Optimal transportation between unequal dimensions

TL;DR

This paper introduces a novel nonlocal Monge-Ampère equation to solve optimal transportation problems between manifolds of different dimensions, establishing conditions for solution smoothness and regularity.

Contribution

It formulates a new nonlocal analog of the Monge-Ampère equation for unequal dimensions and analyzes solution regularity under various conditions, including less restrictive criteria for one-dimensional targets.

Findings

Equivalence between the optimal transport problem and a new nonlocal Monge-Ampère equation.

Conditions under which solutions are smooth and regular.

Less restrictive criteria for regularity in one-dimensional target cases.

Abstract

We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Amp\`ere equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution. We show that this local equation is elliptic, and $C^{2,\alpha}$ solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoothness of the corresponding differential operator, which we prove under simplifying assumptions. For one-dimensional targets, our sufficient criteria for regularity of solutions to the resulting ODE are considerably less restrictive than those required by earlier works.