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.

翻译：我们研究了一类由奇异形式的伍德伯里矩阵恒等式导出的矩阵。针对该矩阵，我们提出了广义逆与伪行列式恒等式，这些恒等式可直接应用于高斯过程回归，特别是其似然表示和精度矩阵。我们将精度矩阵的定义推广至协方差矩阵的Bott-Duffin逆，从而保留了与条件独立性、条件精度和边际精度相关的性质。我们还为所提出的行列式恒等式提供了高效的算法及数值分析，并展示了这些恒等式在与计算高斯过程回归似然函数中对数行列式项相关的特定条件下的优势。