Convergence rates in ℓ1-regularization if the sparsity assumption fails

Burger M, Flemming J, Hofmann B (2013)

Publication Language: English

Publication Type: Journal article

Publication year: 2013

Journal

Inverse Problems Institute of Physics: Hybrid Open Access

Book Volume: 29

Issue: 2

DOI: 10.1088/0266-5611/29/2/025013

Abstract

Variational sparsity regularization based on ℓ1-norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization parameter and noise strength. For this sake, specific error measures such as Bregman distances and specific conditions on the solution such as source conditions or variational inequalities have been developed and used. In this paper we provide, for a certain class of ill-posed linear operator equations, a convergence analysis that works for solutions that are not completely sparse, but have a fast-decaying nonzero part. This case is not covered by standard source conditions, but surprisingly can be treated with an appropriate variational inequality. As a consequence, the paper also provides the first examples where the variational inequality approach, which was often believed to be equivalent to appropriate source conditions, can indeed go farther than the latter. © 2013 IOP Publishing Ltd.

Authors with CRIS profile

Martin Burger

Involved external institutions

Technische Universität Chemnitz DE Germany (DE) Westfälische Wilhelms-Universität (WWU) Münster DE Germany (DE)

How to cite

APA:

Burger, M., Flemming, J., & Hofmann, B. (2013). Convergence rates in ℓ1-regularization if the sparsity assumption fails. Inverse Problems, 29. https://dx.doi.org/10.1088/0266-5611/29/2/025013

MLA:

Burger, Martin, Jens Flemming, and Bernd Hofmann. "Convergence rates in ℓ1-regularization if the sparsity assumption fails." Inverse Problems 29 (2013).

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