Rapid mixing for Gibbs measures in Riemannian manifolds

2026-06-11

AI Summary

This work addresses the challenge of achieving rapid mixing of Gibbs measures on Riemannian manifolds to escape saddle points, flat regions, and spurious local minima. To this end, it introduces— for the first time—the technique of Riemannian submersions to establish a rigorous connection between Langevin dynamics on the manifold and its projected process in the base space. The study systematically analyzes how curvature, temperature, and geometric structure jointly influence the mixing rate. Under suitable geometric and thermal conditions, the authors prove the validity of a logarithmic Sobolev inequality, which yields a polynomial mixing time bound in terms of the manifold’s dimension. This result significantly enhances sampling efficiency in non-Euclidean settings.

Abstract

Langevin dynamics on Riemannian manifolds is analyzed. Conditions ensuring the existence of a suitable logarithmic Sobolev inequality (rapid mixing to the Gibbs measure) are identified. These conditions involve the curvature of the manifold, the inverse temperature, escaping directions from saddle points, and exclude barren plateaus and spurious local minima. We show that when these conditions are met, mixing times polynomial in the dimension of the manifold are achievable. This result is obtained through a relation between Langevin processes in the domain and in the image of a Riemannian submersion. Such a relation can be of independent interest.

Problem

• rapid mixing
• Gibbs measure
• Riemannian manifold
• Langevin dynamics
• logarithmic Sobolev inequality

Innovation

• Langevin dynamics
• Riemannian manifold
• logarithmic Sobolev inequality
• rapid mixing
• Riemannian submersion