Characterization of the Gray-Wyner Rate Region for Multivariate Gaussian Sources: Optimality of Gaussian Auxiliary RV

TL;DR

This paper characterizes the Gray-Wyner rate region for multivariate Gaussian sources, demonstrating the optimality of Gaussian auxiliary random variables and providing explicit parametrizations and structural insights.

Contribution

It proves that Gaussian auxiliary variables are optimal for the Gray-Wyner problem with Gaussian sources, and provides explicit parametrizations of the achievable rate region.

Findings

Gaussian auxiliary RVs achieve the Gray-Wyner rate region.

Explicit parametrization of the rate region via conditional covariances.

Simplified rate region when auxiliary RV makes sources conditionally independent.

Abstract

Examined in this paper, is the Gray and Wyner achievable lossy rate region for a tuple of correlated multivariate Gaussian random variables (RVs) $X_1 : \Omega \rightarrow {\mathbb R}^{p_1}$ and $X_2 : \Omega \rightarrow {\mathbb R}^{p_2}$ with respect to square-error distortions at the two decoders. It is shown that among all joint distributions induced by a triple of RVs $(X_1,X_2, W)$, such that $W : \Omega \rightarrow {\mathbb W} $ is the auxiliary RV taking continuous, countable, or finite values, the Gray and Wyner achievable rate region is characterized by jointly Gaussian RVs $(X_1,X_2, W)$ such that $W $ is an $n$-dimensional Gaussian RV. It then follows that the achievable rate region is parametrized by the three conditional covariances $Q_{X_1,X_2|W}, Q_{X_1|W}, Q_{X_2|W}$ of the jointly Gaussian RVs. Furthermore, if the RV $W$ makes $X_1$ and $X_2$ conditionally independent, then the corresponding subset of the achievable rate region, is simpler, and parametrized by only the two conditional covariances $Q_{X_1|W}, Q_{X_2|W}$. The paper also includes the characterization of the Pangloss plane of the Gray-Wyner rate region along with the characterizations of the corresponding rate distortion functions, their test-channel distributions, and structural properties of the realizations which induce these distributions.