Ribbons from Independence Structure: Hypercontractivity, $Φ$-Mutual Information, and Matrix $Φ$-Entropy
Chenyu Wang, Amin Gohari
TL;DR
This work characterizes hypercontractivity and $\Phi$-ribbons under partial independence constraints by modeling independence with a hypergraph and deriving sharp inner bounds. It provides a concrete convex-hull description for general independence structures and extends non-Shannon-type inequalities to the $\Phi$-information setting, including multipartite Zhang–Yeung-type bounds. The paper further introduces a matrix $\Phi$-ribbon using matrix $\Phi$-entropy, proving tensorization and data-processing properties and obtaining the exact DSPSI constant for DSBS, thereby connecting classical and quantum-information perspectives. Overall, the results offer precise region characterizations and new tools for analyzing information-processing under partial independence, with potential implications for information inequalities and data-processing analyses in complex networks.
Abstract
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.