Characterization of the Gray-Wyner Rate Region for Multivariate Gaussian Sources: Optimality of Gaussian Auxiliary RV
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Examined in this paper, is the Gray and Wyner achievable lossy rate region for a tuple of correlated multivariate Gaussian random variables (RVs) X_1 : Ω→ℝ^p_1 and X_2 : Ω→ℝ^p_2 with respect to square-error distortions at the two decoders. It is shown that among all joint distributions induced by a triple of RVs (X_1,X_2, W), such that W : Ω→𝕎 is the auxiliary RV taking continuous, countable, or finite values, the Gray and Wyner achievable rate region is characterized by jointly Gaussian RVs (X_1,X_2, W) such that W is an n-dimensional Gaussian RV. It then follows that the achievable rate region is parametrized by the three conditional covariances Q_X_1,X_2|W, Q_X_1|W, Q_X_2|W of the jointly Gaussian RVs. Furthermore, if the RV W makes X_1 and X_2 conditionally independent, then the corresponding subset of the achievable rate region, is simpler, and parametrized by only the two conditional covariances Q_X_1|W, Q_X_2|W. The paper also includes the characterization of the Pangloss plane of the Gray-Wyner rate region along with the characterizations of the corresponding rate distortion functions, their test-channel distributions, and structural properties of the realizations which induce these distributions.
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