Computationally-efficient initialisation of GPs: The generalised variogram method

We present a computationally-efficient strategy to find the hyperparameters of a Gaussian process (GP) avoiding the computation of the likelihood function. The found hyperparameters can then be used directly for regression or passed as initial conditions to maximum-likelihood (ML) training. Motivated by the fact that training a GP via ML is equivalent (on average) to minimising the KL-divergence between the true and learnt model, we set to explore different metrics/divergences among GPs that are computationally inexpensive and provide estimates close to those of ML. In particular, we identify the GP hyperparameters by projecting the empirical covariance or (Fourier) power spectrum onto a parametric family, thus proposing and studying various measures of discrepancy operating on the temporal or frequency domains. Our contribution extends the Variogram method developed by the geostatistics literature and, accordingly, it is referred to as the Generalised Variogram method (GVM). In addition to the theoretical presentation of GVM, we provide experimental validation in terms of accuracy, consistency with ML and computational complexity for different kernels using synthetic and real-world data.