共同信息回顾
Common information revisited
摘要：
信息论的主要概念之一是两个信息中的互信息（香农信息论中的两个随机变量或算法信息论中的两个二进制字符串）。$x$和$y$中的互信息测量如果发送方和接收方都预先知道$y$，则$x$的传输可以被简化多少。Gács和Körner给出了一个例子，其中互信息不能作为公共信息（可以从$x$和$y$中轻松提取的第三个信息）。在此基础上，An. Muchnik和A. Romashchenko发现了许多这种类型的其他例子。K.马卡雷切夫和于。Makarychev发现了一个新的证明Gács-Körner结果通过条件独立的随机变量。关于互信息和公共信息之间的差异的问题可以定量地研究：对于给定的$x$和$y$，我们寻找三个消息$a$、$b$、$c$，使得$a$和$c$足以重建$x$，而$b$和$c$足以重建$y$。在本文中：我们的状态和证明（使用产品空间的超压缩性）的Gács-Körner定理的一个定量版本;我们研究之间的权衡$\abs{a}, \abs{b}, \abs{c}$的随机对$(x, y)$，$x$和$y$之间的汉明距离是$\eps n$（我们的界限几乎是紧的）;我们构造“最坏的可能”分布$(x, y)$之间的权衡$\abs{a}, \abs{b}, \abs{c}$。
英文摘要：
One of the main notions of information theory is the notion of mutual information in two messages (two random variables in Shannon information theory or two binary strings in algorithmic information theory). The mutual information in $x$ and $y$ measures how much the transmission of $x$ can be simplified if both the sender and the recipient know $y$ in advance. Gács and Körner gave an example where mutual information cannot be presented as common information (a third message easily extractable from both $x$ and $y$). Then this question was studied in the framework of algorithmic information theory by An. Muchnik and A. Romashchenko who found many other examples of this type. K. Makarychev and Yu. Makarychev found a new proof of Gács--Körner results by means of conditionally independent random variables. The question about the difference between mutual and common information can be studied quantitatively: for a given $x$ and $y$ we look for three messages $a$, $b$, $c$ such that $a$ and $c$ are enough to reconstruct $x$, while $b$ and $c$ are enough to reconstruct $y$. In this paper: We state and prove (using hypercontractivity of product spaces) a quantitative version of Gács--Körner theorem; We study the tradeoff between $\abs{a}, \abs{b}, \abs{c}$ for a random pair $(x, y)$ such that Hamming distance between $x$ and $y$ is $\eps n$ (our bounds are almost tight); We construct "the worst possible" distribution on $(x, y)$ in terms of the tradeoff between $\abs{a}, \abs{b}, \abs{c}$.
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。
