From Eckart and Young approximation to Moreau envelopes and vice versa

Jean-Baptiste Hiriart-Urruty; Hai Yen Le

RAIRO - Operations Research - Recherche Opérationnelle (2013)

  • Volume: 47, Issue: 3, page 299-310
  • ISSN: 0399-0559

Abstract

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In matricial analysis, the theorem of Eckart and Young provides a best approximation of an arbitrary matrix by a matrix of rank at most r. In variational analysis or optimization, the Moreau envelopes are appropriate ways of approximating or regularizing the rank function. We prove here that we can go forwards and backwards between the two procedures, thereby showing that they carry essentially the same information.

How to cite

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Hiriart-Urruty, Jean-Baptiste, and Le, Hai Yen. "From Eckart and Young approximation to Moreau envelopes and vice versa." RAIRO - Operations Research - Recherche Opérationnelle 47.3 (2013): 299-310. <http://eudml.org/doc/275069>.

@article{Hiriart2013,
abstract = {In matricial analysis, the theorem of Eckart and Young provides a best approximation of an arbitrary matrix by a matrix of rank at most r. In variational analysis or optimization, the Moreau envelopes are appropriate ways of approximating or regularizing the rank function. We prove here that we can go forwards and backwards between the two procedures, thereby showing that they carry essentially the same information.},
author = {Hiriart-Urruty, Jean-Baptiste, Le, Hai Yen},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {Eckart and Young theorem; moreau envelopes; rank minimization problems; Moreau envelope},
language = {eng},
number = {3},
pages = {299-310},
publisher = {EDP-Sciences},
title = {From Eckart and Young approximation to Moreau envelopes and vice versa},
url = {http://eudml.org/doc/275069},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Hiriart-Urruty, Jean-Baptiste
AU - Le, Hai Yen
TI - From Eckart and Young approximation to Moreau envelopes and vice versa
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 3
SP - 299
EP - 310
AB - In matricial analysis, the theorem of Eckart and Young provides a best approximation of an arbitrary matrix by a matrix of rank at most r. In variational analysis or optimization, the Moreau envelopes are appropriate ways of approximating or regularizing the rank function. We prove here that we can go forwards and backwards between the two procedures, thereby showing that they carry essentially the same information.
LA - eng
KW - Eckart and Young theorem; moreau envelopes; rank minimization problems; Moreau envelope
UR - http://eudml.org/doc/275069
ER -

References

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  1. [1] U. Helmke and J.B. Moore, Optimization and Dynamical Systems. Spinger Verlag (1994). Zbl0943.93001MR1299725
  2. [2] 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. Zbl0681.65029MR1041063
  3. [3] J.-B. Hiriart-Urruty and H.Y. Le, A variational approach of the rank function. TOP (2013) DOI: 10.1007/s11750-013-0283-y. Zbl1269.49019MR3068480
  4. [4] J.-B. Hiriart-Urruty and J. Malick, A fresh variational analysis look at the world of the positive semidefinite matrices. J. Optim. Theory Appl.153 (2012) 551–577. Zbl1254.90166MR2915584
  5. [5] J.-J. Moreau, 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. Zbl0118.10502MR144188
  6. [6] J.-J. Moreau, Propriétés des applications “prox”. C. R. Acad. Sci. Paris256 (1963) 1069–1071. Zbl0115.10802MR149244
  7. [7] R.T. Rockafellar and R.J.-B. Wets, Variational analysis. Springer (1998). Zbl0888.49001MR1491362
  8. [8] G.W. Stewart, Matrix algorithms, Basic decompositions, Vol. I. Society for Industrial and Applied Mathematics, Philadelphia, PA (1998). Zbl0910.65012MR1653546

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