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In this paper, we consider the frame-based block sparse signal recovery via a l2/l1-synthesis method. A new kind of null space property based on the given dictionary D(block D-NSP) is proposed. It is proved that sensing matrices satisfying the block D-NSP is not just a sufficient and necessary condition for the l2/l1-synthesis method to exactly recover signals that are block sparse in frame D, but also a sufficient and necessary condition for the l2/l1-synthesis to stably recover signals which are block-compressible in frame D. To the best of our knowledge, this new property is the first sufficient and necessary condition for successful signal recovery via the l2/l1-synthesis. In addition, we also characterize the theoretical performance of recovering signals via the l2/l1-synthesis in the case of the measurements are disturbed.

In this paper, we generalize the conditions for the exact or stable recovery of weighted k -sparse signals in weighted sparse phase retrieval in our previous work [ 11 ] from the weighted ℓ 1 minimization to the weighted ℓ q ( 0 < q ≤ 1 ) minimization in a broad sense. Specifically, we first present that the weighted null space property (WNSP) is a sufficient and necessary condition to guarantee the exact recovery of a weighted k -sparse signal from its noiseless phaseless measurements via the weighted ℓ q ( 0 < q ≤ 1 ) minimization in both the real and complex cases. In addition, we establish a general strong weighted restricted isometry property (SWRIP) condition for the stable recovery of a weighted k -sparse signal from its noisy phaseless measurements via the weighted ℓ q ( 0 < q ≤ 1 ) minimization in the real case.

We present a detailed analysis of the unconstrained ℓ 1 -weighted LASSO method for recovery of sparse data from its observation by randomly generated matrices, satisfying the restricted isometry property (RIP) with constant δ < 1 , and subject to negligible measurement and compressibility errors. We prove that if the data are k -sparse, then the size of support of the LASSO minimizer, s , maintains a comparable sparsity, s ⩽ C δ k . For example, if δ = 0.7 then s < 11 k and a slightly smaller δ = 0.4 yields s < 4 k . We also derive new ℓ 2 / ℓ 1 error bounds which highlight precise dependence on k and on the LASSO parameter λ , before the error is driven below the scale of negligible measurement/ and compressiblity errors.

In the context of sparse recovery, it is known that most of the existing regularizers such as ℓ 1 suffer from some bias incurred by some leading entries (in magnitude) of the associated vector. To neutralize this bias, we propose a class of models with partial regularizers for recovering a sparse solution of a linear system. We show that every local minimizer of these models is substantially sparse or the magnitude of all of its nonzero entries is above a uniform constant depending only on the data of the linear system. Moreover, for a class of partial regularizers, any global minimizer of these models is a sparsest solution to the linear system. We also establish some sufficient conditions for local or global recovery of the sparsest solution to the linear system, among which one of the conditions is weaker than the best-known restricted isometry property condition for sparse recovery by ℓ 1 . In addition, a first-order augmented Lagrangian (FAL) method is proposed for solving these models, in which each subproblem is solved by a nonmonotone proximal gradient (NPG) method. Despite the complication of the partial regularizers, we show that each proximal subproblem in NPG can be solved as a certain number of one-dimensional optimization problems, which usually have a closed-form solution. We also show that any accumulation point of the sequence generated by FAL is a first-order stationary point of the models. Numerical results on compressed sensing and sparse logistic regression demonstrate that the proposed models substantially outperform the widely used ones in the literature in terms of solution quality.

Compressed sensing has captured considerable attention of researchers in the past decades. In this paper, with the aid of the powerful null space property, some deterministic recovery conditions are established for the previous ℓ1–ℓ1 method and the ℓ1–ℓ2 method to guarantee the exact sparse recovery when the side information of the desired signal is available. These obtained results provide a useful and necessary complement to the previous investigation of the ℓ1–ℓ1 and ℓ1–ℓ2 methods that are based on the statistical analysis. Moreover, one of our theoretical findings also shows that the sharp conditions previously established for the classical ℓ1 method remain suitable for the ℓ1–ℓ1 method to guarantee the exact sparse recovery. Numerical experiments on both the synthetic signals and the real-world images are also carried out to further test the recovery performance of the above two methods.

Sparse principal component analysis (SPCA) has achieved great success in improving interpretable ability of the derived results and has become a powerful technique for modern data analysis. It presents that principal component can be modified to produce sparse loadings by imposing sparsity-induced penalty, which is often l1-regularized constraint. In order to analyze the l1-regularized sparsity-induced model, in this paper, we propose a general null space property of a matrix A relative to a index set S and give a necessary and sufficient condition for the exact or approximate sparse principal components. Meanwhile, the conclusions with respect to the stable and robust situations are given in the case of exact or approximate sparse principal components, respectively.

This paper introduces a novel capped separable difference of two norms (CSDTN) method for sparse signal recovery, which generalizes the well-known minimax concave penalty (MCP) method. The CSDTN method incorporates two shape parameters and one scale parameter, with their appropriate selection being crucial for ensuring robustness and achieving superior reconstruction performance. We provide a detailed theoretical analysis of the method and propose an efficient iteratively reweighted ℓ1 (IRL1)-based algorithm for solving the corresponding optimization problem. Extensive numerical experiments, including electrocardiogram (ECG) and synthetic signal recovery tasks, demonstrate the effectiveness of the proposed CSDTN method. Our method outperforms existing methods in terms of recovery accuracy and robustness. These results highlight the potential of CSDTN in various signal-processing applications.