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

摘要

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 log detC is of O(N³) operations where N is the matrix dimension. The approximation of log detC 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 detC approximation is investigated. The presented approximation scheme requires 50N²operations, generating an average computational error of 9% as shown by a large number of numerical experiments.