Kantorovich vs. Monge: A Numerical Classification of Extremal Multi-Marginal Mass Transports on Finite State Spaces

Daniela Vögler

Published 2019-01-14Version 1

We analyze the validity of Monge's ansatz regarding the symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces, with uniform-marginal constraint. The class of Monge states reduces the number of unknowns from combinatorial in both $N$ and $\ell$ to linear in $N$ and $\ell$, where $N$ is the number of marginals and $\ell$ the number of states. Unfortunately, this low-dimensional ansatz space is insufficient, i.e., there exist cost functions such that the corresponding multi-marginal optimal transport problem does not admit a Monge-type optimizer. We will analyze this insufficiency numerically by utilizing the convex geometry of the set of admissible trial states for symmetric respectively pairwise-symmetric cost functions. We further consider a model problem of optimally coupling $N$ marginals on three states with respect to a pairwise-symmetric cost function. The restriction to a state space of three elements allows us to visually compare Kantorovich's to Monge's ansatz space. This visual comparison includes a consideration of the volumetric ratio.