Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations

Adcock, Ben; Bao, Anyi; Jakeman, John Davis; Naryan, Akil

doi:10.2172/1434573

Cited by 6 publications

(14 citation statements)

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“…These are presented in Theorem 5.1 and Theorem 5.2 respectively. These results determine an error bound in certain weighted 1 -norms depending on the approximate sparsity in levels, and the noise level. They directly generalize known results for the sparse case, and require no stricter assumptions on the corresponding restricted isometry constant.…”

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confidence: 87%

“…For instance, it can be used to model so-called sparse and distributed or sparse and balanced vectors, which occur in parallel acquisition problems [16,17] and radar [20]. The specific case of two levels also arises in the sparse corruptions problem [1,26], in which, rather than standard Gaussian or uniformly bounded noise, a small fraction of the measurements of a signal is substantially corrupted. Another natural context of interest is the problem of compressive imaging, where sparse in levels vectors model the wavelet coefficients of natural images.…”

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confidence: 99%

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The benefits of acting locally: Reconstruction algorithms for sparse in levels signals with stable and robust recovery guarantees

Adcock,

Brugiapaglia,

King-Roskamp

2020

Preprint

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The sparsity in levels model recently inspired a new generation of effective acquisition and reconstruction modalities for compressive imaging. Moreover, it naturally arises in various areas of signal processing such as parallel acquisition, radar, and the sparse corruptions problem. Reconstruction strategies for sparse in levels signals usually rely on a suitable convex optimization program. Notably, although iterative and greedy algorithms can outperform convex optimization and have been studied extensively in the case of standard sparsity, little is known about their generalizations to the sparse in levels setting. In this paper, we bridge this gap by showing new stable and robust uniform recovery guarantees for sparse in level variants of the iterative hard thresholding and the CoSaMP algorithms. Our theoretical analysis generalizes recovery guarantees currently available in the case of standard sparsity and favorably compare to sparse in levels guarantees for weighted 1 minimization. In addition, we also propose and numerically test an extension of the orthogonal matching pursuit algorithm for sparse in levels signals.

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confidence: 87%

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confidence: 99%

The benefits of acting locally: Reconstruction algorithms for sparse in levels signals with stable and robust recovery guarantees

Adcock,

Brugiapaglia,

King-Roskamp

2020

Preprint

Self Cite

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“…This problem arises in many practical applications of interest, such as face recognition [1], subspace clustering [2], sensor network [3], latent variable modeling [4], principle component analysis [5], source separation [6], and so on. The theoretical aspects of this problem have also been studied under different scenarios in the literature, important examples include sparse signal recovery from sparse corruption [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], low-rank matrix recovery from sparse corruption [4], [5], [19], [20], [21], [22], and structured signal recovery from structured corruption [23], [24], [25], [26], [27], [28], [29].…”

Section: Introductionmentioning

confidence: 99%

“…where τ 1 , τ 2 > 0 are some tradeoff parameters. A large number of numerical results in the literature have suggested that phase transitions emerge in all above three recovery procedures (under random measurements), see e.g., [10], [11], [13], [15], [16], [17], [23], [24], [25], [26], [27]. Concretely, for a specific recovery procedure, when the number of the measurements exceeds a threshold, this procedure can faithfully reconstruct both signal and corruption with high probability, when the number of the measurements is below the threshold, this procedure fails with high probability.…”

Section: Introductionmentioning

confidence: 99%

Phase Transitions in Recovery of Structured Signals from Corrupted Measurements

Sun¹,

Cui²,

Liu³

2021

Preprint

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This paper is concerned with the problem of recovering a structured signal from a relatively small number of corrupted random measurements. Sharp phase transitions have been numerically observed in practice when different convex programming procedures are used to solve this problem. This paper is devoted to presenting theoretical explanations for these phenomenons by employing some basic tools from Gaussian process theory. Specifically, we identify the precise locations of the phase transitions for both constrained and penalized recovery procedures. Our theoretical results show that these phase transitions are determined by some geometric measures of structure, e.g., the spherical Gaussian width of a tangent cone and the Gaussian (squared) distance to a scaled subdifferential. By utilizing the established phase transition theory, we further investigate the relationship between these two kinds of recovery procedures, which also reveals an optimal strategy (in the sense of Lagrange theory) for choosing the tradeoff parameter in the penalized recovery procedure. Numerical experiments are provided to verify our theoretical results.

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“…9 The specific case of two levels also arises in the sparse corruptions problem. 10,11 The focus of past work on this model has been on convex optimization-based decoders such as Quadratically-Constrained Basis Pursuit (QCBP)…”

Section: Introductionmentioning

confidence: 99%

Iterative and greedy algorithms for the sparsity in levels model in compressed sensing

Adcock

Brugiapaglia

King-Roskamp

2019

Wavelets and Sparsity XVIII

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Motivated by the question of optimal functional approximation via compressed sensing, we propose generalizations of the Iterative Hard Thresholding and the Compressive Sampling Matching Pursuit algorithms able to promote sparse in levels signals. We show, by means of numerical experiments, that the proposed algorithms are successfully able to outperform their unstructured variants when the signal exhibits the sparsity structure of interest. Moreover, in the context of piecewise smooth function approximation, we numerically demonstrate that the structure promoting decoders outperform their unstructured variants and the basis pursuit program when the encoder is structure agnostic.

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Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations

Cited by 6 publications

References 0 publications

The benefits of acting locally: Reconstruction algorithms for sparse in levels signals with stable and robust recovery guarantees

The benefits of acting locally: Reconstruction algorithms for sparse in levels signals with stable and robust recovery guarantees

Phase Transitions in Recovery of Structured Signals from Corrupted Measurements

Iterative and greedy algorithms for the sparsity in levels model in compressed sensing

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