Calculus of Variations and Geometric Measure Theory

E. Caglioti - M. Goldman - F. Pieroni - D. Trevisan

Subadditivity and optimal matching of unbounded samples

created by goldman on 08 Jul 2024



Inserted: 8 jul 2024
Last Updated: 8 jul 2024

Year: 2024


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.