Joint Rate Distortion Function of a Tuple of Correlated Multivariate Gaussian Sources with Individual Fidelity Criteria

TL;DR

This paper characterizes the joint rate distortion function for correlated Gaussian sources with individual fidelity criteria, providing structural insights, convex optimization formulations, and closed-form solutions verified by semidefinite programming.

Contribution

It introduces new structural properties of optimal test channels for Gaussian sources and derives closed-form RDF expressions, advancing understanding of rate-distortion limits under multiple fidelity constraints.

Findings

Structural properties of optimal test channels for Gaussian sources

Convex optimization formulation via semidefinite programming

Closed-form expressions matching Gray's bounds

Abstract

In this paper we analyze the joint rate distortion function (RDF), for a tuple of correlated sources taking values in abstract alphabet spaces (i.e., continuous) subject to two individual distortion criteria. First, we derive structural properties of the realizations of the reproduction Random Variables (RVs), which induce the corresponding optimal test channel distributions of the joint RDF. Second, we consider a tuple of correlated multivariate jointly Gaussian RVs, $X_1 : \Omega \rightarrow {\mathbb R}^{p_1}, X_2 : \Omega \rightarrow {\mathbb R}^{p_2}$ with two square-error fidelity criteria, and we derive additional structural properties of the optimal realizations, and use these to characterize the RDF as a convex optimization problem with respect to the parameters of the realizations. We show that the computation of the joint RDF can be performed by semidefinite programming. Further, we derive closed-form expressions of the joint RDF, such that Gray's [1] lower bounds hold with equality, and verify their consistency with the semidefinite programming computations.