概率测度上的Toscani-Fourier距离：Wasserstein控制、模型类上的拓扑等价性与对偶性
Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
- University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
研究Toscani-Fourier距离作为傅里叶侧差异，刻画其有限性窗口、完备性、对偶性及与Wasserstein距离的全局界和拓扑等价性，并揭示其在p=2时与能量距离的精确联系。
中文摘要 AI 辅助
在机器学习中比较概率测度需要在传输几何与计算成本之间权衡：Wasserstein距离编码了$\mathbb R^d$的几何结构，但需要求解传输问题，而核差异评估成本低，却对其测试类高度敏感。我们研究Toscani-Fourier族$\mathrm T_{s,p}$，即两个特征函数之差的加权$L^p$范数，作为$\mathbb R^d$上的连续傅里叶侧差异。对于$1\le p<\infty$，我们证明$d/p<s<1+d/p$正是$\mathrm T_{s,p}$在$\mathcal P_p(\mathbb R^d)$上有限的精确窗口，两个端点对一对Dirac测度已经失效，并建立了所得空间的度量、嵌入和紧致性结构，我们证明该空间是完备的。对偶性将$\mathrm T_{s,p}$识别为齐次Fourier-Lebesgue球上的积分概率度量，当$1<p<\infty$时具有显式极值元。我们证明全局界$\mathrm T_{s,p}\lesssim W_p^{\\,s-d/p}$，其指数是尖锐的，表明任何形式的全局逆命题都不成立，并在有界支撑和一致尾部类上恢复了与$W_p$的拓扑等价性，连同有界支撑类上的显式反向模量，在$p=2$时改进了引入的能量核指数。在$p=2$时，$\mathrm T_{s,2}$是经典能量距离的常数倍，这为经验差异的均值提供了精确的有限样本恒等式；数值实验除此之外仅起诊断作用。
英文摘要
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.
