Efficient Fourier representations of families of Gaussian processes

09/28/2021

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by   Philip Greengard, et al.

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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(NlogN + m^3) 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 Matérn kernels with ν∈ [3/2, 7/2] and ρ∈ [0.1, 0.5]. The algorithms of this paper generalize mathematically to higher dimensions, though they suffer from the standard curse of dimensionality.