Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks
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The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) X_1 : Ω→R^p_1 and X_2 : Ω→R^p_2, with respect to square-error distortion at the two decoders is re-examined using (1) Hotelling's geometric approach of Gaussian RVs-the canonical variable form, and (2) van Putten's and van Schuppen's parametrization of joint distributions P_X_1, X_2, W by Gaussian RVs W : Ω→R^n which make (X_1,X_2) conditionally independent, and the weak stochastic realization of (X_1, X_2). Item (2) is used to parametrize the lossy rate region of the Gray and Wyner source coding problem for joint decoding with mean-square error distortions E{||X_i-X̂_i||_R^p_i^2 }≤Δ_i ∈ [0,∞], i=1,2, by the covariance matrix of RV W. From this then follows Wyner's common information C_W(X_1,X_2) (information definition) is achieved by W with identity covariance matrix, while a formula for Wyner's lossy common information (operational definition) is derived, given by C_WL(X_1,X_2)=C_W(X_1,X_2) = 1/2∑_j=1^n ln( 1+d_j/1-d_j), for the distortion region 0≤Δ_1 ≤∑_j=1^n(1-d_j), 0≤Δ_2 ≤∑_j=1^n(1-d_j), and where 1 > d_1 ≥ d_2 ≥...≥ d_n>0 in (0,1) are the canonical correlation coefficients computed from the canonical variable form of the tuple (X_1, X_2). The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.
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