Geometry of Kantorovich polytopes and support of optimizers for repulsive multi-marginal optimal transport on finite state spaces

Abstract

We consider symmetric multi-marginal Kantorovich optimal transport problems on finite state spaces with uniform-marginal constraint. Hereby the symmetry of the problem refers to an assumption on the cost function as well as a corresponding restriction of the set of admissible trial states where the former enables the latter. Note that the symmetry of this setting forces us to pick for each of the considered marginals one and the same probability measure. The said problems consist of minimizing a linear objective function over a high-dimensional polytope, here referred to as Kantorovich polytope. The presented results are of split nature, computational and theoretical. Within the computational part only small numbers of marginals N and marginal sites ℓ are considered. This restriction allows us to computationally determine all extreme points of the Kantorovich polytope and investigate how many of them are in compliance with the in optimal transport typical Monge ansatz. Singling out the results for ℓ=3 discretization points and pairwise symmetric cost functions enables us to visually compare Kantorovich's to Monge's ansatz space for a varying number of marginals. Finally we present a necessary support-condition for optimizers which is inspired by the insights the said model problem on three sites provided. This result is not limited to the case of ℓ=3 sites and applies to symmetric pair-costs whose diagonal entries lie above a cost-specific threshold. In case N and ℓ display certain relationships the discussed condition provides an optimizer in Monge-form and implies its uniqueness as a solution of the considered Kantorovich optimal transport problem.