The Fourier Discrepancy Function

TL;DR

This paper introduces the Fourier Discrepancy Function, a novel measure for comparing discrete probability measures that considers geometric properties and offers explicit formulas and bounds.

Contribution

It presents the Fourier Discrepancy Function, demonstrating its convexity, differentiability, explicit gradient, and statistical interpretation, along with bounds related to Total Variation distance.

Findings

Fourier Discrepancy is convex and twice differentiable.

Explicit formula for the gradient of Fourier Discrepancy.

Bounds established between Fourier Discrepancy and Total Variation distance.

Abstract

In this paper, we propose the Fourier Discrepancy Function, a new discrepancy to compare discrete probability measures. We show that this discrepancy takes into account the geometry of the underlying space. We prove that the Fourier Discrepancy is convex, twice differentiable, and that its gradient has an explicit formula. We also provide a compelling statistical interpretation. Finally, we study the lower and upper tight bounds for the Fourier Discrepancy in terms of the Total Variation distance.