  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.
