Large scale Gaussian processes with Matheron's update rule and Karhunen-Loève expansion

International audience ; Gaussian processes have become essential for non-parametric function estimation and widely used in many fields like machine learning. In this paper, large scale Gaussian process regression (GPR) is investigated. This problem is related to the simulation of high dimensional Gaussian vectors truncated on the intersection of a set of hyperplanes. The main idea is to combine both Matheron's update rule (MUR) and Karhunen-Lovève expansion (KLE). First, by the MUR we show how simulating from the posterior distribution is possible without computing the posterior covariance matrix and its decomposition. Second, by splitting the input domain in smallest nonoverlapping subdomains, the KLE coefficients are conditioned in order to guarantee the correlation structure in the entire domain. The parallelization of this technique is developed and the advantages are highlighted. By this, the computational complexity is drastically reduced. The mean-square block error is computed. It provides accurate results when using a family of covariance functions with compact support. Some numerical examples to study the performance of the proposed approach are included.