提出黎曼流形上SDE的几何欧拉-马鲁雅玛算法，证明其强收敛阶为1/2。
Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme
- 设计适用于黎曼流形的几何欧拉-马鲁雅玛离散方案
- 在几何与正则条件下实现1/2阶强收敛
- 适用于流形采样，可给出Wasserstein距离上界
真实世界数据的低维结构对生成模型的成功至关重要，这推动了定义在数据内在流形上的扩散模型的发展。这类模型由流形上的随机微分方程（SDE）驱动，因此亟需对流形值SDE数值方法的收敛性理论。在欧氏空间中，欧拉-马鲁雅玛（EM）方案具有1/2阶强收敛，但一般流形离散化下的类似结果尚不明确。本文研究了黎曼流形上SDE的几何版EM方案，在几何与正则条件下证明了强收敛阶为1/2。作为应用，我们获得了通过黎曼朗之万动力学的几何EM离散化进行流形采样时的Wasserstein距离上界。
原文摘要 · Abstract (English)
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.
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