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 for some function is the Langevin Algorithm (LA). Recently, there has been a lot of progress in showing fast convergence of LA even in cases where is non-convex, notably \cite{VW19}, \cite{MoritaRisteski} in which the former paper focuses on functions defined in 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 is defined on a manifold rather than . From technical point of view, we show that KL decreases in a geometric rate whenever the distribution satisfies a log-Sobolev inequality on .
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