Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures

TL;DR

This paper proves the concentration of multioverlaps in high-dimensional log-concave Gibbs measures, leading to simplified representations of the asymptotic measures and variable decoupling, with potential applications in machine learning and inference.

Contribution

It establishes strong replica symmetry results for high-dimensional disordered log-concave Gibbs measures, including concentration of multioverlaps and simplified measure representations.

Findings

Proves concentration of multioverlaps in high-dimensional log-concave measures.

Provides a simple representation of asymptotic Gibbs measures.

Demonstrates variable decoupling in the system.

Abstract

We consider a generic class of log-concave, possibly random, (Gibbs) measures. We prove the concentration of an infinite family of order parameters called multioverlaps. Because they completely parametrise the quenched Gibbs measure of the system, this implies a simple representation of the asymptotic Gibbs measures, as well as the decoupling of the variables in a strong sense. These results may prove themselves useful in several contexts. In particular in machine learning and high-dimensional inference, log-concave measures appear in convex empirical risk minimisation, maximum a-posteriori inference or M-estimation. We believe that they may be applicable in establishing some type of "replica symmetric formulas" for the free energy, inference or generalisation error in such settings.