Exponential Convergence Guarantees for Iterative Markovian Fitting
This work addresses a theoretical gap for researchers in computational optimal transport and generative modeling, offering incremental but rigorous convergence analysis.
The paper tackled the lack of quantitative convergence guarantees for Iterative Markovian Fitting in the Schrödinger Bridge problem, providing the first non-asymptotic exponential convergence guarantees under mild structural assumptions, with results covering log-concave and weakly log-concave marginals.
The Schrödinger 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 Schrödinger 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.