BM$^2$: Coupled Schr\"{o}dinger Bridge Matching

The Schrödinger bridge problem seeks a process, the Schrödinger bridge, with prescribed initial and terminal distributions, such that the distribution of the Schrödinger bridge minimizes the Kullback-Leibler (KL) divergence to the distribution of a reference process. Schrödinger bridges play a central role in measure transport theory (Marzouk et al., 2016). Notably, it is known that the initial-terminal distribution of a Schrödinger bridge provides a solution to a corresponding entropic optimal transport problem (Peyré & Cuturi, 2020). Schrödinger bridges thus provide an effective framework for finding an alignment between samples from two target distributions. Furthermore, diffusion-based generative models (Ho et al., 2020; Song et al., 2021) can be interpreted as solving trivial instances of the Schrödinger bridge problem (Peluchetti, 2023). Consequently, Schrödinger bridges offer a more general approach to contemporary generative applications. We consider the setting where samples are readily available from both target distributions, and where the reference process is a diffusion process solution to a stochastic differential equation (SDE).