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arXiv · 2407.06352

Subadditivity and optimal matching of unbounded samples

Emanuele Caglioti·Michael Goldman·Francesca Pieroni·Dario Trevisan
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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.

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Emanuele Caglioti, Michael Goldman, Francesca Pieroni, Dario Trevisan. 2024-07-08. Subadditivity and optimal matching of unbounded samples. https://arxiv.org/abs/2407.06352

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