Linear convergence rates for Tikhonov regularization with positively homogeneous functionals

Grasmair, Markus

doi:10.1088/0266-5611/27/7/075014

Cited by 39 publications

(45 citation statements)

References 34 publications

(63 reference statements)

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“…Also the verification of convergence rates has been addressed, but mostly in the case of sparse solutions (cf. [1][2][3][4][5][6][7][8][9][10][11][12]). For nonsparse solutions, a first convergence rate result can be found in [13].…”

Section: Introductionmentioning

confidence: 99%

Convergence rates inℓ¹-regularization when the basis is not smooth enough

Flemming

Hegland

2014

Applicable Analysis

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Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation with respect to a fixed basis. We drop this sparsity assumption and provide error estimates for nonsparse solutions. After discussing a result in this direction published earlier by one of the authors and co-authors, we prove a similar error estimate under weaker assumptions. Two examples illustrate that this set of weaker assumptions indeed covers additional situations which appear in applications.

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Section: Introductionmentioning

confidence: 99%

Convergence rates inℓ¹-regularization when the basis is not smooth enough

Flemming

Hegland

2014

Applicable Analysis

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“…In [31,32] it is shown that the RIP implies the conditions in Theorem 3.1. Moreover, the smaller supp@hA, the easier the conditions in Theorems are satisfied.…”

Section: Background From`1-minimizationmentioning

confidence: 96%

Deep Learning Versus <tex>$\ell^{1}$</tex> -Minimization for Compressed Sensing Photoacoustic Tomography

Antholzer

Schwab

Haltmeier

2018

2018 IEEE International Ultrasonics Symposium (IUS)

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We investigate compressed sensing (CS) techniques for reducing the number of measurements in photoacoustic tomography (PAT). High resolution imaging from CS data requires particular image reconstruction algorithms. The most established reconstruction techniques for that purpose use sparsity and`1-minimization. Recently, deep learning appeared as a new paradigm for CS and other inverse problems. In this paper, we compare a recently invented joint`1-minimization algorithm with two deep learning methods, namely a residual network and an approximate nullspace network. We present numerical results showing that all developed techniques perform well for deterministic sparse measurements as well as for random Bernoulli measurements. For the deterministic sampling, deep learning shows more accurate results, whereas for Bernoulli measurements thè 1 -minimization algorithm performs best. Comparing the implemented deep learning approaches, we show that the nullspace network uniformly outperforms the residual network in terms of the mean squared error (MSE).

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“…The estimation in Bregman distances has been widely accepted since then and extended to various other situations (cf. [27,28,88,98,99,107,127,160,161]). We shall here mainly recall the results obtained in [47], and start by defining Bregman distances.…”

Section: Stability Estimatesmentioning

confidence: 99%

A Guide to the TV Zoo

Burger

Osher²

2013

Level Set and PDE Based Reconstruction Methods in Imaging

118

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Total variation methods and similar approaches based on regularizations with`1-type norms (and seminorms) have become a very popular tool in image processing and inverse problems due to peculiar features that cannot be realized with smooth regularizations. In particular total variation techniques had particular success due to their ability to realize cartoon-type reconstructions with sharp edges. Due to an explosion of new developments in this field within the last decade it is a difficult task to keep an overview of the major results in analysis, the computational schemes, and the application fields. With these lectures we attempt to provide such an overview, of course biased by our major lines of research.We are focusing on the basic analysis of total variation methods and the extension of the original ROF-denoising model due various application fields. Furthermore we provide a brief discussion of state-of-the art computational methods and give an outlook to applications in different disciplines.

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Linear convergence rates for Tikhonov regularization with positively homogeneous functionals

Cited by 39 publications

References 34 publications

Convergence rates inℓ¹-regularization when the basis is not smooth enough

Convergence rates inℓ¹-regularization when the basis is not smooth enough

Deep Learning Versus <tex>$\ell^{1}$</tex> -Minimization for Compressed Sensing Photoacoustic Tomography

A Guide to the TV Zoo

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Linear convergence rates for Tikhonov regularization with positively homogeneous functionals

Cited by 39 publications

References 34 publications

Convergence rates inℓ1-regularization when the basis is not smooth enough

Convergence rates inℓ1-regularization when the basis is not smooth enough

Deep Learning Versus <tex>$\ell^{1}$</tex> -Minimization for Compressed Sensing Photoacoustic Tomography

A Guide to the TV Zoo

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Convergence rates inℓ¹-regularization when the basis is not smooth enough

Convergence rates inℓ¹-regularization when the basis is not smooth enough