积分概率度量遇上神经网络：Radon-Kolmogorov-Smirnov检验
Integral Probability Metrics Meet Neural Networks: The Radon-Kolmogorov-Smirnov Test
AI总结：
本研究提出Radon-Kolmogorov-Smirnov检验，将经典KS检验推广到多维与高光滑度场景，证明其见证函数为单神经元岭样条，可借助神经网络优化求解，验证了其渐近全功效并对比了与核MMD检验的优劣。
摘要：
积分概率度量（IPMs）是一类通用的非参数双样本检验方法，其核心是在属于某个函数空间$\u2702\fancyscript{F}$的所有数据变换$f$中，最大化来自分布$P$与分布$Q$的样本之间的均值差。受近期将Radon有界变差（RBV）函数与神经网络建立关联的研究（Parhi和Nowak，2021,2023）启发，我们研究了将$\u2702\fancyscript{F}$取为光滑度$k \u2265 0$的RBV空间单位球时所定义的IPM。我们将该检验称为Radon-Kolmogorov-Smirnov（RKS）检验，它可被视为经典著名的Kolmogorov-Smirnov（KS）检验向多维空间和更高光滑阶数的推广。该检验还与神经网络密切相关：我们证明RKS检验中的见证函数——即实现最大均值差的函数$f$——始终是$k$阶岭样条，也就是神经网络中的单个神经元。因此，我们可以利用现代神经网络优化工具包的能力来（近似）最大化RKS检验的核心准则。我们证明RKS检验在区分任意一对不同分布$P \u2260 Q$时具有渐近全功效，推导了其渐近零分布，并开展实验阐明了RKS检验相较于更传统的核MMD检验的优缺点。
英文摘要：
Integral probability metrics (IPMs) constitute a general class of nonparametric two-sample tests that are based on maximizing the mean difference between samples from one distribution $P$ versus another $Q$, over all choices of data transformations $f$ living in some function space $\mathcal{F}$. Inspired by recent work that connects what are known as functions of $\textit{Radon bounded variation}$ (RBV) and neural networks (Parhi and Nowak, 2021, 2023), we study the IPM defined by taking $\mathcal{F}$ to be the unit ball in the RBV space of a given smoothness degree $k \geq 0$. This test, which we refer to as the $\textit{Radon-Kolmogorov-Smirnov}$ (RKS) test, can be viewed as a generalization of the well-known and classical Kolmogorov-Smirnov (KS) test to multiple dimensions and higher orders of smoothness. It is also intimately connected to neural networks: we prove that the witness in the RKS test -- the function $f$ achieving the maximum mean difference -- is always a ridge spline of degree $k$, i.e., a single neuron in a neural network. We can thus leverage the power of modern neural network optimization toolkits to (approximately) maximize the criterion that underlies the RKS test. We prove that the RKS test has asymptotically full power at distinguishing any distinct pair $P \not= Q$ of distributions, derive its asymptotic null distribution, and carry out experiments to elucidate the strengths and weaknesses of the RKS test versus the more traditional kernel MMD test.
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