Iterative thresholding algorithms

Massimo Fornasier; Holger Rauhut

doi:10.1016/j.acha.2007.10.005

Iterative thresholding algorithms

Massimo Fornasier, Holger Rauhut

Research output: Contribution to journal › Article › peer-review

154 Scopus citations

Abstract

This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an algorithm recently studied by the authors. We prove stability of minimizers with respect to the parameters of the functional by means of Γ-convergence. All investigations are done in the general setting of vector-valued (multi-channel) data.

Original language	English
Pages (from-to)	187-208
Number of pages	22
Journal	Applied and Computational Harmonic Analysis
Volume	25
Issue number	2
DOIs	https://doi.org/10.1016/j.acha.2007.10.005
State	Published - Sep 2008
Externally published	Yes

Keywords

Frames
Joint sparsity
Linear inverse problems
Thresholded Landweber iterations
Variational calculus on sequence spaces
Γ-convergence

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10.1016/j.acha.2007.10.005

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title = "Iterative thresholding algorithms",

abstract = "This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an algorithm recently studied by the authors. We prove stability of minimizers with respect to the parameters of the functional by means of Γ-convergence. All investigations are done in the general setting of vector-valued (multi-channel) data.",

keywords = "Frames, Joint sparsity, Linear inverse problems, Thresholded Landweber iterations, Variational calculus on sequence spaces, Γ-convergence",

author = "Massimo Fornasier and Holger Rauhut",

note = "Funding Information: * Corresponding author. E-mail addresses: [email protected] (M. Fornasier), [email protected] (H. Rauhut). 1 M.F. acknowledges the financial support provided by the European Union{\textquoteright}s Human Potential Programme under contract MOIF-CT-2006-039438. 2 H.R. was supported by the Individual Marie-Curie Fellowship MEIF-CT 2006-022811.",

year = "2008",

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doi = "10.1016/j.acha.2007.10.005",

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AU - Rauhut, Holger

N1 - Funding Information: * Corresponding author. E-mail addresses: [email protected] (M. Fornasier), [email protected] (H. Rauhut). 1 M.F. acknowledges the financial support provided by the European Union’s Human Potential Programme under contract MOIF-CT-2006-039438. 2 H.R. was supported by the Individual Marie-Curie Fellowship MEIF-CT 2006-022811.

PY - 2008/9

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N2 - This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an algorithm recently studied by the authors. We prove stability of minimizers with respect to the parameters of the functional by means of Γ-convergence. All investigations are done in the general setting of vector-valued (multi-channel) data.

AB - This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an algorithm recently studied by the authors. We prove stability of minimizers with respect to the parameters of the functional by means of Γ-convergence. All investigations are done in the general setting of vector-valued (multi-channel) data.

KW - Frames

KW - Joint sparsity

KW - Linear inverse problems

KW - Thresholded Landweber iterations

KW - Variational calculus on sequence spaces

KW - Γ-convergence

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DO - 10.1016/j.acha.2007.10.005

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VL - 25

SP - 187

EP - 208

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 2

ER -

Iterative thresholding algorithms

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Keywords

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