Categorical Schrödinger Bridge Matching

The Schr\"odinger 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 Schr\"odinger Bridge Matching (CSBM). We show the performance of CSBM via a series of experiments with synthetic data and VQ representations of images.

Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting

The Iterative Markovian Fitting (IMF) procedure based on iterative reciprocal and Markovian projections has recently been proposed as a powerful method for solving the Schr\"odinger Bridge problem. However, it has been observed that for the practical implementation of this procedure, it is crucial to alternate between fitting a forward and backward time diffusion at each iteration. Such implementation is thought to be a practical heuristic, which is required to stabilize training and obtain good results in applications such as unpaired domain translation. In our work, we show that this heuristic closely connects with the pioneer approaches for the Schr\"odinger Bridge based on the Iterative Proportional Fitting (IPF) procedure. Namely, we find that the practical implementation of IMF is, in fact, a combination of IMF and IPF procedures, and we call this combination the Iterative Proportional Markovian Fitting (IPMF) procedure. We show both theoretically and practically that this combined IPMF procedure can converge under more general settings, thus, showing that the IPMF procedure opens a door towards developing a unified framework for solving Schr\"odinger Bridge problems.