Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev
This work provides a theoretical foundation for faster sampling in machine learning applications involving manifold-structured data, but it is incremental as it generalizes prior results from Euclidean space to manifolds.
The paper tackles the problem of sampling from high-dimensional distributions on manifolds using the Langevin Algorithm, showing that KL divergence decreases at a geometric rate when the distribution satisfies a log-Sobolev inequality on the manifold.
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 $M$.