Exponential Convergence Guarantees for Iterative Markovian Fitting

TL;DR

This paper establishes the first non-asymptotic exponential convergence guarantees for Iterative Markovian Fitting in the Schr"odinger Bridge problem, covering log-concave and weakly log-concave marginals, under mild assumptions.

Contribution

It provides the first quantitative convergence analysis for IMF, extending theoretical understanding beyond asymptotic results to non-asymptotic exponential guarantees.

Findings

Proves exponential convergence for IMF under mild conditions

Includes analysis for log-concave and weakly log-concave marginals

Lays groundwork for theoretical guarantees of DSBM

Abstract

The Schr\"odinger Bridge (SB) problem has become a fundamental tool in computational optimal transport and generative modeling. To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed-alongside practical approximations like Diffusion Schr\"odinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, a quantitative, non-asymptotic understanding remains unknown. In this paper, we provide the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a sufficiently large time horizon. Our results encompass two key regimes: one where the marginals are log-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM.