On Multimarginal Partial Optimal Transport: Equivalent Forms and Computational Complexity

TL;DR

This paper introduces new equivalent formulations for multimarginal partial optimal transport problems, leveraging graph theory and cost tensor extensions, and develops an efficient approximation algorithm with provable complexity bounds.

Contribution

It provides novel equivalence forms for multimarginal partial optimal transport and proposes the ApproxMPOT algorithm with complexity analysis.

Findings

Two equivalence forms of multimarginal POT are established.

The ApproxMPOT algorithm approximates the optimal value with a specific complexity bound.

The methods extend cost tensors and use graph theory for mass movement procedures.

Abstract

We study the multi-marginal partial optimal transport (POT) problem between $m$ discrete (unbalanced) measures with at most $n$ supports. We first prove that we can obtain two equivalence forms of the multimarginal POT problem in terms of the multimarginal optimal transport problem via novel extensions of cost tensor. The first equivalence form is derived under the assumptions that the total masses of each measure are sufficiently close while the second equivalence form does not require any conditions on these masses but at the price of more sophisticated extended cost tensor. Our proof techniques for obtaining these equivalence forms rely on novel procedures of moving mass in graph theory to push transportation plan into appropriate regions. Finally, based on the equivalence forms, we develop optimization algorithm, named ApproxMPOT algorithm, that builds upon the Sinkhorn algorithm for solving the entropic regularized multimarginal optimal transport. We demonstrate that the ApproxMPOT algorithm can approximate the optimal value of multimarginal POT problem with a computational complexity upper bound of the order $\tilde{\mathcal{O}}(m^3(n+1)^{m}/ \varepsilon^2)$ where $\varepsilon > 0$ stands for the desired tolerance.