The Fourier discrepancy function

In this paper, we introduce the p-Fourier Discrepancy Functions, a new family of metrics for comparing discrete probability measures, inspired by the χr-metrics. Unlike the χr-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