关于可加性与张量化的对偶性
On the Duality of Additivity and Tensorization
摘要：
如果一个函数与互信息类似，在$n$ i.i.d.上计算时扩展因子为$n$，则称该函数是可加的。重复的源或通道。另一方面，如果一个函数在i.i.d.上计算时保持不变，则称该函数满足张量化性质。重复可加速率域是网络信息论中的一个重要概念，它是网络容量域或容量上界。相关性的张量化测量也已经在分布式信源和信道编码问题以及分布式仿真问题中找到了应用。在我们的工作之前，只有两种相关性的度量，即超收缩带和最大相关性（及其衍生物），已知具有张量化性质。在本文中，我们提供了一个一般的框架，以获得一个区域的张量化性质的任何添加剂率区域。我们观察到超收缩带确实来自于Gray-Wyner信源编码问题的速率区域的对偶，并将其推广到多部情况。然后，我们定义了其他措施的相关性与类似的性质，从其他信源编码问题。我们还介绍了我们的结果的一些应用。
英文摘要：
A function is said to be additive if, similar to mutual information, expands by a factor of $n$, when evaluated on $n$ i.i.d. repetitions of a source or channel. On the other hand, a function is said to satisfy the tensorization property if it remains unchanged when evaluated on i.i.d. repetitions. Additive rate regions are of fundamental importance in network information theory, serving as capacity regions or upper bounds thereof. Tensorizing measures of correlation have also found applications in distributed source and channel coding problems as well as the distribution simulation problem. Prior to our work only two measures of correlation, namely the hypercontractivity ribbon and maximal correlation (and their derivatives), were known to have the tensorization property. In this paper, we provide a general framework to obtain a region with the tensorization property from any additive rate region. We observe that hypercontractivity ribbon indeed comes from the dual of the rate region of the Gray-Wyner source coding problem, and generalize it to the multipartite case. Then we define other measures of correlation with similar properties from other source coding problems. We also present some applications of our results.
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