The weighted$\ell _{r}-\ell _{1}$minimization with weight$\alpha$has been extensively employed to robustly estimate a high-dimensional sparse signal$x$coded by the underdetermined linear measurements$y=Ax+z$, where$A$and$z$are the measurement matrix and noise, respectively. In this paper, we demonstrate that if the restricted isometry constant (RIC)$\delta _{s}$of$A$fulfills\begin{align*} \delta _{s}< 1/\left(1+3t/\sqrt{5}\right), \end{align*}where$t$relies on sparsity level$s$for known model parameters$\alpha$and$r$, then any sparse signal$x$are ensured to be robustly reconstructed through solving the weighted$\ell _{r}-\ell _{1}$minimization in the noisy situation. The gained condition is testified to be much better that the state-of-art ones.

In recent years, tensor decomposition-based approaches forhyperspectral anomaly detection(HAD) have gained significant attention in the field of remote sensing. However, existing methods often fail to flexibly and effectively extract both the global correlations and local smoothness of the background components inhyperspectral images(HSIs). To mitigate this critical issue, we put forward a novel HAD method named HAD-EUNTRFR, which incorporates an enhanced unified nonconvex tensor ring (TR) factors regularization. In the HAD-EUNTRFR framework, the raw HSIs are first decomposed into background and anomaly components using the idea of tensor robust principal component analysis. The TR decomposition is then employed to capture the spatial-spectral correlations within the background component. Additionally, we introduce a unified and efficient nonconvex regularizer, induced bytensor singular value decomposition(T-SVD), to simultaneously encode the low-rankness and sparsity of the 3-D gradient TR factors into a unique concise form. The above characterization scheme enables the interpretable gradient TR factors to inherit the low-rankness and smoothness of the original background. To further enhance anomaly detection, we design a generalized nonconvex regularization term to exploit the group sparsity of the anomaly component. Based upon the above, we ultimately propose a scalable and reliable nonconvex HAD model. To solve the resulting doubly nonconvex model, we develop a highly efficient optimization algorithm based on thealternating direction method of multipliers(ADMM) framework. Theoretical results on convergence analysis for the proposed algorithm are derived. Experimental results on several benchmark datasets demonstrate that our proposed method outperforms existingstate-of-the-art(SOTA) approaches in terms of detection accuracy.

Anomaly detection in hyperspectral images (HSIs) aims to identify sparse, interesting anomalies against the background, which has become a significant topic in remote sensing. Although the existing tensor-based methods have achieved commendable performance to some extent, there is still room for further improvement. In combination with three key techniques, i.e., gradient map-based modeling, circular tensor ring (TR) unfolding, and nonconvex regularization, this article proposes a novel generalized nonconvex method for hyperspectral anomaly detection (HAD) tasks within the TR framework. For the implementation of our proposed approach, abbreviated as TR-GNHAD, we first develop an effective and reliable HAD model in virtue of two newly unified nonconvex regularizers. The first regularizer is devised under a new prior characterization paradigm, which has a strong ability to encode two insightful prior information underlying the HSI’s background simultaneously, i.e., global low rankness and local smoothness. The other regularizer can well capture the structured sparsity of the abnormal component. Then, we derive an efficient optimization algorithm to solve the proposed model based on the alternating direction method of multipliers (ADMMs) framework. Experiments conducted on 12 HSI datasets illustrate that the proposed approach achieves highly competitive performance in both qualitative and quantitative metrics compared with several state-of-the-art HAD methods.

Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering. Nevertheless, these methods involve the T-SVD based low-rank approximation, which suffers from high computational costs when dealing with large-scale tensor data. Moreover, most of them are only applicable to third-order tensors. Against these issues, in this article, two efficient low-rank tensor approximation approaches fusing random projection techniques are first devised under the order-d ( d ≥ 3 ) T-SVD framework. Theoretical results on error bounds for the proposed randomized algorithms are provided. On this basis, we then further investigate the robust high-order tensor completion problem, in which a double nonconvex model along with its corresponding fast optimization algorithms with convergence guarantees are developed. Experimental results on large-scale synthetic and real tensor data illustrate that the proposed method outperforms other state-of-the-art approaches in terms of both computational efficiency and estimated precision.

In this article, we bring forward a completely perturbed nonconvex Schatten p -minimization to address a model of completely perturbed low-rank matrix recovery (LRMR). This article based on the restricted isometry property (RIP) and the Schatten- p null space property (NSP) generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, and it gives the RIP condition and the Schatten- p NSP assumption that guarantee the recovery of low-rank matrix and the corresponding reconstruction error bounds. In particular, the analysis of the result reveals that in the case that p decreases 0 and for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition (Recht et al., 2010). In addition, we study the connection between RIP and Schatten- p NSP and discern that Schatten- p NSP can be inferred from the RIP. The numerical experiments are conducted to show better performance and provide outperformance of the nonconvex Schatten p -minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.

In the real world, a large amount of high-order tensor data (order>3) exists, such as color videos, multispectral videos, and light-field images. However, these data often face challenges in transportation, storage, and susceptibility to damage. Meanwhile, most existing tensor-based information processing methods only concentrate on third-order tensors, which may not meet the complex requirements of high-dimensional data processing. In this paper, to better address the high-order tensor recovery issue, we propose a novel method that couples multilayer subspace priors with high-order tensor recovery techniques for tensor completion and robust tensor principal component analysis. Moreover, we provide theoretical guarantees for our approach's recovery and demonstrate that it achieves comparable performance under weaker incoherent conditions. Additionally, we develop two efficient and interpretable algorithms based on the alternating direction method of multipliers (ADMM) to solve our model. Owing to the adaptability of subspace prior information, our method demonstrates superior performance in recovering various types of data, including color videos and multispectral videos, compared with various advanced algorithms currently available.

Currently, low-rank tensor recovery employing the subspace prior information is an emerging topic, which has attracted considerable attention. However, existing studies cannot flexibly and fully utilize the accessible subspace prior information, thereby leading to suboptimal restored performance. Aiming at addressing this issue, based on the tensor singular value decomposition (t-SVD), this article presents a novel strategy that integrates more than two layers of subspace knowledge about columns and rows of target tensor into one unified recovery framework. Specially, we first design a multilayer subspace prior learning scheme, and then apply it to two common low-rank tensor recovery problems, i.e., tensor completion and tensor robust component principal analysis. Crucially, we prove that our approach can achieve exact recovery of tensors under a significantly weaker incoherence assumption than the analogous conditions previously proposed. Furthermore, two efficient algorithms with convergence guarantees based on alternating direction method of multipliers (ADMM) are proposed to solve the corresponding models. The experimental results on synthetic and real tensor data show that the proposed algorithms outperform other state-of-the-art algorithms in terms of both qualitative and quantitative metrics.

Low-rank tensor recovery (LRTR) is a natural extension of low-rank matrix recovery (LRMR) to high-dimensional arrays, which aims to reconstruct an underlying tensor from incomplete linear measurements M(X). However, LRTR ignores the error caused by quantization, limiting its application when the quantization is low-level. In this work, we take into account the impact of extreme quantization and suppose the quantizer degrades into a comparator that only acquires the signs of M(X). We still hope to recover X from these binary measurements. Under the tensor Singular Value Decomposition (t-SVD) framework, two recovery methods are proposedthe first is a tensor hard singular tube thresholding method; the second is a constrained tensor nuclear norm minimization method. These methods can recover a real n1 n2 n3 tensor X with tubal rank r from m random Gaussian binary measurements with errors decaying at a polynomial speed of the oversampling factor := m/((n1+ n2)n3r). To improve the convergence rate, we develop a new quantization scheme under which the convergence rate can be accelerated to an exponential function of . Numerical experiments verify our results, and the applications to real-world data demonstrate the promising performance of the proposed methods.

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$\delta _{s}+\nu (s)\theta _{s,s}< 1$, 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.

Recently, tensor Singular Value Decomposition (t-SVD)-based low-rank tensor completion (LRTC) has achieved unprecedented success in addressing various pattern analysis issues. However, existing studies mostly focus on third-order tensors while order- d ( d ≥ 4 ) tensors are commonly encountered in real-world applications, like fourth-order color videos, fourth-order hyper-spectral videos, fifth-order light-field images, and sixth-order bidirectional texture functions. Aiming at addressing this critical issue, this paper establishes an order- d tensor recovery framework including the model, algorithm and theories by innovatively developing a novel algebraic foundation for order- d t-SVD, thereby achieving exact completion for any order- d low t-SVD rank tensors with missing values with an overwhelming probability. Emperical studies on synthetic data and real-world visual data illustrate that compared with other state-of-the-art recovery frameworks, the proposed one achieves highly competitive performance in terms of both qualitative and quantitative metrics. In particular, as the observed data density becomes low, i.e., about 10%, the proposed recovery framework is still significantly better than its peers. The code of our algorithm is released at https://github.com/Qinwenjinswu/TIP-Code.

The tensor-tensor product-induced tensor nuclear norm (t-TNN) (Lu et al., 2020) minimization for low-tubal-rank tensor recovery attracts broad attention recently. However, minimizing the t-TNN faces some drawbacks. For example, the obtained solution could be suboptimal to the original problem due to its loose approximation. In this article, we extract a unified nonconvex surrogate of the tensor tubal rank as a tighter regularizer, which involves many popular nonconvex penalty functions. An iterative reweighted t-TNN algorithm is proposed to solve the resulting generalized nonconvex tubal rank minimization for tensor recovery. It converges to a critical point globally with rigorous proofs based on the Kurdyka-Łojasiwicz property. Furthermore, we provide the theoretical guarantees for exact and robust recovery by developing the tensor null space property. Extensive experiments demonstrate that our approach markedly enhances recovery performance compared with several state-of-the-art convex and nonconvex methods.

Tensor principal component pursuit (TPCP) is a powerful approach in the tensor robust principal component analysis (TRPCA), where the goal is to decompose a data tensor to a low-tubal-rank part plus a sparse residual. TPCP is shown to be effective under certain tensor incoherence conditions, which can be restrictive in practice. In this paper, we propose a Modified-TPCP, which incorporates the prior subspace information in the analysis. With the aid of prior info, the proposed method is able to recover the low-tubal-rank and the sparse components under a significantly weaker incoherence assumption. We further design an efficient algorithm to implement Modified-TPCP based upon the alternating direction method of multipliers (ADMM). The promising performance of the proposed method is supported by simulations and real data applications.

This paper focuses on the recovery of low-tubal-rank tensors from binary measurements under the frame of tensor Singular Value Decomposition. We show that the direction of a tubal-rank-r tensor X ∈ ℝn1×n2×n3can be approximated from Ω((n1+ n2)n3r) random Gaussian measurements. In addition, incorporating nonadaptive thresholds in the measurements, it is proved that the full X can be recovered. As we will see, under this nonadaptive measurement scheme, recovery errors decay at the rate of polynomial of the oversampling factor λ := m/(n1+ n2)n3r, i.e., O(λ-1/6). In order to obtain faster decay rate, we introduce a recursive strategy which generates thresholds according to previous estimates for each iteration. Under this quantization scheme, An iterative recovery algorithm is proposed which establishes recovery errors decaying at the rate of exponent of λ. Numerical experiments are conducted to demonstrate our results.

The recently proposed Tensor Nuclear Norm (TNN) minimization has been widely used for tensor completion. However, previous works didn’t consider the structural difference between the observed data and missing data, which widely exists in many applications. In this paper, we propose to incorporate a constraint item on the missing values into low-tubal-rank tensor completion to promote the structural hypothesis of the missing values such as sparsity. Theoretically, the proposed model has lower recovery error than classical model, and the target tensor can be recovered exactly with overwhelming probability provided low-tubal-rankness on whole area and sparsity on missing area. Algorithmically, an efficient algorithm by Alternating Direction Method of Multiplier (ADMM) is presented. Extensive experiments on both synthetic and real-world data demonstrate its superiority compared with several state-of-the-art methods.

This letter considers the recovery of a low-tubal-rank tensor from incomplete linear observations. It is shown that the unknown tensor Z ∈ Rn1×n2×n3of tubal-rank r can be reconstructed as a unique solution of a tractable method - tensor nuclear norm (TNN) minimization, provided that the number of Gaussian observations m ≥ 3r(n1+ n2- r)n3+ 1. In this work, we examine the fundamental question of the minimal number of linear observations needed to reconstruct the tensor Z from these observations, regardless of the practicality of the reconstruction scheme. Consequently, we provide two benchmark results so that different reconstruction schemes including TNN minimization can be compared to each other. Specifically, we conclude that m ≥ 2r(n1+ n2- 2r)n3and m ≥ r(n1+ n2- r)n3+ 1 Gaussian observations are necessary and sufficient to guarantee uniform recovery and nonuniform recovery using tensor tubalrank minimization method, respectively.

This study addresses the issue of block‐sparse recovery in compressive sensing in the presence of non‐Gaussian measurement noise. By using the generalised ‐norm noise constraint for to replace the popular ‐norm, in this study, the authors put forward a truncated model for recovering block‐sparse signal. A theoretical analysis is first presented to guarantee the validity of proposed method. If the measurement matrix satisfies an extended block restricted isometry property, the reconstruction error is bounded in the optimisation. Moreover, in order to solve the induced optimisation problem effectively, they present an alternating direction method of multipliers via embedding Karush–Kuhn–Tucker system of ‐norm functions into the frame structure of augmented Lagrangian methods. When compared with some of the state‐of‐the‐art methods, the proposed method becomes more competitive.

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 l 2 / l 1 minimisation method is robust and stable for the recovery of block sparse signals.

Here, the authors discuss the recovery of signals from under‐sampled data in which signals are nearly block sparse via a truncated method with redundant dictionaries. The authors show that the obtained results are better than the previous recovery result in the existence of noise. Furthermore, the authors conduct an alternating direction method of multipliers algorithm to solve the signals recovery problem. Moreover, the numerical experiments prove the strong robustness and stability of truncated method with redundant dictionaries ( t ‐ D ‐ block ‐ ) in the presence of noise.