论文标题
langevin动力学在歧管上的快速收敛：测量学满足log-sobolev
Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev
论文作者
论文摘要
采样是一项基本的，可以说是非常重要的任务，并且在机器学习中进行了许多应用。来自高维分布的样品$ e^{ - f} $的一种方法是langevin算法（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.我们的工作概括了[53]的结果，其中$ f $是在歧管$ m $而不是$ \ mathbb {r}^n $上定义的。从技术角度来看，我们表明，每当分布$ e^{ - f} $满足$ m $上的log-sobolev不平等时，KL的几何速率就会下降。
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 $M$.
