Uniform approximation of common Gaussian process kernels using equispaced Fourier grids

Alex Barnett; Philip Greengard; Manas Rachh

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arXiv · 2305.11065

Uniform approximation of common Gaussian process kernels using equispaced Fourier grids

Alex Barnett·Philip Greengard·Manas Rachh

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Abstract

The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels -- Mat\'ern and squared exponential -- in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Mat\'ern case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.

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Alex Barnett, Philip Greengard, Manas Rachh. 2023-05-18. Uniform approximation of common Gaussian process kernels using equispaced Fourier grids. https://arxiv.org/abs/2305.11065

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