A simple fourier analytic proof of the AKT optimal matching theorem

Sergey G. Bobkov; Michel Ledoux

doi:10.1214/20-AAP1656

A simple fourier analytic proof of the AKT optimal matching theorem

Sergey G. Bobkov, Michel Ledoux

Mathematics

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Abstract

We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.

Original language	English (US)
Pages (from-to)	2567-2584
Number of pages	18
Journal	Annals of Applied Probability
Volume	31
Issue number	6
DOIs	https://doi.org/10.1214/20-AAP1656
State	Published - Dec 2021

Bibliographical note

Funding Information:
Research of S.B. was partially supported by NSF Grant DMS-1855575.

Publisher Copyright:
© Institute of Mathematical Statistics, 2021

Keywords

Ajtai–Komlós–Tusnády theorem
Empirical measure
Fourier analysis
Heat kernel smoothing
Optimal matching

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abstract = "We present a short and elementary proof of the Ajtai–Koml{\'o}s–Tusn{\'a}dy (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.",

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AU - Ledoux, Michel

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AB - We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.

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KW - Empirical measure

KW - Fourier analysis

KW - Heat kernel smoothing

KW - Optimal matching

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A simple fourier analytic proof of the AKT optimal matching theorem

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