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

Zhang, Y. and Leithead, W.E. and Leith, D.J. and Walshe, L. (2008) Log-det approximation based on uniformly distributed seeds and its application to Gaussian process regression. Journal of Computational and Applied Mathematics, 220 (1-2). pp. 198-214. ISSN 0377-0427

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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 Chereafter). In general, the exact computation of log del C is of O(N-3) operations where N is the matrix dimension. The approximation of log del C 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 log det C 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.