A Splitting Method for SDE Terminal-Law Estimation

2026-09-11

AI Summary

本文研究了通过分裂部分路径生成树状路径的方法来提高SDE终端分布估计的准确性，使用Kolmogorov-Smirnov距离作为精度度量，并在多个场景中观察到平均误差降低10-25%。

Abstract

In many settings involving stochastic differential equations, including in diffusion based generative AI, our aim is to accurately generate samples from a terminal distribution. Typically, this is done by generating i.i.d. samples of diffusion paths. Given a fixed simulation budget, a reasonable way to gain efficiency may be to instead generate a tree of paths through appropriately split partial paths. This suggests improved performance, but one worries about the injected dependence. In this paper, we study this issue comprehensively. With Kolmogorov-Smirnov distance as a measure of accuracy, we identify the limiting errors of the associated empirical distributions as the simulation budget increases to infinity. We characterize a splitting strategy motivated by a corresponding asymptotic optimization problem. The theoretical results bring out the elegant underlying structure in the problem. Practical implementation involves two phases, an initial estimation phase and a final inference phase. Overall, we observe a 10-25% improvement in mean error over i.i.d. samples in many settings. In an exploratory CIFAR-10 study, our method reduces the maximum mean discrepancy by 8-13%.

Problem

• stochastic differential equations
• terminal distribution
• path splitting
• dependence

Innovation

• Splitting Method
• Stochastic Differential Equations
• Kolmogorov-Smirnov Distance
• Empirical Distributions
• Asymptotic Optimization