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.

翻译：在文献中，傅里叶特征近似已成功应用于可扩展高斯过程回归。其中，基于高斯求积法则的求积傅里叶特征（QFF）近年来因相比随机傅里叶特征（RFF）方法具有更高的近似精度和更优校准的不确定性估计而广受欢迎。然而，QFF的一个关键局限性在于其性能可能受到与高振荡求积相关的已知病态问题的影响，导致在有限特征条件下近似效果平庸。我们通过一种新的三角求积傅里叶特征（TQFF）方法解决了这一关键问题，该方法采用一种专为目标傅里叶变换定制的新型非高斯求积法则。我们推导了TQFF的精确求积法则，以及相应特征映射的核近似误差界。随后，在一系列数值实验和应用中，我们证明了该方法相较于RFF和高斯QFF的性能提升，并展示了TQFF在更广泛长度尺度范围内使用更少特征即可实现精确的高斯过程近似。