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

2026-03-03

AI Summary

This work addresses the lack of rigorous theoretical guarantees for the strong convergence of the geometric Euler–Maruyama scheme for stochastic differential equations on Riemannian manifolds. Under general geometric and regularity conditions, the authors establish, for the first time, that the scheme achieves strong convergence of order 1/2 and derive a Wasserstein error bound. By integrating tools from Riemannian geometry, stochastic analysis, and geometric numerical integration, this study provides a solid theoretical foundation for manifold-based sampling algorithms such as Riemannian Langevin dynamics, thereby filling a critical gap in the convergence analysis of stochastic numerical methods on nonlinear manifolds.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Riemannian manifold
• stochastic differential equations
• Euler-Maruyama scheme
• strong convergence
• generative models

Innovation

• Riemannian Langevin Dynamics
• Geometric Euler-Maruyama
• Strong Convergence
• Manifold-valued SDEs
• Wasserstein bound