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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

Ya-Ping Hsieh (ETH)
Charlotte Bunne (ETH Zurich)
Marco Cuturi (ENSAE / CREST)
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. 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 Microsoft Research Faculty Fellow and a Kavli Frontiers Fellow of the US National Academy of Sciences. He received ERC Starting Investigator and ERC Consolidator grants, the Deutscher Mustererkennungspreis, an NSF CAREER award, the Okawa Foundation Research Grant recognizing top young researchers in telecommunications as well as the ETH Golden Owl teaching award. His research on machine learning and adaptive systems 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 is regularly serving as Area Chair or Senior Program Committee member for ICML, NeurIPS, AAAI and IJCAI, and as Action Editor for the Journal of Machine Learning Research.

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