Interpolating Log-Determinant and Trace of the Powers of Matrix $\mathbf{A} + t \mathbf{B}$
This work addresses computational bottlenecks in statistics and machine learning for tasks such as parameter estimation, but it is incremental as it builds on known bounds without introducing a fundamentally new approach.
The paper tackles the problem of efficiently approximating log-determinant and trace functions of matrix powers, which are computationally expensive in many applications, by developing heuristic interpolation methods that modify existing bounds and demonstrate accuracy in numerical examples like Gaussian process regression and ridge regression.
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.