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

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