Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
Abstract
Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of 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 Ts,p, the weighted Lp norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on ℝd. For 1≤ p<∞ we show that d/p<s<1+d/p is exactly the window in which Ts,p is finite on 𝒫p(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 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 Ts,p≲ 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, 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.
Related
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BibTeX
@misc{mohammadi2026toscanifourier,
title = {{Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality}},
author = {Mehrdad Mohammadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23163},
eprint = {2609.23163},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23163},
}
RIS
TY - UNPB
TI - Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
AU - Mehrdad Mohammadi
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23163
UR - https://arxiv.org/abs/2609.23163
AB - Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of 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 T_s,p, the weighted L^p norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on ℝ^d. For 1≤ p<∞ we show that d/p<s<1+d/p is exactly the window in which T_s,p is finite on 𝒫_p(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 T_s,p as an integral probability metric over a homogeneous Fourier--Lebesgue ball, with an explicit extremizer when 1<p<∞. We prove the global bound T_s,p≲ 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, 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.
ER -