Multi-marginal temporal Schr\"odinger Bridge Matching from unpaired data

Abstract

Many natural dynamic processes -- such as in vivo cellular differentiation or disease progression -- can only be observed through the lens of static sample snapshots. While challenging, reconstructing their temporal evolution to decipher underlying dynamic properties is of major interest to scientific research. Existing approaches enable data transport along a temporal axis but are poorly scalable in high dimension and require restrictive assumptions to be met. To address these issues, we propose Multi-Marginal temporal Schr\"odinger Bridge Matching (MMtSBM) from unpaired data, extending the theoretical guarantees and empirical efficiency of Diffusion Schr\"odinger Bridge Matching (arXiv:2303.16852) by deriving the Iterative Markovian Fitting algorithm to multiple marginals in a novel factorized fashion. Experiments show that MMtSBM retains theoretical properties on toy examples, achieves state-of-the-art performance on real-world datasets such as transcriptomic trajectory inference in 100 dimensions, and, for the first time, recovers couplings and dynamics in very high-dimensional image settings. Our work establishes multi-marginal Schr\"odinger bridges as a practical and principled approach for recovering hidden dynamics from static data.