Reproduce “Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schrödinger Bridge Estimation” on OpenResearch

About this paper

Ibuki Maeda, Rentian Yao, Atsushi Nitanda

This research provides a rigorous statistical and algorithmic analysis for the Sinkhorn bridge method to estimate Schrödinger Bridges in the more practical two-sample setting, where both initial and terminal distributions are empirical. The work establishes convergence rates for both statistical error due to finite samples and algorithmic error from finite Sinkhorn iterations, also demonstrating how these findings unify various deep learning-based Schrödinger Bridge solvers.

Key insights

• The Sinkhorn bridge estimator achieves a sharper statistical convergence rate of $O(1/m + 1/n)$ in squared total variation distance for the two-sample problem, where $m$ and $n$ are sample sizes, outperforming the previously known $O(1/n)$ for the one-sample case.
• The algorithmic error of using intermediate Sinkhorn iterations decreases exponentially with the number of iterations $k$, providing a clear theoretical basis for the computational efficiency of the Sinkhorn algorithm in this context.
• Several prominent deep learning-based Schrödinger Bridge solvers, including [SF]²M, DSBM-IMF, and BM2, are theoretically unified under the Sinkhorn bridge framework, demonstrating their underlying equivalence or direct connection to its iterative steps.

Summary by alphaXiv.