Subspace Pursuit for Compressive Sensing Signal Reconstruction

Dai, Wei; Milenković, Olgica

doi:10.1109/tit.2009.2016006

Cited by 2,094 publications

(1,645 citation statements)

References 21 publications

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“…IP address: 52.183.12.225, on 11 Apr 2019 at 13:59:57, subject to the Cambridge Core of exact recovery of sparse signals and the Lebesgue-type inequalities for these algorithms: the regularized orthogonal matching pursuit (see [12]), the compressive sampling matching pursuit (CoSaMP; see [11]), and the subspace pursuit (SP; see [1]). The OMP is simpler than the CoSaMP and the SP; however, at the time of invention of the CoSaMP and the SP these algorithms provided exact recovery of sparse signals and the Lebesgue-type inequalities for dictionaries satisfying the restricted isometry property (RIP) (see [11] and [1]). The corresponding results for the OMP were not known at that time.…”

Section: Discussionmentioning

confidence: 99%

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Temlyakov

2014

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We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in L p , 2 p < ∞.

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Section: Discussionmentioning

confidence: 99%

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Temlyakov

2014

Forum math. Sigma

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“…Note that many greedy algorithms have these properties (e.g. [12,19,20]), but relaxation techniques such as ℓ 1 -minimization [5] or the Dantzig selector [21] are not guaranteed to give a k-sparse result.…”

Section: Rip Dictionariesmentioning

confidence: 99%

Greedy signal space methods for incoherence and beyond

Giryes

Needell

2015

Applied and Computational Harmonic Analysis

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ABSTRACT. Compressive sampling (CoSa) has provided many methods for signal recovery of signals compressible with respect to an orthonormal basis. However, modern applications have sparked the emergence of approaches for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary. Recently, several "signal-space" greedy methods have been proposed to address signal recovery in this setting. However, such methods inherently rely on the existence of fast and accurate projections which allow one to identify the most relevant atoms in a dictionary for any given signal, up to a very strict accuracy. When the dictionary is highly overcomplete, no such projections are currently known; the requirements on such projections do not even hold for incoherent or well-behaved dictionaries. In this work, we provide an alternate analysis for signal space greedy methods which enforce assumptions on these projections which hold in several settings including those when the dictionary is incoherent or structurally coherent. These results align more closely with traditional results in the standard CoSa literature and improve upon previous work in the signal space setting.

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“…In sparse coding stage, large number of l 1 minimization and greedy algorithms (e.g. BP [1], OMP [3] and SP [4]) have been designed in order to solve the same question: fix dictionary D and update coefficient X by min X Y − DX 2 F . However algorithms on dictionary update concern about different targets to update.…”

Section: Dictionary Learning and The Framework Of Simcomentioning

confidence: 99%

“…Algorithms developed for sparse coding include Basis Pursuit (BP) [1], Matching Pursuit (MP) [2], Orthogonal Matching Pursuit (OMP) [3], Subspace Pursuit (SP) [4], Gradient Pursuit (GP) [5], etc.…”

Section: Introductionmentioning

confidence: 99%

Weighted SimCO: a novel algorithm for dictionary update

Zhao

Zhou

Dai

et al. 2012

Sensor Signal Processing for Defence (SSPD 2012)

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Abstract-Algorithms aiming at solving dictionary learning problem usually involve iteratively performing two stage operations: sparse coding and dictionary update. In the dictionary update stage, codewords are updated based on a given sparsity pattern. In the ideal case where there is no noise and the true sparsity pattern is known a priori, dictionary update should produce a dictionary that precisely represent the training samples. However, we analytically show that benchmark algorithms, including MOD, K-SVD and regularized SimCO, could not always guarantee this property: they may fail to converge to a global minimum. The key behind the failure is the singularity in the objective function. To address this problem, we propose a weighted technique based on the SimCO optimization framework, hence the term weighted SimCO. Decompose the overall objective function as a sum of atomic functions. The crux of weighted SimCO is to apply weighting coefficients to atomic functions so that singular points are zeroed out. A second order method is implemented to solve the corresponding optimization problem. We numerically compare the proposed algorithm with the benchmark algorithms for noiseless and noisy scenarios. The empirical results demonstrate the significant improvement in the performance.

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Subspace Pursuit for Compressive Sensing Signal Reconstruction

Cited by 2,094 publications

References 21 publications

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Greedy signal space methods for incoherence and beyond

Weighted SimCO: a novel algorithm for dictionary update

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