Entropy factorization via curvature

TL;DR

This paper introduces a geometric framework based on curvature to analyze entropy factorization, leading to new and simplified proofs of key inequalities in probability and statistical physics.

Contribution

It develops a novel curvature-based approach for entropy factorization, unifying and extending several fundamental inequalities across continuous and discrete spaces.

Findings

Provides a unified geometric method for entropy inequalities

Simplifies proofs of known inequalities like log-Sobolev and Brascamp-Lieb

Derives new inequalities for particle systems and permutation measures

Abstract

We develop a new framework for establishing approximate factorization of entropy on arbitrary probability spaces, using a geometric notion known as non-negative sectional curvature. The resulting estimates are equivalent to entropy subadditivity and generalized Brascamp-Lieb inequalities, and provide a sharp modified log-Sobolev inequality for the Gibbs sampler of several particle systems in both continuous and discrete settings. The method allows us to obtain simple proofs of known results, as well as some new inequalities. We illustrate this through various applications, including discrete Gaussian free fields on arbitrary networks, the down-up walk on uniform $n$-sets, the uniform measure over permutations, and the uniform measure on the unit sphere in $\R^n$. Our method also yields a simple, coupling-based proof of the celebrated logarithmic Sobolev inequality for Langevin diffusions in a convex potential, which is one of the most emblematic applications of the Bakry-\'Emery criterion.