Categorical Schr\"odinger Bridge Matching

The Schrödinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. Most SB-related research focuses on continuous data space $\mathbb{R}^{D}$ and leaves open theoretical and algorithmic questions about applying SB methods to discrete data, e.g, on finite spaces $\mathbb{S}^{D}$. Notable examples of such sets $\mathbb{S}$ 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 algorithmic foundation for solving SB in discrete spaces using the recently introduced Iterative Markovian Fitting (IMF) procedure. Specifically, we theoretically 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 Schrödinger Bridge Matching (CSBM). We show the performance of CSBM via a series of experiments with synthetic data and VQ representations of images.

@article{ksenofontov2025_2502.01416,
  title={ Categorical Schr\"odinger Bridge Matching },
  author={ Grigoriy Ksenofontov and Alexander Korotin },
  journal={arXiv preprint arXiv:2502.01416},
  year={ 2025 }
}