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

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 ellcdot(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 ellcdot(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.