Computational Optimal Transport and Filtering on Riemannian manifolds

Understand

In this paper we extend recent developments in computational optimal transport to the setting of Riemannian manifolds.

• In particular, we show how to learn optimal transport maps from samples that relate probability distributions defined on manifolds.
• Specializing these maps for sampling conditional probability distributions provides an ensemble approach for solving nonlinear filtering problems defined on such geometries.
• The proposed computational methodology is illustrated with examples of transport and nonlinear filtering on Lie groups, including the circle $S^1$, the special Euclidean group $SE(2)$, and the special orthogonal group $SO(3)$.