Computationally-efficient initialisation of GPs: The generalised variogram method

TL;DR

This paper introduces a computationally efficient initialisation method for Gaussian process hyperparameters, called the generalised variogram method (GVM), which approximates maximum likelihood training without intensive likelihood computations.

Contribution

The paper extends the variogram approach from geostatistics to Gaussian processes, providing a new pretraining strategy that is both accurate and computationally efficient.

Findings

GVM closely approximates ML hyperparameters across various kernels.

GVM reduces computational complexity compared to traditional ML training.

Experimental results validate GVM's accuracy and efficiency on synthetic and real data.

Abstract

We present a computationally-efficient strategy to initialise the hyperparameters of a Gaussian process (GP) avoiding the computation of the likelihood function. Our strategy can be used as a pretraining stage to find initial conditions for maximum-likelihood (ML) training, or as a standalone method to compute hyperparameters values to be plugged in directly into the GP model. 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 hyperparameter values that are close to those found via ML. In practice, 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 and 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.