Approximate implementation of logarithm of the matrix determinant in Gaussian process regression

Maximum likelihood estimation of hyperparameters in Gaussian processes (GPs) as well as other spatial regression models usually requires the evaluation of the logarithm of the matrix determinant, in short, log det. When using matrix decomposition techniques, the exact implementation of log det is of O(N3) operations, where N is the matrix dimension. In this paper, a power-series expansion-based framework is presented for approximating the log det of general positive-definite matrices. Three novel compensation schemes are proposed to further improve the approximation accuracy and computational efficiency. The proposed log det approximation requires only 50N2 operations. The theoretical analysis is substantiated by a large number of numerical experiments, including tests on randomly generated positive-definite matrices, randomly generated covariance matrices, and sequences of covariance matrices generated online in two GP regression examples. The average approximation error is ∼9%.