Gaussian quadratures for state space approximation of scale mixtures of squared exponential covariance functions

Stationary one-dimensional Gaussian process models in machine learning can be reformulated as state space equations. This reduces the cubic computational complexity of the naive full GP solution to linear with respect to the number of training data points. For infinitely differentiable covariance functions the representation is an approximation. In this paper, we study a class of covariance functions that can be represented as a scale mixture of squared exponentials. We show how the generalized Gauss-Laguerre quadrature rule can be employed in a state space approximation in this class. The explicit form of the rational quadratic covariance function approximation is written out, and we demonstrate the results in a regression and log-Gaussian Cox process study.