# Efficient Sampling on Riemannian Manifolds via Langevin MCMC

arXiv: https://arxiv.org/abs/2402.10357 · HTML: https://mlsanity.com/paper/2402.10357/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 15 Feb 2024.

## Builds on (most cited first, 15 of 15)

- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [Efficient constrained sampling via the mirror-Langevin algorithm](https://mlsanity.com/paper/2010.16212/) (2020)
- [Exponential ergodicity of mirror-Langevin diffusions](https://mlsanity.com/paper/2005.09669/) (2020)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [Escaping from saddle points on Riemannian manifolds](https://mlsanity.com/paper/1906.07355/) (2019)
- [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017)
- [Measuring Sample Quality with Diffusions](https://mlsanity.com/paper/1611.06972/) (2016)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Riemannian Langevin Algorithm for Solving Semidefinite Programs](https://mlsanity.com/paper/2010.11176/) (2020)
- [Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev](https://mlsanity.com/paper/2010.05263/) (2020)
- [Geodesic Walks in Polytopes](https://mlsanity.com/paper/1606.04696/) (2016)
- [Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature](https://mlsanity.com/paper/1609.02901/) (2016)

## Built on by (most cited first, 6 of 6)

- [Convergence of Kinetic Langevin Monte Carlo on Lie groups](https://mlsanity.com/paper/2403.12012/) (2024)
- [Exponential Convergence Guarantees for Iterative Markovian Fitting](https://mlsanity.com/paper/2510.20871/) (2025)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Riemannian Neural Geodesic Interpolant](https://mlsanity.com/paper/2504.15736/) (2025)
- [Diffusion Processes on Implicit Manifolds](https://mlsanity.com/paper/2604.07213/) (2026)
- [Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature](https://mlsanity.com/paper/2609.36568/) (2026)

## Abstract (arXiv)

We study the task of efficiently sampling from a Gibbs distribution d π^* = e-h d volg over a Riemannian manifold M via (geometric) Langevin MCMC; this algorithm involves computing exponential maps in random Gaussian directions and is efficiently implementable in practice. The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming nabla h is Lipschitz and M has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence. Combined with a contraction guarantee for the geometric Langevin Diffusion under Kendall-Cranston coupling, we prove that the Langevin MCMC iterates lie within ε-Wasserstein distance of π^* after tildeO(ε-2) steps, which matches the iteration complexity for Euclidean Langevin MCMC. Our results apply in general settings where h can be nonconvex and M can have negative Ricci curvature. Under additional assumptions that the Riemannian curvature tensor has bounded derivatives, and that π^* satisfies a CD(·,∞) condition, we analyze the stochastic gradient version of Langevin MCMC, and bound its iteration complexity by tildeO(ε-2) as well.