Kalman filtering and smoothing solutions to temporal Gaussian process regression models

In this paper, we show how temporal (i.e., time-series) Gaussian process regression models in machine learning can be reformulated as linear-Gaussian state space models, which can be solved exactly with classical Kalman filtering theory. The result is an efficient non-parametric learning algorithm, whose computational complexity grows linearly with respect to number of observations. We show how the reformulation can be done for Matérn family of covariance functions analytically and for squared exponential covariance function by applying spectral Taylor series approximation. Advantages of the proposed approach are illustrated with two numerical experiments.