A simple Fourier analytic proof of the AKT optimal matching theorem
Sergey G. Bobkov, Michel Ledoux
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.
Topics & Concepts
MathematicsSimple (philosophy)Matching (statistics)Fourier transformElementary proofAnalytic proofSmoothingFourier analysisUpper and lower boundsDiscrete mathematicsDimension (graph theory)Applied mathematicsCombinatoricsMathematical analysisStatisticsMathematical proofGeometryEpistemologyPhilosophyMarkov Chains and Monte Carlo MethodsNonlinear Partial Differential EquationsGeometric Analysis and Curvature Flows