On the design of scalable, high-precision spherical-radial Fourier features

TL;DR

This paper introduces a new family of quadrature rules for Fourier features that efficiently approximate Gaussian kernels in high dimensions, improving accuracy and scalability in kernel methods.

Contribution

It proposes a novel tensor product quadrature rule combining radial and spherical components, addressing high-dimensional approximation challenges.

Findings

Improved approximation bounds for Gaussian kernels

Effective high-dimensional Fourier feature construction

Enhanced scalability of kernel methods

Abstract

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.