From Eckart and Young approximation to Moreau envelopes and vice versa
RAIRO - Operations Research - Recherche Opérationnelle, Tome 47 (2013) no. 3, pp. 299-310.

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.

DOI : 10.1051/ro/2013040
Classification : 15A, 46N10, 65K10, 90C
Mots-clés : Eckart and Young theorem, moreau envelopes, rank minimization problems 
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     author = {Hiriart-Urruty, Jean-Baptiste and Le, Hai Yen},
     title = {From {Eckart} and {Young} approximation to {Moreau} envelopes and \protect\emph{vice versa}},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {299--310},
     publisher = {EDP-Sciences},
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     url = {http://www.numdam.org/articles/10.1051/ro/2013040/}
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Hiriart-Urruty, Jean-Baptiste; Le, Hai Yen. From Eckart and Young approximation to Moreau envelopes and vice versa. RAIRO - Operations Research - Recherche Opérationnelle, Tome 47 (2013) no. 3, pp. 299-310. doi : 10.1051/ro/2013040. http://www.numdam.org/articles/10.1051/ro/2013040/

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