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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 24, Issue 2


A unified approach to convergence rates for ℓ1-regularization and lacking sparsity

Jens Flemming / Bernd Hofmann / Ivan Veselić
Published Online: 2015-10-13 | DOI: https://doi.org/10.1515/jiip-2015-0058


In ℓ1-regularization, which is an important tool in signal and image processing, one usually is concerned with signals and images having a sparse representation in some suitable basis, e.g., in a wavelet basis. Many results on convergence and convergence rates of sparse approximate solutions to linear ill-posed problems are known, but rate results for the ℓ1-regularization in case of lacking sparsity had not been published until 2013. In the last two years, however, two articles appeared providing sufficient conditions for convergence rates in case of non-sparse but almost sparse solutions. In the present paper, we suggest a third sufficient condition, which unifies the existing two and, by the way, also incorporates the well-known restricted isometry property.

Keywords: Linear ill-posed problems; Tikhonov-type regularization; sparsity constraints; convergence rates; variational inequalities; restricted isometry property

MSC: 65J20; 47A52; 49N45

About the article

Received: 2015-06-04

Accepted: 2015-09-19

Published Online: 2015-10-13

Published in Print: 2016-04-01

Funding Source: German Research Foundation (DFG)

Award identifier / Grant number: FL 832/1-1, HO 1454/8-2, VE 253/6-1

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 24, Issue 2, Pages 139–148, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2015-0058.

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