Estimation of block sparsity in compressive sensing

Zhou, Zhiyong; Yu, Jun

doi:10.48550/arxiv.1701.01055

Cited by 7 publications

(18 citation statements)

References 39 publications

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“…The traditional block sparsity measure • 2,0 has a severe practical drawback of being not sensitive to blocks with small ℓ 2 norm. As a soft version, the entropy based block sparsity measure named block q-ratio sparsity was proposed in [34], which possesses many nice properties including continuity, scale-invariance, non-increasing with respect to q and range equal to [1, M ]. For more detailed arguments about this block sparsity measure, readers can refer to [34].…”

Section: Minimization Of the Block Q-ratio Sparsitymentioning

confidence: 99%

“…As a soft version, the entropy based block sparsity measure named block q-ratio sparsity was proposed in [34], which possesses many nice properties including continuity, scale-invariance, non-increasing with respect to q and range equal to [1, M ]. For more detailed arguments about this block sparsity measure, readers can refer to [34]. To be self-contained, here we give the full definition of block q-ratio sparsity.…”

Section: Minimization Of the Block Q-ratio Sparsitymentioning

confidence: 99%

“…Definition 1. ( [34]) For any non-zero z ∈ R N and non-negative q / ∈ {0, 1, ∞}, the block q-ratio sparsity level of z is defined as…”

Section: Minimization Of the Block Q-ratio Sparsitymentioning

confidence: 99%

“…However, previous published studies on this kind of scale invariant approaches such as [20,24,37] are limited to the case of non-block sparse signal recovery. The present paper sets out to investigate the minimization of the block q-ratio sparsity measure given in [34] for block sparse signal recovery.…”

Section: Introductionmentioning

confidence: 99%

“…The benefit of this novel method is that it enjoys a superior performance when highly coherent measurement matrices are confronted. In the present paper, we extend this method to the framework of block sparse signal recovery via minimizing the block version of q-ratio sparsity given in [34] (namely the block q-ratio sparsity). Our main contributions are three folds:…”

Section: Introductionmentioning

confidence: 99%

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Block sparse signal recovery via minimizing the block $q$-ratio sparsity

Zhou

2021

Preprint

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In this paper, we propose a method for block sparse signal recovery that minimizes the block q-ratio sparsity ( z 2,1 / z 2,q ) q q−1 with q ∈ [0, ∞]. For the case of 1 < q ≤ ∞, we present the theoretical analyses and the computing algorithms for both cases of the ℓ 2 -bounded and ℓ 2,∞ -bounded noises. The corresponding unconstrained model is also investigated. Its superior performance in block sparse signal reconstruction is demonstrated by numerical experiments.

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Section: Minimization Of the Block Q-ratio Sparsitymentioning

confidence: 99%

Section: Minimization Of the Block Q-ratio Sparsitymentioning

confidence: 99%

“…Definition 1. ( [34]) For any non-zero z ∈ R N and non-negative q / ∈ {0, 1, ∞}, the block q-ratio sparsity level of z is defined as…”

Section: Minimization Of the Block Q-ratio Sparsitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Block sparse signal recovery via minimizing the block $q$-ratio sparsity

Zhou

2021

Preprint

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Point process simulation of generalised hyperbolic Lévy processes

Kındap,

Godsill

2023

Stat Comput

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Generalised hyperbolic (GH) processes are a class of stochastic processes that are used to model the dynamics of a wide range of complex systems that exhibit heavy-tailed behavior, including systems in finance, economics, biology, and physics. In this paper, we present novel simulation methods based on subordination with a generalised inverse Gaussian (GIG) process and using a generalised shot-noise representation that involves random thinning of infinite series of decreasing jump sizes. Compared with our previous work on GIG processes, we provide tighter bounds for the construction of rejection sampling ratios, leading to improved acceptance probabilities in simulation. Furthermore, we derive methods for the adaptive determination of the number of points required in the associated random series using concentration inequalities. Residual small jumps are then approximated using an appropriately scaled Brownian motion term with drift. Finally the rejection sampling steps are made significantly more computationally efficient through the use of squeezing functions based on lower and upper bounds on the Lévy density. Experimental results are presented illustrating the strong performance under various parameter settings and comparing the marginal distribution of the GH paths with exact simulations of GH random variates. The new simulation methodology is made available to researchers through the publication of a Python code repository.

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Sparse recovery based on q-ratio constrained minimal singular values

Zhou

2019

Signal Processing

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We study verifiable sufficient conditions and computable performance bounds for sparse recovery algorithms such as the Basis Pursuit, the Dantzig selector and the Lasso estimator, in terms of a newly defined family of quality measures for the measurement matrices. With high probability, the developed measures for subgaussian random matrices are bounded away from zero as long as the number of measurements is reasonably large. Comparing to the restricted isotropic constant based performance analysis, the arguments in this paper are much more concise and the obtained bounds are tighter.Numerical experiments are presented to illustrate our theoretical results. Index TermsCompressive sensing; q-ratio sparsity; q-ratio constrained minimal singular values; Convex-concave procedure. z =0, z 2 1 / z 2 2 ≤s Az 2 z 2 and obtained the error 2 bounds in terms of this quality measure of the measurement matrix. Similarly, in [16], the authors definedz ∞ with · ♦ denoting a general norm, and derived the performance bounds on the ∞ norm of the recovery error vector based on this quality measure. This kind of measures has also been used in establishing results for block sparsity recovery [17] and low-rank matrix recovery [18]. In this paper we generalize these two quantities to a more general quantity called q-ratio CMSV with 1 < q ≤ ∞, and establish the performance bounds for both q norm and 1 norm of the reconstruction error. A. ContributionsOur contribution mainly has four aspects. First, we proposed a sufficient condition based on a q-ratio sparsity level for the exact recovery using 1 minimization in the noise free case, and designed a convexconcave procedure to solve the corresponding non-convex problem, leading to an acceptable verification algorithm. Second, we introduced q-ratio CMSV and derived concise bounds on both q norm and 1 norm of the reconstruction error for the Basis Pursuit (BP) [19], the Dantzig selector (DS) [20], and the Lasso estimator [21] in terms of q-ratio CMSV. We established the corresponding stable and robust recovery results involving both sparsity defect and measurement error. Third, we demonstrated that for subgaussion random matrices, the q-ratio CMSVs are bounded away from zero with high probability, as long as the number of measurement is large enough. Finally, we presented algorithms to compute the q-ratio CMSV for an arbitrary measurement matrix, and studied the effects of different parameters on the proposed q-ratio CMSV. Moreover, we illustrated that q-ratio CMSV based bound is tighter than the RIC based one. B. Organization and NotationsThe paper is organized as follows. In Section II, we present the definitions of q-ratio sparsity and q-ratio CMSV, and give a sufficient condition for unique noiseless recovery based on the q-ratio sparsity and an inequality for the q-ratio CMSV. In Section III, we derive performance bounds on both q norm and 1 norm of the reconstruction errors for several convex recovery algorithms in terms of q-ratio CMSVs. In Section IV, we demonstrate that the subgaussi...

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Estimation of block sparsity in compressive sensing

Cited by 7 publications

References 39 publications

Block sparse signal recovery via minimizing the block $q$-ratio sparsity

Block sparse signal recovery via minimizing the block $q$-ratio sparsity

Point process simulation of generalised hyperbolic Lévy processes

Sparse recovery based on q-ratio constrained minimal singular values

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