We present scEGOT, a comprehensive single-cell trajectory inference framework based on entropic Gaussian mixture optimal transport. The main advantage of scEGOT allows us to go back and forth between continuous and discrete problems, and it provides a versatile trajectory inference method including reconstructions of the underlying vector fields at a low computational cost. Applied to the human primordial germ cell-like cell (PGCLC) induction system, scEGOT identified the PGCLC progenitor population and bifurcation time of segregation. Our analysis shows TFAP2A is insufficient for identifying PGCLC progenitors, requiring NKX1-2.

The Gromov-Wasserstein (GW) distance is a generalization of the standard Wasserstein distance between two probability measures on a given ambient metric space. The GW distance assumes that these two probability measures might live on different ambient metric spaces and therefore implements an actual comparison of pairs of metric measure spaces. A metric-measure space is a triple (X,dX,muX) where (X,dX) is a metric space and muX is a fully supported Borel probability measure over X. In Machine Learning and Data Science applications, this distance is estimated either directly via gradient based optimization approaches, or through the computation of lower bounds which arise from distributional invariants of metric-measure spaces. One particular such invariant is the so-called ‘global distance distribution’ which precisely encodes the distribution of pairwise distances between points in a given metric measure space. This invariant has been used in many applications yet its classificatory power is not yet well understood. This talk will overview the construction of the GW distance, the stability of distributional invariants, and will also discuss some results regarding the injectivity of the global distribution of distances for smooth planar curves, hypersurfaces, and metric trees.