ICML 2025

Categorical Schr¨odinger Bridge Matching
Grigoriy Ksenofontov 1 2 Alexander Korotin 1 3
Abstract
The Schr¨odinger Bridge (SB) is a powerful frame-
work for solving generative modeling tasks such
as unpaired domain translation. Most SB-related
research focuses on continuous data space RD and
leaves open theoretical and algorithmic questions
about applying SB methods to discrete data, e.g,
on finite spaces SD. Notable examples of such
sets S are codebooks of vector-quantized (VQ)
representations of modern autoencoders, tokens
in texts, categories of atoms in molecules, etc. In
this paper, we provide a theoretical and algorith-
mic foundation for solving SB in discrete spaces
using the recently introduced Iterative Markovian
Fitting (IMF) procedure. Specifically, we theoreti-
cally justify the convergence of discrete-time IMF
(D-IMF) to SB in discrete spaces. This enables
us to develop a practical computational algorithm
for SB, which we call Categorical Schr¨odinger
Bridge Matching (CSBM). We show the perfor-
mance of CSBM via a series of experiments with
synthetic data and VQ representations of images.
The code of CSBM is available at this repository.
1
Introduction
The Schr¨odinger bridge (Schr¨odinger, 1931, SB) problem
has recently attracted the attention of the machine learning
community due to its relevance to modern challenges in gen-
erative modeling and unpaired learning. Recently, a variety
of methods have been proposed to solve SB in continuous
spaces; see (Gushchin et al., 2023b) for a recent survey.
One modern approach to solving SB is the Iterative Marko-
vian Fitting (IMF) framework (Peluchetti, 2023; Shi et al.,
2023; Gushchin et al., 2024b). Specifically, within this
framework, the discrete-time IMF procedure (Gushchin
1Skoltech,
Moscow,
Russia
2MIPT, Moscow,
Russia
3AIRI, Moscow,
Russia.
Correspondence to:
Grigoriy
Ksenofontov <g.ksenofontov@skoltech.ru>, Alexander Korotin
<a.korotin@skoltech.ru>.
Proceedings of the 42 nd International Conference on ...