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This paper concentrates on the recovery of block-sparse signals, which are not only sparse but also nonzero elements are arrayed into some blocks (clusters) rather than being arbitrary distributed all over the vector, from linear measurements. We establish high-order sufficient conditions based on block RIP, which could ensure the exact recovery of every block s-sparse signal in the noiseless case via mixed l 2 / l p minimization method, and the stable and robust recovery in the case that signals are not accurately block-sparse in the presence of noise. Additionally, a lower bound on necessary number of random Gaussian measurements is gained for the condition to be true with overwhelming probability. Furthermore, a series of numerical experiments are conducted to demonstrate the performance of the mixed l 2 / l p minimization. To the best of the authors' knowledge, the recovery guarantees established in this paper are superior to those currently available. [ABSTRACT FROM AUTHOR]

The null space property (NSP), which relies merely on the null space of the sensing matrix column space, has drawn numerous interests in sparse signal recovery. This paper studies NSP of the weighted ℓ r − ℓ 1 (r ∈ (0 , 1 ]) minimization. Several versions of NSP of the weighted ℓ r − ℓ 1 minimization including the weighted ℓ r − ℓ 1 NSP, the weighted ℓ r − ℓ 1 stable NSP, the weighted ℓ r − ℓ 1 robust NSP and the ℓ q weighted ℓ r − ℓ 1 robust NSP for 1 ≤ q ≤ 2 , are proposed, as well as the associating considerable results are derived. Under these NSPs, sufficient conditions for the recovery of (sparse) signals with the weighted ℓ r − ℓ 1 minimization are established. Furthermore, we show that to some extent, the weighted ℓ r − ℓ 1 stable NSP is weaker than the restricted isometric property (RIP). And the RIP condition we obtained is better than that of Zhou (2022). [ABSTRACT FROM AUTHOR]

Previous work regarding low-rank matrix recovery has concentrated on the scenarios in which the matrix is noise-free and the measurements are corrupted by noise. However, in practical application, the matrix itself is usually perturbed by random noise preceding to measurement. This paper concisely investigates this scenario and evidences that, for most measurement schemes utilized in compressed sensing, the two models are equivalent with the central distinctness that the noise associated with double noise model is larger by a factor to m n / M , where m , n are the dimensions of the matrix and M is the number of measurements. Additionally, this paper discusses the reconstruction of low-rank matrices in the setting, presents sufficient conditions based on the associating null space property to guarantee the robust recovery and obtains the number of measurements. Furthermore, for the non-Gaussian noise scenario, we further explore it and give the corresponding result. The simulation experiments conducted, on the one hand show effect of noise variance on recovery performance, on the other hand demonstrate the verifiability of the proposed model. [ABSTRACT FROM AUTHOR]

In this letter, we discuss the reconstruction of sparse signals from undersampled data, which belongs to the core content of compressed sensing. A new sufficient condition in terms of the restricted isometry constant (RIC) and restricted orthogonality constant (ROC) is first established for the performance guarantee of recently proposed non-convex weighted $\ell _{r}-\ell _{1}$ minimization in recovering (approximately) sparse signals that may be polluted by noise. To be specific, it is shown that if the RIC $\delta _{s}$ and ROC $\theta _{s,s}$ of measurement matrix obey \begin{equation*} \delta _{s}+\nu (s)\theta _{s,s}< 1, \end{equation*} where $\nu (s)$ depends on $s$ for given quantities, then any $s$ -sparse signals in noiseless setting are guaranteed to be recovered accurately via solving the constrained weighted $\ell _{r}-\ell _{1}$ minimization optimization problem and any (approximately) $s$ -sparse signals can be estimated robustly in the noisy case. In addition, we provide several pivotal remarks which indicate the recovery guarantee is much less restricted than the existing one. The results obtained contribute to proving the fidelity of the excellent weighted $\ell _{r}-\ell _{1}$ minimization method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the recovery of group sparse signals corrupted by impulsive noise. In some recent literature, researchers have utilized stable data fitting models, like l 1 -norm, Huber penalty function and Lorentzian-norm, to substitute the l 2 -norm data fidelity model to obtain more robust performance. In this paper, a stable model is developed, which exploits the generalized l p -norm as the measure for the error for sparse reconstruction. In order to address this model, we propose an efficient alternative direction method of multipliers, which includes the proximity operator of l p -norm functions to the framework of Lagrangian methods. Besides, to guarantee the convergence of the algorithm in the case of 0 ≤ p < 1 (nonconvex case), we took advantage of a smoothing strategy. For both 0 ≤ p < 1 (nonconvex case) and 1 ≤ p ≤ 2 (convex case), we have derived the conditions of the convergence for the proposed algorithm. Moreover, under the block restricted isometry property with constant δ τ k 0 < τ / (4 − τ) for 0 < τ < 4 / 3 and δ τ k 0 < (τ − 1) / τ for τ ≥ 4 / 3 , a sharp sufficient condition for group sparse recovery in the presence of impulsive noise and its associated error upper bound estimation are established. Numerical results based on the synthetic block sparse signals and the real-world FECG signals demonstrate the effectiveness and robustness of new algorithm in highly impulsive noise. [ABSTRACT FROM AUTHOR]

One-bit compressed sensing aims to recover unknown sparse signals from extremely quantized linear measurements which just capture their signs. In this paper, we propose a nonconvex ℓp(0 < p < 1) minimization model for one-bit compressed sensing problem and define the set of ℓp effectively s-sparse signals which contains genuinely s-sparse signals. Utilizing properties of covering number, we show that our method can recover the direction of ℓp effectively s-sparse signals with error order. We also employ thresholded one-bit measurements to estimate the magnitude of signals and prove that any ℓp effectively s-sparse bounded signal x can be estimated using augmented ℓp minimization model and empirical distribution function method respectively. Especially, to recover ℓp effectively s-sparse signals in practice, we introduce an adaptive binary iterative thresholding algorithm which can be utilized without knowing the sparsity of underlying signals. Numerical experiments on both synthetic and real-world data sets are conducted to demonstrate the superiority of our algorithm. [ABSTRACT FROM AUTHOR]

This work gains a sharp sufficient condition on the block restricted isometry property for the recovery of sparse signal and corresponding upper bound estimate of error. Under the certain assumption, the signal with block structure can be stably recovered in the presence of noisy case and the block sparse signal can be exactly reconstructed in the noise‐free case. Besides, an example is proposed to exhibit the condition is sharp. Numerical simulations are carried out to demonstrate that authors' results are verifiable and l2 /l1 minimisation method is robust and stable for the recovery of block sparse signals. [ABSTRACT FROM AUTHOR]

Highlights • We propose a novel penalty function for CS using integral convolution approximation. • Our criterion dose not underestimate the large component in signal recovery. • Our methods perform well under both the Gaussian random sensing matrix satisfying RIP and the highly coherent sensing matrix. • We carry out a series of experiments to verify our analysis. Abstract In this paper, we propose a novel nonconvex penalty function for compressed sensing using integral convolution approximation. It is well known that an unconstrained optimization criterion based on ℓ 1 -norm easily underestimates the large component in signal recovery. Moreover, most methods either perform well only under the Gaussian random measurement matrix satisfying restricted isometry property or the highly coherent measurement matrix, which both can not be established at the same time. We introduce a new solver to address both of these concerns by adopting a frame of the difference between two convex functions with integral convolution approximation. What's more, to better boost the recovery performance, a weighted version of it is also provided. Experimental results suggest the effectiveness and robustness of our methods through several signal reconstruction examples in term of success rate and signal-to-noise ratio. [ABSTRACT FROM AUTHOR]