A variational regularity theory for Optimal Transportation, and its application to matching

The optimal matching of two large point clouds is a special case of optimal transportation between measures, a ubiquitous variational problem. In statistics, it is natural to consider random points clouds that arise from sampling from a given distribution, like the uniform distribution. The setting in two space dimension is known to be critical,  and its finer behavior has been predicted by Parisi et al. The predictions rely on the connection between the Monge-Ampere equation, which is the Euler-Lagrange equation for optimal transport, and its linearization, the Poisson equation from electrostatics. Ambrosio et al. established these predictions rigorously on a macroscopic level. A variational regularity theory, used as a large-scale regularity theory, allows to establish this connection down to the microscopic level.  It mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center.