A Realization Approach to Lossy Network Compression of a Tuple of Correlated Multivariate Gaussian RVs

Abstract

Examined in this paper is the Gray and Wyner source coding for a simple network of correlated multivariate Gaussian random variables, Y1:Ω→Rp1 and Y2:Ω→Rp2. The network consists of an encoder that produces two private rates R1 and R2, and a common rate R0, and two decoders, where decoder 1 receives rates (R1,R0) and reproduces Y1 by Y^1, and decoder 2 receives rates (R2,R0) and reproduces Y2 by Y^2, with mean-square error distortions E||Yi−Y^i||Rpi2≤Δi∈[0,∞],i=1,2. Use is made of the weak stochastic realization and the geometric approach of such random variables to derive test channel distributions, which characterize the rates that lie on the Gray and Wyner rate region. Specific new results include: (1) A proof that, among all continuous or finite-valued random variables, W:Ω→W, Wyner’s common information, C(Y1,Y2)=infPY1,Y2,W:PY1,Y2|W=PY1|WPY2|WI(Y1,Y2;W), is achieved by a Gaussian random variable, W:Ω→Rn of minimum dimension n, which makes the two components of the tuple (Y1,Y2) conditionally independent according to the weak stochastic realization of (Y1,Y2), and a the formula C(Y1,Y2)=12∑j=1nln1+dj1−dj, where di∈(0,1),i=1,…,n are the canonical correlation coefficients of the correlated parts of Y1 and Y2, and a realization of (Y1,Y2,W) which achieves this. (2) The parameterization of rates that lie on the Gray and Wyner rate region, and several of its subsets. The discussion is largely self-contained and proceeds from first principles, while connections to prior literature is discussed.