Rényi散度测度下的Wyner公共信息
Wyner's Common Information under Rényi Divergence Measures
摘要：
我们研究了Wyner公共信息问题（也创造了分布式源模拟问题）的广义版本。原始的共同信息问题在于理解共同输入到独立处理器的最小速率，以生成联合分布的近似，当用于量化合成分布和目标分布之间的差异的距离度量是归一化的相对熵时。我们的推广涉及到改变的距离措施的非正规化和正规化的Rényi分歧的顺序$α=1+s\in[0,2]$。我们表明，所需的最小速率，以确保由代码引起的分布和目标分布之间的雷尼分歧消失保持相同的Wyner的设置，除了当顺序$α=1+s=0$。这意味着，Wyner的共同信息是相当强大的距离测量的选择。作为用于建立上述结果的证明的副产品，在全变差距离测度下建立了公共信息问题的指数强逆。
英文摘要：
We study a generalized version of Wyner's common information problem (also coined the distributed source simulation problem). The original common information problem consists in understanding the minimum rate of the common input to independent processors to generate an approximation of a joint distribution when the distance measure used to quantify the discrepancy between the synthesized and target distributions is the normalized relative entropy. Our generalization involves changing the distance measure to the unnormalized and normalized Rényi divergences of order $α=1+s\in[0,2]$. We show that the minimum rate needed to ensure the Rényi divergences between the distribution induced by a code and the target distribution vanishes remains the same as the one in Wyner's setting, except when the order $α=1+s=0$. This implies that Wyner's common information is rather robust to the choice of distance measure employed. As a byproduct of the proofs used to the establish the above results, the exponential strong converse for the common information problem under the total variation distance measure is established.
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