Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T10:43:39.129Z Has data issue: false hasContentIssue false
  • English
  • Français

From Eckart and Young approximation to Moreau envelopes andvice versa

Published online by Cambridge University Press:  26 August 2013

Jean-Baptiste Hiriart-Urruty  and
Hai Yen Le
Jean-Baptiste Hiriart-Urruty
Affiliation:
Institute of Mathematics, Paul Sabatier University, 118 Route de Narbonne, 31400 Toulouse, France.. [email protected]; [email protected] ; http://www.math.univ-toulouse.fr/˜jbhu/
Hai Yen Le
Affiliation:
Institute of Mathematics, Paul Sabatier University, 118 Route de Narbonne, 31400 Toulouse, France.. [email protected]; [email protected] ; http://www.math.univ-toulouse.fr/˜jbhu/
Get access

Abstract

In matricial analysis, the theorem of Eckart and Young provides a best approximation ofan arbitrary matrix by a matrix of rank at most r. In variationalanalysis or optimization, the Moreau envelopes are appropriate ways of approximating orregularizing the rank function. We prove here that we can go forwards and backwardsbetween the two procedures, thereby showing that they carry essentially the sameinformation.

Keywords

Eckart and Young theoremmoreau envelopesrank minimization problems
Type
Research Article
Information
RAIRO - Operations Research , Volume 47 , Issue 3 , July 2013 , pp. 299 - 310
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

U. Helmke and J.B. Moore, Optimization and Dynamical Systems. Spinger Verlag (1994).
N. Higham, Matrix nearness problems and applications, in M.J.C Gover and S. Barnett, eds., Applications of Matrix Theory. Oxford University Press (1989) 1–27.
J.-B. Hiriart-Urruty and H.Y. Le, A variational approach of the rank function. TOP (2013) DOI: 10.1007/s11750-013-0283-y.
Hiriart-Urruty, J.-B. and Malick, J., A fresh variational analysis look at the world of the positive semidefinite matrices. J. Optim. Theory Appl. 153 (2012) 551–577. Google Scholar
Moreau, J.-J., Fonctions convexes duales et points proximaux dans un espace hilbertien. (French) C. R. Acad. Sci. Paris 255 (1962) 2897–2899 (Reviewer: I.G. Amemiya) 46.90. Google Scholar
Moreau, J.-J., Propriétés des applications “prox”. C. R. Acad. Sci. Paris 256 (1963) 1069–1071.Google Scholar
R.T. Rockafellar and R.J.-B. Wets, Variational analysis. Springer (1998).
G.W. Stewart, Matrix algorithms, Basic decompositions, Vol. I. Society for Industrial and Applied Mathematics, Philadelphia, PA (1998).