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

Zhiyuan Zhan, Masashi Sugiyama

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.

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

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 , 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 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.
Paper Structure (51 sections, 52 theorems, 404 equations)

This paper contains 51 sections, 52 theorems, 404 equations.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Riemannian Submanifold
  5. SDE on Riemannian Manifold
  6. p-Strong Convergence of GEM
  7. Main Results
  8. Ideas for Proving Theorem \ref{['thm:ext_p_strong_conv_GEM']}
  9. Well-posed Extension on Euclidean Space
  10. Comparison of Extrinsic and Intrinsic Discretization
  11. Proof of Theorem \ref{['thm:ext_p_strong_conv_GEM']}
  12. p--Wasserstein Convergence of RLD
  13. Conclusion
  14. Notation
  15. Further Preliminaries
  16. ...and 36 more sections

Key Result

Theorem 1

Let $1 \leq p < \infty$. Assume that $\mathcal{M} \subset \mathbb{R}^n$ is a Riemannian submanifold with bounded extrinsic curvatures and is globally well-embedded, and that $V$ satisfies smoothness conditions. For $\bm{X}^h_k$ constructed by the GEM, we have

Theorems & Definitions (69)

  • Theorem 1: $p$-strong convergence of GEM, see Theorem \ref{['thm:intr_p_strong_conv_GEM']}
  • Theorem 2: $p$-Wasserstein convergence of RLD, see Theorem \ref{['thm:intr_convergence_RLD']}
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Theorem 7
  • Corollary 8
  • Lemma 9
  • Proposition 10
  • ...and 59 more