Uniformly Distributed Seeds for Randomized Trace Estimator on O(N2)-operation log-det Approximation in Gaussian Process Regression

Maximum likelihood estimation (MLE) of hyper-parameters 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 log det C is of O(N³) operations where N is the matrix dimension. The approximation of log det C could be developed with O(N²) 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 is investigated. The presented approximation scheme requires 50N² operations, generating an average computational error of 9% as shown by a large number of numerical experiments