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A Local Inverse Formula and a Factorization

  • Gilbert StrangEmail author
  • Shev MacNamara
Chapter

Abstract

When a matrix has a banded inverse there is a remarkable formula that quickly computes that inverse, using only local information in the original matrix. This local inverse formula holds more generally, for matrices with sparsity patterns that are examples of chordal graphs or perfect eliminators. The formula has a long history going back at least as far as the completion problem for covariance matrices with missing data. Maximum entropy estimates, log-determinants, rank conditions, the Nullity Theorem and wavelets are all closely related, and the formula has found wide applications in machine learning and graphical models. We describe that local inverse and explain how it can be understood as a matrix factorization.

Notes

Acknowledgements

The authors gratefully acknowledge a grant from The Mathworks that made this work possible.

References

  1. 1.
    Bartlett, P.: Undirected graphical models: chordal graphs, decomposable graphs, junction trees, and factorizations (2009). https://people.eecs.berkeley.edu/~bartlett/courses/2009fall-cs281a/
  2. 2.
    Blair, J.R.S., Peyton, B.: An Introduction to Chordal Graphs and Clique Trees. In: Graph Theory and Sparse Matrix Computation. The IMA Volumes in Mathematics and Its Applications, vol. 56, pp. 1–29. Springer, New York (1993)Google Scholar
  3. 3.
    Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992)CrossRefGoogle Scholar
  4. 4.
    Dempster, A.P.: Covariance selection. Biometrics 28, 157–175 (1972)CrossRefGoogle Scholar
  5. 5.
    Dym, H., Gohberg, I.: Extensions of band matrices with band inverses. Linear Algebra Appl. 36, 1–24 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Eidelman, Y., Gohberg, I., Haimovici, I.: Separable Type Representations of Matrices and Fast Algorithms, vol. 1. Springer, Basel (2013)zbMATHGoogle Scholar
  7. 7.
    Friedman, J., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3), 432–441 (2008)CrossRefGoogle Scholar
  8. 8.
    Johnson, C.R.: Matrix Completion Problems: A Survey. In: Johnson, C.R. (ed.) Matrix Theory and Applications, pp. 69–87. American Mathematical Society, Providence (1989)Google Scholar
  9. 9.
    Johnson, C.R., Lundquist, M.: Local inversion of matrices with sparse inverses. Linear Algebra Appl. 277, 33–39 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge (2009)zbMATHGoogle Scholar
  11. 11.
    Lauritzen, S.: Graphical Models. Oxford University Press, Oxford (1996)zbMATHGoogle Scholar
  12. 12.
    Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, Boston (1998)zbMATHGoogle Scholar
  13. 13.
    Ravikumar, P., Wainwright, M.J., Raskutti, G., Yu, B., et al.: High-dimensional covariance estimation by minimizing l1-penalized log-determinant divergence. Electron. J. Stat. 5, 935–980 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rose, D.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32, 597–609 (1970)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Speed, T.P., Kiiveri, H.T.: Gaussian Markov distributions over finite graphs. Ann. Stat. 14(1), 138–150 (1986)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Strang, G.: Fast transforms: banded matrices with banded inverses. Proc. Natl. Acad. Sci. U. S. A. 107(28), 12413–12416 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Strang, G.: Introduction to Linear Algebra. Cambridge Press, Wellesley (2016)zbMATHGoogle Scholar
  18. 18.
    Strang, G., Nguyen, T.: Wavelets and Filter Banks. Cambridge Press, Wellesley (1996)zbMATHGoogle Scholar
  19. 19.
    Strang, G., Nguyen, T.: The interplay of ranks of submatrices. SIAM Rev. 46(4), 637–646 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Vandebril, R., van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices, vol. 1. Johns Hopkins, Baltimore (2007)zbMATHGoogle Scholar
  21. 21.
    Wathen, A.J.: An analysis of some Element-by-Element techniques. Comput. Methods Appl. Mech. Eng. 74, 271–287 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.University of Technology SydneyUltimoAustralia

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