A simple Fourier analytic proof of the AKT optimal matching theorem

TL;DR

This paper provides an elementary Fourier analytic proof of the AKT optimal matching theorem in two dimensions, extending its applicability to dependent samples and connecting to PDE methods for bounds.

Contribution

It introduces a simple, Fourier-based proof of the AKT theorem and adapts it for dependent samples, broadening the theorem's scope and understanding.

Findings

Elementary Fourier proof of AKT theorem in 2D

Extension to dependent sample distributions

Connection to PDE approaches for bounds

Abstract

We present a short and elementary proof of the Ajtai-Koml\'os-Tusn\'ady (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.