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.

Noise Estimation in Gaussian Process Regression

We develop a computational procedure to estimate the covariance hyperparameters for semiparametric Gaussian process regression models with additive noise. Namely, the presented method can be used to efficiently estimate the variance of the correlated error, and the variance of the noise based on maximizing a marginal likelihood function. Our method involves suitably reducing the dimensionality of the hyperparameter space to simplify the estimation procedure to a univariate root-finding problem. Moreover, we derive bounds and asymptotes of the marginal likelihood function and its derivatives, which are useful to narrowing the initial range of the hyperparameter search. Using numerical examples, we demonstrate the computational advantages and robustness of the presented approach compared to traditional parameter optimization.

Interpolating the Trace of the Inverse of Matrix $\mathbf{A} + t \mathbf{B}$

We develop heuristic interpolation methods for the function $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$, where the matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive definite and $t$ is a real variable. This function is featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of a sharp upper bound that we derive for this function, which is a new trace inequality for matrices. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for linear Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.