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

arXiv: https://arxiv.org/abs/2609.23163 · HTML: https://mlsanity.com/paper/2609.23163/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2026.

## Abstract (arXiv)

Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of mathbb Rd 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 Ts,p, the weighted Lp norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on mathbb Rd. For 1≤ p<∞ we show that d/p<s<1+d/p is exactly the window in which mathrm Ts,p is finite on mathcal Pp(mathbb Rd), 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 Ts,p as an integral probability metric over a homogeneous Fourier--Lebesgue ball, with an explicit extremizer when 1<p<∞. We prove the global bound mathrm Ts,plesssim Wp,s-d/p, whose exponent is sharp, show that no global converse of any form can hold, and recover topological equivalence with Wp 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 Ts,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.