Skip to main content
Springer Nature Link
Log in

Approximation of rank function and its application to the nearest low-rank correlation matrix

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The rank function rank(.) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(.), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization problems with positive semidefinite cone constraints, and illustrate its application by computing the nearest low-rank correlation matrix. Numerical results indicate that this convex relaxation method is comparable with the sequential semismooth Newton method (Li and Qi in SIAM J Optim 21:1641–1666, 2011) and the majorized penalty approach (Gao and Sun, 2010) in terms of the quality of solutions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bhatia R.: Matrix Analysis. Springer, New York (1997)

    Book  Google Scholar 

  2. Boyd S., Xiao L.: Least-squares covariance matrix adjustment. SIAM J. Matrix Anal. Appl. 27, 532–546 (2005)

    Article  Google Scholar 

  3. Brigo, D.: A note on correlation and rank reduction. http://www.damianobrigo.it (2002)

  4. Burer S., Monteiro R.D.C., Zhang Y.: Maximum stable set formulations and heuristics based on continuous optimization. Math. Program. 94, 137–166 (2002)

    Article  Google Scholar 

  5. Candès E.J., Recht B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)

    Article  Google Scholar 

  6. Cai J.F., Candès E.J., Shen Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956–1982 (2010)

    Article  Google Scholar 

  7. Chao, D., Sun, D.F., Toh K.-C.: An introduction to a class of matrix cone programming. Technical report, http://www.math.nus.edu.sg/ (2010)

  8. Chen C., Mangasarian O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97–138 (1996)

    Article  Google Scholar 

  9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) (reprinted by SIAM, Philadelphia, PA, 1990)

  10. Eldén L.: Matrix Methods in Data Mining and Pattern Recognition (Fundamentals of Algorithms). SIAM, Philadelphia, PA, USA (2007)

    Book  Google Scholar 

  11. Fazel, M.: Matrix rank minimization with applications. PhD thesis, Stanford University (2002)

  12. Fazel, M., Hindi, H.,Boyd, S.: Log-det Heuirstic for Matrix Rank Minimization with Applicatios to Hankel and Euclidean Distance Matrices. ACC03-IEEE06884 (2003)

  13. Fazel, M., Hindi, H., Boyd, S.: Rank minimization and applications in system theory. In: Proceeding of the American Control Conference, vol. 4, pp. 3273–3278 (2004)

  14. Gao, Y., Sun, D.F.: A majorized penalty approach for calibrating rank constrained correlation matrix problems. Technical report (2010)

  15. Goemans, M.X., Williamson, D.P.: 0.878-approximation algorithms for MAX CUT and MAX 2SAT. Lecture Notes Computer Science, pp. 422–431 (1994)

  16. Grubiŝić I., Pietersz R.: Efficient rank reduction of correlation matrices. Linear Algebra Appl. 422, 629–653 (2007)

    Article  Google Scholar 

  17. Hestenes M.R., Stiefel E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bureau Stand. 49, 409–436 (1952)

    Article  Google Scholar 

  18. Higham N.J.: Computing the nearest correlation matrix-a problem from finance. IMA J. Numer. Anal. 22, 329–343 (2002)

    Article  Google Scholar 

  19. Hiriart-Urruty J.B., Lemaréchal C.: Convex Analysis and Minimization Algorithm I. Springer, Berlin Heidelberg (1993)

    Google Scholar 

  20. Lewis A.S.: Derivatives of spectral functions. Math. Oper. Res. 21, 576–588 (1996)

    Article  Google Scholar 

  21. Lewis A.S.: The convex analysis of unitarily invariant matrix functions. J. Convex Anal. 2, 173–183 (1995)

    Google Scholar 

  22. Li Q.N., Qi H.D.: A sequential semismooth Newton method for the nearest low-rank correlation matrix problem. SIAM J. Optim. 21, 1641–1666 (2011)

    Article  Google Scholar 

  23. Linial N., London E., Rabinovich Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 215–245 (1995)

    Article  Google Scholar 

  24. Liu Z., Vandenberghe L.: Interior-point method for nuclear norm approximation with application to system identification. SIAM J. Matrix Anal. Appl. 31, 1235–1256 (2009)

    Article  Google Scholar 

  25. Liu Y.-J., Sun D.F., Toh K.C.: An implementable proximal point algorithmic framework for nuclear norm minimization. Math. Program. 133, 399–436 (2012)

    Article  Google Scholar 

  26. Ma S., Goldfarb D., Chen L.: Fixed point and Bregman iterative methods for matrix rank minimization. Math. Program. 128, 321–353 (2011)

    Article  Google Scholar 

  27. Malick J.: A dual approach to semidefinite least-squares problems. SIAM J. Matrix Anal. Appl. 26, 272–284 (2004)

    Article  Google Scholar 

  28. Mesbahi M.: On the rank minimization problem and its control applications. Syst. Control Lett. 33, 31–36 (1998)

    Article  Google Scholar 

  29. Morita M., Kanade T.: A sequential factorization method for recovering shape and motion from image streams. IEEE Trans. Pattern Anal. Mach. Intell. 19, 858–867 (1997)

    Article  Google Scholar 

  30. Natsoulis G., Pearson C.I., Gollub J., Eynon B.P., Ferng J., Nair R., Idury R., Lee M.D., Fielden M.R., Brennan R.J., Roter A.H., Jarnagin K.: The liver pharmacological and xenobiotic gene response repertoire. Mol. Syst. Biol. 4, 1–12 (2008)

    Article  Google Scholar 

  31. Pietersz R., Grubiŝić I.: Rank reduction of correlation matrices by majorization. Quant. Financ. 4, 649–662 (2004)

    Article  Google Scholar 

  32. Qi H.D., Sun D.F.: A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM J. Matrix Anal. Appl. 28, 360–385 (2006)

    Article  Google Scholar 

  33. Rebonato R.: Modern Princing and Interest-Rate Derivatives. Princeton University Press, New Jersey (2002)

    Google Scholar 

  34. Rebonato R.: Interest-rate term-structure pricing models: a review. Proc. R. Soc. Lond. Ser. A 460, 667–728 (2004)

    Article  Google Scholar 

  35. Recht B., Fazel M., Parrilo P.: Guaranteed minimum rank solutions of matrix equations via nuclear norm minimization. SIAM Rev. 52, 471–501 (2010)

    Article  Google Scholar 

  36. Rennie, J.D.M., Srebro, N.: Fast maximum margin matrix factorization for collaborative prediction. In: Proceeding of the 22nd International Conference on Machine, pp. 713–719 (2005)

  37. Sun D.F., Sun J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002)

    Article  Google Scholar 

  38. Sun D.F.: The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761–776 (2006)

    Article  Google Scholar 

  39. Sun D.F., Sun J: Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems. SIAM J. Numer. Anal. 40, 2352–2367 (2003)

    Article  Google Scholar 

  40. Toh K.C.: An inexact path-following algorithm for convex quadratic SDP. Math. Program. 112, 221–254 (2008)

    Article  Google Scholar 

  41. Toh K.C., Yun S.W.: An accelerated proximal gradient algorithm for nuclear norm regularized linear squares problems. Pac. J. Optim. 6, 615–640 (2010)

    Google Scholar 

  42. Tomasi C., Kanade T: Shape and motion from image streams under orthography: a factorization method. Int. J. Comput. Vis. 9, 137–154 (1992)

    Article  Google Scholar 

  43. Wu L.: Fast at-the-money calibration of the LIBOR market model using Lagrangian multipliers. J. Comput. Financ. 6, 39–77 (2003)

    Google Scholar 

  44. Zhang Z., Wu L.: Optimal low-rank approximation to a correlation matrix. Linear Algebra Appl. 364, 161–187 (2003)

    Article  Google Scholar 

  45. Zhao Y.B.: Approximation theory of matrix rank minimization and its application to quadratic equations. Linear Algebra Appl. 437, 77–93 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Tianhe District, Guangzhou, China

    Shujun Bi, Le Han & Shaohua Pan

Authors
  1. Shujun Bi
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Le Han
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Shaohua Pan
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Shujun Bi.

Additional information

This work was supported by National Young Natural Science Foundation (No. 10901058) and the Fundamental Research Funds for the Central Universities, and Project of Liaoning Innovative Research Team in University (WT2010004).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bi, S., Han, L. & Pan, S. Approximation of rank function and its application to the nearest low-rank correlation matrix. J Glob Optim 57, 1113–1137 (2013). https://doi.org/10.1007/s10898-012-0007-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-0007-0

Keywords