Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces

TL;DR

This paper introduces a new low-dimensional ansatz space for multi-marginal Kantorovich optimal transport on finite state spaces, enabling efficient solutions even for large numbers of marginals and addressing limitations of the Monge class.

Contribution

The authors propose a novel ansatz space that reduces unknowns and guarantees the existence of minimizers, improving computational feasibility for high-dimensional problems.

Findings

Reduces unknowns from combinatorial to linear in marginals and states

Ensures existence of minimizers within the enlarged ansatz class

Applicable to Coulomb cost problems in quantum physics

Abstract

We present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from $\tbinom{N+\ell-1}{\ell-1}$ to $\ell\cdot(N+1)$, where $\ell$ is the number of marginal states and $N$ the number of marginals. The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to $\ell\cdot(N-1)$ unknowns, and cures the insufficiency of the Monge ansatz, i.e. we show that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions. Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which has received much recent interest due to its emergence as the strongly correlated limit of Hohenberg-Kohn density functional theory. In this context $N$ corresponds to the number of particles, motivating the interest in large $N$.