Fast Kernel Approximations for Latent Force Models and Convolved Multiple-Output Gaussian processes

A latent force model is a Gaussian process with a covariance function inspired by a differential operator. Such covariance function is obtained by performing convolution integrals between Green’s functions associated to the differential operators, and covariance functions associated to latent functions. In the classical formula- tion of latent force models, the covariance func- tions are obtained analytically by solving a dou- ble integral, leading to expressions that involve numerical solutions of different types of error functions. In consequence, the covariance ma- trix calculation is considerably expensive, be- cause it requires the evaluation of one or more of these error functions. In this paper, we use random Fourier features to approximate the so- lution of these double integrals obtaining sim- pler analytical expressions for such covariance functions. We show experimental results using ordinary differential operators and provide an extension to build general kernel functions for convolved multiple output Gaussian processes.