Interpolating log-determinant and trace of the powers of matrix [Formula: see text]

We develop heuristic interpolation methods for the functions [Formula: see text] and [Formula: see text] where the matrices [Formula: see text] and [Formula: see text] are Hermitian and positive (semi) definite and [Formula: see text] and [Formula: see text] are real variables. These functions are featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of sharp bounds for these functions. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.