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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.
@article{RO_2013__47_3_299_0,
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},
volume = {47},
number = {3},
year = {2013},
doi = {10.1051/ro/2013040},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ro/2013040/}
}
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 SP - 299 EP - 310 VL - 47 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ro/2013040/ DO - 10.1051/ro/2013040 LA - en ID - RO_2013__47_3_299_0 ER -
%0 Journal Article %A Hiriart-Urruty, Jean-Baptiste %A Le, Hai Yen %T From Eckart and Young approximation to Moreau envelopes and vice versa %J RAIRO - Operations Research - Recherche Opérationnelle %D 2013 %P 299-310 %V 47 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ro/2013040/ %R 10.1051/ro/2013040 %G en %F RO_2013__47_3_299_0
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
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