Valentin De Bortoli: Diffusion Schrödinger Bridge Matching

Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs) implement such a transport through a Stochastic Differential Equation (SDE) or an Ordinary Differential Equation (ODE).

However, while it is desirable in many applications to approximate the deterministic dynamic Optimal Transport (OT) map which admits attractive properties, DDMs and FMMs are not guaranteed to provide transports close to the OT map. In contrast, Schrödinger bridges (SBs) compute stochastic dynamic mappings which recover entropy-regularized versions of OT. Unfortunately, existing numerical methods approximating SBs either scale poorly with dimension or accumulate errors across iterations.

In this work, we introduce Iterative Markovian Fitting (IMF), a new methodology for solving SB problems, and Diffusion Schrödinger Bridge Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM significantly improves over previous SB numerics and recovers as special/limiting cases various recent transport methods. We demonstrate the performance of DSBM on a variety of problems. This is a joint work with Yuyang Shi, Andrew Campbell and Arnaud Doucet.

Valentin De Bortoli is a research scientist at Google DeepMind in London, working with Arnaud Doucet. Valentin’s research interests are in stochastic methods for generative modelling. In particular he studies theoretical/practical aspects of deep learning for data generation. He defended his PhD at ENS (École normale supérieure Paris-Saclay), and has worked as a PostDoc at University of Oxford and a research scientist at CNRS (Centre national de la recherche scientifique).