Ribbons from Independence Structure: Hypercontractivity, -Mutual Information, and Matrix -Entropy
arXiv:2601.18516
Abstract
We study the hypercontractivity ribbon and the -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 -ribbon described by a simple convex hull of incidence vectors. We also provide a new multipartite generalization version and a -mutual information analogue of the Zhang--Yeung inequality, which implies nontrivial points in the hypercontractivity ribbon and the -ribbon respectively. Finally, we propose the matrix -ribbon based on matrix -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.
17 pages