We consider correlated and distributed sources without cooperation at the encoder. For these sources, we derive the best achievable performance in the rate-distortion sense of any distributed compressed sensing scheme, under the constraint of high--rate quantization. Moreover, under this model we derive a closed--form expression of the rate gain achieved by taking into account the correlation of the sources at the receiver and a closed--form expression of the average performance of the oracle receiver for independent and joint reconstruction. Finally, we show experimentally that the exploitation of the correlation between the sources performs close to optimal and that the only penalty is due to the missing knowledge of the sparsity support as in (non distributed) compressed sensing. Even if the derivation is performed in the large system regime, where signal and system parameters tend to infinity, numerical results show that the equations match simulations for parameter values of practical interest.
Consider the problem of estimating a latent signal from a lossy compressed version of the data when the compressor is agnostic to the relation between the signal and the data. This situation arises in a host of modern applications when data is transmitted or stored prior to determining the downstream inference task. Given a bitrate constraint and a distortion measure between the data and its compressed version, let us consider the joint distribution achieving Shannon's rate-distortion (RD) function. Given an estimator and a loss function associated with the downstream inference task, define th…
We consider vector-quantized (VQ) transmission of compressed sensing (CS) measurements over noisy channels. Adopting mean-square error (MSE) criterion to measure the distortion between a sparse vector and its reconstruction, we derive channel-optimized quantization principles for encoding CS measurement vector and reconstructing sparse source vector. The resulting necessary optimal conditions are used to develop an algorithm for training channel-optimized vector quantization (COVQ) of CS measurements by taking the end-to-end distortion measure into account.
Modern compression algorithms exploit complex structures that are present in signals to describe them very efficiently. On the other hand, the field of compressed sensing is built upon the observation that "structured" signals can be recovered from their under-determined set of linear projections. Currently, there is a large gap between the complexity of the structures studied in the area of compressed sensing and those employed by the state-of-the-art compression codes. Recent results in the literature on deterministic signals aim at bridging this gap through devising compressed sensing decod…
Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-opti…
Joint Research Institute for Signal and Image Processing, Edinburgh University（爱丁堡大学信号与图像处理联合研究所）；Heriot-Watt University（赫瑞-瓦特大学）
We propose the notion of a sample distortion (SD) function for independent and identically distributed (i.i.d) compressive distributions to fundamentally quantify the achievable reconstruction performance of compressed sensing for certain encoder-decoder pairs at a given sampling ratio. Two lower bounds on the achievable performance and the intrinsic convexity property is derived. A zeroing procedure is then introduced to improve non convex SD functions. The SD framework is then applied to analyse compressed imaging with a multi-resolution statistical image model using both the generalized Gau…
A framework is developed using techniques from rate distortion theory in statistical testing. The idea is first to do optimal compression according to a certain distortion function and then use information divergence from the compressed empirical distribution to the compressed null hypothesis as statistic. Only very special cases have been studied in more detail, but they indicate that the approach can be used under very general conditions.
In this paper, the `Approximate Message Passing' (AMP) algorithm, initially developed for compressed sensing of signals under i.i.d. Gaussian measurement matrices, has been extended to a multi-terminal setting (MAMP algorithm). It has been shown that similar to its single terminal counterpart, the behavior of MAMP algorithm is fully characterized by a `State Evolution' (SE) equation for large block-lengths. This equation has been used to obtain the rate-distortion curve of a multi-terminal memoryless source. It is observed that by spatially coupling the measurement matrices, the rate-distortio…
RDD: Pareto Analysis of the Rate-Distortion-Distinguishability Trade-off
Department of Electrical, Electronic, and Information Engineering, University of Bologna（电子工程系，博洛尼亚大学）；Advanced Research Center on Electronic Systems, University of Bologna（电子系统高级研究中心，博洛尼亚大学）；CEMSE, King Abdullah University of Science and Technology (KAUST)（科学与技术国王阿卜杜勒阿齐兹大学（KAUST）的CEMSE）；Department of Electronics and Telecommunications (DET), Politecnico di Torino（电子与电信系（DET），托里尼 Politecnico）
The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an even smaller number of samples is sufficient when there exists prior knowledge about the distribution of the unknown vector, or when only partial recovery is needed. An information-theoretic lower bound with connections to free probability theory and an upper bound corresponding to a computationally simple thresholding estimator are derived. It is shown that…
This paper studies the problem of reconstructing sparse or compressible signals from compressed sensing measurements that have undergone nonuniform quantization. Previous approaches to this Quantized Compressed Sensing (QCS) problem based on Gaussian models (bounded l2-norm) for the quantization distortion yield results that, while often acceptable, may not be fully consistent: re-measurement and quantization of the reconstructed signal do not necessarily match the initial observations. Quantization distortion instead more closely resembles heteroscedastic uniform noise, with variance dependin…
The indirect source-coding problem in which a Bernoulli process is compressed in a lossy manner from its noisy observations is considered. These noisy observations are obtained by passing the source sequence through a The indirect source-coding problem in which a Bernoulli process is compressed in a lossy manner from its noisy observations is considered. These noisy observations are obtained by passing the source sequence through a binary symmetric channel so that the channel crossover probability controls the amount of information available about the source realization at the encoder. We use …
We consider a multiterminal source coding problem in which a source is estimated at a central processing unit from lossy-compressed remote observations. Each lossy-encoded observation is produced by a remote sensor which obtains a noisy version of the source and compresses this observation minimizing a local distortion measure which depends only on the marginal distribution of its observation. The central node, on the other hand, has knowledge of the joint distribution of the source and all the observations and produces the source estimate which minimizes a different distortion measure between…
Today's networks are controlled assuming pre-compressed and packetized data. For video, this assumption of data packets abstracts out one of the key aspects - the lossy compression problem. Therefore, first, this paper develops a framework for network control that incorporates both source-rate and source-distortion. Next, it decomposes the network control problem into an application-layer compression control, a transport-layer congestion control and a network-layer scheduling. It is shown that this decomposition is optimal for concave utility functions. Finally, this paper derives further insi…
In this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. We consider two settings: output noise models where the noise enters after the projection and input noise models where the noise enters before the projection. We consider two types of distortion for reconstruction: support errors and mean-squared errors. Our goal is to relate the number of measurements, $m$, and $\snr$, to signal sparsity, $k$, distortion level, $d$, and signal dimension, $n$. We consider support errors in a worst-case setting. We employ di…
Too high sampling rate is the bottleneck to wideband spectrum sensing for cognitive radio in mobile communication. Compressed sensing (CS) is introduced to transfer the sampling burden. The standard sparse signal recovery of CS does not consider the distortion in the analogue-to-information converter (AIC). To mitigate performance degeneration casued by the mismatch in least square distortionless constraint which doesn't consider the AIC distortion, we define the sparse signal with the sampling distortion as a bounded additive noise, and An anti-sampling-distortion constraint (ASDC) is deduced…
The rate-distortion dimension (RDD) of an analog stationary process is studied as a measure of complexity that captures the amount of information contained in the process. It is shown that the RDD of a process, defined as two times the asymptotic ratio of its rate-distortion function $R(D)$ to $\log {1\over D}$ as the distortion $D$ approaches zero, is equal to its information dimension (ID). This generalizes an earlier result by Kawabata and Dembo and provides an operational approach to evaluate the ID of a process, which previously was shown to be closely related to the effective dimension o…
Can compression algorithms be employed for recovering signals from their underdetermined set of linear measurements? Addressing this question is the first step towards applying compression algorithms for compressed sensing (CS). In this paper, we consider a family of compression algorithms $\mathcal{C}_r$, parametrized by rate $r$, for a compact class of signals $\mathcal{Q} \subset \mathds{R}^n$. The set of natural images and JPEG at different rates are examples of $\mathcal{Q}$ and $\mathcal{C}_r$, respectively. We establish a connection between the rate-distortion performance of $\mathcal{C…
We consider a Shannon cipher system for memoryless sources, in which distortion is allowed at the legitimate decoder. The source is compressed using a rate distortion code secured by a shared key, which satisfies a constraint on the compression rate, as well as a constraint on the exponential rate of the excess-distortion probability at the legitimate decoder. Secrecy is measured by the exponential rate of the exiguous-distortion probability at the eavesdropper, rather than by the traditional measure of equivocation. We define the perfect secrecy exponent as the maximal exiguous-distortion exp…
This is a three part paper. Optimality of source-channel separation for communication with a fidelity criterion when the channel is compound as defined by Csiszar and Korner in their book and general as defined by Verdu and Han, is proved in Part I. It is assumed that random codes are permitted. The word "universal" in the title of this paper refers to the fact that the channel model is compound. The proof uses a layered black-box or a layered input-output view-point. In particular, only the end-to-end description of the channel as being capable of communicating a source to within a certain di…
Modern image and video compression codes employ elaborate structures existing in such signals to encode them into few number of bits. Compressed sensing recovery algorithms on the other hand use such signals' structures to recover them from few linear observations. Despite the steady progress in the field of compressed sensing, structures that are often used for signal recovery are still much simpler than those employed by state-of-the-art compression codes. The main goal of this paper is to bridge this gap through answering the following question: Can one employ a given compression code to bu…
University of Toronto（多伦多大学）；McMaster University（麦克马斯特大学）
In the context of lossy compression, Blau & Michaeli (2019) adopt a mathematical notion of perceptual quality and define the information rate-distortion-perception function, generalizing the classical rate-distortion tradeoff. We consider the notion of universal representations in which one may fix an encoder and vary the decoder to achieve any point within a collection of distortion and perception constraints. We prove that the corresponding information-theoretic universal rate-distortion-perception function is operationally achievable in an approximate sense. Under MSE distortion, we show th…
Consider a discrete memoryless multiple source with $m$ components of which $k \leq m$ possibly different sources are sampled at each time instant and jointly compressed in order to reconstruct all the $m$ sources under a given distortion criterion. A new notion of sampling rate distortion function is introduced, and is characterized first for the case of fixed-set sampling. Next, for independent random sampling performed without knowledge of the source outputs, it is shown that the sampling rate distortion function is the same regardless of whether or not the decoder is informed of the sequen…
In this paper, we design and analyze distributed vector quantization (VQ) for compressed measurements of correlated sparse sources over noisy channels. Inspired by the framework of compressed sensing (CS) for acquiring compressed measurements of the sparse sources, we develop optimized quantization schemes that enable distributed encoding and transmission of CS measurements over noisy channels followed by joint decoding at a decoder. The optimality is addressed with respect to minimizing the sum of mean-square error (MSE) distortions between the sparse sources and their reconstruction vectors …
A distributed lossy compression network with $L$ encoders and a decoder is considered. Each encoder observes a source and sends a compressed version to the decoder. The decoder produces a joint reconstruction of target signals with the mean squared error distortion below a given threshold. It is assumed that the observed sources can be expressed as the sum of target signals and corruptive noises which are independently generated from two symmetric multivariate Gaussian distributions. The minimum compression rate of this network versus the distortion threshold is referred to as the rate-distort…
In this work, we investigate an instance of the Heegard-Berger problem with two sources and arbitrarily correlated side information sequences at two decoders, in which the reconstruction sets at the decoders are degraded. Specifically, two sources are to be encoded in a manner that one of the two is reproduced losslessly by both decoders, and the other is reproduced to within some prescribed distortion level at one of the two decoders. We establish a single-letter characterization of the rate-distortion function for this model. The investigation of this result in some special cases also sheds …
A randomized covering-packing duality between source and channel coding will be discussed by considering the source coding problem of coding a source with a certain distortion level and by considering a channel which communicates the source within a certain distortion level. An operational view of source-channel separation for communication with a fidelity criterion will be discussed in brief.
This paper is concerned with a rate-distortion theory for sequences of i.i.d. random variables with general distribution supported on general sets including manifolds and fractal sets. Manifold structures are prevalent in data science, e.g., in compressed sensing, machine learning, image processing, and handwritten digit recognition. Fractal sets find application in image compression and in modeling of Ethernet traffic. We derive a lower bound on the (single-letter) rate-distortion function that applies to random variables X of general distribution and for continuous X reduces to the classical…
Consider a generalized multiterminal source coding system, where $\ell\choose m$ encoders, each observing a distinct size-$m$ subset of $\ell$ ($\ell\geq 2$) zero-mean unit-variance symmetrically correlated Gaussian sources with correlation coefficient $ρ$, compress their observations in such a way that a joint decoder can reconstruct the sources within a prescribed mean squared error distortion based on the compressed data. The optimal rate-distortion performance of this system was previously known only for the two extreme cases $m=\ell$ (the centralized case) and $m=1$ (the distributed case)…
We consider the recovery of a continuous-time Wiener process from a quantized or lossy compressed version of its uniform samples under limited bitrate and sampling rate. We derive a closed form expression for the optimal tradeoff among sampling rate, bitrate, and quadratic distortion in this setting. This expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite-rank operators associated with the optimal estimator of the Wiener process from its samples. We show that the ratio between this expression and the standard distort…
This article introduces a novel communication scheme, termed coded compressed sensing, for unsourced multiple-access communication. The proposed divide-and-conquer approach leverages recent advances in compressed sensing and forward error correction to produce a novel uncoordinated access paradigm, along with a computationally efficient decoding algorithm. Within this framework, every active device partitions its data into several sub-blocks and, subsequently, adds redundancy using a systematic linear block code. Compressed sensing techniques are then employed to recover sub-blocks up to a per…
The problem of performing similarity queries on compressed data is considered. We focus on the quadratic similarity measure, and study the fundamental tradeoff between compression rate, sequence length, and reliability of queries performed on compressed data. For a Gaussian source, we show that queries can be answered reliably if and only if the compression rate exceeds a given threshold - the identification rate - which we explicitly characterize. Moreover, when compression is performed at a rate greater than the identification rate, responses to queries on the compressed data can be made exp…
In this paper, we propose a method of structured construction of the optimal measurement matrix for noiseless compressed sensing (CS), which achieves the minimum number of measurements which only needs to be as large as the sparsity of the signal itself to be recovered to guarantee almost error-free recovery, for sufficiently large dimension. To arrive at the results, we employ a duality between noiseless CS and analog coding across sparse additive noisy channel (SANC). Extending Renyi Information Dimension to Mutual Information Dimension (MID), we show the operational meaning of MID to be the…
This paper advocates the use of the distributed compressed sensing (DCS) paradigm to deploy energy harvesting (EH) Internet of Thing (IoT) devices for energy self-sustainability. We consider networks with signal/energy models that capture the fact that both the collected signals and the harvested energy of different devices can exhibit correlation. We provide theoretical analysis on the performance of both the classical compressive sensing (CS) approach and the proposed distributed CS (DCS)-based approach to data acquisition for EH IoT. Moreover, we perform an in-depth comparison of the propos…
Consider a lossy compression system with $\ell$ distributed encoders and a centralized decoder. Each encoder compresses its observed source and forwards the compressed data to the decoder for joint reconstruction of the target signals under the mean squared error distortion constraint. It is assumed that the observed sources can be expressed as the sum of the target signals and the corruptive noises, which are generated independently from two symmetric multivariate Gaussian distributions. Depending on the parameters of such distributions, the rate-distortion limit of this system is characteriz…
University of Maryland, College Park（马里兰大学帕克分校）；Institute for Systems Research（系统研究所）
Consider a Gaussian memoryless multiple source with $m$ components with joint probability distribution known only to lie in a given class of distributions. A subset of $k \leq m$ components are sampled and compressed with the objective of reconstructing all the $m$ components within a specified level of distortion under a mean-squared error criterion. In Bayesian and nonBayesian settings, the notion of universal sampling rate distortion function for Gaussian sources is introduced to capture the optimal tradeoffs among sampling, compression rate and distortion level. Single-letter characterizat…
It is known that modeling an information source via a symbolic dynamical system evolving over the unit interval, leads to a natural lossless compression scheme attaining the entropy rate of the source, under general conditions. We extend this notion to the lossy compression regime assuming a feedforward link is available, by modeling a source via a two-dimensional symbolic dynamical system where one component corresponds to the compressed signal, and the other essentially corresponds to the feedforward signal. For memoryless sources and an arbitrary bounded distortion measure, we show this app…
Consider a pair of correlated Gaussian sources (X1,X2). Two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder. The decoder is interested in reconstructing a linear combination of X1 and X2 to within a mean-square distortion of D. We obtain an inner bound to the optimal rate-distortion region for this problem. A portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components X1 and X2 first. This results in a better rate region for cert…
The objective of this paper is to further investigate various applications of information Nonanticipative Rate Distortion Function (NRDF) by discussing two working examples, the Binary Symmetric Markov Source with parameter $p$ (BSMS($p$)) with Hamming distance distortion, and the multidimensional partially observed Gaussian-Markov source. For the BSMS($p$), we give the solution to the NRDF, and we use it to compute the Rate Loss (RL) of causal codes with respect to noncausal codes. For the multidimensional Gaussian-Markov source, we give the solution to the NRDF, we show its operational meani…
This work is concerned with the problem of recovering high-dimensional signals $\mathbf{x} \in \mathbb{R}^n$ which belong to a convex set of low-complexity from a small number of quantized measurements. We propose to estimate the signals via a convex program based on rectified linear units (ReLUs) for two different quantization schemes, namely one-bit and uniform multi-bit quantization. Assuming that the linear measurement process can be modelled by a sensing matrix with i.i.d. subgaussian rows, we obtain for both schemes near-optimal uniform reconstruction guarantees by adding well-designed n…
In this paper we study the problem of recovering sparse or compressible signals from uniformly quantized measurements. We present a new class of convex optimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment $p$ (BPDQ$_p$), that model the quantization distortion more faithfully than the commonly used Basis Pursuit DeNoise (BPDN) program. Our decoders proceed by minimizing the sparsity of the signal to be reconstructed subject to a data-fidelity constraint expressed in the $\ell_p$-norm of the residual error for $2\leq p\leq \infty$. We show theoretically that, (i) the re…
Paris Research Center, Huawei Technologies France（华为技术法国巴黎研究中心）；Université Paris-Est（巴黎东大学）
In this paper, we use tools from rate-distortion theory to establish new upper bounds on the generalization error of statistical distributed learning algorithms. Specifically, there are $K$ clients whose individually chosen models are aggregated by a central server. The bounds depend on the compressibility of each client's algorithm while keeping other clients' algorithms un-compressed, and leverage the fact that small changes in each local model change the aggregated model by a factor of only $1/K$. Adopting a recently proposed approach by Sefidgaran et al., and extending it suitably to the d…
We study a distributed learning problem in which Alice sends a compressed distillation of a set of training data to Bob, who uses the distilled version to best solve an associated learning problem. We formalize this as a rate-distortion problem in which the training set is the source and Bob's cross-entropy loss is the distortion measure. We consider this problem for unsupervised learning for batch and sequential data. In the batch data, this problem is equivalent to the information bottleneck (IB), and we show that reduced-complexity versions of standard IB methods solve the associated rate-d…
Many information systems employ lossy compression as a crucial intermediate stage among other processing components. While the important distortion is defined by the system's input and output signals, the compression usually ignores the system structure, therefore, leading to an overall sub-optimal rate-distortion performance. In this paper we propose a compression methodology for an operational rate-distortion optimization considering a known system layout, modeled using linear operators and noise. Using the alternating direction method of multipliers (ADMM) technique, we show that the design…
Wyner's common information was originally defined for a pair of dependent discrete random variables. Its significance is largely reflected in, hence also confined to, several existing interpretations in various source coding problems. This paper attempts to both generalize its definition and to expand its practical significance by providing a new operational interpretation. The generalization is two-folded: the number of dependent variables can be arbitrary, so are the alphabet of those random variables. New properties are determined for the generalized Wyner's common information of N dependen…
University of Electronic Science and Technology of China（电子科技大学）
Compressed sensing (CS) shows that a signal having a sparse or compressible representation can be recovered from a small set of linear measurements. In classical CS theory, the sampling matrix and representation matrix are assumed to be known exactly in advance. However, uncertainties exist due to sampling distortion, finite grids of the parameter space of dictionary, etc. In this paper, we take a generalized sparse signal model, which simultaneously considers the sampling and representation matrix uncertainties. Based on the new signal model, a new optimization model for robust sparse signal …
