O(N2)-operation approximation of covariance matrix inverse in Gaussian process regression based on quasi-Netwon BFGS method

Gaussian process (GP) is a Bayesian nonparametric regression model, showing good performance in various applications. However, during its model-tuning procedure, the GP implementation suffers from numerous covariance-matrix inversions of expensive O(N-3) operations, where N is the matrix dimension. In this article, we propose using the quasi-Newton BFGS O(N-2)-operation formula to approximate/replace recursively the inverse of covariance matrix at every iteration. The implementation accuracy is guaranteed carefully by a matrix-trace criterion and by the restarts technique to generate good initial guesses. A number of numerical tests are then performed based on the sinusoidal regression example and the Wiener - Hammerstein identification example. It is shown that by using the proposed implementation, more than 80% O(N-3) operations could be eliminated, and a typical speedup of 5 - 9 could be achieved as compared to the standard maximum-likelihood-estimation (MLE) implementation commonly used in Gaussian process regression.