Efficient Fourier Representations of Families of Gaussian Processes

We introduce a class of algorithms for constructing Fourier representations of Gaussian processes in 1 dimension that are valid over ranges of hyperparameter values. The scaling and frequencies of the Fourier basis functions are evaluated numerically via generalized Gaussian quadratures. The representations introduced allow for O (NN + m³) inference via the non-uniform FFT where N is the number of data points and m is the number of basis functions. Numerical results are provided for Matern kernels with 3/2, 7/2 and 0. 1, 0. 5. The algorithms of this paper generalize mathematically to higher dimensions, though they suffer from the standard curse of dimensionality.