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

arXiv: https://arxiv.org/abs/2207.08038 · HTML: https://mlsanity.com/paper/2207.08038/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 51 references that are in the map.
- First posted on arXiv 16 Jul 2022.

## Builds on (most cited first, 3 of 3)

- [A Sherman-Morrison-Woodbury Identity for Rank Augmenting Matrices with Application to Centering](https://mlsanity.com/paper/1803.10405/) (2018)
- [Noise Estimation in Gaussian Process Regression](https://mlsanity.com/paper/2206.09976/) (2022)
- [Interpolating Log-Determinant and Trace of the Powers of Matrix A + t B](https://mlsanity.com/paper/2009.07385/) (2020)

## Built on by (most cited first, 1 of 1)

- [Determinant Estimation under Memory Constraints and Neural Scaling Laws](https://mlsanity.com/paper/2503.04424/) (2025)

## Abstract (arXiv)

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.