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This paper concentrates on the recovery of block-sparse signals, which is 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 to 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, the numerical experiments conducted demonstrate the performance of the proposed algorithm.

In this paper, we bring forward a completely perturbed nonconvex Schatten $p$-minimization to address a model of completely perturbed low-rank matrix recovery. The paper that based on the restricted isometry property generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, gives the restricted isometry property condition that guarantees the recovery of low-rank matrix and the corresponding reconstruction error bound. In particular, the analysis of the result reveals that in the case that $p$ decreases $0$ and $a>1$ for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition $\delta_{2r}<1$ \cite{Recht et al 2010}. The numerical experiments are conducted to show better performance, and provides outperformance of the nonconvex Schatten $p$-minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.

Compressed sensing shows that a sparse signal can stably be recovered from incomplete linear measurements. But, in practical applications, some signals have additional structure, where the nonzero elements arise in some blocks. We call such signals as block-sparse signals. In this paper, the $\ell_2/\ell_1-\alpha\ell_2$ minimization method for the stable recovery of block-sparse signals is investigated. Sufficient conditions based on block mutual coherence property and associating upper bound estimations of error are established to ensure that block-sparse signals can be stably recovered in the presence of noise via the $\ell_2/\ell_1-\alpha\ell_2$ minimization method. For all we know, it is the first block mutual coherence property condition of stably reconstructing block-sparse signals by the $\ell_2/\ell_1-\alpha\ell_2$ minimization method. Additionally, the numerical experiments implemented verify the performance of the $\ell_2/\ell_1-\alpha\ell_2$ minimization., Comment: 14 pages, 8 figures

In this paper we investigate the reconstruction conditions of nuclear norm minimization for low-rank matrix recovery. We obtain sufficient conditions $\delta_{tr}

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 (\ref{eq.3}) is larger by a factor to $mn/M$, where $m,~n$ are the dimension 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.

The essential task of tensor data analysis focuses on the tensor decomposition and the corresponding notion of rank. In this paper, by introducing the notion of tensor Singular Value Decomposition (t-SVD), we establish a Regularized Tensor Nuclear Norm Minimization (RTNNM) model for low-tubal-rank tensor recovery. As we know that many variants of the Restricted Isometry Property (RIP) have proven to be crucial analysis tools for sparse recovery. In the t-SVD framework, we initiatively define a novel tensor Restricted Isometry Property (t-RIP). Furthermore, we show that any third-order tensor $\boldsymbol{\mathcal{X}}$ can stably be recovered from few linear noise measurements under some certain t-RIP conditions via the RTNNM model. We note that, as far as the authors are aware, such kind of result has not previously been reported in the literature.