Equispaced Fourier representations for efficient Gaussian process regression from a billion data points

We introduce a Fourier-based fast algorithm for Gaussian process regression in low dimensions. It approximates a translationally-invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of $M$ nodes. This results in a weight-space $M\times M$ system matrix with Toeplitz structure, which can thus be applied to a vector in ${\mathcal O}(M \log{M})$ operations via the fast Fourier transform (FFT), independent of the number of data points $N$. The linear system can be set up in ${\mathcal O}(N + M \log{M})$ operations using nonuniform FFTs. This enables efficient massive-scale regression via an iterative solver, even for kernels with fat-tailed spectral densities (large $M$). We provide bounds on both kernel approximation and posterior mean errors. Numerical experiments for squared-exponential and Mat\'ern kernels in one, two and three dimensions often show 1-2 orders of magnitude acceleration over state-of-the-art rank-structured solvers at comparable accuracy. Our method allows 2D Mat\'ern-\mbox{\frac{3}{2}} regression from $N=10^9$ data points to be performed in 2 minutes on a standard desktop, with posterior mean accuracy $10^{-3}$. This opens up spatial statistics applications 100 times larger than previously possible.