Toward a mathematical theory of trajectory inference

We devise a theoretical framework and a numerical method to infer tra- jectories of a stochastic process from samples of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data, which pro- vide high dimensional measurements of cell states but cannot track the trajec- tories 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 algo- rithm to do so in practice. The method we develop, Global Waddington-OT (gWOT), boils down to a smooth convex optimization problem posed glob- ally 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.