Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs

TL;DR

This paper introduces a novel, efficient, and exact algorithm for solving multimarginal optimal transport problems with pairwise costs, leveraging tree representations and gradient methods to improve accuracy and stability.

Contribution

The paper develops a new algorithm for MMOT with pairwise costs using tree structures and dual gradient methods, addressing limitations of entropy regularization.

Findings

The proposed method achieves accurate solutions for MMOT problems.

It provides a stable alternative to entropy-regularized approaches.

The algorithm is applicable to various fields like image processing and machine learning.

Abstract

In this paper, we address the numerical solution to the multimarginal optimal transport (MMOT) with pairwise costs. MMOT, as a natural extension from the classical two-marginal optimal transport, has many important applications including image processing, density functional theory and machine learning, but yet lacks efficient and exact numerical methods. The popular entropy-regularized method may suffer numerical instability and blurring issues. Inspired by the back-and-forth method introduced by Jacobs and L\'{e}ger, we investigate MMOT problems with pairwise costs. First, such problems have a graphical representation and we prove equivalent MMOT problems that have a tree representation. Second, we introduce a noval algorithm to solve MMOT on a rooted tree, by gradient based method on the dual formulation. Last, we obtain accurate solutions which can be used for the regularization-free applications.