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Register a new userStructural Properties of Optimal Test Channels for Distributed Source Coding with Decoder Side Information for Multivariate Gaussian Sources with Square-Error Fidelity
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Added by
Michail Gkagkos
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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This paper focuses on the structural properties of test channels, of Wyners operational information rate distortion function (RDF), $overline{R}(Delta_X)$, of a tuple of multivariate correlated, jointly independent and identically distributed Gaussian random variables (RVs), ${X_t, Y_t}_{t=1}^infty$, $X_t: Omega rightarrow {mathbb R}^{n_x}$, $Y_t: Omega rightarrow {mathbb R}^{n_y}$, with average mean-square error at the decoder, $frac{1}{n} {bf E}sum_{t=1}^n||X_t - widehat{X}_t||^2leq Delta_X$, when ${Y_t}_{t=1}^infty$ is the side information available to the decoder only. We construct optimal test channel realizations, which achieve the informational RDF, $overline{R}(Delta_X) triangleqinf_{{cal M}(Delta_X)} I(X;Z|Y)$, where ${cal M}(Delta_X)$ is the set of auxiliary RVs $Z$ such that, ${bf P}_{Z|X,Y}={bf P}_{Z|X}$, $widehat{X}=f(Y,Z)$, and ${bf E}{||X-widehat{X}||^2}leq Delta_X$. We show the fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF, $overline{R}(Delta_X)$, satisfy conditional independence, $ {bf P}_{X|widehat{X}, Y, Z}={bf P}_{X|widehat{X},Y}={bf P}_{X|widehat{X}}, hspace{.2in} {bf E}Big{XBig|widehat{X}, Y, ZBig}={bf E}Big{XBig|widehat{X}Big}=widehat{X} $ and (2) similarly for the conditional RDF, ${R}_{X|Y}(Delta_X) triangleq inf_{{bf P}_{widehat{X}|X,Y}:{bf E}{||X-widehat{X}||^2} leq Delta_X} I(X; widehat{X}|Y)$, when ${Y_t}_{t=1}^infty$ is available to both the encoder and decoder, and the equality $overline{R}(Delta_X)={R}_{X|Y}(Delta_X)$.
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In this paper, we analyze the operational information rate distortion function (RDF) ${R}_{S;Z|Y}(Delta_X)$, introduced by Draper and Wornell, for a triple of jointly independent and identically distributed, multivariate Gaussian random variables (RVs), $(X^n, S^n, Y^n)= {(X_{t}, S_t, Y_{t}): t=1,2, ldots,n}$, where $X^n$ is the source, $S^n$ is a measurement of $X^n$, available to the encoder, $Y^n$ is side information available to the decoder only, $Z^n$ is the auxiliary RV available to the decoder, with respect to the square-error fidelity, between the source $X^n$ and its reconstruction $widehat{X}^n$. We also analyze the RDF ${R}_{S;widehat{X}|Y}(Delta_X)$ that corresponds to the above set up, when side information $Y^n$ is available to the encoder and decoder. The main results include, (1) Structural properties of test channel realizations that induce distributions, which achieve the two RDFs, (2) Water-filling solutions of the two RDFs, based on parallel channel realizations of test channels, (3) A proof of equality ${R}_{S;Z|Y}(Delta_X) = {R}_{S;widehat{X}|Y}(Delta_X)$, i.e., side information $Y^n$ at both the encoder and decoder does not incur smaller compression, and (4) Relations to other RDFs, as degenerate cases, which show past literature, contain oversights related to the optimal test channel realizations and value of the RDF ${R}_{S;Z|Y}(Delta_X)$.
Index coding is a source coding problem in which a broadcaster seeks to meet the different demands of several users, each of whom is assumed to have some prior information on the data held by the sender. If the sender knows its clients requests and their side-information sets, then the number of packet transmissions required to satisfy all users demands can be greatly reduced if the data is encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance of the index coding with side-information (ICSI) problem. The encoding function is called the index code of the instance, and the number of transmissions employed by the code is referred to as its length. The main ICSI problem is to determine the optimal length of an index code for and instance. As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. Two interesting generalizations of the problem that have appeared in the literature are the subject of this work. The first of these is the case of index coding with coded side information, in which linear combinations of the source data are both requested by and held as users side-information. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise. In this paper we characterize the optimal length of a scalar or vector linear index code with coded side information (ICCSI) over a finite field in terms of a generalized min-rank and give bounds on this number based on constructions of random codes for an arbitrary instance. We furthermore consider the length of an optimal error correcting code for an instance of the ICCSI problem and obtain bounds on this number, both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors.
An encoder, subject to a rate constraint, wishes to describe a Gaussian source under squared error distortion. The decoder, besides receiving the encoders description, also observes side information consisting of uncompressed source symbol subject to slow fading and noise. The decoder knows the fading realization but the encoder knows only its distribution. The rate-distortion function that simultaneously satisfies the distortion constraints for all fading states was derived by Heegard and Berger. A layered encoding strategy is considered in which each codeword layer targets a given fading state. When the side-information channel has two discrete fading states, the expected distortion is minimized by optimally allocating the encoding rate between the two codeword layers. For multiple fading states, the minimum expected distortion is formulated as the solution of a convex optimization problem with linearly many variables and constraints. Through a limiting process on the primal and dual solutions, it is shown that single-layer rate allocation is optimal when the fading probability density function is continuous and quasiconcave (e.g., Rayleigh, Rician, Nakagami, and log-normal). In particular, under Rayleigh fading, the optimal single codeword layer targets the least favorable state as if the side information was absent.
This letter investigates a new class of index coding problems. One sender broadcasts packets to multiple users, each desiring a subset, by exploiting prior knowledge of linear combinations of packets. We refer to this class of problems as index coding with coded side-information. Our aim is to characterize the minimum index code length that the sender needs to transmit to simultaneously satisfy all user requests. We show that the optimal binary vector index code length is equal to the minimum rank (minrank) of a matrix whose elements consist of the sets of desired packet indices and side- information encoding matrices. This is the natural extension of matrix minrank in the presence of coded side information. Using the derived expression, we propose a greedy randomized algorithm to minimize the rank of the derived matrix.
A rateless code-i.e., a rate-compatible family of codes-has the property that codewords of the higher rate codes are prefixes of those of the lower rate ones. A perfect family of such codes is one in which each of the codes in the family is capacity-achieving. We show by construction that perfect rateless codes with low-complexity decoding algorithms exist for additive white Gaussian noise channels. Our construction involves the use of layered encoding and successive decoding, together with repetition using time-varying layer weights. As an illustration of our framework, we design a practical three-rate code family. We further construct rich sets of near-perfect rateless codes within our architecture that require either significantly fewer layers or lower complexity than their perfect counterparts. Variations of the basic construction are also developed, including one for time-varying channels in which there is no a priori stochastic model.
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