Approximation using Fourier features is a popular technique for scaling kernel methods to large-scale problems, with myriad applications in machine learning and statistics. This method replaces the integral representation of a shift-invariant kernel with a sum using a quadrature rule. The design of the latter is meant to reduce the number of features required for high-precision approximation. Specifically, for the squared exponential kernel, one must design a quadrature rule that approximates the Gaussian measure on $\mathbb{R}^d$. Previous efforts in this line of research have faced difficulties in higher dimensions. We introduce a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions by exploiting its isotropy. These rules are constructed as a tensor product of a radial quadrature rule and a spherical quadrature rule. Compared to previous work, our approach leverages a thorough analysis of the approximation error, which suggests natural choices for both the radial and spherical components. We demonstrate that this family of Fourier features yields improved approximation bounds.

翻译：傅里叶特征逼近是一种将核方法扩展至大规模问题的流行技术，在机器学习与统计学领域有着广泛的应用。该方法通过数值积分规则，将平移不变核的积分表示替换为求和形式。后者的设计旨在减少高精度逼近所需的特征数量。具体而言，对于平方指数核，需要设计一个能够逼近 $\mathbb{R}^d$ 上高斯测度的数值积分规则。该研究方向先前的工作在高维情形下面临困难。本文提出了一类新的数值积分规则族，其通过利用高斯测度的各向同性性质，能够精确逼近高维情形下的高斯测度。这些规则被构造为径向积分规则与球面积分规则的张量积。与先前工作相比，我们的方法基于对逼近误差的深入分析，从而为径向与球面分量提出了自然的选择方案。我们证明，此类傅里叶特征族能够获得更优的逼近界。