Structural Properties of Optimal Test Channels for Gaussian Multivariate Partially Observable Distributed Sources

In this paper, we analyze the operational information rate distortion function (RDF) ${R}_{S;Z|Y}(\Delta_X)$, introduced by Draper and Wornell \cite{draper-wornell2004}, for a triple of jointly independent and identically distributed, multivariate Gaussian random variables (RVs), $(X^n, S^n, Y^n)= \{(X_{t}, S_t, Y_{t}): t=1,2, \ldots,n\}$, where $X^n$ is the source, $S^n$ is a measurement of $X^n$, available to the encoder, $Y^n$ is side information available to the decoder only, $Z^n$ is the auxiliary RV available to the decoder, with respect to the square-error fidelity, between the source $X^n$ and its reconstruction $\widehat{X}^n$. We also analyze the RDF ${R}_{S;\widehat{X}|Y}(\Delta_X)$ that corresponds to the above set up, when side information $Y^n$ is available to the encoder and decoder. The main results include, (1) Structural properties of test channel realizations that induce distributions, which achieve the two RDFs, (2) Water-filling solutions of the two RDFs, based on parallel channel realizations of test channels, (3) A proof of equality ${R}_{S;Z|Y}(\Delta_X) = {R}_{S;\widehat{X}|Y}(\Delta_X)$, i.e., side information $Y^n$ at both the encoder and decoder does not incur smaller compression, and (4) Relations to other RDFs, as degenerate cases, which show past literature, contain oversights related to the optimal test channel realizations and value of the RDF ${R}_{S;Z|Y}(\Delta_X)$.