Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality

arXiv:2609.23163v1 Announce Type: cross Abstract: Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of mathbb R^d but require solving a transport problem, while kernel discrepancies are cheap to evaluate yet depend delicately on their test class. We study the Toscani--Fourier family mathrm T_{s,p}, the weighted L^p norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on mathbb R^d. For 1le p<infty we show that d/p<s<1+d/p is exactly the window in which mathrm T_{s,p} is finite on mathcal P_p(mathbb R^d), both endpoints already failing for a pair of Dirac measures, and we establish the metric, embedding, and compactness structure of the resulting space, which we prove to be complete. Duality identifies mathrm T_{s,p} as an integral probability metric over a homogeneous Fourier--Lebesgue ball, with an explicit extremizer when 1<p<infty. We prove the global bound mathrm T_{s,p}lesssim W_p^{,s-d/p}, whose exponent is sharp, show that no global converse of any form can hold, and recover topological equivalence with W_p on bounded-support and uniform-tail classes, together with explicit reverse moduli on bounded-support classes that improve the imported energy-kernel exponent at p=2. There, mathrm T_{s,2} is a constant multiple of the classical energy distance, which yields an exact finite-sample identity for the mean of the empirical discrepancy; the numerical experiments are otherwise diagnostic.

Source: arXiv cs.LG | 2026-09-22