Communications in Mathematical Sciences

Volume 21 (2023)

Number 3

The Fourier Discrepancy Function

Pages: 627 – 639

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n3.a2

Authors

Gennaro Auricchio (Department of Mathematics, University of Pavia, Pavia, Italy)

Andrea Codegoni (Department of Mathematics, University of Pavia, Pavia, Italy)

Stefano Gualandi (Department of Mathematics, University of Pavia, Pavia, Italy)

Lorenzo Zambon (Department of Mathematics, University of Pavia, Pavia, Italy)

Abstract

In this paper, we introduce the $p$-Fourier Discrepancy Functions, a new family of metrics for comparing discrete probability measures, inspired by the $\chi_r$-metrics. Unlike the $\chi_r$-metrics, the $p$-Fourier Discrepancies are well-defined for any pair of measures. We prove that the $p$-Fourier Discrepancies are convex, twice differentiable, and that their gradient has an explicit formula. Moreover, we study the lower and upper tight bounds for the $p$-Fourier Discrepancies in terms of the Total Variation distance.

Keywords

Fourier metrics, discrete discrepancy, tight bounds

2010 Mathematics Subject Classification

60A10, 60E10, 60E15, 94A17

Received 5 January 2022

Received revised 2 June 2022

Accepted 4 July 2022

Published 27 February 2023