Log-det approximation based on uniformly distributed seeds and its application to Gaussian process regression

Authors:
Yunong Zhang;W. E. Leithead;D. J. Leith;L. Walshe
Affiliations:
Department of Electronics and Communication Engineering, Sun Yat-Sen University, Guangzhou 510275, PR China;Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G1 1QE, UK;Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland;Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

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Abstract

Maximum likelihood estimation (MLE) of hyperparameters in Gaussian process regression as well as other computational models usually and frequently requires the evaluation of the logarithm of the determinant of a positive-definite matrix (denoted by C hereafter). In general, the exact computation of logdetC is of O(N^3) operations where N is the matrix dimension. The approximation of logdetC could be developed with O(N^2) operations based on power-series expansion and randomized trace estimator. In this paper, the accuracy and effectiveness of using uniformly distributed seeds for logdetC approximation are investigated. The research shows that uniform-seed based approximation is an equally good alternative to Gaussian-seed based approximation, having slightly better approximation accuracy and smaller variance. Gaussian process regression examples also substantiate the effectiveness of such a uniform-seed based log-det approximation scheme.