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

W.E. Leithead, Yunong Zhang, Science Foundation Ireland grant, 00/PI.1/C067 (Funder), EPSRC, GR/M76379/01 (Funder)

Research output: Contribution to journalArticlepeer-review

57 Citations (Scopus)


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(N3) operations, where N is the matrix dimension. In this article, we propose using the quasi-Newton BFGS O(N2)-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(N3) 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.
Original languageEnglish
Pages (from-to)367-380
Number of pages13
JournalCommunications in Statistics - Simulation and Computation
Issue number2
Publication statusPublished - 2 Mar 2007


  • Gaussian process regression
  • matrix inverse
  • optimization
  • O(N2) operations
  • quasi-Newton BFGS method

Cite this