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From Eckart and Young approximation to Moreau envelopes and vice versa

Published online by Cambridge University Press:  26 August 2013

Jean-Baptiste Hiriart-Urruty
Affiliation:
Institute of Mathematics, Paul Sabatier University, 118 Route de Narbonne, 31400 Toulouse, France.. jbhu@math.univ-toulouse.fr; hyle@math.univ-toulouse.fr ; http://www.math.univ-toulouse.fr/˜jbhu/
Hai Yen Le
Affiliation:
Institute of Mathematics, Paul Sabatier University, 118 Route de Narbonne, 31400 Toulouse, France.. jbhu@math.univ-toulouse.fr; hyle@math.univ-toulouse.fr ; http://www.math.univ-toulouse.fr/˜jbhu/
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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.

Type
Research Article
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

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