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

arXiv: https://arxiv.org/abs/2010.05263 · HTML: https://mlsanity.com/paper/2010.05263/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2020.

## Built on by (most cited first, 16 of 16)

- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Sampling in Constrained Domains with Orthogonal-Space Variational Gradient Descent](https://mlsanity.com/paper/2210.06447/) (2022)
- [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024)
- [Constrained Langevin Algorithms with L-mixing External Random Variables](https://mlsanity.com/paper/2205.14192/) (2022)
- [Efficient Sampling on Riemannian Manifolds via Langevin MCMC](https://mlsanity.com/paper/2402.10357/) (2024)
- [Convergence of Kinetic Langevin Monte Carlo on Lie groups](https://mlsanity.com/paper/2403.12012/) (2024)
- [Sampling and estimation on manifolds using the Langevin diffusion](https://mlsanity.com/paper/2312.14882/) (2023)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Convergence Error Analysis of Reflected Gradient Langevin Dynamics for Globally Optimizing Non-Convex Constrained Problems](https://mlsanity.com/paper/2203.10215/) (2022)
- [Learning Rate Free Sampling in Constrained Domains](https://mlsanity.com/paper/2305.14943/) (2023)
- [Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks](https://mlsanity.com/paper/2111.14762/) (2021)
- [Blessing of Dimensionality for Approximating Sobolev Classes on Manifolds](https://mlsanity.com/paper/2408.06996/) (2024)
- [Accelerating Convergence in Global Non-Convex Optimization with Reversible Diffusion](https://mlsanity.com/paper/2305.11493/) (2023)
- [Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management](https://mlsanity.com/paper/2502.19210/) (2025)

## Abstract (arXiv)

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 ℝn 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 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.