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

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.