This paper focuses on the structural properties of test channels, of Wyner's operational information rate distortion function (RDF), $\overline{R}(\Delta_X)$, of a tuple of multivariate correlated, jointly independent and identically distributed Gaussian random variables (RVs), $\{X_t, Y_t\}_{t=1}^\infty$, $X_t: \Omega \rightarrow {\mathbb R}^{n_x}$, $Y_t: \Omega \rightarrow {\mathbb R}^{n_y}$, with average mean-square error at the decoder, $\frac{1}{n} {\bf E}\sum_{t=1}^n||X_t - \widehat{X}_t||^2\leq \Delta_X$, when $\{Y_t\}_{t=1}^\infty$ is the side information available to the decoder only. We construct optimal test channel realizations, which achieve the informational RDF, $\overline{R}(\Delta_X) \triangleq\inf_{{\cal M}(\Delta_X)} I(X;Z|Y)$, where ${\cal M}(\Delta_X)$ is the set of auxiliary RVs $Z$ such that, ${\bf P}_{Z|X,Y}={\bf P}_{Z|X}$, $\widehat{X}=f(Y,Z)$, and ${\bf E}\{||X-\widehat{X}||^2\}\leq \Delta_X$. We show the fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF, $\overline{R}(\Delta_X)$, satisfy conditional independence, $ {\bf P}_{X|\widehat{X}, Y, Z}={\bf P}_{X|\widehat{X},Y}={\bf P}_{X|\widehat{X}}, \hspace{.2in} {\bf E}\Big\{X\Big|\widehat{X}, Y, Z\Big\}={\bf E}\Big\{X\Big|\widehat{X}\Big\}=\widehat{X} $ and (2) similarly for the conditional RDF, ${R}_{X|Y}(\Delta_X) \triangleq \inf_{{\bf P}_{\widehat{X}|X,Y}:{\bf E}\{||X-\widehat{X}||^2\} \leq \Delta_X} I(X; \widehat{X}|Y)$, when $\{Y_t\}_{t=1}^\infty$ is available to both the encoder and decoder, and the equality $\overline{R}(\Delta_X)={R}_{X|Y}(\Delta_X)$.

中文翻译：

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具有平方误差保真度的多元高斯源的带有解码器边信息的分布式源编码的最佳测试通道的结构特性

本文重点介绍测试通道的结构特性，Wyner的操作信息率失真函数（RDF），$ \ overline {R}（\ Delta_X）$，多元相关，共同独立且均布的高斯随机变量元组（ RVs），$ \ {X_t，Y_t \} _ {t = 1} ^ \ infty $，$ X_t：\ Omega \ rightarrow {\ mathbb R} ^ {n_x} $，$ Y_t：\ Omega \ rightarrow {\ mathbb R} ^ {n_y} $，解码器处的均方误差为$ \ frac {1} {n} {\ bf E} \ sum_ {t = 1} ^ n || X_t-\ widehat {X} _t || ^ 2 \ leq \ Delta_X $，当$ \ {Y_t \} _ {t = 1} ^ \ infty $仅是解码器可用的辅助信息时。我们构造了最佳的测试通道实现，该实现实现了信息RDF：$ \ overline {R}（\ Delta_X）\ triangleq \ inf _ {{\ cal M}（\ Delta_X）} I（X; Z | Y）$，其中$ {\ cal M}（\ Delta_X）$是辅助RV $ Z $的集合，使得$ {\ bf P} _ {Z | X，Y} = {\ bf P} _ {Z | X} $，$ \ widehat {X} = f（Y，Z）$和$ {\ bf E} \ {|| X- \ widehat {X} | | ^ 2 \} \ leq \ Delta_X $。我们显示了基本的结构特性：（1）实现RDF的最佳测试通道实现$ \ overline {R}（\ Delta_X）$满足条件独立性$ {\ bf P} _ {X | \ widehat {X} ，Y，Z} = {\ bf P} _ {X | \ widehat {X}，Y} = {\ bf P} _ {X | \ widehat {X}}，\ hspace {.2in} {\ bf E } \ Big \ {X \ Big | \ widehat {X}，Y，Z \ Big \} = {\ bf E} \ Big \ {X \ Big | \ widehat {X} \ Big \} = \ widehat {X } $和（2）类似，对于条件RDF，$ {R} _ {X | Y}（\ Delta_X）\ triangleq \ inf _ {{\ bf P} _ {\ widehat {X} | X，Y}：{ \ bf E} \ {|| X- \ widehat {X} || ^^ 2 \} \ leq \ Delta_X} I（X; \ widehat {X} | Y）$，当$ \ {Y_t \} _ {t = 1} ^ \ infty $可用于编码器和解码器，并且等式$ \ overline {R}（\ Delta_X）= {R} _ {X | Y}（\ Delta_X）$。