Collision-based Dynamics for Multi-Marginal Optimal Transport

TL;DR

This paper introduces a collision-based Monte Carlo method inspired by Boltzmann kinetics for solving multi-marginal optimal transport problems efficiently, especially in high-dimensional settings.

Contribution

It presents a novel collision-based dynamics approach with linear scaling in samples, improving computational efficiency over existing methods.

Findings

Method scales linearly with number of samples

Demonstrates superior efficiency in high-dimensional examples

Applicable to complex multi-marginal optimal transport problems

Abstract

Inspired by the Boltzmann kinetics, we propose a collision-based dynamics with a Monte Carlo solution algorithm that approximates the solution of the multi-marginal optimal transport problem via randomized pairwise swapping of sample indices. The computational complexity and memory usage of the proposed method scale linearly with the number of samples, making it highly attractive for high-dimensional settings. In several examples, we demonstrate the efficiency of the proposed method compared to the state-of-the-art methods.