关于精确信息与$\infty$ -Rényi公共信息
On Exact and $\infty$-Rényi Common Informations
摘要：
最近，Kumar，Li，El Gamal（KLE）和本文作者分别给出了Wyner公共信息\textemdash exact和Rényi公共信息\textemdash的两个推广.这类公共信息问题涉及确定两个独立处理器的公共输入的最小速率，以精确或近似地生成目标联合分布。对于精确公共信息问题，需要精确生成目标分布;对于Wyner公共信息问题和$α$ -Rényi公共信息问题，分别用相对熵和Rényi散度来量化合成分布与目标分布之间的差异.精确的公共信息大于或等于Wyner的公共信息。然而，它是迄今未知的，前者是否严格大于后者的一些联合分布。本文首先建立了精确信息量与$\infty$ -Rényi公共信息量之间的等价性，然后给出了这两个量的单字母上下界.对于双对称二进制源，我们证明了上下界是一致的，这意味着对于这样的源，精确和$\infty$ -Rényi公共信息是完全特征化的。有趣的是，我们观察到，对于这样的来源，这两个共同的信息是严格大于Wyner的。这回答了KLE提出的一个开放问题。此外，我们扩展Wyner的，$\infty$ -Rényi，和确切的共同信息源与可数无限或连续字母，包括高斯源。
英文摘要：
Recently, two extensions of Wyner's common information\textemdash exact and Rényi common informations\textemdash were introduced respectively by Kumar, Li, and El Gamal (KLE), and the present authors. The class of common information problems involves determining the minimum rate of the common input to two independent processors needed to exactly or approximately generate a target joint distribution. For the exact common information problem, exact generation of the target distribution is required, while for Wyner's and $α$-Rényi common informations, the relative entropy and Rényi divergence with order $α$ were respectively used to quantify the discrepancy between the synthesized and target distributions. The exact common information is larger than or equal to Wyner's common information. However, it was hitherto unknown whether the former is strictly larger than the latter for some joint distributions. In this paper, we first establish the equivalence between the exact and $\infty$-Rényi common informations, and then provide single-letter upper and lower bounds for these two quantities. For doubly symmetric binary sources, we show that the upper and lower bounds coincide, which implies that for such sources, the exact and $\infty$-Rényi common informations are completely characterized. Interestingly, we observe that for such sources, these two common informations are strictly larger than Wyner's. This answers an open problem posed by KLE. Furthermore, we extend Wyner's, $\infty$-Rényi, and exact common informations to sources with countably infinite or continuous alphabets, including Gaussian sources.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
