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

TL;DR

This paper introduces BM$^2$, a non-iterative neural network method for learning Schr"{o}dinger bridges that efficiently transports between distributions, supported by theoretical analysis and numerical experiments.

Contribution

BM$^2$ is a novel, simple, non-iterative approach for Schr"{o}dinger bridge learning that leverages available samples and tractable diffusion dynamics.

Findings

Effective in learning transport maps between distributions

Supports theoretical convergence analysis

Demonstrates strong empirical performance

Abstract

A Schr\"{o}dinger bridge establishes a dynamic transport map between two target distributions via a reference process, simultaneously solving an associated entropic optimal transport problem. We consider the setting where samples from the target distributions are available, and the reference diffusion process admits tractable dynamics. We thus introduce Coupled Bridge Matching (BM$^2$), a simple non-iterative approach for learning Schr\"{o}dinger bridges with neural networks. A preliminary theoretical analysis of the convergence properties of BM$^2$ is carried out, supported by numerical experiments that demonstrate the effectiveness of our proposal.