A data-driven linear-programming methodology for optimal transport

TL;DR

This paper introduces a data-driven linear programming method for optimal transport that uses adaptive meshes and mixture models to efficiently solve large-scale problems, including Wasserstein barycenters, with potential for parallelization.

Contribution

It proposes a novel adaptive, mixture-based linear programming framework for optimal transport that decomposes complex problems into manageable sub-problems and enables efficient computation of Wasserstein barycenters.

Findings

Efficient solution of optimal transport problems using adaptive meshes.

Decoupling of marginal distributions and coupling via mixture models.

Applicability to parallel computing for large-scale problems.

Abstract

A data-driven formulation of the optimal transport problem is presented and solved using adaptively refined meshes to decompose the problem into a sequence of finite linear programming problems. Both the marginal distributions and their unknown optimal coupling are approximated through mixtures, which decouples the problem into the the optimal transport between the individual components of the mixtures and a classical assignment problem linking them all. A factorization of the components into products of single-variable distributions makes the first sub-problem solvable in closed form. The size of the assignment problem is addressed through an adaptive procedure: a sequence of linear programming problems which utilize at each level the solution from the previous coarser mesh to restrict the size of the function space where solutions are sought. The linear programming approach for pairwise optimal transportation, combined with an iterative scheme, gives a data driven algorithm for the Wasserstein barycenter problem, which is well suited to parallel computing.