Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schrödinger Bridge Estimation

TL;DR

This work analyzes the statistical performance of Sinkhorn-based Schrödinger bridge estimators in a two-sample setting with finite samples from both source and target distributions. It establishes a finite-sample total-variation bound for intermediate Sinkhorn iterations, showing $\mathbb E[\mathrm{TV}^2] = \mathcal{O}\left(\frac{1}{m}+\frac{1}{n}+r^{4k}\right)$ for $r\in(0,1)$, and proves exponential convergence in $k$ for the iterates of the dual potentials. The results unify and extend prior one-sample guarantees to two-sample scenarios, and connect Sinkhorn-bridge estimators with several representative Schrödinger-bridge solvers such as [SF]$^2$M, DSBM-IMF, BM2, and LightSB(-M), including conditions under which their $k$-iterates coincide. The paper also investigates neural-network drift estimation to improve computational efficiency and provides extensive experiments validating the theoretical rates, offering practical guidance on sample sizes, iteration counts, and regularization for Schrödinger-bridge estimation in practice.

Abstract

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.

Figures (3)

• Figure 1: Heatmaps illustrating $\text{MSE}_{\text{sample}}(m,n, t)$ as a function of sample sizes $m$ and $n$ for various time points $t$. The error decreases roughly proportionally to $(m^{-1} + n^{-1})$.
• Figure 2: Integrated mean squared error as a function of Sinkhorn iterations $k$ for multiple integration intervals $[0, \tau]$.
• Figure 3: From left to right, the simulation results of the Schrödinger bridge using the estimated drifts $b_{m,n}^{(1)}$, $b_{m,n}^{(5)}$, and $b_{m,n}^{(10)}$ obtained by terminating the Sinkhorn iteration after 1, 5, and 10 iterations, respectively, the optimal drift $b_{m,n}^*$, and the neural network–approximated drift $b_{\theta}$.

Theorems & Definitions (38)

• Theorem 1: Statistical convergence rate
• Remark 1
• Theorem 2: Algorithmic convergence rate
• Proposition 1
• Proposition 2
• Theorem 2: Statistical convergence rate
• proof : Proof of Theorem \ref{['theorem:entropy_of_(infty,infty)_and_(m,n)']}
• Lemma 1
• proof : Proof of Lemma \ref{['lem:drift_of_bar(infty,n)_and_(m,n)']}
• Theorem 2: Algorithmic convergence rate