[2603.03626] Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme

Statistics > Machine Learning arXiv:2603.03626 (stat) [Submitted on 4 Mar 2026] Title:Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme Authors:Zhiyuan Zhan, Masashi Sugiyama View a PDF of the paper titled Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme, by Zhiyuan Zhan and Masashi Sugiyama View PDF 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. Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Numerical Analysis (math.NA); Probability (math.PR) Cite as: arXiv:2603.03626 [stat.ML]   (or arXiv:2603.03626v1 [stat.ML] for this version)   https://doi.org/10.48550/arXiv.2603.03626 Fo...