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

Optimal transport (OT) theory underlies many emerging 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.