Elastic-net regularization versus l(1)-regularization for linear inverse problems with quasi-sparse solutions
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AbstractWe consider the ill-posed operator equation Ax = y with an injective and bounded linear operator A mapping between l(2) and a Hilbert space Y, possessing the unique solution x={x(k)}(k-1)(infinity). For the cases that sparsity x is an element of l(0) is expected but often slightly violated in practice, we investigate in comparison with the l(1)-regularization the elastic-net regularization, where the penalty is a weighted superposition of the l(1)-norm and the l(2)-norm square, under the assumption that x is an element of l(1). There occur two positive parameters in this approach, the weight parameter. and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in l(1)-regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator A and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay x(k) -> 0 for k -> infinity and the classical smoothness properties of x as an element in l(2).
Acceptance Date21/11/2016
All Author(s) ListDe-Han Chen, Bernd Hofmann, Jun Zou
Journal nameInverse Problems
Year2017
Month1
Volume Number33
Issue Number1
PublisherIOP PUBLISHING LTD
Pages015004
ISSN0266-5611
eISSN1361-6420
LanguagesEnglish-United Kingdom
Keywordslinear ill-posed problems,sparsity constraints,elastic-net regularization,l(1)-regularization,convergence rates,source conditions
Web of Science Subject CategoriesMathematics, Applied;Physics, Mathematical;Mathematics;Physics

Last updated on 2021-07-12 at 23:32