Abstract
We devise a theoretical framework and a numerical method to infer trajectories of a stochastic process from samples of its temporal marginals. This problem arises in the analysis of single-cell RNA-sequencing data, which provide high-dimensional measurements of cell states but cannot track the trajectories of the cells over time. We prove that for a class of stochastic processes it is possible to recover the ground truth trajectories from limited samples of the temporal marginals at each time-point, and provide an efficient algorithm to do so in practice. The method we develop, Global Waddington-OT (gWOT), boils down to a smooth convex optimization problem posed globally over all time-points involving entropy-regularized optimal transport. We demonstrate that this problem can be solved efficiently in practice and yields good reconstructions, as we show on several synthetic and real data sets.
Funding Statement
This work was supported in part by a UBC Affiliated Fellowship to S.Z., an Exploration Grant to G.S. and Y.H.K. from the New Frontiers in Research Fund (NFRF), a Career Award at the Scientific Interface from the Burroughs Wellcome Fund to G.S. and NSERC Discovery Grants to Y.H.K. and G.S. Part of this work was done while H.L. was supported by the Pacific Institute for the Mathematical Sciences (PIMS) through a PIMS postdoctoral fellowship.
Acknowledgments
The authors wish to thank Aymeric Baradat and Jonathan Niles-Weed for stimulating discussions, as well as Igor Prünster and Giacomo Zanella for valuable comments on a earlier draft of the present work.
Citation
Hugo Lavenant. Stephen Zhang. Young-Heon Kim. Geoffrey Schiebinger. "Toward a mathematical theory of trajectory inference." Ann. Appl. Probab. 34 (1A) 428 - 500, February 2024. https://doi.org/10.1214/23-AAP1969
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