Low-Rank Matrix Recovery via Modified Schatten-p Norm Minimization with Convergence Guarantees

Anchor Graph Network for Incomplete Multiview Clustering

Incomplete multiview clustering (IMVC) has received extensive attention in recent years. However, existing works still have several shortcomings: 1) some works ignore the correlation of sample pairs in the global structural distribution; 2) many methods are computational expensive, thus cannot be applicable to the large-scale incomplete data clustering tasks; and 3) some methods ignore the refinement of the bipartite graph structure. To address the above issues, we propose a novel anchor graph network for IMVC, which includes a generative model and a similarity metric network. Concretely, the method uses a generative model to construct bipartite graphs, which can mine latent global structure distributions of sample pairs. Later, we use graph convolution network (GCN) with the constructed bipartite graphs to learn the structural embeddings. Notably, the introduction of bipartite graphs can greatly reduce the computational complexity and thus enable our model to handle large-scale data. Unlike previous works based on bipartite graph, our method employs bipartite graphs to guide the learning process in GCNs. In addition, an innovative adaptive learning strategy that can construct robust bipartite graphs is incorporated into our method. Extensive experiments demonstrate that our method achieves the comparable or superior performance compared with the state-of-the-art methods.

Iteratively Capped Reweighting Norm Minimization with Global Convergence Guarantee for Low-Rank Matrix Learning

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.

Accurate Concentration Recovery for Quantitative Magnetic Particle Imaging Reconstruction via Nonconvex Regularization

Magnetic particle imaging (MPI) uses nonlinear response signals to noninvasively detect magnetic nanoparticles in space, and its quantitative properties hold promise for future precise quantitative treatments. In reconstruction, the system matrix based method necessitates suitable regularization terms, such as Tikhonov or non-negative fused lasso (NFL) regularization, to stabilize the solution. While NFL regularization offers clearer edge information than Tikhonov regularization, it carries a biased estimate of the l1 penalty, leading to an underestimation of the reconstructed concentration and adversely affecting the quantitative properties. In this paper, a new nonconvex regularization method including min-max concave (MC) and total variation (TV) regularization is proposed. This method utilized MC penalty to provide nearly unbiased sparse constraints and adds the TV penalty to provide a uniform intensity distribution of images. By combining the alternating direction multiplication method (ADMM) and the two-step parameter selection method, a more accurate quantitative MPI reconstruction was realized. The performance of the proposed method was verified on the simulation data, the Open-MPI dataset, and measured data from a homemade MPI scanner. The results indicate that the proposed method achieves better image quality while maintaining the quantitative properties, thus overcoming the drawback of intensity underestimation by the NFL method while providing edge information. In particular, for the measured data, the proposed method reduced the relative error in the intensity of the reconstruction results from 28% to 8%.

A continuous-time neurodynamic approach in matrix form for rank minimization

This article proposes a continuous-time neurodynamic approach for solving the rank minimization under affine constraints. As opposed to the traditional neurodynamic approach, the proposed neurodynamic approach extends the form of the variables from the vector form to the matrix form. First, a continuous-time neurodynamic approach with variables in matrix form is developed by combining the optimal rank r projection and the gradient. Then, the optimality of the proposed neurodynamic approach is rigorously analyzed by demonstrating that the objective function satisfies the functional property which is called as (2r,4r)-restricted strong convexity and smoothness ((2r,4r)-RSCS). Furthermore, the convergence and stability analysis of the proposed neurodynamic approach is rigorously conducted by establishing appropriate Lyapunov functions and considering the relevant restricted isometry property (RIP) condition associated with the affine transformation. Finally, through experiments involving low-rank matrix recovery under affine transformations and the completion of low-rank real image, the effectiveness of this approach has been demonstrated, along with its superiority compared to the vector-based approach.

Iteratively Reweighted Minimax-Concave Penalty Minimization for Accurate Low-rank Plus Sparse Matrix Decomposition

Low-rank plus sparse matrix decomposition (LSD) is an important problem in computer vision and machine learning. It has been solved using convex relaxations of the matrix rank and l₀-pseudo-norm, which are the nuclear norm and l₁-norm, respectively. Convex approximations are known to result in biased estimates, to overcome which, nonconvex regularizers such as weighted nuclear-norm minimization and weighted Schatten p-norm minimization have been proposed. However, works employing these regularizers have used heuristic weight-selection strategies. We propose weighted minimax-concave penalty (WMCP) as the nonconvex regularizer and show that it admits an equivalent representation that enables weight adaptation. Similarly, an equivalent representation to the weighted matrix gamma norm (WMGN) enables weight adaptation for the low-rank part. The optimization algorithms are based on the alternating direction method of multipliers technique. We show that the optimization frameworks relying on the two penalties, WMCP and WMGN, coupled with a novel iterative weight update strategy, result in accurate low-rank plus sparse matrix decomposition. The algorithms are also shown to satisfy descent properties and convergence guarantees. On the applications front, we consider the problem of foreground-background separation in video sequences. Simulation experiments and validations on standard datasets, namely, I2R, CDnet 2012, and BMC 2012 show that the proposed techniques outperform the benchmark techniques.

Nonconvex Structural Sparsity Residual Constraint for Image Restoration

This article proposes a novel nonconvex structural sparsity residual constraint (NSSRC) model for image restoration, which integrates structural sparse representation (SSR) with nonconvex sparsity residual constraint (NC-SRC). Although SSR itself is powerful for image restoration by combining the local sparsity and nonlocal self-similarity in natural images, in this work, we explicitly incorporate the novel NC-SRC prior into SSR. Our proposed approach provides more effective sparse modeling for natural images by applying a more flexible sparse representation scheme, leading to high-quality restored images. Moreover, an alternating minimizing framework is developed to solve the proposed NSSRC-based image restoration problems. Extensive experimental results on image denoising and image deblocking validate that the proposed NSSRC achieves better results than many popular or state-of-the-art methods over several publicly available datasets.

Weighted Schatten p-Norm Low Rank Error Constraint for Image Denoising

Traditional image denoising algorithms obtain prior information from noisy images that are directly based on low rank matrix restoration, which pays little attention to the nonlocal self-similarity errors between clear images and noisy images. This paper proposes a new image denoising algorithm based on low rank matrix restoration in order to solve this problem. The proposed algorithm introduces the non-local self-similarity error between the clear image and noisy image into the weighted Schatten p-norm minimization model using the non-local self-similarity of the image. In addition, the low rank error is constrained by using Schatten p-norm to obtain a better low rank matrix in order to improve the performance of the image denoising algorithm. The results demonstrate that, on the classic data set, when comparing with block matching 3D filtering (BM3D), weighted nuclear norm minimization (WNNM), weighted Schatten p-norm minimization (WSNM), and FFDNet, the proposed algorithm achieves a higher peak signal-to-noise ratio, better denoising effect, and visual effects with improved robustness and generalization.

Low-Rank Matrix Recovery from Noise via an MDL Framework-Based Atomic Norm

The recovery of the underlying low-rank structure of clean data corrupted with sparse noise/outliers is attracting increasing interest. However, in many low-level vision problems, the exact target rank of the underlying structure and the particular locations and values of the sparse outliers are not known. Thus, the conventional methods cannot separate the low-rank and sparse components completely, especially in the case of gross outliers or deficient observations. Therefore, in this study, we employ the minimum description length (MDL) principle and atomic norm for low-rank matrix recovery to overcome these limitations. First, we employ the atomic norm to find all the candidate atoms of low-rank and sparse terms, and then we minimize the description length of the model in order to select the appropriate atoms of low-rank and the sparse matrices, respectively. Our experimental analyses show that the proposed approach can obtain a higher success rate than the state-of-the-art methods, even when the number of observations is limited or the corruption ratio is high. Experimental results utilizing synthetic data and real sensing applications (high dynamic range imaging, background modeling, removing noise and shadows) demonstrate the effectiveness, robustness and efficiency of the proposed method.