Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme

TL;DR

This paper establishes the strong convergence order of 1/2 for a geometric Euler-Maruyama scheme applied to SDEs on Riemannian manifolds, with implications for sampling in manifold-based generative models.

Contribution

It extends the convergence theory of Euler-Maruyama schemes to Riemannian manifolds, providing a rigorous foundation for manifold-valued stochastic simulations.

Findings

Proves strong convergence order 1/2 for the geometric EM scheme on manifolds.

Derives a Wasserstein bound for sampling via Riemannian Langevin dynamics.

Establishes conditions under which the scheme converges strongly.

Abstract

Low-dimensional structure in real-world data plays an important role in the success of generative models, which motivates diffusion models defined on intrinsic data manifolds. Such models are driven by stochastic differential equations (SDEs) on manifolds, which raises the need for convergence theory of numerical schemes for manifold-valued SDEs. In Euclidean space, the Euler--Maruyama (EM) scheme achieves strong convergence with order $1/2$, but an analogous result for manifold discretizations is less understood in general settings. In this work, we study a geometric version of the EM scheme for SDEs on Riemannian manifolds and prove strong convergence with order $1/2$ under geometric and regularity conditions. As an application, we obtain a Wasserstein bound for sampling on manifolds via the geometric EM discretization of Riemannian Langevin dynamics.