Dalhousie-AARMS AAMP Seminar: Alex Barnett (Flatiron Institute, NYC)

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

Abstract: Gaussian process regression is widely used in geostatistics, time-series analysis, and machine learning. It infers an unknown continuous function in a principled fashion from noisy measurements at $N$ scattered data points.  The prior on the function is Gaussian, with covariance given by some user-chosen translationally invariant
kernel.  Yet $N$ has been limited to about $10^6$, even with modern low-rank methods.  Focusing on low spatial dimension (1–3), we present a GP regression method using kernel approximation by an equispaced quadrature grid in the Fourier domain.  This enables the iterative solution of a smaller Toeplitz linear system, exploiting both the FFT and the nonuniform FFT to give ${\mathcal O}(N)$ cost. The result is often one to two orders of magnitude faster than state of the art methods, and enables cheap massive-scale regressions. For example, for a 2D Mat\’ern-3/2 kernel and $N = 10^9$ points, the posterior mean function is found to 3-digit accuracy in two minutes on a desktop.

Joint work with Philip Greengard (Columbia) and Manas Rachh (Flatiron Institute)