  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 datasets.
