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تسجيل مستخدم جديدStructural Properties of Optimal Test Channels for Gaussian Multivariate Partially Observable Distributed Sources
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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 ${R}_{S;Z|Y}(Delta_X)$.
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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 Gaussia
n 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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