Interpolating log-determinant and trace of the powers of matrix A plus tB

We develop heuristic interpolation methods for the functions t bar right arrow log det (A + tB) and t bar right arrow trace ((A + tB)(p)) where the matrices A and B are Hermitian and positive (semi) definite and p and t 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.

Automated summary

In this paper, heuristic interpolation methods are developed for calculating two specific functions. By modifying sharp bounds, accurate computation is achieved. Experimental results validate the accuracy and performance of the proposed method.