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首页> 外文会议>IEEE International Symposium on Information Theory >Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks
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Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks

机译：多元高斯随机变量的条件独立性表征和弱实现：在网络中的应用

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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) ${X_1}:Omega o {mathbb{R}^{{p_1}}}$ and ${X_2}:Omega o {mathbb{R}^{{p_2}}}$, with respect to square-error distortion at the two decoders is reexamined 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 PX1,X2,W by Gaussian RVs $W:Omega o {mathbb{R}^n}$ which make (X1,X2) conditionally independent, and the weak stochastic realization of (X1,X2). 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 ${mathbf{E}}left{ {left| {{X_i} - {{hat X}_i}} ight|_{{mathbb{R}^p}i}^2} ight} leq {Delta _i} in [0,infty ]$,i=1,2, by the covariance matrix of RV W. From this then follows Wyner’s common information CW(X1,X2) (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 ${C_{WL}}left( {{X_1},{X_2}} ight) = {C_W}left( {{X_1},{X_2}} ight) = rac{1}{2}sumolimits_{j = 1}^n {ln } left( {rac{{1 + {d_j}}}{{1 - {d_j}}}} ight)$, for the distortion region 0 ≤ ∆1 ≤ n(1−d1), 0 ≤ ∆2 ≤ n(1−d1), and where 1 > d1 ≥ d2 ≥ … ≥ dn > 0 in (0,1) are the canonical correlation coefficients computed from the canonical variable form of the tuple (X1,X2)The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.
机译：Gray和Wyner有损源代码，用于为生成生成联合高斯随机变量（RV）$ {X_1}：\ Omega \ to {\ mathbb {R} ^ {{p_1}}} $和$的元组的简单网络进行编码对于{X_2}：\ Omega \ to {\ mathbb {R} ^ {{p_2}}} $，使用（1）Hotelling的高斯RVs几何方法（规范变量）重新检查两个解码器的平方误差失真（2）van Putten和van Schuppen对联合分布P的参数化 X1 ， X2 ， W 由高斯RV $ W：\ Omega \至{\ mathbb {R} ^ n} $ 1 ，X 2 ）有条件地独立，并且（X的弱随机实现 1 ，X 2 ）。项（2）用于对Gray和Wyner源编码问题的有损率区域进行参数化，以进行均方误差失真$ {\ mathbf {E}} \ left \ {{\ left \ | {{X_i}-{{\ hat X} _i}} \ right \ | _ {{\ mathbb {R} ^ p} i} ^ 2} \ right \} \ leq {\ Delta _i} \ in [0， \ infty] $，i = 1,2，通过RV W的协方差矩阵。然后从中得出Wyner的公共信息C W （X 1 ，X 2 ）（信息定义）是由W使用单位协方差矩阵实现的，而Wyner有损公共信息的公式（操作定义）则由$ {C_ {WL}} \ left（{{X_1}，{X_2}}给出\ right）= {C_W} \ left（{{X_1}，{X_2}} \ right）= \ frac {1} {2} \ sum \ nolimits_ {j = 1} ^ n {\ ln} \ left（{ \ frac {{1 + {d_j}}} {{1-{d_j}}}} \ right）$，对于失真区域0≤∆ 1 ≤n（1-d 1 ），0≤∆ 2 ≤n（1-d 1 ），其中1> d 1 ≥d 2 ≥…≥d n >（0,1）中的0是从元组（X 1 ，X 2 这些方法对于将条件独立性作为约束的多用户通信的其他问题至关重要。

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