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

Abstract

Examined in this paper, is the Gray and Wyner achievable lossy rate region for a tuple of correlated multivariate Gaussian random variables (RVs) $X_1:\Omega \to {\mathbb{R}}^{p_1}$ and $X_2:\Omega \to {\mathbb{R}}^{p_2}$ 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 $W:\Omega \to \mathbb{W}$ 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.