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

Source: [Diffusion & Adversarial Schr\\"odinger Bridges via Iterative Proportional Markovian Fitting](<https://www.sota2.com/research/paper/68fa79821e209131cea7de26>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-03-06T05:00:05.533Z

Authors: Sergei Kholkin, Grigoriy Ksenofontov, David Li, Nikita Kornilov, Nikita Gushchin, Alexandra Suvorikova, Alexey Kroshnin, Evgeny Burnaev, Alexander Korotin

Published: 2024-10-03T15:43:17.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Image-to-Image Translation (male-to-female) on CelebA 64 x 64 (test)](<https://www.sota2.com/research/sota/image-to-image-translation-male-to-female-on-celeba-64-x-64-test.md>): Image-to-Image Translation (male-to-female); CelebA 64 × 64 (test); FID: 0.

- [Image-to-Image Translation on AFHQ cat→wild (test)](<https://www.sota2.com/research/sota/image-to-image-translation-on-afhq-cat-wild-test.md>): Image-to-Image Translation; AFHQ cat→wild (test); FID: 53.42.

- [Male-to-Female Translation on CelebA 64 x 64 (test)](<https://www.sota2.com/research/sota/male-to-female-translation-on-celeba-64-x-64-test.md>): male-to-female translation; CelebA 64 x 64 (test); CMMD: 0.28.

- [Static Schrödinger Bridge Approximation on SB Benchmark](<https://www.sota2.com/research/sota/static-schr-dinger-bridge-approximation-on-sb-benchmark.md>): Static Schrödinger Bridge Approximation; SB Benchmark; $BW^2\_2$-UVP (ε=0.1, D=2) Error: 5.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2410.02601>)

[PDF](<https://arxiv.org/pdf/2410.02601.pdf>)