Non-convex Low-rank Approximation Research Articles

Tensor Robust Principal Component Analysis via Non-convex Low-Rank Approximation Based on the Laplace Function

Tensor Robust Principal Component Analysis via Non-convex Low-Rank Approximation Based on the Laplace Function

Multitemporal image cloud removal using group sparsity and nonconvex low-rank approximation

Multitemporal image cloud removal using group sparsity and nonconvex low-rank approximation

Robust Low-Rank Graph Multi-View Clustering via Cauchy Norm Minimization

Graph-based multi-view clustering methods aim to explore the partition patterns by utilizing a similarity graph. However, many existing methods construct a consensus similarity graph based on the original multi-view space, which may result in the lack of information on the underlying low-dimensional space. Additionally, these methods often fail to effectively handle the noise present in the graph. To address these issues, a novel graph-based multi-view clustering method which combines spectral embedding, non-convex low-rank approximation and noise processing into a unit framework is proposed. In detail, the proposed method constructs a tensor by stacking the inner product of normalized spectral embedding matrices obtained from each similarity matrix. Then, the obtained tensor is decomposed into a low-rank tensor and a noise tensor. The low-rank tensor is constrained via nonconvex low-rank tensor approximation and a novel Cauchy norm with an upper bound is proposed to handle the noise. Finally, we derive the consensus similarity graph from the denoised low-rank tensor. The experiments on five datasets demonstrate that the proposed method outperforms other state-of-the-art methods on five datasets.

An Unsupervised Image Denoising Method Using a Nonconvex Low-Rank Model with TV Regularization

In real-world scenarios, images may be affected by additional noise during compression and transmission, which interferes with postprocessing such as image segmentation and feature extraction. Image noise can also be induced by environmental variables and imperfections in the imaging equipment. Robust principal component analysis (RPCA), one of the traditional approaches for denoising images, suffers from a failure to efficiently use the background’s low-rank prior information, which lowers its effectiveness under complex noise backgrounds. In this paper, we propose a robust PCA method based on a nonconvex low-rank approximation and total variational regularization (TV) to model the image denoising problem in order to improve the denoising performance. Firstly, we use a nonconvex γ-norm to address the issue that the traditional nuclear norm penalizes large singular values excessively. The rank approximation is more accurate than the nuclear norm thanks to the elimination of matrix elements with substantial approximation errors to reduce the sparsity error. The method’s robustness is improved by utilizing the low sensitivity of the γ-norm to outliers. Secondly, we use the l1-norm to increase the sparsity of the foreground noise. The TV norm is used to improve the smoothness of the graph structure in accordance with the sparsity of the image in the gradient domain. The denoising effectiveness of the model is increased by employing the alternating direction multiplier strategy to locate the global optimal solution. It is important to note that our method does not require any labeled images, and its unsupervised denoising principle enables the generalization of the method to different scenarios for application. Our method can perform denoising experiments on images with different types of noise. Extensive experiments show that our method can fully preserve the edge structure information of the image, preserve important features of the image, and maintain excellent visual effects in terms of brightness smoothing.

RFI Localization Using Jointly Non-Convex Low-Rank Approximation and Expanded Virtual Array in Microwave Interferometric Radiometry

The scientific goal of the Soil Moisture and Ocean Salinity (SMOS) mission is to retrieve the geophysical parameter from brightness temperature (TB) maps. However, radio frequency interference (RFI) significantly influences the interpretation of TB maps, leading to a deteriorated retrieval performance. RFI localization is essential for switching off these illegal emitters and mitigating their impacts on TB maps. This article proposes a novel high-resolution RFI localization method via jointly non-convex low-rank approximation and expanded virtual array (EVA). Concretely, the RFI localization problem is first formulated from the perspective of non-convex low-rank recovery, which better approximates the rank of the covariance matrix collecting visibility samples. Then, we propose the EVA concept by relaxing the size constraint on the physical antenna array. Moreover, we employ a new algorithm based on the joint Schatten-p and Lp (JSL) norms to solve the above non-convex low-rank recovery problem. This JSL algorithm can improve the spatial resolution for RFI localization. Combining the JSL algorithm and the EVA can further improve detection performance and enhance the spatial resolution for RFI localization. The experimental results using synthetic data and real SMOS data prove that the proposed method shows enhanced spatial resolution, better detection performance, and competitive or better localization accuracy compared with the currently existing methods.

Moving object detection via RPCA framework using non-convex low-rank approximation and total variational regularization

Moving object detection via RPCA framework using non-convex low-rank approximation and total variational regularization

Weighted Schatten [formula omitted]-norm minimization with logarithmic constraint for subspace clustering

Rank minimization-based subspace clustering methods have been widely developed in the past decades. Although some smooth surrogates, such as the nuclear norm and Schatten-p norm mitigate the NP-hard issue to some extend, these existing methods may yield unsatisfactory results, due to the singular values of the coefficient matrix not being further suppressed. To tackle this, in this paper, we propose a novel non-convex low-rank approximation based on weighted Schatten-p norm jointed logarithmic constraint, which can suppress the small and large singular values flexibly with a tighter relaxation. Specifically, we firstly proposed a low-rank approximation termed SLog by utilizing the logarithmic to tighten the Schatten-p norm, which can shrink the large singular values in a similar trend of the real rank minimization. Furthermore, to suppress the small singular values simultaneously, usually considered noise, we propose a weighted Schatten p-norm minimization named WSLog based on SLog by introducing the weight w, which behaves more robust to the sparse noise, especially w<1. Compared with recent-proposed methods, extensive experiments in subspace clustering on real datasets demonstrate the effective performance of our methods.

Hyperspectral Image Denoising Using Factor Group Sparsity-Regularized Nonconvex Low-Rank Approximation

Hyperspectral image (HSI) mixed noise removal is a fundamental problem and an important preprocessing step in remote sensing fields. The low-rank approximation-based methods have been verified effective to encode the global spectral correlation for HSI denoising. However, due to the large scale and complexity of real HSI, previous low-rank HSI denoising techniques encounter several problems, including coarse rank approximation (such as nuclear norm), the high computational cost of singular value decomposition (SVD) (such as Schatten <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -norm), and adaptive rank selection (such as low-rank factorization). In this article, two novel factor group sparsity-regularized nonconvex low-rank approximation (FGSLR) methods are introduced for HSI denoising, which can simultaneously overcome the mentioned issues of previous works. The FGSLR methods capture the spectral correlation via low-rank factorization, meanwhile utilizing factor group sparsity regularization to further enhance the low-rank property. It is SVD-free and robust to rank selection. Moreover, FGSLR is equivalent to Schatten <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -norm approximation ( <xref ref-type="theorem" rid="theorem1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorem 1</xref> ), and thus FGSLR is tighter than the nuclear norm in terms of rank approximation. To preserve the spatial information of HSI in the denoising process, the total variation regularization is also incorporated into the proposed FGSLR models. Specifically, the proximal alternating minimization is designed to solve the proposed FGSLR models. Experimental results have demonstrated that the proposed FGSLR methods significantly outperform existing low-rank approximation-based HSI denoising methods.

Low-Rank Approximation and Multiple Sparse Constraint Modeling for Infrared Low-Flying Fixed-Wing UAV Detection

Infrared dim small target detection is one of the important contents in the research of military applications such as remote sensing intelligence reconnaissance, long-range precision strike, aerospace offense–defense confrontation, etc. In this article, we focus on the detection of low-flying fixed-wing unmanned aerial vehicle target based on infrared imaging. To this end, we propose a simple and effective detection model, which can be viewed as a combination of low-rank approximation and multiple sparse constraints. We first model the infrared image that to be detected as a sum of three patch matrices called background, target, and noise. Then, we put a nonconvex low-rank approximation on the background patch matrix to suppress the background edges and put a reweighted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{1,1}$</tex-math></inline-formula> -norm constraint on the target matrix to better preserve the dim small target. Moreover, in order to eliminate the strong residual edges left in the target image under complex background, both the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{1,1}$</tex-math></inline-formula> matrix norm and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2,1}$</tex-math></inline-formula> matrix norm are used to constrain the noise patch. Finally, we develop an alternating optimization algorithm to solve the associated minimization problem. Extensive experiments carried out on a recently released real low-flying UAV database show that the proposed approach works well in detecting infrared dim small target measured by qualitative analysis and quantitative analysis.

Downlink channel estimation for millimeter wave communication combining low-rank and sparse structure characteristics

The acquisition of channel state information (CSI) is essential in millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems. The mmWave channel exhibits sparse scattering characteristics and a meaningful low-rank structure, which can be simultaneously employed to reduce the complexity of channel estimation. Most existing works recover the low-rank structure of channels using nuclear norm theory. However, solving the nuclear norm-based convex problem often leads to a suboptimal solution of the rank minimization problem, thus degrading the accuracy of channel estimation. Previous contributions recover the channel using over-complete dictionary with the assumption that the mmWave channel can be sparsely represented under some dictionary. While over-complete dictionary may increase the computational complexity. To address these problems, we propose a channel estimation framework based on non-convex low-rank approximation and dictionary learning by exploring the joint low-rank and sparse representations of wireless channels. We surrogate the widely used nuclear norm theory with non-convex low-rank approximation method and design a dictionary learning algorithm based on channel feature classification employing deep neural network (DNN). Our simulation results reveal the proposed scheme outperform the conventional dictionary learning algorithm, Bayesian framework algorithm, and compressed sensing-based algorithms.

Robust subspace clustering based on non-convex low-rank approximation and adaptive kernel

As a relatively advanced method, the low-rank kernel space clustering method shows good performance in dealing with nonlinear structure of high-dimensional data. Unfortunately, this method is sensitive to large corruptions and doesn’t balance the contribution of all singular values. To solve the above problems, the low-rank kernel method is modified, and a robust subspace clustering method (LAKRSC) based on non-convex low-rank approximation and adaptive kernel is proposed. In our model, the weighted Schatten p-norm is introduced to balance the importance of different singular values, which can more accurately approximate the rank function and be more flexible in practical applications. Therefore, applying weighted Schatten p-norm to adaptive kernel can approximate the original low rank hypothesis better when the data is mapped into the feature space. In addition, our model uses correntropy to handle complex noise which enhances the robustness of the model. A new algorithm HQ&ADMM, combined by Half-Quadratic technique (HQ) and ADMM, is studied to solve our model. Experiments on four real-world datasets show that the clustering performance of LAKRSC is significantly better than that of several more advanced methods.

Nonconvex Low Rank Approximation With Phase Congruency Regularization for Mixed Noise Removal

Mixed noise removal from a natural image is a challenging task since the complex noise distribution usually is inestimable. Many noise removal methods based on the low rank approximation have an excellent image denoising performance and are effective for recovering the images corrupted by Gaussian noise. These methods based on the additive white Gaussian noise(AWGN) model are sensitive to the outliers and non-Gaussian noise, such as the salt-and-pepper impulse noise (SPIN) and random valued impulse noise (RVIN). Such methods for mixed noise removal, however, are less effective in preserving image structures and tend to undesired staircase artifacts. This paper presents a novel Nonconvex Low Rank Model with Phase congruency and overlapping Group sparsity regularization (NLRM-PG) for mixed noise removal. Moreover, an efficient optimization algorithm under the alternating direction method of multipliers and majorization minimization framework is proposed to solve the NLRM-PG model. We demonstrate that the proposed method is effective for preserving local irregular structures and it reduces staircase artifacts with the two types of mixture noise, namely, AWGN+SPIN and AWGN+RVIN. Both qualitative and quantitative experiment results on synthetic noisy images and real noisy images illustrate that the proposed method can remove mixture noise in images more efficiently than the existing methods can do. And the results also outperform those obtained by using the competing state-of-the-art methods, particularly for the removal of high-density impulse noise.

Matrix Completion Based on Non-convex Low Rank Approximation.

Without any prior structure information, Nuclear Norm Minimization (NNM), a convex relaxation for Rank Minimization (RM), is a widespread tool for matrix completion and relevant low rank approximation problems. Nevertheless, the result derivated by NNM generally deviates the solution we desired, because NNM ignores the difference between different singular values. In this paper, we present a non-convex regularizer and utilize it to construct two matrix completion models. In order to solve the constructed models efficiently, we develop an efficient optimization method with convergence guarantee, which can achieve faster convergence speed compared to conventional approaches. Particularly, we show that the proposed regularizer as well as optimization method are suitable for other RM problems, such as subspace clustering based on low rank representation. Extensive experimental results on real images demonstrate that the constructed models provide significant advantages over several state-of-the-art matrix completion algorithms. In addition, we implement numerous experiments to investigate the convergence speed of developed optimization method.

Image compressed sensing based on non-convex low-rank approximation

Nonlocal sparsity and structured sparsity have been evidenced to improve the reconstruction of image details in various compressed sensing (CS) studies. The nonlocal processing is achieved by grouping similar patches of the image into the groups. To exploit these nonlocal self-similarities in natural images, a non-convex low-rank approximation is proposed to regularize the CS recovery in this paper. The nuclear norm minimization, as a convex relaxation of rank function minimization, ignores the prior knowledge of the matrix singular values. This greatly restricts its capability and flexibility in dealing with many practical problems. In order to make a better approximation of the rank function, the non-convex low-rank regularization namely weighted Schatten p-norm minimization (WSNM) is proposed. In this way, both the local sparsity and nonlocal sparsity are integrated into a recovery framework. The experimental results show that our method outperforms the state-of-the-art CS recovery algorithms not only in PSNR index, but also in local structure preservation.

Reference Information Based Remote Sensing Image Reconstruction with Generalized Nonconvex Low-Rank Approximation

Because of the contradiction between the spatial and temporal resolution of remote sensing images (RSI) and quality loss in the process of acquisition, it is of great significance to reconstruct RSI in remote sensing applications. Recent studies have demonstrated that reference image-based reconstruction methods have great potential for higher reconstruction performance, while lacking accuracy and quality of reconstruction. For this application, a new compressed sensing objective function incorporating a reference image as prior information is developed. We resort to the reference prior information inherent in interior and exterior data simultaneously to build a new generalized nonconvex low-rank approximation framework for RSI reconstruction. Specifically, the innovation of this paper consists of the following three respects: (1) we propose a nonconvex low-rank approximation for reconstructing RSI; (2) we inject reference prior information to overcome over smoothed edges and texture detail losses; (3) on this basis, we combine conjugate gradient algorithms and a single-value threshold (SVT) simultaneously to solve the proposed algorithm. The performance of the algorithm is evaluated both qualitatively and quantitatively. Experimental results demonstrate that the proposed algorithm improves several dBs in terms of peak signal to noise ratio (PSNR) and preserves image details significantly compared to most of the current approaches without reference images as priors. In addition, the generalized nonconvex low-rank approximation of our approach is naturally robust to noise, and therefore, the proposed algorithm can handle low resolution with noisy inputs in a more unified framework.

Non-Convex Low-Rank Approximation for Image Denoising and Deblurring

Non-Convex Low-Rank Approximation for Image Denoising and Deblurring