Trigonometric Quadrature Fourier Features for Scalable Gaussian Process Regression

TL;DR

This paper introduces Trigonometric Quadrature Fourier Features (TQFF), a novel method for scalable Gaussian Process regression that improves approximation accuracy and efficiency over existing Fourier feature methods by addressing oscillatory quadrature issues.

Contribution

The paper proposes a new TQFF method with a tailored quadrature rule, providing exactness and error bounds, leading to better GP approximations with fewer features.

Findings

TQFF outperforms RFF and Gaussian QFF in numerical experiments.

TQFF achieves accurate GP approximations across various length-scales.

Fewer features are needed for comparable or better accuracy with TQFF.

Abstract

Fourier feature approximations have been successfully applied in the literature for scalable Gaussian Process (GP) regression. In particular, Quadrature Fourier Features (QFF) derived from Gaussian quadrature rules have gained popularity in recent years due to their improved approximation accuracy and better calibrated uncertainty estimates compared to Random Fourier Feature (RFF) methods. However, a key limitation of QFF is that its performance can suffer from well-known pathologies related to highly oscillatory quadrature, resulting in mediocre approximation with limited features. We address this critical issue via a new Trigonometric Quadrature Fourier Feature (TQFF) method, which uses a novel non-Gaussian quadrature rule specifically tailored for the desired Fourier transform. We derive an exact quadrature rule for TQFF, along with kernel approximation error bounds for the resulting feature map. We then demonstrate the improved performance of our method over RFF and Gaussian QFF in a suite of numerical experiments and applications, and show the TQFF enjoys accurate GP approximations over a broad range of length-scales using fewer features.