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rank minimization problems

RAIRO - Operations Research (1)

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From Eckart and Young approximation to Moreau envelopes andvice versa
Jean-Baptiste Hiriart-Urruty, Hai Yen Le
Journal:

RAIRO - Operations Research / Volume 47 / Issue 3 / July 2013

Published online by Cambridge University Press:

26 August 2013, pp. 299-310

Print publication:

July 2013
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In matricial analysis, the theorem of Eckart and Young provides a best approximation ofan arbitrary matrix by a matrix of rank at most r. In variationalanalysis or optimization, the Moreau envelopes are appropriate ways of approximating orregularizing the rank function. We prove here that we can go forwards and backwardsbetween the two procedures, thereby showing that they carry essentially the sameinformation.