On the impact of prior distributions on efficiency of sparse Gaussian process regression

Abstract

Gaussian process regression (GPR) is a kernel-based learning model, which unfortunately suffers from computational intractability for irregular domain and large datasets due to the full kernel matrix. In this paper, we propose a novel method to produce a sparse kernel matrix using the compact support radial kernels (CSRKs) to efficiently learn the GPR from large datasets. The CSRKs can effectively avoid the ill-conditioned and full kernel matrix during GPR training and prediction, consequently reducing computational costs and memory requirements. In practice, the interest in CSRKs waned slightly as it became evident that, there is a trade-off principle (conflict between accuracy and sparsity) for compactly supported kernels. Hence, when using kernels with compact support, during GPR training, the main focus will be on providing a high level of accuracy. In this case, the advantage of achieving a sparse covariance matrix for CSRKs will almost disappear, as we will see in the numerical results. This trade-off has led authors to search for an “optimal” value of the scale parameter. Accordingly, by selecting the suitable priors on the kernel hyperparameters, and simply estimating the hyperparameters using a modified version of the maximum likelihood estimation (MLE), the GPR model derived from the CSRKs yields maximal accuracy while still maintaining a sparse covariance matrix. In fact, in GPR training, modified version of the MLE will be proportional to the product of MLE and a given suitable prior distribution for the hyperparameters that provides an efficient method for learning. The misspecification of prior distributions and their impact on the predictability of the sparse GPR models are also comprehensively investigated using several empirical studies. The proposed new approach is applied to some irregular domains with noisy test functions in 2D data sets in a comparative study. We finally investigate the effect of prior on the predictability of GPR models based on the real dataset. The derived results suggest the proposed method leads to more sparsity and well-conditioned kernel matrices in all cases.