Trigonometric Quadrature Fourier Features for Scalable Gaussian Process
  Regression
Kevin Li Max Balakirsky Simon Mak
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.