Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
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 $1\le p<\infty$ 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 esta...
| Authors | Mehrdad Mohammadi |
|---|---|
| Primary category | stat.ML |
| Published | 19 September 2026 |
| Last updated at source | 19 September 2026 |
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