Rahimi and Recht (2007) introduced the idea of decomposing positive definite shift-invariant kernels by randomly sampling from their spectral distribution for machine learning applications. This famous technique, known as Random Fourier Features (RFF), is in principle applicable to any such kernel whose spectral distribution can be identified and simulated. In practice, however, it is usually applied to the Gaussian kernel because of its simplicity, since its spectral distribution is also Gaussian. Clearly, simple spectral sampling formulas would be desirable for broader classes of kernels. In this paper, we show that the spectral distribution of positive definite isotropic kernels in $\mathbb{R}^{d}$ for all $d\geq1$ can be decomposed as a scale mixture of $α$-stable random vectors, and we identify the mixing distribution as a function of the kernel. This constructive decomposition provides a simple and ready-to-use spectral sampling formula for many multivariate positive definite shift-invariant kernels, including exponential power kernels, and generalized Cauchy kernels, as well as newly introduced kernels such as the generalized Matérn, Tricomi, and Fox $H$ kernels. In particular, we retrieve the fact that the spectral distributions of these kernels, which can only be explicited in terms of the Fox $H$ special function, are scale mixtures of the multivariate Gaussian distribution, along with an explicit mixing distribution formula. This result has broad applications for support vector machines, kernel ridge regression, Gaussian processes, and other kernel-based machine learning techniques for which the random Fourier features technique is applicable.

翻译：Rahimi 与 Recht (2007) 为机器学习应用提出了一种通过从谱分布中随机采样来分解正定平移不变核的思想。这一著名技术被称为随机傅里叶特征 (RFF)，原则上适用于任何谱分布可识别且可模拟的此类核。然而，在实践中，由于其简单性（其谱分布亦为高斯分布），该技术通常应用于高斯核。显然，对于更广泛的核类，简单的谱采样公式是理想的。本文证明，对于所有 $d\geq1$，$\mathbb{R}^{d}$ 中正定各向同性核的谱分布可分解为 $α$-稳定随机向量的尺度混合，并将该混合分布确定为核的函数。这一构造性分解为许多多元正定平移不变核提供了简单且即用的谱采样公式，包括指数幂核、广义柯西核，以及新引入的核，如广义 Matérn 核、Tricomi 核和 Fox $H$ 核。特别地，我们重新得到了以下事实：这些核的谱分布（仅能以 Fox $H$ 特殊函数显式表示）是多元高斯分布的尺度混合，并给出了显式的混合分布公式。该结果对于支持向量机、核岭回归、高斯过程以及其他适用随机傅里叶特征技术的基于核的机器学习方法具有广泛的应用价值。