Optimal Transport with Proximal Splitting

TL;DR

This paper reviews and develops proximal splitting algorithms for solving discretized dynamic optimal transport problems, including extensions to general cost functions and Riemannian manifolds, improving computational efficiency.

Contribution

It introduces a staggered grid discretization and applies proximal splitting schemes, extending optimal transport computation to more general settings and manifolds.

Findings

Effective proximal splitting algorithms for large-scale optimal transport.

Extension to general cost functions and Riemannian manifolds.

Connection to initial algorithms by Benamou and Brenier.

Abstract

This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the $L^2$ optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold.