Twisted Schrödinger Bridge Matching
Machine Learning
Summary
The authors developed a new method called Twisted Schrödinger Bridge Matching (TSBM) to better model the flow between two data distributions using diffusion processes influenced by time-varying potentials. Their work generalizes previous approaches by including these potentials, improving accuracy in estimating optimal transport paths. They also introduced techniques to reduce randomness during training, making the method more stable. The authors tested TSBM on complex tasks like crowd movement and single-cell analysis, showing it performs well in higher dimensions.
Authors
Maxence Noble, Marie Scheid, Yazid Janati, Eric Moulines, Alain Durmus
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.