A Singular Woodbury and Pseudo-Determinant Matrix Identities and Application to Gaussian Process Regression

TL;DR

This paper introduces new identities for a singular matrix related to the Woodbury identity, with applications to Gaussian process regression, improving the computation of likelihoods and precision matrices.

Contribution

It generalizes inverse and pseudo-determinant identities for a singular matrix, extending Gaussian process likelihood computations and providing efficient algorithms.

Findings

New pseudo-determinant identities for singular matrices

Extended precision matrix definition using Bott-Duffin inverse

Enhanced algorithms for Gaussian process likelihood calculations

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.