# Sampling and estimation on manifolds using the Langevin diffusion

arXiv: https://arxiv.org/abs/2312.14882 · HTML: https://mlsanity.com/paper/2312.14882/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 86 references that are in the map.
- First posted on arXiv 22 Dec 2023.

## Builds on (most cited first, 10 of 10)

- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [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)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [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)
- [Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature](https://mlsanity.com/paper/1609.02901/) (2016)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [A diffusion approach to Stein's method on Riemannian manifolds](https://mlsanity.com/paper/2003.11497/) (2020)

## Built on by (most cited first, 5 of 5)

- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Score-based constrained generative modeling via Langevin diffusions with boundary conditions](https://mlsanity.com/paper/2510.23985/) (2025)
- [Diffusion Processes on Implicit Manifolds](https://mlsanity.com/paper/2604.07213/) (2026)
- [Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms](https://mlsanity.com/paper/2604.20761/) (2026)
- [Time-Inhomogeneous Preconditioned Langevin Dynamics](https://mlsanity.com/paper/2605.06091/) (2026)

## Abstract (arXiv)

Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure dμφ propto e-φ dvolg  on a compact Riemannian manifold. Two estimators of linear functionals of μφ  based on the discretized Markov process are considered: a time-averaging estimator based on a single trajectory and an ensemble-averaging estimator based on multiple independent trajectories. Imposing no restrictions beyond a nominal level of smoothness on φ, first-order error bounds, in discretization step size, on the bias and variance/mean-square error of both estimators are derived. The order of error matches the optimal rate in Euclidean and flat spaces, and leads to a first-order bound on distance between the invariant measure μφ and a stationary measure of the discretized Markov process. This order is preserved even upon using retractions when exponential maps are unavailable in closed form, thus enhancing practicality of the proposed algorithms. Generality of the proof techniques, which exploit links between two partial differential equations and the semigroup of operators corresponding to the Langevin diffusion, renders them amenable for the study of a more general class of sampling algorithms related to the Langevin diffusion. Conditions for extending analysis to the case of non-compact manifolds are discussed. Numerical illustrations with distributions, log-concave and otherwise, on the manifolds of positive and negative curvature elucidate on the derived bounds and demonstrate practical utility of the sampling algorithm.