Abstract:
Take a family of random points (X_1, … ,X_n) uniformly distributed in the square.
The random matching problem concerns the study of the optimal matching cost/map between the Lebesgue measure and these points, namely the minimization of \int |x-T(x)|^2 dx among all maps T that send the Lebesgue measure into the empirical measure associated to the family of points, i.e. T_#(dx) = (δ_{X_1} + … + δ_{X_n})/n.
In 2016, L. Ambrosio, F. Stra and D. Trevisan derived precise asymptotics for the expected value of the matching cost.
Building upon their work, after an introduction to the subject, we will show that (with high probability) the optimal map T can be recovered from the solution of a stochastic Poisson equation. We will discuss how to generalize the result to a curved ambient space (a Riemannian manifold instead of the square) and why we do not expect the result to hold in dimension higher than 2.
The proof is a mix of concentration inequalities, a heat-flow regularization technique and methods from optimal transport.
This is joint work with L. Ambrosio and D. Trevisan.