2024/08/23 by Ayoub Belhadji, Belhadji, Ayoub, Qianyu Julie Zhu +3
Computer Science · Engineering · #Advanced Measurement and Metrology Techniques #Advanced Numerical Analysis Techniques #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optical measurement and interference techniques
paper · pdf · doi:10.48550/arxiv.2408.13231
openalex publication_date 2024/08/23 · openalex created_date 2024/09/21 · openalex updated_date 2026/07/28
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 ℝd. 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.