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

@article{Ameli2022ASW,
  title={A Singular Woodbury and Pseudo-Determinant Matrix Identities and Application to Gaussian Process Regression},
  author={Siavash Ameli and Shawn C. Shadden},
  journal={ArXiv},
  year={2022},
  volume={abs/2207.08038}
}
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… 
2 Citations

Figures from this paper

Interpolating log-determinant and trace of the powers of matrix bfA + tbfB

The presented interpolation functions are based on the modification of sharp bounds for these functions, and the accuracy and performance of the proposed method is demonstrated with numerical examples.

Interpolating log-determinant and trace of the powers of matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textb

We develop heuristic interpolation methods for the functions t↦logdetA+tB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}

References

SHOWING 1-10 OF 45 REFERENCES

Interpolating log-determinant and trace of the powers of matrix bfA + tbfB

The presented interpolation functions are based on the modification of sharp bounds for these functions, and the accuracy and performance of the proposed method is demonstrated with numerical examples.

Matrix Algebra From a Statistician's Perspective

Preface. - Matrices. - Submatrices and partitioned matricies. - Linear dependence and independence. - Linear spaces: row and column spaces. - Trace of a (square) matrix. - Geometrical considerations.

Matrix analysis

This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.

Updating the Inverse of a Matrix

The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed.

Computing Moore–Penrose Inverses with Polynomials in Matrices

  • I. Bajo
  • Mathematics
    Am. Math. Mon.
  • 2021
A method for computing the Moore–Penrose inverse of a complex matrix using polynomials in matrices is proposed, valid for all matrices and does not involve spectral calculation, which could be infeasible when the size of the matrix is large.