# 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

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topHiriart-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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- [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] 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] 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] J.-J. Moreau, Propriétés des applications “prox”. C. R. Acad. Sci. Paris256 (1963) 1069–1071. Zbl0115.10802MR149244
- [7] R.T. Rockafellar and R.J.-B. Wets, Variational analysis. Springer (1998). Zbl0888.49001MR1491362
- [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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