Diffusion & Adversarial Schr\"odinger Bridges via Iterative Proportional Markovian Fitting

TL;DR

This paper introduces the IPMF procedure, combining IMF and IPF methods, to solve Schr"odinger Bridge problems efficiently, with proven convergence and practical benefits for image translation tasks.

Contribution

It reveals the connection between IMF and IPF, proposes the IPMF method, and demonstrates its convergence and practical advantages in applications.

Findings

IPMF effectively integrates IMF and IPF procedures.

Theoretical convergence of IPMF is established.

IPMF offers flexible trade-offs in image translation quality.

Abstract

The Iterative Markovian Fitting (IMF) procedure, which iteratively projects onto the space of Markov processes and the reciprocal class, successfully solves the Schr\"odinger Bridge (SB) problem. However, an efficient practical implementation requires a heuristic modification -- alternating between fitting forward and backward time diffusion at each iteration. This modification is crucial for stabilizing training and achieving reliable results in applications such as unpaired domain translation. Our work reveals a close connection between the modified version of IMF and the Iterative Proportional Fitting (IPF) procedure -- a foundational method for the SB problem, also known as Sinkhorn's algorithm. Specifically, we demonstrate that the heuristic modification of the IMF effectively integrates both IMF and IPF procedures. We refer to this combined approach as the Iterative Proportional Markovian Fitting (IPMF) procedure. Through theoretical and empirical analysis, we establish the convergence of the IPMF procedure under various settings, contributing to developing a unified framework for solving SB problems. Moreover, from a practical standpoint, the IPMF procedure enables a flexible trade-off between image similarity and generation quality, offering a new mechanism for tailoring models to specific tasks.