Twisted Schrödinger Bridge Matching

2026-07-18

AI Summary

This work addresses the problem of constructing optimal transport trajectories between prescribed initial and terminal distributions under time-varying potential fields. Building upon the generalized Schrödinger bridge framework, it presents the first rigorous extension of Iterative Markov Fitting (IMF) to settings involving time-dependent potentials. By introducing a twisted Brownian motion as the reference process and leveraging the Feynman–Kac transformation, the authors formulate a bridge-matching loss that explicitly depends on the gradient of the potential. To enhance optimization stability, they further devise a trajectory variance reduction technique. Empirical evaluations on high-dimensional trajectory inference tasks—such as crowd navigation and single-cell dynamics modeling—demonstrate substantial improvements over existing methods, confirming the approach’s effectiveness and scalability in complex, dynamically evolving potential landscapes.

Abstract

Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

Problem

• Schrödinger bridge
• optimal transport
• twisted Brown日晚间 motion
• Feynman-Kac transform
• generative modeling

Innovation

• Twisted Schrödinger Bridge
• Feynman-Kac transform
• Iterative Markovian Fitting
• diffusion-based generative modeling
• trajectory inference