An ODE characterisation of multi-marginal optimal transport with pairwise cost functions

TL;DR

This paper introduces an ODE-based numerical method for multi-marginal optimal transport problems with pairwise costs, reducing computational complexity by interpolating between a simplified and the original cost function.

Contribution

It presents a novel ODE characterization that enables efficient solution of complex multi-marginal optimal transport problems through interpolation.

Findings

The method reduces complexity from exponential to linear in the number of marginals.

Simulations demonstrate the effectiveness of explicit Euler and Runge-Kutta schemes.

The approach successfully recovers solutions to the original problem via ODE integration.

Abstract

The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one parameter family of cost functions that interpolates between the original and a special cost function for which the problem's complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $\epsilon$, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge-Kutta schemes to compute solutions to the ODE, and, as a result, the multi-marginal optimal transport problem.