Subadditivity and optimal matching of unbounded samples

TL;DR

This paper establishes new bounds and asymptotic rates for the optimal matching cost of empirical measures with unbounded support, especially for radially symmetric, rapidly decaying distributions like the Gaussian.

Contribution

It provides the first asymptotic rate of convergence for all power exponents and dimensions, including the exact prefactor when p ≤ d, using a novel geometric decomposition approach.

Findings

Derived new bounds for unbounded support measures.

Proved asymptotic convergence rates for a broad class of distributions.

Identified the exact prefactor for p ≤ d in optimal matching costs.

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 $p\le d$. 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 the density, such as its radial symmetry and its decay at infinity.