On the Duality of Additivity and Tensorization

TL;DR

This paper explores the relationship between additivity and tensorization in information measures, providing a framework to derive tensorization properties from additive rate regions and extending known measures to multipartite cases.

Contribution

It introduces a general method to obtain tensorization properties from additive rate regions and generalizes hypercontractivity and correlation measures to multipartite scenarios.

Findings

Hypercontractivity ribbon is dual to Gray-Wyner rate region.

General framework for tensorization from additive regions.

Extension of correlation measures to multipartite cases.

Abstract

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.