Structural Properties of the Wyner–Ziv Rate Distortion Function: Applications for Multivariate Gaussian Sources

Abstract

The main focus of this paper is the derivation of the structural properties of the test channels of Wyner’s operational information rate distortion function (RDF), R¯(ΔX), for arbitrary abstract sources and, subsequently, the derivation of additional properties for a tuple of multivariate correlated, jointly independent, and identically distributed Gaussian random variables, {Xt,Yt}t=1∞, Xt:Ω→Rnx, Yt:Ω→Rny, with average mean-square error at the decoder and the side information, {Yt}t=1∞, available only at the decoder. For the tuple of multivariate correlated Gaussian sources, we construct optimal test channel realizations which achieve the informational RDF, R¯(ΔX)=▵infM(ΔX)I(X;Z|Y), where M(ΔX) is the set of auxiliary RVs Z such that PZ|X,Y=PZ|X, X^=f(Y,Z), and E{||X−X^||2}≤ΔX. We show the following fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF and satisfy conditional independence, PX|X^,Y,Z=PX|X^,Y=PX|X^,EX|X^,Y,Z=EX|X^=X^. (2) Similarly, for the conditional RDF, RX|Y(ΔX), when the side information is available to both the encoder and the decoder, we show the equality R¯(ΔX)=RX|Y(ΔX). (3) We derive the water-filling solution for RX|Y(ΔX).

Keywords

• Wyner’s side information
• multivariate Gaussian sources
• test channel distributions