Structural Properties of Optimal Test Channels for Distributed Source Coding with Decoder Side Information for Multivariate Gaussian Sources with Square-Error Fidelity (2011.10941v1)

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), ${X_t, Y_t}{t=1}^\infty$, $X_t: \Omega \rightarrow {\mathbb R}^{n_x}$, $Y_t: \Omega \rightarrow {\mathbb R}^{n_y}$, with average mean-square error at the decoder, $\frac{1}{n} {\bf E}\sum{t=1}^n||X_t - \widehat{X}t||^2\leq \Delta_X$, 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, $\overline{R}(\Delta_X) \triangleq\inf_{{\cal M}(\Delta_X)} I(X;Z|Y)$, 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{X\Big|\widehat{X}, Y, Z\Big}={\bf E}\Big{X\Big|\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 $\overline{R}(\Delta_X)={R}_{X|Y}(\Delta_X)$.