Unbalanced Multi-Marginal Optimal Transport

TL;DR

This paper extends unbalanced optimal transport to the multi-marginal setting, introducing a new problem formulation, proving existence and uniqueness of solutions, and generalizing the Sinkhorn algorithm with convergence guarantees, applicable to various practical problems.

Contribution

It introduces the unbalanced multi-marginal optimal transport problem, develops a dual formulation, and generalizes the Sinkhorn algorithm with proven convergence for this setting.

Findings

Existence and uniqueness of the optimal transport plan under mild conditions.

A generalized Sinkhorn algorithm for unbalanced multi-marginal optimal transport with convergence proof.

Efficient computation when the cost function decouples according to a tree structure.

Abstract

Entropy regularized optimal transport and its multi-marginal generalization have attracted increasing attention in various applications, in particular due to efficient Sinkhorn-like algorithms for computing optimal transport plans. However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual, and show that a unique optimal transport plan exists under mild assumptions. Further, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. If the cost function decouples according to a tree, the iterates can be computed efficiently. At the end, we discuss three applications of our framework, namely two barycenter problems and a transfer operator approach, where we establish a relation between the barycenter problem and the multi-marginal optimal transport with an appropriate tree-structured cost function.