Riemannian Metric Learning via Optimal Transport
This work addresses metric learning for evolving probability measures on manifolds, with applications in computational biology and ecology.
The authors tackled the problem of learning Riemannian metrics from cross-sectional data using optimal transport, enabling nonlinear interpolation between probability measures and geodesic computation. Their method improved trajectory inference quality on scRNA and bird migration data with minimal additional data requirements.
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.