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

Examined in this paper is the Gray and Wyner source coding for a simple network of correlated multivariate Gaussian random variables, [Formula: see text] and [Formula: see text]. The network consists of an encoder that produces two private rates [Formula: see text] and [Formula: see text] , and a common rate [Formula: see text] , and two decoders, where decoder 1 receives rates [Formula: see text] and reproduces [Formula: see text] by [Formula: see text] , and decoder 2 receives rates [Formula: see text] and reproduces [Formula: see text] by [Formula: see text] , with mean-square error distortions [Formula: see text]. 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, [Formula: see text] , Wyner’s common information, [Formula: see text] , is achieved by a Gaussian random variable, [Formula: see text] of minimum dimension n, which makes the two components of the tuple [Formula: see text] conditionally independent according to the weak stochastic realization of [Formula: see text] , and a the formula [Formula: see text] where [Formula: see text] are the canonical correlation coefficients of the correlated parts of [Formula: see text] and [Formula: see text] , and a realization of [Formula: see text] 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.