On Multimarginal Partial Optimal Transport: Equivalent Forms and Computational Complexity
Abstract
We study the multimarginal 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.07992
 Bibcode:
 2021arXiv210807992L
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Machine Learning;
 Mathematics  Optimization and Control;
 Statistics  Computation
 EPrint:
 20 pages, 3 figures. Khang Le and Huy Nguyen contributed equally to this work