Interpolating Log-Determinant and Trace of the Powers of Matrix $\mathbf{A} + t \mathbf{B}$

TL;DR

This paper introduces heuristic interpolation methods for efficiently estimating the log-determinant and trace functions of matrix sums, which are crucial in statistics, machine learning, and physics.

Contribution

It proposes novel interpolation techniques based on modified bounds for these matrix functions, improving computational efficiency.

Findings

High accuracy in numerical experiments

Effective in Gaussian process regression

Useful for ridge regression parameter estimation

Abstract

We develop heuristic interpolation methods for the functions $t \mapsto \log \det \left( \mathbf{A} + t \mathbf{B} \right)$ and $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{p} \right)$ where the matrices $\mathbf{A}$ and $\mathbf{B}$ 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.