A singular Woodbury and pseudo-determinant matrix identities and application to Gaussian process regression

We study a matrix that arises from a singular form of the Woodbury matrix identity. We present generalized inverse and pseudo-determinant identities for this matrix, which have direct applications for Gaussian process regression, specifically its likelihood representa-tion and precision matrix. We extend the definition of the precision matrix to the Bott- Duffin inverse of the covariance matrix, preserving properties related to conditional inde-pendence, conditional precision, and marginal precision. We also provide an efficient al-gorithm and numerical analysis for the presented determinant identities and demonstrate their advantages under specific conditions relevant to computing log-determinant terms in likelihood functions of Gaussian process regression.(c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Automated summary

This paper studies a matrix derived from a singular form of the Woodbury matrix identity. Generalized inverse and pseudo-determinant identities for this matrix are presented, with direct applications for Gaussian process regression, especially in likelihood representation and precision matrix. The definition of precision matrix is extended to the Bott-Duffin inverse of the covariance matrix, preserving properties related to conditional independence, conditional precision, and marginal precision. An efficient algorithm and numerical analysis for the presented determinant identities are provided, demonstrating their advantages in computing log-determinant terms in likelihood functions of Gaussian process regression.