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

The Iterative Markovian Fitting (IMF) procedure, which iteratively projects onto the space of Markov processes and their reciprocal class, successfully solves the Schrödinger Bridge 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 Schrödinger Bridge problem, also known as Sinkhorn's algorithm. Specifically, we demonstrate that this 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 IPMF procedure under various settings, contributing to developing a unified framework for solving Schrödinger Bridge problems.

@article{kholkin2025_2410.02601,
  title={ Diffusion & Adversarial Schr\"odinger Bridges via Iterative Proportional Markovian Fitting },
  author={ Sergei Kholkin and Grigoriy Ksenofontov and David Li and Nikita Kornilov and Nikita Gushchin and Alexandra Suvorikova and Alexey Kroshnin and Evgeny Burnaev and Alexander Korotin },
  journal={arXiv preprint arXiv:2410.02601},
  year={ 2025 }
}