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

Zhiyuan Zhan; Masashi Sugiyama

stat.MLMar 4, 2026

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

Authors: Zhiyuan Zhan, Masashi Sugiyama

Organizations: The University of Tokyo, Japan · RIKEN Center for AIP, Japan

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.

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Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature

Yuhao Liu, Longbo Huang

Institute for Interdisciplinary Information Sciences, Tsinghua University

Diffusion Model Sampling Diffusion Models

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Diffusion models have emerged as state-of-the-art generative models, with recent extensions from Euclidean spaces to Riemannian manifolds. However, existing convergence guarantees for Riemannian diffusion models typically require

\tilde{O}(\mathrm{poly}(d,T)/ε^2)

score evaluations, with potentially unfavorable dependence on the dimension. In this work, we develop a general framework that separates score discretization from Brownian-motion simulation and allows multiple geodesic random-walk steps per score evaluation. Under nonnegative Ricci curvature assumption and an exact Brownian-motion simulation oracle, we show that

\tilde{O}(d/ε^2)

score evaluations suffice to achieve an

ε^2

KL divergence from the target distribution, matching the existing convergence rate of Euclidean diffusion models. We further show that

\tilde{O}(d^4T/ε^2)

geodesic random-walk steps suffice to approximate the required drifted Brownian motion to

ε

total variation error. Combining these results yields a sampling scheme with

\tilde{O}(d/ε^2)

score evaluations and

\tilde{O}(d^4T/ε^2)

geodesic random-walk steps, motivating multiple random-walk steps between consecutive score evaluations. Our results provide a sharper characterization of the convergence and sampling complexity of Riemannian diffusion models.

Abstract

Explore similar work

Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature

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