A Singular Woodbury and Pseudo-Determinant Matrix Identities and Application to Gaussian Process Regression
This work addresses computational bottlenecks in Gaussian process regression for practitioners, though it is incremental as it builds on existing matrix identities.
The paper tackles the problem of computing log-determinant terms in Gaussian process regression by deriving generalized inverse and pseudo-determinant identities from a singular Woodbury matrix, resulting in an efficient algorithm with advantages under specific conditions.
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.