Efficient Fourier Representations of Families of Gaussian Processes
Philip Greengard · SIAM Journal on Scientific Computing · 2025
Abstract. 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 quadratures. The representations introduced allow for [Formula: see text] inference, independent of [Formula: see text], for all hyperparameters in the user-specified range after [Formula: see text] precomputation, where [Formula: see text], the number of data points, is usually significantly larger than [Formula: see text], the number of basis functions. Inference independent of [Formula: see text] for various hyperparameters is facilitated by the generalized quadratures, and the [Formula: see text] precomputation is achieved with the nonuniform FFT. Numerical results are provided for Matérn kernels with [Formula: see text] and lengthscale [Formula: see text] and squared-exponential kernels with lengthscale [Formula: see text]. The algorithms of this paper generalize mathematically to higher dimensions, although they suffer from the standard curse of dimensionality.