分析了两样本情形下Sinkhorn迭代在薛定谔桥估计中的统计性能。
Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schrödinger Bridge Estimation
- 通过分析中间迭代步骤，建立误差上界
- 误差随样本量增大而减小，迭代次数影响可控
- 适用于多种现有方法，指导实际参数选择
薛定谔桥问题旨在寻找连接两个给定概率分布的最优随机过程，且能量变化最小。虽然Sinkhorn算法广泛用于求解静态最优传输问题，但近期工作（Pooladian和Niles-Weed, 2024）提出了基于最优传输构建时变漂移的随机微分方程的Sinkhorn桥方法，并在单样本估计场景中提供了统计保证。本文进一步在两样本估计设置下研究该方法的统计表现，即仅从源和目标分布中获得有限样本的情形。我们建立了对中间Sinkhorn迭代的平方总变差误差的统计界：$O(1/m+1/n + r^{4k})~(r ∈ (0,1))$，其中 $m$ 与 $n$ 分别为源与目标分布的样本量，$k$ 为Sinkhorn迭代次数。该结果为薛定谔桥估计器的有限样本性能提供了理论支持，并可为样本量与迭代次数的选择提供实用指导。值得注意的是，我们的理论结果可通过此前未被注意的联系，适用于[SF]$^2$M、DSBM-IMF、BM2和LightSB(-M)等多种代表性方法，在特定设置下。
原文摘要 · Abstract (English)
The Schrödinger bridge problem seeks the optimal stochastic process that connects two given probability distributions with minimal energy modification. While the Sinkhorn algorithm is widely used to solve the static optimal transport problem, a recent work (Pooladian and Niles-Weed, 2024) proposed the Sinkhorn bridge, which estimates Schrödinger bridges by plugging optimal transport into the time-dependent drifts of SDEs, with statistical guarantees in the one-sample estimation setting where the true source distribution is fully accessible. In this work, to further justify this method, we study the statistical performance of intermediate Sinkhorn iterations in the two-sample estimation setting, where only finite samples from both source and target distributions are available. Specifically, we establish a statistical bound on the squared total variation error of Sinkhorn bridge iterations: $O(1/m+1/n + r^{4k})~(r \in (0,1))$, where $m$ and $n$ are the sample sizes from the source and target distributions, respectively, and $k$ is the number of Sinkhorn iterations. This result provides a theoretical guarantee for the finite-sample performance of the Schrödinger bridge estimator and offers practical guidance for selecting sample sizes and the number of Sinkhorn iterations. Notably, our theoretical results apply to several representative methods such as [SF]$^2$M, DSBM-IMF, BM2, and LightSB(-M) under specific settings, through the previously unnoticed connection between these estimators.
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