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

Siavash Ameli, Shawn C. Shadden

Abstract: Highlights•We examine a matrix derived from a singular form of the Woodbury matrix identity, which is commonly used in machine learning and statistics.•In the framework of generalized inverse, we proposed Woodbury-like identities and pseudo-determinant relations with direct applications in Gaussian process regression.•The definition of the precision matrix was extended to the Bott–Duffin inverse of the covariance matrix, maintaining key properties such as conditional independence, conditional precision, and marginal precision.•We analyzed the computational complexity of the pseudo-determinant identities and demonstrated their numerical advantages in computing log-determinant terms in Gaussian process regression.