Structural Properties of Optimal Test Channels for Distributed Source Coding with Decoder Side Information for Multivariate Gaussian Sources with Square-Error Fidelity

Abstract

This paper focuses on the structural properties of test channels, of Wyner's 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), $\{X_t,Y_t{\}}_{t=1}^{\infty }$, $X_t:\Omega \to {\mathbb{R}}^{n_x}$, $Y_t:\Omega \to {\mathbb{R}}^{n_y}$, with average mean-square error at the decoder, $\frac{1}{n}\mathbf{E}{\sum }_{t=1}^n||X_t-{\hat{X}}_t|{|}^2\le {\Delta }_X$, 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, $\overline{R}({\Delta }_X)\triangleq {\inf }_{\mathcal{M}({\Delta }_X)}I(X;Z|Y)$, where $\mathcal{M}({\Delta }_X)$ is the set of auxiliary RVs $Z$ such that, ${\mathbf{P}}_{Z|X,Y}={\mathbf{P}}_{Z|X}$, $\hat{X}=f(Y,Z)$, and $\mathbf{E}\{||X-\hat{X}|{|}^2\}\le {\Delta }_X$. We show the fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF, $\overline{R}({\Delta }_X)$, satisfy conditional independence, ${\mathbf{P}}_{X|\hat{X},Y,Z}={\mathbf{P}}_{X|\hat{X},Y}={\mathbf{P}}_{X|\hat{X}},\mathbf{E}\left\{X\right|\hat{X},Y,Z\}=\mathbf{E}\{X\left|\hat{X}\right\}=\hat{X}$ and (2) similarly for the conditional RDF, $R_{X|Y}({\Delta }_X)\triangleq {\inf }_{{\mathbf{P}}_{\hat{X}|X,Y}:\mathbf{E}\{||X-\hat{X}|{|}^2\}\le {\Delta }_X}I(X;\hat{X}|Y)$, when $\{Y_t{\}}_{t=1}^{\infty }$ is available to both the encoder and decoder, and the equality $\overline{R}({\Delta }_X)=R_{X|Y}({\Delta }_X)$.