# Convergence of the Riemannian Langevin Algorithm

arXiv: https://arxiv.org/abs/2204.10818 · HTML: https://mlsanity.com/paper/2204.10818/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 27 references that are in the map.
- First posted on arXiv 22 Apr 2022.

## Builds on (most cited first, 14 of 14)

- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [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)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [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)
- [Volesti: Volume Approximation and Sampling for Convex Polytopes in R](https://mlsanity.com/paper/2007.01578/) (2020)
- [Sampling and Optimization on Convex Sets in Riemannian Manifolds of Non-Negative Curvature](https://mlsanity.com/paper/1907.10531/) (2019)

## Built on by (most cited first, 25 of 25)

- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Equivariant Descriptor Fields: SE(3)-Equivariant Energy-Based Models for End-to-End Visual Robotic Manipulation Learning](https://mlsanity.com/paper/2206.08321/) (2022)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [Query lower bounds for log-concave sampling](https://mlsanity.com/paper/2304.02599/) (2023)
- [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023)
- [Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators](https://mlsanity.com/paper/2210.07219/) (2022)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [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)
- [Simplicity Bias via Global Convergence of Sharpness Minimization](https://mlsanity.com/paper/2410.16401/) (2024)
- [Mirror Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2505.02621/) (2025)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Total Variation Rates for Riemannian Flow Matching](https://mlsanity.com/paper/2602.05174/) (2026)
- [Gaussian Cooling and Dikin Walks: The Interior-Point Method for Logconcave Sampling](https://mlsanity.com/paper/2307.12943/) (2023)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [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)
- [Sampling from multi-modal distributions on Riemannian manifolds with training-free stochastic interpolants](https://mlsanity.com/paper/2602.00641/) (2026)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)
- [Time-Inhomogeneous Preconditioned Langevin Dynamics](https://mlsanity.com/paper/2605.06091/) (2026)
- [Rapid mixing for Gibbs measures in Riemannian manifolds](https://mlsanity.com/paper/2606.13453/) (2026)
- [Efficient Mass Matrix Estimation with Gaussian Cooling](https://mlsanity.com/paper/2610.04806/) (2026)

## Abstract (arXiv)

We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density nu with respect to the natural measure on a manifold with metric g. We assume that the target density satisfies a log-Sobolev inequality with respect to the metric and prove that the manifold generalization of the Unadjusted Langevin Algorithm converges rapidly to nu for Hessian manifolds. This allows us to reduce the problem of sampling non-smooth (constrained) densities in bf Rn to sampling smooth densities over appropriate manifolds, while needing access only to the gradient of the log-density, and this, in turn, to sampling from the natural Brownian motion on the manifold. Our main analytic tools are (1) an extension of self-concordance to manifolds, and (2) a stochastic approach to bounding smoothness on manifolds. A special case of our approach is sampling isoperimetric densities restricted to polytopes by using the metric defined by the logarithmic barrier.