In recent years, a large number of studies have shown that low rank matrix learning (LRML) has become a popular approach in machine learning and computer vision with many important applications, such as image inpainting, subspace clustering, and recommendation system. The latest LRML methods resort to using some surrogate functions as convex or nonconvex relaxation of the rank function. However, most of these methods ignore the difference between different rank components and can only yield suboptimal solutions. To alleviate this problem, in this paper we propose a novel nonconvex regularizer called capped reweighting norm minimization (CRNM), which not only considers the different contributions of different rank components, but also adaptively truncates sequential singular values. With it, a general LRML model is obtained. Meanwhile, under some mild conditions, the global optimum of CRNM regularized least squares subproblem can be easily obtained in closed-form. Through the analysis of the theoretical properties of CRNM, we develop a high computational efficiency optimization method with convergence guarantee to solve the general LRML model. More importantly, by using the Kurdyka-Łojasiewicz (KŁ) inequality, its local and global convergence properties are established. Finally, we show that the proposed nonconvex regularizer as well as the optimization approach are suitable for different low rank tasks, such as matrix completion and subspace clustering. Extensive experimental results demonstrate that the constructed models and methods provide significant advantages over several state-of-the-art low rank matrix leaning models and methods.

Hyperspectral images (HSIs) are frequently corrupted by mixed sparse and Gaussian noise during acquisition and transmission processes, which degrades image quality and compromises subsequent analysis. In recent years, low-rank approximation-based methods have achieved promising results in HSI denoising and have garnered widespread attention. However, most of these methods often lack robustness, failing to deliver consistent performance across diverse noise scenarios, and tend to oversmooth fine image details. To address these issues, this paper fully explores low-rank and sparse prior knowledge of HSIs and proposes a novel generalized nonconvex framework for HSI denoising. The proposed method offers two significant advantages. First, the model exhibits high flexibility. By selecting scenario-specific nonconvex surrogates, its generalized nonconvex surrogate ensures optimal singular value recovery performance of HSIs across diverse noise scenarios. Second, by enhancing the distinction between image details and noise, its sparse regularizer further improves singular value recovery accuracy of HSIs. To solve the proposed model, an efficient algorithm with theoretical guarantees is developed by leveraging the well-known inexact augmented Lagrange multiplier (ALM) framework. Experimental results on both simulated and real datasets demonstrate that the proposed method outperforms state-of-the-art approaches and achieves superior results both quantitatively and visually. MATLAB code is available at https://github.com/wangzhi-swu/GNRSR.

Due to the high flexibility and remarkable performance, low-rank approximation has been widely studied for color image denoising. However, existing methods usually ignore the cross-channel difference or the spatial variation of noise, which limits their capacity in the task of real world color image denoising. To overcome these drawbacks, this paper proposes a double-weighted truncated nuclear norm minus truncated Frobenius norm minimization (DtNFM) model, and apply it to color image denoising through exploiting the nonlocal self-similarity prior. The proposed DtNFM model has two merits. First, it models and utilizes both the cross-channel difference and the spatial variation of noise. This provides sufficient flexibility for handling the complex distribution of noise in real world images. Second, the proposed DtNFM model provides a close approximation to the underlying clean matrix since it can treat different rank components flexibly. To solve the DtNFM model, an efficient algorithm is devised through exploiting the framework of alternating directions method of multipliers (ADMM). Meanwhile, the truncated nuclear norm minus truncated Frobenius norm regularized least squares subproblem is discussed in detail, and the results show that its global optimum can be directly obtained in closed form. Therefore, the DtNFM model can be efficiently solved by a single ADMM. Rigorous mathematical derivation proves that the solution sequences generated by our proposed algorithm converge to a single critical point. Extensive experiments on synthetic and real noise datasets demonstrate that the proposed method outperforms many state-of-the-art color image denoising methods. MATLAB code is available at https://github.com/wangzhi-swu/DtNFM.

Robust principal component analysis (RPCA) has gained popularity for handling high-dimensional data. The nuclear norm minimization (NNM) in RPCA is a classical method and has been widely investigated, which can recover low-rank and sparse matrices with high probability under certain conditions. However, NNM shrinks all singular values by the same threshold and over-penalizes larger singular values, resulting in this model being biased. Therefore, we propose a new method based on the truncated$l_{1-2}$norm to solve this problem in this paper, which is unbiased and flexible to capture the low-rank structure of the data matrix more accurately while separating the sparse noise. We also develop a robust and efficient algorithm to solve the proposed nonconvex optimization model, with the computational complexity and convergence discussed. Then the proposed scheme is applied to synthetic data as well as real-world data, including video background subtraction, facial shadow removal, and anomaly detection, for testing. These experimental results demonstrate that our proposed method is effect and superior in accuracy and robustness compared to other state-of-the-art methods.

Most existing image denoising methods commonly assume that the image is contaminated by additive white Gaussian noise (AWGN). However, real‐world color image noise exhibits more complicated distribution properties, making it challenging to develop an accurate model. Consequently, denoising methods designed for AWGN often fail to achieve satisfactory performance on real‐world images. In this paper, we present a novel multi‐channel optimization model for real‐world color images denoising within the multi‐weighted Schatten p‐norm minimization. Our proposed model utilizes the weighted Schatten p‐norm as the regularization term, while the data fidelity term employs two weight matrices to balance the noise level across channels and regions. Besides, it helps to preserve as much detail as possible in the recovered image while removing noise. Although our proposed model is nonconvex and has no analytical solution, an accurate and efficient optimization algorithm is established based on the alternating direction method of multipliers (ADMMs) framework. Finally, we demonstrate the superior performance of our proposed method over existing state‐of‐the‐art models on three real image datasets.

Hyperspectral image (HSI) denoising is a challenging task, not only because it is unavoidably contaminated by severe mixed noises, but also for its hard-to-recover spatial-spectral structure. Since it has been found that HSI has low-rank property, low-rank models have received extensive attention in dealing with the HSI denoising task. However, these models either use nuclear norm, which can only obtain sub-optimal solutions, or require some predefined information that is difficult to determine. To address these issues, in this paper we propose a new HSI denoising model based on non-convex fraction function, which has excellent performance in removing mixed noises. Specifically, the proposed model can capture the rank information of HSI automatically, which allows it to separate clean HSI from noises more accurately. Then, an iterative optimization algorithm is developed by exploiting the framework of the augmented Lagrange multiplier (ALM). Meanwhile, the subproblems at each iteration can be solved by the proximal operator with a closed-form solution. Besides, the convergence of the proposed algorithm is also provided theoretically. Extensive experiments implemented with simulated and real datasets demonstrate that our proposed model performs better than state-of-the-art models in HSI denoising. MATLAB code is available at https://github.com/wangzhi-swu/HSI-Denosing.

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.

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.

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.