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

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 \cite{VW19}, \cite{MoritaRisteski} 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 \cite{VW19} 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 .