Skip to main content
SpringerLink
Log in

A Local Inverse Formula and a Factorization

  • Chapter
  • First Online:
Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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.

With congratulations to Ian Sloan!

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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. 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. Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992)

    Book  Google Scholar 

  4. Dempster, A.P.: Covariance selection. Biometrics 28, 157–175 (1972)

    Article  Google Scholar 

  5. Dym, H., Gohberg, I.: Extensions of band matrices with band inverses. Linear Algebra Appl. 36, 1–24 (1981)

    Article  MathSciNet  Google Scholar 

  6. Eidelman, Y., Gohberg, I., Haimovici, I.: Separable Type Representations of Matrices and Fast Algorithms, vol. 1. Springer, Basel (2013)

    MATH  Google Scholar 

  7. Friedman, J., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3), 432–441 (2008)

    Article  Google Scholar 

  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. Johnson, C.R., Lundquist, M.: Local inversion of matrices with sparse inverses. Linear Algebra Appl. 277, 33–39 (1998)

    Article  MathSciNet  Google Scholar 

  10. Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge (2009)

    MATH  Google Scholar 

  11. Lauritzen, S.: Graphical Models. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  12. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, Boston (1998)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Rose, D.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32, 597–609 (1970)

    Article  MathSciNet  Google Scholar 

  15. Speed, T.P., Kiiveri, H.T.: Gaussian Markov distributions over finite graphs. Ann. Stat. 14(1), 138–150 (1986)

    Article  MathSciNet  Google Scholar 

  16. Strang, G.: Fast transforms: banded matrices with banded inverses. Proc. Natl. Acad. Sci. U. S. A. 107(28), 12413–12416 (2010)

    Article  MathSciNet  Google Scholar 

  17. Strang, G.: Introduction to Linear Algebra. Cambridge Press, Wellesley (2016)

    MATH  Google Scholar 

  18. Strang, G., Nguyen, T.: Wavelets and Filter Banks. Cambridge Press, Wellesley (1996)

    MATH  Google Scholar 

  19. Strang, G., Nguyen, T.: The interplay of ranks of submatrices. SIAM Rev. 46(4), 637–646 (2004)

    Article  MathSciNet  Google Scholar 

  20. Vandebril, R., van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices, vol. 1. Johns Hopkins, Baltimore (2007)

    MATH  Google Scholar 

  21. Wathen, A.J.: An analysis of some Element-by-Element techniques. Comput. Methods Appl. Mech. Eng. 74, 271–287 (1989)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Gilbert Strang

  2. University of Technology Sydney, Ultimo, NSW, Australia

    Shev MacNamara

Authors
  1. Gilbert Strang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shev MacNamara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilbert Strang .

Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Josef Dick

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Frances Y. Kuo

  3. Department of Computer Science, Columbia University, New York, USA

    Henryk Woźniakowski

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Strang, G., MacNamara, S. (2018). A Local Inverse Formula and a Factorization. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_51

Download citation

Publish with us

Policies and ethics

Search

Navigation