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

• URL: http://arxiv.org/abs/2009.07385v3
• Date: Wed, 3 Aug 2022 20:21:59 GMT
• Title: Interpolating Log-Determinant and Trace of the Powers of Matrix $\mathbf{A} + t \mathbf{B}$
• Authors: Siavash Ameli, Shawn C. Shadden
• Abstract summary: We develop methods for the functions $t mapsto log det left( mathbfA + t mathbfB right)$ and $t mapsto nametraceleft (mathbfA + t mathbfB)p right)$ where the matrices $mathbfA$ and $mathbfB$ are Hermitian and positive (semi) definite and $p$ and $t$ are real variables.
• Score: 1.5002438468152661
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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