论文标题
多元高斯随机变量的条件独立性和弱实现的表征：网络应用
Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks
论文作者
论文摘要
为简单网络编码的灰色和Wyner损失源，用于生成一个共同高斯随机变量（RVS）$ x_1：ω\ rightarrow {\ Mathbb r}^{p_1} $和$ x_2：ω\ rightarrow {使用（1）使用（1）Hotelling的Gaussian RVS-The Canonical变量形式的几何方法对解码器进行重新审查，以及（2）Van Putten's和Van Schuppen的联合分布参数$ {\ bf p} _ {\ bf p} _ {x_1，x_1，x_2，x_2，w} $ a $ w： $（x_1，x_2）$有条件独立，$（x_1，x_2）$的弱随机实现。项目（2）用于参数灰色和Wyner源编码问题的损耗率区域，用于与均方根误差扭曲$ {\ bf e} \ big \ \ {|| x_i- \ hat {x_i- \ hat {x} {x} {x} {x} _i _i || \ in [0，\ infty]，i = 1,2 $，由RV $ W $的协方差矩阵。 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 $C_{WL}(X_1,X_2)=C_W(X_1,X_2) = \ frac {1} {2} \ sum_ {j = 1}^n \ ln \左边（ \ frac {1+d_j} {1-d_j} \ right），$对于失真区域$ 0 \leqΔ_1\ leq \ sum_ {j = 1}^n（1-d_j）$，$ 0 \leqΔ_2\ leq leq sum_ { $（0,1）$是{\ em从元组$（x_1，x_2）$计算的规范相关系数}。这些方法对多用户交流的其他问题至关重要，在这种情况下，将有条件的独立性作为约束。
The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) $X_1 : Ω\rightarrow {\mathbb R}^{p_1}$ and $X_2 : Ω\rightarrow {\mathbb R}^{p_2}$, 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 ${\bf P}_{X_1, X_2, W}$ by Gaussian RVs $W : Ω\rightarrow {\mathbb 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 ${\bf E}\big\{||X_i-\hat{X}_i||_{{\mathbb R}^{p_i}}^2 \big\}\leq Δ_i \in [0,\infty], i=1,2$, 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 $C_{WL}(X_1,X_2)=C_W(X_1,X_2) = \frac{1}{2} \sum_{j=1}^n \ln \left( \frac{1+d_j}{1-d_j} \right),$ for the distortion region $ 0\leq Δ_1 \leq \sum_{j=1}^n(1-d_j)$, $0\leq Δ_2 \leq \sum_{j=1}^n(1-d_j)$, and where $1 > d_1 \geq d_2 \geq \ldots \geq d_n>0$ in $(0,1)$ are {\em 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.
