A reduced-order model for parametrized Optimal Transport problems

TL;DR

This paper develops a reduced-order model for parametrized optimal transport problems, enabling efficient solutions with error bounds, and demonstrates its application on image color transfer compared to Sinkhorn.

Contribution

It introduces a novel reduced-order modeling approach with error estimation for parametrized optimal transport problems, improving computational efficiency.

Findings

Reduced-order model simplifies optimal transport to a small linear program.

Two a posteriori error estimators provide bounds on solution accuracy.

Application on image color transfer shows performance advantages over Sinkhorn.

Abstract

In this work, we aim at efficiently solving a parametrized family of optimal transport problems by using model order reduction methods. We propose a reduced-order model by adding to the primal (respectively dual) version of the high-fidelity model the additional constraint to live in a non negative sub cone (resp. in subspaces) of small dimension. The reduced-order model then reads as a linear program with a small number of degrees of freedom and constraints. We identify explicit conditions under which this reduced-order model has at least one solution. We propose two a posteriori error estimations that bounds the error between the optimal values of the high-fidelity problem and the reduced-order model. As one of these estimations requires the computation of non linear terms (with respect to the reduction of dimension), we use an Empirical Interpolation Method (EIM) (see e.g. \cite{maday2007general} or \cite{barrault2004empirical}) to numerically efficiently compute this estimation. We apply the whole methodology on a simple 1D example and on a problem of color transfer between images, and compare its performances to Sinkhorn algorithm.