Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev

TL;DR

This paper proves that Langevin dynamics converge rapidly on manifolds when the target distribution satisfies a log-Sobolev inequality, extending previous results from Euclidean spaces to manifold settings.

Contribution

It generalizes convergence results of Langevin algorithms from Euclidean spaces to manifolds, using log-Sobolev inequalities to establish geometric decay of KL divergence.

Findings

KL divergence decreases geometrically on manifolds under log-Sobolev conditions

Extends Langevin convergence analysis from Euclidean space to manifolds

Provides theoretical foundation for sampling on complex geometric structures

Abstract

Sampling is a fundamental and arguably very important task with numerous applications in Machine Learning. One approach to sample from a high dimensional distribution $e^{-f}$ for some function $f$ is the Langevin Algorithm (LA). Recently, there has been a lot of progress in showing fast convergence of LA even in cases where $f$ is non-convex, notably [53], [39] in which the former paper focuses on functions $f$ defined in $\mathbb{R}^n$ and the latter paper focuses on functions with symmetries (like matrix completion type objectives) with manifold structure. Our work generalizes the results of [53] where $f$ is defined on a manifold $M$ rather than $\mathbb{R}^n$. From technical point of view, we show that KL decreases in a geometric rate whenever the distribution $e^{-f}$ satisfies a log-Sobolev inequality on $M$.