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

Siavash Ameli and Shawn C. Shadden

Applied Mathematics and Computation, 2023, vol. 452, issue C

Abstract: 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 representation 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 independence, conditional precision, and marginal precision. We also provide an efficient algorithm 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.

Keywords: Matrix determinant lemma; Outer inverse; Bott–Duffin inverse; EP matrix; Likelihood function; Precision matrix (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:452:y:2023:i:c:s0096300323002011

DOI: 10.1016/j.amc.2023.128032

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