Factored couplings in multi-marginal optimal transport via difference of convex programming

TL;DR

This paper explores multi-marginal optimal transport (MMOT) in machine learning, proposing a difference of convex programming approach to incorporate structural information into couplings, enabling more flexible modeling.

Contribution

It introduces a novel DC programming formulation for MMOT that unifies various OT methods and allows structural coupling information to be integrated.

Findings

DC optimization yields qualitatively comparable solutions to existing methods.

Incorporating structural information enhances the flexibility of MMOT models.

The proposed approach broadens the applicability of OT in ML tasks.

Abstract

Optimal transport (OT) theory underlies many emerging machine learning (ML) methods nowadays solving a wide range of tasks such as generative modeling, transfer learning and information retrieval. These latter works, however, usually build upon a traditional OT setup with two distributions, while leaving a more general multi-marginal OT formulation somewhat unexplored. In this paper, we study the multi-marginal OT (MMOT) problem and unify several popular OT methods under its umbrella by promoting structural information on the coupling. We show that incorporating such structural information into MMOT results in an instance of a different of convex (DC) programming problem allowing us to solve it numerically. Despite high computational cost of the latter procedure, the solutions provided by DC optimization are usually as qualitative as those obtained using currently employed optimization schemes.