Manifold optimization for non-linear optimal transport problems

TL;DR

This paper introduces a Riemannian manifold optimization framework for solving general non-linear optimal transport problems, leveraging the geometry of doubly stochastic matrices, and provides a practical implementation in Python and Matlab.

Contribution

It presents a novel approach to non-linear OT problems using manifold optimization on doubly stochastic matrices, with accompanying open-source code.

Findings

Effective manifold optimization techniques for non-linear OT.

Implementation of the MOT repository in Python and Matlab.

Potential improvements in computational efficiency for OT problems.

Abstract

Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to computationally approach general non-linear OT problems within the framework of Riemannian manifold optimization. The basis of this is the manifold of doubly stochastic matrices (and their generalization). Even though the manifold geometry is not new, surprisingly, its usefulness for solving general non-linear OT problems has not been popular. To this end, we specifically discuss optimization-related ingredients that allow modeling the OT problem on smooth Riemannian manifolds by exploiting the geometry of the search space. We also discuss extensions where we reuse the developed optimization ingredients. We make available the Manifold optimization-based Optimal Transport, or MOT, repository with codes useful in solving OT problems in Python and Matlab. The codes are available at \url{https://github.com/SatyadevNtv/MOT}.