Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting

This is the official PyTorch implementation of our ICLR'2026 paper “Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting” by Sergei Kholkin*, Grigoriy Ksenofontov*, David Li*, Nikita Kornilov*, Nikita Gushchin*, Alexandra Suvorikova, Alexey Kroshnin, Evgeny Burnaev, Alexander Korotin, where * states for equal contribution.

Diagrams of IPF, IMF, and unified IPMF procedure. All procedures aim to converge to the Schrödinger Bridge.

Abstract

The Iterative Markovian Fitting (IMF) procedure, which iteratively projects onto the space of Markov processes and the reciprocal class, successfully solves the Schrödinger 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.

Repository structure

For the discrete-time (ASBM) IPMF implementation, please see the ASBM folder and corresponding ASBM/README.md

For the continuous-time (DSBM) IPMF implementation, please see the DSBM folder and corresponding DSBM/README.md

For high-dimensional Gaussian analytical, please see Gaussian_IPMF_experiments.ipynb

Citation

@inproceedings{
kholkin2026diffusion,
title={Diffusion \& Adversarial Schr\"odinger Bridges via Iterative Proportional Markovian Fitting},
author={Sergei Kholkin and Grigoriy Ksenofontov and David Li and Nikita Maksimovich Kornilov and Nikita Gushchin and Alexandra Suvorikova and Alexey Kroshnin and Evgeny Burnaev and Alexander Korotin},
booktitle={The Fourteenth International Conference on Learning Representations},
year={2026},
url={https://openreview.net/forum?id=38fGCBhFF5}
}