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Recovering Stochastic Dynamics via Gaussian Schrödinger Bridges
Ya-Ping Hsieh · Charlotte Bunne · Marco Cuturi · Andreas Krause
We propose a new framework to reconstruct a stochastic process $\left\{\mathbb{P}_{t}: t \in[0, T]\right\}$ using only samples from its marginal distributions, observed at start and end times 0 and T. This reconstruction is useful to infer population dynamics, a crucial challenge, e.g., when modeling the time-evolution of cell populations from single-cell sequencing data. Our general framework encompasses the more specific Schrödinger bridge (SB) problem, where $\mathbb{P}_{t}$ represents the evolution of a thermodynamic system at almost equilibrium. Estimating such bridges from scratch is notoriously difficult, motivating our proposal for a novel adaptive scheme called the GSBflow. Our approach is to first perform a Gaussian approximation of the general SB via matching the moments of the data, which proves to significantly stabilize the training of SB. To that end, we solve the SB problem with Gaussian marginals, for which we provide, as a central contribution, a closed-form solution, and SDE representation. We use these formulas to define the reference process used to estimate more complex SBs, and obtain notable numerical improvements when reconstructing both synthetic processes and single-cell genomics.

#### Author Information

##### Andreas Krause (ETH Zurich)

Andreas Krause is a Professor of Computer Science at ETH Zurich, where he leads the Learning & Adaptive Systems Group. He also serves as Academic Co-Director of the Swiss Data Science Center and Chair of the ETH AI Center, and co-founded the ETH spin-off LatticeFlow. Before that he was an Assistant Professor of Computer Science at Caltech. He received his Ph.D. in Computer Science from Carnegie Mellon University (2008) and his Diplom in Computer Science and Mathematics from the Technical University of Munich, Germany (2004). He is a Max Planck Fellow at the Max Planck Institute for Intelligent Systems, an ELLIS Fellow, a Microsoft Research Faculty Fellow and a Kavli Frontiers Fellow of the US National Academy of Sciences. He received the Rössler Prize, ERC Starting Investigator and ERC Consolidator grants, the German Pattern Recognition Award, an NSF CAREER award as well as the ETH Golden Owl teaching award. His research has received awards at several premier conferences and journals, including the ACM SIGKDD Test of Time award 2019 and the ICML Test of Time award 2020. Andreas Krause served as Program Co-Chair for ICML 2018, and currently serves as General Chair for ICML 2023 and as Action Editor for the Journal of Machine Learning Research.