Adaptive Greedy Approximations

Davis, G.

doi:10.1007/s003659900033

Cited by 187 publications

(268 citation statements)

References 0 publications

Supporting

Mentioning

263

Contrasting

Unclassified

Order By: Relevance

“…By assumption of the lemma 11) and sup ρ∈ c |P (2) ρ | ≤ b under condition (5.9) as desired. Also it follows from (5.1) that κ < 1 as S ≥ 1 without loss of generality (if = ∅ then x = 0 and 1 -minimization will clearly recover x.…”

Section: A General Recovery Lemmamentioning

confidence: 97%

See 1 more Smart Citation

Sparsity in Time-Frequency Representations

Pfander

Rauhut

2009

J Fourier Anal Appl

View full text Add to dashboard Cite

We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an S-sparse Gabor representation in C n with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S ≤ Cn/ log(n). Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.

show abstract

Section: A General Recovery Lemmamentioning

confidence: 97%

“…(1.5) Unfortunately, this problem is NP hard in general [11], and hence, is not feasible in practice. In order to avoid this computational bottleneck, several alternative reconstruction methods have been suggested as mentioned above.…”

Section: Objectivementioning

confidence: 99%

Sparsity in Time-Frequency Representations

Pfander

Rauhut

2009

J Fourier Anal Appl

View full text Add to dashboard Cite

show abstract

“…Signals having a sparse B Ljubiša Stanković ljubisa@ac.me 1 University of Montenegro, 81000 Podgorica, Montenegro representation can be reconstructed from a reduced subset of randomly positioned samples. Processing of these signals with a large number of missing/unavailable samples attracted significant interest in the recent years within the theory of compressive sensing (CS) [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]22,30,31,33,34,36]. The number of samples required to reconstruct the signal is related to the number of nonzero coefficients in the sparse domain [4,12,17].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Sparse Signals in Impulsive Disturbance Environments

Stanković

Daković

Vujovic

2016

Circuits Syst Signal Process

View full text Add to dashboard Cite

Sparse signals corrupted by impulsive disturbances are considered. The assumption about disturbances is that they degrade the original signal sparsity. No assumption about their statistical behavior or range of values is made. In the first part of the paper, it is assumed that some uncorrupted signal samples exist. A criterion for selection of corrupted signal samples is proposed. It is based on the analysis of the first step of a gradient-based iterative algorithm used in the signal reconstruction. An iterative extension of the original criterion is introduced to enhance its selection property. Based on this criterion, the corrupted signal samples are efficiently removed. Then, the compressive sensing theory-based reconstruction methods are used for signal recovery, along with an appropriately defined criterion to detect a full recovery event among different realizations. In the second part of the paper, a case when all signal samples are corrupted by an impulsive disturbance is considered as well. Based on the defined criterion, the most heavily corrupted samples are removed. The presented criterion and the reconstruction algorithm are applied on the signal with a Gaussian noise.

show abstract

“…Of course, the key point is the capability to choose the right terminology. Back to mathematical terms, the combination of adaptivity (i.e., the capability to choose the right terminology) and redundancy (i.e., the richness or non-uniqueness of representations) indeed gives rise to compressed and accurate approximations [20,25,29,38,39]. Numerical experiments in [12,13,44] show that frames improve conditioning without increasing the effective dimension of the problem, and that the frame approach has optimal complexity.…”

Section: Introductionmentioning

confidence: 99%

Adaptive frame methods for nonlinear variational problems

2008

View full text Add to dashboard Cite

In this paper we develop adaptive numerical solvers for certain nonlinear variational problems. The discretization of the variational problems is done by a suitable frame decomposition of the solution, i.e., a complete, stable, and redundant expansion. The discretization yields an equivalent nonlinear problem on the space of frame coefficients. The discrete problem is then adaptively solved using approximated nested fixed point and Richardson type iterations. We investigate the convergence, stability, and optimal complexity of the scheme. A theoretical advantage, for example, with respect to adaptive finite element schemes is that convergence and complexity results for the latter are usually hard to prove. The use of frames is further motivated by their redundancy, which, at least numerically, has been shown to improve the M. Fornasier acknowledges the financial support provided through the Intra-European Individual Marie Curie Fellowship Programme, under contract MOIF-CT-2006-039438. All of the authors acknowledge the hospitality of Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma "La Sapienza", Italy, during the early preparation of this work. The authors want to thank Daniele Boffi, Dorina Mitrea, and Karsten Urban for the helpful and fruitful discussions on divergence-free function spaces. 123 46 M. Charina et al.conditioning of the discretization matrices. Also frames are usually easier to construct than Riesz bases. We present a construction of divergence-free wavelet frames suitable for applications in fluid dynamics and magnetohydrodynamics. Mathematics Subject Classification (2000)41A46 · 42C15 · 42C40 · 46E35 · 65J15 · 65N12 · 65N99 · 76D03 · 76D05 · 76W05

show abstract

Adaptive Greedy Approximations

Cited by 187 publications

References 0 publications

Sparsity in Time-Frequency Representations

Sparsity in Time-Frequency Representations

Reconstruction of Sparse Signals in Impulsive Disturbance Environments

Adaptive frame methods for nonlinear variational problems

Contact Info

Product

Resources

About