Quantum Information Journal Club: ‘Rapid mixing for Gibbs measures in Riemannian manifolds’

Speaker: Pablo Páez-Velasco (Universidad Complutense de Madrid)

Venue & Time: Gray Room 2 / 14:30

Abstract: Given some sufficiently smooth, generally non-convex, function F on a Riemannian manifold, and a fixed inverse temperature, the problem of sampling with respect to the associated Gibbs distribution has played a central role in several areas, such as statistical mechanics or non-convex optimization. A natural approach to do so is to simulate the Langevin diffusion associated to F, whose stationary distribution is precisely the Gibbs distribution. However, obtaining quantitative convergence guarantees in this non-convex, geometric setting remains difficult.

In this work, we identify a set of assumptions on F and the underlying manifold under which a log-Sobolev inequality (LSI) holds for the Gibbs distribution in the low-temperature regime. As a consequence, we obtain exponential convergence of the Langevin diffusion to its stationary distribution.