[论文解读] Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme

Zhiyuan Zhan, Masashi Sugiyama|arXiv (Cornell University)|Mar 4, 2026

Markov Chains and Monte Carlo Methods被引用 0

一句话总结

该论文为黎曼 Langevin 动力学建立了一个框架，并证明几何 Euler-Maruyama 离散化的强收敛性，辅以一系列关于不等式与微分几何的技术引理。

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.

研究动机与目标

• Motivate the study of Langevin dynamics on Riemannian manifolds and address convergence properties of discretizations.
• Develop and apply geometric-analytic tools to control stochastic and geometric errors in the discretization.
• Establish auxiliary lemmas that bound moments, norms, and operator mappings required for convergence proofs.

提出的方法

• Introduce and utilize lemmas for stochastic estimates and moment bounds (e.g., Hölder, Minkowski inequalities) relevant to stochastic processes on manifolds.
• Construct smooth cutoff functions and projection operators to manage non-Lipschitz or unbounded behavior.
• Employ differential-geometric identities (e.g., Gauss, second fundamental form) to relate Euclidean derivatives to manifold-valued derivatives.
• Prove Lipschitz-type properties for projection maps and right-inverses to ensure stability of discretized dynamics on manifolds.

实验结果

研究问题

• RQ1 如何为黎曼 Langevin 动力学建立几何 Euler-Maruyama 方案的强收敛性？
• RQ2 为了在流形上控制离散化误差，需要哪些辅助估计（矩、 Lipschitz 性、投影）？
• RQ3 微分几何结构（如二阶本质形式）如何影响流形上离散化随机流的收敛分析？
• RQ4 在收敛证明中，平滑 cutoff 函数在处理非界限或有问题区域中起什么作用？

主要发现

• 提供了一系列不等式和引理，用于在黎曼随机背景下界定矩及范数。
• 投影到球体的算子及其 Lipschitz 性质被刻画，以确保离散化步长的稳定性。
• 黎曼流形及相关双线性映射的性质与收敛分析所需的算子界限相关联。
• 将欧几里得 Hessian 与流形 Hessian 的联系形式化，以在 Gauss 公式下关联微分算子。