Trigonometric Quadrature Fourier Features for Scalable Gaussian Process Regression
This work addresses a critical limitation in scalable Gaussian Process regression for practitioners needing efficient and accurate uncertainty estimates, though it is incremental as it builds on existing QFF methods.
The paper tackles the issue of poor performance in Quadrature Fourier Features (QFF) for Gaussian Process regression due to oscillatory quadrature pathologies, resulting in the Trigonometric Quadrature Fourier Feature (TQFF) method that achieves accurate approximations over a broad range of length-scales using fewer features.
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.