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

We study a matrix that arises in a singular formulation of the Woodbury matrix identity when the Woodbury identity no longer holds. We present generalized inverse and pseudo-determinant identities for such matrix that have direct applications to the Gaussian process regression, in particular, its likelihood representation and its precision matrix. We also provide an efficient algorithm and numerical analysis for the presented determinant identities and demonstrate their advantages in certain conditions which are applicable to computing log-determinant terms in likelihood functions of Gaussian process regression.