Structural Properties of Optimal Test Channels for Distributed Source Coding with Decoder Side Information for Multivariate Gaussian Sources with Square-Error Fidelity
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This paper focuses on the structural properties of test channels, of Wyner's operational information rate distortion function (RDF), R(Δ_X), of a tuple of multivariate correlated, jointly independent and identically distributed Gaussian random variables (RVs), {X_t, Y_t}_t=1^∞, X_t: Ω→ℝ^n_x, Y_t: Ω→ℝ^n_y, with average mean-square error at the decoder, 1/n E∑_t=1^n||X_t - X_t||^2≤Δ_X, when {Y_t}_t=1^∞ is the side information available to the decoder only. We construct optimal test channel realizations, which achieve the informational RDF, R(Δ_X) ≜inf_ M(Δ_X) I(X;Z|Y), where M(Δ_X) is the set of auxiliary RVs Z such that, P_Z|X,Y= P_Z|X, X=f(Y,Z), and E{||X-X||^2}≤Δ_X. We show the fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF, R(Δ_X), satisfy conditional independence, P_X|X, Y, Z= P_X|X,Y= P_X|X, E{X|X, Y, Z}= E{X|X}=X and (2) similarly for the conditional RDF, R_X|Y(Δ_X) ≜inf_ P_X|X,Y: E{||X-X||^2}≤Δ_X I(X; X|Y), when {Y_t}_t=1^∞ is available to both the encoder and decoder, and the equality R(Δ_X)=R_X|Y(Δ_X).
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