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

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.