (182) and (183) yield the identity v u u tE \"\u0012r\u03c3X \u03c3Y Y \u2212\u02c6 X \u00132# = v u u tE \"\u0012r\u03c3Y \u03c3X X \u2212\u02c6 X \u00132# + p D \u2212(\u03c3X \u2212\u03c3Y )2. (184) Since \u02c6 X has density and the Gaussian distribution maximizes differential entropy for a given variance (cf. [17]), we have the inequality h( \u02c6 X) \u22641 2 log (2\u03c0e(\u03c12\u03c3X\u03c3Y + \u03c32 Z)). It follows that I(X; \u02c6 X) \u22641 2 log \u0012\u03c12\u03c3X\u03c3Y + \u03c32 Z \u03c32 Z \u0013 = log \u0012 2\u03c3X\u03c3Y \u03c32 X + \u03c32 Y \u2212D \u0013 = RID(D, N(0, \u03c32 X), N(0, \u03c32 Y )). Thus, for D \u0338= (\u03c3X \u2212\u03c3Y )2, an application of Theorem 3 implies that RID(D, PX, PY ) \u2264RID(D, N(0, \u03c32 X), N(0, \u03c32 Y )). (185) To handle the case where D = (\u03c3X \u2212\u03c3Y )2, we note that RID(D, PX, PY ) is nondecreasing in D. Since lim D\u2193(\u03c3X\u2212\u03c3Y )2 RID(D, N(0, \u03c32 X), N(0, \u03c32 Y )) = 0, (186) inequality (185) implies that we must have RID(D, PX, PY ) = 0 at D = (\u03c3X \u2212\u03c3Y )2. This completes the proof. F. Robust Identi\ufb01cation Schemes Fix R > RID (D, N(0, \u03c32 X), N(0, \u03c32 Y )) and consider the setup described in section III-D. Speci\ufb01cally, let P \u02dc X, P \u02dc Y be zero-mean distributions with variances \u03c32 X and \u03c32 Y , respectively. Recall that, for a given blocklength n, the argument in the achievability proof of Theorem 1 constructs a signature assignment function T (n) : \u02dc x \u2192Rn for which the query g(n) \u0000T (n)(\u02dc x), \u02dc y \u0001 returns \u201cmaybe\u201d only if 1) The angle \u2220(\u02dc y, T (n)(\u02dc x)) is at most \u03b8\u2032, where \u03b8\u2032 < \u03c0/2 is a \ufb01xed constant de\ufb01ned in (68), and \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 35 2) We have \u02dc x \u2208Styp X , where Styp X is the \u201ctypical shell\u201d of \u02dc x vectors de\ufb01ned in (57). We remark that the gap between \u03c0/2 \u2212\u03b8\u2032 and the thickness of the shell Styp X depend on the parameter \u03b7 > 0, which is a function of the gap between R and RID (D, N(0, \u03c32 X), N(0, \u03c32 Y )). In light of the conditions under which g(n) \u0000T (n)(\u02dc x), \u02dc y \u0001 returns \u201cmaybe\u201d, the probability of the event n g(n) \u0010 T (n)( \u02dc X), \u02dc Y \u0011 = maybe o is bounded by Pr n g(n) \u0010 T (n)( \u02dc X), \u02dc Y \u0011 = maybe o \u2264Pr n \u2220( \u02dc Y, T (n)( \u02dc X)) \u2264\u03b8\u2032o + Pr n \u02dc X / \u2208Styp X o . (187) The term Pr n \u02dc X / \u2208Styp X o vanishes by the weak law of large numbers as n \u2192\u221e. Therefore, since \u02dc X and \u02dc Y are independent, it is suf\ufb01cient to show that Pr n \u2220( \u02dc Y, \u03b1) \u2264\u03b8\u2032o vanishes for any given unit vector \u03b1 = (\u03b11, \u03b12, . . . , \u03b1n) and constant \u03b8\u2032 \u2208(0, \u03c0/2). To this end, de\ufb01ne \u03b2n \u225c\u03c3Y 2 \u221an, and observe that Pr n \u2220( \u02dc Y, \u03b1) \u2264\u03b8\u2032o = Pr ( n X i=1 \u03b1i \u02dc Yi \u2265\u2225\u02dc Y\u2225cos \u03b8\u2032 ) (188) \u2264Pr ( n X i=1 \u03b1i \u02dc Yi \u2265\u03b2n cos \u03b8\u2032 ) + Pr n \u2225\u02dc Y\u2225\u2264\u03b2n o . (189) First, note limn\u2192\u221ePr n \u2225\u02dc Y\u2225\u2264\u03b2n o = 0 by the weak law of large numbers. Next, since \u03b1 is a unit vector, we have Pn i=1 \u03b12 i = 1, and it follows that VAR n X i=1 \u03b1i \u02dc Yi ! = \u03c32 Y . (190) Since E[ \u02dc Yi] = 0, Chebyshev\u2019s inequality implies Pr ( n X i=1 \u03b1i \u02dc Yi \u2265\u03b2n cos \u03b8\u2032 ) \u2264 \u03c32 Y \u03b22 n cos2 \u03b8\u2032 = 4 n cos2 \u03b8\u2032, (191) proving that Pr n g(n) \u0010 T (n)( \u02dc X), \u02dc Y \u0011 = maybe o \u21920 as desired. Since the D-admissibility of the scheme (T (n), g(n)) did not depend on the Gaussianity assumption in the proof of Theorem 1, the scheme (T (n), g(n)) continues to be D-admissible for the sources \u02dc X, \u02dc Y. Therefore, we can conclude that a sequence of rate-R, D-admissible schemes {T (n), g(n)}n\u2192\u221e constructed as described in the proof of Theorem 1 exhibit the robustness property explained in Section III-D. V. CONCLUDING REMARKS We studied the problem of answering similarity queries from compressed data from an informationtheoretic perspective. We focused on the setting where the similarity criterion is the (normalized) quadratic distance. For the case of i.i.d. Gaussian data, we gave an explicit characterization of the \f36 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY minimal compression rate which permits reliable queries (i.e., the identi\ufb01cation rate). Furthermore, we characterized the best exponential rate at which the probability for false positives can be made to vanish. For general sources, we derived an upper bound on the identi\ufb01cation rate, and proved that it is at most that of the Gaussian source of the same variance. Finally, we presented a single, robust, scheme that compresses any source at the Gaussian identi\ufb01cation rate, while permitting reliable responses to queries. ACKNOWLEDGEMENT The authors would like to thank Golan Yona for stimulating discussions that motivated this work. APPENDIX A COVERING A SHELL WITH SPHERES Proof of Lemma 2: According to [24, Theorem 1], for any r > \u03c1 > 0 there exists a covering of Sr with balls of radius \u03c1, with density \u03d1 upper bounded by \u03d1 \u2264(n \u22121) log(n \u22121) \u00121 2 + 2 log log(n \u22121) + 5 log(n \u22121) \u0013 (192) \u2264n log(n), (193) where (192) holds for all n \u22654, and (193) holds for n large enough so that 2 log log(n\u22121)+5 log(n\u22121) \u22641 2. This translates to k balls of radius \u03c1 that cover Sr, where k \u2264n log(n) \u2126(\u03b8) , (194) and \u03b8 \u225carcsin(\u03c1/r). We choose r = r0 = \u221a n\u03c32 and \u03c1 = \u03c10 = \u221anD0, so \u03b8 = \u03b80 = arcsin( p D0/\u03c32) < \u03c0/2 and is independent of n. When n is large enough s.t. \u03b8 \u2264arccos(1/\u221an), we may use (42) and get an upper bound on the covering size: k \u2264n log(n) \u2126(\u03b80) (195) \u2264 n log(n) 1 3 \u221a 2\u03c0n cos \u03b80 sinn\u22121 \u03b80 (196) \u22643 \u221a 2\u03c0n3/2 log(n)(\u03c10/r0)n\u22121, (197) which proves (48). Note that for a code that covers a spherical shell, the biggest covering by any single point u \u2208C is obtained when the point u is located at distance p r2 0 \u2212\u03c12 0 from the origin. We therefore can assume, without altering the covering property of C, that \u2225u\u2225= p r2 0 \u2212\u03c12 0 for all u \u2208C (see also [24, Eq. (1)] and the discussion that follows). The intersection of BALL\u03c10(u) and Sr0 is precisely given by CAPr0(u, \u03b80). \fINGBER et al.: COMPRESSION FOR QUADRATIC SIMILARITY QUERIES 37 APPENDIX B Proof of Lemma 3: Let y \u2208\u0393D (T \u22121(u)) \u2229Styp Y . Our goal is to show that y \u2208CONE(u, \u03b8\u2032). In other words, we need to show that \u2220(u, y) \u2264\u03b8\u2032. (198) Since y \u2208\u0393DT \u22121(u), there exists x \u2208T \u22121(u) s.t. d(x, y) \u2264D. By the triangle inequality for the angle operator (which is proportional to the geodesic metric in spherical geometry) we can write \u2220(u, y) \u2264\u2220(u, x) + \u2220(x, y). (199) Since T \u22121(u) \u2286CAPr\u2212,r+(u, \u03b80), we know that \u2220(u, x) \u2264\u03b80. Further, by the law of cosines for the triangle (x, y, 0) we can write \u2220(x, y) = arccos \u0014\u2225x\u22252 + \u2225y\u22252 \u2212\u2225x \u2212y\u22252 2\u2225x\u2225\u2225y\u2225 \u0015 (200) (a) \u2264arccos \" \u03c32 X + \u03c32 Y \u22122\u03b7 \u2212nD 2 p (\u03c32 X + \u03b7)(\u03c32 Y + \u03b7) # (201) = \u03b81, (202) where (a) follows since x \u2208Styp X , y \u2208Styp Y and d(x, y) \u2264D. Therefore by de\ufb01nition we have y \u2208CONE(u, \u03b8\u2032). All there\u2019s left to show is that \u03b8\u2032 < \u03c0 2. This follows immediately since D0 satis\ufb01es (52) by de\ufb01nition, and from the fact that arcsin(\u03c6) + arccos(\u03c6) = \u03c0 2. APPENDIX C Proof of Lemma 4: De\ufb01ne I as the set of indices i for which pi \u2265p\u2217: I \u225c{i : pi \u2265p\u2217}. (203) Clearly \u2126(\u03b8D\u2032\u2032 + \u2126\u22121(pi)) \u2265\u2126\u2217if and only if i \u2208I, so I can be thought of as the set of \u2018bad\u2019 values for i, i.e. those that contribute a lot to the sum in (87). Consider the following sequence of inequalities: c \u00b7 \u2126\u2217\u2265 2nR X i=1 pi \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(pi) \u0001 \u2265 X i\u2208I pi \u00b7 \u2126 \u0000\u03b8D\u2032\u2032 + \u2126\u22121(pi) \u0001 \u2265\u2126\u2217X i\u2208I pi. (204) \f38 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY On the other hand, 1 = X i pi = X i\u2208I pi + X i/ \u2208I pi (a) \u2264c + X i/ \u2208I pi (b) \u2264c + X i/ \u2208I p\u2217 \u2264c + p\u22172nR. (a) follows from (204). (b) follows from the de\ufb01nition of I. Eq. (88) follows immediately. APPENDIX D Proof of Lemma 5: For any t > 0 and a > 0 we have Pr{\u2225X\u22252 > a} (a) \u2264e\u2212t\u00b7aE \u0002 exp \u0000t \u00b7 \u2225X\u22252\u0001\u0003 (b) = e\u2212t\u00b7a(1 \u22122t\u03c32 X)\u2212n/2. (a) is the Chernoff bound. (b) follows since the moment generating function of Z is given by E \u0002 etZ\u0003 = (1 \u22122t)\u2212n/2, for t < 1 2. (205) Here it holds for any t < 1 2\u03c32 X . We choose t = 1 4\u03c32 X and write: Pr{\u2225X\u22252 > a} \u2264e \u2212 a 4\u03c32 X \u00b7 2n/2. (206) Choosing a = n\u03c32 max(n) = n2\u03c32 X results in Pr{\u2225X\u22252 > n\u03c32 max(n)} \u2264e\u22121 4n2+o(n2). (207)" + }, + { + "url": "http://arxiv.org/abs/1103.0171v3", + "title": "Finite Dimensional Infinite Constellations", + "abstract": "In the setting of a Gaussian channel without power constraints, proposed by\nPoltyrev, the codewords are points in an n-dimensional Euclidean space (an\ninfinite constellation) and the tradeoff between their density and the error\nprobability is considered. The capacity in this setting is the highest\nachievable normalized log density (NLD) with vanishing error probability. This\ncapacity as well as error exponent bounds for this setting are known. In this\nwork we consider the optimal performance achievable in the fixed blocklength\n(dimension) regime. We provide two new achievability bounds, and extend the\nvalidity of the sphere bound to finite dimensional infinite constellations. We\nalso provide asymptotic analysis of the bounds: When the NLD is fixed, we\nprovide asymptotic expansions for the bounds that are significantly tighter\nthan the previously known error exponent results. When the error probability is\nfixed, we show that as n grows, the gap to capacity is inversely proportional\n(up to the first order) to the square-root of n where the proportion constant\nis given by the inverse Q-function of the allowed error probability, times the\nsquare root of 1/2. In an analogy to similar result in channel coding, the\ndispersion of infinite constellations is 1/2nat^2 per channel use. All our\nachievability results use lattices and therefore hold for the maximal error\nprobability as well. Connections to the error exponent of the power constrained\nGaussian channel and to the volume-to-noise ratio as a figure of merit are\ndiscussed. In addition, we demonstrate the tightness of the results numerically\nand compare to state-of-the-art coding schemes.", + "authors": "Amir Ingber, Ram Zamir, Meir Feder", + "published": "2011-03-01", + "updated": "2011-09-05", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION Coding schemes over the Gaussian channel are traditionally limited by the average/peak power of the transmitted signal [1]. Without the power restriction (or a similar restriction) the channel capacity becomes in\ufb01nite, since one can space the codewords arbitrarily far apart from each other and achieve a vanishing error probability. However, many coded modulation schemes take an in\ufb01nite constellation (IC) and restrict the usage to points of the IC that lie within some n-dimensional form in Euclidean space (a \u2018shaping\u2019 region). Probably the most important example for an IC is a lattice (see Fig. 1), and examples for the shaping regions include a hypersphere in n dimensions, and a Voronoi region of another lattice [2]. The material in this paper will be presented in part at the IEEE International Symposium on Information Theory (ISIT) 2011 A. Ingber is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. This research was supported in part by the Israel Science Foundation, grant no. 634/09. \f2 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY (a) A lattice (b) A non-lattice in\ufb01nite constellation Fig. 1. Examples for 2-dimensional in\ufb01nite constellations. Only a \ufb01nite section of the IC is shown. In 1994, Poltyrev [3] studied the model of a channel with Gaussian noise without power constraints. In this setting the codewords are simply points of the in\ufb01nite constellation in the n-dimensional Euclidean space. The analog to the number of codewords is the density \u03b3 of the constellation points (the average number of points per unit volume). The analog of the communication rate is the normalized log density (NLD) \u03b4 \u225c 1 n log \u03b3. The error probability in this setting can be thought of as the average error probability, where all the points of the IC have equal transmission probability (precise de\ufb01nitions follow later on in the paper). Poltyrev showed that the NLD \u03b4 is the analog of the rate in classical channel coding, and established the corresponding \u201ccapacity\u201d, the ultimate limit for the NLD denoted \u03b4\u2217(also known as Poltyrev\u2019s capacity), given by 1 2 log 1 2\u03c0e\u03c32, where \u03c32 denotes the noise variance per dimension1. Random coding, expurgation and sphere packing error exponent bounds were derived, which are analogous to Gallager\u2019s error exponents in the classical channel coding setting [4], and to the error exponents of the power-constrained additive white Gaussian noise (AWGN) channel [5], [4]. In classical channel coding, the channel capacity gives the ultimate limit for the rate when arbitrarily small error probability is required, and the error exponent quanti\ufb01es the (exponential) speed at which the error probability goes to zero as the dimension grows, where the rate is \ufb01xed (and below the channel capacity). This type of analysis is asymptotic in nature neither the capacity nor the error exponent theory can tell what is the best achievable error probability with a given rate R and block length n. A big step in the nonasymptotic direction was recently made in a paper by Polyanskiy et al. [6], where explicit bounds for \ufb01nite n were derived. In addition to the error exponent formulation, another asymptotic question can be asked: Suppose that the (codeword) error probability is \ufb01xed 1logarithms are taken w.r.t. to the natural base e and rates are given in nats. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 3 to some value \u03b5. Let R\u03b5(n) denote the maximal rate for which there exist communication schemes with codelength n and error probability at most \u03b5. As n grows, R\u03b5(n) approaches the channel capacity C, and the speed of convergence is quanti\ufb01ed by [7][6] R\u03b5(n) = C \u2212 r V n Q\u22121(\u03b5) + O \u0012log n n \u0013 , (1) where Q\u22121(\u00b7) is the inverse complementary standard Gaussian cumulative distribution function. The constant V , termed the channel dispersion, is the variance of the information spectrum i(x; y) \u225clog PXY (x,y) PX(x)PY (y) for a capacity-achieving distribution. This result holds for discrete memoryless channels (DMC\u2019s), and was recently extended to the (power constrained) AWGN channel [8][6]. More re\ufb01nements of (1) and further details can be found in [6]. In this paper we take an in-depth look at the unconstrained Gaussian channel where the block length (dimension) is \ufb01nite. We give new achievability bounds which enable easy evaluation of the achievable error probability. We then analyze the new achievability bounds and the so-called sphere bound (converse bound), and obtain asymptotic analysis of the lowest achievable error probability for \ufb01xed NLD \u03b4 which greatly re\ufb01nes Poltyrev\u2019s error exponent results. In addition, we analyze the behavior of the highest NLD when the error probability is \ufb01xed. We show that the behavior demonstrated in (1) for DMC\u2019s and the power constrained AWGN channel carries on to the unconstrained AWGN channel as well. We demonstrate the tightness of the results both analytically and numerically, and compare to state-of-the-art coding schemes. The main results in the paper are summarized below. A. New Finite-Dimensional Performance Bounds Poltyrev\u2019s achievability results [3] for the capacity and for the error exponent are based on a bound that holds for \ufb01nite dimensions, but is hard to calculate, as it involves optimizing w.r.t. a parameter and 3-dimensional integration. We derive two new bounds that hold for \ufb01nite dimensions, and are easier to calculate than Poltyrev\u2019s. Like Poltyrev\u2019s bound, we bound the error probability by the sum of the probability that the noise leaves a certain region (a sphere), and the probability of error for noise realization within that sphere. This classic technique is due to Gallager [9], sometimes called \u201cGallager\u2019s \ufb01rst bounding technique\u201d [10]. Our \ufb01rst bound, called the typicality bound, is based on a simple \u2018typicality\u2019 decoder (close in spirit to that used in the standard achievability proofs [11]). It shows that there exist IC\u2019s with NLD \u03b4 and error probability bounded by Pe \u2264P TB e (n, \u03b4) \u225cen\u03b4Vnrn + Pr {\u2225Z\u2225> r} , (2) where Vn denotes the volume of an n-dimensional sphere with unit radius [12] and Z denotes the noise vector. The bound holds for any r > 0, and the value minimizing the bound is given by r = \u03c3 p n(1 + 2\u03b4\u2217\u22122\u03b4). Evaluating this bound only involves 1D integration, and the simple expression is amenable to precise asymptotic analysis. A stronger bound, called the maximum likelihood (ML) bound, which is based on the ML decoder, shows that there exist IC\u2019s with error probability bounded by Pe \u2264P MLB e (n, \u03b4) \u225cen\u03b4Vn Z r 0 fR(\u02dc r)\u02dc rnd\u02dc r + Pr {\u2225Z\u2225> r} , (3) \f4 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY fR(\u00b7) is the pdf of the norm \u2225Z\u2225of the noise vector. The bound holds for any r > 0, and the value minimizing the bound is given by r = re\ufb00\u225ce\u2212\u03b4V \u22121/n n . Note that re\ufb00, called the effective radius of the lattice (or IC), is the radius of a sphere with the same volume as the Voronoi cell of the lattice (or the average volume of the Voronoi cells of the IC2 ). Evaluating the ML bound also involves 1D integration only. We further show that the ML bound gives the exact value of Poltyrev\u2019s bound, therefore the simplicity does not come at the price of a weaker bound. In the achievability part of the results we use lattices (and the Minkowski-Hlawka theorem [13][14]). Because of the regular structure of lattices, all our achievability results hold in the stronger sense of maximal error probability. In the converse part we base our results on the sphere bound [15][3][16], i.e. on the fact that the error probability is lower bounded by the probability that the noise leaves a sphere with the same volume as a Voronoi cell. For lattices (and more generally, for IC\u2019s with equal-volume Voronoi cells), it is given by Pe \u2265P SB e (n, \u03b4) \u225cPr{\u2225Z\u2225> re\ufb00}. (4) We extend the validity of the sphere bound to any IC, and to the stronger sense of average error probability. Therefore our results hold for both average and maximal error probability, and for any IC (lattice or not). Note that since the optimal value for r in the ML bound (3) is exactly re\ufb00, the difference between the ML upper bound and the sphere packing lower bound is the left term in (3). This fact enables a precise evaluation of the best achievable Pe, see Section V. B. Asymptotic Analysis: Fixed NLD The asymptotics of the bounds on the error probability were studied by Poltyrev [3] using large deviation techniques and error exponents. The error exponent for the unconstrained AWGN is de\ufb01ned in the usual manner: E(\u03b4) \u225clim n\u2192\u221e 1 n log Pe(n, \u03b4), (5) (assuming the limit exists), where Pe(n, \u03b4) is the best error probability for any IC with NLD \u03b4. Poltyrev showed that the error exponent is bounded by the random coding and sphere packing exponents Er(\u03b4) and Esp(\u03b4) which are the in\ufb01nite constellation counterparts of the similar exponents in the power constrained AWGN. The random coding and sphere packing exponents coincide when the NLD is above the critical NLD \u03b4cr, de\ufb01ned later on. However, even when the error exponent bounds coincide, the optimal error probability Pe(n, \u03b4) is known only up to an unknown sub-exponential term (which can be, for example n100, or worse, e.g. e \u221an). We present a signi\ufb01cantly tighter asymptotic analysis using a more delicate (and direct) approach. Speci\ufb01cally, we show that the sphere bound is given asymptotically by P SB e (n, \u03b4) \u223c = e\u2212nEsp(\u03b4)(n\u03c0)\u22121 2e2(\u03b4\u2217\u2212\u03b4) e2(\u03b4\u2217\u2212\u03b4) \u22121 , (6) 2Note that the average volume of the Voronoi cells is not always well-de\ufb01ned, as in general there may exist cells with in\ufb01nite volume. See IV-D for more details. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 5 where a \u223c = b means that a b \u21921. We further show that the ML bound is given by P MLB e (n, \u03b4) \u223c = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 e\u2212nEr(\u03b4) 1 \u221a 2\u03c0n, \u03b4 < \u03b4cr; e\u2212nEr(\u03b4) 1 \u221a 8\u03c0n, \u03b4 = \u03b4cr; e\u2212nEr(\u03b4) (n\u03c0)\u22121 2 e2(\u03b4\u2217\u2212\u03b4) (2\u2212e2(\u03b4\u2217\u2212\u03b4))(e2(\u03b4\u2217\u2212\u03b4)\u22121), \u03b4cr < \u03b4 < \u03b4\u2217; (7) As a consequence, for NLD above \u03b4cr, Pe(n, \u03b4) is known asymptotically up to a constant, compared to a sub-exponential term in Poltyrev\u2019s error exponent analysis. The weaker typicality bound is given by P TB e (n, \u03b4) \u223c = e\u2212nEt(\u03b4) 1 \u221an\u03c0 \u00b7 1 + 2(\u03b4\u2217\u2212\u03b4) 2(\u03b4\u2217\u2212\u03b4) (8) where Et(\u03b4) is the typicality exponent, de\ufb01ned later on, which is lower than Er(\u03b4). C. Asymptotic Analysis: Fixed Error Probability For a \ufb01xed error probability value \u03b5, let \u03b4\u03b5(n) denote the maximal NLD for which there exists an IC with dimension n and error probability at most \u03b5. We shall be interested in the asymptotic behavior \u03b4\u03b5(n). This type of analysis for in\ufb01nite constellations has never appeared in literature (to the best of the authors\u2019 knowledge). In the current paper we utilize central limit theorem (CLT) type tools (speci\ufb01cally, the Berry-Esseen theorem) to give a precise asymptotic analysis of \u03b4\u03b5(n), a result analogous to the channel dispersion [7][8][6] in channel coding. Speci\ufb01cally, we show that \u03b4\u03b5(n) = \u03b4\u2217\u2212 r 1 2nQ\u22121(\u03b5) + 1 2n log n + O \u00121 n \u0013 . (9) By the similarity to Eq. (1), we identify the constant 1 2 as the dispersion of in\ufb01nite constellations. This fact can be intuitively explained in several ways: \u2022 The dispersion as the (inverse of the) second derivative of the error exponent: for DMC\u2019s and for the power constrained AWGN channel, the channel dispersion is given by the inverse of the second derivative of the error exponent evaluated at the capacity [6]. Straightforward differentiation of the error exponent E(\u03b4) (which near the capacity is given by Er(\u03b4) = Esp(\u03b4)) veri\ufb01es the value of 1 2. \u2022 The unconstrained AWGN channel as the high-SNR AWGN channel: While the capacity of the power constrained AWGN channel grows without bound with the SNR, the error exponent attains a nontrivial limit. This limit is the error exponent of the unconstrained AWGN channel (as noticed in [2]), where the distance to capacity is replaced by the NLD distance to \u03b4\u2217. By this analogy, we examine the high-SNR limit of the dispersion of the AWGN channel (given in [8][6] by 1 2 (1 \u2212(1 + SNR)\u22122)) and arrive at the expected value of 1 2. D. Volume-to-Noise Ratio (VNR) Another \ufb01gure of merit for lattices (that can be de\ufb01ned for general IC\u2019s as well) is the volume-to-noise ratio (VNR), which generalizes the SNR notion [16] (see also [17]). The VNR quanti\ufb01es how good a lattice is for channel coding over the unconstrained AWGN at some given error probability \u03b5. It is known that for any \u03b5 > 0, the optimal (minimal) \f6 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY VNR of any lattice approaches 1 when the dimension n grows (see e.g. [17]). We note that the VNR and the NLD are tightly connected, and deduce equivalent \ufb01nite-dimensional and asymptotic results for the optimal VNR. The paper is organized as follows. In Section II we de\ufb01ne the notations and in Section III we review previous results. In Section IV we derive the new typicality and ML bounds for the optimal error probability of \ufb01nite dimensional IC\u2019s, and we re\ufb01ne the sphere bound as a lower bound on the average error probability for any \ufb01nite dimensional IC. In Section V the bounds are analyzed asymptotically with the dimension where the NLD is \ufb01xed, to derive asymptotic bounds that re\ufb01ne the error exponent bounds. In Section VI we \ufb01x the error probability and study the asymptotic behavior of the optimal achievable NLD with n. We use normal approximation tools to derive the dispersion theorem for the setting. In Section VII we compare the bounds from previous sections with the performance of some good known in\ufb01nite constellations. In Section VIII we discuss the VNR and its connection to the NLD \u03b4. We conclude the paper in Section IX. II. DEFINITIONS A. Notation We adopt most of the notations of Poltyrev\u2019s paper [3]: Let Cb(a) denote a hypercube in Rn Cb(a) \u225c n x \u2208Rn s.t. \u2200i|xi| < a 2 o . (10) Let Ball(r) denote a hypersphere in Rn and radius r > 0, centered at the origin Ball(r) \u225c{x \u2208Rn s.t. \u2225x\u2225< r}, (11) and let Ball(y, r) denote a hypersphere in Rn and radius r > 0, centered at y \u2208Rn Ball(y, r) \u225c{x \u2208Rn s.t. \u2225x \u2212y\u2225< r}. (12) Let S be an IC. We denote by M(S, a) the number of points in the intersection of Cb(a) and the IC S, i.e. M(S, a) \u225c|S T Cb(a) |. The density of S, denoted by \u03b3(S), or simply \u03b3, measured in points per volume unit, is de\ufb01ned by \u03b3(S) \u225clim sup a\u2192\u221e M(S, a) an . (13) The normalized log density (NLD) \u03b4 is de\ufb01ned by \u03b4 = \u03b4(S) \u225c1 n log \u03b3. (14) It will prove useful to de\ufb01ne the following: De\ufb01nition 1 (Expectation over points in a hypercube): Let Ea[f(s)] denote the expectation of an arbitrary function f(s), f : S \u2192R, where s is drawn uniformly from the code points that reside in the hypercube Cb(a): Ea[f(s)] \u225c 1 M(S, a) X s\u2208S\u2229Cb (a) f(s). (15) \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 7 Throughout the paper, an IC will be used for transmission of information through the unconstrained AWGN channel with noise variance \u03c32 (per dimension). The additive noise shall be denoted by Z = [Z1, ..., Zn]T. An instantiation of the noise vector shall be denoted by z = [z1, ..., zn]T. For s \u2208S, let Pe(s) denote the error probability when s was transmitted. When the maximum likelihood (ML) decoder is used, the error probability is given by Pe(s) = Pr{s + Z / \u2208W(s)}, (16) where W(s) is the Voronoi region of s, i.e. the convex polytope of the points that are closer to s than to any other point s\u2032 \u2208S. The maximal error probability is de\ufb01ned by P max e (S) \u225csup s\u2208S Pe(s), (17) and the average error probability is de\ufb01ned by Pe(S) \u225clim sup a\u2192\u221eEa[Pe(s)]. (18) The following related quantities, de\ufb01ne the optimal performance limits for IC\u2019s. De\ufb01nition 2 (Optimal Error Probability and Optimal NLD): \u2022 Given NLD value \u03b4 and dimension n, Pe(n, \u03b4) denotes the optimal error probability that can be obtained by any IC with NLD \u03b4 and a \ufb01nite dimension n. \u2022 Given an error probability value \u03b5 and dimension n, \u03b4\u03b5(n) denotes the maximal NLD for which there exists an IC with dimension n and error probability at most \u03b5. Clearly, these two quantities are tightly connected, and any nonasymptotic bound for either quantity gives a bound for the other. However, their asymptotic analysis (with n \u2192\u221e) is different: for \ufb01xed \u03b4 < \u03b4\u2217, it is known that Pe(n, \u03b4) vanishes exponentially with n. In this paper we will re\ufb01ne these results. For a \ufb01xed error probability \u03b5, it is known that \u03b4\u03b5(n) goes to \u03b4\u2217when n \u2192\u221e. In this paper we will show that the gap to \u03b4\u2217vanishes like O (1/\u221an), see Section VI. fn = O(gn) shall mean that there exist a constant c s.t. for all n > n0 for some n0, |fn| \u2264c \u00b7 gn. Similarly, fn \u2264O(gn) shall mean that there exist c, n0 s.t. for all n > n0, fn \u2264c\u00b7gn. fn \u2265O(gn) means \u2212fn \u2264O(\u2212gn). fn = \u0398(gn) shall mean that both fn = O(gn) and gn = O(fn) hold. B. Measuring the Gap from Capacity Suppose we are given an IC S with a given density \u03b3 (and NLD \u03b4 = 1 n log \u03b3), used for information transmission over the unconstrained AWGN with noise variance \u03c32. The gap from optimality can be quanti\ufb01ed in several ways. Knowing that the optimal NLD (for n \u2192\u221e) is \u03b4\u2217, we may consider the difference \u2206\u03b4 = \u03b4\u2217\u2212\u03b4, (19) which gives the gap to capacity in nats, where a zero gap means that we are working at capacity. An equivalent alternative would be to measure the ratio between the noise variance \f8 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY that is tolerable (in the capacity sense) with the given NLD \u03b4, given by e\u22122\u03b4 2\u03c0e , and the actual noise variance \u03c32 (equal to e\u22122\u03b4\u2217 2\u03c0e ). This ratio is given by \u00b5 \u225ce\u22122\u03b4/(2\u03c0e) \u03c32 = e2(\u03b4\u2217\u2212\u03b4). (20) For lattices, the term e\u22122\u03b4 is equal to v2/n, where v is the volume of a Voronoi cell of the lattice. Therefore \u00b5 was termed the Volume-to-Noise Ratio (VNR) by Forney et al. [16] (where it is denoted by \u03b12(\u039b, \u03c32)). The VNR can be de\ufb01ned for general IC\u2019s as well. It is generally above 1 (below capacity) and approaches 1 at capacity. It is often expressed in dB 3, i.e. 10 log10 e\u22122\u03b4/(2\u03c0e) \u03c32 = 10 log10 e2(\u03b4\u2217\u2212\u03b4) \u223c = 8.6859\u2206\u03b4. (21) Note that the VNR appears under different names and scalings in the literature. Poltyrev [3] de\ufb01ned the quantity e\u22122\u03b4 \u03c32 and called it the Generalized SNR (and also denoted it by \u00b5). In certain cases the latter de\ufb01nition is bene\ufb01cial, as it can be viewed as the dual of the normalized second moment (NSM), which at n \u2192\u221eapproaches 1 2\u03c0e [17]. An alternative way to quantify the gap from optimal performance is based on the fact that the Voronoi regions of an optimal IC (at n \u2192\u221e) becomes sphere-like. For example, the sphere bound (the converse bound) is based on a sphere with the same volume as the Voronoi cells of the IC (i.e. a sphere with radius re\ufb00). As n grows, the Voronoi regions of the optimal IC (that achieves capacity) becomes closer to a sphere with squared radius that is equal to the mean squared radius of the noise, n\u03c32. Therefore a plausible way to measure the gap from optimality would be to measure the ratio between the squared effective radius of the IC and the expected squared noise amplitude, i.e. \u03c1 \u225cr2 e\ufb00 n\u03c32 = e\u22122\u03b4V \u22122/n n n\u03c32 . (22) This quantity was called \u201cLattice SNR\u201d in [15], and \u201cVoronoi-to-Noise Effective Radius Ratio\u201d (squared) in [18]. Similarly to the VNR \u00b5, this ratio also approaches 1 at capacity, and is also often expressed in dB. However, the two measures (20) and (22) are not equivalent. For a given gap in dB, different IC densities (and NLD\u2019s) are derived, and only as n \u2192\u221e the measures coincide (this can be seen by approximating Vn, see Appendix F). In the current paper, whenever we state a gap from capacity in dB, we refer to the gap (21). In the current paper we shall be interested in the gap to capacity in the forms of (19) and (20). The \ufb01nite-dimensional results in Section IV are speci\ufb01c for each n and can be written as a function of either the NLD \u03b4 or the ratio (22). However, the asymptotic analysis in Sections V and VI depends on the selected measure. Speci\ufb01cally, in Section V we study the behavior of the error probability with n \u2192\u221ewhere \u03b4 is \ufb01xed. This is equivalent to \ufb01xing the ratio (20) (but not (22)). While the exponential behavior of the bounds on the error probability is the same whether we \ufb01x (20) or (22), the sub-exponential behavior differs. In Section VI we are interested in the behavior of the gap (19) with n \u2192\u221efor \ufb01xed error probability. Equivalent results in terms of the ratio (22) can be derived using the same tools4. 3For \u2206\u03b4 measured in bits we would get the familiar 6.02 dB/bit instead of 8.6859 dB/nat in (21). 4It is interesting to note that although we choose to stick with the gap in nats and to the ratio (20), the term (22) will pop out in the asymptotic analysis in Section V. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 9 III. PREVIOUS RESULTS A. Known Bounds on Pe(n, \u03b4) Here we review existing non-asymptotic bounds on Pe(n, \u03b4), and discuss how easy are they for evaluation and asymptotic analysis. The following non-asymptotic achievability bound can be distilled from Poltyrev\u2019s paper [3]: Theorem 1 (Poltyrev\u2019s achievability): For any r > 0, Pe(n, \u03b4) \u2264en\u03b4nVn Z 2r 0 wn\u22121 Pr{Z \u2208D(r, w)}dw + Pr{\u2225Z\u2225> r}, (23) where D(r, w) denotes the section of the sphere with radius r that is cut off by a hyperplane at a distance w 2 from the origin. In [3] it is stated that the optimal value for r (the one that minimizes the upper bound) is given by the solution to an integral equation, and it is shown that as n \u2192\u221e, the optimal r satis\ufb01es r2 n \u2192\u03c32e2(\u03b4\u2217\u2212\u03b4). However, no explicit expression for the optimal r is given, so in order to compute the bound for \ufb01nite values of n one has to numerically optimize w.r.t. r (in addition to the numerical integration). In order to derive the error exponent result, Poltyrev [3] used the asymptotic (but suboptimal) r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4. The converse bound used in [3], which will be used in the current paper as well, is based on the following simple fact: Theorem 2 (Sphere bound): Let W(s) be the Voronoi region of an IC point s, and let SW (s) denote a sphere with the same volume as W(s). Then the error probability Pe(s) is lower bounded by Pe(s) \u2265Pr{Z / \u2208SW (s)}, (24) where Z denotes the noise vector. This simple but important bound (see, e.g. [15][19]) is based on the fact that the pdf of the noise vector has spherical symmetry and decreases with the radius. An immediate corollary is the following bound for lattices (or more generally, any IC with equal-volume Voronoi cells): Pe(n, \u03b4) \u2265P SB e (n, \u03b4) \u225cPr{\u2225Z\u2225> re\ufb00} = Z \u221e re\ufb00 fR(r\u2032)dr\u2032, (25) where re\ufb00is the radius of a hypersphere with the same volume as a Voronoi cell, and fR(r) is the pdf of the norm of the noise vector, i.e. a (normalized) Chi distribution with n degrees of freedom. Note that this bound holds for any point s in the IC, therefore it holds for the average error probability Pe(n, \u03b4) (and trivially for the maximal error probability as well). In [3] the argument is extended to IC\u2019s which not necessarily obey the constant volume condition in the following manner: \ufb01rst, it is claimed that there must exist a Voronoi region with volume that is at less than the average volume \u03b3\u22121, so the bound holds for P max e (S). In order to apply the bound to the average error probability, a given IC S with average error probability \u03b5 is expurgated to get another IC S\u2032 with maximal error probability at most 2\u03b5. Applying the previous argument for the maximal error probability of S\u2032 gives a bound on the average error probability of S. The expurgation process, in addition to the factor of 2 in the error probability, also incurs a factor of 2 loss in the density \u03b3. When evaluating \f10 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY the asymptotic exponential behavior of the error probability these factors have no meaning, but if we are interested (as in the case in this paper) in the bound values for \ufb01nite n, and in the asymptotic behavior of \u03b4\u03b5(n), these factors weaken the sphere bound signi\ufb01cantly. In Section IV we show that (25) holds verbatim for any \ufb01nite dimensional IC, and for the average error probability as well. The sphere bound (25) includes a simple (but with no known closed-form solution) 1D integral and can be evaluated numerically. An alternative for the numerical integration was proposed in [15], where the integral was transformed into a sum of n/2 elements to allow the exact calculation of the bound. While the result gives an alternative to numeric integration, it does not shed any light on the asymptotic behavior of the bound with growing n. B. Known Asymptotic Bounds at Fixed \u03b4 (Error Exponent) The error exponent E(\u03b4) for the unconstrained AWGN was de\ufb01ned in (5). The nonasymptotic bounds in the previous subsection can lead to upper and lower bounds on the exponent. The asymptotic evaluation of Poltyrev\u2019s achievability bound (Theorem 1) is hard: in [3], in order to provide a lower bound on the error exponent, a suboptimal value for r is chosen for \ufb01nite n \u0000r = \u221an\u03c3e\u2212(\u03b4\u2217\u2212\u03b4)\u0001 . The resulting bound is the random coding exponent for this setting Er(\u03b4), given by Er(\u03b4) = \uf8f1 \uf8f2 \uf8f3 \u03b4\u2217\u2212\u03b4 + log e 4, \u03b4 \u2264\u03b4cr; 1 2 \u0002 e2(\u03b4\u2217\u2212\u03b4) \u22121 \u22122(\u03b4\u2217\u2212\u03b4) \u0003 , \u03b4cr \u2264\u03b4 < \u03b4\u2217; 0, \u03b4 \u2265\u03b4\u2217, (26) where \u03b4cr = 1 2 log 1 4\u03c0e\u03c32. Poltyrev also provided an expurgation-type argument to improve the error exponent at low NLD values (below \u03b4ex \u225c\u03b4\u2217\u2212log 2). This NLD region is outside the focus of the current paper. An upper bound on the error exponent is the sphere packing exponent. It is given by [3]: Esp(\u03b4) = 1 2 \u0002 e2(\u03b4\u2217\u2212\u03b4) \u22121 \u22122(\u03b4\u2217\u2212\u03b4) \u0003 , (27) which is derived from the sphere bound (see [3, Appendix C]). The upper and lower bounds on the error exponent only hint on the value of Pe(n, \u03b4): e\u2212n(Esp(\u03b4)+o(1)) \u2264Pe(n, \u03b4) \u2264e\u2212n(Er(\u03b4)+o(1)). (28) Even when the error exponent bounds coincide (above the critical NLD \u03b4cr), the optimal error probability Pe(n, \u03b4) is known only up to an unknown sub-exponential term. In Section V we present a signi\ufb01cantly tighter asymptotic analysis and show, for example, that at NLD above \u03b4cr, Pe(n, \u03b4) is known, asymptotically, up to a constant. IV. BOUNDS FOR FINITE DIMENSIONAL IC\u2019S In this section we analyze the optimal performance of \ufb01nite dimensional in\ufb01nite constellations in Gaussian noise. We describe two new achievability bounds, both based on lattices: The \ufb01rst bound is based on a simple \u2018typicality\u2019 decoder, and the second one based on the ML decoder. Both bounds result in simpler expressions than Poltyrev\u2019s bound (Theorem 1). The \ufb01rst bound is simpler to derive but proves to be weak. The second bound gives the exact value of the bound as Poltyrev\u2019s (Theorem 1), without the need for 3D integration and an additional numeric optimization, but only a single 1D integral (which can be analyzed \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 11 further see Section V). As for converse bounds, we extend the validity of the sphere bound to the most general case of IC\u2019s (not only those with equal-volume Voronoi cells) and average error probability. A. Typicality Decoder Based Bound Theorem 3: For any r > 0, Pe(n, \u03b4) \u2264P TB e \u225cen\u03b4Vnrn + Pr {\u2225Z\u2225> r} , (29) and the optimal value for r is given by r\u2217= \u03c3 p n(1 + 2\u03b4\u2217\u22122\u03b4). (30) Proof: Let \u039b be a lattice that is used as an IC for transmission over the unconstrained AWGN. We consider a suboptimal decoder, and therefore the performance of the optimal ML decoder can only be better. The decoder, called a typicality decoder, shall operate as follows. Suppose that \u03bb \u2208\u039b is sent, and the point y = \u03bb + z is received, where z is the additive noise. Let r be a parameter for the decoder, which will be determined later on. If there is only a single point in the ball Ball(y, r), then this will be the decoded word. If there are no codewords in the ball, or more than one codeword in the ball, an error is declared (one of the code points is chosen at random). Lemma 1: The average error probability of a lattice \u039b (with the typicality decoder) is bounded by Pe(\u039b) \u2264Pr {Z / \u2208Ball(r)} + X \u03bb\u2208\u039b\\{0} Pr {Z \u2208Ball(\u03bb, r) \u2229Ball(r)} , (31) where Z denotes the noise vector. Proof: Since \u039b is a lattice we can assume without loss of generality that the zero point was sent. We divide the error events to two cases. First, if the noise falls outside the ball of radius r (centered at the origin), then there surely will be erroneous decoding since the transmitted (0) point is outside the ball. The remaining error cases are where the noise Z is within Ball(r), and the noise falls in the typical ball of some other lattice point (that is different than the transmitted zero point). We therefore get Pe(\u039b) \u2264Pr {Z / \u2208Ball(r)} + Pr \uf8f1 \uf8f2 \uf8f3Z \u2208Ball(r) \\ \uf8ee \uf8f0 [ \u03bb\u2208\u039b\\{0} Ball(\u03bb, r) \uf8f9 \uf8fb \uf8fc \uf8fd \uf8fe = Pr {Z / \u2208Ball(r)} + Pr \uf8f1 \uf8f2 \uf8f3Z \u2208 [ \u03bb\u2208\u039b\\{0} Ball(\u03bb, r) \u2229Ball(r) \uf8fc \uf8fd \uf8fe \u2264Pr {Z / \u2208Ball(r)} + X \u03bb\u2208\u039b\\{0} Pr {Z \u2208Ball(\u03bb, r) \u2229Ball(r)} , (32) where the last inequality follows from the union bound. We use the Minkowski-Hlawka theorem [14][13]: 5 5 The MH theorem is usually written as (33) with an \u01eb added to the RHS that is arbitrarily small (e.g. [13, Lemma 3, p. 65], and [14, Theorem 1, p. 200]). The version (33) follows from a slightly improved version of the theorem due to Siegel, often called the Minkowski-Hlawka-Siegel (MHS) theorem, see [14, Theorem 5, p. 205]. \f12 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Theorem 4 (MH): Let f : Rn \u2192R+ be a nonnegative integrable function with bounded support. Then for every \u03b3 > 0, there exist a lattice \u039b with det \u039b = \u03b3\u22121 that satis\ufb01es X \u03bb\u2208\u039b\\{0} f(\u03bb) \u2264\u03b3 Z Rn f(\u03bb)d\u03bb. (33) Since Pr {Z \u2208Ball(\u03bb, r) \u2229Ball(r)} = 0 for any \u03bb s.t. \u2225\u03bb\u2225> 2r we may apply the MH theorem to the sum in (32). We deduce that for any \u03b3 > 0, there must exist a lattice \u039b with density \u03b3, s.t. X \u03bb\u2208\u039b\\{0} Pr {Z \u2208Ball(\u03bb, r) \u2229Ball(r)} \u2264\u03b3 Z Rn Pr {Z \u2208Ball(\u03bb, r) \u2229Ball(r)} d\u03bb. (34) We further examine the resulting integral: Z Rn Pr {Z \u2208Ball(\u03bb, r) \u2229Ball(r)} d\u03bb = Z Rn Z Ball (\u03bb,r)\u2229Ball (r) fZ(z)dzd\u03bb \u2264 Z Rn Z Ball (\u03bb,r) fZ(z)dzd\u03bb = Z Rn Z Ball (r) fZ(z\u2032 + \u03bb)dz\u2032d\u03bb = Z Ball (r) 1dz\u2032 = Vnrn. (35) Combined with (31) we get that there exist a lattice \u039b with density \u03b3, for which Pe(\u039b) \u2264\u03b3Vnrn + Pr {\u2225Z\u2225> r} , (36) where r > 0 and \u03b3 = en\u03b4 can be chosen arbitrarily. The optimal value for r follows from straightforward optimization of the RHS of (36): we \ufb01rst write Pr {\u2225Z\u2225> r} = Pr ( 1 \u03c32 n X i=1 Z2 i > r2 \u03c32 ) . We note that the sum 1 \u03c32 Pn i=1 Z2 i is a sum of n i.i.d. standard Gaussian RV\u2019s, which is exactly a \u03c72 random variable with n degrees of freedom. The pdf of this RV is well known, and given by f\u03c72 n(x) = 2\u2212n/2 \u0393(n/2)xn/2\u22121e\u2212x/2, where \u0393(\u00b7) is the Gamma function. Equipped with this, the RHS of (36) becomes en\u03b4Vnrn + Z \u221e r2 \u03c32 2\u2212n/2 \u0393(n/2)xn/2\u22121e\u2212x/2. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 13 Differentiating w.r.t. r and equating to zero gives nen\u03b4Vnrn\u22121 \u22122r \u03c32 2\u2212n/2 \u0393(n/2)(r2/\u03c32)n/2\u22121e\u2212r2 2\u03c32 = 0. We plug in the expression for Vn = \u03c0n/2 n 2 \u0393(n/2) and get nen\u03b4 \u03c0n/2 n 2\u0393(n/2)rn\u22121 \u22122r \u03c32 2\u2212n/2 \u0393(n/2)(r2/\u03c32)n/2\u22121e\u2212r2 2\u03c32 = 0, which simpli\ufb01es to the required r = \u03c3 p n(1 + 2\u03b4\u2217\u22122\u03b4). B. ML Decoder Based Bound The second achievability bound is based on the ML decoder (using a different technique than Poltyrev [3]): Theorem 5: For any r > 0 and dimension n, there exist a lattice \u039b with error probability Pe(n, \u03b4) \u2264P MLB e (n, \u03b4) \u225cen\u03b4Vn Z r 0 fR(\u02dc r)\u02dc rnd\u02dc r + Pr {\u2225Z\u2225> r} , (37) and the optimal value for r is given by r\u2217= re\ufb00= e\u2212\u03b4V \u22121/n n . (38) Before the proof, note that this speci\ufb01c value for r gives a new interpretation to the bound: the term Pr {\u2225Z\u2225> r} is exactly the sphere bound (24), and the other term can be thought of as a \u2018redundancy\u2019 term. Making this value small results in tightening of the gap between the bounds. Proof: Suppose that the zero lattice point was sent, and the noise vector is z \u2208Rn. An error event occurs (for a ML decoder) when there is a nonzero lattice point \u03bb \u2208\u039b whose Euclidean distance to z is less than the distance between the zero point and noise vector. We denote by E the error event, condition on the radius R of the noise vector and get Pe(\u039b) = Pr{E} = = ER [Pr {E | \u2225Z\u2225= R}] = Z \u221e 0 fR(r) Pr {E | \u2225Z\u2225= r} dr \u2264 Z r\u2217 0 fR(r) Pr {E | \u2225Z\u2225= r} dr + Pr{\u2225Z\u2225> r\u2217}, (39) where the last inequality follows by upper bounding the probability by 1. It holds for any r\u2217> 0. We examine the conditional error probability Pr {E | \u2225z\u2225= r}: Pr {E | \u2225Z\u2225= r} = Pr \uf8f1 \uf8f2 \uf8f3 [ \u03bb\u2208\u039b\\{0} \u2225Z \u2212\u03bb\u2225\u2264\u2225Z\u2225 \f \f \f \f \f \f \u2225Z\u2225= r \uf8fc \uf8fd \uf8fe \u2264 X \u03bb\u2208\u039b\\{0} Pr {\u2225Z \u2212\u03bb\u2225\u2264\u2225Z\u2225| \u2225Z\u2225= r} = X \u03bb\u2208\u039b\\{0} Pr {\u03bb \u2208Ball(Z, \u2225Z\u2225) | \u2225Z\u2225= r} , (40) \f14 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY where the inequality follows from the union bound. Plugging into the left term in (39) gives Z r\u2217 0 fR(r) X \u03bb\u2208\u039b\\{0} Pr {\u03bb \u2208Ball(Z, \u2225Z\u2225) | \u2225Z\u2225= r} dr = X \u03bb\u2208\u039b\\{0} Z r\u2217 0 fR(r) Pr {\u03bb \u2208Ball(Z, \u2225Z\u2225) | \u2225Z\u2225= r} dr. (41) Note that the last integral has a bounded support (w.r.t. \u03bb) it is always zero if \u2225\u03bb\u2225\u22652r\u2217. Therefore we can apply the Minkowski-Hlawka theorem as in Theorem 3 and get that for any \u03b3 > 0 there exists a lattice \u039b with density \u03b3, whose error probability is upper bounded by Pe(\u039b) \u2264\u03b3 Z \u03bb\u2208Rn Z r\u2217 0 fR(r) Pr {\u03bb \u2208Ball(Z, \u2225Z\u2225) | \u2225Z\u2225= r} drd\u03bb + Pr{\u2225Z\u2225> r\u2217}. We continue with Z \u03bb\u2208Rn Z r\u2217 0 fR(r) Pr {\u03bb \u2208Ball(Z, \u2225Z\u2225) | \u2225Z\u2225= r} drd\u03bb = Z r\u2217 0 fR(r) Z \u03bb\u2208Rn Pr {\u03bb \u2208Ball(Z, \u2225Z\u2225) | \u2225Z\u2225= r} d\u03bbdr = Z r\u2217 0 fR(r) Z \u03bb\u2208Rn E \u0002 1{\u03bb\u2208Ball (Z,\u2225Z\u2225)} \f \f \u2225Z\u2225= r \u0003 d\u03bbdr = Z r\u2217 0 fR(r)E \u0014Z \u03bb\u2208Rn 1{\u03bb\u2208Ball (Z,\u2225Z\u2225)}d\u03bb \f \f \f \f \u2225Z\u2225= r \u0015 dr = Z r\u2217 0 fR(r)E [\u2225Z\u2225nVn| \u2225Z\u2225= r] dr = Vn Z r\u2217 0 fR(r)rndr, and we obtain (37). To \ufb01nd the optimal value for r (the one that minimizes the RHS of (37)), we see that: Pr {\u2225Z\u2225> r} = Z \u221e r fR(\u02dc r)d\u02dc r. (42) Differentiating the RHS of (37) w.r.t. r in order to \ufb01nd the minimum gives en\u03b4VnfR(r)rn \u2212fR(r) = 0, (43) and r\u2217= re\ufb00= e\u2212\u03b4V \u22121/n n immediately follows. C. Equivalence of the ML bound with Poltyrev\u2019s bound In Theorems 3 and 5 we provided a new upper bounds on the error probability that were simpler than Poltyrev\u2019s original bound (Theorem 1). For example, in order to compute Poltyrev\u2019s bound, one has to apply 3D numerical integration, and numerically optimize w.r.t. r. In contrast, both new bounds requires only a single integration, and the optimal value for r has a closed-form expression so no numerical optimization is required. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 15 It appears that the simplicity of the bound in Theorem 5 does not come at a price of a weaker bound. In fact, it proves to be equivalent to Poltyrev\u2019s bound: Theorem 6: Poltyrev\u2019s bound (Theorem 1) for the error probability, for the optimal value of r, is equal to the ML bound from Theorem 5: min r>0 \u001a en\u03b4nVn Z 2r 0 wn\u22121 Pr{Z \u2208D(r, w)}dw + Pr{\u2225Z\u2225> r} \u001b =en\u03b4Vn Z r\u2217 0 fR(\u03c1)\u03c1nd\u03c1 + Pr {\u2225Z\u2225> r\u2217} , (44) where r\u2217= re\ufb00= e\u2212\u03b4V \u22121/n n . In fact, we can strengthen (44) and show that \u03b3nVn Z 2r 0 wn\u22121 Pr{Z \u2208D(r, w)}dw = \u03b3Vn Z r 0 fR(\u03c1)\u03c1nd\u03c1 (45) for any r > 0. Proof: Appendix A. Note that proving (45) shows that both bounds are equivalent, regardless of the value of r. Consequently, the optimal value for r in Poltyrev\u2019s bound is also found. In [3] the optimal value (denoted there d \u2217 c(n, \u03b4)) was given as the solution to an integral equation, and was only evaluated asymptotically. D. The Sphere Bound for Finite Dimensional In\ufb01nite Constellations The sphere bound (25) applies to in\ufb01nite constellations with \ufb01xed Voronoi cell volume. Poltyrev [3] extended it to general IC\u2019s with the aid of an expurgation process, without harming the tightness of the error exponent bound. When the dimension n is \ufb01nite, the expurgation process incurs a non-negligible loss (a factor of 2 in the error probability and in the density). In this section we show that the sphere bound applies without any loss to general \ufb01nite dimensional IC\u2019s and average error probability. We \ufb01rst concentrate on IC\u2019s with some mild regularity assumptions: De\ufb01nition 3 (Regular IC\u2019s): An IC S is called regular, if: 1) There exists a radius r0 > 0, s.t. for all s \u2208S, the Voronoi cell W(s) is contained in Ball(s, r0). 2) The density \u03b3(S) is given by lima\u2192\u221e M(S,a) an (rather than lim sup in the original de\ufb01nition). For s \u2208S, we denote by v(s) the volume of the Voronoi cell of s, |W(s)|. De\ufb01nition 4 (Average Voronoi cell volume): For a regular IC S, the average Voronoi cell volume is de\ufb01ned by v(S) \u225clim sup a\u2192\u221e Ea[v(s)]. (46) Lemma 2: For a regular IC S, the average volume is given by the inverse of the density: \u03b3(S) = 1 v(S). (47) Proof: Appendix B. \f16 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY For brevity, let SPB(v) denote the probability that the noise vector Z leaves a sphere of volume v. With this notation, the sphere bound reads Pe(s) \u2265SPB(v(s)), (48) and holds for any individual point s \u2208S. We also note the following: Lemma 3: The equivalent sphere bound SPB(v) is convex in the Voronoi cell volume v. Proof: Appendix C. We now show that the above equation holds for the average volume and error probability as well. Theorem 7: Let S be a regular (\ufb01nite dimensional) IC with NLD \u03b4, and let v(S) be the average Voronoi cell volume of S (so the density of S is \u03b3 = v(S)\u22121). Then the average error probability of S is lower bounded by Pe(S) \u2265SPB(v(S)) = SPB(\u03b3\u22121) = P SB e (n, \u03b4). (49) Proof: We start with the de\ufb01nition of the average error probability and get Pe(S) = lim sup a\u2192\u221eEa[Pe(s)] (a) \u2265lim sup a\u2192\u221e Ea[SPB(v(s))] (b) \u2265lim sup a\u2192\u221eSPB(Ea[v(s)]) (c) = SPB(lim sup a\u2192\u221eEa[v(s)]) = SPB(v(S)). (50) (a) follows from the sphere bound for each individual point s \u2208S, (b) follows from the Jensen inequality and the convexity of SPB(\u00b7) (Lemma 3), and (c) follows from the fact that SPB(\u00b7) is continuous. As a consequence, we get that the sphere bound holds for regular IC\u2019s as well, without the need for expurgation (as in [3]). So far the discussion was constrained to regular IC\u2019s only. This excludes constellations with in\ufb01nite Voronoi regions (e.g. contains points only in half of the space), and also constellations in which the density oscillates with the cube size a (and the formal limit \u03b3 does not exist). We now extend the proof of the converse for any IC, without the regularity assumptions. The proof is based on the following regularization process: Lemma 4 (Regularization): Let S be an IC with density \u03b3 and average error probability Pe(S) = \u03b5. Then for any \u03be > 0 there exists a regular IC S\u2032 with density \u03b3\u2032 \u2265\u03b3/(1 + \u03be), and average error probability Pe(S\u2032) = \u03b5\u2032 \u2264\u03b5(1 + \u03be). Proof: Appendix D. Theorem 8 (Sphere Bound for Finite Dimensional IC\u2019s): Let S be a \ufb01nite dimensional IC with density \u03b3. Then the average error probability of S is lower bounded by Pe(S) \u2265SPB(\u03b3\u22121) = P SB e (n, \u03b4) (51) Proof: Let \u03be > 0. By the regularization lemma (Lemma 4) there exists a regular IC S\u2032 with \u03b3\u2032 \u2265\u03b3/(1 + \u03be), and Pe(S\u2032) \u2264Pe(S)(1 + \u03be). We apply Theorem 7 to S\u2032 and get that Pe(S)(1 + \u03be) \u2265Pe(S\u2032) \u2265SPB(\u03b3\u2032\u22121) \u2265SPB((1 + \u03be)\u03b3\u22121), (52) \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 17 0 200 400 600 800 1000 10-4 0.001 0.01 0.1 1 Dimension n Error probability Sfrag replacements P SB e P TB e P MLB e P MLB e with r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4 Fig. 2. Numerical evaluation of the bounds for \u03b4 = \u22121.5nat with \u03c32 = 1 (0.704db from capacity). From bottom to top: Solid the sphere bound (Theorem 2). Gray the ML bound (Theorem 5). Dashed Poltyrev\u2019s bound (Theorem 1). Dot-dashed the typicality-based achievability bound (Theorem 3). or Pe(S) \u2265 1 1 + \u03beSPB((1 + \u03be)\u03b3\u22121), (53) for all \u03be > 0. Since SPB(\u00b7) is continuous, we may take the limit \u03be \u21920 and get to (51). E. Numerical Comparison Here we numerically compare the bounds in this section with Poltyrev\u2019s achievability bound (Theorem 1). As shown in the previous subsection, the bounds in Theorems 1 and 5 are equivalent. However, as discussed following the statement of Theorem 1 above, in [3] the suboptimal value for r is used. We therefore refer to the achievability bound in Theorem 1 (or Theorem 5) with r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4 as \u2018Poltyrev\u2019s bound\u2019. The results are shown in Figures 2 and 3. The exponential behavior of the bounds (the asymptotic slope of the curves in the log-scale graph) is clearly seen in the \ufb01gures: at NLD above \u03b4cr, the sphere bound and the ML and Poltyrev\u2019s achievability bounds have the same exponent, while for NLD below \u03b4cr the exponent of the sphere bound is better. In both cases the typicality bound has a weaker exponent. These observations are corroborated analytically in Section V below. V. ANALYSIS AND ASYMPTOTICS AT FIXED NLD \u03b4 In this section we analyze the bounds presented in the previous section with two goals in mind: \f18 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 5 10 15 20 25 30 35 40 10-9 10-7 10-5 0.001 0.1 Dimension n Error probability g replacements P SB e P TB e P MLB e P MLB e with r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4 Fig. 3. Numerical evaluation of the bounds for \u03b4 = \u22122nat with \u03c32 = 1 (5.05db from capacity). From bottom to top: Solid the sphere bound (Theorem 2). Gray the ML-based achievability bound (Theorem 5). Dashed Poltyrev\u2019s bound (Theorem 1). Dot-dashed the typicality-based achievability bound (Theorem 3). 1) To derive tight analytical bounds (that require no integration) that allow easy evaluation of the bounds, both upper and lower. 2) To analyze the bounds asymptotically (for \ufb01xed \u03b4) and re\ufb01ne the error exponent results for the setting. In V-A we present the re\ufb01ned analysis of the sphere bound. While the sphere bound P SB e will present the same asymptotic form for any \u03b4, the ML bound P MLB e has a different behavior above and below \u03b4cr. In V-B we focus on the ML bound above \u03b4cr. The tight results from V-A and V-B reveal that (above \u03b4cr) the optimal error probability Pe(n, \u03b4) is known asymptotically up to a constant. This is discussed in V-C. In V-D we focus on the ML bound below \u03b4cr, and in V-E we consider the special case of \u03b4 = \u03b4cr. In V-F we study the asymptotics of the typicality bound P T e (n, \u03b4) and in V-G we analyze Poltyrev\u2019s bound, i.e. the ML bound with r set to r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4 instead of re\ufb00. The fact that the ML bound behaves differently above and below \u03b4cr can be explained by the following. Consider the \ufb01rst term in the ML bound, en\u03b4Vn R re\ufb00 0 fR(r)rndr. Loosely speaking, the value of this integral is determined (for large n) by the value of the integrand with the most dominant exponent. When \u03b4 > \u03b4cr, the dominating value for the integral is at r = re\ufb00. For \u03b4 < \u03b4cr, the dominating value is approximately at r = \u221a 2n\u03c32. Note that this value does not depend on \u03b4, so the dependence in \u03b4 comes from the term en\u03b4 alone, and the exponential behavior of the bound is of a straight line. Since we are interested in more than merely the exponential behavior of the bound, we use more re\ufb01ned machinery \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 19 in order to analyze the bounds. Poltyrev [3] used an expurgation technique in order to improve the error exponent for lower NLD values (below \u03b4ex = \u03b4\u2217\u2212log 2). The re\ufb01ned tools used here can also be applied to the expurgation bound in order to analyze its sub-exponential behavior. However, in this region the ratio between the upper and lower bounds grows exponentially, and therefore the sub-exponential analysis of the expurgation bound is of little interest and is not included in this paper. A. Analysis of the Sphere Bound The sphere bound (Theorem 2) is a simple bound based on the geometry of the coding problem. However, the resulting expression, given by an integral that has no elementary form, is generally hard to evaluate. There are several approaches for evaluating this bound: \u2022 Numeric integration is only possible for small moderate values of n. Moreover, the numeric evaluation does not provide any hints about the asymptotical behavior of the bound. \u2022 Tarokh et al. [15] were able to represent the integral in the bound as a sum of n/2 elements. This result indeed helps in numerically evaluating the bound, but does not help in understanding its asymptotics. \u2022 Poltyrev [3] used large-deviation techniques to derive the sphere packing error exponent, i.e. lim n\u2192\u221e\u22121 n log Pe(n, \u03b4) \u2264Esp(\u03b4) = 1 2 \u0002 e2(\u03b4\u2217\u2212\u03b4) \u22121 \u22122(\u03b4\u2217\u2212\u03b4) \u0003 . (54) The error exponent, as its name suggests, only hints on the exponential behavior of the bound, but does not aid in evaluating the bound itself or in more precise asymptotics. Here we derive non-asymptotic, analytical bounds based on the sphere bound. These bounds allow easy evaluation of the bound, and give rise to more precise asymptotic analysis for the error probability (where \u03b4 is \ufb01xed). Theorem 9: Let r\u2217\u225cre\ufb00= e\u2212\u03b4V \u22121/n n , \u03c1\u2217\u225c r2 e\ufb00 n\u03c32 and \u03a5 \u225c n(\u03c1\u2217\u22121+ 2 n) \u221a 2(n\u22122) . Then for any NLD \u03b4 < \u03b4\u2217and for any dimension n > 2, the sphere bound P SB e (n, \u03b4) is lower bounded by P SB e (n, \u03b4) \u2265en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n 2 \u03c1\u2217\u00b7 e \u03a52 2 r n2\u03c0 n \u22122Q (\u03a5) (55) \u2265en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n 2 \u03c1\u2217 \u03c1\u2217\u22121 + 2 n \u0012 1 1 + \u03a5\u22122 \u0013 , (56) upper bounded by P SB e (n, \u03b4) \u2264en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n 2 \u03c1\u2217 \u03c1\u2217\u22121 + 2 n , (57) and for \ufb01xed \u03b4, given asymptotically by P SB e (n, \u03b4) = e\u2212nEsp(\u03b4)(n\u03c0)\u22121 2 e2(\u03b4\u2217\u2212\u03b4) e2(\u03b4\u2217\u2212\u03b4) \u22121 \u0012 1 + O \u0012log2 n n \u0013\u0013 . (58) Some notes regarding the above results: \f20 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY \u2022 Eq. (55) provides a lower bound in terms of the Q function, and (56) gives a slightly looser bound, but is based on elementary functions only. \u2022 The upper bound (57) on the sphere bound has no direct meaning in terms of bounding the error probability Pe(n, \u03b4) (since the sphere bound is a lower bound). However, it used for evaluating the sphere bound itself (i.e. to derive (58)), and it will prove useful in upper bounding Pe(n, \u03b4) in Theorem 10 below. \u2022 A bound of the type (57), i.e. an upper bound on the probability that the noise leaves a sphere, can be derived using the Chernoff bound as was done by Poltyrev [3, Appendix B]. However, while Poltyrev\u2019s technique indeed gives the correct exponential behavior, it falls short of attaining the sub-exponential terms, and therefore (57) is tighter. Moreover, (57) leads to the exact precise asymptotics (58). \u2022 (58) gives an asymptotic bound that is signi\ufb01cantly tighter than the error exponent term alone. The asymptotic form (58) applies to (55), (56) and (57) as well. \u2022 Note that \u03c1\u2217is a measure that can also quantify the gap from capacity (see II-B). It is an alternative to \u2206\u03b4 = \u03b4\u2217\u2212\u03b4 (or to \u00b5 = e2\u2206\u03b4). The measures are not equivalent, but as n \u2192\u221ewe have \u03c1\u2217= e2(\u03b4\u2217\u2212\u03b4) + o(1), see (65) and (66) below. Proof: We write the sphere bound explicitly: P SB e (n, \u03b4) = Pr{\u2225Z\u2225> r\u2217} = Z \u221e r\u2217fR(r\u2032)dr\u2032 = Z \u221e r\u22172/\u03c32 f\u03c72 n(\u03c1)d\u03c1 = 2\u2212n 2 \u0393 \u0000n 2 \u0001 Z \u221e r\u22172/\u03c32 \u03c1 n 2 \u22121e\u2212\u03c1/2d\u03c1 = 2\u2212n 2 nn/2 \u0393 \u0000n 2 \u0001 Z \u221e \u03c1\u2217\u03c1 n 2 \u22121e\u2212n\u03c1/2d\u03c1. (59) In order to evaluate the integral in (59) we require the following lemma. Lemma 5: Let n > 2 and x > 1\u22122 n. Then the integral R \u221e x \u03c1 n 2 \u22121e\u2212n\u03c1/2d\u03c1 can be bounded from above by Z \u221e x \u03c1 n 2 \u22121e\u2212n\u03c1/2d\u03c1 \u2264 2x n 2 e\u2212nx 2 n(x \u22121 + 2 n) (60) and from below by Z \u221e x \u03c1 n 2 \u22121e\u2212n\u03c1/2d\u03c1 \u22652x n 2 e\u2212nx 2 exp \u0014\u03a52 2 \u0015 r \u03c0 n \u22122Q (\u03a5) (61) \u2265 2x n 2 e\u2212nx 2 n(x \u22121 + 2 n) \u0012 1 1 + \u03a5\u22122 \u0013 , (62) \u2265 2x n 2 e\u2212nx 2 n(x \u22121 + 2 n) \u0012 1 \u22121 \u03a52 \u0013 , (63) where \u03a5 \u225c n(x\u22121+ 2 n) \u221a 2(n\u22122) . \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 21 Proof: Appendix E. We continue the proof of the theorem: (55) follows by plugging (61) into (59) with x = \u03c1\u2217. It can be shown that \u03c1\u2217\u22651 for all \u03b4 < \u03b4\u2217so the condition x > 1 \u22122 n is met. (56) follows similarly using (62) and the de\ufb01nition of \u03b4\u2217. The upper bound (57) follows using (60). To derive (58) we \ufb01rst note the following asymptotic results: Vn = \u03c0n/2 n 2\u0393( n 2) = \u00122\u03c0e n \u0013n/2 1 \u221an\u03c0 \u0012 1 + O \u00121 n \u0013\u0013 , (64) \u03c1\u2217= e\u22122\u03b4V \u22122/n n n\u03c32 = e2(\u03b4\u2217\u2212\u03b4)(n\u03c0)1/n \u0012 1 + O \u0012 1 n2 \u0013\u0013 (65) = e2(\u03b4\u2217\u2212\u03b4) \u0012 1 + 1 n log(n\u03c0) + O \u0012log2 n n2 \u0013\u0013 , (66) \u03a5 = n(\u03c1\u2217\u22121 + 2 n) p 2(n \u22122) = rn 2 \u0000e2(\u03b4\u2217\u2212\u03b4) \u22121 \u0001 \u0012 1 + O \u0012log n n \u0013\u0013 = \u0398(\u221an). (67) For (64) see Appendix F. (65) follows from (64) and the de\ufb01nition of \u03b4\u2217. (66) follows by writing (n\u03c0)1/n = e 1 n log(n\u03c0) and the Taylor approximation. (67) follows directly from (66). We evaluate the term e\u2212n 2 \u03c1\u2217in (56) and (57): e\u2212n 2 \u03c1\u2217= exp \u0014 \u2212n 2 e2(\u03b4\u2217\u2212\u03b4) \u0012 1 + 1 n log(n\u03c0) + O \u0012log2 n n2 \u0013\u0013\u0015 = e\u2212n 2 e2(\u03b4\u2217\u2212\u03b4) exp \u0014 \u22121 2e2(\u03b4\u2217\u2212\u03b4) log(n\u03c0) + O \u0012log2 n n \u0013\u0015 = e\u2212n 2 e2(\u03b4\u2217\u2212\u03b4)(n\u03c0)\u22121 2e2(\u03b4\u2217\u2212\u03b4) \u0012 1 + O \u0012log2 n n \u0013\u0013 . (68) Plugging (66), (67) and (68) into (56) and (57), along with the de\ufb01nition of Esp(\u03b4), leads to the desired (58). In Fig. 4 we demonstrate the tightness of the bounds and precise asymptotics of Theorem 9. In the \ufb01gure the sphere bound is presented with its bounds and approximations. The lower bound (55) is the tightest lower bound (but is based on the non-analytic Q function). The analytic lower bound (56) is slightly looser than (55), but is tight enough in order to derive the precise asymptotic form (58). The upper bound (57) of the sphere bound is also tight. The error exponent itself (without the sub-exponential terms) is clearly way off, compared to the precise asymptotic form (58). B. Analysis of the ML Bound Above \u03b4cr In order to derive the random coding exponent Er(\u03b4), Poltyrev\u2019s achievability bound (Theorem 1) was evaluated asymptotically by setting a suboptimal value \u221an\u03c3e\u2212(\u03b4\u2217\u2212\u03b4) for the parameter r. While setting this value still gives the correct exponential behavior of the bound, a more precise analysis (in the current and following subsections) using the optimal value for r as in Theorem 5 gives tighter analytical and asymptotic results. \f22 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 100 200 300 400 500 0.001 0.002 0.005 0.010 0.020 0.050 0.100 0.200 Dimension n Error probability g replacements Sphere bound P SB e Upper bound (57) Lower bound (56) Lower bound (55) Asymptotic form (58) e\u2212nEsp(\u03b4) Fig. 4. Numerical evaluation of the sphere bound and its bounds and approximation in Theorem 9 vs the dimension n. Here \u03b4 = \u22121.5nat and \u03c32 = 1 (0.704db from capacity). The tight bounds (55), (56) and (57) lead to the asymptotic form (58). The error exponent term alone is evidently way off compared to (58). Theorem 10: Let r\u2217\u225cre\ufb00= e\u2212\u03b4V \u22121/n n , \u03c1\u2217\u225c r2 e\ufb00 n\u03c32, \u03a5 \u225c n(\u03c1\u2217\u22121+ 2 n ) \u221a 2(n\u22122) and \u03a8 \u225c \u221an \u0010 2\u2212\u03c1\u2217+ 2 n \u0011 2\u221a\u03c1\u2217 . Then for any NLD \u03b4 and for any dimension n > 2 where 1 \u22122 n < \u03c1\u2217< 2 \u22122 n, the ML bound P MLB e (n, \u03b4) is upper bounded by P MLB e (n, \u03b4) \u2264 en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n 2 \u03c1\u2217 \u00002 \u2212\u03c1\u2217\u22122 n \u0001 \u0000\u03c1\u2217\u22121 + 2 n \u0001, (69) lower bounded by P MLB e (n, \u03b4) \u2265en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n\u03c1\u2217/2 \" e\u03a82/2 r n\u03c0 2\u03c1\u2217Q(\u03a8) + e\u03a52/2 r n2\u03c0 n \u22122Q (\u03a5) # (70) \u2265en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n\u03c1\u2217/2 \u0014 1 2 \u2212\u03c1\u2217+ 2 n \u00b7 1 1 + \u03a8\u22122 + 1 \u03c1\u2217\u22121 + 2 n \u00b7 1 1 + \u03a5\u22122 \u0015 , (71) and for \u03b4cr < \u03b4 < \u03b4\u2217, given asymptotically by P MLB e (n, \u03b4) = e\u2212nEr(\u03b4)(n\u03c0)\u22121 2e2(\u03b4\u2217\u2212\u03b4) (2 \u2212e2(\u03b4\u2217\u2212\u03b4)) (e2(\u03b4\u2217\u2212\u03b4) \u22121) \u0012 1 + O \u0012log2 n n \u0013\u0013 . (72) Some notes regarding the above results: \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 23 \u2022 For large n, the condition \u03c1\u2217< 2 \u22122 n translates to the fact that \u03b4cr < \u03b4. \u03c1\u2217> 1 \u22122 n holds for all \u03b4 < \u03b4\u2217. The case of \u03b4 \u2264\u03b4\u2217is addressed later on in the current section. \u2022 The lower bounds (70) and (71) have no direct meaning in terms of bounding the error probability Pe(n, \u03b4) (since they lower bound an upper bound). However, they are useful for evaluating the achievability bound itself (i.e. to derive (72)). \u2022 (72) gives an asymptotic bound that is signi\ufb01cantly tighter than the error exponent term alone. It holds above the \u03b4cr only, where below \u03b4cr and exactly at \u03b4cr we have Theorems 11 and 12 below. The asymptotic form (72) applies to (69), (70) and (71) as well. Proof: The proof relies on a precise analysis of the ML bound: en\u03b4Vn Z r\u2217 0 fR(r)rndr + Pr {\u2225Z\u2225> r\u2217} . (73) The second term is exactly the sphere bound, which allows the usage of the analysis of Theorem 9. We therefore proceed with analyzing the \ufb01rst term: en\u03b4Vn Z r\u2217 0 fR(r)rndr = en\u03b4Vn\u03c3n Z r\u2217 \u03c3 0 f\u03c7n(y)\u03c1ndy = en\u03b4Vn\u03c3n 21\u2212n 2 \u0393 \u0002n 2 \u0003 Z r\u2217 \u03c3 0 e\u2212y2/2y2n\u22121dy = en\u03b4Vn\u03c3n 2\u2212n 2 \u0393 \u0002n 2 \u0003 Z r\u22172 \u03c32 0 e\u2212t/2tn\u22121dt = n 2en(\u03b4+\u03b4\u2217)V 2 n en/2\u03c32nnn Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 (74) We need the following lemma: Lemma 6: Let 0 < x < 2 \u22122 n. Then the integral R x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 is upper bounded by Z x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 \u2264 2xne\u2212nx/2 n \u00002 \u2212x \u22122 n \u0001 \u0012 1 \u2212e \u2212n \u0010 1\u22121 n \u2212x 2 \u0011\u0013 , (75) and is lower bounded by Z x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 \u2265xne\u2212nx/2e\u03a82/2 r 2\u03c0 nxQ(\u03a8) (76) \u2265 2xne\u2212nx/2 n \u00002 \u2212x + 2 n \u0001 \u00b7 1 1 + \u03a8\u22122, (77) where \u03a8 \u225c \u221an \u0010 2\u2212x+ 2 n \u0011 2\u221ax . Proof: Appendix E. \f24 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY To prove the upper bound (69) we use (75) with x = \u03c1\u2217to bound (74): en\u03b4Vn Z r\u2217 0 fR(r)rndr = n 2en(\u03b4+\u03b4\u2217)V 2 n en/2\u03c32nnn Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 (78) \u2264n 2en(\u03b4+\u03b4\u2217)V 2 n en/2\u03c32nnn \u00b7 2\u03c1\u2217ne\u2212n 2 \u03c1\u2217 n \u00002 \u2212\u03c1\u2217\u22122 n \u0001 (79) = en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n 2 \u03c1\u2217 2 \u2212\u03c1\u2217\u22122 n . (80) We combine the above with the upper bound (57) on the sphere bound and get en\u03b4Vn Z r\u2217 0 fR(r)rndr + Pr {\u2225Z\u2225> r\u2217} \u2264en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n 2 \u03c1\u2217 2 \u2212\u03c1\u2217\u22122 n + en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n 2 \u03c1\u2217 \u03c1\u2217\u22121 + 2 n , (81) which immediately leads to (69). In order to attain the lower bound (70) we use (76) with x = \u03c1\u2217and get that en\u03b4Vn R r\u2217 0 fR(r)rndr is lower bounded by n 2en(\u03b4+\u03b4\u2217)V 2 n en/2\u03c32nnn \u00b7 \u03c1\u2217ne\u2212n\u03c1\u2217/2e\u03a82/2 r 2\u03c0 n\u03c1\u2217Q(\u03a8) = en(\u03b4\u2217\u2212\u03b4)en/2e\u2212n\u03c1\u2217/2 \u00b7 e\u03a82/2 r n\u03c0 2\u03c1\u2217Q(\u03a8). Eq. (70) follows by using the lower bound (55) on the sphere bound. The analytic bound (71) follows from (77). The asymptotic form (72) follows by the fact that \u03a8 = \u0398(\u221an), and by plugging (66) and (67) into the analytical bounds (69) and (71). In Fig. 5 we demonstrate the tightness of the bounds and precise asymptotics in Theorem 10. In the \ufb01gure the ML bound is presented with its bounds and approximations. The image is similar to the Fig. 4, referring to the sphere bound. The lower bound (70) is the tightest lower bound (but is based on the non-analytic Q function). The analytic lower bound (71) is slightly looser than (70), but is tight enough in order to derive the precise asymptotic form (72). The upper bound (69) of the sphere bound is also tight. The error exponent itself (without the sub-exponential terms) is clearly way off, compared to the precise asymptotic form (72). C. Tightness of the Bounds Above \u03b4cr Corollary 1: For \u03b4cr < \u03b4 < \u03b4\u2217the ratio between the upper and lower bounds on Pe(n, \u03b4) converges to a constant, i.e. P MLB e (n, \u03b4) P SB e (n, \u03b4) = 1 (2 \u2212e2(\u03b4\u2217\u2212\u03b4)) + O \u0012log n n \u0013 . (82) Proof: The proof follows from Theorems 9 and 10. Note that the result is tighter than the ratio of the asymptotic forms (58) and (72) (i.e. O( log n n ) and not O( log2 n n )) since the term that contributes the log2 n term is e\u2212n 2 \u03c1\u2217which is common for both upper and lower bounds. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 25 100 200 300 400 500 0.001 0.005 0.010 0.050 0.100 Dimension n Error probability Sfrag replacements ML bound P MLB e Upper bound (69) Lower bound (71) Lower bound (70) Asymptotic form (72) e\u2212nEr(\u03b4) Fig. 5. Numerical evaluation of the ML bound and its bounds and approximation in Theorem 10 vs the dimension n. Here \u03b4 = \u22121.5nat (0.704db from capacity). The tight bounds (69), (70) and (71) lead to the asymptotic form (72). The error exponent term alone is evidently way off compared to (72). D. The ML Bound Below \u03b4cr Here we provide the asymptotic behavior of the ML bound at NLD values below \u03b4cr. Theorem 11: For any \u03b4 < \u03b4cr, the ML bound can be approximated by P MLB e (n, \u03b4) = e\u2212nEr(\u03b4) \u221a 2\u03c0n \u00001 + O \u0000 1 n \u0001\u0001 . (83) Proof: We start as in the proof of Theorem 5 to have en\u03b4Vn Z r\u2217 0 fR(r)rndr = n 2en\u03b4V 2 n \u03c3n(2\u03c0)\u2212n 2 nn Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1. (84) We continue by approximating the integral as follows: Lemma 7: Let x > 2. Then the integral R x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 can be approximated by Z x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = r 2\u03c0 n e\u2212n2n \u00001 + O \u0000 1 n \u0001\u0001 . (85) Proof: The proof relies on the fact that the integrand is maximized at the interior of the interval [0, x]. Note that the result does not depend on x. We \ufb01rst rewrite the integral to the form Z x 0 1 \u03c1e\u2212n(\u03c1/2\u2212log \u03c1)d\u03c1 = Z x 0 g(\u03c1)e\u2212nG(\u03c1)d\u03c1, (86) \f26 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY where g(\u03c1) \u225c1 \u03c1 and G(\u03c1) \u225c\u03c1/2 \u2212log \u03c1. When n grows, the asymptotical behavior of the integral is dominated by the value of the integrand at \u02dc \u03c1 = 2 (which minimizes G(\u03c1)). This is formalized by Laplace\u2019s method of integration (see, e.g. [20, Sec. 3.3]): Z x 0 g(\u03c1)e\u2212nG(\u03c1)d\u03c1 = g(\u02dc \u03c1)e\u2212nG(\u02dc \u03c1) s 2\u03c0 n\u22022G(\u02dc \u03c1) \u2202\u03c12 |\u03c1=\u02dc \u03c1 \u00001 + O \u0000 1 n \u0001\u0001 = 1 2en(1\u2212log 2) s 2\u03c0 n \u00b7 1 4 \u00001 + O \u0000 1 n \u0001\u0001 , which leads to (85). Before we apply the result of the lemma to (84), we note that whenever \u03b4 is below the critical \u03b4cr, \u03c1\u2217> e2(\u03b4\u2217\u2212\u03b4) = 2e2(\u03b4cr\u2212\u03b4) > 2 for all n. Therefore for all n we have Z 2e2(\u03b4cr\u2212\u03b4) 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 \u2264 Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 \u2264 Z \u221e 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1. (87) We apply Lemma 7 to both sides of the equation and conclude that Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = r 2\u03c0 n e\u2212n2n \u00001 + O \u0000 1 n \u0001\u0001 . (88) The proof of the theorem is completed using the approximation (64) for Vn. It should be noted that the sphere bound part of the achievability bound vanishes with a stronger exponent (Esp(\u03b4)), and therefore does not contribute to the asymptotic value. In Fig. 6 we demonstrate the tightness of the precise asymptotics in Theorem 11. Here too the precise asymptotic form is signi\ufb01cantly tighter than the error exponent only. E. The ML Bound at \u03b4cr In previous subsections we provided asymptotic forms for the upper bound on Pe(n, \u03b4), for \u03b4 > \u03b4cr and for \u03b4 < \u03b4cr (Theorems 10 and 11 respectively). Unfortunately, neither theorem holds for \u03b4cr exactly. We now analyze the upper bound at \u03b4cr, and show that its asymptotic form is different at this point. As a consequence, at the critical NLD, the ratio between the upper and lower bounds on Pe(n, \u03b4) is of the order of \u221an (this ratio above \u03b4cr is a constant, and below \u03b4cr the ratio increases exponentially since the error exponents are different). Theorem 12: At \u03b4 = \u03b4cr, the ML bound is given asymptotically by P MLB e (n, \u03b4cr) = e\u2212nEr(\u03b4cr) 1 2\u03c0 \u0014r \u03c0 2n + log(n\u03c0e2) n \u0015 \u0010 1 + O \u0010 log2 n n \u0011\u0011 (89) = e\u2212nEr(\u03b4cr) 1 \u221a 8\u03c0n \u0010 1 + O \u0010 log n \u221an \u0011\u0011 (90) Proof: Appendix G. In Fig. 7 we demonstrate the tightness of the precise asymptotics of Theorem 12. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 27 20 40 60 80 100 10-9 10-7 10-5 0.001 0.1 Dimension n Error probability Sfrag replacements ML bound P MLB e Asymptotic form (83) e\u2212nEr(\u03b4) Fig. 6. Numerical evaluation of the ML bound and its approximation in Theorem 11 vs the dimension n. Here \u03b4 = \u22121.8nat (3.31db from capacity). The precise asymptotic form (83) is clearly tighter than the error exponent only. F. Asymptotic Analysis of the Typicality Bound The typicality upper bound on Pe(n, \u03b4) (Theorem 3) is typically weaker than the MLbased bound (Theorem 5). In fact, it admits a weaker exponential behavior than the random coding exponent Er(\u03b4). De\ufb01ne the typicality exponent Et(\u03b4) as Et(\u03b4) \u225c\u03b4\u2217\u2212\u03b4 \u22121 2 log(1 + 2(\u03b4\u2217\u2212\u03b4)). (91) Theorem 13: For any \u03b4 < \u03b4\u2217, the typicality upper bound is given asymptotically by P TB e (n, \u03b4) = e\u2212nEt(\u03b4) \u221an\u03c0 \u00b7 1 + 2(\u03b4\u2217\u2212\u03b4) 2(\u03b4\u2217\u2212\u03b4) \u0012 1 + O \u00121 n \u0013\u0013 (92) Proof: Appendix H. The error exponent Et(\u03b4) is illustrated in Figure 8. As seen in the \ufb01gure, Et(\u03b4) is lower than Er(\u03b4) for all \u03b4. G. Asymptotic Analysis of P MLB e with Poltyrev\u2019s r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4 In Poltyrev\u2019s proof of the random coding exponent [3], the suboptimal value for r was used, cf. Section IV above. Instead of the optimal r = re\ufb00= e\u2212\u03b4V 1/n n , he chose r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4. In Figures 2 and 3 above we demonstrated how this suboptimal choice of r affects the ML bound at \ufb01nite n. In the \ufb01gures, it is shown that for \u03b4 = \u22121.5nat (above \u03b4cr) the loss is more signi\ufb01cant than for \u03b4 = \u22122nat (below \u03b4cr). Here we utilize the \f28 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 20 40 60 80 100 10-9 10-7 10-5 0.001 0.1 Dimension n Error probability g replacements ML bound P MLB e Tight asymptotic form (89) Loose asymptotic form (90) e\u2212nEr(\u03b4cr) Fig. 7. Numerical evaluation of the ML bound at \u03b4 = \u03b4cr (3.01db from capacity) and its approximations in Theorem 12 vs the dimension n. The asymptotic form (89) is tighter than the simpler (90). Both forms approximate the true value of the ML bound better than the error exponent term alone. -2 -1.9 -1.8 -1.7 -1.6 -1.5 \u2206cr \u2206* 0.0 0.1 0.2 0.3 0.4 0.5 -2 -1.9 -1.8 -1.7 -1.6 -1.5 \u2206cr \u2206* The normalized log density \u2206 Exponent @natsD ag replacements Esp(\u03b4) Er(\u03b4) Et(\u03b4) Fig. 8. Error exponents for the unconstrained AWGN channel. The typicality error exponent Et(\u03b4) (dot-dashed) vs. the random coding exponent Er(\u03b4) (dotted) and the sphere packing Esp(\u03b4) (solid). The noise variance \u03c32 is set to 1. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 29 techniques used in the current section in order to provide asymptotic analysis of the ML bound with the suboptimal r, and by that explain this phenomenon. Theorem 14: The ML bound P MLB e , with r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4, denoted \u02dc P MLB e (n, \u03b4), is given asymptotically as follows: For \u03b4cr < \u03b4 < \u03b4\u2217: \u02dc P MLB e (n, \u03b4) = e\u2212nEr(\u03b4) \u0014 1 n\u03c0(2 \u2212e2(\u03b4\u2217\u2212\u03b4)) + 1 \u221an\u03c0(e2(\u03b4\u2217\u2212\u03b4) \u22121) \u0015 \u00001 + O \u0000 1 n \u0001\u0001 (93) = e\u2212nEr(\u03b4) 1 \u221an\u03c0(e2(\u03b4\u2217\u2212\u03b4) \u22121) \u0010 1 + O \u0010 1 \u221an \u0011\u0011 . (94) For \u03b4 < \u03b4cr: \u02dc P MLB e (n, \u03b4) = e\u2212nEr(\u03b4) 1 \u221a 2\u03c0n \u00001 + O \u0000 1 n \u0001\u0001 . (95) For \u03b4 = \u03b4cr: \u02dc P MLB e (n, \u03b4cr) = e\u2212nEr(\u03b4cr) 1 \u221a\u03c0n \u0014 1 + 1 \u221a 8 \u0015 \u00001 + O \u0000 1 n \u0001\u0001 (96) Notes: \u2022 For \u03b4 > \u03b4cr, \u02dc P MLB e (n, \u03b4) is indeed asymptotically worse than P MLB e with the optimal r = re\ufb00(37), see (72). Speci\ufb01cally, the choice of r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4 only balances the exponents of the two expressions of the bound (37), while leaving the sub-exponential terms unbalanced see (93). The optimal selection r = re\ufb00balances the sub-exponential terms to the order of n\u22121 2e2(\u03b4\u2217\u2212\u03b4), see Theorem 10. This in fact quanti\ufb01es the asymptotic gap between the bounds, as seen in the Fig. 2. \u2022 For \u03b4 < \u03b4cr, the selection of the suboptimal r has no asymptotic effect, as seen by comparing (95) and (83). This corroborates the numerical \ufb01ndings presented in Fig. 3. \u2022 For \u03b4 = \u03b4cr the asymptotic form of the bound is changes by a constant (compare (96) and (89),(90)), and the correction term in the approximation tighter. Proof: Appendix I. VI. ASYMPTOTICS FOR FIXED ERROR PROBABILITY In the previous section we were interested in the asymptotic behavior of Pe(n, \u03b4) when the NLD \u03b4 is \ufb01xed. We now turn to look at a related scenario where the error probability \u03b5 is \ufb01xed, and we are interested in the asymptotic behavior of the optimal achievable NLD, denoted \u03b4\u03b5(n), with n \u2192\u221e. This setting parallels the channel dispersion type results [7][6][21, Problem 2.1.24], and is strongly related to the dispersion of the power constrained AWGN channel [8][6]. A. The Dispersion of In\ufb01nite Constellations Let \u03b5 > 0 denote a \ufb01xed error probability value. Clearly, for any \u03b5, \u03b4\u03b5(n) approaches the optimal NLD \u03b4\u2217as n \u2192\u221e. Here we study the asymptotic behavior of this convergence. Theorem 15: For a \ufb01xed error probability \u03b5, the optimal NLD \u03b4\u03b5(n) is given, for large enough n, by \u03b4\u03b5(n) = \u03b4\u2217\u2212 r 1 2nQ\u22121(\u03b5) + 1 2n log n + O \u00121 n \u0013 . (97) \f30 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 50 100 150 200 250 300 350 400 450 500 \u22121.8 \u22121.75 \u22121.7 \u22121.65 \u22121.6 \u22121.55 \u22121.5 \u22121.45 \u22121.4 NLD \u03b4 [nats] Dimension n 50 100 150 200 250 300 350 400 450 500 \u22121.8 \u22121.75 \u22121.7 \u22121.65 \u22121.6 \u22121.55 \u22121.5 \u22121.45 \u22121.4 NLD \u03b4 [nats] Dimension n nts Poltyrev capacity \u03b4\u2217 Critical rate \u03b4cr Converse Achievability (ML) Achievability (Typicality) Tight approximation (97) \u03b4\u2217\u2212 q 1 2n Q\u22121(\u03b5) + 1 2n log n Loose approximation (98) \u03b4\u2217\u2212 q 1 2nQ\u22121(\u03b5) Fig. 9. Bounds and approximations of the optimal NLD \u03b4\u03b5(n) for error probability \u03b5 = 0.01. Here the noise variance \u03c32 is set to 1. The proof is based on an asymptotic analysis of the \ufb01nite-dimensional bounds derived in Section IV. Speci\ufb01cally, the converse bound (an upper bound in (97)) is based on the sphere bound (4). The achievability part (a lower bound in (97)) is based on the ML bound (37). The weaker typicality bound is also useful for deriving a result of the type (97), but in a slightly weaker form the typicality bound can only lead to \u03b4\u03b5(n) \u2265\u03b4\u2217\u2212 r 1 2nQ\u22121(\u03b5) + O \u00121 n \u0013 . (98) In Fig. 9 we show the bounds on \u03b4\u03b5(n) that are derived from the \ufb01nite dimensional bounds on Pe(n, \u03b4) given in Sec. IV, along with the asymptotic form (97), derived in this section, which tightly approximates \u03b4\u03b5(n). In addition, the term (98) is also depicted, which only loosely approximates \u03b4\u03b5(n). The chosen error probability for the \ufb01gure is \u03b5 = 0.01. Before proving the theorem, let us discuss the result. By the similarity of Equations (1) and (97) we can isolate the constant 1 2 and identify it as the dispersion of the unconstrained AWGN setting. This fact can be intuitively explained from several directions. One interesting property of the channel dispersion theorem (1) is the following connection to the error exponent. Under some mild regularity assumptions, the error exponent can be approximated near the capacity by E(R) \u223c = (C \u2212R)2 2V , (99) \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 31 where V is the channel dispersion. The fact that the error exponent can be approximated by a parabola with second derivative 1 V was already known to Shannon (see [6, Fig. 18]). This property holds for DMC\u2019s and for the power constrained AWGN channel and is conjectured to hold in more general cases. Note, however, that while the parabolic behavior of the exponent hints that the gap to the capacity should behave as O \u0010 1 \u221an \u0011 , the dispersion theorem cannot be derived directly from the error exponent theory. Even if the error probability was given by e\u2212nE(R) exactly, (1) cannot be deduced from (99) (which holds only in the Taylor approximation sense). Analogously to (99), we examine the error exponent for the unconstrained Gaussian setting. For NLD values above the critical NLD \u03b4cr (but below \u03b4\u2217), the error exponent is given by [3]: E(\u03b4) = e\u22122\u03b4 4\u03c0e\u03c32 + \u03b4 + 1 2 log 2\u03c0\u03c32. (100) By straightforward differentiation we get that the second derivative (w.r.t. \u03b4) of E(\u03b4, \u03c32) at \u03b4 = \u03b4\u2217is given by 2, so according to (99), it is expected that the dispersion for the unconstrained AWGN channel will be 1 2. This agrees with our result (97) and its similarity to (1), and extends the correctness of the conjecture (99) to the unconstrained AWGN setting as well. It should be noted, however, that our result provides more than just proving the conjecture: there also exist examples where the error exponent is well de\ufb01ned (with second derivative), but a connection of the type (99) can only be achieved asymptotically with \u03b5 \u21920 (see, e.g. [22]). Our result (97) holds for any \ufb01nite \u03b5, and also gives the exact 1 n log n term in the expansion. Another indication that the dispersion for the unconstrained setting should be 1 2 comes from the connections to the power constrained AWGN. While the capacity 1 2 log(1 + P), where P denotes the channel SNR, is clearly unbounded with P, the form of the error exponent curve does have a nontrivial limit as P \u2192\u221e. In [2] it was noticed that this limit is the error exponent of the unconstrained AWGN channel (sometimes termed the \u2018Poltyrev exponent\u2019), where the distance to the capacity is replaced by the NLD distance to \u03b4\u2217. By this analogy we examine the dispersion of the power constrained AWGN channel at high SNR. In [6] the dispersion was found, given (in nat2 per channel use) by VAW GN = P(P + 2) 2(P + 1)2. (101) This term already appeared in Shannon\u2019s 1959 paper on the AWGN error exponent [5], where its inverse is exactly the second derivative of the error exponent at the capacity (i.e. (99) holds for the AWGN channel). It is therefore no surprise that by taking P \u2192\u221e, we get the desired value of 1 2, thus completing the analogy between the power constrained AWGN and its unconstrained version. This convergence is quite fast, and is tight for SNR as low as 10dB (see Fig. 10). B. A Key Lemma In order to prove Theorem 15 we need the following lemma regarding the norm of a Gaussian vector. Lemma 8: Let Z = [Z1, ..., Zn]T be a vector of n zero-mean, independent Gaussian random variables, each with mean \u03c32. Let r > 0 be a given arbitrary radius. Then the \f32 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY -30 -20 -10 0 10 20 30 0.0 0.1 0.2 0.3 0.4 0.5 0.6 SNR @dBD Dispersion @nats2\u0090channel useD Fig. 10. The power-constrained AWGN dispersion (101) (solid) vs. the unconstrained dispersion (dashed) following holds for any dimension n: \f \f \f \fPr{\u2225Z\u2225> r} \u2212Q \u0012r2 \u2212n\u03c32 \u03c32\u221a 2n \u0013\f \f \f \f \u22646T \u221an, (102) where T = E \"\f \f \f \f X2 \u22121 \u221a 2 \f \f \f \f 3# \u22483.0785, (103) for a Standard Gaussian RV X. Proof: The proof relies on the convergence of a sum of independent random variables to a Gaussian random variable, i.e. the central limit theorem. We \ufb01rst note that Pr{\u2225Z\u2225> r} = Pr ( n X i=1 Z2 i > r2 ) . (104) Let Yi = Z2 i \u2212\u03c32 \u03c32\u221a 2 . It is easy to verify that E[Yi] = 0 and that VAR[Yi] = 1. Let Sn \u225c 1 \u221an Pn i=1 Yi. Note that Sn also has zero mean and unit variance. It follows that Pr ( n X i=1 Z2 i \u2265r2 ) = Pr ( n X i=1 Z2 i \u2212\u03c32 \u03c32\u221a 2 \u2265r2 \u2212n\u03c32 \u03c32\u221a 2 ) = Pr ( n X i=1 Yi \u2265r2 \u2212n\u03c32 \u03c32\u221a 2 ) = Pr \u001a Sn \u2265r2 \u2212n\u03c32 \u03c32\u221a 2n \u001b . (105) \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 33 Sn is a normalized sum of i.i.d. variables, and by the central limit theorem converges to a standard Gaussian random variables. The Berry-Esseen theorem (see Appendix J) quanti\ufb01es the rate of convergence in the cumulative distribution function sense. In the speci\ufb01c case discussed in the lemma we get \f \f \f \fPr \u001a Sn \u2265r2 \u2212n\u03c32 \u03c32\u221a 2n \u001b \u2212Q \u0014r2 \u2212n\u03c32 \u03c32\u221a 2n \u0015\f \f \f \f \u22646T \u221an, (106) where T = E[|Yi|3]. Note that T is independent of \u03c32, \ufb01nite, and can be evaluated numerically to about 3.0785. C. Proof of Theorem 15 Proof of Direct part: Let \u03b5 denote the required error probability. We shall prove the existence of an IC (more speci\ufb01cally, a lattice) with error probability at most \u03b5 and NLD satisfying (97). It is instructive to \ufb01rst prove a slightly weaker version of (97) based on the typicality decoder (Theorem 3). While easier to derive, this will show the existence of lattices with NLD \u03b4 = \u03b4\u2217\u2212 q 1 2nQ\u22121(\u03b5) + O \u0000 1 n \u0001 . Proving the stronger result (97) is more technical and will proven afterwards using the ML achievability bound (Theorem 5). Recall the achievability bound in Theorem 3: for any r > 0 there exist lattices with NLD \u03b4 and error probability Pe that is upper bounded by Pe \u2264\u03b3Vnrn + Pr {\u2225Z\u2225> r} . (107) We determine r s.t. Pr(\u2225Z\u2225> r) = \u03b5 h 1 \u2212 1 \u221an i and \u03b3 s.t. \u03b3Vnrn = \u03b5 \u221an. This way it is assured that the error probability is not greater than the required \u03b5 h 1 \u2212 1 \u221an i + \u03b5 \u221an = \u03b5. Now de\ufb01ne \u03b1n s.t. r2 = n\u03c32(1 + \u03b1n) (note that r implicitly depends on n as well). Lemma 9: \u03b1n, de\ufb01ned above, is given by \u03b1n = r 2 nQ\u22121(\u03b5) + O \u00121 n \u0013 . (108) Proof: By construction, r is chosen s.t. Pr(\u2225Z\u22252 > r2) = \u03b5 \u0014 1 \u22121 \u221an \u0015 . (109) By the de\ufb01nition of \u03b1n, Pr(\u2225Z\u22252 > n\u03c32(1 + \u03b1n)) = \u03b5 \u0014 1 \u22121 \u221an \u0015 . (110) By Lemma 8, Pr(\u2225Z\u22252 > n\u03c32(1 + \u03b1n)) = Q \u0012n\u03c32(1 + \u03b1n) \u2212n\u03c32 \u03c32\u221a 2n \u0013 + O \u0012 1 \u221an \u0013 = Q \u0012rn 2\u03b1n \u0013 + O \u0012 1 \u221an \u0013 . (111) \f34 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Combined with (110), we get \u03b5 \u0014 1 \u22121 \u221an \u0015 = Q \u0012rn 2\u03b1n \u0013 + O \u0012 1 \u221an \u0013 , (112) or \u03b5 + O \u0012 1 \u221an \u0013 = Q \u0012rn 2\u03b1n \u0013 . (113) Taking Q\u22121(\u00b7) of both sides, we get rn 2\u03b1n = Q\u22121 \u0012 \u03b5 + O \u0012 1 \u221an \u0013\u0013 . (114) By the Taylor approximation of Q\u22121(\u03b5 + x) around x = 0, we get rn 2\u03b1n = Q\u22121 (\u03b5) + O \u0012 1 \u221an \u0013 , (115) or \u03b1n = r 2 nQ\u22121 (\u03b5) + O \u00121 n \u0013 , (116) as required. So far, we have shown the existence of a lattice \u039b with error probability at most \u03b5. The NLD is given by \u03b4 = 1 n log \u03b3 = 1 n log \u03b5 Vnrn\u221an = \u22121 n log Vn \u2212log r \u2212log n 2n + 1 n log \u03b5 = \u22121 n log Vn \u22121 2 log[n\u03c32(1 + \u03b1n)] \u2212log n 2n + 1 n log \u03b5. Vn can be approximated by (see Appendix F) by 1 n log Vn = 1 2 log 2\u03c0e n \u22121 2n log n + O \u00121 n \u0013 . (117) We therefore have \u03b4 = \u22121 2 log(2\u03c0e\u03c32) \u22121 2 log(1 + \u03b1n) + O \u00121 n \u0013 (118) (a) = \u03b4\u2217\u22121 2 log(1 + \u03b1n) + O \u00121 n \u0013 (119) (b) = \u03b4\u2217\u22121 2\u03b1n + O \u00121 n \u0013 (120) (c) = \u03b4\u2217\u2212 r 1 2nQ\u22121(\u03b5) + O \u00121 n \u0013 , (121) \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 35 where (a) follows from the de\ufb01nition of \u03b4\u2217, (b) follows from the Taylor approximation for log(1 + \u03b1n) around \u03b1n = 0 and from the fact that \u03b1n = O(1/\u221an), and (c) follows from Lemma 9. This completes the achievability part based on the typicality decoder. In order to prove the stronger achievability result that \ufb01ts (97) we follow the same steps with the ML achievability bound. By Theorem 5 we get that for any r > 0 there exist a lattice with density \u03b3 and error probability upper bounded by Pe \u2264\u03b3Vn Z r 0 fR(\u02dc r)\u02dc rnd\u02dc r + Pr {\u2225Z\u2225> r} . (122) Now determine r s.t. Pr(\u2225Z\u2225> r) = \u03b5 h 1 \u2212 1 \u221an i and \u03b3 s.t. \u03b3Vn R r 0 fR(\u02dc r)\u02dc rnd\u02dc r = \u03b5 \u221an. Again, it is assured that the error probability is upper bounded by \u03b5. De\ufb01ne \u03b1n s.t. r2 = n\u03c32(1 + \u03b1n). The resulting density is given by \u03b3 = \u03b5 \u221anVn R r 0 fR(\u02dc r)\u02dc rnd\u02dc r, (123) and the NLD by \u03b4 = 1 n log \u03b3 = 1 n log \u03b5 \u22121 n log \u0014\u221anVn Z r 0 fR(\u02dc r)\u02dc rnd\u02dc r \u0015 = 1 n log \u03b5 \u22121 2n log n \u22121 n log Vn \u22121 n log Z r 0 fR(\u02dc r)\u02dc rnd\u02dc r = \u22121 2 log 2\u03c0e n \u22121 n log Z \u221a n(1+\u03b1n) 0 fR(\u02dc r)\u02dc rnd\u02dc r + O \u00121 n \u0013 . (124) where the last equality follows from the approximation (117) for Vn. We repeat the derivation as in Eq. (74) where r\u2217is replaced by r = p n\u03c32(1 + \u03b1n) and have Z \u221a n\u03c32(1+\u03b1n) 0 fR(\u02dc r)\u02dc rnd\u02dc r = \u03c3n2\u2212n/2nn \u0393 \u0002n 2 \u0003 Z 1+\u03b1n 0 e\u2212n\u02dc r/2\u02dc rn\u22121d\u02dc r \u2264\u03c3n2\u2212n/2nn \u0393 \u0002n 2 \u0003 2(1 + \u03b1n)ne\u2212n(1+\u03b1n)/2 n \u00001 \u2212\u03b1n \u22122 n \u0001 = \u03c3n2\u2212n/2nn \u0393 \u0002n 2 \u0003 2en log(1+\u03b1n)e\u2212n(1+\u03b1n)/2 n \u00001 \u2212\u03b1n \u22122 n \u0001 , where the inequality follows from Lemma 6. Therefore the term in (124) can be bounded \f36 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY by 1 n log Z \u221a n\u03c32(1+\u03b1n) 0 fR(\u02dc r)\u02dc rnd\u02dc r \u22641 2 log \u03c32 \u22121 2 log 2 + log n + log(1 + \u03b1n) \u22121 2(1 + \u03b1n) + 1 n log 1 n 2\u0393 \u0002n 2 \u0003 1 \u00001 \u2212\u03b1n \u22122 n \u0001 = 1 2 log \u03c32 \u22121 2 log 2 + log n + log(1 + \u03b1n) \u22121 2(1 + \u03b1n) + 1 n log 1 n 2\u0393 \u0002n 2 \u0003 + O \u0000 1 n \u0001 = 1 2 log \u03c32 \u22121 2 log 2 + log n + log(1 + \u03b1n) \u22121 2(1 + \u03b1n) \u22121 n \u0000 1 2 log(\u03c0n) + n 2 log n 2e \u0001 + O \u0000 1 n \u0001 = 1 2 log \u03c32 + 1 2 log n + log(1 + \u03b1n) \u22121 2\u03b1n \u2212 1 2n log n + O \u0000 1 n \u0001 (a) = 1 2 log \u03c32 + 1 2 log n + 1 2\u03b1n \u2212 1 2n log n + O \u0000 1 n \u0001 (a) follows from the Taylor expansion of log(1 + \u03be) at \u03be = 0 and from the fact that \u03b1n = O( 1 \u221an). Plugging back to (124) combined with Lemma 9 completes the proof of the direct part. Proof of Converse Part: Let \u03b5 > 0, and let {Sn}n\u2208N be a series of IC\u2019s, where for each n, Pe(Sn) \u2264\u03b5. Our goal is to upper bound the NLD \u03b4n of Sn for growing n. By the sphere bound we have \u03b5 \u2265Pe(Sn) \u2265Pr{\u2225Z\u2225> r\u2217}, (125) where r\u2217= e\u2212\u03b4nV \u22121/n n . By Lemma 8, \u03b5 \u2265Pr{\u2225Z\u2225> r\u2217} \u2265Q \u0012r\u22172 \u2212n\u03c32 \u03c32\u221a 2n \u0013 \u22126T \u221an, (126) where T is a constant. It follows by algebraic manipulations that \u03b4n \u2264\u22121 2 log 1 + r 2 nQ\u22121 \u0012 \u03b5 + 6T \u221an \u0013! \u22121 n log Vn \u22121 2 log(n\u03c32). (127) By the Taylor approximation of log(1 + x) at x = 0 and of Q\u22121(y) at y = \u03b5, and by the approximation (117) for Vn, \u03b4n \u2264\u2212 r 1 2nQ\u22121 (\u03b5) \u22121 2 log 2\u03c0e n + 1 2n log n \u22121 2 log(n\u03c32) + O \u00121 n \u0013 . (128) By the de\ufb01nition of \u03b4\u2217we \ufb01nally get \u03b4n \u2264\u03b4\u2217\u2212 r 1 2nQ\u22121 (\u03b5) + 1 2n log n + O \u00121 n \u0013 , (129) as required. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 37 5 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Gap to capacity [dB] Dimension n 5 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Gap to capacity [dB] Dimension n Integer Lattice Hexagonal Lattice (d=2) Checkerboard Lattice D4 Gosset Lattice E8 Barnes\u2212Wall Lattice BW16 Leech Lattice \u039b24 cements Poltyrev capacity \u03b4\u2217 rate \u03b4cr Converse Achievability (ML) Approximation (97) Fig. 11. Low-dimensional IC\u2019s for coding over the unconstrained AWGN. The error probability is set to \u03b5 = 0.01. VII. COMPARISON WITH KNOWN INFINITE CONSTELLATIONS In this section we compare the \ufb01nite dimensional bounds of Section IV with the performance of some known IC\u2019s. We start with the low dimensional IC\u2019s, which include classic sphere packings: the integer lattice, the hexagonal lattice, the packings D4, E8, BW16 and the leech lattice \u039b24 (see Conway and Sloane [12]). In low dimensions we present Monte Carlo simulation results based on the ML decoder. In higher dimensions we consider low density lattice codes (LDLC) [23] with dimensions n = 100, 300, 500 and 1000 (designed by Y. Yona). In dimension n = 127 we present the results for the packing S127 [24]. In Fig. 11 we show the gap to (Poltyrev\u2019s) capacity of the low dimensional IC\u2019s, where the error probability is set to \u03b5 = 0.01. As seen in the \ufb01gure, these low dimensional IC\u2019s outperform the best achievability bound (Theorem 5). At n = 1, the integer lattice achieves the sphere bound (and is, essentially, the only lattice for n = 1). From the presentation of Fig. 11 it is dif\ufb01cult to compare IC\u2019s with different dimensions. For example, the hexagonal lattice closer to the capacity than the lattice D4, and also the gap to the sphere bound is smaller. Obviously this does not mean that D4 is inferior. To facilitate the comparison between different dimensions we propose the following comparison: Set a \ufb01xed value for the error probability for n = 1 denoted \u03b51. Then de\ufb01ne, for each n, the normalized error probability \u03b5n \u225c1 \u2212(1 \u2212\u03b51)n. \f38 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Using this normalization enables the true comparison between IC\u2019s with different dimensions. The achieved gap to capacity with a normalized error probability remains the same when a scheme is used say k times, and the block length becomes k \u00b7 n. For example, the integer lattice maintains a constant \u03b4 for any n with the normalized error probability, as opposed to the case presented in Fig. 11, where the performance decreases. In Fig. 12 we plot the same data as in Fig. 11 for normalized error probability with \u03b51 = 10\u22125. We also plot the normalized error probability itself for reference. In Fig. 13 we present the performance of IC\u2019s in higher dimensions (again, with normalized error probability and \u03b51 = 10\u22125). The included constellations are the leech lattice again (for reference), LDLC with n = 100, 300, 500, 1000 and degrees 5, 6, 7, 7 respectively (cf. [23] and [25] for more details on the construction of LDLC and the degree). For LDLC\u2019s, the \ufb01gure shows simulation results based on a suboptimal low complexity parametric iterative decoder [25]. In addition, we present the performance of the packing S127[24] (which is a multilevel coset code[16]). Notes: \u2022 At higher dimensions, the performance of the presented IC\u2019s no longer outperforms the achievability bound. \u2022 It is interesting to note that the Leech lattice replicated 4 times (resulting in an IC at n = 96) outperforms the LDLC with n = 100 (but remember that the LDLC performance is based on a low complexity suboptimal decoder where the Leech lattice performance is based on the ML decoder). \u2022 The approximation (97) no longer holds formally for the case of normalized error probability. This follows since the correction term in (97) depends on the error probability. Nevertheless, as appears in Fig. 13, the approximation appears to still hold. VIII. VOLUME-TO-NOISE RATIO ANALYSIS The VNR \u00b5, de\ufb01ned in (20), can describe the distance from optimality for a given IC and noise variance, and we say that an IC S operating at noise level \u03c32 is in fact operating at VNR \u00b5. Equivalently, we can de\ufb01ne the VNR as a function of the IC and the error probability: For a given IC S and error probability \u03b5, let \u00b5(S, \u03b5) be de\ufb01ned as follows: \u00b5(S, \u03b5) \u225c e\u22122\u03b4(S) 2\u03c0e\u03c32(\u03b5), (130) where \u03c32(\u03b5) is the noise variance s.t. the error probability is exactly \u03b5. Note that \u00b5(S, \u03b5) does not depend on scaling of the IC S, and therefore can be thought of as a quantity that depends only on the \u2018shape\u2019 of the IC. We now de\ufb01ne a related fundamental quantity \u00b5n(\u03b5), as the minimal value of \u00b5(S, \u03b5) among all n-dimensional IC\u2019s. It is known that for any 0 < \u03b5 < 1, \u00b5n(\u03b5) \u21921 as n \u2192\u221e [17]. We now quantify this convergence, based on the analysis of \u03b4\u03b5(n). It follows from the de\ufb01nitions of \u00b5n(\u03b5) and \u03b4\u03b5(n) that the following relation holds for any \u03c32: \u00b5n(\u03b5) = e\u22122\u03b4\u03b5(n) 2\u03c0e\u03c32 = e2(\u03b4\u2217\u2212\u03b4\u03b5(n)). (131) (note that \u03b4\u03b5(n) implicitly depends on \u03c32 as well). We may therefore use the results in the paper to understand the behavior of \u00b5n(\u03b5). For example, any of the bounds in Theorem 3, \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 39 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 Gap to capacity [dB] Dimension n 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 Gap to capacity [dB] Dimension n Integer Lattice (d=1) Hexagonal Lattice (d=2) Checkerboard Lattice D4 Gosset Lattice E8 Barnes\u2212Wall Lattice BW16 Leech Lattice \u039b24 5 10 15 20 25 30 35 40 45 50 10 \u22125 10 \u22124 10 \u22123 Dimension n Normalized Error Probability 5 10 15 20 25 30 35 40 45 50 10 \u22125 10 \u22124 10 \u22123 Dimension n Normalized Error Probability cements Poltyrev capacity \u03b4\u2217 rate \u03b4cr Converse Achievability (ML) Approximation (97) Fig. 12. Top: performance of different constellations (dimensions 1\u221224) for normalized error probability, with \u03b51 = 10\u22125. Bottom: the normalized error probability. \f40 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 100 200 300 400 500 600 700 800 900 1000 1100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Gap to capacity [dB] Dimension n 100 200 300 400 500 600 700 800 900 1000 1100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Gap to capacity [dB] Dimension n Leech Lattice \u039b24 LDLC n=100 (deg=5) S127 (EGMD) LDLC n=300 (deg=6) LDLC n=500 (deg=7) LDLC n=1000 (deg=7) nts Poltyrev capacity \u03b4\u2217 cr Converse Achievability (ML) Approximation (97) Fig. 13. Performance of different constellations (dimensions 24\u22121000) for normalized error probability, with \u03b51 = 10\u22125. Theorem 5 or the sphere bound (Theorem 2) can be applied in order to bound \u00b5n(\u03b5) for \ufb01nite n and \u03b5. Furthermore, the asymptotic behavior of \u00b5n(\u03b5) is characterized by the following: Theorem 16: For a \ufb01xed error probability 0 < \u03b5 < 1, The optimal VNR \u00b5n(\u03b5) is given by \u00b5n(\u03b5) = 1 + r 2 nQ\u22121(\u03b5) \u22121 n log n + O \u00121 n \u0013 . (132) Proof: In Theorem 15 we have shown that for given \u03b5 and \u03c32, the optimal NLD \u03b4 is given by \u03b4\u03b5(n) = \u03b4\u2217\u2212 r 1 2nQ\u22121(\u03b5) + 1 2n log n + O \u00121 n \u0013 , (133) where \u03b4\u2217= 1 2 log 1 2\u03c0e\u03c32. According to (131) we write \u00b5n(\u03b5) = exp \"r 2 nQ\u22121(\u03b5) \u22121 n log n + O \u00121 n \u0013# = 1 + r 2 nQ\u22121(\u03b5) \u22121 n log n + O \u00121 n \u0013 (134) where the last step follows from the Taylor expansion of ex. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 41 IX. SUMMARY In this paper we examined the unconstrained AWGN channel setting in the \ufb01nite dimension regime. We provided two new achievability bounds and extended the converse bound (sphere bound) to \ufb01nite dimensional IC\u2019s. We then analyzed these bounds asymptotically in two settings. In the \ufb01rst setting where the NLD (which is equivalent to the rate in classic channel coding) was \ufb01xed, we evaluated the (bounds on the) error probability when the dimension n grows, and provided asymptotic expansions that are signi\ufb01cantly tighter than those in the existing error exponent analysis. In the second setting, the error probability \u03b5 is \ufb01xed, and we investigated the optimal achievable NLD for growing n. We showed that the optimal NLD can be tightly approximated by a closed-form expression, and the gap to the optimal NLD vanishes as the inverse of the square root of the dimension n. The result is analogous to the channel dispersion theorem in classical channel coding, and agrees with the interpretation of the unconstrained setting as the high-SNR limit of the power constrained AWGN channel. The approach and tools developed in this paper can be used to extend the results to more general noise models, and also to \ufb01nite constellations. ACKNOWLEDGMENT The authors are grateful to Y. Yona for assistance in providing the simulation results for low-density lattice codes in Section VII. APPENDIX A PROOF OF THE BOUNDS EQUIVALENCE Proof of Theorem 6: It remains to show that n Z 2r 0 wn\u22121 Pr{Z \u2208D(r, w)}dw = Z r\u2217 0 fR(\u03c1)\u03c1nd\u03c1. (135) Lemma 10: For Z \u223cN(0, I\u03c32), and any r \u2265w/2 \u22650, Pr{Z \u2208D(r, w)} = Z r w/2 fZ(z) Z \u221a r2\u2212z2 \u03c3 0 f\u03c7n\u22121(t)dtdz, (136) where D(w, r) was de\ufb01ned after Eq. (23), fZ(z) = 1 \u221a 2\u03c0\u03c32e\u2212z2/(2\u03c32) is the pdf of a N(0, \u03c32) random variable, and f\u03c7n\u22121(t) = tn\u22122e\u2212t2/2 2 n\u22121 2 \u22121\u0393( n\u22121 2 ) is the pdf of a \u03c7 random variable with n\u22121 degrees of freedom. Proof: By the spherical symmetry of the Gaussian pdf, we may assume w.l.o.g. that \f42 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY the hyperplane at distance w 2 is perpendicular to the Z1 axis. We therefore have Pr{Z \u2208D(w, r)} = Pr{Z1 > w 2 , \u2225Z\u2225\u2264r} = Pr ( Z1 > w 2 , n X i=1 Z2 i \u2264r2 ) = Z r w/2 fZ(z) Pr ( n X i=1 Z2 i \u2264r2|Z1 = z ) dz = Z r w/2 fZ(z) Pr ( n X i=2 Z2 i \u2264r2 \u2212z2 ) dz = Z r w/2 fZ(z) Pr \uf8f1 \uf8f2 \uf8f3 1 \u03c3 v u u t n X i=2 Z2 i \u2264 \u221a r2 \u2212z2 \u03c3 \uf8fc \uf8fd \uf8fedz = Z r w/2 fZ(z) Z \u221a r2\u2212z2 \u03c3 0 f\u03c7n\u22121(t)dtdz, where the last equality follows from the fact that a \u03c7n\u22121 random variable is equivalent to the square root of a sum of n\u22121 independent squared standard Gaussian random variables. We use the result of the lemma and get n Z 2r 0 wn\u22121 Pr{Z \u2208D(r, w)}dw =n Z 2r 0 wn\u22121 Z r w/2 fZ(z) Z \u221a r2\u2212z2 \u03c3 0 f\u03c7n\u22121(t)dtdzdw =n Z 2r 0 Z r w/2 wn\u22121fZ(z) Z \u221a r2\u2212z2 \u03c3 0 f\u03c7n\u22121(t)dtdzdw (a) =n Z r 0 Z 2z 0 wn\u22121fZ(z) Z \u221a r2\u2212z2 \u03c3 0 f\u03c7n\u22121(t)dtdwdz = Z r 0 Z \u221a r2\u2212z2 \u03c3 0 (2z)nfZ(z)f\u03c7n\u22121(t)dtdz, were (a) follows from changing the order of integration. We set u = z/\u03c3 and get Z r/\u03c3 0 Z \u221a r2/\u03c32\u2212u2 0 (2\u03c3u)n\u03c3fZ(\u03c3u)f\u03c7n\u22121(t)dtdu = Z r/\u03c3 0 Z \u221a r2/\u03c32\u2212u2 0 (2\u03c3u)n 1 \u221a 2\u03c0e\u2212u2/2 tn\u22122e\u2212t2/2 2 n\u22121 2 \u22121\u0393 \u0000n\u22121 2 \u0001dtdu = 2 n 2 +1\u03c3n \u221a\u03c0\u0393 \u0000n\u22121 2 \u0001 Z r/\u03c3 0 Z \u221a r2/\u03c32\u2212u2 0 untn\u22122e\u2212(u2+t2)/2dtdu. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 43 We switch to polar coordinates and set u = \u03c1 cos \u03b8 and t = \u03c1 sin \u03b8. The expression becomes 2 n 2 +1\u03c3n \u221a\u03c0\u0393 \u0000 n\u22121 2 \u0001 Z r/\u03c3 0 Z \u03c0/2 0 \u03c12n\u22121 cosn \u03b8 sinn\u22122 \u03b8e\u2212\u03c12/2d\u03b8d\u03c1 = 2 n 2 +1\u03c3n \u221a\u03c0\u0393 \u0000 n\u22121 2 \u0001 Z r/\u03c3 0 \u03c12n\u22121e\u2212\u03c12/2d\u03c1 Z \u03c0/2 0 cosn \u03b8 sinn\u22122 \u03b8d\u03b8. (137) It can be shown using e.g. [26, Eqs. 18.32 and 25.9] that Z \u03c0/2 0 cosn \u03b8 sinn\u22122 \u03b8d\u03b8 = 2\u2212n\u221a\u03c0\u0393 \u0000n\u22121 2 \u0001 \u0393 \u0000n 2 \u0001 . (138) Equation (137) now simpli\ufb01es to 21\u2212n 2 \u03c3n \u0393 \u0000n 2 \u0001 Z r/\u03c3 0 \u03c12n\u22121e\u2212\u03c12/2d\u03c1 =\u03c3n Z r/\u03c3 0 f\u03c7n(\u03c1)\u03c1nd\u03c1 =\u03c3n Z r/\u03c3 0 \u03c3fR(\u03c3\u03c1)\u03c1nd\u03c1 = Z r 0 fR(\u03c1\u2032)\u03c1\u2032nd\u03c1\u2032 = Z r 0 fR(\u03c1)\u03c1nd\u03c1, where f\u03c7n(\u00b7) is the pdf of a \u03c7 random variable with n degrees of freedom, and fR(\u00b7) is the pdf of \u2225Z\u2225. This completes the proof of the theorem. APPENDIX B PROPERTIES OF REGULAR IC\u2019S Proof of Lemma 2: Let S be a regular IC with a given r0. Let V(a) denote the union of all the Voronoi cells of code points in Cb(a): V(a) \u225c [ s\u2208S\u2229Cb (a) W(s). (139) Since all Voronoi cells are bounded in spheres of radius r0, we note the following (for a > 2r0): \u2022 All the Voronoi cells of the code points in Cb(a) are contained in Cb(a + 2r0), and therefore V(a) \u2286Cb(a + 2r0). \u2022 Any point in Cb(a \u22122r0) must be in a Voronoi cell of some code point. These code points cannot be outside Cb(a) because the Voronoi cells are bounded in spheres of radius r0, so they must lie within Cb(a), and we get that Cb(a \u22122r0) \u2286V(a). It follows that (a \u22122r0)n \u2264|V(r)| \u2264(a + 2r0)n, or (a \u22122r0)n \u2264 X s\u2208S\u2229Cb (a) v(s) \u2264(a + 2r0)n. (140) \f44 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Dividing by an and taking the limit of a \u2192\u221egives lim a\u2192\u221e P s\u2208S\u2229Cb (a) v(s) an = 1. (141) Since, by assumption, the limit of the density \u03b3(S) exists, we get \u03b3(S) = lim a\u2192\u221e M(S, a) an = lim a\u2192\u221e M(S, a) P s\u2208S\u2229Cb (a) v(s) = 1 lima\u2192\u221eEa[v(S)] = 1 v(S). (142) As a corollary, we get that for regular IC\u2019s the average volume v(S) exists in the limit (and not only in the lim sup sense). APPENDIX C CONVEXITY OF THE EQUIVALENT SPHERE BOUND Proof of Lemma 3: Suppose v is the volume of the Voronoi cell. The radius of the equivalent sphere is given by r = v1/nV \u22121/n n . The equivalent sphere bound is given by SPB(v) = Pr ( n X i=1 Z2 i \u2265r2 ) = Pr ( n X i=1 (Zi/\u03c3)2 \u2265 v2/n V 2/n n \u03c32 ) \u225cPr ( n X i=1 (Zi/\u03c3)2 \u2265(C1 \u00b7 v)2/n ) , (143) where C1 is a constant. We note that Pn i=1(Zi/\u03c3)2 is a sum of n i.i.d. squared Gaussian RV\u2019s with zero mean and unit variance, which is exactly a \u03c72 distribution with n degrees of freedom. We therefore get: SPB(v) = 1 \u0393(n/2)2n/2 Z \u221e (C1\u00b7v)2/n xn/2\u22121e\u2212x/2dx = C2 Z \u221e (C1\u00b7v)2/n xn/2\u22121e\u2212x/2dx \u225cC2F(C1 \u00b7 v), (144) where C2 is a constant and F(t) \u225c R \u221e t2/n xn/2\u22121e\u2212x/2dx. It can be veri\ufb01ed by straightforward differentiation that \u22022 \u2202t2 F(t) = \u22022 \u2202t2 Z \u221e t2/n xn/2\u22121e\u2212x/2dx = 2 n2t 2 n \u22121 exp \u0012 \u22121 2t2/n \u0013 , (145) which is strictly positive for all t > 0. Therefore F(t) is convex, and the equivalent sphere bound SPB(v) = C2F(C1 \u00b7 v) is a convex function of v. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 45 APPENDIX D PROOF OF THE REGULARIZATION LEMMA Proof of Lemma 4: Our \ufb01rst step will be to \ufb01nd a hypercube Cb(a\u2217), so that the density of the points in S \u2229Cb(a\u2217) and the error probability of codewords in S \u2229Cb(a\u2217) are close enough to \u03b3 and \u03b5, respectively. We then replicate this cube in order to get a regular IC with the desired properties. The idea is similar to that used in [3, Appendix C], where it was used for expurgation purposes. As discussed in IV-D above, we wish to avoid expurgation process that weakens the bound for \ufb01nite dimensional IC\u2019s. By the de\ufb01nition of Pe(S) and \u03b3(S), \u03b3(S) = lim sup a\u2192\u221e M(S, a) an = lim a\u2192\u221esup b>a M(S, b) bn (146) \u03b5 = Pe(S) = lim sup a\u2192\u221e 1 M(S, a) X s\u2208S\u2229Cb (a) Pe(s) = lim a\u2192\u221esup b>a 1 M(S, b) X s\u2208S\u2229Cb (b) Pe(s). (147) Let \u03c4\u03b3 = \u221a1 + \u03be and \u03c4\u03b5 = 1 + \u03be 2. By de\ufb01nition of the limit, there must exist a0 large enough s.t. for every a > a0, both hold: sup b>a M(S, b) bn > \u03b3 \u00b7 1 \u03c4\u03b3 , (148) and sup b>a 1 M(S, b) X s\u2208S\u2229Cb (b) Pe(s) < \u03b5 \u00b7 \u03c4\u03b5. (149) De\ufb01ne \u2206s.t. Q(\u2206/\u03c3) = \u03b5 \u00b7 \u03be 2, and de\ufb01ne a\u2206as the solution to \u0012a\u2206+ 2\u2206 a\u2206 \u0013n = p 1 + \u03be. (150) Let amax = max{a0, a\u2206}. According to (148), there must exist a\u2217> amax s.t. M(S, a\u2217) an \u2217 > \u03b3 \u00b7 1 \u03c4\u03b3 . (151) By (149) we get that 1 M(S, a\u2217) X s\u2208S\u2229Cb (a\u2217) Pe(s) \u2264sup b>amax 1 M(S, b) X s\u2208S\u2229Cb (b) Pe(s) < \u03b5 \u00b7 \u03c4\u03b5. (152) Now consider the \ufb01nite constellation G = S \u2229Cb(a\u2217). For s \u2208G, denote by P G e (s) the error probability of s when G is used for transmission with Gaussian noise. Since G \u2282S, clearly P G e (s) \u2264Pe(s) for all s \u2208G. The average error probability for G is bounded by Pe(G) \u225c1 |G| X s\u2208G P G e (s) \u22641 |G| X s\u2208G Pe(s) \u2264\u03b5 \u00b7 \u03c4\u03b5. (153) We now turn to the second part constructing an IC from the code G. De\ufb01ne the IC S\u2032 as an in\ufb01nite replication of G with spacing of 2\u2206between every two copies as follows: S\u2032 \u225c{s + I \u00b7 (a\u2217+ 2\u2206) : s \u2208G, I \u2208Zn} , (154) \f46 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY where Zn denotes the integer lattice of dimension n. Now consider the error probability of a point s \u2208S\u2032 denoted by P S\u2032 e (s). This error probability equals the probability of decoding by mistake to another codeword from the same copy of G or to a codeword in another copy. By the union bound, we get that P S\u2032 e (s) \u2264P G e (s) + Q(\u2206/\u03c3). (155) The right term follows from the fact that in order to make a mistake to a codeword in a different copy of G, the noise must have an amplitude of at least \u2206. The average error probability over S\u2032 is bounded by Pe(S\u2032) \u2264Pe(G) + Q(\u2206/\u03c3) \u2264\u03b5 \u00b7 \u03c4\u03b5 + Q(\u2206/\u03c3) = \u03b5(1 + \u03be) (156) as required, where the last equality follows from the de\ufb01nition of \u03c4\u03b5 and \u2206. The density of points in the new IC enclosed within a cube of edge size a\u2217+2\u2206is given by |G|(a\u2217+ 2\u2206)\u2212n. De\ufb01ne \u02dc ak = (a\u2217+ 2\u2206)(2k \u22121) for any integer k. Note that for any k > 0, Cb(\u02dc ak) contains exactly (2k \u22121)n copies of G, and therefore M(S\u2032, \u02dc ak) \u02dc an k = |G|(2k \u22121)n \u02dc an k = |G| (a\u2217+ 2\u2206)n. (157) For any a > 0, let k\u2217be the minimal integer k s.t. \u02dc ak \u2265a. Clearly, \u02dc ak\u2217\u22121 = \u02dc ak\u2217\u2212(a\u2217+ 2\u2206) < a \u2264\u02dc ak\u2217. (158) Therefore M(S\u2032, \u02dc ak\u2217\u22121) an < M(S\u2032, a) an \u2264M(S\u2032, \u02dc ak\u2217) an , (159) and |G| (a\u2217+ 2\u2206)n \u02dc an k\u2217\u22121 an < M(S\u2032, a) an \u2264 |G| (a\u2217+ 2\u2206)n \u02dc an k\u2217 an . (160) By taking the limit a \u2192\u221eof (160), we get that the limit exists and is given by \u03b3(S\u2032) = lim a\u2192\u221e M(S\u2032, a) an = |G| (a\u2217+ 2\u2206)n. (161) It follows that \u03b3(S\u2032) = |G| (a\u2217+ 2\u2206)n = |G| an \u2217 an \u2217 (a\u2217+ 2\u2206)n (a) \u2265\u03b3(S) 1 \u03c4\u03b3 \u0012 a\u2217 a\u2217+ 2\u2206 \u0013n (b) \u2265\u03b3(S) 1 1 + \u03be. (162) where (a) follows from (151) and (b) follows from the de\ufb01nitions of \u03c4\u03b3, a\u2206and from the fact that a\u2206\u2264a\u2217. It remains to show that the resulting IC S\u2032 is regular, i.e. that all the Voronoi cells can be bounded in a sphere with some \ufb01xed radius r0. The fact that the average density is achieved in the limit (ant not only in the lim sup sense) was already established in (161). \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 47 Let s be an arbitrary point in S\u2032. By construction (see (154)), the points {s \u00b1 (a\u2217+ 2\u2206)ei|i = 1, ..., n} are also in S\u2032 (where ei denotes the vector of 1 in the i-th coordinate, and the rest are zeros). We therefore conclude that the Voronoi cell W(s) is contained in the hypercube s + Cb(a\u2217+ 2\u2206), and is clearly bounded within a sphere of radius r0 \u225c\u221an(a\u2217+ 2\u2206). APPENDIX E PROOF OF INTEGRAL BOUNDING LEMMAS Proof of Lemma 5: De\ufb01ne F(\u03c1) \u225clog \u0002 \u03c1 n 2 \u22121e\u2212n\u03c1/2\u0003 = \u0010n 2 \u22121 \u0011 log \u03c1 \u2212n\u03c1 2 , so that \u03c1 n 2 \u22121e\u2212n\u03c1/2 = exp[F(\u03c1)]. Let F1(\u03c1) and F2(\u03c1) be the \ufb01rst and second order Taylor series of F(\u03c1) around \u03c1 = x, respectively, i.e. F1(\u03c1) = \u03b1 + \u03b2(\u03c1 \u2212x); F2(\u03c1) = \u03b1 + \u03b2(\u03c1 \u2212x) \u2212\u03c4 2(\u03c1 \u2212x)2 (163) where \u03b1 \u225c \u0010n 2 \u22121 \u0011 log x \u2212nx 2 ; \u03b2 \u225c n 2 \u22121 x \u2212n 2; \u03c4 \u225c r n 2 \u22121 2x2 . We note that for any \u03be > 0, \u03be \u2212\u03be2 2 \u2264log(1 + \u03be) \u2264\u03be. (164) It follows that for all \u03c1 > x, F2(\u03c1) \u2264F(\u03c1) \u2264F1(\u03c1), (165) and the integral is bounded from above and below by Z \u221e x eF2(\u03c1)d\u03c1 \u2264 Z \u221e x \u03c1 n 2 \u22121e\u2212n\u03c1/2d\u03c1 \u2264 Z \u221e x eF1(\u03c1)d\u03c1. (166) To prove the upper bound (60) we continue with Z \u221e x eF (\u03c1)d\u03c1 \u2264 Z \u221e x eF1(\u03c1)d\u03c1 = e\u03b1 Z \u221e x exp [\u03b2(\u03c1 \u2212x)] d\u03c1 = e\u03b1 \u2212\u03b2 , where the last equality follows from the assumption x > 1 \u22122 n. Plugging the values for \u03b1 and \u03b2 yields (60). \f48 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY To prove the lower bound (60) we write Z \u221e x eF (\u03c1)d\u03c1 \u2265 Z \u221e x eF2(\u03c1)d\u03c1 = Z \u221e x exp \u0002 \u03b1 + \u03b2(\u03c1 \u2212x) \u2212\u03c4 2(\u03c1 \u2212x)2\u0003 d\u03c1 = Z \u221e 0 exp \u0002 \u03b1 + \u03b2\u03c1 \u2212\u03c4 2\u03c12\u0003 d\u03c1 = exp \u0012 \u03b1 + \u03b22 4\u03c4 2 \u0013 Z \u221e 0 exp \" \u2212 \u0012 \u03c4\u03c1 \u2212\u03b2 2\u03c4 \u00132# d\u03c1 = exp \u0012 \u03b1 + \u03b22 4\u03c4 2 \u0013 \u221a\u03c0 \u03c4 Z \u221e 0 \u03c4 \u221a\u03c0 exp \" \u2212\u03c4 2 \u0012 \u03c1 \u2212\u03b2 2\u03c4 2 \u00132# d\u03c1 = exp \u0012 \u03b1 + \u03b22 4\u03c4 2 \u0013 \u221a\u03c0 \u03c4 Q \u0012 \u2212\u03b2 \u03c4 \u221a 2 \u0013 . Plugging back the values for \u03b1, \u03b2 and \u03c4 leads to (61). We continue with a well known lower bound for the Q function: Q(z) \u2265 1 \u221a 2\u03c0z e\u2212z2/2 \u0012 1 1 + z\u22122 \u0013 \u2200z > 0. (167) Recalling the de\ufb01nition of \u03a5, we write Z \u221e x \u03c1 n 2 \u22121e\u2212n\u03c1/2d\u03c1 \u22652x n 2 e\u2212nx 2 exp \u0002 \u03a52/2 \u0003 r \u03c0 n \u22122Q (\u03a5) \u2265 2x n 2 e\u2212nx 2 n(x \u22121 + 2 n) \u0012 1 1 + \u03a5\u22122 \u0013 , to arrive at (62). Eq. (63) follows immediately since 1 \u2212\u03be \u2264 1 1+\u03be for all \u03be \u2208R. Proof of Lemma 6: We rewrite the integrand as eG(\u03c1) where G(\u03c1) \u225c\u2212n\u03c1/2 + (n \u2212 1) log \u03c1. Since G(\u03c1) is concave, it is upper bounded its \ufb01rst order Taylor approximation at any point. We choose the tangent at \u03c1 = x. We denote by G1(\u03c1) the \ufb01rst order Taylor approximation at that point, and get G(\u03c1) \u2264G1(\u03c1) \u225cG(x) + G\u2032(x)(\u03c1 \u2212x), (168) where G\u2032(\u03c1) = \u2202G(\u03c1) \u2202\u03c1 = \u2212n 2 + n\u22121 \u03c1 . It follows that G1(\u03c1) = (n \u22121)(log x \u22121) + \u0012 \u2212n 2 + n \u22121 x \u0013 \u03c1. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 49 Since G(\u03c1) \u2264G1(\u03c1) for all \u03c1, we have Z x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = Z x 0 eG(\u03c1)d\u03c1 \u2264 Z x 0 eG1(\u03c1)d\u03c1 = xn\u22121e\u2212(n\u22121) Z x 0 exp \u0014\u0012 \u2212n 2 + n \u22121 x \u0013 \u03c1 \u0015 d\u03c1 = 2xn n \u00002 \u2212x \u22122 n \u0001 \u0010 e\u2212n 2 x \u2212e\u2212(n\u22121)\u0011 , (169) which gives (75). Some extra effort is required in order to prove the lower bound (76). We \ufb01rst switch variables and getZ x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = Z \u221e 1/x e\u2212n 2u u\u2212n\u22121du (170) = Z \u221e 1/x exp \u0010 \u2212n 2u \u2212(n + 1) log u \u0011 du. (171) We lower bound the exponent as follows: \u2212n 2u \u2212(n + 1) log u = \u2212n 2u + (n + 1)(log x \u2212log(ux)) = \u2212n 2u + (n + 1)(log x \u2212log(1 + ux \u22121)) (a) \u2265\u2212n 2u + (n + 1)(log x \u2212(ux \u22121)) (b) \u2265\u2212nx 2 (u2x2 \u22123ux + 3) + (n + 1)(log x \u2212(ux \u22121)) = \u2212nx 2 [x2(u \u22121/x)2 \u2212x(u \u22121/x) + 1] + (n + 1)(log x \u2212x(u \u22121/x)). (a) follows from the fact that log(1 + \u03be) \u2264\u03be for all \u03be \u2208R. (b) follows from the fact that 1 \u03be \u2264\u03be2 \u22123\u03be + 3 for all \u03be > 1 (which follows from the fact that (\u03be \u22121)3 \u22650). So far the integral R x 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 is lower bounded by Z \u221e 1/x exp(\u03b1 + \u03b2(u \u22121/x) \u2212\u03c4 2(u \u22121/x)2)du, (172) where \u03b1 \u225c(n + 1) log x \u2212nx 2 ; \u03b2 \u225cnx2 2 \u2212(n + 1)x; \u03c4 \u225c r nx3 2 . Following the same steps as in the proof of (55) in Lemma 5 gives Z \u221e 1/x exp(\u03b1 + \u03b2(u \u22121/x) \u2212\u03c4 2(u \u22121/x)2) \u2265exp \u0012 \u03b1 + \u03b22 4\u03c4 2 \u0013 \u221a\u03c0 \u03c4 Q \u0012 \u2212\u03b2 \u03c4 \u221a 2 \u0013 . (173) \f50 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Plugging the values for \u03b1, \u03b2 and \u03c4 yields (76). (77) follows by applying the lower bound (167) on the Q function. APPENDIX F APPROXIMATING Vn Here we derive the required approximations for Vn, used in Sections V and VI. We \ufb01rst derive (64). The volume of a hypersphere of unit radius Vn is given by \u03c0n/2 \u0393(n/2+1) (see, e.g. [12, p. 9]). We use the Stirling approximation for the Gamma function for z \u2208R (see, e.g. [27, Sec. 5.11]). \u0393(z + 1) = z\u0393(z) = z r 2\u03c0 z \u0010z e \u0011z \u0012 1 + O \u00121 z \u0013\u0013 = \u221a 2\u03c0z \u0010z e \u0011z \u0012 1 + O \u00121 z \u0013\u0013 . (174) Vn becomes Vn = \u03c0n/2 \u0393(n/2 + 1) = \u00122\u03c0e n \u0013n/2 1 \u221an\u03c0 \u0012 1 + O \u00121 n \u0013\u0013 . (175) Eq. (117) follows by taking 1 n log(\u00b7) and the Taylor expansion of log(1+x) around x = 0: 1 n log Vn = 1 2 log \u03c0 \u22121 2n log n \u22121 2 log n 2e + O \u00121 n \u0013 = 1 2 log 2\u03c0e n \u22121 2n log n + O \u00121 n \u0013 . (176) APPENDIX G EVALUATING THE ML BOUND AT \u03b4cr Proof of Theorem 12: We start as in the proof of Theorem 5 to have en\u03b4crVn Z r\u2217 0 fR(r)rndr = n 2 en\u03b4V 2 n \u03c3n(2\u03c0)\u2212n 2 nn Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1. (177) We evaluate the integral in two parts: Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = Z 2 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 + Z \u03c1\u2217 2 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1. (178) The term R 2 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 can be evaluated by the Laplace method, as in the proof of Lemma 7. The difference is that the exponent is minimized with zero \ufb01rst derivative at the boundary point \u03c1 = 2, which causes the integral to be evaluated to half the value of the integral in Lemma 7, i.e. Z 2 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = r \u03c0 2ne\u2212n2n \u00001 + O \u0000 1 n \u0001\u0001 . (179) \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 51 The second term in (178) requires some extra effort. We \ufb01rst upper bound it as follows: Z \u03c1\u2217 2 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = Z \u03c1\u2217 2 1 \u03c1e\u2212n\u03c1/2\u03c1nd\u03c1 (a) \u2264 Z \u03c1\u2217 2 1 2e\u2212n\u03c1/2\u03c1nd\u03c1 (b) \u2264 Z \u03c1\u2217 2 1 2e\u2212n2nd\u03c1 = 1 2e\u2212n2n(\u03c1\u2217\u22122), where (a) follows since in the integration interval, \u03c1 > 2. (b) follows since e\u2212n\u03c1/2\u03c1n is maximized at \u03c1 = 2. With (66) we have Z \u03c1\u2217 2 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 \u22641 2e\u2212n2n(\u03c1\u2217\u22122) = 1 2e\u2212n2n \u0010 2 n log(n\u03c0) + O \u0010 log2 n n2 \u0011\u0011 = e\u2212n2nlog(n\u03c0) n \u00001 + O \u0000 log n n \u0001\u0001 . The integral can also be lower bounded as follows: Z \u03c1\u2217 2 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 (a) \u22651 \u03c1\u2217 Z \u03c1\u2217 2 e\u2212n\u03c1/2\u03c1nd\u03c1 (b) \u22651 \u03c1\u2217 Z \u03c1\u2217 2 en log 2 e \u2212n 8 (\u03c1\u22122)2d\u03c1 = 1 \u03c1\u22172ne\u2212n Z \u03c1\u2217 2 e\u2212n 8 (\u03c1\u22122)2d\u03c1 = 1 \u03c1\u22172ne\u2212n Z \u03c1\u2217\u22122 0 e\u2212n 8 \u03c12d\u03c1 = 1 \u03c1\u22172ne\u2212n r 8\u03c0 n \u0012 Q(0) \u2212Q \u0012 \u03c1\u2217\u22122 \u221a 4/n \u0013\u0013 (c) = 1 \u03c1\u22172ne\u2212n r 8\u03c0 n \u0012 1 2 \u2212 \u0012 1 2 \u2212 1 \u221a 2\u03c0 \u03c1\u2217\u22122 \u221a 4/n + O \u0010 log2 n n \u0011\u0013\u0013 = 1 \u03c1\u22172ne\u2212n \u0010 \u03c1\u2217\u22122 + O \u0010 log2 n n\u221an \u0011\u0011 = 2ne\u2212nlog(n\u03c0) n \u0010 1 + O \u0010 log n \u221an \u0011\u0011 . (a) follows since \u03c1 \u2264\u03c1\u2217. (b) follows from the fact that n\u03c1/2+n log \u03c1 \u2265n log 2 e \u2212n 8(\u03c1\u22122)2 for all \u03c1 > 2 (which follows from (164)). (c) follows from the Taylor expansion Q(\u03be) = 1 2 \u2212 \u03be \u221a 2\u03c0 + O(\u03be2) and since \u03c1\u2217\u22122 = O( log n n ). In total we get Z \u03c1\u2217 2 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = 2ne\u2212nlog(n\u03c0) n \u0010 1 + O \u0010 log n \u221an \u0011\u0011 . \f52 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY Combined with (179) we have Z \u03c1\u2217 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = Z 2 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 + Z \u03c1\u2217 2 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = 2ne\u2212n \u0014r \u03c0 2n + log(n\u03c0) n \u0015 \u0010 1 + O \u0010 log2 n n \u0011\u0011 . The approximation (64) for Vn \ufb01nally yields en\u03b4crVn Z r\u2217 0 fR(r)rndr = e\u2212nEr(\u03b4cr) 1 2\u03c0 \u0014r \u03c0 2n + log(n\u03c0) n \u0015 \u0010 1 + O \u0010 log2 n n \u0011\u0011 , (180) and the proof is completed by adding the asymptotic form (58) of the sphere bound at \u03b4 = \u03b4cr. APPENDIX H Proof of Theorem 13: The typicality bound is given by Pe(n, \u03b4) \u2264\u03b3Vnrn + Pr {\u2225Z\u2225> r} , (181) where r = \u03c3 p n(1 + 2\u03b4\u2217\u22122\u03b4). The rightmost term can be written as (see the proof of Theorem 9): Pr {\u2225Z\u2225> r} = (n/2)n/2 \u0393 \u0002n 2 \u0003 Z \u221e 1+2(\u03b4\u2217\u2212\u03b4) \u03c1 n 2 \u22121e\u2212n 2 \u03c1d\u03c1. (182) The above integral can be bounded according to Lemma 5 by 2x n 2 e\u2212nx 2 n(x \u22121 + 2 n) 1 1 + \u03a5\u22122 \u2264 Z \u221e x \u03c1 n 2 \u22121e\u2212n 2 \u03c1d\u03c1 \u2264 2x n 2 e\u2212nx 2 n(x \u22121 + 2 n) (183) where x = 1 + 2(\u03b4\u2217\u2212\u03b4) and \u03a5 \u225c n(x\u22121+ 2 n) \u221a 2(n\u22122) = \u0398(\u221an). Eq. (92) then follows using the approximation (64) for Vn. APPENDIX I Proof of Theorem 14: We \ufb01rst prove (93) (and (94) follows immediately). Let \u02dc \u03c1 = e2(\u03b4\u2217\u2212\u03b4). The ML bounds for r = \u221an\u03c3e\u03b4\u2217\u2212\u03b4 can be written as (see (59) and (74)): P MLB e = n 2en(\u03b4\u2217+\u03b4)V 2 n en/2\u03c32nnn Z \u02dc \u03c1 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 + 2\u2212n/2nn/2 \u0393(n/2) Z \u221e \u02dc \u03c1 e\u2212n\u03c1/2\u03c1n/2\u22121d\u03c1. Using Lemma 5 we get that Z \u221e \u02dc \u03c1 e\u2212n\u03c1/2\u03c1n/2\u22121d\u03c1 = 2\u02dc \u03c1n/2e\u2212n\u02dc \u03c1/2 n(\u02dc \u03c1 \u22121) \u0012 1 + O \u00121 n \u0013\u0013 , (184) and using Lemma 6 we get that Z \u02dc \u03c1 0 e\u2212n\u03c1/2\u03c1n\u22121d\u03c1 = 2\u02dc \u03c1n/2e\u2212n\u02dc \u03c1/2 n(2 \u2212\u02dc \u03c1) \u0012 1 + O \u00121 n \u0013\u0013 . (185) (93) then follows by simple algebra. \fINGBER ET AL.: FINITE DIMENSIONAL INFINITE CONSTELLATIONS 53 To show (95), repeat the proof of Theorem 11 with \u02dc \u03c1 instead of \u03c1\u2217. To show (96), repeat the proof of Theorem 12 with \u02dc \u03c1 instead of \u03c1\u2217. Here \u02dc \u03c1 = 2, therefore there is no need to split the integral into two parts as in (178). Therefore the term en\u03b4Vn R r 0 fR(\u02dc r)\u02dc rnd\u02dc r contributes the 1 \u221a 8 part of the expression. The contribution of the sphere bound term (the term Pr{\u2225Z\u2225> r}) is approximated as in (184). Note that the result here is different than the re\ufb00case, where the contributions of the two terms in the bound are of different order (see Eq. (89)). APPENDIX J CENTRAL LIMIT THEOREM AND THE BERRY-ESSEEN THEOREM By the central limit theorem (CLT), A normalized sum of n independent random variables converges (in distribution) to a Gaussian random variable. The Berry-Esseen theorem shows the speed of the convergence (see [28, Ch. XVI.5]). We write here the version for i.i.d. random variables, which is suf\ufb01cient for this paper. Theorem 17 (Berry-Esseen for i.i.d. RV\u2019s [28]): Let {Yi}n i=1 be i.i.d. random variables with zero mean and unit variance. Let T \u225cE[|Yi|3] and assume it is \ufb01nite. Let Sn \u225c 1 \u221an Pn i=1 Yi be the normalized sum. Note that Sn also has zero mean and unit variance. Then for all \u03b1 \u2208R and for all n \u2208N, | Pr{Sn \u2265\u03b1} \u2212Q(\u03b1)| \u22646T \u221an. (186)" + }, + { + "url": "http://arxiv.org/abs/1102.2598v1", + "title": "The Dispersion of Lossy Source Coding", + "abstract": "In this work we investigate the behavior of the minimal rate needed in order\nto guarantee a given probability that the distortion exceeds a prescribed\nthreshold, at some fixed finite quantization block length. We show that the\nexcess coding rate above the rate-distortion function is inversely proportional\n(to the first order) to the square root of the block length. We give an\nexplicit expression for the proportion constant, which is given by the inverse\nQ-function of the allowed excess distortion probability, times the square root\nof a constant, termed the excess distortion dispersion. This result is the dual\nof a corresponding channel coding result, where the dispersion above is the\ndual of the channel dispersion. The work treats discrete memoryless sources, as\nwell as the quadratic-Gaussian case.", + "authors": "Amir Ingber, Yuval Kochman", + "published": "2011-02-13", + "updated": "2011-02-13", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION Rate-distortion theory [1] tells us that in the limit of large block-length n, a discrete memoryless source (DMS) with distribution p can be represented with some average distortion D by a code of any rate greater than the rate-distortion function (RDF) R(p, D) = min W :Ep,W [d(X, \u02c6 X)]\u2264D I(p, W), (1) where d(x, \u02c6 x) is the distortion measure, W(\u02c6 x|x) is any channel from the source to the reproduction alphabet and I(\u00b7, \u00b7) denotes the mutual information. However, beyond the expected distortion, one may be interested in ensuring that the distortion for one source block is below some threshold. To that end, we de\ufb01ne an excess distortion event E(D) as E(D) \u225c{d(x, \u02c6 x) > D}, (2) where d(x, \u02c6 x) \u225c 1 n Pn i=1 d(xi, \u02c6 xi) is the distortion between the source and reproduction words x and \u02c6 x. A natural question to ask is how fast can the probability of such event be made to decay as a function of the block length. An asymptotic answer is given by Marton\u2019s excess distortion exponent [2]: for the best code of rate R, lim n\u2192\u221e\u22121 n log Pr{E(D)} = min q:R(q,D)\u2265R D(q\u2225p) \u225cF(R, p, D), (3) assuming the limit exists. D(\u00b7\u2225\u00b7) is the divergence between the two distributions.1 Intuitively speaking, this result means that, asymptotically, the error probability is governed by the \ufb01rstorder empirical statistics of the source sequence; if the sequence happens to be \u201ctoo rich\u201d to be quantized with rate R, en error (excess distortion event) will occur. 1Throughout the paper logarithms are taken with the natural base e and rates are given in nats. \fWe are interested in the following related question: for a given excess distortion probability \u03b5, what is the optimal (minimal) rate required to achieve it? This question is unanswered by Marton\u2019s exponent, and even the asymptotical behavior of the optimal rate is unknown. A similar question can be asked in the context of channel coding: for a given error probability \u03b5, what is the maximal communication rate that can be achieved. Again, this question is unanswered by the channel error exponent [3]. The asymptotics of the rate behavior was \ufb01rst studied in the 1960\u2019s [4] using the normal approximation. This result was recently tightened and extended to the Gaussian channel, along with nonasymptotic results, in a comprehensive work by Polyanskiy et al. [5]. In channel coding the maximal rate that can be achieved over a channel W is approximately given by R \u223c = C(W) \u2212 r V (W) n Q\u22121(\u03b5), (4) where C(W) is the channel capacity, Q is the complementary Gaussian cumulative distribution function, and the quantity V (W) is a constant that depends on the channel only, termed the channel dispersion. See [5] for details and more re\ufb01nements of (4). Our main result is the following. Suppose the source p is to be quantized with distortion threshold D, and a \ufb01xed probability for excess distortion \u03b5 > 0. Then the minimal rate R needed for quantization in blocks of length n is given by R \u223c = R(p, D) + r V (p, D) n Q\u22121(\u03b5), (5) where V (p, D) is a constant which we call the excess distortion dispersion, given in detail later on. We show that (5) holds for any DMS under some smoothness conditions on R(p, D), and for a Gaussian source with quadratic distortion measure, see Theorems 1 and 2 respectively. It is worth noting that that there is a large body of previous work regarding the redundancy of lossy source coding in related setting. However, these works are mostly concerned with two questions: the behavior of the word-length of variable-rate codes where the distortion should always be below some threshold (a.k.a. D-semifaithful codes) [6], or the average excess distortion of \ufb01xed-rate codes; see e.g. [7],[8] and the references therein. We consider the excess-distortion probability, thus bridging between these works and the concepts of excess-distortion exponent and dispersion discussed above. In this context, the work by Kontoyiannis [8] is of special interest, since it introduces a constant which equals V (p, D), see in the sequel. II. MAIN RESULT FOR DISCRETE MEMORYLESS SOURCES Let the source X be drawn from an i.i.d. distribution p over the alphabet X = {1, ..., L}, and let the reproduction alphabet be \u02c6 X = {1, ..., K}. The distribution p can be seen as a vector p = [p1, ..., pL]T \u2208PL, where pi = Pr(X = i) and PL is the probability simplex: PL \u225c ( q \u2208RL|qi \u22650\u2200i \u2208{1..L}; L X i=1 qi = 1 ) . (6) Let d : X \u00d7 \u02c6 X \u2192R+ denote a general nonnegative single-letter distortion measure, bounded by some \ufb01nite Dmax. Denote the rate distortion function for the source p and the distortion \fmeasure d(\u00b7, \u00b7) at some level D by R(p, D). Whenever this function is differentiable w.r.t. its coordinates pi, de\ufb01ne the partial derivatives by R\u2032(i) \u225c \u2202 \u2202qi R(q, D) \f \f \f \f q=p . (7) Note that R\u2032(i) implicitly depends on p and D as well. For a random source symbol X, we may look at R\u2032(i) as the values that a random variable R\u2032(X) takes. Also note that in order to de\ufb01ne the derivative, we extend the de\ufb01nition of the RDF R(p, D) to general vectors in (0, 1)L (cf. [6, Theorem 2]). In any case, we will only be interested in the value of this derivative for values of p within the simplex, i.e. that represent probability distributions. Let x \u2208X n and \u02c6 x \u2208\u02c6 X n denote the source and reproduction words respectively. Recalling (2), let Rp,D,\u03b5(n) be the optimal (minimal) code rate at length n s.t. the probability of an excess distortion event E(D) is at most \u03b5. It is known that Rp,D,\u03b5(n) \u2192R(p, D) as n \u2192\u221e. This can be deduced e.g. by Marton\u2019s excess distortion exponent [2]. Our main result quanti\ufb01es the rate of this convergence. Theorem 1: A DMS with probability p is to be quantized with distortion threshold D, block length n and excess distortion probability \u03b5. Assume that R(q, D) is differentiable w.r.t. D and twice differentiable w.r.t. q in some neighborhood of (p, D). Then Rp,D,\u03b5(n) = R(p, D) + r V (p, D) n Q\u22121(\u03b5) + O \u0012log n n \u0013 , (8) where V (p, D) is the excess distortion dispersion, given by V (p, D) \u225cVar[R\u2032(X)] = L X i=1 pi(R\u2032(i))2 \u2212 \" L X x=1 piR\u2032(i) #2 . (9) This result is closely related to the following central-limit theorem (CLT) result of [8]. If we allow a code with variable rate r(x) \u225cl(x)/n, where l(x) is the length of the codeword needed to describe the source word x, then for the best code: r(x) = R(p, D) + Gn \u221an + O \u0012log n n \u0013 , where {Gn} converge in distribution to a Gaussian random variable of variance V (p, D).2 If Gn are exactly Gaussian, and then we truncate this variable-length code by assuming an excess-distortion event at each time that the length is over nR, then the excess distortion probability exactly satis\ufb01es the achievability bound of Theorem 1. However, this is not immediate, as one needs to take into account the rate of convergence of the sequence {Gn}. We follow a different direction, which is closer in spirit to the derivation of the excess distortion exponent in [2]. Speci\ufb01cally, we show that the O(1/\u221an) redundancy term comes only from the probability that the source will produce a sequence whose type is too complex to be covered with rate R. The proof is based on the method of types. We adopt the notation of Csisz\u00b4 ar and K\u00a8 orner [9]: The type of a sequence x \u2208X n is the vector Px \u2208PL whose elements are the relative frequencies of the alphabet letters in X . Tn denotes all the types of sequences of length n. 2The variance has a different expression in [8], we show in Section III-B that the forms are equivalent. \fWe say that a sequence x has type q \u2208Tn if Px = q. The type class of the type q \u2208Tn, denoted Tq, is the set of all sequences x \u2208X n with type q. For a reconstruction word \u02c6 x \u2208\u02c6 X, we say that x is D-covered by \u02c6 x if d(x, \u02c6 x) \u2264D. Proposition 1 (Type covering): Let q \u2208Tn with a corresponding type class Tq. Let A(q, C, D) be the intersection of Tq with the set of source sequences x \u2208X n which are D-covered by at least one of the words in a codebook C with rate R (i.e. |C| = enR). Then: 1) If |\u2202R(q, D)/\u2202D| is bounded in some neighborhood of q, then there exists a codebook Cq that completely D-covers Tq (i.e. A(q, Cq, D) = Tq), where for large enough n, 1 n log |Cq| = R \u2264R(q, D) + J1 log n n , (10) where J1 = J1(L, K) is a constant. 2) For any type q \u2208Tn s.t. R(q, D) > R, the fraction of the type class that is D-covered by any code with rate R is bounded by |A(q, Cn, D)| |Tq| \u2264exp \u001a \u2212n \u0014 R(q, D) \u2212R + J2 log n n \u0015\u001b , (11) where J2 = J2(L, K) is a constant. The \ufb01rst part of this proposition is a re\ufb01nement of Berger\u2019s type-covering lemma [1], found in [6]. The second part is a corollary of [7, Lemma 3]. Both parts of the proposition are stronger versions than needed in [2], due to the non-exponential treatment of the excess distortion probability.3 Equipped with this, the missing ingredient is an analysis of the relation between the rate R and the probability of the source to produce a type which requires a description rate higher than R. It is given in the following lemma which is proved in Section V. Lemma 1 (Rate Redundancy): Consider a DMS p and a distortion threshold D. Assume that R(p, D) is differentiable w.r.t. D and twice differentiable w.r.t. p at some neighborhood of (p, D). A random source word is denoted by x and its type by Px. Let \u03b5 be a given probability and let \u2206R be chosen s.t. Pr{R(Px, D) \u2212R(p, D) > \u2206R} = \u03b5. Then, as n grows, \u2206R = r V (p, D) n Q\u22121(\u03b5) + O \u0012log n n \u0013 , (12) where V (p, D) is given by (9). The same holds even if we replace \u03b5 with \u03b5 + gn, as long gn = O \u0010 log n \u221an \u0011 . Proof of Theorem 1: Achievability part. Let \u2206R > 0. We construct a code C as follows. The code shall consist of the union of the codes that cover all the types q \u2208\u03a6(n, D, \u2206R), where \u03a6(n, D, \u2206R) = {q : R(q, D) \u2264R(p, D) + \u2206R} \u2229\u2126n, (13) where \u2126n = \b q : \u2225p \u2212q\u22252 \u2264Llog n n \t . 3For the \ufb01rst part, Marton uses Berger\u2019s original lemma, while for the second part it is proved that the ratio between |Tq| and |A(q, C, D)| is upper-bounded by a constant. \fLemma 2: For a source word x drawn from the p, we have Pr{Px / \u2208\u2126n} \u22642L n2 . The proof for this technical lemma is omitted. It can be proved using techniques similar to those in [6, Theorem 2]. The size of the code is bounded by |C| \u2264 X q\u2208\u03a6(n,D,\u2206R) |Cq| \u2264|Tn||Cq\u2217| \u2264(n + 1)L|Cq\u2217|, (14) where q\u2217is the largest type class that is covered. Since we assumed that R(p, D) is differentiable w.r.t. D at p, the derivative is bounded over any small enough neighborhood of p. In particular, it is bounded over \u2126n for large enough n, thus for all types covered by the codebook. We can thus apply part 1 of Proposition 1 and we get a bound on the rate: R = 1 n log |C| \u2264L n log(n + 1) + 1 n log |Cq\u2217| (15) \u2264R(p, D) + \u2206R + O \u0012log n n \u0013 . (16) Since we completely cover all the types in \u03a6(n, D, \u2206R), we have that the probability of excess distortion (2) satis\ufb01es Pr{E(D)} = Pr n Px / \u2208\u03a6(n, D, \u2206R) o \u2264Pr n R(Px, D) \u2264R(p, D) + \u2206R o + Pr{Px / \u2208\u2126n} (17) \u2264Pr n R(Px, D) \u2264R(p, D) + \u2206R o + 2L n2 . (18) where (17) follows from the union bound, and (18) is justi\ufb01ed by Lemma 2. We select \u2206R s.t. the probability for {R(Px, D) > R(p, D) + \u2206R} is exactly \u03b5 \u22122L n2 , and get a code with excess distortion probability at most \u03b5. By Lemma 1 we have \u2206R = r V (p, D) n Q\u22121(\u03b5) + O \u0012log n n \u0013 , and by plugging into (16) the rate R is bounded by the RHS of (8), as required. Converse part. Let C be a code with rate R, and suppose that its excess distortion probability is \u03b5. Our goal is to lower bound \u2206R = R \u2212R(p, D). Again, the source word is x and its type is Px. The following holds for any \u03a8: \u03b5 = Pr{E(D)} = Pr n E(D)|R(Px, D) \u2264R + \u03a8 o Pr n R(Px, D) \u2264R + \u03a8 o + Pr n E(D)|R(Px, D) > R + \u03a8 o Pr n R(Px, D) > R + \u03a8 o \u2265Pr n E(D)|R(Px, D) > R + \u03a8 o Pr n R(Px, D) > R + \u03a8 o . (19) Take a type q \u2208Tn, and assume that R(q, D) > R + \u03a8. By the second part of Proposition 1, the fraction of the type class Tq that is covered by the code C is at most exp \u001a \u2212n \u0014 R(q, D) \u2212R + J2 log n n \u0015\u001b \u2264exp {\u2212n\u03a8 + J2 log n} (20) \fBy setting \u03a8 = (J2 + 1) log n n we get that the fraction is bounded by 1/n. Since the source sequences within a given type are uniformly distributed, we get that the probability of covering a sequence from a type that its R(Px, D) is too high is at most 1/n. We therefore have \u03b5 \u2265 \u0012 1 \u22121 n \u0013 Pr n R(Tx, D) > R + \u03a8 o \u2265 1 1 + 2 n Pr n R(Tx, D) > R + \u03a8 o , (21) where the last inequality follows since 1 \u2212x \u2265 1 1\u22122x for all x \u2208[0, 1/2]. We rewrite (21) and get that \u2206R must satisfy \u03b5 \u0012 1 + 2 n \u0013 \u2265Pr n R(Tx, D) \u2212R(p, D) > \u2206R + \u03a8 o . (22) By Lemma 1 and the fact that \u03a8 = O \u0000log n n \u0001 , we get \u2206R \u2265 r V (p, D) n Q\u22121(\u03b5) + O \u0012log n n \u0013 , (23) as required. III. EXCESS DISTORTION DISPERSION: PROPERTIES AND EVALUATION A. Differentiability of the RDF In the results above, we assumed differentiability of the RDF R(p, D) with respect to D (once) and p (twice). In general, the RDF is not differentiable w.r.t. either. However, it is differentiable \u201calmost always\u201d in the following sense. Let K\u2032(p, D) be the \u201ceffective reproduction alphabet size\u201d, i.e., the number of reproduction letters of positive probability for the channel minimizing (1). Then, if K\u2032(p, D) is constant in a neighborhood of D, then R(p, D) is differentiable w.r.t. D and twice differentiable w.r.t. p at that point. When keeping p \ufb01xed and changing D, such points may represent \u201cjumps\u201d in the excess distortion dispersion V (p, D). In these points, we can not specify the exact behavior of the excess rate, but careful derivation should verify that it is between V (p, D\u2212) and V (p, D+). However, in the process we will encounter at most L \u22122 such points. B. Alternative Representations The evaluation of the the excess distortion dispersion seems to be a dif\ufb01cult task, as it involves derivatives of the RDF w.r.t. the source distribution. However, we have the following alternative representations. First we connect the dispersion to the excess-distortion exponent (3), much in the same way that the channel dispersion constant is related to the channel error exponent; See [5] for details on the early origins of this approximation by Shannon. Proposition 2: If R(p, D) is differentiable at distortion level D, then V (p, D) = \" \u22022 \u2202R2F(R, p, D) \f \f \f \f R=R(p,D) #\u22121 . The proof, not included in this version, follows by directly considering the exponent de\ufb01nition (3) in the limit of small excess rate. \fWe further show equivalence to the variance of the excess rate in [8], which is close in spirit to the dispersion as discussed in Section II: Proposition 3: If R(p, D) is differentiable at distortion level D, then V (p, D) = Var[f(X)] where f(i) = \u2212log E \u02c6 X exp{\u2212\u03bb[d(xi, \u02c6 x) \u2212D]}, where the expectation is taken according to the reproduction distribution induced by the channel minimizing (1) for p and D, and \u03bb = \u2202R(p, D)/\u2202D at that point. This form is especially appealing, since it can also be shown that R(p, D) = E{f(X)}, thus presenting the dispersion as a \u201csecond-order RDF\u201d. The equivalence can be proven by starting from the RDF presentation above. Applying (9), V (p, D) = Var \uf8f1 \uf8f2 \uf8f3 \u2202 \u2202qi L X j=1 qjf(j) \f \f \f \f \f q=p \uf8fc \uf8fd \uf8fe= Var ( f(i) + L X j=1 pj \u00b7 \u2202f(j) \u2202qi \f \f \f \f q=p ) . Straightforward derivation shows that the term to the right of the addition in the last form is constant in i, thus it does not effect the variance, as required. C. Some Special Cases In some cases the evaluation may be simpli\ufb01ed, as follows. 1) Zero distortion. Whenever R(p, 0) = H(p), we have R\u2032(i) = \u2202 \u2202qi H(q) \f \f \f \f q=p = \u22121 \u2212log pi. Thus, V (p, 0) = Var{log pi}. (24) This is in agreement with the long known lossless dispersion result [4]. 2) Difference distortion measure with low distortion. Assume that d(x, \u02c6 x) = d([x \u2212\u02c6 x] mod L) \u225cd(z). Since we assumed that each source letter has positive probability, there exists some D0(p) > 0 s.t. for all D \u2264D0 the optimum backword channel is x = \u02c6 x + z. The RDF is then given by R(p, D) = H(p) \u2212H(wz) (25) where wz is the maximum-entropy distribution such that E{d(z)} \u2264D [1, Sec. 4.3.1]. Since this distribution is D-independent as long as D < D0(p), we have that the second term in (25) is \ufb01xed in p in a neighborhood of the source distribution. Consequently the derivatives only come from the \ufb01rst term, and (24) holds for all 0 \u2264D < D0. 3) Hamming distortion measure. In this special case of a difference distortion measure, the optimum backward channel is modulo-additive also above D0, where the modulo is taken over a reduced alphabet. Consequently, the dispersion is the variance of the logarithm of a normalized smaller-alphabet distribution. 4) Zero dispersion. The dispersion becomes zero when the source distribution maximizes the RDF over all possible source distributions among the input alphabet (thus the rate \fredundancy in Lemma 1 is zero). Note that this is in agreement with the fact that for this case the excess-distortion exponent \u201cjumps\u201d from zero to in\ufb01nity at zero excess rate. For difference measures, this happens if and only if the source is uniform, in agreement with the observation in [8]. However, in general p need not be uniform. IV. GAUSSIAN SOURCE WITH QUADRATIC DISTORTION MEASURE In this section we part with the assumption that the source is discrete. While the derivation of the excess distortion dispersion for general continuous-amplitude sources is left for future work, we solve the important special case of Gaussian source with MSE (quadratic) distortion measure. Let the source X be i.i.d. zero-mean Gaussian with variance \u03c32. The distortion measure is given by: d(x, y) = (x \u2212y)2. For D \u2264\u03c32, the quadratic-Gaussian RDF is given by: R(\u03c32, D) = 1 2 log \u0012\u03c32 D \u0013 . (26) In this case, the excess distortion exponent (3) is given by [10]: F(R, \u03c32, D) = 1 2 \u0014 D \u03c32e2R \u22121 \u2212log \u0012 D \u03c32e2R \u0013\u0015 = e2\u2206R \u22121 \u22122\u2206R 2 , (27) where \u2206R = R \u2212R(\u03c32, D). As in the \ufb01nite alphabet case, we de\ufb01ne R\u03c32,D,\u03b5(n) to be the minimal code rate at length n s.t. the excess distortion probability is at most \u03b5. From the excess distortion exponent (27) it follows that R\u03c32,D,\u03b5(n) \u2192R(\u03c32, D) as n \u2192\u221e. We are interested in the behavior of R\u03c32,D,\u03b5(n) as n grows. We show that the quadraticGaussian case behaves according to (5) just like the \ufb01nite-alphabet one. Recalling Proposition 2, one expects the dispersion constant to be V (\u03c32, D) = \" \u22022 \u2202R2 F(R, \u03c32, D) \f \f \f \f R=R(\u03c32,D) #\u22121 = 1 2. It can also be shown that the value of 1 2 can be obtained by a continuous version of (9). We now show that this is the case indeed. Theorem 2: Let \u03b5 > 0 be a given excess distortion probability. Then the rate R\u03c32,D,\u03b5(n) satis\ufb01es O \u00121 n \u0013 \u2264R \u2212R(\u03c32, D) \u2212 r 1 2nQ\u22121(\u03b5) \u22645 2n log n + O \u0012 1 n \u0013 (28) Proof outline: The proof is similar in spirit to the proof of Theorem 1, where spheres take the part of types. The type class of types near the source distribution is analogous to a sphere with radius r, where r2 is close to n\u03c32. For the achievability part, we de\ufb01ne a \u201ctypical\u201d sphere with radius p n\u03c32(1 + \u03b1n) with \u03b1n \u21920 as n \u2192\u221e. \u03b1n is chosen s.t. the probability that the source falls outside the sphere is exactly \u03b5, so our code needs to D-cover the entire sphere. Note that the radius is just over the typical radius of the source. We use a sphere covering result by Rogers [11, Theorem 3], and \ufb01nd a code that can D-cover the entire typical sphere with no more than cn5/2 (\u03c32(1 + \u03b1n)/D)n/2 reconstruction words for some constant c. By arguments similar to those used in the proof of Lemma 1 we get \u03b1n = p 2/nQ\u22121 (\u03b5) + O \u0000 1 n \u0001 , so the rate R is bounded according to (28). \fFor the converse part, we follow the proof of the converse to the excess distortion exponent in [10]. We get that the excess distortion probability is lower bounded by the probability to leave a sphere that has a volume of enR times the volume of a single D-ball around a reconstruction point. Again, using the Berry-Esseen theorem we connect excess distortion probability and the ratio of the radiuses, and get that the rate R is lower bounded according to (28). V. PROOF OF THE RATE REDUNDANCY LEMMA Proof of Lemma 1: Let x be a source word with type Px, drawn from the source p. We prove the more general version of the lemma, with \u03b5 + gn being the given probability. The relation between \u03b5 and \u2206R is given by \u03b5 + gn = Pr {R(Px, D) > Rp(D) + \u2206R} . (29) By the regularity assumptions on R(p, D), we use the Taylor approximation and write R(Px, D) = R(p, D) + L X i=1 (Px(i) \u2212pi)R\u2032(i) + \u03b3(Px, p), (30) where R\u2032(\u00b7) was de\ufb01ned in (7), and \u03b3(Px, p) is the correction term for the approximation. Equation (29) now becomes \u03b5 + gn = Pr ( L X i=1 (Px(i) \u2212pi)R\u2032(i) + \u03b3(Px, p) > \u2206R ) . (31) By the Taylor approximation theorem, and by the assumption of \ufb01nite second derivatives of R(p, D), we have that the correction term \u03b3(Px, p) = O(\u2225Px \u2212p\u22252). This means that there exists a constant \u03b7, s.t. for large enough n, \u03b3(Px, p) < \u03b7\u2225Px \u2212p\u22252. By Lemma 2 there exists \u0393 = O(log n/n) s.t. Pr{\u03b3(Px, p) > \u0393} \u22642L n2 . Using simple probability rules, for any random variables A and B and a constant c, we have that for any \u03931, \u03932 the following holds: Pr n A + B > c o \u2264Pr n A > c \u2212\u03931 o + Pr n B > \u03931 o (32) Pr n A + B > c o \u2265Pr n A > c + \u03932 o \u2212Pr n B < \u2212\u03932 o (33) In our case, we use (32) (resp. (33)) to show the upper (resp. lower) bound on \u2206R. By selecting \u03931 = \u03932 = \u0393, we get \u03b5 + gn \u2264Pr ( L X i=1 (Px(i) \u2212pi)R\u2032(i) > \u2206R \u2212\u0393 ) + O \u0012 1 n2 \u0013 , (34) \u03b5 + gn \u2265Pr ( L X i=1 (Px(i) \u2212pi)R\u2032(i) > \u2206R + \u0393 ) \u2212O \u0012 1 n2 \u0013 . (35) Now consider the probability expression in (34): Pr ( L X i=1 (Px(i) \u2212pi)R\u2032(i) > \u2206R \u2212\u0393 ) = Pr ( 1 n n X k=1 R\u2032(xk) \u2212 L X i=1 piR\u2032(i) > \u2206R \u2212\u0393 ) . \f1 n Pn k=1 R\u2032(xk) can be interpreted as an average of n i.i.d. random variables R\u2032(X), whose expectation is given by E[R\u2032(X)] = PL i=1 piR\u2032(i). Their variance is given by V (p, D), de\ufb01ned in (9). By the central limit theorem, the sum of i.i.d. random variables normalized by \u221an converges to a Gaussian random variable as n grows. Speci\ufb01cally, by the BerryEsseen theorem (see, e.g. [12, Ch. XVI.5]), we get Pr ( 1 \u221an n X k=1 (R\u2032(xk) \u2212E[R\u2032(X)]) > \u221an(\u2206R \u2212\u0393) ) = Q \u0012 (\u2206R \u2212\u0393) r n V (p, D) \u0013 \u00b1 6\u03be \u221an, (36) where \u03be = E [|R\u2032(X) \u2212E[R\u2032(X)]|3]. By applying the same derivation \u2206R + \u0393, (34) and (35) can be written together as \u03b5 + O \u0012log n \u221an \u0013 = Q \u0012 (\u2206R \u00b1 \u0393) r n V (p, D) \u0013 . (37) By the smoothness of Q\u22121(\u00b7) around \u03b5 and the Taylor approximation we have \u2206R = r V (p, D) n Q\u22121 (\u03b5) + O \u0012log n n \u0013 , (38) as required." + }, + { + "url": "http://arxiv.org/abs/1007.1407v2", + "title": "Parallel Bit Interleaved Coded Modulation", + "abstract": "A new variant of bit interleaved coded modulation (BICM) is proposed. In the\nnew scheme, called Parallel BICM, L identical binary codes are used in parallel\nusing a mapper, a newly proposed finite-length interleaver and a binary dither\nsignal. As opposed to previous approaches, the scheme does not rely on any\nassumptions of an ideal, infinite-length interleaver. Over a memoryless\nchannel, the new scheme is proven to be equivalent to a binary memoryless\nchannel. Therefore the scheme enables one to easily design coded modulation\nschemes using a simple binary code that was designed for that binary channel.\nThe overall performance of the coded modulation scheme is analytically\nevaluated based on the performance of the binary code over the binary channel.\nThe new scheme is analyzed from an information theoretic viewpoint, where the\ncapacity, error exponent and channel dispersion are considered. The capacity of\nthe scheme is identical to the BICM capacity. The error exponent of the scheme\nis numerically compared to a recently proposed mismatched-decoding exponent\nanalysis of BICM.", + "authors": "Amir Ingber, Meir Feder", + "published": "2010-07-08", + "updated": "2010-08-17", + "primary_cat": "cs.IT", + "cats": [ + "cs.IT", + "math.IT" + ], + "main_content": "INTRODUCTION Bit interleaved coded modulation (BICM) is a pragmatic approach for coded modulation [1]. It enables the construction of nonbinary communication schemes from binary codes by using a long bit interleaver that separates the coding and the modulation. BICM has drawn much attention in recent years, because of its ef\ufb01ciency for wireless and fading channels. The information-theoretic properties of BICM were \ufb01rst studied by Caire et. al. in [2]. BICM was modeled as a binary channel with a random state that is known at the receiver. The state determines how the input bit is mapped to the channel, along with the other bits that are assumed to be random. Under the assumption of an in\ufb01nite-length, ideal interleaver, the BICM scheme is modeled by parallel uses of independent instances of this binary channel. This model is referred to as the independent parallel channel model. Using this model the capacity of the BICM scheme could be calculated. It was further shown that BICM suffers from a gap from the full channel capacity, and that when Gray mapping is used this gap is generally small. In [2], methods for evaluating the error probability of BICM were proposed, which rely on the properties of the speci\ufb01c binary codes that were used (e.g. Hamming weight of error events). A basic information-theoretic quantity other than the channel capacity is the error exponent [3], which quanti\ufb01es the speed at which the error probability decreases to zero with the block length n. Another tool for evaluating the performance at \ufb01nite block length is the channel dispersion, which was presented in 1962 [4] and was given more attention only in recent years [5], [6]. It would therefore be interesting to analyze BICM at \ufb01nite block length from the information-theoretic viewpoint. Several attempts have been made to provide error exponent results for BICM. In their work on multilevel codes, Wachsmann et. al. [7] have considered the random coding error exponent of BICM, by relying on the independent parallel channels model. However, there were several \ufb02aws in the derivation: \u2022 The independent parallel channels model is justi\ufb01ed by an in\ufb01nite-length interleaver. Therefore it might be problematic to use its properties for evaluating the \ufb01nite length performance of the BICM scheme. In the current paper we address this point and propose a scheme with a \ufb01nite-length interleaver for that purpose. \u2022 There was a technical \ufb02aw in the derivation, which resulted in an inaccurate expression for the random coding error exponent. We discuss this point in detail in Theorem 4. \u2022 As noticed in [8], the error exponent result obtained in [7] sometimes may even exceed that of unconstrained coding over the channel (called in [8] the \u201ccoded modulation exponent\u201d). We therefore agree with [8] in the claim that \u201cthe independent parallel channel model fails to capture the statistics of the channel\u201d. However, by properly designing the communication scheme the model can become valid in a rigorous way, as we show in Theorem 1. In [8] (see also [9]), Martinez et al. have considered the BICM decoder as a mismatched decoder, which has access only to the log-likelihood values (LLR) of each bit, where the LLR calculation assumes that the other bits are random, independent and equiprobable (as in the classical BICM scheme [2]). Using results from mismatched decoding, they presented the generalized error exponent and the generalized mutual information, and pinpointed the loss of BICM that incurs from using the mismatched LLRs. Note that when a binary code of length n is used, the scheme requires only n/L channel uses. While this result is valid for any block size and any interleaver length, achieving this error exponent in practice requires complex code design. \u2020 A. Ingber is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. \fFor example, one cannot design a good binary code for a binary memoryless channel and have any guarantee that the BICM scheme will perform well with that code. In fact, the code design for this scheme requires taking into account the memory within the levels, or equivalently, nonbinary codes, which is what we wish to avoid when choosing BICM. On the theoretical side, another drawback of existing approaches is the lack of converse results (for either capacity or error exponent). The initial discussion of BICM information theory in [2] assumes the model of independent channels, and any converse result based on this model must assume that an in\ufb01nite, ideal, interleaver. Therefore the converse results (such as upper bound on the achievable rate with BICM) do not hold for \ufb01nite-length interleavers. The authors in [8] provide no converse results for their model. In this paper we propose the parallel BICM (PBICM) scheme, which has the following properties. First, the scheme includes an explicit, \ufb01nite length interleaver. Second, in order to attain good performance on any memoryless channel, PBICM allows one to design a binary code for a binary memoryless channel, and guarantees good performance on the nonbinary channel. Third, because the scheme does not rely on the use of an in\ufb01nite-length interleaver, the error exponent and the dispersion of the scheme can be calculated (both achievability and converse results) as means to evaluate the PBICM performance at \ufb01nite block length. The comparison between PBICM and the mismatched decoding approach [8] should be done with care. With PBICM, when the binary codeword length is n the scheme requires n channel uses. Therefore when the latency kept equal for both schemes, PBICM uses a codeword length that is L times shorter than the codeword used in the mismatched decoder. A fair comparison would be to \ufb01x the binary codeword length n for both schemes, resulting in different latency, but equal decoder complexity. The results presented in the paper are summarized as follows: \u2022 The PBICM communication framework is presented. Over a memoryless channel, it is shown to be equivalent to a binary memoryless channel (Theorem 1). \u2022 In Theorem 2, the capacity of PBICM is shown to be equal to the BICM capacity, as calculated in [2]. \u2022 PBICM is analyzed at \ufb01nite block length. The error exponent of PBICM is de\ufb01ned and bounded by error exponent bounds of the underlying binary channel (Theorems 3 and 4). \u2022 The PBICM dispersion is de\ufb01ned as an alternative measure for \ufb01nite-length performance. It is calculated by the dispersion of the underlying binary channel (Theorems 5 and 6). \u2022 The error exponent of PBICM is numerically compared to the mismatched-decoding error exponent of BICM [8]. The additive white Gaussian noise (AWGN) channel and the Rayleigh fading channel are considered. When the latency of both schemes is equal, the mismatched-decoding is generally better. However, when the complexity is equal (or where the codeword length of the underlying binary code is equal), the PBICM exponent is better in many cases. The paper is organized as follows. In Section II we review the classical BICM model and its properties, under the assumption of an in\ufb01nite-length, ideal interleaver. In Section III the parallel BICM scheme is presented and the equivalence to a memoryless binary channel is established. In Section IV parallel BICM is studied from an information-theoretical viewpoint. Numerical examples and summary follow in Sections V and VI respectively. II. THE BICM COMMUNICATION MODEL Notation: letters in bold (x, y...) denote row vectors, capital letters (X, Y ...) denote random variables, and tilde denotes interleaved signals (\u02dc b, \u02dc z). PX(x) denotes the probability that the random variable (RV) X will get the value x, and similarly PY |X(y|x) denotes the probability Y will get the value y given that the RV X is equal to x. E[\u00b7] denotes statistical expectation. log means log2. A. Channel model Let W denote a memoryless channel with input and output alphabets X and Y respectively. The transition probabilities are de\ufb01ned by W(y|x) for y \u2208Y and x \u2208X. We assume that \u2225X\u2225= 2L. We consider equiprobable signaling only over the channel W. An (n, R) code C \u2286X n is a set of M = 2nR codewords c \u2208X n. The encoder wishes to convey one of M equiprobable messages. The error probability of interest shall be the codeword error probability. An (n, R) code with codeword error probability pe will sometimes be called an (n, R, pe) code. B. Classical BICM encoding and decoding In BICM, a binary code is used to encode information messages [m1, m2, ...] into binary codewords [b1, b2, ...]. The binary codewords are then interleaved using a long interleaver \u03c0(\u00b7), which applies a permutation on the coded bits. The interleaved bit stream \u02dc b is partitioned into groups of L consecutive bits and inserted into a mapper \u00b5 : {0, 1}L \u2192X. The mapper output, denoted x, is fed into the channel. The encoding process is described in Figure 1. \fm1,m2.. Binary encoder b1, b2.. \u03c0 \u00b5 x \u02dc b Fig. 1. BICM encoding process PSfrag replacements \u02c6 m1, \u02c6 m2.. Binary dec. LLR calc. z1, z2.. \u03c0\u2212 1 W x y \u02dc z Fig. 2. BICM decoding process The decoding process of BICM proceeds as follows. The channel output y is fed into a bit metric calculator, which calculates the log-likelihood ratio (LLR) of each input bit b given the corresponding output sample y (L LLR values for each output sample). These LLR values (or bit metrics) denoted \u02dc z are de-interleaved and partitioned into bit metrics [z1, z2, ...] that correspond to the binary input codewords. Finally, the binary decoder decodes the messages [ \u02c6 m1, \u02c6 m2, ...] from [z1, z2, ...]. The decoding process is described in Figure 2. The LLR of the jth bit in a symbol given the output value y is calculated as follows: LLRj(y) \u225clog PY |Bj(y|0) PY |Bj(y|1), (1) where PY |Bj(y|b) is the conditional probability of the channel output getting the value y given that the jth bit at the mapper input was b, and the other (L \u22121) bits are equiprobable independent binary random variables (RVs). C. Classical BICM analysis: ideal interleaving In classical BICM (e.g. [2]) the LLR calculation is motivated by the assumption of a very long (ideal) interleaver \u03c0, so the coded bits go through essentially independent channels. These binary channels are de\ufb01ned as follows: De\ufb01nition 1: Let Wi be a binary channel with transition probability Wi(y|b) \u225cE [W(y|X = \u00b5(B1, ..., BL))|Bi = b] (2) = 1 2L\u22121 X bj; i\u0338=j bi=b W(y|\u00b5(b1, ..., bL)). (3) The channel Wi(y|bi) can be thought of as the original channel W where the input is x = \u00b5(b1...bL), where the bits {bj}j\u0338=i are equiprobable independent RVs (see Fig. 3). In [2], Caire et al. have proposed the following channel model for BICM called the independent parallel channel model. In this model the channel has a binary input b. A channel state s is selected at random from S \u225c{1, ..., L} with equal probability (and independently of b). Given a state s, the input bit b is fed into the channel Ws. The channel outputs are the state s and the output y of the channel Ws. The channel, denoted by f W, is depicted in Figure 4. The transition probability function of f W is given by f W(y, s|b) = PY,S|B(y, s|b) = PY |S,B(y|s, b)PS(s) = 1 LWs(y, b). (4) PSfrag replacements \u00b5 B1 B2 Bi BL X Y W Wi : : Fig. 3. The binary channel Wi. The bits {Bj}j\u0338=i are equiprobable independent RVs. \fPSfrag replacements B S: random state S X Y Ws Fig. 4. The binary channel f W. The random state S is known at the receiver. Note that both outputs can be combined into a single output, the LLR, which is a suf\ufb01cient statistic for optimal decoding over any binary-input channel. The LLR calculation for the channel f W is given by LLRf W (y, s) = LLRs(y), (5) where LLRs is given in (1). Therefore the independent parallel channel model transforms the original nonbinary channel W to a simple, memoryless channel. Using an in\ufb01nite-length interleaver and a binary code that was designed for the simple binary channel f W, reliable communication for the original channel W can be attained. Let C(\u00b7) denote the Shannon capacity of a channel (with equiprobable input). Lemma 1 (following [2]): Let CBICM(W) denote the capacity of the channel W with BICM, a given mapping \u00b5(\u00b7) and an in\ufb01nite-length interleaver (according to the independent parallel channel model). Denote by C(Ws) the capacity of the channel Ws. Then CBICM(W) = L X s=1 C(Ws). (6) Proof: Since the independent parallel channel model assumes L independent uses of the channel f W, we get that CBICM(W) = L \u00b7 C(f W). The capacity of f W is given by C \u0010 f W \u0011 = I(B; Y, S) = I(B; Y |S) = ESI(B; Y |S = s) = ESC(Ws) = 1 L L X s=1 C(Ws). (7) It is known that CBICM(W) is generally smaller than the full channel capacity C(W), as opposed to other schemes, most notably multilevel coding and multistage decoding (MLC-MSD) [7], in which C(W) can be achieved. However, for Gray mapping the gap is small and can sometimes be tolerated. For example, for 8-PSK signaling over the AWGN channel with SNR = 5dB, C(W) = 1.86bit where CBICM(W) = 1.84bit. III. THE PARALLEL BICM SCHEME In this section we propose an explicit BICM-type communication scheme which we call parallel BICM (PBICM), which allows the usage of binary codes on nonbinary channels at \ufb01nite blocklength. The main features of the scheme include the following: \u2022 Binary codewords are used in parallel to construct a codeword that enters the channel. \u2022 A new \ufb01nite-length interleaver. \u2022 A random binary signal (binary dither) that is added to the binary codewords. With the proposed scheme, we rigorously show how the original channel W relates to the channel f W, thus allowing exact analysis and design of codes at \ufb01nite block lengths. A. Interleaver Design We wish to design a \ufb01nite length interleaver, where: \u2022 The length of the interleaver is minimal, \u2022 The interleaver should be as simple as possible, \u2022 The binary codewords will go through a binary memoryless channel. \fb1 bL m1 mL ENC1 ENCL . . . . . . . . . s \u03c0 \u00b5 x \u02dc B Fig. 5. Interleaving scheme viewed as parallel encoders PSfrag replacements y LLR calc. \u02dc Z s \u03c0\u22121 z1 zL DEC1 DECL . . . . . . . . . \u02c6 m1 \u02c6 mL x Fig. 6. De-interleaving scheme viewed as parallel decoders In order for the binary codewords to experience a memoryless channel, each binary codeword must be spread over n channel uses of W, so the interleaver output length cannot be less than n channel uses. The newly proposed interleaver has of output length of exactly n, which satis\ufb01es the above requirements. Let ENC and DEC be an encoder-decoder pair for a binary code. Let b1, ..., bL be L consecutive codewords from the output of ENC, bunched together to a matrix B: B = \uf8eb \uf8ec \uf8ed b1 . . . bL \uf8f6 \uf8f7 \uf8f8= \uf8eb \uf8ec \uf8ed b11 . . . b1n . . . blk . . . bL1 . . . bLn \uf8f6 \uf8f7 \uf8f8. (8) Let s be a vector of i.i.d. random states drawn from Sn = {1..L}n. s shall be the interleaving signal. Each column in B shall be shifted cyclically by the corresponding element sk, so the interleaved signal \u02dc B is de\ufb01ned as \u02dc B = \uf8eb \uf8ec \uf8ed b(1+s1)L1 . . . b(1+sn)Ln . . . b(l+sk)Lk . . . b(L+s1)L1 . . . b(1+sn)Ln \uf8f6 \uf8f7 \uf8f8, where (\u03be)L \u225c(\u03be modulo L) + 1. Each column vector of interleaved signal \u02dc B is mapped to a single channel symbol: xk = \u00b5(b(1+sk)Lk, . . . , b(L+sk)Lk), (9) and we call x = [x1, ..., xn] the channel codeword. At the decoder an LLR value is calculated for every bit b in \u02dc B from y. The LLR values are denoted by \u02dc Z. We assume that s is known at the decoder (utilizing common randomness), therefore the de-interleaving operation is simply sorting back the columns of \u02dc Z according to s by reversing the modulo operation. The de-interleaver output is a vector of LLR values z for each transmitted codeword b, according to (1). Each codeword is decoded independently by DEC. B. Binary dither Since the decoder decodes each binary codeword independently, the communication scheme employing the above interleaver can be viewed as as set of parallel encoder-decoder pairs, which we denote by ENC1, ..., ENCL and DEC1, ..., DECL (see Figures 5 and 6). We do not assume any independence between the effective channels between each encoder-decoder pair. Consider the \ufb01rst encoder-decoder pair, ENC1 and DEC1. Since the input of DEC1 depends on the codewords transmitted by ENC2,...,ENCL, the channel between ENC1 and DEC1 is not strictly memoryless. If, somehow, the decoders DEC2,...,DECL were forced to send i.i.d. equiprobable binary codewords, then the channel between ENC1 and DEC1 would be exactly the channel f W (which is a binary memoryless channel) with the accurate LLR calculation (1). In order to achieve the goal of L binary memoryless channels between each encoder-decoder pair simultaneously, we add a binary dither an i.i.d. equiprobable binary signal to each encoder-decoder pair as follows. Let the dither signals dl = [dl1, ..., dln], l \u2208{1, ..., L} be L random vectors, each of length n, that are drawn independently from a memoryless equiprobable binary source. The output of each encoder ENCl, bl, goes through a component-wise XOR \fb1 bL b\u2032 1 b\u2032 L m1 mL d1 dL + + ENC1 ENCL . . . . . . . . . s \u03c0 \u00b5 x \u02dc B Fig. 7. PBICM encoding scheme. \u2018+\u2019 denoted modulo-2 addition (XOR). y LLR calc. \u02dc Z s \u03c0\u22121 \u00d7 \u00d7 z1 zL z\u2032 1 z\u2032 L \u03b41 \u03b4L DEC1 DECL . . . . . . . . . \u02c6 m1 \u02c6 mL x Fig. 8. PBICM decoding scheme. \u03b4l \u225c1 \u22122 \u00b7 dl, \u2018\u00d7\u2019 denotes element-wise multiplication. operation with the dither vector dl. The output of the XOR operation, denoted b\u2032 l, is fed into the interleaver \u03c0. The full PBICM encoding scheme is shown in Fig. 7. We let each decoder DECl know the value of the dither used by its corresponding encoder ENCl, dl (in practice the dither signals are generated using a pseudo-random generator which allows the common randomness). In order to compensate for the dither at the decoder, the LLR values are modi\ufb01ed by \ufb02ipping their sign for each dither value of 1 (and maintaining the sign where the dither is 0). Formally, denote the LLR values at the de-interleaver output by z\u2032 l = [z\u2032 l1 ... z\u2032 ln]. The LLR values at the decoders input shall be denoted by zl = [zl1 ... zln] and calculated as follows: zlj = z\u2032 lj(1 \u22122dlj), j = 1, ..., n. (10) The PBICM decoding scheme is shown in Fig. 8. C. Model equivalence Before we analyze the channel between each encoder-decoder pair in PBICM, let us de\ufb01ne a binary memoryless channel that is related to f W, that will prove useful in the analysis of PBICM. De\ufb01nition 2: Let W be a memoryless binary channel with input B and output \u27e8Y, S, D\u27e9: S is drawn at random from {1, ..., L}, D is drawn at random from {0, 1} (S and D are independent, and both do not depend on the input B). Y is the output of the channel WS with input B \u2295D (\u2295is the XOR operation). Note that the channel W is the channel f W where the input is XORed with a binary RV D (see Fig. 9). Note that the LLR calculation for the channel W is given by LLRW(y, s, d) = (\u22121)dLLRf W(y, s) = (\u22121)dLLRs(y), (11) where LLRf W and LLRs are given in (5) and (1), respectively. Theorem 1: In parallel BICM, the channel between every encoder-decoder pair is exactly the binary memoryless channel W, with its exact LLR output. Proof: Consider the pair ENC1 and DEC1. Let b1 be the codeword sent from ENC1. After adding the dither d1, the dithered codeword b\u2032 1 enters the interleaver. The other codewords b2, ..., bL are dithered using d2, ..., dL. Since the dither of these codewords is unknown at DEC1, the dithered codewords b\u2032 2, ..., b\u2032 L are truly random i.i.d. signals. The interleaving signal s interleaves the dithered codewords according to (8). The interleaved signal enters the mapper \u00b5 and the channel W, PSfrag replacements B S \u2208{1, ..L} S D \u2208{0, 1} D X Y Ws f W W Fig. 9. The binary channel W. The random state S and the dither D are known at the receiver. \fresulting in an output y. Since the dithered codewords b\u2032 2, ..., b\u2032 L are i.i.d., the equivalent channel from b\u2032 1 to \u27e8y, s\u27e9is exactly the channel f W. The LLR calculation at the PBICM receiver along with the interleaver produce z\u2032 1 , which is exactly the LLR calculation that \ufb01ts the channel f W (cf. (5)). Recalling that the channel W is nothing but the channel f W with its input XORed with a binary RV, and that the LLR of the channel f W can be easily modi\ufb01ed by the dither according to Eq. (11) to produce the LLR of the channel W, we conclude that the channel between b1 to z1 is exactly the channel W with LLR calculation. Since by symmetry the above holds for any encoder-decoder pair ENCl-DECl, the proof is concluded. An important note should be made: Parallel BICM allows the decomposition of the nonbinary channel W to L binary channels of the type W. These L channels are not independent. For example, if W is an additive noise channel, and at some point the noise instance is very strong, this will affect all the decoders and they will fail in decoding together. However, since in the PBICM scheme the channels are used independently, the operation of each decoder depends only on the marginal distribution of the relevant channel outputs. The outputs of these decoders will inevitably be statistically dependent, and we take this into consideration when analyzing the performance of coding using PBICM in the following. D. Error Probability Analysis We wish to analyze the performance of PBICM, and speci\ufb01cally, we are interested in the overall codeword error probability. Let C be a binary (n, R) code, used in the PBICM scheme. To assure a fair comparison, we regard each L consecutive information messages (m1, ..., mL) as a single message m, and regard the scheme as a code of length n on the channel input alphabet X. We de\ufb01ne the following error events: Let El be the event of a codeword error in DECl, and let E be the event of an error in any of the messages {m1, ..., mL}, i.e. E = S l El. Denote the corresponding error probabilities by pel and pe respectively. Corollary 1: Let pe(W) be the codeword error probability of the code C over the channel W. Then the overall error probability pe of the code C used with PBICM can be bounded by pe(W ) \u2264pe \u2264L \u00b7 pe(W ). (12) Proof: Since the error events El in codewords that are mapped to the same channel codeword together are dependent, we can only bound the overall error probability pe using the union bound. pe can be also lower bounded by the minimum of the error probabilities in any of the channels: min{pe1, ..., peL} \u2264pe \u2264 X l pel. (13) Since by Theorem 1 the channel between each of the encoder-decoder pairs is W, we get that the error probabilities must be all equal to the error probability of the code C over the channel W. Setting pe1 = pe2 = ... = peL = pe(W) in (13) completes the proof. In many cases the bit error rate (BER) is of interest. Suppose that each of the messages (m1, ..., mL) represents k information bits and the entire message m represents L \u00b7 k information bits. Let Eb lk\u2032 denote the error in the k\u2032-th bit of the information message ml. The average BER for the encoder-decoder pair ENCl-DECl is de\ufb01ned by pb el \u225c1 k k X k\u2032=1 Pr{Eb lk\u2032}. (14) Similarly, de\ufb01ne the overall average BER as pb e \u225c 1 L \u00b7 k L X l=1 k X k\u2032=1 Pr{Eb lk\u2032} = 1 L L X l=1 pb el. (15) Corollary 2: Let pb e(W ) be the average BER of a binary code C over the channel W. Then the average BER pb e of the code C used with PBICM is equal to pb e(W ). Proof: Follows directly from Theorem 1 and from the de\ufb01nition of the average BER in (15). IV. PARALLEL BICM: INFORMATION THEORETICAL ANALYSIS In the previous section we de\ufb01ned the PBICM scheme and analyzed its basic error probability properties. The equivalence of the channel between each encoder-decoder pair that was established in Theorem 1 enables a full information-theoretical analysis of the scheme. We show that the highest achievable rate by PBICM (the PBICM capacity) is equal to the BICM capacity as in Equation (7), which should not be a surprise. At the \ufb01nite-length regime, we derive error exponent and channel dispersion results as information-theoretical measures for optimal PBICM performance at \ufb01nite-length. \fA. Capacity Let the PBICM capacity of W, CPBICM(W), be the highest achievable rate for reliable communication over the channel W with PBICM and a given mapping \u00b5. (As usual, reliable communication means a vanishing codeword error probability as the codelength n goes to in\ufb01nity.) Theorem 2: The PBICM capacity is given by CPBICM(W) = L \u00b7 C(W ) = L X s=1 C(Ws) = CBICM(W). (16) Proof: Achievability: Let C(n) be a series of (binary) capacity-achieving codes for the channel W, and let p(n) e (W) be the corresponding (vanishing) codeword error probabilities. By Corollary 1, the overall error probability of PBICM with a binary code is upper bounded by L times the error probability of the same code over the channel W, therefore when the codes C(n) are used with PBICM, the overall error probability is bounded by L \u00b7 p(n) e (W) and also vanish with n. Since there are L instances of the channel W, we get that the rate of L \u00b7 C(W) is achievable by PBICM. Converse: Let C(n) be a series of binary codes that are used with PBICM and achieve a vanishing overall error probability p(n) e , and suppose that the overall PBICM rate is given by L \u00b7 R (a rate of R at each encoder-decoder pair). By Corollary 1, the codeword error probability of a code over W is upper bounded by the overall error probability of the same code used in PBICM. Therefore, if p(n) e vanishes as n \u2192\u221e, then the error probability over W must also vanish, and therefore the communication rate between each encoder-decoder pair must be upper bounded by C(W), and the overall rate cannot surpass L \u00b7 C(W ). All that remains is to calculate the capacity of W: C(W) = I(B; Y, S, D) = I(B; Y, S|D) = 1 2 (I(B; Y, S|D = 0) + I(B; Y, S|D = 1)) . (17) When D = 0, we get the channel f W exactly, and when D = 1 we get the channel f W with its input symbols always switched. In either way, the expression I(B; Y, S|D = d) is equal to the capacity of f W. Using Lemma 1, we get that C(W) = C(f W) = 1 L L X s=1 C(Ws). (18) A note regarding the capacity proof: one might me tempted to try and prove the capacity theorem for PBICM without dither, since with random coding, the code C is merely an i.i.d. binary random vector. This approach fails because of the following. In the decoding of each codeword, the correctness of the model f W relies on the fact that the other codewords are i.i.d. signals. Since PBICM requires a single code for all the L levels, such a condition can never be met. It it possible to prove the achievability without dither when using a different random code at each level, but such an approach will not guarantee the existence of a single code, as required by PBICM. B. Error Exponent The error exponent of a channel W is de\ufb01ned by E(R) \u225clim n\u2192\u221e\u22121 n log (pe(n)) , (19) where pe(n) is the average codeword error probability for the best code of length n. A lower bound on the error exponent for memoryless channels is the random coding error exponent [3], which is given by Er(R) = max \u03c1\u2208[0,1] max PX(\u00b7){E0(\u03c1, PX) \u2212\u03c1R}, (20) where E0(\u03c1, PX) \u225c\u2212log \uf8ee \uf8f0X y\u2208Y X x\u2208X PX(x)W(y|x)1/(1+\u03c1) !1+\u03c1\uf8f9 \uf8fb. (21) Since we consider equiprobable inputs only we omit the dependence of E0(\u03c1) in PX, and omit the maximization w.r.t. PX in (20). Others known bounds on the error exponent include the expurgation error exponent lower bound, the sphere packing error exponent (an upper bound) and others [3]. The random coding and sphere packing exponents coincide for rates above the critical rate, and therefore the error exponent is known precisely at these rates. \f1) PBICM error exponent: Similarly to (19), we de\ufb01ne the PBICM error exponent: De\ufb01nition 3: For a given channel W and a mapping \u00b5, let EPBICM(R) be de\ufb01ned as EPBICM(R) \u225clim n\u2192\u221e\u22121 n log (pe(n)) , (22) where pe(n) is the average codeword error probability for the best PBICM scheme with block length of n. Using Corollary 1, we can calculate the PBICM exponent using the error exponent of W: Theorem 3: The PBICM error exponent of a channel W is given by EPBICM(R) = E(R/L), (23) where E(\u00b7) is the error exponent function of the binary channel W. Proof: Let C(n) be a series of the binary codes. Denote their codeword error probabilities over the channel W by p(n) e (W). Let p(n) e be the error probabilities of the corresponding PBICM schemes with C(n) used as underlying codes. It follows from (12) that \u22121 n log(L \u00b7 p(n) e (W)) \u2264\u22121 n log p(n) e \u2264\u22121 n log p(n) e (W ). (24) By taking n \u2192\u221ethe factor of L vanishes and we get that for any series of codes, lim n\u2192\u221e\u22121 n log p(n) e = lim n\u2192\u221e\u22121 n log p(n) e (W). (25) The above equation holds for the series of best codes for the channel W, as well as for the series of the best codes for PBICM. Therefore the equality holds for the sequence of best codes on either side. Since the rate for PBICM is L times the rate for coding on W, the proof is concluded. 2) The error exponent of W: The channel W has a special structure, and is related to the binary sub-channels Wi. We now calculate two basic bounds for the error exponent of W in terms of the sub-channels Wi. By Theorem 3, the PBICM error exponent of the channel W can be bounded accordingly. Theorem 4: Let E(R) be the error exponent of the channel W. It can be bounded as follows: Random coding: E(R) \u2265Er(R) = max \u03c1\u2208[0,1]{E0(\u03c1) \u2212\u03c1R}, (26) where E0(\u03c1) = \u2212log E h 2\u2212E( S) 0 (\u03c1)i , (27) E( s) 0 (\u03c1) is the E0 function for the channel Ws, and the expectation is w.r.t. the state S which is drawn uniformly from {1..L}. Sphere packing: E(R) \u2264Esp(R) = max \u03c1>0 {E0(\u03c1) \u2212\u03c1R}, (28) where E0(\u03c1) is given in (27). Proof: The bounds in the theorem are the original random coding and sphere packing exponents [3]. The proof, therefore, boils down to the simpli\ufb01cation of the E0 function to the form of (27). Consider the channel W (De\ufb01nition 2) with binary input B and outputs \u27e8Y, S, D\u27e9. Since W is equivalent to the channel f W with input B \u2295D, where D is an equiprobable binary RV (and known at the receiver), we get that W(y, s, d|b) = 1 2 f W(y, s|b \u2295d). (29) The channel f W, in turn, is nothing more than the channel Ws with the additional output S. This yields 1 2 f W(y, s|b \u2295d) = 1 2LWs(y|b \u2295d). (30) \fCombining the above, the function E0 of W is therefore given by E0(\u03c1) = \u2212log X y\u2208Y s\u2208{1..L} d\u2208{0,1} \uf8ee \uf8f0X b\u2208{0,1} PB(b)W (y, s, d|b) 1 1+\u03c1 \uf8f9 \uf8fb 1+\u03c1 = \u2212log X y\u2208Y s\u2208{1..L} d\u2208{0,1} \uf8ee \uf8f0X b\u2208{0,1} 1 2 \u0012 1 2LWs(y|b \u2295d) \u0013 1 1+\u03c1 \uf8f9 \uf8fb 1+\u03c1 (a) = \u2212log X y\u2208Y s\u2208{1..L} \uf8ee \uf8f0X b\u2032\u2208{0,1} 1 2 \u0012 1 LWs(y|b\u2032) \u0013 1 1+\u03c1 \uf8f9 \uf8fb 1+\u03c1 (31) = \u2212log X s\u2208{1..L} 1 L X y\u2208Y \uf8ee \uf8f0X b\u2032\u2208{0,1} 1 2Ws(y|b\u2032) 1 1+\u03c1 \uf8f9 \uf8fb 1+\u03c1 (b) = \u2212log X s\u2208{1..L} 1 L2\u2212E( s) 0 (\u03c1) = \u2212log E h 2\u2212E( s) 0 (\u03c1)i (32) (a) follows by setting b\u2032 \u225cb \u2295d, and by noting that the summation result is independent of the value of d. (b) follows from the de\ufb01nition of E( s) 0 (\u03c1) (the E0 function for the channel Ws). Several notes can be made: \u2022 It is well known that the random coding and sphere packing exponents coincide at rates above the critical rate. Therefore the exact error exponent of W is known at rates above the critical rate of W, RW cr . It follows that the exact PBICM error exponent is known at rates above RPBICM cr \u225cL \u00b7 RW cr , which we de\ufb01ne to be the PBICM critical rate. \u2022 In theorem 4 we have shown that the random coding and the sphere packing bounds have a compact form because of the special structure of the channel W. Clearly, following Theorem 3, every bound on E(R) of W serves as a bound on the PBICM error exponent. However, for other bounds (such as the expurgation error exponent [3]), no compact form could be found. Such bounds, of course, can still be applied to bound EPBICM(R). \u2022 The E0 function of the channel f W is equal to the E0 function of the channel W. This can easily be seen from the proof above: E0 for f W is given in (31) by de\ufb01nition. \u2022 In [7], the authors offered the model of f W for calculating the error exponent of BICM. It is claimed that E0 of the channel f W is given by [7, Eq. (37)]: E h E( S) 0 (\u03c1) i = 1 L L X s=1 E( s) 0 (\u03c1). (33) As we have just shown in Theorem 4, this is not the exact expression. In fact, it can be shown that E0(\u03c1) \u2264E h E( S) 0 (\u03c1) i . This follows directly from the convexity of the function 2\u2212(\u00b7) and the Jensen inequality. Therefore the incorrect expression in [7, Eq. (37)] always overestimates the value of E0(\u03c1), and therefore the resulting Er(R) expression also overestimates the true random coding expression. C. Channel Dispersion An alternative information theoretical measure for quantifying coding performance with \ufb01nite block lengths is the channel dispersion. Suppose that a \ufb01xed codeword error probability pe and a codeword length n are given. We can then seek the maximal achievable rate R given pe and n. It appears that for \ufb01xed pe and n, the gap to the channel capacity is approximately proportional to Q\u22121(pe)/\u221an (where Q(\u00b7) is the complementary Gaussian cumulative distribution function). The proportion constant (squared) is called the channel dispersion. Formally, de\ufb01ne the (operational) channel dispersion as follows [6]: De\ufb01nition 4: The dispersion V(W) of a channel W with capacity C is de\ufb01ned as V(W) = lim pe\u21920 lim sup n\u2192\u221e n \u00b7 \u0012C \u2212R(n, pe) Q\u22121(pe) \u00132 , (34) \fwhere R(n, pe) is the highest achievable rate for codeword error probability pe and codeword length n. In 1962 , Strassen [4] used the Gaussian approximation to derive the following result for DMCs1: R(n, pe) = C \u2212 p V/nQ\u22121(pe) + O \u0012log n n \u0013 , (35) where C is the channel capacity, and the new quantity V is the (information-theoretic) dispersion , which is given by V \u225cVAR(i(X; Y )), (36) where i(x; y) is the information spectrum, given by i(x; y) \u225clog PXY (x, y) PX(x)PY (y), (37) and the distribution of X is the capacity-achieving distribution that minimizes V . Strassen\u2019s result proves that the dispersion of DMCs is equal to VAR(i(X; Y )). This result was recently tightened (and extended to the power-constrained AWGN channel) in [6]. It is also known that the channel dispersion and the error exponent are related as follows. For a channel with capacity C and dispersion V , the error exponent can be approximated by E(R) \u223c = (C\u2212R)2 2V ln 2 . See [6] for details on the early origins of this approximation by Shannon. 1) PBICM dispersion: In order to estimate the \ufb01nite-block performance of PBICM schemes we extend the dispersion de\ufb01nition as follows: De\ufb01nition 5: The PBICM dispersion VPBICM(W) of a channel W with a given mapping \u00b5 and PBICM capacity CPBICM(W) is de\ufb01ned as VPBICM(W) = lim pe\u21920 lim sup n\u2192\u221e n \u00b7 \u0012CPBICM(W) \u2212R(n, pe) Q\u22121(pe) \u00132 , (38) where R(n, pe) is the highest achievable rate for any PBICM scheme with a given n and pe. Relying on the relationship between the PBICM scheme and the binary channel W, we can show the following: Theorem 5: Let n be a given block length and let pe be a given codeword error probability. The highest achievable rate attained using PBICM, RPBICM(n, pe) is bounded from above and below by: RPBICM(n, pe) \u2265 CPBICM(W) \u2212 s L2V(W ) n Q\u22121 \u0010pe L \u0011 + O \u0012 1 n \u0013 , (39) RPBICM(n, pe) \u2264 CPBICM(W) \u2212 s L2V(W ) n Q\u22121(pe) + O \u0012log n n \u0013 . (40) As a result, the PBICM dispersion is given by VPBICM(W) = L2V(W ). (41) Proof: Direct: From the achievability proof of (35) [6, Theorem 45], there must exist an (n, R\u2032, p\u2032 e = pe/L) binary code for W that satis\ufb01es R\u2032 \u2265C(W) \u2212 s V(W) n Q\u22121(pe/L) + O \u0012 1 n \u0013 . (42) By Theorem 1 and Corollary 1, it follows that the PBICM scheme based on this code is not greater than Lp\u2032 e = pe. The rate of the PBICM scheme satis\ufb01es R = L \u00b7 R\u2032 \u2265L \uf8ee \uf8f0C(W) \u2212 s V(W) n Q\u22121(p\u2032 e) + O \u0012 1 n \u0013\uf8f9 \uf8fb (43) = CBICM(W) \u2212 s L2V(W ) n Q\u22121 \u0010pe L \u0011 + O \u0012 1 n \u0013 . (44) 1see Appendix B for the big-O notation. \fConverse: Suppose we have a (n, R, pe) PBICM scheme. According to Corollary 1, the codeword error probability p\u2032 e of the underlying binary code is not greater than than pe. By Equation (35), the rate R\u2032 of the underlying binary code is bounded by R\u2032 \u2264C(W) \u2212 s V(W ) n Q\u22121(p\u2032 e) + O \u0012log n n \u0013 . (45) Since Q\u22121(\u00b7) is a decreasing function, the bound loosens by replacing p\u2032 e with the higher pe. Therefore the overall rate R is bounded by R = L \u00b7 R\u2032 \u2264L \uf8ee \uf8f0C(W) \u2212 s V(W ) n Q\u22121(p\u2032 e) + O \u0012log n n \u0013\uf8f9 \uf8fb (46) \u2264 CPBICM(W) \u2212 s L2V(W ) n Q\u22121 (pe) + O \u0012log n n \u0013 . (47) PBICM dispersion: Rewriting Equations (39) and (40), we get the following: s L2V(W ) n Q\u22121(pe) + O \u0012log n n \u0013 \u2264CPBICM(W) \u2212R \u2264 s L2V(W) n Q\u22121 \u0010pe L \u0011 + O \u0012 1 n \u0013 (48) q L2V(W ) + O \u0012log n \u221an \u0013 \u2264\u221an \u0012CPBICM(W) \u2212R Q\u22121(pe) \u0013 \u2264 q L2V(W ) \u00b7 Q\u22121 \u0000 pe L \u0001 Q\u22121(pe) + O \u0012 1 \u221an \u0013 (49) Taking the limit w.r.t. n yields q L2V(W ) \u2264lim sup n\u2192\u221e \u221an \u0012CPBICM(W) \u2212R Q\u22121(pe) \u0013 \u2264 q L2V(W) \u00b7 Q\u22121 \u0000 pe L \u0001 Q\u22121(pe) , (50) or L2V(W) \u2264lim sup n\u2192\u221en \u0012CPBICM(W) \u2212R Q\u22121(pe) \u00132 \u2264L2V(W) Q\u22121 \u0000 pe L \u0001 Q\u22121(pe) !2 . (51) By noting that lim\u03b5\u21920+ Q\u22121(\u03b5)2 2 ln 1 \u03b5 = 1 (see Appendix A), we get that lim pe\u21920 Q\u22121 \u0000 pe L \u0001 Q\u22121(pe) !2 = lim pe\u21920 ln(L/pe) ln(1/pe) = lim pe\u21920 ln(1/pe) + ln L ln(1/pe) = 1, (52) which leads to the desired result: VPBICM(W) = lim pe\u21920 lim sup n\u2192\u221e n \u00b7 \u0012CPBICM(W) \u2212R(n, pe) Q\u22121(pe) \u00132 = L2V(W). (53) Note that the PBICM dispersion result is not as tight as the bound for general coding schemes as in (35). The reason is the unavoidable use of the union bound when estimating the overall error probability of PBICM in Theorem 1. In the dispersion proof for DMCs, the value of the dispersion is obtained even without taking the limit w.r.t. pe. However, the gap between Q\u22121(pe) and Q\u22121(pe/L) for values of interest is not very large. 2) The dispersion of W: As in the error exponent case, the PBICM dispersion of a channel is related to the dispersion of the binary channel W. We now calculate it explicitly from the dispersions of the sub-channels Wi. Theorem 6: The dispersion of the channel W is given by V(W) = V(f W ) = E[V(WS)] + VAR [C(WS)] = \" 1 L L X s=1 V(Ws) # + VAR(C(WS)) (54) where VAR(C(WS)) is the statistical variance of the capacity of Ws, i.e. VAR(C(WS)) \u225cE[C2(WS)] \u2212E2[C(WS)]. (55) \fProof: Consider the channel W (De\ufb01nition 2) with binary input B and outputs \u27e8Y, S, D\u27e9, and recall that PY SD|B(y, s, d|b) = W(y, s, d|b) = 1 2 f W(y, s|b \u2295d) = 1 2LWs(y|b \u2295d). (56) We \ufb01rst calculate the dispersion of f W. Since S and the channel input B are independent, the information spectrum is given by i(b; y, s) \u225c log PY SB(y, s, b) PY S(y, s)PB(b) = log PY |SB(y|s, b)PS(s)PB(b) PY S(y, s)PB(b) (57) = log PY |SB(y|s, b) PY |S(y|s) \u225ci(b; y|s). (58) Using this notation, the dispersion of the channel Ws is given by V(Ws) = VAR(i(B; Y |s)|S = s) = E \u0002 i2(B; Y |s)|S = s \u0003 \u2212C(Ws)2. Next, the dispersion of the channel f W is given as follows: V(f W ) = VAR(i(B; Y, S)) = VAR(i(B; Y |S)) (a) = E [VAR[i(B; Y |s)|S = s]] + VAR [E[i(B; Y |S)|S = s]] = E[V(WS)] + VAR [C(WS)] = \" 1 L L X s=1 V(Ws) # + VAR(C(WS)). (59) (a) follows from the law of total variance. Finally, the dispersion of the channel W is calculated as follows: Let us combine the outputs of the channel f W to a single output Z = \u27e8Y, S\u27e9. We therefore end up with a channel with input B and outputs Z and D (see Fig. 9). Similarly to (57), we get that the information spectrum is given by i(b; z, d) \u225clog PZDB(z, d, b) PZD(z, d)PB(b) = i(b; z|d). (60) Following (59), we get that V(W) = E[V(f WD)] + VAR h C(f WD) i = 1 2 X d={0,1} V(f Wd) + VAR(C(f WD)), (61) where f Wd is the channel f W with its input XORed with the value d. Since only equiprobable inputs are considered, it follows that C(f W0) = C(f W1) = C(f W), and that V(f W0) = V(f W1) = V(f W ). It therefore follows that VAR(C(f WD)) = 0, and consequently, V(W ) = V(f W ), as required. Note that since large dispersion means higher backoff from the capacity (see (35)), the term VAR(C(WS)) can be thought of as a penalty factor for the dispersion, over the expected dispersion over the channels Ws, E[V(WS)]. This factor grows as the capacities of the sub-channels Wi are more spread. V. NUMERICAL RESULTS In this section we evaluate numerically the information-theoretical quantities for PBICM. In particular, we calculate the PBICM random coding error exponent (see Theorems 3 and 4) in order to compare with the mismatched decoding approach [8]. We consider the AWGN channel and the Rayleigh fading channel (with perfect channel state information at the receiver) over a wide range of SNR values and constellations. Gray mapping was used throughout all the examples. A. Normalization: latency vs. complexity One way to compare the PBICM error exponent with the mismatched decoding exponent is to compare the error probability when the block length n is \ufb01xed, which gives a simple comparison between the exponent values. Such an approach makes sense, since both schemes have the same latency of n channel uses. As will be seen in the coming examples, for \ufb01xed n the PBICM error exponent is inferior to that of the mismatched decoding. This can also be seen by observing that the PBICM random coding exponent has a slope of \u22121/L (in its straight-line region), where the mismatched decoding exponent has a slope of \u22121. \fHowever, it should be taken into consideration that when the block length is n, the mismatched decoder is working with a binary code of length n\u00b7L. The complexity of the maximum-metric decoder is proportional to the number of codewords 2n\u00b7L\u00b7R [8], where R is the rate of the binary code. On the other hand, the number of codewords in the PBICM scheme is L \u00b7 2n\u00b7R only. In order to assure a fair comparison from the complexity point of view, one has to allow the PBICM scheme to use a block length that is L time the block length of the mismatched decoding scheme. Comparing the error probabilities of both schemes gives nLEPBICM r = nEMismatched r . We therefore de\ufb01ne the normalized PBICM error exponent as L times the PBICM error exponent. We conclude that when the complexity is more important (and the latency is less important), the normalized PBICM exponent is the quantity of interest. It could be claimed, of course, that practical codes used today (such as low-density parity check (LDPC) codes ) will be used and they do not have exponential decoding complexity. On the other hand, such codes do not guarantee an exponentially decaying error probability. B. Comparison with the Mismatched Decoding Exponent In the following \ufb01gures we show the comparison between the PBICM error exponent and the mismatched decoding error exponent [8]. The \ufb01gures show the (unconstrained) random coding error exponent of the channel, along with the mismatched error exponent and the PBICM random coding error exponent (both normalized and un-normalized). Figure 10 compares the exponents of 16QAM signaling over the Rayleigh fading channel at SNR = 5dB. Figure 11 shows the same graph, zoomed-in on the capacity region. It can be seen that throughout the entire range of rates between zero and the BICM capacity, the normalized PBICM random coding exponent is higher (better) than the mismatched decoding exponent. Both BICM exponents are above zero for rates below the BICM capacity, and the unconstrained random coding exponent reaches zero at the full channel capacity, as expected. A fact that might be somewhat surprising at \ufb01rst glance is that the normalized PBICM exponent is better than the unconstrained random coding exponent in some rates. While this may seem contradictory, recall that we consider coding schemes with the same maximum-likelihood (or maximum metric) complexity. When normalizing the schemes complexity, PBICM operates with a block length that is L times the block length of the unconstrained scheme, and therefore there is no contradiction. The mismatched decoder never attains higher values than the unconstrained exponent, a fact that is known as the data processing inequality for exponents (see e.g. [8, Proposition 3.2]). Figure 12 shows a similar picture (zoomed on the capacity in Figure 13). Again, the normalized PBICM outperforms the mismatched decoding exponent for all rates. In this case, the BICM capacity is very close to the full channel capacity, which enables the normalized PBICM to outperform the unconstrained exponent for essentially all rates. On the Rayleigh fading channel, the same behavior was observed for the range of all practical ranges of SNR for 8PSK, 16QAM and 64QAM signaling: the normalized PBICM exponent outperformed the mismatched decoding exponent. On the AWGN channel it cannot be claimed that the normalized PBICM exponent outperforms the mismatched exponent, and the other way around is also not true: for 16QAM signaling and a SNR of 0dB (Fig. 14) the normalized PBICM exponent was better, while for a SNR of 5dB the mismatched exponent was better (Fig. 15). VI. DISCUSSION In this paper we have presented parallel bit-interleaved coded modulation (PBICM). The scheme is based on a \ufb01nite-length interleaver and adding binary dither to the binary codewords. The scheme is shown to be equivalent to a binary memoryless channel, therefore the scheme allows easy code design and exact analysis. The scheme was analyzed from an informationtheoretical viewpoint, and the capacity, error exponent and the dispersion of the PBICM scheme were calculated. Another approach for analyzing BICM at \ufb01nite block length was proposed in [8], where BICM is thought of as a mismatched decoder. Since this BICM setting uses \ufb01nite length, the random coding error exponent of the scheme can be calculated. In the previous section we have compared the error exponents of PBICM and of the mismatched decoding approach. When the two schemes have the same latency (same block length) the PBICM exponent is inferior to that of the mismatched decoding approach. However, when the complexity of the scheme is considered (or equivalently, when codeword length of the underlying code is the same), PBICM becomes comparable, and generally better over the Rayleigh fading channel. An important merit of the PBICM scheme is that it allows an easy code design. In PBICM, one has to design a binary code for a memoryless binary channel. In recent years there have developed methods to design very ef\ufb01cient binary codes, such as LDPC codes [10]. When designing LDPC codes, A desired property of a binary channel is that its output will be symmetric. It appears that no matter what channel W we have at hand, the resulting binary channel W is always output-symmetric (when the output is the LLR). Because of its simplicity and easy code design, we conclude that PBICM is an attractive practical communication scheme, which also allows exact theoretical analysis. \f0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Capacity Capacity (BICM) Er(R) Er(R) [PBICM] Er(R) [PBICM, normalized] Er(R) [Mismatched decoding] Fig. 10. Random coding exponents over the Rayleigh fading channel with 16-QAM signaling and SNR of 5dB. 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 R[bits] Er(R) Capacity Capacity (BICM) Er(R) Er(R) [PBICM] Er(R) [PBICM, normalized] Er(R) [Mismatched decoding] Fig. 11. Random coding exponents over the Rayleigh fading channel with 16-QAM signaling and SNR of 5dB (zoomed on the capacity) \f0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 R[bits] Er(R) Capacity Capacity (BICM) Er(R) Er(R) [PBICM] Er(R) [PBICM, normalized] Er(R) [Mismatched decoding] Fig. 12. Random coding exponents over the Rayleigh fading channel with 64-QAM signaling and SNR of 20dB. 3 3.5 4 4.5 5 5.5 0 0.2 0.4 0.6 0.8 1 1.2 R[bits] Er(R) Capacity Capacity (BICM) Er(R) Er(R) [PBICM] Er(R) [PBICM, normalized] Er(R) [Mismatched decoding] Fig. 13. Random coding exponents over the Rayleigh fading channel with 64-QAM signaling and SNR of 20dB (zoomed on the capacity) \f0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R[bits] Er(R) Capacity Capacity (BICM) Er(R) Er(R) [PBICM] Er(R) [PBICM, normalized] Er(R) [Mismatched decoding] Fig. 14. Random coding exponents over the AWGN channel with 16QAM signaling and SNR of 0dB 0 0.5 1 1.5 2 0 0.5 1 1.5 R[bits] Er(R) Capacity Capacity (BICM) Er(R) Er(R) [PBICM] Er(R) [PBICM, normalized] Er(R) [Mismatched decoding] Fig. 15. Random coding exponents over the AWGN channel with 16QAM signaling and SNR of 5dB \fSeveral additional notes can be made: \u2022 The analysis holds for any mapping \u00b5. Finding the mapping that yields the optimal performance at \ufb01nite lengths is an open question (although Gray mapping is expected to perform well). \u2022 PBICM scheme is composed of, among other things, binary dither. Such tool is used in some cases as a theoretical tool for proving achievability in some problems. In PBICM, it is an essential part of the scheme itself, and even the random capacity proof becomes impossible without it. The main role of the dither is to validate the equivalence of the PBICM scheme to a binary memoryless channel. In addition, the binary dither is the element that symmetrizes the binary channel, which makes the code design easier. This symmetrization property was also noticed by [11] where a similar dither is used with BICM (and termed \u2019channel adapters\u2019). The code design proposed in [11] rely on the assumption of an ideal interleaver. \u2022 The channel is assumed to be memoryless. This captures many interesting channels, including the AWGN channel, and the memoryless fading channel with and without state known at the receiver (ergodic fading). For slow-fading channels, another interleaver (symbol interleaver) is required in order to transform the slowly fading channel into a fast-fading channel (cf. [2]). APPENDIX A APPROXIMATION OF THE INVERSE Q-FUNCTION The following is a useful approximation for the inverse Q-function. Lemma 2: lim \u03b5\u21920 \u0014(Q\u22121(\u03b5))2 2 ln 1 \u03b5 \u0015 = 1. (62) Proof: We start with the well known bound on the Q function: 1 \u221a 2\u03c0x \u0012 1 + 1 x2 \u0013 e\u2212x2 2 \u2264Q(x) \u2264 1 \u221a 2\u03c0xe\u2212x2 2 (63) Dividing by the upper bound yields \u0012 1 + 1 x2 \u0013 \u2264 Q(x) 1 \u221a 2\u03c0xe\u2212x2 2 \u22641. (64) Taking the limit x \u2192\u221egives lim x\u2192\u221e Q(x) 1 \u221a 2\u03c0xe\u2212x2 2 = 1. (65) Since the limit exists, we may take the natural logarithm: lim x\u2192\u221eln Q(x) 1 \u221a 2\u03c0xe\u2212x2 2 = 0. (66) lim x\u2192\u221eln Q(x) \u2212ln 1 \u221a 2\u03c0x \u2212ln e\u2212x2 2 = 0. (67) Since limx\u2192\u221eln Q(x) = \u2212\u221e, we get lim x\u2192\u221e ln Q(x) \u2212ln 1 \u221a 2\u03c0x \u2212ln e\u2212x2 2 ln Q(x) = 0, (68) which leads to lim x\u2192\u221e ln e\u2212x2 2 ln Q(x) = lim x\u2192\u221e \u2212x2 2 ln Q(x) = 1. (69) Since lim\u03b5\u21920 Q\u22121(\u03b5) = \u221e, we may substitute x with Q\u22121(\u03b5), and write lim \u03b5\u21920 \u2212(Q\u22121(\u03b5))2 2 ln \u03b5 = 1, (70) which leads to (62). \fAPPENDIX B BIG-O NOTATION: As usual, f(n) = O(\u03b5n) means that there exist c > 0 and n0 > 0 s.t. for all n > n0, |f(n)| \u2264\u03b5n or equivalently, that \u2212c\u03b5n \u2264f(n) \u2264c\u03b5n. (71) fn = gn + O(\u03b5n) will mean that fn \u2212gn = O(\u03b5n), which means that fn can be approximated by gn, up to a factor that is not greater in absolute value than c \u00b7 \u03b5n for some constant c. Sometimes we will be interested in only one of the sides in (71). For that purpose, f(n) \u2264O(\u03b5n) means that there exist c > 0 and n0 > 0 s.t. for all n > n0, f(n) \u2264c \u00b7 \u03b5n, and f(n) \u2265O(\u03b5n) will mean that there exist c > 0 and n0 > 0 s.t. for all n > n0, \u2212f(n) \u2264c \u00b7 \u03b5n. The different combinations of usages of the O notation are listed in the table below. Notation Meaning fn = O(\u03b5n) \u2203c>0,n0>0\u2200n>n0 |fn| \u2264c \u00b7 \u03b5n fn = gn + O(\u03b5n) fn \u2212gn = O(\u03b5n) fn \u2264O(\u03b5n) \u2203c>0,n0>0\u2200n>n0 fn \u2264c \u00b7 \u03b5n fn \u2264gn + O(\u03b5n) fn \u2212gn \u2264O(\u03b5n) fn \u2265O(\u03b5n) \u2212fn \u2264O(\u03b5n), or \u2203c>0,n0>0\u2200n>n0 \u2212fn \u2264c \u00b7 \u03b5n fn \u2265gn + O(\u03b5n) fn \u2212gn \u2265O(\u03b5n) Note that fn \u2264O(\u03b5n) with fn \u2265O(\u03b5n) is equivalent to fn = O(\u03b5n), as expected. ACKNOWLEDGMENT Interesting discussions with A. G. i F` abregas are acknowledged." + } + ], + "Bang Cong Vu": [ + { + "url": "http://arxiv.org/abs/1110.1697v1", + "title": "A splitting algorithm for dual monotone inclusions involving cocoercive operators", + "abstract": "We consider the problem of solving dual monotone inclusions involving sums of\ncomposite parallel-sum type operators. A feature of this work is to exploit\nexplicitly the cocoercivity of some of the operators appearing in the model.\nSeveral splitting algorithms recently proposed in the literature are recovered\nas special cases.", + "authors": "Bang Cong Vu", + "published": "2011-10-08", + "updated": "2011-10-08", + "primary_cat": "math.OC", + "cats": [ + "math.OC", + "47H05, 49M29, 49M27, 90C25" + ], + "main_content": "Introduction Monotone operator splitting methods have found many applications in applied mathematics, e.g., evolution inclusions [2], partial di\ufb00erential equations [1, 20, 23], mechanics [21], variational inequalities [6, 19], Nash equilibria [8], and various optimization problems [7, 9, 10, 14, 15, 17, 25, 29]. In such formulations, cocoercivity often plays a central role; see for instance [2, 6, 11, 13, 19, 20, 21, 23, 28, 29, 30]. Recall that an operator C : H \u2192H is cocoercive with constant \u03b2 \u2208]0, +\u221e[ if its inverse is \u03b2-strongly monotone, that is, (\u2200x \u2208H)(\u2200y \u2208H) \u27e8x \u2212y | Cx \u2212Cy\u27e9\u2265\u03b2\u2225Cx \u2212Cy\u22252. (1.1) \u2217This work was supported by the Agence Nationale de la Recherche under grant ANR-08-BLAN-0294-02 and the Vietnam National Foundation for Science and Technology Development. 1 \fIn this paper, we revisit a general primal-dual splitting framework proposed in [16] in the presence Lipschitzian operators in the context of cocoercive operators. This will lead to a new type of splitting technique and provide a unifying framework for some algorithms recently proposed in the literature. The problem under investigation is the following, where the parallel sum operation is denoted by \u25a1(see (2.4)). Problem 1.1 Let H be a real Hilbert space, let z \u2208H, let m be a strictly positive integer, let (\u03c9i)1\u2264i\u2264m be real numbers in ]0, 1] such that Pm i=1 \u03c9i = 1, let A: H \u21922H be maximally monotone, and let C : H \u2192H be \u00b5-cocoercive for some \u00b5 \u2208]0, +\u221e[. For every i \u2208{1, . . . , m}, let Gi be a real Hilbert space, let ri \u2208Gi, let Bi : Gi \u21922Gi be maximally monotone, let Di : Gi \u21922Gi be maximally monotone and \u03bdi-strongly monotone for some \u03bdi \u2208]0, +\u221e[, and suppose that Li : H \u2192Gi is a nonzero bounded linear operator. The problem is to solve the primal inclusion \ufb01nd x \u2208H such that z \u2208Ax + m X i=1 \u03c9iL\u2217 i \u0000(Bi \u25a1Di)(Lix \u2212ri) \u0001 + Cx, (1.2) together with the dual inclusion \ufb01nd v1 \u2208G1, . . . , vm \u2208Gm such that (\u2203x \u2208H) ( z \u2212Pm i=1 \u03c9iL\u2217 i vi \u2208Ax + Cx \u0000\u2200i \u2208{1, . . . , m} \u0001 vi \u2208(Bi \u25a1Di)(Lix \u2212ri). (1.3) We denote by P and D the sets of solutions to (1.2) and (1.3), respectively. In the case when (D\u22121 i )1\u2264i\u2264m and C are general monotone Lipschitzian operators, Problem 1.1 was investigated in [16]. Here are a couple of special cases of Problem 1.1. Example 1.2 In Problem 1.1, set z = 0 and \u0000\u2200i \u2208{1, . . . , m} \u0001 Bi : v 7\u2192{0} and Di : v 7\u2192 ( Gi if v = 0, 0 if v \u0338= 0. (1.4) The primal inclusion (1.2) reduces to \ufb01nd x \u2208H such that 0 \u2208Ax + Cx. (1.5) This problem is studied in [2, 11, 13, 17, 23, 28, 29]. Example 1.3 Suppose that in Problem 1.1 the operators (Di)1\u2264i\u2264m are as in (1.4), and that A: x 7\u2192{0} and C : x 7\u21920. (1.6) Then we obtain the primal-dual pair \ufb01nd x \u2208H such that z \u2208 m X i=1 \u03c9iL\u2217 i \u0000Bi(Lix \u2212ri) \u0001 , (1.7) and \ufb01nd v1 \u2208G1, . . . , vm \u2208Gm such that (Pm i=1 \u03c9iL\u2217 i vi = z, (\u2203x \u2208H) \u0000\u2200i \u2208{1, . . . , m} \u0001 vi \u2208Bi(Lix \u2212ri). (1.8) 2 \fThis framework is considered in [7], where further special cases will be found. In particular, it contains the classical Fenchel-Rockafellar [27] and Mosco [24] duality settings, as well as that of [3]. The paper is organized as follows. Section 2 is devoted to notation and background. In Section 3, we present our algorithm, prove its convergence, and compare it to existing work. Applications to minimization problems are provided in Section 4, where further connections with the state-of-the-art are made. 2 Notation and background We recall some notation and background from convex analysis and monotone operator theory (see [6] for a detailed account). Throughout, H, G, and (Gi)1\u2264i\u2264m are real Hilbert spaces. The scalars product and the associated norms of both H and G are denoted respectively by \u27e8\u00b7 | \u00b7\u27e9and \u2225\u00b7 \u2225. For every i \u2208{1, . . . , m}, the scalar product and associated norm of Gi are denoted respectively by \u27e8\u00b7 | \u00b7\u27e9Gi and \u2225\u00b7 \u2225Gi. We denote by B(H, G) the space of all bounded linear operators from H to G. The symbols \u21c0and \u2192denote respectively weak and strong convergence. Let A: H \u21922H be a set-valued operator. The domain and the graph of A are respectively de\ufb01ned by dom A = \b x \u2208H | Ax \u0338= \u2205 \t and gra A = \b (x, u) \u2208H \u00d7 H | u \u2208Ax \t . We denote by zer A = \b x \u2208H | 0 \u2208Ax \t the set of zeros of A, and by ran A = \b u \u2208H | (\u2203x \u2208H) u \u2208Ax \t the range of A. The inverse of A is A\u22121 : H 7\u21922H : u 7\u2192 \b x \u2208H | u \u2208Ax \t . The resolvent of A is JA = (Id +A)\u22121, (2.1) where Id denotes the identity operator on H. Moreover, A is monotone if (\u2200(x, y) \u2208H \u00d7 H) (\u2200(u, v) \u2208Ax \u00d7 Ay) \u27e8x \u2212y | u \u2212v\u27e9\u22650, (2.2) and maximally monotone if it is monotone and there exists no monotone operator B : H \u21922H such that gra B properly contains gra A. We say that A is uniformly monotone at x \u2208dom A if there exists an increasing function \u03c6: [0, +\u221e[ \u2192[0, +\u221e] vanishing only at 0 such that \u0000\u2200u \u2208Ax \u0001\u0000\u2200(y, v) \u2208gra A \u0001 \u27e8x \u2212y | u \u2212v\u27e9\u2265\u03c6(\u2225x \u2212y\u2225). (2.3) If A \u2212\u03b1 Id is monotone for some \u03b1 \u2208]0, +\u221e[, then A is said to be \u03b1-strongly monotone. The parallel sum of two set-valued operators A and B from H to 2H is A \u25a1B = (A\u22121 + B\u22121)\u22121. (2.4) The class of all lower semicontinuous convex functions f : H \u2192]\u2212\u221e, +\u221e] such that dom f = \b x \u2208H | f(x) < +\u221e \t \u0338= \u2205is denoted by \u03930(H). Now, let f \u2208\u03930(H). The conjugate of f is the function f \u2217\u2208\u03930(H) de\ufb01ned by f \u2217: u 7\u2192supx\u2208H(\u27e8x | u\u27e9\u2212f(x)), and the subdi\ufb00erential of f \u2208\u03930(H) is the maximally monotone operator \u2202f : H \u21922H : x 7\u2192 \b u \u2208H | (\u2200y \u2208H) \u27e8y \u2212x | u\u27e9+ f(x) \u2264f(y) \t (2.5) 3 \fwith inverse given by (\u2202f)\u22121 = \u2202f \u2217. (2.6) Moreover, the proximity operator of f is proxf : H \u2192H: x 7\u2192argmin y\u2208H f(y) + 1 2\u2225x \u2212y\u22252. (2.7) We have J\u2202f = proxf . (2.8) The in\ufb01mal convolution of two functions f and g from H to ]\u2212\u221e, +\u221e] is f \u25a1g: H \u2192]\u2212\u221e, +\u221e] : x 7\u2192inf y\u2208H(f(x) + g(x \u2212y)). (2.9) Finally, let S be a convex subset of H. The relative interior of S, i.e., the set of points x \u2208S such that the cone generated by x + S is a vector subspace of H, is denoted by ri S. 3 Algorithm and convergence Our main result is the following theorem, in which we introduce our splitting algorithm and prove its convergence. Theorem 3.1 In Problem 1.1, suppose that z \u2208ran \u0012 A + m X i=1 \u03c9iL\u2217 i \u0000(Bi \u25a1Di)(Li \u00b7 \u2212ri) \u0001 + C \u0013 . (3.1) Let \u03c4 and (\u03c3i)1\u2264i\u2264m be strictly positive numbers such that 2\u03c1 min{\u00b5, \u03bd1, . . . , \u03bdm} > 1, where \u03c1 = min n \u03c4 \u22121, \u03c3\u22121 1 , . . . , \u03c3\u22121 m o 1 \u2212 v u u t\u03c4 m X i=1 \u03c3i\u03c9i\u2225Li\u22252 ! . (3.2) Let \u03b5 \u2208]0, 1[, let (\u03bbn)n\u2208N be a sequence in [\u03b5, 1], let x0 \u2208H, let (a1,n)n\u2208N and (a2,n)n\u2208N be absolutely summable sequences in H. For every i \u2208{1, . . . , m}, let vi,0 \u2208Gi, and let (bi,n)n\u2208N and (ci,n)n\u2208N be absolutely summable sequences in Gi. Let (xn)n\u2208N and (v1,n, . . . , vm,n)n\u2208N be sequences generated by the following routine (\u2200n \u2208N) \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 pn = J\u03c4A \u0010 xn \u2212\u03c4 \u0010 Pm i=1 \u03c9iL\u2217 i vi,n + Cxn + a1,n \u2212z \u0011\u0011 + a2,n yn = 2pn \u2212xn xn+1 = xn + \u03bbn(pn \u2212xn) for i = 1, . . . , m $ qi,n = J\u03c3iB\u22121 i \u0010 vi,n + \u03c3i \u0010 Liyn \u2212D\u22121 i vi,n \u2212ci,n \u2212ri \u0011\u0011 + bi,n vi,n+1 = vi,n + \u03bbn(qi,n \u2212vi,n). (3.3) Then the following hold for some x \u2208P and (v1, . . . , vm) \u2208D. 4 \f(i) xn \u21c0x and (v1,n, . . . , vm,n) \u21c0(v1, . . . , vm). (ii) Suppose that C is uniformly monotone at x. Then xn \u2192x. (iii) Suppose that D\u22121 j is uniformly monotone at vj for some j \u2208{1, . . . , m}. Then vj,n \u2192vj. Proof. We de\ufb01ne G as the real Hilbert space obtained by endowing the Cartesian product G1 \u00d7 . . . \u00d7 Gm with the scalar product and the associated norm respectively de\ufb01ned by \u27e8\u00b7 | \u00b7\u27e9G : (v, w) 7\u2192 m X i=1 \u03c9i \u27e8vi | wi\u27e9Gi and \u2225\u00b7 \u2225G : v 7\u2192 v u u t m X i=1 \u03c9i\u2225vi\u22252 Gi, (3.4) where v = (v1, . . . , vm) and w = (w1, . . . , wm) denote generic elements in G. Next, we let K be the Hilbert direct sum K = H \u2295G. (3.5) Thus, the scalar product and the norm of K are respectively de\ufb01ned by \u27e8\u00b7 | \u00b7\u27e9K : \u0000(x, v), (y, w) \u0001 7\u2192\u27e8x | y\u27e9+ \u27e8v | w\u27e9G and \u2225\u00b7 \u2225K : (x, v) 7\u2192 q \u2225x\u22252 + \u2225v\u22252 G. (3.6) Let us set M : K \u21922K (x, v1, . . . , vm) 7\u2192 \u0000\u2212z + Ax \u0001 \u00d7 \u0000r1 + B\u22121 1 v1 \u0001 \u00d7 . . . \u00d7 \u0000rm + B\u22121 m vm \u0001 . (3.7) Since the operators A and (Bi)1\u2264i\u2264m are maximally monotone, M is maximally monotone [6, Propositions 20.22 and 20.23]. We also introduce S : K \u2192K (3.8) (x, v1, . . . , vm) 7\u2192 \u0012 m X i=1 \u03c9iL\u2217 i vi, \u2212L1x, . . . , \u2212Lmx \u0013 . (3.9) Note that S is linear, bounded, and skew (i.e, S\u2217= \u2212S). Hence, S is maximally monotone [6, Example 20.30]. Moreover, since dom S = K, M + S is maximally monotone [6, Corollary 24.24(i)]. Since, for every i \u2208{1, . . . , m}, Di is \u03bdi-strongly monotone, D\u22121 i is \u03bdi-cocoercive. Let us prove that Q: K \u2192K (x, v1, . . . , vm) 7\u2192 \u0000Cx, D\u22121 1 v1, . . . , D\u22121 m vm \u0001 (3.10) is \u03b2-cocoercive with \u03b2 = min{\u00b5, \u03bd1, . . . , \u03bdm}. (3.11) 5 \fFor every (x, v1, . . . , vm) and every (y, w1, . . . , wm) in K, we have \u27e8(x, v1, . . . , vm) \u2212(y, w1, . . . , wm) | Q(x, v1, . . . , vm) \u2212Q(y, w1, . . . , wm)\u27e9K = \u27e8x \u2212y | Cx \u2212Cy\u27e9+ m X i=1 \u03c9i vi \u2212wi | D\u22121 i vi \u2212D\u22121 i wi \u000b Gi \u2265\u00b5\u2225Cx \u2212Cy\u22252 + m X i=1 \u03bdi\u03c9i\u2225D\u22121 i vi \u2212D\u22121 i wi\u22252 Gi \u2265\u03b2 \u0012 \u2225Cx \u2212Cy\u22252 + m X i=1 \u03c9i\u2225D\u22121 i vi \u2212D\u22121 i wi\u22252 Gi \u0013 = \u03b2\u2225Q(x, v1, . . . , vm) \u2212Q(y, w1, . . . , wm)\u22252 K. (3.12) Therefore, by (1.1), Q is \u03b2-cocoercive. It is shown in [16, Eq. (3.12)] that under the condition (3.1), zer(M + S + Q) \u0338= \u2205. Moreover, [16, Eq. (3.21)] and [16, Eq. (3.22)] yield (x, v) \u2208zer(M + S + Q) \u21d2x \u2208P and v \u2208D. (3.13) Now, de\ufb01ne V : K \u2192K (x, v1, . . . , vm) 7\u2192 \u0012 \u03c4 \u22121x \u2212 m X i=1 \u03c9iL\u2217 i vi, \u03c3\u22121 1 v1 \u2212L1x, . . . , \u03c3\u22121 m vm \u2212Lmx \u0013 . (3.14) Then V is self-adjoint. Let us check that V is \u03c1-strongly positive. To this end, de\ufb01ne T : H \u2192G : x 7\u2192 \u0010\u221a\u03c31L1x, . . . , \u221a\u03c3mLmx \u0011 . (3.15) Then, (\u2200x \u2208H) \u2225T x\u22252 G = m X i=1 \u03c9i\u03c3i\u2225Lix\u22252 Gi \u2264\u2225x\u22252 m X i=1 \u03c9i\u03c3i\u2225Li\u22252, (3.16) which implies that \u2225T \u22252 \u2264 m X i=1 \u03c9i\u03c3i\u2225Li\u22252. (3.17) Now set \u03b4 = v u u t\u03c4 m X i=1 \u03c3i\u03c9i\u2225Li\u22252 !\u22121 \u22121. (3.18) Then, it follows from (3.2) that \u03b4 > 0. Moreover, (3.17) and (3.18) yield \u03c4\u2225T \u22252(1 + \u03b4) \u2264\u03c4(1 + \u03b4) m X i=1 \u03c9i\u03c3i\u2225Li\u22252 = (1 + \u03b4)\u22121. (3.19) 6 \fFor every x = (x, v1, . . . , vm) in K, by using (3.19), we obtain \u27e8x | V x\u27e9K = \u03c4 \u22121\u2225x\u22252 + m X i=1 \u03c3\u22121 i \u03c9i\u2225vi\u22252 Gi \u22122 m X i=1 \u03c9i \u27e8Lix | vi\u27e9Gi = \u03c4 \u22121\u2225x\u22252 + m X i=1 \u03c3\u22121 i \u03c9i\u2225vi\u22252 Gi \u22122 m X i=1 \u03c9i D\u221a\u03c3iLix | \u221a\u03c3i \u22121vi E Gi = \u03c4 \u22121\u2225x\u22252 + m X i=1 \u03c3\u22121 i \u03c9i\u2225vi\u22252 Gi \u22122 D T x | (\u221a\u03c31 \u22121v1, . . . , \u221a\u03c3m \u22121vm) E G \u2265\u03c4 \u22121\u2225x\u22252 + m X i=1 \u03c3\u22121 i \u03c9i\u2225vi\u22252 Gi \u2212 \u2225T x\u22252 G \u03c4(1 + \u03b4)\u2225T \u22252 + \u03c4(1 + \u03b4)\u2225T \u22252 m X i=1 \u03c3\u22121 i \u03c9i\u2225vi\u22252 Gi ! \u2265 \u0010 1 \u2212(1 + \u03b4)\u22121\u0011\u0012 \u03c4 \u22121\u2225x\u22252 + m X i=1 \u03c3\u22121 i \u03c9i\u2225vi\u22252 Gi \u0013 \u2265 \u0010 1 \u2212(1 + \u03b4)\u22121\u0011 min{\u03c4 \u22121, \u03c3\u22121 1 , . . . , \u03c3\u22121 m }\u2225x\u22252 K = \u03c1\u2225x\u22252 K. (3.20) Therefore, V is \u03c1-strongly positive. Furthermore, it follows from (3.20) that V \u22121 exists and \u2225V \u22121\u2225\u2264\u03c1\u22121. (3.21) (i): We \ufb01rst observe that (3.3) is equivalent to (\u2200n \u2208N) \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 \u03c4 \u22121(xn \u2212pn) \u2212Pm i=1 \u03c9iL\u2217 i vi,n \u2212Cxn \u2208 \u2212z + A(pn \u2212a2,n) + a1,n \u2212\u03c4 \u22121a2,n xn+1 = xn + \u03bbn(pn \u2212xn) for i = 1, . . . , m \uf8ef \uf8ef \uf8ef \uf8f0 \u03c3\u22121 i (vi,n \u2212qi,n) \u2212Li(xn \u2212pn) \u2212D\u22121 i vi,n \u2208 ri + B\u22121 i (qi,n \u2212bi,n) \u2212Lipn + ci,n \u2212\u03c3\u22121 i bi,n vi,n+1 = vi,n + \u03bbn(qi,n \u2212vi,n). (3.22) Now set \u0000\u2200n \u2208N \u0001 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xn = (xn, v1,n, . . . , vm,n) yn = (pn, q1,n, . . . , qm,n) an = (a2,n, b1,n, . . . , bm,n) cn = (a1,n, c1,n, . . . , cm,n) dn = (\u03c4 \u22121a2,n, \u03c3\u22121 1 b1,n, . . . , \u03c3\u22121 m bm,n). (3.23) We have X n\u2208N \u2225an\u2225K < +\u221e, X n\u2208N \u2225cn\u2225K < +\u221e, and X n\u2208N \u2225dn\u2225K < +\u221e. (3.24) Furthermore, (3.22) yields (\u2200n \u2208N) \u0016 V (xn \u2212yn) \u2212Qxn \u2208(M + S)(yn \u2212an) + San + cn \u2212dn xn+1 = xn + \u03bbn(yn \u2212xn). (3.25) 7 \fNext, we set (\u2200n \u2208N) bn = V \u22121\u0000(S + V )an + cn \u2212dn \u0001 . (3.26) Then (3.24) implies that X n\u2208N \u2225bn\u2225K < +\u221e. (3.27) Moreover, using (3.21) and (3.26), we have (\u2200n \u2208N) V (xn \u2212yn) \u2212Qxn \u2208(M + S)(yn \u2212an) + San + cn \u2212dn \u21d4(\u2200n \u2208N) (V \u2212Q)xn \u2208(M + S + V )(yn \u2212an) + (S + V )an + cn \u2212dn \u21d4(\u2200n \u2208N) yn = \u0000M + S + V \u0001\u22121\u0010 (V \u2212Q)xn \u2212(S + V )an \u2212cn + dn \u0011 + an \u21d4(\u2200n \u2208N) yn = \u0010 Id + V \u22121(M + S) \u0011\u22121\u0010\u0000Id \u2212V \u22121Q \u0001 xn \u2212bn \u0011 + an. (3.28) We derive from (3.25) that (\u2200n \u2208N) xn+1 = xn + \u03bbn \u0010\u0000Id + V \u22121(M + S) \u0001\u22121\u0000xn \u2212V \u22121Qxn \u2212bn \u0001 + an \u2212xn \u0011 = xn + \u03bbn \u0010 JA \u0000xn \u2212Bxn \u2212bn \u0001 + an \u2212xn \u0011 , (3.29) where A = V \u22121(M + S) and B = V \u22121Q. (3.30) Algorithm (3.29) has the structure of the forward-backward splitting algorithm [13]. Hence, it is su\ufb03cient to check the convergence conditions of the forward-backward splitting algorithm [13, Corollary 6.5] to prove our claims. To this end, let us introduce the real Hilbert space KV with scalar product and norm de\ufb01ned by \u0000\u2200(x, y) \u2208K \u00d7 K \u0001 \u27e8x | y\u27e9V = \u27e8x | V y\u27e9K and \u2225x\u2225V = q \u27e8x | V x\u27e9K, (3.31) respectively. Since V is a bounded linear operator, it follows from (3.24) and (3.27) that X n\u2208N \u2225an\u2225V < +\u221e and X n\u2208N \u2225bn\u2225V < +\u221e. (3.32) Moreover, since M + S is monotone on K, we have \u0000\u2200(x, y) \u2208K \u00d7 K \u0001 \u27e8x \u2212y | Ax \u2212Ay\u27e9V = \u27e8x \u2212y | V Ax \u2212V Ay\u27e9K = \u27e8x \u2212y | (M + S)x \u2212(M + S)y\u27e9K (3.33) \u22650. (3.34) Hence, A is monotone on KV . Likewise, B is monotone on KV . Since V is strongly positive, and since M + S is maximally monotone on K, A is maximally monotone on KV . Next, let us 8 \fshow that B is (\u03b2\u03c1)-cocoercive on KV . Using (3.12), (3.20) and (3.21), we have \u0000\u2200(x, y) \u2208KV \u00d7 KV \u0001 \u27e8x \u2212y | Bx \u2212By\u27e9V = \u27e8x \u2212y | V Bx \u2212V By\u27e9K = \u27e8x \u2212y | Qx \u2212Qy\u27e9K \u2265\u03b2\u2225Qx \u2212Qy\u22252 K = \u03b2\u2225Qx \u2212Qy\u2225K\u2225Qx \u2212Qy\u2225K = \u03b2\u2225V \u22121\u2225\u22121\u2225V \u22121\u2225\u2225Qx \u2212Qy\u2225K\u2225Qx \u2212Qy\u2225K \u2265\u03b2\u2225V \u22121\u2225\u22121\u2225V \u22121Qx \u2212V \u22121Qy\u2225K\u2225Qx \u2212Qy\u2225K \u2265\u03b2\u2225V \u22121\u2225\u22121 V \u22121Qx \u2212V \u22121Qy | Qx \u2212Qy \u000b K = \u03b2\u2225V \u22121\u2225\u22121 \u27e8Bx \u2212By | Qx \u2212Qy\u27e9K = \u03b2\u2225V \u22121\u2225\u22121\u2225Bx \u2212By\u22252 V \u2265\u03b2\u03c1\u2225Bx \u2212By\u22252 V . (3.35) Hence, by (1.1), B is (\u03b2\u03c1)-cocoercive on KV . Moreover, it follows from our assumption that 2\u03b2\u03c1 > 1. Altogether, by [13, Corollary 6.5] the sequence (xn)n\u2208N converges weakly in KV to some x = (x, v1, . . . , vm) \u2208zer(A + B) = zer(M + S + Q). Since V is self-adjoint and V \u22121 exists, the weak convergence of the sequence (xn)n\u2208N to x in KV is equivalent to the weak convergence of (xn)n\u2208N to x in K. Hence, xn \u21c0x \u2208zer(M + S + Q). It follows from (3.13) that x \u2208P and (v1, . . . , vm) \u2208D. This proves (i). (ii)&(iii): It follows from [13, Remark 3.4] that X n\u2208N \u2225Bxn \u2212Bx\u22252 V < +\u221e. (3.36) On the other hand, from (3.20) and (3.36) yield Bxn\u2212Bx = V \u22121(Qxn\u2212Qx) \u21920, which implies that Qxn \u2212Qx \u21920. Hence, Cxn \u2192Cx and \u0000\u2200i \u2208{1, . . . , m} \u0001 D\u22121 i vi,n \u2192D\u22121 i vi. (3.37) If C is uniformly monotone at x, then there exists an increasing function \u03c6C : [0, +\u221e[ \u2192[0, +\u221e] vanishing only at 0 such that \u03c6C(\u2225xn \u2212x\u2225) \u2264\u27e8xn \u2212x | Cxn \u2212Cx\u27e9\u2264\u2225xn \u2212x\u2225\u2225Cxn \u2212Cx\u2225. (3.38) Notice that (xn\u2212x)n\u2208N is bounded. It follows from (3.37) and (3.38) that xn \u2192x. This proves (ii), and (iii) is proved in a similar fashion. Remark 3.2 Here are some remarks concerning the connections between our framework and existing work. (i) The strategy used in the proof of Theorem 3.1(i) is to reformulate algorithm (3.3) as a forward-backward splitting algorithm in a real Hilbert space endowed with a suitable norm. This renorming technique was used in [22] for a minimization problem in \ufb01nite-dimensional spaces. The same technique is also used in the primal-dual minimization problem of [18]. 9 \f(ii) Consider the special case when z = 0, and (Bi)1\u2264i\u2264m and (Di)1\u2264i\u2264m are as in (1.4). Then algorithm (3.3) reduces to (\u2200n \u2208N) xn+1 = xn + \u03bbn \u0012 J\u03c4A \u0010 xn \u2212\u03c4(Cxn + a1,n) \u0011 + a2,n \u2212xn \u0013 , (3.39) which is the standard forward-backward splitting algorithm [13, Algorithm 6.4] where the sequence (\u03b3n)n\u2208N in [13, Eq. (6.3)] is constant. (iii) The inclusions (1.7) and (1.8) in Example 1.3 can be solved by [7, Theorem 3.8]. However, the algorithm resulting from (3.3) in this special case is di\ufb00erent from that of [7, Theorem 3.8]. (iv) In Problem 1.1, since C and (D\u22121 i )1\u2264i\u2264m are cocoercive, they are Lipschitzian. Hence, Problem 1.1 can be solved by the algorithm proposed in [16, Theorem 3.1], which has a di\ufb00erent structure from the present algorithm. (v) Consider the special case when z = 0 and (\u2200i \u2208{1, . . . , m}) Gi = H, Li = Id, D\u22121 i = 0, ri = 0. Then the primal inclusion (1.2) reduces to \ufb01nd x \u2208H such that 0 \u2208Ax + m X i=1 \u03c9iBix + Cx. (3.40) This inclusion can be solved by the algorithm proposed in [26], which is not designed as a primal-dual scheme. 4 Application to minimization problems We provide an application of the algorithm (3.3) to minimization problems, by revisiting [16, Problem 4.1]. Problem 4.1 Let H be a real Hilbert space, let z \u2208H, let m be a strictly positive integer, let (\u03c9i)1\u2264i\u2264m be real numbers in ]0, 1] such that Pm i=1 \u03c9i = 1, let f \u2208\u03930(H), and let h: H \u2192R be convex and di\ufb00erentiable with a \u00b5\u22121-Lipschitzian gradient for some \u00b5 \u2208]0, +\u221e[. For every i \u2208{1, . . . , m}, let Gi be a real Hilbert space, let ri \u2208Gi, let gi \u2208\u03930(Gi), let \u2113i \u2208\u03930(Gi) be \u03bdi-strongly convex, for some \u03bdi \u2208]0, +\u221e[, and suppose that Li : H \u2192Gi is a nonzero bounded linear operator. Consider the primal problem minimize x\u2208H f(x) + m X i=1 \u03c9i(gi \u25a1\u2113i)(Lix \u2212ri) + h(x) \u2212\u27e8x | z\u27e9, (4.1) and the dual problem minimize v1\u2208G1,...,vm\u2208Gm (f \u2217\u25a1h\u2217) \u0012 z \u2212 m X i=1 \u03c9iL\u2217 i vi \u0013 + m X i=1 \u03c9i \u0000g\u2217 i (vi) + \u2113\u2217 i (vi) + \u27e8vi | ri\u27e9Gi \u0001 . (4.2) We denote by P1 and D1 the sets of solutions to (4.1) and (4.2), respectively. 10 \fCorollary 4.2 In Problem 4.1, suppose that z \u2208ran \u0012 \u2202f + m X i=1 \u03c9iL\u2217 i \u0000(\u2202gi \u25a1\u2202\u2113i)(Li \u00b7 \u2212ri) \u0001 + \u2207h \u0013 . (4.3) Let \u03c4 and (\u03c3i)1\u2264i\u2264m be strictly positive numbers such that 2\u03c1 min{\u00b5, \u03bd1, . . . , \u03bdm} > 1, where \u03c1 = min n \u03c4 \u22121, \u03c3\u22121 1 , . . . , \u03c3\u22121 m o 1 \u2212 v u u t\u03c4 m X i=1 \u03c3i\u03c9i\u2225Li\u22252 ! . (4.4) Let \u03b5 \u2208]0, 1[ and let (\u03bbn)n\u2208N be a sequence in [\u03b5, 1], let x0 \u2208H, let (a1,n)n\u2208N and (a2,n)n\u2208N be absolutely summable sequences in H. For every i \u2208{1, . . . , m}, let vi,0 \u2208Gi, and let (bi,n)n\u2208N and (ci,n)n\u2208N be absolutely summable sequences in Gi. Let (xn)n\u2208N and (v1,n, . . . , vm,n)n\u2208N be sequences generated by the following routine (\u2200n \u2208N) \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 pn = prox\u03c4f \u0010 xn \u2212\u03c4 \u0010 Pm i=1 \u03c9iL\u2217 i vi,n + \u2207h(xn) + a1,n \u2212z \u0011\u0011 + a2,n yn = 2pn \u2212xn xn+1 = xn + \u03bbn(pn \u2212xn) for i = 1, . . . , m $ qi,n = prox\u03c3ig\u2217 i \u0010 vi,n + \u03c3i \u0010 Liyn \u2212\u2207\u2113\u2217 i (vi,n) + ci,n \u2212ri \u0011\u0011 + bi,n vi,n+1 = vi,n + \u03bbn(qi,n \u2212vi,n). (4.5) Then the following hold for some x \u2208P1 and (v1, . . . , vm) \u2208D1. (i) xn \u21c0x and (v1,n, . . . , vm,n) \u21c0(v1, . . . , vm). (ii) Suppose that h is uniformly convex at x. Then xn \u2192x. (iii) Suppose that \u2113\u2217 j is uniformly convex at vj for some j \u2208{1, . . . , m}. Then vj,n \u2192vj. Proof. The connection between Problem 4.1 and Problem 1.1 is established in the proof of [16, Theorem 4.2]. Since \u2207h is \u00b5\u22121-Lipschitz continuous, by the Baillon-Haddad Theorem [4, 5], it is \u00b5-cocoercive. Moreover since, for every i \u2208{1, . . . , m}, \u2113i is \u03bdi-strongly convex, \u2202\u2113i is \u03bdi-strongly monotone. Hence, by applying Theorem 3.1(i) with A = \u2202f, J\u03c4A = prox\u03c4f, C = \u2207h and for every i \u2208{1, . . . , m}, D\u22121 i = \u2207\u2113\u2217 i , Bi = \u2202gi, J\u03c3iB\u22121 i = prox\u03c3ig\u2217 i , we obtain that the sequence (xn)n\u2208N converges weakly to some x \u2208H such that z \u2208\u2202f(x) + m X i=1 \u03c9iL\u2217 i \u0000(\u2202gi \u25a1\u2202\u2113i)(Lix \u2212ri) \u0001 + \u2207h(x), (4.6) and the sequence ((v1,n, . . . , vm,n))n\u2208N converges weakly to some (v1, . . . , vm) such that \u0000\u2203x \u2208H \u0001 ( z \u2212Pm i=1 \u03c9iL\u2217 i vi \u2208\u2202f(x) + \u2207h(x) (\u2200i \u2208{1, . . . , m}) vi \u2208(\u2202gi \u25a1\u2202\u2113i)(Lix \u2212ri). (4.7) As shown in the proof of [16, Theorem 4.2], x \u2208P1 and (v1, . . . , vm) \u2208D1. This proves (i). Now, if h is uniformly convex at x, then \u2207h is uniformly monotone at x. Hence, (ii) follows from Theorem 3.1(ii). Similarly, (iii) follows from Theorem 3.1(iii). 11 \fRemark 4.3 Here are some observations on the above results. (i) If a function \u03d5: H \u2192R is convex and di\ufb00erentiable function with a \u03b2\u22121-Lipschitzian gradient, then \u2207\u03d5 is \u03b2-cocoercive [4, 5]. Hence, in the context of convex minimization problems, the restriction of cocoercivity made in Problem 1.1 with respect to the problem considered in [16] disappears. Yet, the algorithm we obtain is quite di\ufb00erent from that proposed in [16, Theorem 4.2]. (ii) Su\ufb03cient conditions which ensure that (4.3) is satis\ufb01ed are provided in [16, Proposition 4.3]. For instance, if (4.1) has at least one solution, and if H and (Gi)1\u2264i\u2264m are \ufb01nite-dimensional, and there exists x \u2208ri dom f such that \u0000\u2200i \u2208{1, . . . , m} \u0001 Lix \u2212ri \u2208ri dom gi + ri dom \u2113i, (4.8) then (4.3) holds. (iii) Consider the special case when z = 0 and, for every i \u2208{1, . . . , m}, ri = 0, \u03c3i = \u03c3 \u2208]0, +\u221e[, and \u2113i : v 7\u2192 ( 0 if v = 0, +\u221e otherwise. (4.9) Then, (4.5) reduces to (\u2200n \u2208N) \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 pn = prox\u03c4f \u0010 xn \u2212\u03c4 \u0010 Pm i=1 \u03c9iL\u2217 i vi,n + \u2207h(xn) + a1,n \u0011\u0011 + a2,n yn = 2pn \u2212xn xn+1 = xn + \u03bbn(pn \u2212xn) for i = 1, . . . , m $ qi,n = prox\u03c3g\u2217 i \u0010 vi,n + \u03c3 \u0000Liyn + ci,n \u0001\u0011 + bi,n vi,n+1 = vi,n + \u03bbn(qi,n \u2212vi,n), (4.10) which is the method proposed in [18, Eq. (36)]. However, in this setting, the conditions (4.4) and (4.3) are di\ufb00erent from the conditions [18, Eq. (38)] and [18, Eq. (39)], respectively. Moreover, the present paper provides the strong convergence conditions. (iv) In \ufb01nite-dimensional spaces, with exact implementation of the operators, and with the further restriction that m = 1, h: x 7\u21920, \u21131 is as in (4.9), r1 = 0, and z = 0, (4.5) remains convergent if \u03bbn \u2261\u03bb \u2208]0, 2[ under the same condition presented here [22, Remark 5.4]. If we further impose the restriction \u03bbn \u22611, then (4.5) reduces to the method proposed in [10, Algorithm 1]. An alternative primal-dual algorithm for this problem is proposed in [12]. Acknowledgement. I thank Professor Patrick L. Combettes for bringing this problem to my attention and for helpful discussions." + } + ], + "Kun Yuan": [ + { + "url": "http://arxiv.org/abs/2307.16813v1", + "title": "Capturing Co-existing Distortions in User-Generated Content for No-reference Video Quality Assessment", + "abstract": "Video Quality Assessment (VQA), which aims to predict the perceptual quality\nof a video, has attracted raising attention with the rapid development of\nstreaming media technology, such as Facebook, TikTok, Kwai, and so on. Compared\nwith other sequence-based visual tasks (\\textit{e.g.,} action recognition), VQA\nfaces two under-estimated challenges unresolved in User Generated Content (UGC)\nvideos. \\textit{First}, it is not rare that several frames containing serious\ndistortions (\\textit{e.g.,}blocking, blurriness), can determine the perceptual\nquality of the whole video, while other sequence-based tasks require more\nframes of equal importance for representations. \\textit{Second}, the perceptual\nquality of a video exhibits a multi-distortion distribution, due to the\ndifferences in the duration and probability of occurrence for various\ndistortions. In order to solve the above challenges, we propose \\textit{Visual\nQuality Transformer (VQT)} to extract quality-related sparse features more\nefficiently. Methodologically, a Sparse Temporal Attention (STA) is proposed to\nsample keyframes by analyzing the temporal correlation between frames, which\nreduces the computational complexity from $O(T^2)$ to $O(T \\log T)$.\nStructurally, a Multi-Pathway Temporal Network (MPTN) utilizes multiple STA\nmodules with different degrees of sparsity in parallel, capturing co-existing\ndistortions in a video. Experimentally, VQT demonstrates superior performance\nthan many \\textit{state-of-the-art} methods in three public no-reference VQA\ndatasets. Furthermore, VQT shows better performance in four full-reference VQA\ndatasets against widely-adopted industrial algorithms (\\textit{i.e.,} VMAF and\nAVQT).", + "authors": "Kun Yuan, Zishang Kong, Chuanchuan Zheng, Ming Sun, Xing Wen", + "published": "2023-07-31", + "updated": "2023-07-31", + "primary_cat": "cs.CV", + "cats": [ + "cs.CV" + ], + "main_content": "INTRODUCTION User Generated Content (UGC) has brought evolution to the dailylife consumer domain, which empowers amateurs to become active producers more than consumers. Lower video production cost leads to an explosion of UGC videos on video-sharing platforms, such as FaceBook, Kwai, and so on, which aims to deliver high-quality Quality of Experience (QoE) / Quality of Service (QoS) experience \u2020Authors contributed equally to this research. blocky out-of-focus motion blur \u2026 \u2026 Quality Score 4.0 4.1 4.2 AVG=4.19 \u2206 MOS=3.68 Frame Index 30 60 90 120 150 180 Figure 1: An example of UGC videos of No.13079468475 in the KoNViD-1k [20]. This video contains multiple distortions along the temporal dimension, including motion-related types (i.e. out-of-focus and blurriness) and compressionrelated types (i.e. blocking artifacts). There exists a large difference in perceptual quality among frames. And those frames with low quality determine the overall quality instead of an arithmetic mean. Therefore, a more appropriate strategy is needed for spatiotemporal representation in VQA. to users. Compared with Professionally Generated Content (PGC), UGC videos inevitably have worse conditions of shooting, poor capturing equipment, and unstable transmission links [54]. For video streaming services, VQA has attracted more attention to filter out videos with low perceptual qualities [14, 17]. Furthermore, VQA is also used to conduct content-aware video encoding [6] or enhancement [12, 21, 66], resulting in lower bandwidth cost and better viewing experience. Therefore, it is of great economic value to rate the perceptual quality of UGC videos through VQA. A large number of studies on image/video quality assessment (QA) are studied in the previous literature. According to the availability of reference videos, QA measures can be categorized into full-reference (FR) [16, 26], reduced-reference (RR) [37] and noreference (NR) [25, 30, 40, 57]. Since distortion-free reference videos are often hard to obtain, NR-VQA is widely adopted in the UGC scenarios. According to the feature generation types, QA methods are divided into traditional hand-crafted [38, 40] and learning-based [4, 18, 23, 31] ones. With the rapid development of deep learning, arXiv:2307.16813v1 [cs.CV] 31 Jul 2023 \fMM\u201923, Oct. 29\u2013Nov. 3, 2023, Ottawa, Canada Kun Yuan\u2020, Zishang Kong\u2020, Chuanchuan Zheng, Ming Sun, and Xing Wen convolutional neural networks (CNN) [29, 30, 67] and Transformers [25, 57, 59] are also used to boost the VQA domain. As shown in Fig. 1, the common phenomenon is that multiple distortions co-occur within a UGC video, where different distortions begin to appear at different frames and own different time-span. Such a phenomenon casts two challenges for a better fit of human perceptual quality. First, the perceptual quality of a video is determined by the keyframes that contain particular distortions. Excessively dense sampling brings an unbalanced distribution of frames and may disturb the learning process of distortion characteristics. While relatively current sparse sampling may ignore keyframes. How to select frames efficiently is an essential problem to be solved. Second, due to the differences in temporal duration of different distortions, the perceptual quality of frames within a video exhibits a multi-distribution mode. Take some distorted characteristics for example, blocking artifact [23], dirty lens [19], and noise [33, 70] are usually easy-detected given an individual frame. But out-of-focus and motion blurriness [34] can only be recognized using multiple frames. These factors put forward a higher request for VQA methods with the ability to perform frames analysis under different durations simultaneously. To overcome the aforementioned annoying challenges, we propose Visual Quality Transformer (VQT) to extract quality-aware features focusing on multi-distortions more efficiently. Specifically, to solve the first challenge, a Sparse Temporal Attention (STA) is proposed to sample keyframes via analyzing the temporal correlation between frames. It reformulates self-attention from the perspective of sparse sampling and adopts a proper sampling ratio according to the Johnson-Lindenstrauss (JL) lemma [22]. The keyframes can be selected by comparing the Kullback\u2013Leibler (KL) difference between the Uniform distribution and its cosine similarity with other frames. As for model efficiency, compared with vanilla temporal attention, STA reduces the computational complexity from \ud835\udc42(\ud835\udc472) to \ud835\udc42(\ud835\udc47log\ud835\udc47). To solve the second challenge, owing to the efficiency of the STA module, Multi-Pathway Temporal Network (MPTN) is adopted to capture co-existing distortions in a video simultaneously, which stacks multiple STA modules with different degrees of sparsity. Finally, the aggregated features are used for the representation of a video in VQA. Our contributions are summarized as follows: \u2022 We propose an effective and efficient Visual Quality Transformer (VQT) for the NR-VQA tasks, where the proposed STA selects key frames containing particular distortions and the MPTN helps capture different distorted characteristics simultaneously in UGC scenarios. \u2022 VQT demonstrates superior performance than many stateof-the-art (SoTA) methods in three NR-VQA datasets, raising the performance by 2.14% of PLCC in KoNViD-1k and 2.17% of PLCC in YouTube-UGC over the best results. Furthermore, VQT obtains better results in four FR-VQA datasets (crossdataset evaluation in three of them) against widely-adopted industrial algorithms (e.g., VMAF and AVQT). \u2022 VQT can act as a plug-in module used for general computervision tasks and shows good generalization ability in the video classification task. Compared with the original dense attention mechanism (e.g., TimeSformer), the computational cost decreases from 197 TFLOPs to 154 TFLOPs (-22%). 2 RELATED WORK 2.1 Perceptual Quality Assessment According to the accessibility of the reference images or videos, QA is divided into FR-QA, RR-QA, and NR-QA tasks. FR-QA and RR-QA tasks require a full and partial reference respectively. While the NR-QA method only takes distorted images or videos as input, which is often more challenging, but also more practical in most scenarios. In this paper, we focus on the NR-VQA domain. In the early period of NR-VQA, most works [1, 36, 56] focused on identifying specific types of distortions(e.g., blur, blocky). Then, more methods [38, 62] have been proposed to focus on multiple distortions jointly to carry out comprehensive QA. With the rapid progress of deep learning, learning-based methods [27, 49, 63, 67\u2013 69] have suppressed the performance of traditional hand-crafted ones, due to their versatility and generalization. RAPIQUE [49] combined quality-aware features of scene statistics and semanticsaware deep convolutional features. A combination of 3D-CNN and LSTM was adopted [64] to extract local spatiotemporal features from small clips in the video. Patch-VQA [63] devised a local-toglobal patch-based architecture, and extracted both 2D and 3D video features using a temporal DNN to predict the quality. STDAM [60] introduced using the graph convolution and attention module to extract and enhance the quality-related features. 2BiVQA [44] proposed using two Bi-directional Long Short Term Memory (Bi-LSTM) to conduct the quality assessment. One is for capturing short-range dependencies between image patches, and the other is for capturing long-range dependencies between frames. However, how to efficiently select keyframes to model quality-related features is still an open problem in the VQA domain. Recently, DisCoVQA [57] designs a Transformer-based Spatial-Temporal Distortion Extraction module to tackle temporal quality attention. 2.2 Video Transformer Architecture Visual Transformers [10] are the most popular alternatives applied in various vision tasks, due to their good ability in modeling long-term dependency of sequential data. Following ViT, many Transformer-based models [2, 3, 13, 35, 65] were developed for video classification tasks. TimeSformer [3] explored the factorization of spatial-temporal dimension for efficient computation. Video Swin Transformer [35] globally connected patches across the spatial and temporal dimensions, and advocated an inductive bias of locality for a better speed-accuracy trade-off. There also exists some work exploring the applicability of Transformers in the field of VQA. B-VQA [28] combined GRU unit with Transformer encoder to model the temporal information, which further boosts the performance. StarVQA [59] transferred the divided space-time attention in TimeSformer directly into VQA, showing well generalization ability in the regression task. But these architectures are not designed specifically for VQA, especially for efficiently modeling co-existing distortions. In this paper, we deeply analyze the problems faced by VQA and design the VQT in a targeted manner. \fCapturing Co-existing Distortions in User-Generated Content for No-reference Video Quality Assessment MM\u201923, Oct. 29\u2013Nov. 3, 2023, Ottawa, Canada 0 0 0 1 1 0 0 0 \ud835\udc47\u00d7\ud835\udc51 \ud835\udc3e$ Sparse Temporal Attention \u00d7 \ud835\udc26 Multi-Pathway Temporal Network \u2026 \u2026 Spatial Attn. Multi-Pathway Temporal Network \ud835\udc3f\u00d7 MLP Head Quality Score Clip Tokenization N MLP 2 1 0 qs LayerNorm LayerNorm LayerNorm \ud835\udc44$ \ud835\udc49 $ \ud835\udc47\u00d7\ud835\udc51 \ud835\udc47\u00d7\ud835\udc51 Sparse Query Sampling log\ud835\udc47 log\ud835\udc47\u00d72 A ` \ud835\udc47\u00d7\ud835\udc51 \ud835\udc3e$ \ud835\udc44$ \ud835\udc49 $ \ud835\udc47\u00d7\ud835\udc51 \ud835\udc47\u00d7\ud835\udc51 Sparse Query Sampling 0 1 1 1 1 0 0 0 2!log\ud835\udc47 2!log\ud835\udc47\u00d72 A 2!log\ud835\udc47\u00d7\ud835\udc51 log\ud835\udc47\u00d7\ud835\udc51 \u2026 \u2026 \u2026 C \ud835\udc49 \" # \ud835\udc49 $ # (1 + \u22ef+ 2!)log\ud835\udc47\u00d7\ud835\udc51 P \ud835\udc47\u00d7\ud835\udc51 \ud835\udc4d% A C P Patch Alignment Sparse Value Concate Mean Padding Input frames 0 Position Embedding qs Quality Score Token Clip Embedding Fig2 \ud835\udc47\u00d7\ud835\udc51 ` log\ud835\udc47\u00d7\ud835\udc47 ` ` \ud835\udc47\u00d7\ud835\udc51 2!log\ud835\udc47\u00d7\ud835\udc47 \u2026 Figure 2: Illustration of the proposed Visual Quality Transformer (VQT). It receives a clip as input and reshapes each frame into patches for tokenization. Then the sequence of tokens is fed into a stacked encoder, performing the spatiotemporal attention. In the temporal dimension, VQT utilizes a Multi-Pathway Temporal Network to capture different distorted characteristics simultaneously with stacking Sparse Temporal Attention blocks. In each block, STA conducts sparse query sampling to locate keyframes for distortion representation. To further enhance STA with spatial relationships, temporal offsets across frames are assigned to patches for alignment. Different blocks contribute to the final representation within an encoder block. To predict the video quality, the corresponding quality score token is used for the final representation. 3 METHOD 3.1 Revisiting Video Transformer Different from image Transformers, video Transformers receive a sequence of frames as input. There exist some differences in basic architecture modules, including clip tokenization and spatiotemporal attention. We first briefly revisit them for a better understanding. Clip Tokenization. The video Transformer takes a clip as input denoted as X \u2208R\ud835\udc47\u00d7\ud835\udc3b\u00d7\ud835\udc4a\u00d73, which composes of \ud835\udc47RGB frames with the size of \ud835\udc3b\u00d7\ud835\udc4aunder equal temporal interval sampled from original videos. Following ViT, video Transformers reshape each frame into \ud835\udc41non-overlapping patches, where each size is \ud835\udc43\u00d7 \ud835\udc43 and \ud835\udc41= \ud835\udc3b\u00d7\ud835\udc4a \ud835\udc432 . Then the sequence of frames can be flattened into X \u2208R\ud835\udc47\u00d7\ud835\udc41\u00d7(3\ud835\udc432). Besides, an extra learnable positional embedding E\ud835\udc5d\ud835\udc5c\ud835\udc60is added to encode the spatiotemporal position of each patch. Then the input embedding Z \u2208R\ud835\udc47\u00d7\ud835\udc41\u00d7\ud835\udc51is calculated as: Z = WX\u22a4+ E\ud835\udc5d\ud835\udc5c\ud835\udc60, (1) where W \u2208R\ud835\udc51\u00d73\ud835\udc432 is the mapping weight and \ud835\udc51is the embedding dimension of each frame token. Divided Space-Time Attention. To process spatiotemporal information, TimeSformer utilizes the \u201cDivided Space-Time Attention\" module, where the temporal attention and the spatial attention are performed sequentially. In each encoder block \u2113, the temporal attention first computes the relationship among all patches in the same spatial location from different frames, expressed as: Z\u2113 time = Softmax Q\u2113\u22121 \ud835\udc61\u2032 \u221a \ud835\udc51 K\u2113\u22121 \ud835\udc61\u2032 \u22a4 ! V\u2113\u22121, (2) where Q, K, V are the query, key, and value. And \ud835\udc61\u2032 denotes that the inner products are computed on the temporal dimension. Then the generated Z\u2113 time is fed back for the spatial attention, computing as: Z\u2113space = Softmax Q\u2113\u22121 \ud835\udc5d\u2032 \u221a \ud835\udc51 K\u2113\u22121 \ud835\udc5d\u2032 \u22a4 ! V\u2113\u22121, (3) where \ud835\udc5d\u2032 denotes inner products on the spatial dimension. 3.2 Visual Quality Transformer The illustration of VQT is given in Fig. 2. The core components are STA and MPTN, which we will describe in more detail. Sparse Temporal Attention. Video clips own widespread information redundancy in the temporal dimension [55, 61], both in the frame level and feature level. Frames containing distorted characteristics largely reflect the perceptual quality of the whole video. The upper part of Fig. 3 demonstrates this phenomenon in the VQA domain, where the computed temporal attention map shows that all frames have a strong correlation with the 5-\ud835\udc61\u210eframe (i.e., , an over-exposed distorted image). And the factor of overexposure in \fMM\u201923, Oct. 29\u2013Nov. 3, 2023, Ottawa, Canada Kun Yuan\u2020, Zishang Kong\u2020, Chuanchuan Zheng, Ming Sun, and Xing Wen Query frame index Key frame index Patch 1 Patch 2 Patch 3 Patch 4 Patch 5 Patch 6 Patch 7 Patch 8 Patch 9 Input patch column Input patch row Temporal Attention Map 1 0 Cosine Similarity Frame Index Input patch column Input patch row Temporal Attention Map 1 0 Cosine Similarity Frame Index Frame index 1 2 3 4 5 6 7 8 Query frames Key frames Temporal attention maps of TimeSformer Cosine similarity Correlation 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 0.827 Cosine similarity Key frame index Calculated similarity Fitted distribution Normal distribution D =0.794 KL 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 0.09234 Cosine similarity Key frame index Calculated similarity Fitted distribution Normal distribution D =0.027 KL Figure 3: The upper images show that the quality of the video is largely affected by frames containing distortions. The below curves give the distributions of \ud835\udc37\ud835\udc3e\ud835\udc3ffor the 5-th (left) and 3-th (right) frame. this frame deteriorates the quality of this video. Based on the observations in Fig. 1 and Fig. 3, we propose a Sparse Temporal Attention, aiming to sample key-frames containing distortions in a video. Specifically, the attention mechanism is first reformatted from the perspective of sparse sampling. Given the query Q \u2208R\ud835\udc47\u00d7\ud835\udc51, key K \u2208R\ud835\udc47\u00d7\ud835\udc51, and value V \u2208R\ud835\udc47\u00d7\ud835\udc51, the self-attention is computed as V\u2032 = softmax( QK\u22a4 \u221a \ud835\udc51)V, where V\u2032 \u2208R\ud835\udc47\u00d7\ud835\udc51. According to the JL lemma, there exists a linear transformation that projects V into V\u2032 \u2208Rlog\ud835\udc47\u00d7\ud835\udc51with low-distortion embedding, where log\ud835\udc47is the minimal number of frames sampled from \ud835\udc47frames. Specifically, we let \ud835\udc5e\ud835\udc56and \ud835\udc58\ud835\udc57represent the feature of \ud835\udc56-th row and \ud835\udc57-th row in Q and K, respectively. Then we calculate the cosine similarity between \ud835\udc5e\ud835\udc56and \ud835\udc58\ud835\udc57,\ud835\udc57=1,..,\ud835\udc47. And the computed distribution can be noted as \ud835\udc5d(K|\ud835\udc5e\ud835\udc56), representing the correlation of the \ud835\udc56-th frame with all frames. Then we compute the KL divergence between \ud835\udc5d(K|\ud835\udc5e\ud835\udc56) and the uniform distribution \ud835\udc48: \ud835\udc37KL(\ud835\udc5d(K|\ud835\udc5e\ud835\udc56)\u2225\ud835\udc48) = \ud835\udc47 \u2211\ufe01 \ud835\udc57=1 1 \ud835\udc47log 1 \ud835\udc47\u22121 \ud835\udc47log \ud835\udc5e\ud835\udc56\ud835\udc58\ud835\udc47 \ud835\udc57 \u00cd\ud835\udc47 \ud835\udc57=1 \ud835\udc5e\ud835\udc56\ud835\udc58\ud835\udc47 \ud835\udc57 ! (4) = \u2212log 1 \ud835\udc47\u2212 \ud835\udc47 \u2211\ufe01 \ud835\udc56=1 1 \ud835\udc47 \u00a9 \u00ad \u00ab ln exp \ud835\udc5e\ud835\udc56\ud835\udc58\ud835\udc47 \ud835\udc57 \u221a \ud835\udc51 \u2212ln \ud835\udc47 \u2211\ufe01 \ud835\udc57=1 \ud835\udc52 \ud835\udc5e\ud835\udc56\ud835\udc58\ud835\udc47 \ud835\udc57 \u221a \ud835\udc51\u00aa \u00ae \u00ac . (5) For simplicity, we ignore the constant term of \u2212log 1 \ud835\udc47. And the equation can be rewritten as: D\ud835\udc3e\ud835\udc3f(\ud835\udc5d(K|\ud835\udc5e\ud835\udc56)||\ud835\udc48) = ln \ud835\udc47 \u2211\ufe01 \ud835\udc57=1 \ud835\udc52 \ud835\udc5e\ud835\udc56\ud835\udc58\u22a4 \ud835\udc57 \u221a \ud835\udc51 \u22121 \ud835\udc47 \ud835\udc47 \u2211\ufe01 \ud835\udc57=1 \ud835\udc5e\ud835\udc56\ud835\udc58\u22a4 \ud835\udc57 \u221a \ud835\udc51 ! . (6) Algorithm 1 Pseudo-code of selection of keyframes 1: \ud835\udc44\ud835\udc60\ud835\udc4e\ud835\udc5a\ud835\udc5d\ud835\udc59\ud835\udc52= \ud835\udc5f\ud835\udc4e\ud835\udc5b\ud835\udc51\ud835\udc5c\ud835\udc5a.\ud835\udc60\ud835\udc4e\ud835\udc5a\ud835\udc5d\ud835\udc59\ud835\udc52(\ud835\udc44,\ud835\udc54\ud835\udc4e\ud835\udc5a\ud835\udc5a\ud835\udc4e) \u22b2gamma:logT/T 2: \ud835\udc5a\ud835\udc62= \ud835\udc3e\ud835\udc3f-\ud835\udc37\ud835\udc56\ud835\udc63\ud835\udc52\ud835\udc5f\ud835\udc54\ud835\udc52\ud835\udc5b\ud835\udc50\ud835\udc52(\ud835\udc44\ud835\udc60\ud835\udc4e\ud835\udc5a\ud835\udc5d\ud835\udc59\ud835\udc52, \ud835\udc3e).\ud835\udc5a\ud835\udc52\ud835\udc4e\ud835\udc5b(\ud835\udc51\ud835\udc56\ud835\udc5a= \u22121) 3: \ud835\udc60\ud835\udc56\ud835\udc54\ud835\udc5a\ud835\udc4e= \ud835\udc3e\ud835\udc3f-\ud835\udc37\ud835\udc56\ud835\udc63\ud835\udc52\ud835\udc5f\ud835\udc54\ud835\udc52\ud835\udc5b\ud835\udc50\ud835\udc52(\ud835\udc44\ud835\udc60\ud835\udc4e\ud835\udc5a\ud835\udc5d\ud835\udc59\ud835\udc52, \ud835\udc3e).\ud835\udc60\ud835\udc61\ud835\udc51(\ud835\udc51\ud835\udc56\ud835\udc5a= \u22121) 4: init \ud835\udc56= 0; \ud835\udc4e\ud835\udc61\ud835\udc61\ud835\udc5b= \ud835\udc67\ud835\udc52\ud835\udc5f\ud835\udc5c\ud835\udc60(\ud835\udc59\ud835\udc5c\ud835\udc54\ud835\udc47,\ud835\udc47) 5: for \ud835\udc5e\ud835\udc56\ud835\udc5b\ud835\udc44do 6: if \ud835\udc56< \ud835\udc59\ud835\udc5c\ud835\udc54\ud835\udc47and \ud835\udc3e\ud835\udc3f-\ud835\udc37\ud835\udc56\ud835\udc63\ud835\udc52\ud835\udc5f\ud835\udc54\ud835\udc52\ud835\udc5b\ud835\udc50\ud835\udc52(\ud835\udc5e, \ud835\udc3e) > \ud835\udc5a\ud835\udc62+ \ud835\udc60\ud835\udc56\ud835\udc54\ud835\udc5a\ud835\udc4ethen 7: \ud835\udc4e\ud835\udc61\ud835\udc61\ud835\udc5b[\ud835\udc56, :] = \ud835\udc5e@\ud835\udc3e.\ud835\udc61\ud835\udc5f\ud835\udc4e\ud835\udc5b\ud835\udc60\ud835\udc5d\ud835\udc5c\ud835\udc60\ud835\udc52() 8: \ud835\udc56+=1 9: end if 10: end for 11: \ud835\udc49= \ud835\udc4f\ud835\udc5a\ud835\udc5a(\ud835\udc4e\ud835\udc61\ud835\udc61\ud835\udc5b,\ud835\udc49) \u22b2bmm:batch matrix multiplication Larger values of D\ud835\udc3e\ud835\udc3f(\ud835\udc5d(K|\ud835\udc5e\ud835\udc56)||\ud835\udc48) indicate a stronger correlation with other frames, which can be used for the selection of keyframes. Examples are given in Fig. 3, where frames containing distortions own larger (e.g., , the 5-\ud835\udc61\u210eframe) values and vice versa. To reduce the complexity of computing the divergence of all frames, log\ud835\udc47frames are sampled randomly. The distribution of divergence of all frames can be estimated by calculating the mean \ud835\udf07and variance \ud835\udf0eof log\ud835\udc47frames. Then log\ud835\udc47query frames can be selected, whose divergence meets D\ud835\udc3e\ud835\udc3f(\ud835\udc5d(K|\ud835\udc5e\ud835\udc56)||\ud835\udc48) > \ud835\udf07+ \ud835\udf0e. The dimension of sampled query \u02c6 Q is reduced to log\ud835\udc47\u00d7 \ud835\udc51. And the pseudocode is shown in Alg. 1. To enhance the spatial relationship of queries between different frames, STA further performs a spatial shift for alignment. Specifically, we reshape the spatial dimension of selected query features \u02c6 Q \u2208Rlog\ud835\udc47\u00d7\ud835\udc41\u00d7\ud835\udc51into Rlog\ud835\udc47\u00d7 \u221a \ud835\udc41\u00d7 \u221a \ud835\udc41\u00d7\ud835\udc51(the spatial dimension \ud835\udc41is not shown in Fig. 2 for simplify). Then a linear projection layer is attached on the reshaped features to predict 2D offsets P for each token, where P \u2208Rlog\ud835\udc47\u00d7 \u221a \ud835\udc41\u00d7 \u221a \ud835\udc41\u00d72. Then the shifted query features \u02c6 Q\u2032 can be obtained by bilinear interpolation. Then the weighted value features can be computed by: V\u2032 = softmax( \u02c6 Q\u2032K\u22a4 \u221a \ud835\udc51 )V. (7) Multi-Pathway Temporal Network. To capture different distorted characteristics simultaneously, multiple STA modules with different degrees of sparsity are stacked in parallel. Given a video clip with \ud835\udc47frames, the number \ud835\udc5aof pathway is determined by \ud835\udc5a= \u230alog( \ud835\udc47 \u2308log\ud835\udc47\u2309+ 1)\u230b\u22121, where \u2308\u00b7\u2309and \u230a\u00b7\u230brepresents the ceiling and flooring operation respectively. The minimal number of frames for different STA modules is log\ud835\udc47, and the maximal is 2\ud835\udc5alog\ud835\udc47. Take an input clip with 96 frames, for example, its \ud835\udc5ais 3, containing 3 pathways. And each pathway selects 7, 14, and 28 keyframes respectively. Each block performs temporal attention over selected frames, resulting in a weighted value of V\u2032 \ud835\udc5a\u2208R2\ud835\udc5alog\ud835\udc47\u00d7\ud835\udc51. Then the values generated by different blocks are concatenated in the temporal dimension, resulting in \u02c6 V \u2208R(1+\u00b7\u00b7\u00b7+2\ud835\udc5a) log\ud835\udc47\u00d7\ud835\udc51. To align with succeeding encoder blocks, we use the mean padding operation to fill \u02c6 V in the temporal dimension, generating Z\ud835\udc59\u2208R\ud835\udc47\u00d7\ud835\udc51: \u02c6 V = Concate(V\u2032 1, \u00b7 \u00b7 \u00b7 , V\u2032 \ud835\udc5a) Z\ud835\udc59= Mean-Padding( \u02c6 V). (8) \fCapturing Co-existing Distortions in User-Generated Content for No-reference Video Quality Assessment MM\u201923, Oct. 29\u2013Nov. 3, 2023, Ottawa, Canada 3.3 Optimization Objective A smooth L1 loss is adopted to train VQT models. Let F (\u00b7) represent the mapping function of the VQT model. Given mini-batch videos during training, the objective function can be denoted as: min 1 |B| B \u2211\ufe01 \ud835\udc56=1 L1\u2212\ud835\udc60\ud835\udc5a\ud835\udc5c\ud835\udc5c\ud835\udc61\u210e(F (X\ud835\udc56),\ud835\udc66\ud835\udc56), (9) where X\ud835\udc56and \ud835\udc66\ud835\udc56is the input video and corresponding labeled Mean Opinion Score (MOS). B indicates the size of the mini-batch. 3.4 Computational Efficiency Computational Complexity of STA. For an input clip with\ud835\udc47frames, the computational cost of the original temporal attention module is \ud835\udc42(\ud835\udc472). The STA module applies sparse computation among the temporal dimension with a cost of \ud835\udc42(2\ud835\udc5alog\ud835\udc47\u00b7\ud835\udc47). And 2\ud835\udc5alog\ud835\udc47is smaller than \ud835\udc47as mentioned above. Computational Complexity of MPTN. MPTN is composed of multiple STA blocks, whose total computational cost is computed by combining each one of \ud835\udc42(\u00cd\ud835\udc5a \ud835\udc4e=0 2\ud835\udc4elog\ud835\udc47\u00b7\ud835\udc47\u00b7\ud835\udc51). Since \u00cd\ud835\udc5a \ud835\udc4e=0 2\ud835\udc4elog\ud835\udc47 is smaller than \ud835\udc47, the computational cost of MPTN is still less than that of the original temporal attention module. Measured Inference Speed. Further evaluations of efficiency are given in the following experiments. In Tab. 5 and 6, compared with the original version of dense attention (i.e., , TimeSformer used in StarVQA), VQT has a less computational cost (-22%), faster inference speed (+13%) and higher performance in PLCC (+3.49%). 4 EXPERIMENTS 4.1 Datasets and Evaluation NR datasets. We leverage three NR-VQA datasets to evaluate VQT models: LIVE Video Quality Challenge Database (LIVE-VQC) [43], Konstanz Natural Video Database (KoNViD-1k) [20], and Blind Video Quality Assessment for User Generated Content (YoutubeUGC) [53]. Subjective quality scores are provided in the form of MOS. LIVE-VQC contains 585 videos labeled by MOS with a resolution from 240P to 1080P. KoNViD-1k comprises a total of 1,200 videos with a resolution of 960 \u00d7 540 that are fairly sampled from a large public video dataset, YFCC100M[45]. The duration of videos is 8s with 24/25/30FPS, whose MOS ranges from 1.22 to 4.64. YoutubeUGC is composed of 1,500 videos that are sampled from millions of YouTube videos belonging to 15 categories annotated by a knowledge graph. The resolutions of videos are from 360P to 4K. For the Youtube-UGC, we follow the default training and testing splits [53]. For LIVE-VQC and KoNViD-1k, following [49], 80% of the dataset is used for training, and the remaining 20% is used for testing. FR Datasets. To verify the generalization ability to video codec field, four more FR-VQA datasets are tested, including ICME-FR \u2020, VQEG HD3 \u2020, NFLX Video dataset \u2020 and the Waterloo IVC 4k Video Quality database [32]. The VQEG HD3 database is composed of 9 source clips with a resolution of 1080P. And the source clips are encoded into 63 distorted videos for evaluation. The NFLX Video \u2020http://2021.ieeeicme.org/2021.ieeeicme.org \u2020https://www.cdvl.org \u2020https://github.com/Netflix/vmaf consists of 34 source clips, whose duration is 6s. They are sampled from popular TV shows and movies on Netflix. Source clips are encoded at resolutions ranging from 384 \u00d7 288 to 1920 \u00d7 1080, resulting in about 300 distorted videos. The Waterloo IVC 4k Video Quality database is created from 20 pristine 4K videos. Each video is encoded by five encoders: HEVC, H264, VP9, AV1, and AVS2, and divided into three solutions (960 \u00d7 540, 1920 \u00d7 1080 and 3840 \u00d7 2160 with four distortion levels, resulting in 1,200 encoded videos. Evaluation Criteria. Pearson\u2019s Linear Correlation Coefficient (PLCC), Spearman\u2019s Rank-Order Correlation Coefficient (SROCC), Kendall\u2019s Rank-Order Correlation Coefficient (KROCC), and Root Mean Square Error (RMSE) are used for evaluation. PLCC and RMSE measure the prediction accuracy, SROCC, and KROCC indicate the prediction monotonicity. Better VQA methods should have larger PLCC/SROCC/KROCC and smaller RMSE values. 4.2 Implementation Details Our implementation is based on PyTorch [41] and MMAction2 [8]. All models are trained using 4 NVIDIA Tesla V100. The number of encoders follows the original setting of TimeSformer. We set a patch size of 16 in clip tokenization. The embedding dimension \ud835\udc51is 768. We use models that have been pre-trained on ImageNet [9] and Kinetics-400 [24] for training. During the optimization procedure, we use the AdamW optimizer with a learning rate of 1e-5 decayed by a factor of 0.1 every 30 epochs, minimizing the L1 loss. All models are trained for 90 epochs. By default, the checkpoint generated by the last iteration is used for evaluation. The batch size of video clips is set to 4 with a clip length of 96. Other training settings are the same with [3]. The median result of 10 repeat runs is used for Tab. 1 with different random splits. 4.3 Comparison with SoTA Methods Extensive experiments are conducted to compare with SoTA QA methods. As given in Tab. 1, we report the PLCC and SROCC performance in KoNViD-1k, LIVE-VQC and Youtube-UGC datasets. Besides, weighted average scores are reported based on the number of videos of three datasets. Some observations and conclusions can be found here. First, compared with IQA methods (i.e. NIQE [40], BRISQUE [38], CORNIA [62]), VQT obtains a large performance lead (+31.54%, +24.24%, +26.04% of PLCC in KoNViD-1k), showing the effectiveness of fusion strategy of frames over an arithmetic mean. Second, compared with hand-crafted features (i.e. [39]), VQT demonstrates the utility of learning-based methods over prior knowledge. Third, compared with CNN models (i.e. [28, 49, 60, 63]), VQT shows the advantage of Transformer models in building longrange dependencies. Fourth, compared with current Transformer models (i.e. StarVQA [59], TimeSformer), VQT also evaluate the gains from architecture modification by large margins (+7.24% of PLCC in KoNViD-1k, +2.77% of PLCC in LIVE-VQC). Compared with current SOTA method, VQT also outperforms STDAM [60] in three datasets(+2.59% of PLCC in KoNViD-1k, +1.53% of PLCC in LIVE-VQC, and +2.17% of PLCC in Youtube-UGC). Since the size of LIVE-VQC is the smallest (only containing 585 videos), BVQA [28] achieves the highest performance by introducing extra QA training data. However, VQT still surpasses it in the weighted scores (+2.39% \fMM\u201923, Oct. 29\u2013Nov. 3, 2023, Ottawa, Canada Kun Yuan\u2020, Zishang Kong\u2020, Chuanchuan Zheng, Ming Sun, and Xing Wen Table 1: Quantitative results of different methods on three public NR-VQA datasets. Larger PLCC and SROCC indicate better performance. Besides, weighted average scores are reported based on the number of videos of three datasets. The best and second best performances are highlighted and underlined. The mark \u201c-\" denotes that results are not reported originally. The \u201c*\" mark indicates using extra training data for QA tasks. The VQT models outperform almost all SoTA methods by large margins. Method KoNViD-1k LIVE-VQC Youtube-UGC Weighted Average PLCC \u2191 SROCC \u2191 PLCC \u2191 SROCC \u2191 PLCC \u2191 SROCC \u2191 PLCC \u2191 SROCC \u2191 VIIDEO [39] 0.303 0.298 0.2164 0.0332 0.1534 0.0580 0.2230 0.1459 NIQE [40] 0.5530 0.5417 0.6286 0.5957 0.2776 0.2379 0.4500 0.4225 CORNIA [62] 0.608 0.610 BRISQUE [38] 0.626 0.654 0.638 0.592 0.395 0.382 0.5299 0.5265 VBLIINDS [42] 0.6576 0.6947 0.7120 0.7015 0.5551 0.5590 0.6431 0.6427 GRU-VQA [29] 0.744 0.755 TLVQM [11] 0.7688 0.7729 0.8025 0.7988 0.6590 0.6693 0.7284 0.7337 MDTVSFA [30] 0.7856 0.7812 0.7728 0.7382 UGC-VQA [48] 0.7803 0.7832 0.7514 0.7522 0.7733 0.7787 0.7719 0.7754 PVQ [63] 0.786 0.791 0.837 0.827 RAPIQUE [49] 0.8175 0.8031 0.7863 0.7548 0.7684 0.7591 0.7907 0.7753 StarVQA [59] 0.796 0.812 0.808 0.732 BVQA\u2217[28] 0.8335 0.8362 0.8415 0.8412 0.8194 0.8312 0.8290 0.8350 STDAM [60] 0.8415 0.8448 0.8204 0.7931 0.8297 0.8341 0.8320 0.8305 2BiVQA [44] 0.835 0.815 0.832 0.761 0.790 0.771 0.794 0.800 DisCoVQA [57] 0.847 0.847 0.826 0.820 Our TimeSformer 0.8293 0.8342 0.8017 0.7845 0.8279 0.8133 0.8235 0.8159 VQT 0.8684 0.8582 0.8357 0.8238 0.8514 0.8357 0.8529 0.8421 of PLCC), showing strong generalization ability. Compared with recent DisCoVQA, VQT obtain a higher result of PLCC by 2.14%. 4.4 Comparing with VMAF/AVQT We further compare the effectiveness of our method with the two industrial standards, i.e. Netflix\u2019s VMAF \u2020 and Apple\u2019s AVQT \u2020 on the four widely adopted open datasets. The VMAF and AVQT have been used as standard deals to their simplicity in computations and consistent performance across different types of videos. As shown in Tab. 2, their performance is consistently high across the four datasets except for the Waterloo IVC 4k since both VMAF and AVQT were developed before 4K videos were becoming popular. The performance decrease on Waterloo IVC 4k indicates that the generability of these two algorithms is not always satisfying when a new video format is induced. We train our VQT model with data solely from the ICME-FR train split. However, we not only test it on the ICME-FR test split but also directly evaluate its performance on the aforementioned three datasets without any fine-tuning (i.e., cross-dataset evaluation). From Tab. 2, we observe that our proposed VQT method further improves the performance. This enhancement is not only attributed to the fact that our model is learned from a large-scale dataset but also thanks to our carefully designed architecture that can effectively extract and integrate spatiotemporal features to better model users\u2019 Mean Opinion Scores (MOS). It is worth noting that our method is efficient and can be easily integrated and substituted by existing \u2020https://github.com/Netflix/vmaf \u2020https://developer.apple.com/videos/play/wwdc2021/10145 Table 2: Comparisons with industrial standards. The VQT model is trained on the ICME-FR datasets with the supervision of DMOS. And direct inference results are reported on the other three datasets without fine-tuning. VQT shows strong generalization ability and practical prospects. Datasets Method PLCC \u2191 SROCC \u2191 ICME-FR VMAF 0.9423 0.9137 AVQT 0.9730 0.9334 VQT 0.9867 0.9364 NFLX Video Dataset VMAF 0.9351 0.9173 AVQT 0.9571 0.9420 VQT 0.9715 0.9532 VQEG HD3 VMAF 0.9266 0.9238 AVQT 0.9481 0.9417 VQT 0.9603 0.9576 Waterloo IVC 4k VMAF 0.7324 0.7325 AVQT 0.7749 0.7738 VQT 0.7885 0.7821 methods such as VMAF/AVQT for evaluating the performance of different video encoding strategies. 4.5 Ablation Studies and Visualization To verify the rationality of the proposed modules, ablation studies are conducted in the following aspects. \fCapturing Co-existing Distortions in User-Generated Content for No-reference Video Quality Assessment MM\u201923, Oct. 29\u2013Nov. 3, 2023, Ottawa, Canada Table 3: Ablation study of individual modules in VQT, conducted in KoNViD-1k, Youtube-UGC, and Kinetics-400. Modules KoNViD-1k YoutuUGC K400 STA MPTN PLCC \u2191SROCC \u2191PLCC \u2191SROCC \u2191Top-1 Acc \u2191 \u2713 \u2713 0.8684 0.8582 0.8514 0.8357 80.3 \u2713 \u00d7 0.8530 0.8510 0.8429 0.8280 79.0 \u00d7 \u00d7 0.8293 0.8342 0.8279 0.8133 78.0 Table 4: Ablation study on the type of reduction and the number of frames used in the STA module. Experiments are conducted in the KoNViD-1k. Sampling Strategy Frames PLCC\u2191 SROCC\u2191 our KL-based log\ud835\udc47 0.8684 0.8582 our KL-based 0.5 log\ud835\udc47 0.8320 0.8309 our KL-based 2 log\ud835\udc47 0.8691 0.8575 Random[7] log\ud835\udc47 0.8067 0.7798 Linear[50] log\ud835\udc47 0.8142 0.8091 Conv[51] log\ud835\udc47 0.8140 0.8090 Clustering log\ud835\udc47 0.8379 0.8203 The rationale of the proposed STA module. The rationale of STA is verified theoretically and experimentally. Theoretically, according to the JL lemma, the lower bound of the error coefficient \ud835\udf00after projection is measured by \ud835\udc51> 8 ln(\ud835\udc47)/\ud835\udf002, where \ud835\udc47is the number of frames, and \ud835\udc51is the embedding size. In our setting of \ud835\udc47= 96 and \ud835\udc51= 768. So \ud835\udf00can be calculated as 0.215, which means more than 78.5% of temporal information or more is maintained by selected log\ud835\udc47keyframes. Experimentally, sufficient ablation studies also confirm the effectiveness of STA as shown in Tab. 3 and 4. Simply adding the STA module to the baseline can bring consistent promotion on two VQA datasets (KoNViD-1k and Youtube-UGC) and one general video classification dataset (Kinetics-400). Reduction types in the STA module. We conducted an evaluation of various linear reduction methods, including random[7] (log\ud835\udc47 frames are randomly selected from \ud835\udc47frames), linear reduction[50] (features are transformed using a matrix of log\ud835\udc47\u00d7\ud835\udc47, resulting in the representation of log\ud835\udc47frames), conv reduction[51] (features are transformed using a Conv/BN/ReLU module, which reduces the channel from \ud835\udc47to log\ud835\udc47), clustering (features are clustered into log\ud835\udc47centers according to cosine similarity), and STA. The results, presented in Tab. 4, demonstrate that the STA reduction method has the most significant positive impact on performance, effectively removing redundant information. Additionally, we found that the log\ud835\udc47setting was the optimal sparsity setting through experimentation involving increasing and decreasing the number of frames used to represent keyframes. Different combinations of proposed modules. We conduct ablation experiments for different proposed modules, including STA and MPTN. Results are given in Tab. 3. The best performance is Table 5: Comparison of inference cost with academic and industrial methods. VQT shows high efficiency. Type TFLOPs Device Time Speed Ratio VMAF CPU 9.85s 1.0\u00d7 AVQT CPU 4.61s 2.1\u00d7 MDTSVFA 231 GPU 7.07s 1.4\u00d7 StarVQA 197 GPU 0.57s 17.3\u00d7 BVQA 89 GPU 0.51s 19.3\u00d7 STDAM 106 GPU 2.12s 4.6\u00d7 VQT 154 GPU 0.50s 19.7\u00d7 obtained by combining them. We also verify individual modules on Kinetics-400, where a combination of modules improves 1.3% on Top-1 accuracy which proves the ability of generalization on general classification tasks. For better understanding, we plot the visualization of the learned temporal attention maps in Fig. 4. The visualization shows that different STA modules in the MPTN pay attention to different frames in a clip. It means that the proposed MPTN can capture different distortions simultaneously. Visualization. To further validate the effectiveness of VQT, we visualize the frames with the highest response and analyze the corresponding quality scores for each individual frame. As shown in Fig. 5, 6 and 7, VQT is capable of effectively perceiving co-existing low-quality features in videos, such as interlace, motion blur, out-ofdefocus and blocking artifacts. Furthermore, compared to averaging the predicted results for all frames, VQT-based temporal processing yields prediction results that are closer to the labeled MOS values. Efficiency comparison. Computation Efficiency is compared with SoTA methods under 1080P/30FPS/30s videos, as shown in Tab. 5, including academic algorithms and industrial algorithms. Due to commercial confidentiality, we cannot obtain open-source model information (e.g., , FLOPs) from Netflix/Apple. For a fair comparison, in GPU, VQT shows higher efficiency than image-based (MDTSVAF, STDAM) and video-based SoTA methods (StarVQA, BVQA). That 0.5s inference time can fulfill the real-time monitoring of the serviceside quality variation. Further, speed-up can be investigated in future work (e.g., , knowledge distillation and quantization). 4.6 Generalization in Video Classification To further evaluate the generalization ability of VQT to other general semantic task, the performance on the video classification dataset of Kinetics-400[5] with SoTA video classification models is evaludated, including R(2+1)D [47], I3D [5], I3D+NL [52], ip-CSN152 [46], SlowFast [15], TimeSformer [3] and Video Swin Transformer [35]. We follow the default training setting in MMAction2 for a fair comparison. As shown in Tab. 6, VQT achieves a Top-1 accuracy by 80.3%, outperforming CNN models [46, 47, 58] and very recent Transformer-based models, e.g., TimeSformer[3] and Video Swin Transformer[35]. This proves the effectiveness and generalization ability of VQT in the general video classification task. We hope this VQT module can achieve more satisfactory performance when used in more general computer vision tasks. \fMM\u201923, Oct. 29\u2013Nov. 3, 2023, Ottawa, Canada Kun Yuan\u2020, Zishang Kong\u2020, Chuanchuan Zheng, Ming Sun, and Xing Wen Time V! \" 2 5 6 3 7 4 8 1 Cosine Similarity V# \" Figure 4: Visualization results of the temporal attention maps generated by an MPTN consisting of two STA modules (No.5956265529 in the KoNViD-1k). By the comparison between V\u2032 1 and V\u2032 2, these two STA modules concentrate on different distortions. The STA with fewer frames pays much attention to compression distortions that can be detected by spatial information, and the STA with more frames focuses on camera movement that can be detected by temporal information. Youtube-UGC/TelevisionClip-0604 1 Frame Index=336 2 Frame Index=360 3 Frame Index=432 4 Frame Index=464 AVG: average image quality score PRS: predicted video quality score MOS: mean opinion score AVG=4.24 PRS=3.68 MOS=3.76 1 2 3 4 Figure 5: Visualization results of the No.TelevisionClip-0604 video sampled from Youtube-UGC. The mixture of interlace and motion blur deteriorates the video quality. Compared to the average image quality score, VQT focuses more on the clips containing distortions, as shown by 4 sampled frames. LIVE-VQC/B174 AVG: average image quality score PRS: predicted video quality score MOS: mean opinion score 1 Frame Index=52 2 Frame Index=74 3 Frame Index=216 4 Frame Index=232 AVG=1.42 PRS=0.90 MOS=0.67 1 2 3 4 Figure 6: Visualization results of the No.B174 video sampled from LIVE-VQC. Our analysis indicates that the primary factors contributing to the degradation are motion blur (exhibited in the 1st and 2nd frames) and out-of-focus (exhibited in the 3rd and 4th frames). 5" + }, + { + "url": "http://arxiv.org/abs/2210.07863v1", + "title": "Revisiting Optimal Convergence Rate for Smooth and Non-convex Stochastic Decentralized Optimization", + "abstract": "Decentralized optimization is effective to save communication in large-scale\nmachine learning. Although numerous algorithms have been proposed with\ntheoretical guarantees and empirical successes, the performance limits in\ndecentralized optimization, especially the influence of network topology and\nits associated weight matrix on the optimal convergence rate, have not been\nfully understood. While (Lu and Sa, 2021) have recently provided an optimal\nrate for non-convex stochastic decentralized optimization with weight matrices\ndefined over linear graphs, the optimal rate with general weight matrices\nremains unclear.\n This paper revisits non-convex stochastic decentralized optimization and\nestablishes an optimal convergence rate with general weight matrices. In\naddition, we also establish the optimal rate when non-convex loss functions\nfurther satisfy the Polyak-Lojasiewicz (PL) condition. Following existing lines\nof analysis in literature cannot achieve these results. Instead, we leverage\nthe Ring-Lattice graph to admit general weight matrices while maintaining the\noptimal relation between the graph diameter and weight matrix connectivity.\nLastly, we develop a new decentralized algorithm to nearly attain the above two\noptimal rates under additional mild conditions.", + "authors": "Kun Yuan, Xinmeng Huang, Yiming Chen, Xiaohan Zhang, Yingya Zhang, Pan Pan", + "published": "2022-10-14", + "updated": "2022-10-14", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG", + "math.OC" + ], + "main_content": "Introduction 1.1 Motivation. Decentralized optimization is an emerging paradigm for large-scale machine learning. By letting each node average with its neighbors, decentralized algorithms save great communication overhead compared to traditional approaches with a central server. While numerous effective decentralized algorithms (see, e.g., [49, 11, 60, 37, 5, 39, 76]) have been proposed in literature showing both theoretical guarantees and empirical successes, the performance limits in decentralized optimization have not been fully clari\ufb01ed. This paper provides further understandings in optimal convergence rate in decentralized optimization and develops algorithms to achieve these rates. Weight matrix and connectivity measure. Assume n computing nodes are connected by some network topology (graph). We associate the topology with a weight matrix W = [wij]n i,j=1 \u2208Rn\u00d7n in which wij \u2208(0, 1) if node j is connected to node i otherwise wij = 0. We introduce \u03b2 := \u2225W \u22121 n1n1T n\u2225\u2208(0, 1), where 1n = (1, . . . , 1)T \u2208Rn has all entries being 1, as the connectivity measure to gauge how well the network topology is connected. Quantity \u03b2 \u21920 (which implies W \u21921 n1n1T n) indicates a well-connected topology while \u03b2 \u21921 (which implies W \u2192I) indicates a badly-connected topology. It is the weight matrix W and its connectivity measure \u03b2 that bring major challenges to convergence analysis in decentralized optimization. \u2217Equal Contribution. Corresponding Author: Kun Yuan 36th Conference on Neural Information Processing Systems (NeurIPS 2022). arXiv:2210.07863v1 [cs.LG] 14 Oct 2022 \fTable 1: Rate comparison between different algorithms in smooth and non-convex stochastic decentralized optimization. Parameter n denotes the number of all computing nodes, \u03b2 \u2208[0, 1) denotes the connectivity measure of the weight matrix, \u03c32 measures the gradient noise, b2 denotes data heterogeneity, and T is the number of iterations. Other constants such as the initialization f(x(0)) \u2212f \u22c6and smoothness constant L are omitted for clarity. Detailed complexities with constant L are listed in Table 6 in Appendix B. The de\ufb01nition of transient iteration complexity can be found in Remark 6 (the smaller the better). \u201cLB\u201d is lower bound while \u201cUB\u201d is upper bound. Notation \u02dc O(\u00b7) hides all logarithm factors." + } + ], + "Jialin Liu": [ + { + "url": "http://arxiv.org/abs/2305.18577v1", + "title": "Towards Constituting Mathematical Structures for Learning to Optimize", + "abstract": "Learning to Optimize (L2O), a technique that utilizes machine learning to\nlearn an optimization algorithm automatically from data, has gained arising\nattention in recent years. A generic L2O approach parameterizes the iterative\nupdate rule and learns the update direction as a black-box network. While the\ngeneric approach is widely applicable, the learned model can overfit and may\nnot generalize well to out-of-distribution test sets. In this paper, we derive\nthe basic mathematical conditions that successful update rules commonly\nsatisfy. Consequently, we propose a novel L2O model with a mathematics-inspired\nstructure that is broadly applicable and generalized well to\nout-of-distribution problems. Numerical simulations validate our theoretical\nfindings and demonstrate the superior empirical performance of the proposed L2O\nmodel.", + "authors": "Jialin Liu, Xiaohan Chen, Zhangyang Wang, Wotao Yin, HanQin Cai", + "published": "2023-05-29", + "updated": "2023-05-29", + "primary_cat": "cs.LG", + "cats": [ + "cs.LG", + "math.OC", + "stat.ML" + ], + "main_content": "Introduction Solving mathematical problems with the help of artificial intelligence, particularly machine learning techniques, has gained increasing interest recently (Davies et al., 2021; Charton, 2021; Polu et al., 2022; Drori et al., 2021). Optimization problems, a type of math problem that finds a point with minimal objective function value in a given space, can also be solved with machine learning models (Gregor & LeCun, 2010; Andrychowicz et al., 2016; Chen et al., 2021a; Bengio et al., 2021). Such technique is coined as Learning to Optimize (L2O). As an example, we consider an unconstrained optimization problem minx\u2208Rn F(x) where F is differentiable. A classic *Equal contribution 1Alibaba Group (U.S.) Inc, Bellevue, WA, USA 2Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX, USA 3Department of Statistics and Data Science and Department of Computer Science, University of Central Florida, Orlando, FL, USA. Correspondence to: HanQin Cai . Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s). algorithm to solve this problem is gradient descent: xk+1 = xk \u2212\u03b1k\u2207F(xk), k = 0,1,2,\u22ef, where the estimate of x is updated in an iterative manner, \u03b1k > 0 is a positive scalar named as step size, and the update direction \u03b1k\u2207F(xk) is aligned with the gradient of F at xk. Instead of the vanilla gradient descent, (Andrychowicz et al., 2016) proposes to parameterize the update rule into a learnable model that suggests the update directions by taking the current estimate and the gradient of F as inputs xk+1 = xk \u2212dk(xk,\u2207F(xk);\u03d5), k = 0,1,\u22ef,K \u22121, (1) where \u03d5 is the learnable parameter that can be trained by minimizing a loss function: min \u03d5 L(\u03d5) \u2236= EF \u2208F[ K \u2211 k=1 wkF(xk)], (2) where F is the problem set we concern and {wk}K k=1 is a set of hand-tuned weighting coefficients. Such loss function aims at finding an update rule of xk such that the objective values {F(xk)} are as small as possible for all F \u2208F. This work and its following works (Lv et al., 2017; Wichrowska et al., 2017; Wu et al., 2018; Metz et al., 2019; Chen et al., 2020; Shen et al., 2021; Harrison et al., 2022) show that modeling dk with a deep neural network and learning a good update rule from data is doable. To train such models, they randomly pick some training samples from F and build estimates of the loss function defined in (2). Such learned rules are able to generalize to unseen instances from F, i.e., the problems similar to the training samples. This method is quite generic and we can use it as long as we can access the gradient or subgradient of F. For simplicity, we name the method in (1) as generic L2O. Generic L2O is flexible and applicable to a broad class of problems. However, generalizing the learned update rules to out-of-distribution testing problems is quite challenging and a totally free dk usually leads to overfitting (Metz et al., 2020; 2022). In this paper, we propose an approach to explicitly regularize the update rule dk. Our motivation comes from some common properties that basic optimization algorithms should satisfy. For example, if an iterate xk reaches one of the minimizers of the objective F(x), the next iterate xk+1 should be fixed. Such condition is satisfied by many 1 arXiv:2305.18577v1 [cs.LG] 29 May 2023 \fTowards Constituting Mathematical Structures for Learning to Optimize basic algorithms like gradient descent, but not necessarily satisfied if dk is free to choose. In this paper, we refer to an update rule as a good rule if it fulfills these conditions. Our main contributions are three-fold: 1. We strictly describe some basic mathematical conditions that a good update rule should satisfy on convex optimization problems. 2. Based on these conditions, we derive a math-inspired structure of the explicit update rule on xk. 3. We numerically validate that our proposed scheme has superior generalization performance. An update rule trained with randomly generated data can even perform surprisingly well on real datasets. Organization. The rest of this paper is organized as follows. In Section 2, we derive mathematical structures for L2O models. In Section 3, we propose a novel L2O model and discuss its relationship with other L2O models. In Section 4, we verify the empirical performance of the proposed model via numerical experiments. In Section 5, we conclude the paper with some discussions on the future directions. 2. Deriving Mathematical Structures for L2O Update Rule In this study, we consider optimization problems in the form of minx\u2208Rn F(x) = f(x) + r(x), where f(x) is a smooth convex function with Lipschitz continuous gradient, and r(x) is a convex function that may be non-smooth. More rigorously, we write that f \u2208FL(Rn) and r \u2208F(Rn) where the function spaces F(Rn) and FL(Rn) are defined below. Definition 1 (Spaces of Objective Functions). We define function spaces F(Rn) and FL(Rn) as F(Rn) = {r \u2236Rn \u2192R \u2223r is proper, closed and convex}, FL(Rn) = {f \u2236Rn \u2192R \u2223f is convex, differentiable, and \u2225\u2207f(x) \u2212\u2207f(y)\u2225\u2264L\u2225x \u2212y\u2225,\u2200x,y \u2208Rn}. The first derivative of F plays an important role in the update rule (1). For differentiable function f \u2208FL(Rn), we can access to its gradient \u2207f(x). For non-differentiable function r \u2208F(Rn), we have to use the concepts of subgradient and subdifferential that are described below. Definition 2 (Subdifferential and Subgradient). For r \u2208 F(Rn), its subdifferential at x is defined as \u2202r(x) = {g \u2208Rn \u2223r(y) \u2212r(x) \u2265g\u22ba(y \u2212x), \u2200y \u2208Rn}. Each element in the subdifferential, i.e., each g \u2208\u2202r(x), is a subgradient of function r at point x. Settings of dk. We clarify some definitions about the update direction dk. A general parameterized update rule can be written as xk+1 = xk \u2212dk(zk;\u03d5), (3) where zk \u2208Z is the input vector and Z is the input space. The input vector may involve dynamic information such as {xk,F(xk),\u2207F(xk)}. Take (Andrychowicz et al., 2016) as an example, as described in (1), the input vector is zk = [x\u22ba k,\u2207F(xk)\u22ba]\u22baand the input space is Z = R2n. In our theoretical analysis, we relax the structure of (3) and use a general update rule dk \u2236Z \u2192Rn instead of the parameterized rule dk(zk;\u03d5) and write (3) as: xk+1 = xk \u2212dk(zk). (4) To facilitate the theoretical analysis, we assume the update direction dk is differentiable with respect to the input vector, and its derivatives are bounded. Specifically, dk is taken from the space DC(Z) which is defined below. Definition 3 (Space of Update Rules). Let Jd(z) denote the Jacobian matrix of operator d \u2236Z \u2192Rn and \u2225\u22c5\u2225F denote the Frobenius norm, we define the space: DC(Z) = {d \u2236Z \u2192Rn \u2223d is differentiable, \u2225Jd(z)\u2225F \u2264C, \u2200z \u2208Z}. In practice, training the deep network that is parameterized from dk will usually need the derivatives of dk. Thus, the differentiability and bounded Jacobian of dk are important for this study. Note that dk \u2208DC(Z) has been commonly used and satisfied in many existing parameterization approaches, e.g., Long Short-Term Memory (LSTM), which is one of the most popular models adopted in L2O (Andrychowicz et al., 2016; Lv et al., 2017). 2.1. Smooth Case In the smooth case, \u2207F(x) equals to \u2207f(x) as the nonsmooth part r(x) = 0. Thus, (4) can be written as: xk+1 = xk \u2212dk(xk,\u2207f(xk)). (5) We refer (5) to be a good update rule if it satisfies the following two assumptions: Asymptotic Fixed Point Condition. We assume that xk+1 = x\u2217as long as xk = x\u2217, where x\u2217\u2208arg minx f(x). In other words, if xk is exactly a solution, the next iterate should be fixed. Substituting both xk and xk+1 with x\u2217, we obtain: x\u2217= x\u2217\u2212dk(x\u2217,\u2207f(x\u2217)), for all k = 0,1,2,\u22ef. 2 \fTowards Constituting Mathematical Structures for Learning to Optimize Convex analysis theory tells us \u2207f(x\u2217) = 0, and we obtain dk(x\u2217,0) = 0. Instead of using this strong assumption, we relax it and assume dk(x\u2217,0) \u21920 as k \u2192\u221e. Formally, it is written as below and coined as (FP1): For any x\u2217\u2208arg min x\u2208Rn f(x), lim k\u2192\u221edk(x\u2217,0) = 0. (FP1) Such a condition can be viewed as an extension to the Fixed Point Encoding (Ryu & Yin, 2022) in optimization, which is useful guidance for designing efficient convex optimization algorithms. Global Convergence. We assume that, the sequence {xk}\u221e k=0 converges to one of the minimizers of the objective function f(x), as long as it yields the update rule (5). Formally, it is written as (GC1): For any sequences {xk}\u221e k=0 generated by (5), there exists x\u2217\u2208arg min x\u2208Rn f(x) such that lim k\u2192\u221exk = x\u2217. (GC1) Actually, assumptions (FP1) and (GC1) are fundamental in the field of optimization and can be satisfied by many basic update schemes. For example, as long as f \u2208FL(Rn), gradient descent satisfies (FP1) unconditionally and satisfies (GC1) with a properly chosen step size. To outperform the vanilla update rules like gradient descent, a learned update rule dk should also satisfy (FP1) and (GC1). The following theorem provides an analysis on dk under (FP1) and (GC1). Note that proofs of all theorems in this section are deferred to the Appendix. Theorem 1. Given f \u2208FL(Rn), we pick a sequence of operators {dk}\u221e k=0 with dk \u2208DC(R2n) and generate {xk}\u221e k=0 by (5). If both (FP1) and (GC1) hold, then for all k = 0,1,2,\u22ef, there exist Pk \u2208Rn\u00d7n and bk \u2208Rn satisfying dk(xk,\u2207f(xk)) = Pk\u2207f(xk) + bk, with Pk is bounded and bk \u21920 as k \u2192\u221e. Theorem 1 illustrates that an update rule dk is not completely free under assumptions (FP1) and (GC1). It suggests the following structured update rule instead of the free update rule (5): xk+1 = xk \u2212Pk\u2207f(xk) \u2212bk, (6) where Pk is named as a preconditioner and bk is named as a bias. The scheme (6) covers several classical algorithms. For example, with Pk = \u03b1I and bk = \u03b2(xk \u2212\u03b1\u2207f(xk) \u2212 xk\u22121 \u2212\u03b1\u2207f(xk\u22121)), it reduces to Nesterov accelerated gradient descent (Nesterov, 1983); with bk = 0 and properly chosen Pk, it covers Newton\u2019s method and quasi-Newton method like L-BFGS (Liu & Nocedal, 1989). Furthermore, Theorem 1 implies that, as long as (FP1) and (GC1) are both satisfied, finding the optimal update direction dk equals to finding the optimal preconditioner and bias. To train an update rule for smooth convex objective functions f \u2208FL, one may parameterize Pk and bk instead of parameterizing the entire dk, that is, xk+1 = xk \u2212Pk(zk;\u03d5)\u2207f(xk) \u2212bk(zk;\u03c8), where the input vector zk = [x\u22ba k,\u2207f(xk)\u22ba]\u22ba. Detailed parameterization and training methods are later described in Section 3. Note that even if the update rule satisfies (6) instead of (5), one cannot guarantee (FP1) and (GC1) hold. Under our assumptions, if one further makes assumptions on the trajectory of {xk}K k=0, then they would obtain a convergence guarantee. This idea falls into a subfield of optimization called Performance Estimation Problem (PEP) (Taylor et al., 2017; Ryu et al., 2020). However, algorithms obtained by PEP are much slower than L2O on specific types of problems that a single user is concerned with. This is because PEP imposes many restrictions on the iteration path, while L2O can find shortcuts for particular problem types. Although we also impose restrictions (Asymptotic Fixed Point Condition and Global Convergence) on L2O, these restrictions target the asymptotic performance and the final fixed point, rather than the path. As a result, the mathstructured L2O proposed in this paper avoids the limitations of convergence guarantees while allowing shortcuts. By analyzing the fixed point, we discover a more effective learnable optimizer structure. 2.2. Non-Smooth Case In the non-smooth case (i.e., f(x) = 0), we use subgradient instead of gradient. A simple extension to (5) is to pick a subgradient gk from subdifferential \u2202r(xk) in each iteration and use it in the update rule: xk+1 = xk \u2212dk(xk,gk), gk \u2208\u2202r(xk). (7) Such an update rule is an extension to subgradient descent method: xk+1 = xk \u2212\u03b1kgk. Compared with gradient descent in the smooth case, the convergence of subgradient descent method is usually unstable, and it may not converge to the solution if a constant step size is used. To guarantee convergence, one has to use certain diminishing step sizes, which may lead to slow convergence. (Bertsekas, 2015). For non-smooth problems, Proximal Point Algorithm (PPA) (Rockafellar, 1976) usually converges faster and more stably than the subgradient descent method. While subgradient descent method uses the explicit update, PPA takes implicit update rule: xk+1 = xk \u2212\u03b1kgk+1, where gk+1 \u2208\u2202r(xk+1). Inspired by PPA, we propose to use the 3 \fTowards Constituting Mathematical Structures for Learning to Optimize following implicit rule instead: xk+1 = xk \u2212dk(xk+1,gk+1), gk+1 \u2208\u2202r(xk+1). (8) Generally speaking, it is hard to calculate xk+1 from the implicit formula (8) given xk. However, the following discussion provides a mathematical structure of dk in (8), and, under some mild assumptions, (8) can be written in a much more practical way. With the same argument in the smooth case, we obtain the math description of the asymptotic fixed point condition: For any x\u2217\u2208arg minx r(x), there exists a g\u2217\u2208\u2202r(x\u2217) such that dk(x\u2217,g\u2217) \u21920 as k \u2192\u221e. With convex analysis theory, it holds that 0 \u2208\u2202r(x\u2217) if and only if x\u2217\u2208arg minx r(x). Thus, it is natural to take g\u2217= 0. Formally, it is written as For any x\u2217\u2208arg min x\u2208Rn r(x), lim k\u2192\u221edk(x\u2217,0) = 0. (FP2) Similar to (GC1), in the non-smooth case, we require the sequence {xk} converges to one of the minimizers of the function r(x). Formally, it is written as For any sequences {xk}\u221e k=0 generated by (8), there exists x\u2217\u2208arg min x\u2208Rn r(x) such that lim k\u2192\u221exk = x\u2217. (GC2) Theorem 2. Given r \u2208F(Rn), we pick a sequence of operators {dk}\u221e k=0 with dk \u2208DC(R2n) and generate {xk}\u221e k=0 by (8). If both (FP2) and (GC2) hold, then for all k = 0,1,2,\u22ef, there exist Pk \u2208Rn\u00d7n and bk \u2208Rn satisfying xk+1 = xk \u2212Pkgk+1 \u2212bk, gk+1 \u2208\u2202r(xk+1), with Pk is bounded and bk \u21920 as k \u2192\u221e. If we further assume Pk is symmetric positive definite, then xk+1 can be uniquely determined through xk+1 = arg min x\u2208Rn r(x) + 1 2\u2225x \u2212xk + bk\u22252 P\u22121 k , (9) where the norm \u2225\u22c5\u2225P\u22121 k is defined as \u2225x\u2225P\u22121 k \u2236= \u221a x\u22baP\u22121 k x. Define the preconditioned proximal operator of r(x) as proxr,P(\u00af x) \u2236= arg min x r(x) + 1 2\u2225x \u2212\u00af x\u22252 P\u22121, (10) where P is a symmetric positive definite preconditioner. Then (9) can be written as xk+1 = proxr,Pk(xk \u2212bk). If we set Pk = I and bk = 0, (9) reduces to the standard PPA. Instead of learning a free update rule as shown in (8), Theorem 2 suggests learning the preconditioner Pk and the bias bk in a structured rule, as illustrated in equation (9). 2.3. Composite Case As special cases, the smooth case and non-smooth case provide important preliminaries to the composite case: F(x) = f(x) + r(x). Inspired by Theorems 1 and 2, we use explicit formula for f and implicit formula for r in the composite case: xk+1 = xk \u2212dk(xk,\u2207f(xk),xk+1,gk+1), (11) where gk+1 \u2208\u2202r(xk+1) and the input vector zk = [x\u22ba k,\u2207f(xk)\u22ba,x\u22ba k+1,g\u22ba k+1]\u22baand input space is Z = R4n. To derive the asymptotic fixed point condition in this case, we use the same arguments in Sections 2.1 and 2.2, and we obtain the following statement for all x\u2217\u2208arg minx F(x): lim k\u2192\u221edk(x\u2217,\u2207f(x\u2217),x\u2217,g\u2217) = 0, for some g\u2217\u2208\u2202r(x\u2217). The convexity of f and r implies that 0 \u2208\u2207f(x\u2217)+\u2202r(x\u2217) if and only if x\u2217\u2208arg minx F(x). Thus, it holds that \u2212\u2207f(x\u2217) \u2208\u2202r(x\u2217). With g\u2217= \u2212\u2207f(x\u2217), one could obtain the formal statement of the fixed point condition. For any x\u2217\u2208arg minx\u2208Rn F(x), it holds that lim k\u2192\u221edk(x\u2217,\u2207f(x\u2217),x\u2217,\u2212\u2207f(x\u2217)) = 0. (FP3) The global convergence is stated as: For any sequences {xk}\u221e k=0 generated by (11), there exists x\u2217\u2208arg min x\u2208Rn F(x) such that lim k\u2192\u221exk = x\u2217. (GC3) Theorem 3. Given f \u2208FL(Rn) and r \u2208F(Rn), we pick a sequence of operators {dk}\u221e k=0 with dk \u2208DC(R4n) and generate {xk}\u221e k=0 by (11). If both (FP3) and (GC3) hold, then for all k = 0,1,2,\u22ef, there exist Pk \u2208Rn\u00d7n and bk \u2208 Rn satisfying xk+1 = xk \u2212Pk(\u2207f(xk) \u2212gk+1) \u2212bk, gk+1 \u2208\u2202r(xk+1), with Pk is bounded and bk \u21920 as k \u2192\u221e. If we further assume Pk is symmetric positive definite, then xk+1 can be uniquely determined given xk through xk+1 = proxr,Pk(xk \u2212Pk\u2207f(xk) \u2212bk). (12) With bk = 0 and Pk = \u03b1I, (12) reduces to a standard Proximal Gradient Descent (PGD). Therefore, scheme (12) is actually an extension of PGD with a preconditioner Pk and a bias bk. Theorem 3 implies that it\u2019s enough to learn an extended PGD instead of a free scheme (11). 2.4. Longer Horizon Those update schemes (5), (8) and (11) introduced in previous sections explicitly depend on only the current status 4 \fTowards Constituting Mathematical Structures for Learning to Optimize xk. Now we introduce an auxiliary variable yk that encodes historical information through operator m: yk = m(xk,xk\u22121,\u22ef,xk\u2212T ). (13) To facilitate parameterization and training, we assume m is differentiable: m \u2208DC(R(T +1)\u00d7n) (see Definition 3). With yk, we could encode more information into the update rule and extend (11) to the following: xk+1 = xk \u2212dk(xk,\u2207f(xk),xk+1,gk+1,yk,\u2207f(yk)), where gk+1 \u2208\u2202r(xk+1). (14) Now let\u2019s derive the asymptotic fixed point condition and the global convergence that (13) and (14) should follow. Since the global convergence requires xk \u2192x\u2217, the continuity of operator m implies the convergence of sequence {yk}. If the limit point of {yk} is denoted by y\u2217, we can assume y\u2217= x\u2217without loss of generality because, for any operator m, we can always construct another operator by shifting the output: \u02c6 m = m \u2212y\u2217+ x\u2217such that the sequence generated by \u02c6 m converges to x\u2217. Roughly speaking, we assume that the sequence {xk,yk} generated by (13) and (14) satisfies xk \u2192x\u2217and yk \u2192x\u2217. By extending (FP3) and (GC3), we obtain the formal statement of our assumptions that (13) and (14) should follow. For any x\u2217\u2208arg minx\u2208Rn F(x), it holds that lim k\u2192\u221edk(x\u2217,\u2207f(x\u2217),x\u2217,\u2212\u2207f(x\u2217),x\u2217,\u2207f(x\u2217)) = 0, m(x\u2217,x\u2217,\u22ef,x\u2217) = x\u2217. (FP4) For any sequences {xk,yk}\u221e k=0 generated by (13) and (14), there exists one x\u2217\u2208arg minx\u2208Rn F(x) such that lim k\u2192\u221exk = lim k\u2192\u221eyk = x\u2217. (GC4) Theorem 4. Suppose T = 1. Given f \u2208FL(Rn) and r \u2208F(Rn), we pick an operator m \u2208DC(R2n) and a sequence of operators {dk}\u221e k=0 with dk \u2208DC(R6n). If both (FP4) and (GC4) hold, for any bounded matrix sequence {Bk}\u221e k=0, there exist P1,k,P2,k,Ak \u2208Rn\u00d7n and b1,k,b2,k \u2208Rn satisfying xk+1 = xk \u2212(P1,k \u2212P2,k)\u2207f(xk) \u2212P2,k\u2207f(yk) \u2212b1,k \u2212P1,kgk+1 \u2212Bk(yk \u2212xk), gk+1 \u2208\u2202r(xk+1), (15) yk+1 = (I \u2212Ak)xk+1 + Akxk + b2,k (16) for all k = 0,1,2,\u22ef, with {P1,k,P2,k,Ak} are bounded and b1,k \u21920,b2,k \u21920 as k \u2192\u221e. If we further assume P1,k is uniformly symmetric positive definite1, then we can 1A sequence of uniformly symmetric positive definite matrices means that the smallest eigenvalues of all symmetric positive definite matrices are uniformly bounded away from zero. substitute P2,kP\u22121 1,k with Bk and obtain \u02c6 xk = xk \u2212P1,k\u2207f(xk), \u02c6 yk = yk \u2212P1,k\u2207f(yk), xk+1 = proxr,P1,k((I \u2212Bk)\u02c6 xk + Bk \u02c6 yk \u2212b1,k), yk+1 = xk+1 + Ak(xk+1 \u2212xk) + b2,k. (17) In the update scheme (17), b1,k and b2,k are biases that play the same role with bk in (12); Ak can be viewed as an extension of Nesterov momentum and we name it as an accelerator; P1,k is the preconditioner that plays a similar role as Pk in (12); Bk is a balancing term between \u02c6 xk and \u02c6 yk. If Bk = 0, then xk+1 merely depends on xk and (17) reduces to (12); and if Bk = I, then xk+1 merely depends on yk explicitly. 3. An Efficient Math-Inspired L2O Model As long as the basic assumptions (FP4) and (GC4) hold, one could derive a math-structured update rule (17) from generic update rule (13) and (14). Moreover, we suggest using diagonal matrices for P1,k,Bk,Ak in practice: P1,k = diag(pk), Bk = diag(bk), Ak = diag(ak), where pk,bk,ak \u2208Rn are vectors. Let 1 be the vector whose all elements are ones and \u2299be the element-wise multiplication. The suggested update rule then becomes: \u02c6 xk = xk \u2212pk \u2299\u2207f(xk), \u02c6 yk = yk \u2212pk \u2299\u2207f(yk), xk+1 = proxr,pk((1 \u2212bk) \u2299\u02c6 xk + bk \u2299\u02c6 yk \u2212b1,k), yk+1 = xk+1 + ak \u2299(xk+1 \u2212xk) + b2,k. (18) We choose diagonal P1,k,Bk,Ak over full matrices for efficiency. On one hand, the diagonal formulation reduces the degree of freedom of the update rule. Therefore, when P1,k,Bk,Ak are parameterized as the output of, or a part of, a learnable model and trained with data, the difficulty of training is decreased and thus the efficiency improved. On the other hand, for a broad class of r(x), the proximal operator proxr,pk has efficient explicit formula with the diagonal preconditioner2 pk. Although the non-diagonal formulation may lead to a (theoretically) better convergence rate, it could increase the computational difficulty of the proximal operator proxr,Pk. An interesting future topic is how to calculate (18) with non-diagonal Pk and how to train deep models to generate such Pk. LSTM Parameterization. Similar to (Andrychowicz et al., 2016; Lv et al., 2017), we model pk, ak, bk, b1,k, 2Examples can be found in Appendix D. 5 \fTowards Constituting Mathematical Structures for Learning to Optimize b2,k as the output of a coordinate-wise LSTM, which is parameterized by learnable parameters \u03d5LSTM and takes the current estimate xk and the gradient \u2207f(xk) as the input: ok,hk = LSTM(xk,\u2207f(xk),hk\u22121;\u03d5LSTM), pk,ak,bk,b1,k,b2,k = MLP(ok;\u03d5MLP). (19) Here, hk is the internal state maintained by the LSTM with h0 randomly sampled from Gaussian distribution. It is common in classic optimization algorithms to take positive pk and ak. Hence we post-process pk and ak with an additional activation function such as sigmoid and softplus. A \u201ccoordinate-wise\u201d LSTM means that the same network is shared across all coordinates of xk, so that this single model can be applied to optimization problems of any scale. (18) and (19) together define an optimization scheme. We call it an L2O optimizer. Training. We train the proposed L2O optimizer, that is to find the optimal \u03d5LSTM and \u03d5MLP in (19), on a dataset F of optimization problems. Each sample in F is an instance of the optimization problem, which we call an optimizee, and is characterized by its objective function F. During training, we apply optimizer to each optimizee F for K iterations to generate a sequence of iterates (y1,...,yK), and optimize \u03d5LSTM and \u03d5MLP by minimizing the following loss function: min \u03d5LSTM,\u03d5MLP L(\u03d5LSTM,\u03d5MLP) \u2236= 1 \u2223F\u2223\u2211 F \u2208F [ 1 K K \u2211 k=1 F(yk)]. Compared with Algorithm Unrolling. Algorithm Unrolling (Monga et al., 2019) is another line of works parallel to generic L2O. It was first proposed to fast approximate the solution of sparse coding (Gregor & LeCun, 2010) which is named Learned ISTA (LISTA). Since then, many efforts have been made to further improve or better understand LISTA (Xin et al., 2016; Metzler et al., 2017; Moreau & Bruna, 2017; Chen et al., 2018; Liu et al., 2019; Ito et al., 2019; Chen et al., 2021b), as well as applying this idea to different optimization problems (Yang et al., 2016; Zhang & Ghanem, 2018; Adler & \u00a8 Oktem, 2018; Mardani et al., 2018; Gupta et al., 2018; Solomon et al., 2019; Xie et al., 2019; Cai et al., 2021). The main difference between our method and algorithmunrolling methods lies at how parameterization is done. Different from the LSTM parameterization (19), algorithmunrolling methods turn pk, ak, bk, b1,k, b2,k themselves as learnable parameters and directly optimize them from data. However, such direct parameterization causes limitations on the flexibility of the model in many ways. It loses the ability to capture dynamics between iterations and tends to memorize more about the datasets. Moreover, direct parameterization means that we need to match the dimensions of the learnable parameters with the problem scale, Table 1. Comparison between different types of methods of their theoretical convergence analysis (Theory), convergence speed (Fast), and Flexibility. Methods Theory Fast Flexibility Classic Algorithms \u2713 \u2013 \u2713 Algorithm Unrolling \u2713 \u2713 \u2013 Generic L2O \u2013 \u2713 \u2713 Math-Inspired (Ours) \u2713 \u2713 \u2713 which implies that the trained model can not be applied to optimization problems of a different scale at all during inference time. Although this can be worked around by reducing the parameters to scalars, it will significantly decrease the capacity of the model. In fact, despite the difference in parameterization, our proposed scheme (18) covers many existing algorithmunrolling methods in the literature. For example, if we use the standard LASSO objective as the objective function F(x) and set ak = bk = b2,k = 0, we will recover StepLISTA (Ablin et al., 2019) with properly chosen pk,b1,k. More details and proofs are provided in the Appendix. Compared with Generic L2O. Our proposed method uses a similar coordinate-wise LSTM parameterization as generic L2O methods. Therefore, both of these two share the flexibility of being applied to optimization problems of any scale. However, we constrain the update rule to have a specific form, i.e., the formulation in (18). The reduced degree of freedom enables the convergence analysis in Section 2 from the theoretical perspective and empirically works as a regularization so that the trained L2O optimizer is more stable and can generalize better, which is validated by our numerical observations in the next section. We summarize in Table 1 the comparison between classic algorithms, algorithm-unrolling methods, generic L2O methods, and our math-inspired method in terms of theoretical convergence analysis, convergence speed, and flexibility. Relationship with Operator Learning. In this paper, we consider the proximal operator as an accessible basic routine and focus on learning the overall update rule that results in fast convergence. However, if the proximal operator is difficult to compute, one might explore another aspect of L2O: learning fast approximations of proximal operators (Zhang et al., 2017; Meinhardt et al., 2017; Li et al., 2022). For instance, if the matrix P1,k in (17) is not diagonal, calculating the proximal operator would be challenging. Additionally, our assumptions (FP4) and (GC4) allow the optimization problem F(x) to have multiple optima. As a result, investigating diverse optima following the idea of (Li et al., 2022) using our proposed scheme could be an 6 \fTowards Constituting Mathematical Structures for Learning to Optimize intriguing topic for future research. Relationship with Meta-Learning. L2O and MetaLearning are closely related topics as they both deal with learning from experience in previous tasks to improve performance on new tasks. L2O treats tasks as optimization problems and aims to discover superior optimization algorithms, while Meta-Learning is a more comprehensive concept that focuses on training a model on a set of related tasks or problems to swiftly adapt to new, unseen tasks using knowledge acquired from prior tasks. For example, in our paper\u2019s equation (1), L2O seeks to learn dk while keeping the initialization x0 fixed or randomized. Meanwhile, a typical Meta-Learning method like (Khodak et al., 2019a) learns a suitable initialization from a series of observed tasks, enabling quick adaptation to unseen tasks. In addition to the initialization, (Khodak et al., 2019b) also learns a meta-learning rate shared among different tasks. Furthermore, one can learn a regularization term based on the distance to a bias vector (Denevi et al., 2019) or even a conditional regularization term (Denevi et al., 2020) from a set of tasks. 4. Experimental Results We strictly follow the setting in (Lv et al., 2017) for experiment setup. More specifically, in all our experiments on the LSTM-based L2O models (including our method and other baseline competitors), we use two-layer LSTM cells with 20 hidden units with sigmoid activation functions. During training, each minibatch contains 64 instances of optimization problems, to which the L2O optimizers will be applied for 100 iterations. The 100 iterations are evenly segmented into 5 periods of 20 iterations. Within each of these, the L2O optimizers are trained with truncated Backpropagation Through Time (BPTT) with an Adam optimizer. All models are trained with 500 minibatches (32,000 optimization problems in total) generated synthetically, but are evaluated on both synthesized testing sets and real-world testing sets. We elaborate more on the data generation in the Appendix. The code is available online at https://github.com/xhchrn/MS4L2O. 4.1. Ablation Study We conduct an ablation study on LASSO to figure out the roles of pk,ak,bk,b1,k,b2,k in our proposed scheme (18). Both the training and testing samples are independently sampled from the same random distribution. The form of LASSO is given below min x\u2208Rn F(x) = 1 2\u2225Ax \u2212b\u22252 + \u03bb\u2225x\u22251, (20) where each tuple of (A,b,\u03bb) determines an objective function and thus a LASSO problem instance. The size of each 0 25 50 75 100 125 150 175 200 Iteration k 10 6 10 4 10 2 100 (F(xk) F*)/F* P A PA PBA PBA1 PBA2 PBA12 Figure 1. Ablation study on LASSO. 0 25 50 75 100 125 150 175 200 Iteration k 10 5 10 3 10 1 ||b1, k|| ||b2, k|| Figure 2. Visualization of the learned b1,k, b2,k. instance is A \u2208R250\u00d7500 and b \u2208R500 and other details are provided in the Appendix. We do not fix A and, instead, let each LASSO instance take an independently generated A. This setting is fundamentally more challenging than those in most of algorithm-unrolling works (Gregor & LeCun, 2010; Liu et al., 2019; Ablin et al., 2019; Behrens et al., 2021). On the benchmark, we compare the following settings: PBA12: pk,ak,bk,b1,k,b2,k are all learnable. PBA1: pk,ak,bk,b1,k are learnable; b2,k is fixed as 0. PBA2: pk,ak,bk,b2,k are learnable; b1,k is fixed as 0. PBA: pk,ak,bk are learnable; b2,k and b1,k are both fixed as 0. PA: pk,ak are learnable; b2,k and b1,k are both fixed as 0; bk is fixed as 1. P: only pk is learnable; ak, b2,k, b1,k are fixed as 0; bk is fixed as 1. A: only ak is learnable; b2,k and b1,k are both fixed as 0; bk is fixed as 1; pk is fixed as (1/L)1. The results are reported in Figure 1. In Figure 1, F\u2217denotes the best objective value of each instance and (F(xk) \u2212F\u2217)/F\u2217measures the average optimality gap on the test set. Each curve describes the convergence performance of a learned optimizer. From Figure 1, one can conclude that, with all of the learned optimizers, convergence can be reached within the machine precision. An interesting observation is that the proposed scheme (18) may not benefit from parameterizing and learning more components. Comparing PBA, PBA1, PBA2 and PBA12, we find that fixing b1,k = b2,k = 0 is a better choice than 7 \fTowards Constituting Mathematical Structures for Learning to Optimize learning them. This phenomenon can be explained by our theory. In Theorem 4, both b2,k and b1,k are expected to converge to zero, otherwise, the convergence would be violated. However, in Figure 2 where we report the average values of \u2225b1,k\u2225and \u2225b2,k\u2225in PBA12, both \u2225b1,k\u2225and \u2225b2,k\u2225converge to a relatively small value after a few (\u226410) iterations, but there is no guarantee that they converge exactly to zero given parameterization (19). Such observation actually validates our theories and suggests to fix b2,k and b1,k as 0 instead of learning them. A Simplified Scheme. Furthermore, comparing PA and PBA in Figure 1, we find that parameterizing bk does not show significant benefits. To reduce computational overhead, we adopt PA and fix b1,k = b2,k = 0 and bk = 1, which reduces (18) and (19) to the following simplified scheme: ok,hk = LSTM(xk,\u2207f(xk),hk\u22121;\u03d5LSTM), pk,ak = MLP(ok;\u03d5MLP), xk+1 = proxr,pk(yk \u2212pk\u2207f(yk)), yk+1 = xk+1 + ak \u2299(xk+1 \u2212xk). (L2O-PA) 4.2. Comparison with Competitors We compare (L2O-PA) with some competitors in two settings. In the first setting (also coined as In-Distribution), the training and testing samples are generated independently with the same distribution. The second setting is much more challenging, where the models are trained on randomly synthetic data but tested on real data, which is coined as Out-of-Distribution or OOD. Solving LASSO Regression. In-Distribution experiments follow the settings in Section 4.1, where the training and testing sets are generated randomly and independently, but share the same distribution. In contrast, Out-of-Distribution (OOD) experiments also generate training samples similarly to In-Distribution experiments, but the test samples are generated based on a natural image benchmark BSDS500 (Martin et al., 2001). Detailed information about the synthetic and real data used can be found in the Appendix. We first choose ISTA, FISTA (Beck & Teboulle, 2009), and Adam (Kingma & Ba, 2014) as baselines since they are classical manually-designed update rules. We choose a state-ofthe-art Algorithm-Unrolling method Ada-LISTA (Aberdam et al., 2021) as another baseline since it is applicable in the settings of various A. Additionally, we choose two generic L2O methods that following (3) as baselines: L2O-DM (Andrychowicz et al., 2016) and L2O-RNNprop (Lv et al., 2017). Finally, we choose AdamHD (Baydin et al., 2017), a hyperparameter optimization (HPO) method that adaptively tunes the learning rate in Adam with online learning, as a baseline. Since Ada-LISTA is not applicable to problems 100 101 102 103 Iteration k 10 8 10 6 10 4 10 2 100 102 (F(xk) F*)/F* ISTA FISTA Ada-LISTA L2O-DM L2O-RNNprop L2O-PA Adam AdamHD Figure 3. LASSO: Train and test on synthetic data. 100 101 102 103 Iteration k 10 8 10 6 10 4 10 2 100 (F(xk) F*)/F* ISTA FISTA L2O-DM L2O-RNNprop L2O-PA Adam AdamHD Figure 4. LASSO: Train on synthetic data and test on real data. that have different sizes with training samples, we only compare our method with Ada-LISTA in In-Distribution experiments. In-Distribution results are reported in Figure 3 and Out-ofDistribution results are reported in Figure 4. In both of the settings, the proposed (L2O-PA) performs competitively. In the OOD setting, the other two learning-based methods, L2O-DM and L2O-RNNprop, both struggle to converge with optimality tolerance 10\u22122, but (L2O-PA) is still able to converge until touching the machine precision. Solving \u21131-regularized Logistic Regression. An \u21131regularized logistic regression for binary classification is characterized by set of training examples {(ai,bi) \u2208Rn \u00d7 {0,1}}m i=1 where ai is an n-dimensional feature vector and bi is a binary label. To solve the \u21131-regularized logistic regression problem is to find an optimal x\u2217so that h(a\u22ba i x\u2217) predicts p(bi\u2223ai) well, where h(c) = 1/(1 + e\u2212c) is the logistic function. The exact formula of the objective function can be found in the Appendix. We train L2O models on randomly generated logistic regression datasets for binary classification. Each dataset contains 1,000 samples that have 50 features. We evaluate the trained models in both the In-Distribution and OOD settings. Results are shown in Figures 5 and 6 respectively. Details about synthetic data generation and real-world datasets are provided in the Appendix. 8 \fTowards Constituting Mathematical Structures for Learning to Optimize 100 101 102 103 Iteration k 10 6 10 4 10 2 (F(xk) F*)/F* ISTA FISTA L2O-DM L2O-RNNprop L2O-PA Adam AdamHD Figure 5. Logistic: Train and test on synthetic data. 100 101 102 103 Iteration k 10 6 10 4 10 2 (F(xk) F*)/F* ISTA FISTA L2O-DM L2O-RNNprop L2O-PA Adam AdamHD Figure 6. Logistic: Train on synthetic data and test on real data. Under the In-Distribution setting (Figure 5), our method, i.e., L2O-PA, can converge to optimal solutions within 10 iterations, almost 100\u00d7 faster than FISTA, while the other two generic L2O baselines fail to converge. When tested on OOD optimization problems (Figure 6), L2O-PA can still converge within 100 iterations, faster than FISTA by more than 10 times. 5." + }, + { + "url": "http://arxiv.org/abs/2003.00804v1", + "title": "Task Augmentation by Rotating for Meta-Learning", + "abstract": "Data augmentation is one of the most effective approaches for improving the\naccuracy of modern machine learning models, and it is also indispensable to\ntrain a deep model for meta-learning. In this paper, we introduce a task\naugmentation method by rotating, which increases the number of classes by\nrotating the original images 90, 180 and 270 degrees, different from\ntraditional augmentation methods which increase the number of images. With a\nlarger amount of classes, we can sample more diverse task instances during\ntraining. Therefore, task augmentation by rotating allows us to train a deep\nnetwork by meta-learning methods with little over-fitting. Experimental results\nshow that our approach is better than the rotation for increasing the number of\nimages and achieves state-of-the-art performance on miniImageNet, CIFAR-FS, and\nFC100 few-shot learning benchmarks. The code is available on\n\\url{www.github.com/AceChuse/TaskLevelAug}.", + "authors": "Jialin Liu, Fei Chao, Chih-Min Lin", + "published": "2020-02-08", + "updated": "2020-02-08", + "primary_cat": "cs.CV", + "cats": [ + "cs.CV" + ], + "main_content": "Introduction Although the machine learning systems have achieved a human-level ability in many \ufb01elds with a large amount of data, learning from a few examples is still a challenge for modern machine learning techniques. Recently, the machine learning community has paid signi\ufb01cant attention to this problem, where few-shot learning is the common task for meta-learning (e.g., [20, 5, 30, 27]). The purpose of few-shot learning is to learn to maximize generalization accuracy across different tasks with few training examples. In a classi\ufb01cation application of the few-shot learning, tasks are generated by sampling from a conventional classi\ufb01cation dataset; then, training samples are randomly selected from several classes in the classi\ufb01cation dataset. In addition, a part of the examples is used as training examples and testing examples. Thus, a tiny learning task is formed by these examples. The meta-learning methods are applied to control the learning process of a base learner, so as to correctly classify on testing examples. Data augmentation is widely used to improve the training of deep learning models. Usually, data augmentation is regarded as an explicit form of regularization [9, 26, 13]. Data augmentation aims at arti\ufb01cially generating the training data by using various translations on existing data, such as: adding noises, cropping, \ufb02ipping, rotation, translation, etc. The general idea of data augmentations is increasing the number of images by change data slightly to be different from original data, but the data still can be recognized by human. The new images involved in the classes are identical to the original data, we call this as Image Aug. However, the minimum units of meta-learning are tasks rather than data, so we should use rotation operation to augment the number of tasks, which is called as task augmentation (referred to Task Aug). Task Aug means increasing the types of task instances by increasing the data that can be clearly recognized as the different classes as the original data and associating them as the novel classes(we show examples in Figure 1). This is important for the metalearning, since meta-learning models require to predict unseen classes during the testing phase, increasing the diverseness of tasks would help models to generate to unseen classes. In experiments, we compared two cases, 1) the new images are converted to the classes of original images and 2) 1 arXiv:2003.00804v1 [cs.CV] 8 Feb 2020 \fthe new images are associated to the novel classes with the method proposed in [3] on CIFAR-FS, FC100, miniImageNet few-shot learning tasks, and showed the second case got better results. Then the proposed method is evaluated by experiments with the state of art meta-learning methods [27, 15, 3] on CIFAR-FS, FC100, miniImageNet fewshot learning tasks, and compare with the results without the data augmentation by rotating. In the comparative experiments, Task Aug by rotating achieves the better accuracy than the original meta-learning methods. Besides, the best results of our experiments exceed the current state-of-art result over a large margin. 2. Related Work Meta-learning involves two hierarchies learning processes: low-level and high-level. The low-level learning process learns to deal with general tasks, often termed as the \u201cinner loop\u201d; and the high-level learning process learns to improve the performance of a low-level task, often termed as the \u201couter loop\u201d. Since models are required to handle sensory data like images, deep learning methods are often applied for the \u201couter loop\u201d. However, the machine learning methods applied for the \u201cinner loop\u201d are very diverse. Based on different methods in the \u201cinner loop\u201d, meta-learning can be applied in image recognition [4, 24, 5, 30, 20], image generation [2, 31, 21], reinforce learning [5, 1], and etc. This work focuses on few-shot learning image recognition based on meta-learning. Therefore, in the experiment, the methods applied in the \u201cinner loop\u201d are able to classify data, and they are K-nearest neighbor (KNN), Support Vector Machine (SVM) and ridge regression, respectively [27, 15, 3]. Previous studies have introduced many popular regularization techniques to few-shot learning from deep learning, such as weight decay, dropout, label smooth [3], and data augmentation. Common data augmentation techniques for image recognition are usually designed manually and the best augmentation strategies depend on dataset. In natural color image datasets, random cropping and random horizontal \ufb02ipping are the most common. Since the few-shot learning tasks consist of natural color images, the random horizontal \ufb02ipping and random cropping are applied in fewshot learning. In addition, color (brightness, contrast, and saturation) jitter is often applied in the works of few-shot learning [7, 19]. Other data augmentation technologies related to fewshot learning include generating samples by few-shot learning and generating samples for few-shot learning. The former tried to synthesize additional examples via transferring, extracting, and encoding to create the data of the new classes, that are intra-class relationships between pairs of reference classes\u2019 data instances [8, 25]. The later tried to apply meta-learning in a few-shot generation to generate samples from other models [2].In addition to these two types of studies, the data augmentation technology most closed to the new proposed approach is applied to Omniglot dataset, which consists of handwritten words [14]. They created the novel classes by rotating the original images 90, 180 and 270 degrees [24]. However, when this approach is applied for the natural color image, it would be slightly changed, and we will explain this in Section 3. 3. Method 3.1. Problem De\ufb01nition We adopt the formulation purposed by [30] to describe the N-way K-shot task. A few-shot task contains many task instances (denoted by Ti), each instance is a classi\ufb01cation problem consisting of the data sampled from N classes. The classes are randomly selected from a classes set. The classes set are split into M tr, M val and M test for a training class set Ctr, a validation classes set Cval, and a test classes set Ctest. In particular, each class does not overlap others (i.e., the classes used during testing are unseen classes during training). Data is randomly sampled from Ctr, Cval and Ctest, so as to create task instances for training meta-set Str, validation meta-set Sval, and test meta-set Stest, respectively. The validation and testing meta-sets are used for model selection and \ufb01nal evaluation, respectively. The data in each task instance, Ti, are divided into training examples Dtr and validation examples Dval. Both of them only contains the data from N classes which sampled from the appropriate classes set randomly (for a task instance applied during training, the classes form a subset of the training classes set Ctr). In most settings, the training set Dtr = {(xk n, yk n)|n = 1 . . . N; k = 1 . . . K} consists of K data instances from each class, this processing usually called as a \u201cshot\u201d. The validation set, Dval, consists of several other data instances from the same classes, this processing is usually called as a \u201cquery\u201d. An evaluation is provided for generalization performance on the N classi\ufb01cation task instance Dtr. Note that: the validation set of a task instance Dval (for optimizing model during \u201couter loop\u201d) is different from the held-out validation classes set Cval and meta-set Sval (for model selection). 3.2. Task Augmentation by Rotating This work is to increase the size of the training classes set, M tr, by rotating all images within the training classes set with 90, 180, 270 degrees. The size, M tr, is increased for three times. In the Omniglot dataset consisting of handwritten words [24], this approach works well, since it can rotate a handwritten word multiple of 90 degrees and treat the new one as another word; in addition, it is really possible that the novel word is similar to some words, which are not included in the training classes but existed. \fOriginal classes Novel classes 2 Original classes Novel classes 1 Novel classes 3 Novel classes 2 Novel classes 1 Novel classes 3 Figure 1: Examples of the novel created classes. Algorithm 1 Task Augmentation by Rotating. Require: Classes set C = {c1, c2, . . . , cM}; Max possibility for Task Aug pmax; The delay to Task Aug T; The current count t; The number of ways, shots and queries N, K, H 1: t \u2190t + 1 2: p \u2190pmax \u2217min{1, t T} 3: n \u223cBinomial(N, p) 4: Dtr, Dval \u2190{}, {} 5: V \u2190Sample N \u2212n from {1, 2, \u00b7 \u00b7 \u00b7 , M} 6: for all v \u2208V do 7: D \u2190Sample K + H from cv 8: Dtr \u2190Dtr\u222aFirst K of D 9: Dval \u2190Dval\u222aLast H of D 10: end for 11: U \u2190Sample n from {M, M + 1, \u00b7 \u00b7 \u00b7 , 4M} 12: for all u \u2208U do 13: v \u2190(u mod M) + 1 14: D \u2190Sample K + H from cv 15: r \u2190\u230au M \u230b 16: Rotate all x \u2208{x|(x, y) \u2208D} 90r degrees 17: Dtr \u2190Dtr\u222aFirst K of D 18: Dval \u2190Dval\u222aLast H of D 19: end for 20: return (Dtr, Dval) c1 c2 cM p 1-p pmax\u2217 min{1, t T } N-n from original classes n from novel classes Dtr Dval Figure 2: The process of generating a task instance with Task Aug by rotating. For natural images, it is obvious that the images generated by rotating is real enough. But should the new generated images be classi\ufb01ed as the novel classes or the original classes? It dependents on the similarity between the new images and the original classes. If the most of they are similar enough, the new images should be classi\ufb01ed as the original classes, and vice versa. This logic shows that one of the two methods must be effective. Since there are almost not works merge the new images into the original classes which worked well, we assume that Task Aug by rotating is effective for meta-learning, and we will compare two methods in experiments. Besides, it is different from in handwritten that we assign the new data smaller weights than the original data, so as to make models prioritize learning the features of the original classes, since the images generated by rotating rarely exist in the original data. This way makes the features of the novel classes as a supplement to prevent the augmented data \ffrom taking up large capacity in the model, which is same as other common data augment methods. The smaller weights are implemented in two ways, 1) lower probability and 2) delaying the probability of selecting the novel classes. For a class in a task instance, the probability of the class coming from the novel classes is p, and the probability coming from the original classes is 1\u2212p. Besides, the initial p is set to 0, then linearly rises from 0 to pmax for the \ufb01rst T tasks. The max probability pmax is set lower than the proportion of the novel classes in all classes to make each novel class have a lower probability than each original class. The whole process of Task Aug on a classes set is summarized in Algorithm 1 and Figure 2. 3.3. Ensemble In this work, we also compare the methods with the training protocol with ensemble method [11] in addition to the standard training protocol, which choosing a model by the validation set. The training protocol with an ensemble method use the models with different training epoch to an ensemble model, in order to better use the models obtained in a single training process, and this approach has been proved to be valid for meta-learning by experiments [16]. We adopt this ensemble method. However, unlike [11] and [16] that we did not use cyclic annealing for learning rate and any methods to select models. We directly took the average of the prediction of all models, which are saved according to an interval of 1 epoch. In Section 4, the methods with this ensemble approach are marked by \u201c+ens\u201d. 4. Experiments We evaluate the proposed method on few-shot learning tasks. In order to ensure fair, both the results of baseline and Task Aug were run in our own environment. The comparative experiment is designed to answer the following questions: (1) Image Aug and Task Aug by rotating which is able to improve the performance of meta-learning? (2) How much should the probably for the novel classes be set? (3) Is Task Aug by rotating able to improve the performance of the current popular meta-learning methods? 4.1. Experimental Con\ufb01guration 4.1.1 Backbone Following [15, 18, 17], we used ResNet-12 network in our experiments. The ResNet-12 network had four residual blocks which contains three 3 \u00d7 3 convolution, batch normalization and Leaky ReLU with 0.1 negative slope. One 2 \u00d7 2 max-pooling layer is used for reducing the size of the feature map. The numbers of the network channels were 64, 160, 320 and 640, respectively. DropBlock regularization [6] is used in the last two residual blocks, the conventional dropout [10] is used in the \ufb01rst two residual blocks. The block sizes of DropBlock were set to 2 and 5 for CIFAR derivatives and ImageNet derivatives, respectively. In all experiments, the dropout possibility was set to 0.1. The global average pooling was not used for the \ufb01nal output of the last residual block. 4.1.2 Base Learners We used ProtoNets [27], MetaOptNet-SVM [15] (we write it as M-SVM) and Ridge Regression Differentiable Discriminator (R2-D2) [3] as basic methods to verify the effective of Task Aug. For ProtoNets, we did not use a higher way for training than testing like [27]. Instead, the equal number of shot and way were used in both training and evaluation, and its output multiplied by a learnable scale before the softmax following [18, 15]. For M-SVM, we set training shot to 5 for CIFAR-FS; 15 for FC100; and 15 for miniImageNet; regularization parameter of SVM was set to 0.1; and a learnable scale was used following [15]. We did not use label smoothing like [15], because we did not \ufb01nd that label smoothing can improve the performance in our environment. This was also af\ufb01rmed from the [15] author\u2019s message on GitHub, that Program language packages and environment might affect results of the meta-learning method. For R2-D2, we set the same training shot as for M-SVM, and used a learnable scale and bias following [3]. It was different from [3] we used a \ufb01xed regularization parameter of ridge regression which was set to 50 because [3] has con\ufb01rmed that making it learnable might not be helpful. Last, for all methods, each class in a task instance contained 6 test (query) examples during training and 15 test (query) examples during testing. 4.1.3 Training Con\ufb01guration Stochastic gradient descent (SGD) was used. Following [29], we set weight decay and Nesterov momentum to 0.0005 and 0.9, respectively. Each mini-batch contained 8 task instances. The meta-learning model was trained for 60 epochs, and 1000 mini-batchs for each epoch. We set the initial learning rate to 0.1, then multiplied it by 0.06, 0.012, and 0.0024 at epochs 20, 40 and 50, respectively, as in [7]. The results, which are marked by \u201c+ens\u201d were used the 60 models saved after each epoch to become an ensemble model. For the \ufb01nal training, the training classes set was augmented by the validation classes set. When we only chose one model, we will chose the model at the epoch where we got the best model during training on the training classes set. The results of the \ufb01nal run are marked by \u201c+val\u201d in this subsection. Since the base idea of \u201c+ens\u201d was proposed by other works and \u201c+val\u201d is popular for metalearning, we do not explain more details about them. \f0.00 0.25 0.50 0.75 The max probability of rotation 80.0 82.5 85.0 87.5 Accuracy (%) CIFAR-FS 5-way 5-shot Baseline 5-shot Image Aug 5-shot Task Aug 5-shot 0.00 0.25 0.50 0.75 The max probability of rotation 54 56 58 Accuracy (%) FC100 5-way 5-shot 0.00 0.25 0.50 0.75 The max probability of rotation 76 78 80 Accuracy (%) miniImageNet 5-way 5-shot 0.00 0.25 0.50 0.75 The max probability of rotation 70 75 Accuracy (%) CIFAR-FS 5-way 1-shot Baseline 1-shot Image Aug 1-shot Task Aug 1-shot 0.00 0.25 0.50 0.75 The max probability of rotation 37.5 40.0 42.5 Accuracy (%) FC100 5-way 1-shot 0.00 0.25 0.50 0.75 The max probability of rotation 60.0 62.5 Accuracy (%) miniImageNet 5-way 1-shot 0.00 0.25 0.50 0.75 The max probability of rotation 85.0 87.5 Accuracy (%) CIFAR-FS 5-way 5-shot with ensemble 0.00 0.25 0.50 0.75 The max probability of rotation 56 58 60 Accuracy (%) FC100 5-way 5-shot with ensemble 0.00 0.25 0.50 0.75 The max probability of rotation 78 80 82 Accuracy (%) miniImageNet 5-way 5-shot with ensemble 0.00 0.25 0.50 0.75 The max probability of rotation 72.5 75.0 77.5 Accuracy (%) CIFAR-FS 5-way 1-shot with ensemble 0.00 0.25 0.50 0.75 The max probability of rotation 42 44 46 Accuracy (%) FC100 5-way 1-shot with ensemble 0.00 0.25 0.50 0.75 The max probability of rotation 62 64 66 Accuracy (%) miniImageNet 5-way 1-shot with ensemble Figure 3: The accuracies (%) on meta-test sets with varying probability pmax for the novel classes.The 95% con\ufb01dence interval is denoted by the shaded region. For data augmentation, we adopted random crop, horizontal \ufb02ip, and color (brightness, saturation, and contrast) jitter data augmentation following the work of [7, 19]. In the experiments of comparing Task Aug and Image Aug by rotating, R2-D2 was applied, and we set T to 80000. In the evaluation of Task Aug for ProtoNets and M-SVM, we set pmax to the value getting the best results for R2-D2. 4.1.4 Dataset The CIFAR-FS [3] containing all 100 classes from CIFAR100 [12] is proposed as few-shot classi\ufb01cation benchmark recently. These classes are randomly divided into training classes, validation classes and test classes. The three types contain 64, 16 and 20 classes, respectively. There are 600 nature color images of size 32 \u00d7 32 in each class. The FC100 [18] are also derived from CIFAR-100 [12], and the 100 classes are grouped into 20 superclasses. The training, validation, and testing classes contain 60 classes from 12 superclasses, 20 classes from 4 superclasses, and 20 classes from 4 superclasses, respectively. The target is to minimize the information overlap between classes to make it more challenging than current few-shot classi\ufb01cation tasks. Same as CIFAR-FS, there are 600 nature color images of size 32 \u00d7 32 in each class. The miniImageNet [30] is one of the most popular benchmark for few-shot classi\ufb01cation, which contains 100 classes randomly selected from ILSVRC-2012 [22]. The classes are randomly divided into training classes, validation classes and test classes, and them contain 64, 16 and 20 classes, respectively. There are 600 nature color images of size 84 \u00d7 84 in each class. Since [30] did not release the class splits, we use the more common split proposed by [20]. 4.2. Comparison between Task Aug and Image Aug To prove our assumption that rotation multi 90 degrees for Task Aug is better than that for Image Aug, we draw the accuracy curves depending on pmax for both Task Aug and Image Aug (curves showed in Figure 3). The linear rising of p was also used for Image Aug, and T = 80000 for both Task Aug and Image Aug. In all the results showed in Figure 3, the training classes set was not augmented by the validation classes set. As shown in Figure 3, the performance of Task Aug on most of the regimes is better than Image Aug and baseline in general. Besides, we observed that: with the increase of pmax, the accuracy rises at \ufb01rst, reaches the peaks between 0.25 and 0.5, then declines and reaches baseline when pmax = 0.75 at the end, which is the proportion of the novel classes in all classes. The accuracy of Task Aug on CIFARFS, FC100 and miniImagNet reach the peaks at 0.5, 0.25 and 0.25 respectively. At the same time, the rotation multi 90 degrees for Image Aug cannot improve or even cause \fTable 1: Comparison to the average accuracies (%) with 95% con\ufb01dence intervals between the methods with and without Task Aug on CIFAR-FS 5-way 1-shot. Method Baseline Task Aug ProtoNets [27] 71.88\u00b10.52 74.15\u00b10.50 ProtoNets (+ens) 73.95\u00b10.51 75.89\u00b10.48 ProtoNets (+val) 73.20\u00b10.51 75.10\u00b10.49 ProtoNets (+ens+val) 76.05\u00b10.49 77.28\u00b10.47 M-SVM [15] 71.52\u00b10.51 72.95\u00b10.48 M-SVM (+ens) 74.12\u00b10.50 75.85\u00b10.47 M-SVM (+val) 72.42\u00b10.50 73.13\u00b10.47 M-SVM (+ens+val) 75.91\u00b10.48 76.75\u00b10.46 R2-D2 [3] 72.27\u00b10.51 74.42\u00b10.48 R2-D2 (+ens) 75.06\u00b10.50 76.51\u00b10.47 R2-D2 (+val) 73.52\u00b10.50 76.02\u00b10.47 R2-D2 (+ens+val) 76.40\u00b10.49 77.66\u00b10.46 worse performance. 4.3. Evaluation of Task Aug In order to further prove the proposed approach can improve the performance of meta-learning, we evaluate Task Aug by rotating on several meta-learning methods in this section. We choose several currently the state of art base learners for experiments, we detail in Section 4.1.2. Besides, the training protocol with ensemble method can get better results than the standard training protocol, we involve it in the experiments. We think this is important, because the proposed method can only be a contribution if it can further improve performance based on the best method available at present. Results. Table 1-6 show the average accuracies (%) with 95% con\ufb01dence intervals of the methods with and without Task Aug, and the best results are highlighted. The tables show that the proposed method can improve the performance in most of cases. We can observe that: some results without the ensemble approach [11] of baseline and Task Aug are close, but the advantage of Task Aug is still obvious on the comparison results with the ensemble approach. We suspect that the scale of backbone limits the performance of the best model. A larger scale backbone is needed for the training process with Task Aug. For the results of ensemble approach, since Task Aug reduces the over-\ufb01tting, more models during the training process have good performance, which provide ensemble with models of higher quality. Last we compare the results of this work with the results proposed by the prior works, in order to show how much Table 2: Comparison to the average accuracies (%) with 95% con\ufb01dence intervals between the methods with and without Task Aug on CIFAR-FS 5-way 5-shot. Method Baseline Task Aug ProtoNets [27] 84.14\u00b10.36 85.37\u00b10.35 ProtoNets (+ens) 85.72\u00b10.35 87.33\u00b10.33 ProtoNets (+val) 85.29\u00b10.35 86.53\u00b10.34 ProtoNets (+ens+val) 86.88\u00b10.34 88.24\u00b10.33 M-SVM [15] 84.01\u00b10.36 85.91\u00b10.36 M-SVM (+ens) 85.85\u00b10.34 87.73\u00b10.33 M-SVM (+val) 84.94\u00b10.36 86.94\u00b10.34 M-SVM (+ens+val) 87.15\u00b10.34 88.38\u00b10.33 R2-D2 [3] 84.60\u00b10.36 86.02\u00b10.35 R2-D2 (+ens) 86.11\u00b10.34 87.63\u00b10.34 R2-D2 (+val) 85.39\u00b10.36 86.73\u00b10.34 R2-D2 (+ens+val) 87.04\u00b10.34 88.33\u00b10.33 Table 3: Comparison to the average accuracies (%) with 95% con\ufb01dence intervals between the methods with and without Task Aug on FC100 5-way 1-shot. Method Baseline Task Aug ProtoNets [27] 37.53\u00b10.40 38.89\u00b10.40 ProtoNets (+ens) 40.04\u00b10.41 42.00\u00b10.43 ProtoNets (+val) 43.63\u00b10.43 44.91\u00b10.46 ProtoNets (+ens+val) 47.16\u00b10.46 48.91\u00b10.47 M-SVM [15] 40.50\u00b10.39 41.17\u00b10.40 M-SVM (+ens) 43.24\u00b10.42 44.38\u00b10.42 M-SVM (+val) 46.72\u00b10.45 47.39\u00b10.44 M-SVM (+ens+val) 49.50\u00b10.46 49.77\u00b10.45 R2-D2 [3] 40.66\u00b10.41 41.47\u00b10.40 R2-D2 (+ens) 43.27\u00b10.42 44.75\u00b10.43 R2-D2 (+val) 47.12\u00b10.44 48.21\u00b10.45 R2-D2 (+ens+val) 49.92\u00b10.45 51.35\u00b10.46 this work raises the baselines after combining several prior methods and the proposed method, and they are showed in Table 7, 8 and 9. The tables show that the highest accuracies of our experiments exceeded the current state-of-art accuracies 2% to 5%. 5." + } + ] + }, + "edge_feat": {} + } +} \ No newline at end of file