Approximate state-space Gaussian processes via spectral transformation

State-space representations of Gaussian process regression use Kalman filtering and smoothing theory to downscale the computational complexity of the regression in the number of data points from cubic to linear. As their exact implementation requires the covariance function to possess rational spectral density, rational approximations to the spectral density must be often used. In this article we introduce new spectral transformation based methods for this purpose: a spectral composition method and a spectral preconditioning method. We study convergence of the approximations theoretically and run numerical experiments to attest their accuracy for different densities, in particular the fractional Matern.