Calculus of Variations and Geometric Measure Theory
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H. Lavenant - S. Zhang - Y. H. Kim - G. Schiebinger

Towards a mathematical theory of trajectory inference

created by lavenant on 19 Feb 2021


Submitted Paper

Inserted: 19 feb 2021

Year: 2021

ArXiv: 2102.09204 PDF


We devise a theoretical framework and a numerical method to infer trajectories of a stochastic process from snapshots 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 datasets.

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