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 Vempala and Wibisono [2019], Moitra and Risteski [2020] in which the former paper focuses on functions f defined in Rⁿ and the latter paper focuses on functions with symmetries (like matrix completion type objectives) with manifold structure. Our work generalizes the results of Vempala and Wibisono [2019] where f is defined on a manifold M rather than Rⁿ. 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.