Subadditivity and optimal matching of unbounded samples

Abstract

We obtain new bounds for the optimal matching cost for empirical measures with unbounded support. For a large class of radially symmetric and rapidly decaying probability laws, we prove for the first time the asymptotic rate of convergence for the whole range of power exponents p and dimensions d . Moreover we identify the exact prefactor when p ≤ d . We cover in particular the Gaussian case, going far beyond the currently known bounds. Our proof technique is based on approximate sub- and super-additivity bounds along a geometric decomposition adapted to some features of the density, such as its radial symmetry and its decay at infinity.