In this paper, we consider the one-shot version of the classical Wyner-Ziv problem where a source is compressed in a lossy fashion when only the decoder has access to a correlated side information. Following the entropy-constrained quantization framework, we assume a scalar quantizer followed by variable length entropy coding. We consider compression of a uniform source, motivated by its role in the compression of processes with low-dimensional features embedded within a high-dimensional ambient space. We find upper and lower bounds to the entropy-distortion functions of the uniform source for quantized and noisy side information, and illustrate tightness of the bounds at high compression rates.
We introduce a distributed source coding scheme called successive Wyner-Ziv coding. We show that any point in the rate region of the quadratic Gaussian CEO problem can be achieved via the successive Wyner-Ziv coding. The concept of successive refinement in the single source coding is generalized to the distributed source coding scenario, which we refer to as distributed successive refinement. For the quadratic Gaussian CEO problem, we establish a necessary and sufficient condition for distributed successive refinement, where the successive Wyner-Ziv coding scheme plays an important role.
We describe and analyze sparse graphical code constructions for the problems of source coding with decoder side information (the Wyner-Ziv problem), and channel coding with encoder side information (the Gelfand-Pinsker problem). Our approach relies on a combination of low-density parity check (LDPC) codes and low-density generator matrix (LDGM) codes, and produces sparse constructions that are simultaneously good as both source and channel codes. In particular, we prove that under maximum likelihood encoding/decoding, there exist low-density codes (i.e., with finite degrees) from our construct…
Distributed source coding (DSC) addresses the compression of correlated sources without communication links among them. This paper is concerned with the Wyner-Ziv problem: coding of an information source with side information available only at the decoder in the form of a noisy version of the source. Both the theoretical analysis and code design are addressed in the framework of multi-dimensional nested lattice coding (NLC). For theoretical analysis, accurate computation of the rate-distortion function is given under the high-resolution assumption, and a new upper bound using the derivative of…
We investigate the Wyner-Ziv coding in which the statistics of the principal source is known but the statistics of the channel generating the side-information is unknown except that it is in a certain class. The class consists of channels such that the distortion between the principal source and the side-information is smaller than a threshold, but channels may be neither stationary nor ergodic. In this situation, we define a new rate-distortion function as the minimum rate such that there exists a Wyner-Ziv code that is universal for every channel in the class. Then, we show an upper bound an…
An $l$-link binary CEO problem is considered in this paper. We present a practical encoding and decoding scheme for this problem employing the graph-based codes. A successive coding scheme is proposed for converting an $l$-link binary CEO problem to the $(2l-1)$ single binary Wyner-Ziv (WZ) problems. By using the compound LDGM-LDPC codes, the theoretical bound of each binary WZ is asymptotically achievable. Our proposed decoder successively decodes the received data by employing the well-known Sum-Product (SP) algorithm and leverages them to reconstruct the source. The sum-rate distortion perf…
We consider the role of Wyner-Ziv binning in compress-forward for relay channels. In the one-way relay channel, we analyze a compress-forward scheme without Wyner- Ziv binning but with joint decoding of both the message and compression index. It achieves the same rate as the original compress-forward scheme with binning and successive decoding. Therefore, binning helps reduce decoding complexity by allowing successive decoding, but has no impact on achievable rate for the one-way relay channel. On the other hand, no binning simplifies relay operation. By extending compress-forward without binn…
We consider both channel coding and source coding, with perfect past feedback/feedforward, in the presence of side information. It is first observed that feedback does not increase the capacity of the Gel'fand-Pinsker channel, nor does feedforward improve the achievable rate-distortion performance in the Wyner-Ziv problem. We then focus on the Gaussian case showing that, as in the absence of side information, feedback/feedforward allows to efficiently attain the respective performance limits. In particular, we derive schemes via variations on that of Schalkwijk and Kailath. These variants, whi…
The $L$-link binary Chief Executive Officer (CEO) problem under logarithmic loss is investigated in this paper. A quantization splitting technique is applied to convert the problem under consideration to a $(2L-1)$-step successive Wyner-Ziv (WZ) problem, for which a practical coding scheme is proposed. In the proposed scheme, low-density generator-matrix (LDGM) codes are used for binary quantization while low-density parity-check (LDPC) codes are used for syndrome generation; the decoder performs successive decoding based on the received syndromes and produces a soft reconstruction of the remo…
The combination of source coding with decoder side-information (Wyner-Ziv problem) and channel coding with encoder side-information (Gel'fand-Pinsker problem) can be optimally solved using the separation principle. In this work we show an alternative scheme for the quadratic-Gaussian case, which merges source and channel coding. This scheme achieves the optimal performance by a applying modulo-lattice modulation to the analog source. Thus it saves the complexity of quantization and channel decoding, and remains with the task of "shaping" only. Furthermore, for high signal-to-noise ratio (SNR),…
In problems of lossy source/noisy channel coding with side information, the theoretical bounds are achieved using "good" source/channel codes that can be partitioned into "good" channel/source codes. A scheme that achieves optimality in channel coding with side information at the encoder using independent channel and source codes was outlined in previous works. In practice, the original problem is transformed into a multiple-access problem in which the superposition of the two independent codes can be decoded using successive interference cancellation. Inspired by this work, we analyze the sup…
Wyner-Ziv coding (WZC) is a compression technique using decoder side information, which is unknown at the encoder, to help the reconstruction. In this paper, we propose and implement a new WZC structure, called residual WZC, for the quadratic-Gaussian Wyner-Ziv problem where side information can be arbitrarily distributed. In our two-stage residual WZC, the source is quantized twice and the input of the second stage is the quantization error (residue) of the first stage. The codebook of the first stage quantizer must be simultaneously good for source and channel coding, since it also acts as a…
In 1985 Kaspi provided a single-letter characterization of the sum-rate-distortion function for a two-way lossy source coding problem in which two terminals send multiple messages back and forth with the goal of reproducing each other's sources. Yet, the question remained whether more messages can strictly improve the sum-rate-distortion function. Viewing the sum-rate as a functional of the distortions and the joint source distribution and leveraging its convex-geometric properties, we construct an example which shows that two messages can strictly improve the one-message (Wyner-Ziv) rate-dist…
We provide a complete characterization of the rate-distortion region for the multistage successive refinement of the Wyner-Ziv source coding problem with degraded side informations at the decoder. Necessary and sufficient conditions for a source to be successively refinable along a distortion vector are subsequently derived. A source-channel separation theorem is provided when the descriptions are sent over independent channels for the multistage case. Furthermore, we introduce the notion of generalized successive refinability with multiple degraded side informations. This notion captures whet…
We show how real-number codes can be used to compress correlated sources and establish a new framework for distributed lossy source coding, in which we quantize compressed sources instead of compressing quantized sources. This change in the order of binning and quantization blocks makes it possible to model correlation between continuous-valued sources more realistically and compensate for the quantization error when the sources are completely correlated. We focus on the asymmetric case, i.e., lossy source coding with side information at the decoder, also known as Wyner-Ziv coding. The encodin…
This paper addresses lossy transmission of a common source over a broadcast channel when there is correlated side information at the receivers, with emphasis on the quadratic Gaussian and binary Hamming cases. A digital scheme that combines ideas from the lossless version of the problem, i.e., Slepian-Wolf coding over broadcast channels, and dirty paper coding, is presented and analyzed. This scheme uses layered coding where the common layer information is intended for both receivers and the refinement information is destined only for one receiver. For the quadratic Gaussian case, a quantity c…
We consider a variation of the Wyner-Ziv problem pertaining to lossy compression of individual sequences using finite-state encoders and decoders. There are two main results in this paper. The first characterizes the relationship between the performance of the best $M$-state encoder-decoder pair to that of the best block code of size $\ell$ for every input sequence, and shows that the loss of the latter relative to the former (in terms of both rate and distortion) never exceeds the order of $(\log M)/\ell$, independently of the input sequence. Thus, in the limit of large $M$, the best rate-dis…
We consider a two-user state-dependent multiaccess channel in which the states of the channel are known non-causally to one of the encoders and only strictly causally to the other encoder. Both encoders transmit a common message and, in addition, the encoder that knows the states non-causally transmits an individual message. We find explicit characterizations of the capacity region of this communication model in both discrete memoryless and memoryless Gaussian cases. The analysis also reveals optimal ways of exploiting the knowledge of the state only strictly causally at the encoder that sends…
National University of Singapore（新加坡国立大学）；Indian Institute of Science（印度科学学院）；Google Research（谷歌研究院）
Communication efficient distributed mean estimation is an important primitive that arises in many distributed learning and optimization scenarios such as federated learning. Without any probabilistic assumptions on the underlying data, we study the problem of distributed mean estimation where the server has access to side information. We propose \emph{Wyner-Ziv estimators}, which are communication and computationally efficient and near-optimal when an upper bound for the distance between the side information and the data is known. As a corollary, we also show that our algorithms provide effici…
This paper investigates the joint source-channel coding problem of sending a memoryless source over a memoryless broadcast channel. An inner bound and several outer bounds on the admissible distortion region are derived, which respectively generalize and unify several existing bounds. As a consequence, we also obtain an inner bound and an outer bound for the degraded broadcast channel case. When specialized to the Gaussian or binary source broadcast, the inner bound and outer bound not only recover the best known inner bound and outer bound in the literature, but also generate some new results…
Noisy network coding is recently proposed for the general multi-source network by Lim, Kim, El Gamal and Chung. This scheme builds on compress-forward (CF) relaying but involves three new ideas, namely no Wyner-Ziv binning, relaxed simultaneous decoding and message repetition. In this paper, using the two-way relay channel as the underlining example, we analyze the impact of each of these ideas on the achievable rate region of relay networks. First, CF without binning but with joint decoding of both the message and compression index can achieve a larger rate region than the original CF scheme …
In this paper we provide a new inequality useful for the proofs of strong converse theorems in the multiterminal information theory. We apply this inequality to the recent work by Tyagi and Watanabe on the strong converse theorem for the Wyner-Ziv source coding problem to obtain a new strong converse outer bound. This outer bound deviates from the Wyner-Ziv rate distortion region with the order $O\left(\frac{1}{\sqrt{n}}\right)$ on the length $n$ of source outputs.
We present a nested lattice-code-based strategy that achieves the random-coding based Compress-and-Forward (CF) rate for the three node Gaussian relay channel. To do so, we first outline a lattice-based strategy for the $(X+Z_1,X+Z_2)$ Wyner-Ziv lossy source-coding with side-information problem in Gaussian noise, a re-interpretation of the nested lattice-code-based Gaussian Wyner-Ziv scheme presented by Zamir, Shamai, and Erez. We use the notation $(X+Z_1,X+Z_2)$ Wyner-Ziv to mean that the source is of the form $X+ Z_1$ and the side-information at the receiver is of the form $X+ Z_2$, for inde…
We consider the rate distortion problem with side information at the decoder posed and investigated by Wyner and Ziv. The rate distortion function indicating the trade-off between the rate on the data compression and the quality of data obtained at the decoder was determined by Wyner and Ziv. In this paper, we study the error probability of decoding at rates below the rate distortion function. We evaluate the probability of decoding such that the estimation of source outputs by the decoder has a distortion not exceeding a prescribed distortion level. We prove that when the rate of the data com…
It is well known that lossless compression of a discrete memoryless source with near-uniform encoder output is possible at a rate above its entropy if and only if the encoder is randomized. This work focuses on deriving conditions for near-uniform encoder output(s) in the Wyner-Ziv and the distributed lossy compression problems. We show that in the Wyner-Ziv problem, near-uniform encoder output and operation close to the WZ-rate limit is simultaneously possible, whereas in the distributed lossy compression problem, jointly near-uniform outputs is achievable in the interior of the distributed l…
Advancing over up-to-date information theoretic results that assume symmetric correlation models, in this work we consider the problem of lossy binary source coding with side information, where the correlation is expressed by a generic binary asymmetric channel. Specifically, we present an in-depth analysis of rate distortion with side information available to both the encoder and decoder (conventional predictive), as well as the Wyner-Ziv problem for this particular setup. Prompted by our recent results for the Z-channel correlation case, we evaluate the rate loss between the Wyner-Ziv and th…
Indian Institute of Technology, Kharagpur（印度理工学院卡拉格普尔分校）；Azure Software Systems Ltd.（Azure软件系统有限公司）
In this paper, we propose a new feedback-channel-free Distributed Video Coding (DVC) algorithm using Local Rank Transform (LRT). The encoder computes LRT by considering selected neighborhood pixels of Wyner-Ziv frame. The ranks from the modified LRT are merged, and their positions are entropy coded and sent to the decoder. In addition, means of each block of Wyner-Ziv frame are also transmitted to assist motion estimation. Using these measurements, the decoder generates side information (SI) by implementing motion estimation and compensation in LRT domain. An iterative algorithm is executed on…
In the first paper of this two part communication, we solved in a unified framework a variety of two terminal source coding problems with noncooperative encoders, thereby consolidating works of Shannon, Slepian-Wolf, Wyner, Ahlswede-Körner, Wyner-Ziv, Berger {\em et al.} and Berger-Yeung. To achieve such unification we made use of a fundamental principle that dissociates bulk of the analysis from the distortion criterion at hand (if any) and extends the typicality arguments of Shannon and Wyner-Ziv. In this second paper, we generalize the fundamental principle for any number of sources and on …
We study error exponents for source coding with side information. Both achievable exponents and converse bounds are obtained for the following two cases: lossless source coding with coded information (SCCSI) and lossy source coding with full side information (Wyner-Ziv). These results recover and extend several existing results on source-coding error exponents and are tight in some circumstances. Our bounds have a natural interpretation as a two-player game between nature and the code designer, with nature seeking to minimize the exponent and the code designer seeking to maximize it. In the Wy…
This article gives a self-contained analysis of the performance of the Lempel-Ziv compression algorithm on (hidden) Markovian sources. Specifically we include a full proof of the assertion that the compression rate approaches the entropy rate of the chain being compressed.
We consider a source coding problem with a network scenario in mind, and formulate it as a remote vector Gaussian Wyner-Ziv problem under covariance matrix distortions. We define a notion of minimum for two positive-definite matrices based on which we derive an explicit formula for the rate-distortion function (RDF). We then study the special cases and applications of this result. We show that two well-studied source coding problems, i.e. remote vector Gaussian Wyner-Ziv problems with mean-squared error and mutual information constraints are in fact special cases of our results. Finally, we ap…
In this work, the problem of transmitting an i.i.d Gaussian source over an i.i.d Gaussian wiretap channel with an i.i.d Gaussian side information available at the intended receiver is considered. The intended receiver is assumed to have a certain minimum SNR and the eavesdropper is assumed to have a strictly lower SNR, compared to the intended receiver. The objective is to minimize the distortion of source reconstruction at the intended receiver. In this work, it is shown that the source-channel separation coding scheme is optimum in the sense of achieving minimum distortion. Two hybrid digita…
A multiterminal lossy coding problem, which includes various problems such as the Wyner-Ziv problem and the complementary delivery problem as special cases, is considered. It is shown that any point in the achievable rate-distortion region can be attained even if the source statistics are not known.
Compression refers to encoding data using bits, so that the representation uses as few bits as possible. Compression could be lossless: i.e. encoded data can be recovered exactly from its representation) or lossy where the data is compressed more than the lossless case, but can still be recovered to within prespecified distortion metric. In this paper, we prove the optimality of Codelet Parsing, a quasi-linear time algorithm for lossy compression of sequences of bits that are independently and identically distributed (\iid) and Hamming distortion. Codelet Parsing extends the lossless Lempel Zi…
Since the publication of Shannon's theory of one terminal source coding, a number of interesting extensions have been derived by researchers such as Slepian-Wolf, Wyner, Ahlswede-Körner, Wyner-Ziv and Berger-Yeung. Specifically, the achievable rate or rate-distortion region has been described by a first order information-theoretic functional of the source statistics in each of the above cases. At the same time several problems have also remained unsolved. Notable two terminal examples include the joint distortion problem, where both sources are reconstructed under a combined distortion criteri…
An important notion of common information between two random variables is due to Wyner. In this paper, we derive a lower bound on Wyner's common information for continuous random variables. The new bound improves on the only other general lower bound on Wyner's common information, which is the mutual information. We also show that the new lower bound is tight for the so-called "Gaussian channels" case, namely, when the joint distribution of the random variables can be written as the sum of a single underlying random variable and Gaussian noises. We motivate this work from the recent variations…
In this paper, we introduce new lower bounds on the distortion of scalar fixed-rate codes for lossy compression with side information available at the receiver. These bounds are derived by presenting the relevant random variables as a Markov chain and applying generalized data processing inequalities a la Ziv and Zakai. We show that by replacing the logarithmic function with other functions, in the data processing theorem we formulate, we obtain new lower bounds on the distortion of scalar coding with side information at the decoder. The usefulness of these results is demonstrated for uniform …
The Sliding Window Lempel-Ziv (SWLZ) algorithm has been studied from various perspectives in information theory literature. In this paper, we provide a general law which defines the asymptotics of match length for stationary and ergodic zero entropy processes. Moreover, we use this law to choose the match length $L_o$ in the almost sure optimality proof of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. First, through an example of stationary and ergodic processes generated by irrational rotation we establish that for a window size of $n_w$ a compression ratio…
One of the most famous and investigated lossless data-compression scheme is the one introduced by Lempel and Ziv about 40 years ago. This compression scheme is known as "dictionary-based compression" and consists of squeezing an input string by replacing some of its substrings with (shorter) codewords which are actually pointers to a dictionary of phrases built as the string is processed. Surprisingly enough, although many fundamental results are nowadays known about upper bounds on the speed and effectiveness of this compression process and references therein), ``we are not aware of any parsi…
Explicit constructions of polar codes and polar lattices for both lossless and lossy Gray-Wyner problems are studied. Polar codes are employed to extract Wyner's common information of doubly symmetric binary source; polar lattices are then extended to extract that of a pair of Gaussian sources or multiple Gaussian sources. With regard to the discrete sources, the entire best-known region of the lossless Gray-Wyner problem are achieved by specifying the test channels to construct polar codes without time-sharing. As a result, we are able to give an interpretation that the Wyner's common informa…
The problem of side-information scalable (SI-scalable) source coding is considered in this work, where the encoder constructs a progressive description, such that the receiver with high quality side information will be able to truncate the bitstream and reconstruct in the rate distortion sense, while the receiver with low quality side information will have to receive further data in order to decode. We provide inner and outer bounds for general discrete memoryless sources. The achievable region is shown to be tight for the case that either of the decoders requires a lossless reconstruction, as…
The classical Gray and Wyner source coding for a simple network for sources that generate a tuple of multivariate, correlated Gaussian random variables $(Y_1,Y_2)$ is re-examined using the geometric approach of Gaussian random variables, and the weak stochastic realization of correlated Gaussian random variables. New results are: (1) The formulation, methods and algorithms to parametrize all random variables $W : Ω\rightarrow {\mathbb R}^n $ which make the two components of the tuple $(Y_1,Y_2)$ conditionally independent, according to the weak stochastic realization of $(Y_1, Y_2)$. (2) The tr…
We consider lossy source compression of a binary symmetric source using polar codes and the low-complexity successive encoding algorithm. It was recently shown by Arikan that polar codes achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a successive decoding strategy. We show the equivalent result for lossy source compression, i.e., we show that this combination achieves the rate-distortion bound for a binary symmetric source. We further show the optimality of polar codes for various problems including the binary Wyner-Ziv and the binary Gelfand-Pinske…
The pressing need for eficient compression schemes for XML documents has recently been focused on stack computation [6, 9], and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. The celebrated …
We investigate the problem of the successive refinement for Wyner-Ziv coding with degraded side information and obtain a complete characterization of the rate region for the quadratic vector Gaussian case. The achievability part is based on the evaluation of the Tian-Diggavi inner bound that involves Gaussian auxiliary random vectors. For the converse part, a matching outer bound is obtained with the aid of a new extremal inequality. Herein, the proof of this extremal inequality depends on the integration of the monotone path argument and the doubling trick as well as information-estimation re…
In this work, a likelihood encoder is studied in the context of lossy source compression. The analysis of the likelihood encoder is based on a soft-covering lemma. It is demonstrated that the use of a likelihood encoder together with the soft-covering lemma gives alternative achievability proofs for classical source coding problems. The case of the rate-distortion function with side information at the decoder (i.e. the Wyner-Ziv problem) is carefully examined and an application of the likelihood encoder to the multi-terminal source coding inner bound (i.e. the Berger-Tung region) is outlined.
The Sliding Window Lempel-Ziv (SWLZ) algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature. In this paper, we undertake a finer study of these quantities under two different scenarios, i) \emph{zero entropy} sources that are characterized by strong long-term memory, and ii) the processes with weak memory as described through various mixing conditions. For zero entropy sources, a general statement on match length is obtained. It is used in the proof of almost sure optimality of Fixed Shift Variant of Lempel-Ziv…
