Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme
This work addresses a theoretical gap for researchers in machine learning and numerical analysis by providing convergence guarantees for manifold-valued SDEs, though it is incremental as it extends known Euclidean results to Riemannian settings.
The paper tackles the problem of proving strong convergence for numerical schemes of stochastic differential equations on Riemannian manifolds, which is crucial for diffusion models on intrinsic data manifolds, and shows that a geometric Euler-Maruyama scheme achieves strong convergence with order 1/2, with applications in sampling via Wasserstein bounds.
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.