Advertisement

Optimization Letters

, Volume 8, Issue 3, pp 939–947 | Cite as

Positive definite matrix approximation with condition number constraint

  • Mirai TanakaEmail author
  • Kazuhide Nakata
Original Paper

Abstract

Positive definite matrix approximation with a condition number constraint is an optimization problem to find the nearest positive definite matrix whose condition number is smaller than a given constant. We demonstrate that this problem can be converted to a simpler one when we use a unitary similarity invariant norm as a metric. We can especially convert it to a univariate piecewise convex optimization problem when we use the Ky Fan p-k norm. We also present an analytical solution to the problem whose metric is the spectral norm and the trace norm.

Keywords

Matrix approximation Covariance estimation Unitary similarity invariant norm Ky Fan p-k norm 

Notes

Acknowledgments

The second author was supported by Grant-in-Aid for Young Scientists (B) 22710136.

References

  1. 1.
    Cornuejols, G., Tütüncü, R.: Optimization Methods in Finance. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  2. 2.
    Daniels, M.J., Kass, R.E.: Shrinkage estimators for covariance matrices. Biometrics 57, 1173–1184 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dey, D.K., Srinivasan, C.: Estimation of a covariance matrix under Stein’s loss. Ann. Stat. 13, 1581–1591 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Haff, L.: The variational form of certain Bayes estimators. Ann. Stat. 19, 1163–1190 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Horn, R.H., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  6. 6.
    Horn, R.H., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  7. 7.
    James, W., Stein, C.: Estimation with quadratic loss. In: Proceedings of Fourth Berkeley Symposium on Mathematical Statistics and Probability. pp. 361–379 (1961)Google Scholar
  8. 8.
    Leodit, O., Wolf, M.: A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal. 88, 365–411 (2004)Google Scholar
  9. 9.
    Won, J.H., Kim, S.-J.: Maximum likelihood covariance estimation with a condition number constraint. In: Proceedings of Fortieth Asilomar Conference on Signals, Systems, and Computers. pp. 1445–1449 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Ookayama, Meguro-kuJapan

Personalised recommendations