Computational Optimal Transport and Filtering on Riemannian manifolds

TL;DR

This paper extends computational optimal transport to Riemannian manifolds, enabling learning of transport maps from samples and applying them to nonlinear filtering on manifold geometries like Lie groups.

Contribution

It introduces a method to learn optimal transport maps on manifolds and applies this to nonlinear filtering problems on Lie groups, expanding the scope of optimal transport techniques.

Findings

Successful transport and filtering on Lie groups such as S^1, SE(2), and SO(3)

Demonstrated effectiveness of the approach through illustrative examples

Extended optimal transport methods to manifold settings

Abstract

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 $SO(3)$.