Ribbons from Independence Structure: Hypercontractivity, $\Phi$-Mutual Information, and Matrix $\Phi$-Entropy

TL;DR

This paper investigates the hypercontractivity and $\Phi$-ribbons for distributions with independence structures, providing bounds, generalizations, and new inequalities, including a matrix $\Phi$-ribbon with tensorization and data processing properties.

Contribution

It introduces explicit bounds for $\Phi$-ribbons in independence structures, generalizes to multipartite cases, and develops a matrix $\Phi$-ribbon with key properties and exact constants.

Findings

Derived tight bounds for $\Phi$-ribbons in basic regimes

Provided a new multipartite generalization of the $\Phi$-ribbon

Established properties and exact constants for the matrix $\Phi$-ribbon

Abstract

We study the hypercontractivity ribbon and the $\Phi$-ribbon for joint distributions that obey a given independence structure, obtaining tight bounds in some basic regimes. For general independence structures, modeled as a hypergraph whose hyperedges specify mutually independent subcollections of random variables, we provide an explicit inner bound on the $\Phi$-ribbon described by a simple convex hull of incidence vectors. We also provide a new multipartite generalization version and a $\Phi$-mutual information analogue of the Zhang--Yeung inequality, which implies nontrivial points in the hypercontractivity ribbon and the $\Phi$-ribbon respectively. Finally, we propose the matrix $\Phi$-ribbon based on matrix $\Phi$-entropy and establish the tensorization and data processing properties, together with the calculation of an exact matrix SDPI constant for the doubly symmetric binary source.