Toscani Fourier distances offer new ways to compare probability data
Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
Machine Learning
Summary
Comparing different probability distributions is important but can be slow or tricky depending on the method. The authors study a family of distances that work by comparing the Fourier transforms (a way to analyze signals) of distributions. They find precise conditions under which these distances are well-defined and how they relate to common distances used in machine learning called Wasserstein distances. Their work also clarifies the mathematical properties and provides formulas that can help compute these distances more efficiently in some cases.
What this means in practice
- •For machine learning engineers: Use Toscani-Fourier distances to efficiently measure how different probability models or datasets are, benefiting tasks like generative modeling and distribution comparison without heavy computation.
- •For signal processing teams: Incorporate Fourier-based distances to analyze and compare signal distributions where geometry and frequency information matter, offering new tools beyond classical metrics.
Authors
Mehrdad Mohammadi
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 $1\le p<\infty$ 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 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.