Tensor robust principal component analysis (TRPCA), as a popular linear low-rank method, has been widely applied to various visual tasks. The mathematical process of the low-rank prior is derived from the linear latent variable model. However, for nonlinear tensor data with rich information, their nonlinear structures may break through the assumption of low-rankness and lead to the large approximation error for TRPCA. Motivated by the latent low-dimensionality of nonlinear tensors, the general paradigm of the nonlinear tensor plus sparse tensor decomposition problem, called tensor robust kernel principal component analysis (TRKPCA), is first established in this paper. To efficiently tackle TRKPCA problem, two novel nonconvex regularizers the kernelized tensor Schatten- $p$ norm (KTSPN) and generalized nonconvex regularization are designed, where the former KTSPN with tighter theoretical support adequately captures nonlinear features (i.e., implicit low-rankness) and the latter ensures the sparser structural coding, guaranteeing more robust separation results. Then by integrating their strengths, we propose a double nonconvex TRKPCA (DNTRKPCA) method to achieve our expectation. Finally, we develop an efficient optimization framework via the alternating direction multiplier method (ADMM) to implement the proposed nonconvex kernel method. Experimental results on synthetic data and several real databases show the higher competitiveness of our method compared with other state-of-the-art regularization methods. The code has been released in our ResearchGate homepage: https://www.researchgate.net/publication/397181729_DNTRKPCA_code.

The field of low-tubal-rank tensor recovery, especially with subspace prior information, has recently garnered significant attention. However, existing methods encounter limitations when dealing with tensor data affected by simultaneous damage and loss. Moreover, they frequently necessitate clean (with no outliers) data to generate subspace prior information, which presents practical challenges. Addressing these issues, this article proposes a generalized subspace coupling (GSC) scheme, equipped with a novel tool to quantify the accuracy of the prior subspace. Building upon this foundation, we delve into the robust low-tubal-rank tensor completion problem, aiming to recover a low-tubal-rank tensor from partially observed data corrupted by sparse noise. Importantly, we theoretically demonstrate that the proposed method achieves exact tensor recovery under significantly weaker incoherence conditions compared to those previously suggested. Additionally, to optimize the proposed model, we design a symmetric Gauss-Seidel-based alternating direction method of multipliers (sGS-ADMM) with guaranteed convergence. Experiments conducted on various datasets, including facial images, medical scans, and video sequences, validate the superiority of our model over existing competitors in both qualitative and quantitative assessments.

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.

A plethora of previous studies indicates that making full use of multifarious intrinsic properties of primordial data is a valid pathway to recover original images from their degraded observations. Typically, both low-rankness and local-smoothness broadly exist in real-world tensor data such as hyperspectral images and videos. Modeling based on both properties has received a great deal of attention, whereas most studies concentrate on experimental performance, and theoretical investigations are still lacking. In this paper, we study the tensor compressive sensing problem based on the tensor correlated total variation, which is a new regularizer used to simultaneously capture both properties existing in the same dataset. The new regularizer has the outstanding advantage of not using a trade-off parameter to balance the two properties. The obtained theories provide a robust recovery guarantee, where the error bound shows that our model certainly benefits from both properties in ground-truth data adaptively. Moreover, based on the ADMM update procedure, we design an algorithm with a global convergence guarantee to solve this model. At last, we carry out experiments to apply our model to hyperspectral image and video restoration problems. The experimental results show that our method is prominently better than many other competing ones. Our code and Supplementary Material are available at https://github.com/fsliuxl/cs-tctv.

Tensor Robust Principal Component Analysis (TRPCA) aims at recovering the low-rank and sparse components from target tensor, which has extensive applications in multi-dimensional data recovery. However, most of the existing methods only exploit the global low-rank of image data, which result in missing local details in the recovered data. To restore the data more accurately, we propose a new TRPCA method which simultaneously combines the model-based method and data-driven approaches to preserve the global structure and fine local information. Specially, we pick the tensor nuclear norm to characterize the global correlation and a convolutional neural network(CNN) denoiser which reserves the local detail. Then, a flexible alternating direction method of multipliers (ADMM) is designed to deal with the proposed optimization model. Extensive experiments on various types of tensor data illustrate that our model enhances performance compared to state-of-the-art methods.

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.

It is known that the decomposition in low-rank and sparse matrices (L+S for short) can be achieved by several Robust PCA techniques. Besides the low rankness, the local smoothness (LSS) is a vitally essential prior for many real-world matrix data such as hyperspectral images and surveillance videos, which makes such matrices have low-rankness and local smoothness property at the same time. This poses an interesting question: Can we make a matrix decomposition in terms of L&LSS +S form exactly? To address this issue, we propose in this paper a new RPCA model based on three-dimensional correlated total variation regularization (3DCTV-RPCA for short) by fully exploiting and encoding the prior expression underlying such joint low-rank and local smoothness matrices. Specifically, using a modification of Golfing scheme, we prove that under some mild assumptions, the proposed 3DCTV-RPCA model can decompose both components exactly, which should be the first theoretical guarantee among all such related methods combining low rankness and local smoothness. In addition, by utilizing Fast Fourier Transform (FFT), we propose an efficient ADMM algorithm with a solid convergence guarantee for solving the resulting optimization problem. Finally, a series of experiments on both simulations and real applications are carried out to demonstrate the general validity of the proposed 3DCTV-RPCA model.

Tensor recovery is a fundamental problem in tensor research field. It generally requires to explore intrinsic prior structures underlying tensor data, and formulate them as certain forms of regularization terms for guiding a sound estimate of the restored tensor. Recent researches have made significant progress by adopting two insightful tensor priors, i.e., global low-rankness (L) and local smoothness (S), which are always encoded as a sum of two separate regularizers into recovery models. However, unlike the primary theoretical developments on low-rank tensor recovery, these joint "L+S" models have no theoretical exact-recovery guarantees yet, making the methods lack reliability in real practice. To this crucial issue, in this work, we build a unique regularizer termed as tensor correlated total variation (t-CTV), which essentially encodes both L and S priors of a tensor simultaneously. Especially, by equipping t-CTV into the recovery models, we can rigorously prove the exact recovery guarantees for two typical tensor recovery tasks, i.e., tensor completion and tensor robust principal component analysis. To the best of our knowledge, this should be the first exact-recovery results among all related "L+S" methods for tensor recovery. We further propose ADMM algorithms with fine convergence to solve the proposed models. Significant recovery accuracy improvements are observed in extensive experiments. Typically, our method achieves a workable performance when the missing rate is extremely large, e.g., 99.5%, for the color image inpainting task, while all its peers totally fail in such a challenging case. Code is released at https://github.com/wanghailin97.

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.

We consider Quantized Tensor Robust Principal Component Analysis (Q-TRPCA), which aims to recover a low-rank tensor and a sparse tensor from noisy, quantized, and sparsely corrupted measurements. A nonconvex constrained maximum likelihood (ML) estimation method is proposed for Q-TRPCA. We provide an upper bound on the Frobenius norm of tensor estimation error under this method. Making use of tools in information theory, we derive a theoretical lower bound on the best achievable estimation error from unquantized measurements. Compared with the lower bound, the upper bound on the estimation error is nearly order-optimal. We further develop an efficient convex ML estimation scheme for Q-TRPCA based on the tensor nuclear norm (TNN) constraint. This method is more robust to sparse noises than the latter nonconvex ML estimation approach. Conducting experiments on both synthetic data and real-world data, we show the effectiveness of the proposed methods.

The latest tensor recovery methods based on tensor Singular Value Decomposition (t-SVD) mainly utilize the tensor nuclear norm (TNN) as a convex surrogate of the rank function. However, TNN minimization treats each rank component equally and tends to over-shrink the dominant ones, thereby usually leading to biased solutions. To handle this critical issue, we put forward a weighted tensor Schantten-p (0q(0 < q ≤ 1) sparse regularization item on the extensively existed noises/outliers is incorporated into the WSTN minimization to enhance its robustness in the impulsive scenarios. Finally, we propose an efficient and scalable robust high-order tensor recovery method solving a double nonconvex optimization with convergence guarantees. Synthetic and real experiments demonstrate that the proposed approach outperforms the state-of-the-art ones in terms of both accuracy and computational complexity.

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.

A dynamically weighted directed network (DWDN) is frequently encountered in various big data-related applications like a terminal interaction pattern analysis system (TIPAS) concerned in this study. It consists of large-scale dynamic interactions among numerous nodes. As the involved nodes increase drastically, it becomes impossible to observe their full interactions at each time slot, making a resultant DWDN High Dimensional and Incomplete (HDI). An HDI DWDN, in spite of its incompleteness, contains rich knowledge regarding involved nodes various behavior patterns. To extract such knowledge from an HDI DWDN, this paper proposes a novel Alternating direction method of multipliers (ADMM)-based Nonnegative Latent-factorization of Tensors (ANLT) model. It adopts three-fold ideas: a) building a data density-oriented augmented Lagrangian function for efficiently handling an HDI tensors incompleteness and nonnegativity; b) splitting the optimization task in each iteration into an elaborately designed subtask series where each one is solved based on the previously solved ones following the ADMM principle to achieve fast convergence; and c) theoretically proving that its convergence is guaranteed with its efficient learning scheme. Experimental results on six DWDNs from real applications demonstrate that the proposed ANLT outperforms state-of-the-art models significantly in both computational efficiency and prediction accuracy.

Recovering low-rank matrices by nuclear norm minimization and local-smooth matrices by total variation seminorm minimization are two common methods in the context of compressive sensing. As a matter of fact, the two properties simultaneously exist in many real-world datasets, typically exampling hyperspectral images. The two methods may not perform well in this situation. To better address this issue, in this letter, we study the correlated total variation norm minimization problem both theoretically and numerically. We obtain an error bound for the robust recovery of our method in theory, which reflects that this model indeed benefits from low-rank and local-smooth properties of the matrix to be restored. Experiments on the recovery of hyperspectral images show that this model is superior to many other competing ones.

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.

Low-rank minimization aims to recover a matrix of minimum rank subject to linear system constraint. It can be found in various data analysis and machine learning areas, such as recommender systems, video denoising, and signal processing. Nuclear norm minimization is a dominating approach to handle it. However, such a method ignores the difference among singular values of target matrix. To address this issue, nonconvex low-rank regularizers have been widely used. Unfortunately, existing methods suffer from different drawbacks, such as inefficiency and inaccuracy. To alleviate such problems, this article proposes a flexible model with a novel nonconvex regularizer. Such a model not only promotes low rankness but also can be solved much faster and more accurate. With it, the original low-rank problem can be equivalently transformed into the resulting optimization problem under the rank restricted isometry property (rank-RIP) condition. Subsequently, Nesterov's rule and inexact proximal strategies are adopted to achieve a novel algorithm highly efficient in solving this problem at a convergence rate of O(1/K) , with K being the iterate count. Besides, the asymptotic convergence rate is also analyzed rigorously by adopting the Kurdyka- ojasiewicz (KL) inequality. Furthermore, we apply the proposed optimization model to typical low-rank problems, including matrix completion, robust principal component analysis (RPCA), and tensor completion. Exhaustively empirical studies regarding data analysis tasks, i.e., synthetic data analysis, image recovery, personalized recommendation, and background subtraction, indicate that the proposed model outperforms state-of-the-art models in both accuracy and efficiency.

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.

In this paper, the generative model is used to introduce the structural properties of the signal to replace the common sparse hypothesis, and a non-convex compressed sensing sparse deviation model based on the generative model (ℓq-Gen) is proposed. By establishing ℓqvariant of the restricted isometry property (q-RIP) and Set-Restricted Eigenvalue Condition (q-S-REC), the error upper bound of the optimal decoder is derived when the recovered signal is within the sparse deviation range of the generator. Furthermore, it is proved that the Gaussian matrix satisfying a certain number of measurements is sufficient to ensure a good recovery for the generating function with high probability. Finally, a series of experiments are carried out to verify the effectiveness and superiority of the ℓq-Gen model.

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.

Recently, deep learning methods have made a remarkable improvement in compressed sensing image recovery stage. In the compressed measurement stage, the existing methods measured by block by block owing to a huge measurement dictionary for the whole images and the high computational complexity. In this work, a novel deep convolutional neural network (DCNN) named Convolutional Measurement Compressed Sensing network (CMCS‐net) is proposed for image compressed sensing considering both convolutional measurement (CM) and sparse prior. Different from existing works, the convolution operation is adopted both in the measurement phase and reconstruction phase, which retains the structure information of images much better. Simultaneously, the size of the measurement matrix is no longer limited by data dimensions. Particularly, by unfolding the CM process to analyse a Toeplitz‐type matrix, the theoretical support of the convolutional compressed measurement is proposed. In addition, in the recovery phase, the authors consider the sparse prior in nature images by embedding the truncated hierarchical projection algorithm into their architecture to solve the problem of multilayered convolutional sparse coding. Furthermore, extensive experiments demonstrate that their proposed CMCS‐net can marvellously reconstruct the images and fully remove the block artefact.

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.

Motivated by the recently emerged method for sparse signal recovery, in this study, the authors make an ongoing effect to extend this methodology to the setting of block sparsity, which directly leads to the proposed method for block‐sparse signal recovery. Some theoretical results are induced to guarantee the validity of proposed method. In particular, the obtained recovery condition rigorously includes the one induced by Yin et al ., and the obtained error estimate can be used to model both the (block‐) sparse and non‐sparse signals, which is more comprehensive than that induced by Yin et al . which applies only to the sparse signals. The authors also derive an alternating direction method of multipliers (ADMM)‐based algorithm to tackle the induced optimisation problem. Some experimental results that are based on the synthetic block‐sparse signals and the real‐world foetal electrocardiogram signals further demonstrate the better performance of the method when it is compared with the state‐of‐the‐art group‐lasso method and method for 0 < q < 1.

Compressed sensing is a novel theory for signal sampling, which breaks through Nyquist/Shannon sampling limitation and makes it into reality that one can efficiently collect and robustly reconstruct a sparse signal. However, some signals exhibit additional structures in some redundant dictionaries, which is called block‐sparse signal. In this study, non‐convex block‐sparse compressed sensing with redundant dictionaries is investigated. Under the block D‐RIP condition , a sufficient condition for robust signal reconstruction with redundant dictionaries by mixed minimisation is established. Furthermore, the authors’ theoretical results show that, under the assumption that , , where urn:x-wiley:17519675:media:sil2bf00440:sil2bf00440-math-0005 then the block k ‐sparse signal can be stably reconstructed via non‐convex ℓ 2 /ℓ p minimisation with redundant dictionaries in the presence of noise. Particularly, this improves the existed result when the block‐sparse signal degenerate to the conventional signal case. Besides, the authors also obtain robust reconstruction condition and error upper bound estimation when the block number is no more than four times the sparsity of the block signal . Moreover, the numerical experiments to some extent testify the performance of non‐convex minimisation with redundant dictionaries.

By embedding an ℓp-norm noise constraint for p ≥ 2 into the recently emerged ℓ1-2method, in this letter, we study theoretically and numerically an ℓ1-2/ℓpmethod for recovery of general noisy signals from highly coherent measurement matrices. In particular, the obtained theoretical results not only improve the condition deduced in [1] for Gaussian noisy signal recovery but also provide a new theoretical guarantee for generally nonGaussian noisy signal recovery. What is more, to better boost the recovery performance, a partial sum ℓ1-2/ℓpmethod is also proposed latter. This improved method, together with the previous ℓ1-2/ℓpmethod, becomes more competitive when compared with some of the state-of-the-art methods in recovering noisy signals from highly coherent measurement matrices.

The mixed l 2 / l p (0 < p ≤ 1) norm minimisation method with partially known support for recovering block‐sparse signals is studied. The authors mainly extend this work on block‐sparse compressed sensing by incorporating some known part of the block support information as a priori and establish sufficient restricted p ‐isometry property ( p ‐RIP) conditions for exact and robust recovery. The authors’ theoretical results show it is possible to recover the block‐sparse signals via l 2 / l p minimisation from reduced number of measurements by applying the partially known support. The authors also derive a lower bound on necessary random Gaussian measurements for the p ‐RIP conditions to hold with high possibility. Finally, a series of numerical experiments are carried out to illustrate that fewer measurements with smaller p are needed to reconstruct the signal.

In this paper, we develop a novel coordinate descent fuzzy twin SVM (CDFTSVM) for classification. The proposed CDFTSVM not only inherits the advantages of twin SVM but also leads to a rapid and robust classification results. Specifically, our CDFTSVM has two distinguished advantages: (1) An effective fuzzy membership function is produced for removing the noise incurred by the contaminant inputs. (2) A coordinate descent strategy with shrinking by active set is used to deal with the computational complexity brought by the high dimensional input. In addition, a series of simulation experiments are conducted to verify the performance of the CDFTSVM, which further supports our previous claims.

This paper presents a new local region-based and local gradient-based active contour model in a variational level set formulation for image segmentation. The model consists of three parts: the local region term, the local gradient term and the regularization term. The local region term is insensitive to noise, while the local gradient term has better capability of localization than the local region. The energy minimization is achieved by the level set evolution in an iterative strategy. In each iteration, the local intensity and local gradient are updated and fed into the level set evolution. Comparative experiments show that our model achieves the better performance on the ultrasound images with noise and low signal-to-noise ratio than the local binary fitting (LBF) energy model.