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

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.

💡 Research Summary

This paper addresses a fundamental gap in the numerical analysis of stochastic differential equations (SDEs) defined on Riemannian manifolds, a setting that has become increasingly relevant for diffusion‑based generative models operating on low‑dimensional data manifolds. While the classical Euler–Maruyama (EM) scheme in Euclidean space enjoys strong convergence of order ½, analogous results for manifold‑valued discretizations have been limited to special cases (e.g., spheres, orthogonal groups) or to weak convergence.

The authors focus on the geometric Euler–Maruyama (GEM) scheme, which updates a point (X_{h,k}) on a manifold (M) by moving along the exponential map: \