  We introduce an optimal transport-based model for learning a metric tensor
from cross-sectional samples of evolving probability measures on a common
Riemannian manifold. We neurally parametrize the metric as a spatially-varying
matrix field and efficiently optimize our model's objective using a simple
alternating scheme. Using this learned metric, we can nonlinearly interpolate
between probability measures and compute geodesics on the manifold. We show
that metrics learned using our method improve the quality of trajectory
inference on scRNA and bird migration data at the cost of little additional
cross-sectional data.
