Advertisement

TOP

, Volume 21, Issue 2, pp 207–240 | Cite as

A variational approach of the rank function

  • Jean-Baptiste Hiriart-Urruty
  • Hai Yen LeEmail author
Invited Paper

Abstract

In the same spirit as the one of the paper (Hiriart-Urruty and Malick in J. Optim. Theory Appl. 153(3):551–577, 2012) on positive semidefinite matrices, we survey several useful properties of the rank function (of a matrix) and add some new ones. Since the so-called rank minimization problems are the subject of intense studies, we adopt the viewpoint of variational analysis, that is the one considering all the properties useful for optimizing, approximating or regularizing the rank function.

Keywords

Rank of matrix Optimization Nonsmooth analysis Moreau–Yosida regularization Generalized subdifferentials 

Mathematics Subject Classification

49-02 65K10 

Notes

Acknowledgements

We thank the various readers or referees who, after the first circulation of this survey paper (July 2012), provided comments, improvements and additional references.

Last but not least, we would like to thank Professor M. Goberna (University of Alicante) for his instrumental role in the publication of this survey paper.

References

  1. Absil P-A, Malick J (2012) Projection-like retractions on matrix manifolds. SIAM J Optim 22(1):135–158 CrossRefGoogle Scholar
  2. Attouch H, Bolte J, Svaiter BF (2011) Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss–Seidel method. Math Program. doi: 10.1007/s10107-011-0484-9 Google Scholar
  3. Barioli F (2002) Completely positive matrices of small and large acute sets of vectors. Talk at the 10th ILAS Conference in Auburn Google Scholar
  4. Barioli F, Berman A (2003) The maximal cp-rank of rank k completely positive matrices. Linear Algebra Appl 363:17–33 CrossRefGoogle Scholar
  5. Behrends E, Geschke S, Natkaniec T (2007) Functions for which all points are local extrema. Real Anal Exch 33(2):467–470 Google Scholar
  6. Berman A, Shaked-Monderer N (1998) Remarks on completely positive matrices. Linear Multilinear Algebra 44(2):149–163 CrossRefGoogle Scholar
  7. Berman A, Shaked-Monderer N (2003) Completely positive matrices. World Scientific, Singapore CrossRefGoogle Scholar
  8. Bernstein DS (2005) Matrix mathematics: theory, facts, and formulas. Princeton University Press, Princeton Google Scholar
  9. Bomze I (2012) Copositive optimization—recent developments and applications. Eur J Oper Res 216:509–520 CrossRefGoogle Scholar
  10. Bruckstein AM, Donoho DL, Elad M (2009) From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev 51(1):34–81 CrossRefGoogle Scholar
  11. Clarke FH (1983) Optimisation and nonsmooth analysis. Wiley, New York Google Scholar
  12. Clarke FH, Ledyaev YS, Stern RT, Wolenski PR (1998) Nonsmooth analysis and control theory. Springer, Berlin Google Scholar
  13. Donoho D, Elad M (2003) Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. Publ Natl Acad Sci 100(5):2197–2202 CrossRefGoogle Scholar
  14. Drew JH, Johnson CR, Loewy R (1994) Completely positive matrices associated with M-matrices. Linear Multilinear Algebra 37:303–310 CrossRefGoogle Scholar
  15. Drusvyatskiy D, Lewis AS (2013) Semi-algebraic functions have small subdifferentials. Math Program Ser B. doi: 10.1007/s10107-012-0624-x. Special issue in honor of C. Lemaréchal Google Scholar
  16. Drusvyatskiy D, Ioffe AD, Lewis AS (2012) The dimension of semialgebraic subdifferential graphs. Nonlinear Anal 75:1231–1245 CrossRefGoogle Scholar
  17. Dür M (2010) Copositive programming—a survey. In: Recent advances in optimization and its applications in engineering, pp 3–20. Part 1. doi: 10.1007/978-3-642-12598-0-1 CrossRefGoogle Scholar
  18. Eckart C, Young G (1936) The approximation of one matrix by another of lower rank. Psychometrika 1:211–218 CrossRefGoogle Scholar
  19. Elad M (2010) Sparse and redundant representations: from theory to applications in signal and image processing. Springer, Berlin CrossRefGoogle Scholar
  20. Fazel M (2002) Matrix rank minimization with applications. Ph.D. thesis, Stanford University Google Scholar
  21. Flores S (2011) Problèmes d’optimisation globale en statistique robuste. Ph.D. thesis, Paul Sabatier University, Toulouse Google Scholar
  22. Hanna J, Laffey TJ (1983) Nonnegative factorization of completely positive matrices. Linear Algebra Appl 55:1–9 CrossRefGoogle Scholar
  23. Higham N (1989) Matrix nearness problems and applications. In: Gover MJC, Barnett S (eds) Applications of matrix theory. Oxford University Press, London, pp 1–27 Google Scholar
  24. Hiriart-Urruty J-B (2009) Deux questions de rang. Working Note, Paul Sabatier University Google Scholar
  25. Hiriart-Urruty J-B (2012) When only global optimization matters. J Glob Optim. doi: 10.1007/s10898-011-9826-7 Google Scholar
  26. Hiriart-Urruty J-B, Le HY (2011) Convexifying the set of matrices of bounded rank: applications to the quasiconvexification and convexification of the rank function. Optim Lett. doi: 10.1007/s11590-011-0304-4 Google Scholar
  27. Hiriart-Urruty J-B, Le HY (2012) From Eckart & Young approximation to Moreau envelopes and vice versa. Preprint (submitted) Google Scholar
  28. Hiriart-Urruty J-B, Lemaréchal C (1993) Convex analysis and minimization algorithms. Springer, Berlin Google Scholar
  29. Hiriart-Urruty J-B, Malick J (2012) A fresh variational analysis look at the world of the positive semidefinite matrices. J Optim Theory Appl 153(3):551–577 CrossRefGoogle Scholar
  30. Hiriart-Urruty J-B, Seeger A (2010) A variational approach to copositive matrices. SIAM Rev 54(4):593–629 CrossRefGoogle Scholar
  31. Horn RA, Johnson CR (1975) Matrix analysis. Cambridge University Press, Cambridge Google Scholar
  32. Jourani A, Thibault L, Zagrodny D (2012) Differential properties of Moreau envelope. Preprint Google Scholar
  33. Kruskal JB (1977) Three-way arrays: rank and uniqueness of trilinear decomposition, with application to arithmetic complexity and statistics. Linear Algebra Appl 18(2):95–138 CrossRefGoogle Scholar
  34. Le HY (2012a) Convexifying the counting function on \(\mathbb{R}^{p}\) for convexifying the rank function on \(\mathcal{M}_{m,n}(\mathbb{R})\). J Convex Anal 19(2) Google Scholar
  35. Le HY (2012b) The generalized subdifferentials of the rank function. Optim Lett. doi: 10.1007/s11590-012-0456-x Google Scholar
  36. Le HY (2013) Ph.D. Thesis. Paul Sabatier University. To be defended in 2013 Google Scholar
  37. Lewis AS (1995) The convex analysis of unitarily invariant matrix functions. J Convex Anal 2(1):173–183 Google Scholar
  38. Lewis AS (1996) Convex analysis on the Hermitian matrices. SIAM J Optim 6:164–177 CrossRefGoogle Scholar
  39. Lewis AS, Mallick J (2008) Alternating projections on manifolds. Math Oper Res 33:216–234 CrossRefGoogle Scholar
  40. Lewis AS, Sendov HS (2005a) Nonsmooth analysis of singular values. Part I: theory. Set-Valued Anal 13:213–241 CrossRefGoogle Scholar
  41. Lewis AS, Sendov HS (2005b) Nonsmooth analysis of singular values. Part II: applications. Set-Valued Anal 13:243–264 CrossRefGoogle Scholar
  42. Lim L-H, Common P (2010) Multiarray signal processing: tensor decomposition meets compressed sensing. Preprint Google Scholar
  43. Liu Y-J, Sun D, Toh K-C (2012) An implementable proximal point algorithmic framework for nuclear norm minimization. Math Program 133:399–436 CrossRefGoogle Scholar
  44. Loewy R, Tam B-S (2003) CP-rank of completely positive matrices of order 5. Linear Algebra Appl 363:161–176 CrossRefGoogle Scholar
  45. Lokam SV (2001) Spectral methods for matrix rigidity with applications to size-depth trade-offs and communication complexity. J Comput Syst Sci 63:449–473 CrossRefGoogle Scholar
  46. Luke DR (2012) Prox-regularity of rank constraint sets and implications for algorithms. J Math Imaging Vis. doi: 10.1007/s10851-012-0406-3 Google Scholar
  47. Malick J (2007) The spherical constraint in Boolean quadratic programs. J Glob Optim 39:609–622 CrossRefGoogle Scholar
  48. Mirsky L (1960) Symmetric gauge functions and unitary invariant norms. Q J Math 11:50–59 CrossRefGoogle Scholar
  49. Parrilo PA (2009) The convex algebraic geometry of rank minimization. Plenary talk at the International symposium on mathematical programming, Chicago Google Scholar
  50. Recht B, Fazel M, Parrilo PA (2010) Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev 52(3):471–501 CrossRefGoogle Scholar
  51. Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin CrossRefGoogle Scholar
  52. Stewart GW (1993) On the early history of the singular value decomposition. SIAM Rev 35(4):551–556 CrossRefGoogle Scholar
  53. Valiant LG (1977) Graph theoretic arguments in low-level complexity. In: Lecture notes in computer science, vol 53. Springer, Berlin, pp 162–176 Google Scholar
  54. Zhao Y-B (2012) An approximation theory of matrix rank minimization and its application to quadratic equations. Linear Algebra Appl 77–93 Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2013

Authors and Affiliations

  1. 1.Institute of MathematicsPaul Sabatier UniversityToulouseFrance

Personalised recommendations