Ribbons from Independence Structure: Hypercontractivity, Φ-Mutual Information, and Matrix Φ-Entropy

We study the hypercontractivity ribbon and the $Φ$-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 $Φ$-ribbon described by a simple convex hull of incidence vectors. We also provide a new multipartite generalization version and a $Φ$-mutual information analogue of the Zhang--Yeung inequality, which implies nontrivial points in the hypercontractivity ribbon and the $Φ$-ribbon respectively. Finally, we propose the matrix $Φ$-ribbon based on matrix $Φ$-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.