This paper proposes a distributed-memory parallel randomized iterative algorithm for solving linear systems, called the parallel randomized kaczmarz projection (PRKP) algorithm. The algorithm has the property of greedy sampling, alternating projection, and lazy approximation. We derive the alternating projection process from the Randomized Kaczmarz algorithm and develop the greedy sampling and lazy approximation process to improve the convergence rate and reduce the per-process communication volume. Moreover, we develop a sampling residual estimation scheme for our proposed algorithm, which greatly reduces the extra computation cost required to obtain residuals. Our experimental results show that the proposed algorithm significantly outperforms previous parallel iterative algorithms on overdetermined linear systems, and yields up to $\mathbf{159}\times$ and $\mathbf{384}\times$ speedups on average respectively compared to the leading pipelined CG and pipelined GMRES algorithms.

Wireless sensor networks (WSNs) are widely used in industrial production, environmental monitoring, and military applications. In the process of using, the node non-uniform distribution strategy can mitigate the energy hole and node suboptimal energy balance technology in wireless sensor networks. This paper discusses this strategy theoretically, proposes a node non-uniform distribution strategy, and it constructs a suboptimal energy balance algorithm, which based on the non-uniform distribution theory system. It has proved that in the circular network with non-uniform distribution of nodes, the uniform distributed method and the random non-uniform distributed method are tested and compared. The experimental results show that the non-uniform distributed method has high efficiency and good scalability, and it can be used to achieve the suboptimal energy balance. The simulation results also show that the nodes in the WSN are almost equal to the energy consumption.

Recently, robust principal component analysis (RPCA) has been widely used in the detection of moving objects. However, this method fails to effectively utilize the low-rank prior information of the background and the spatiotemporal continuity prior of the moving object, and the target extraction effect is often poor when dealing with large-scale complex scenes. To solve the above problems, a new non-convex rank approximate RPCA model based on segmentation constraint is proposed. Firstly, the model adopts the low-rank sparse decomposition method to divide the original video sequence into three parts: low-rank background, moving foreground and sparse noise. Then, a new non-convex function is proposed to better constrain the low-rank characteristic of the video background. Finally, based on the spatiotemporal continuity of the foreground object, the video is segmented by the super-pixel segmentation technology, so as to realize the constraint of the motion foreground region. The augmented Lagrange multiplier method is used to solve the model. Experimental results show that the proposed model can effectively improve the accuracy of moving object detection, and has better visual effect of foreground object detection than existed methods.

Background/foreground separation has become an inevitable step in numerous image/video processing applications, such as image/video inpainting, anomaly detection, motion segmentation, augmented reality, and so on. Recent low-rank based approaches, such as robust principal component analysis separating a data matrix into a low-rank matrix with a sparse matrix, have achieved encouraging performance. However, these approaches usually need relatively high computation cost, mainly due to calculation of full or partial singular value decomposition of large matrices. On the other hand, the nuclear norm is widely exploited as a convex surrogate of the original rank function, while it is not a tighter envelope of the original rank function. To address these above-mentioned issues, this paper proposes a fast background/foreground separation algorithm in which the low-rank constraint is solved by a matrix factorization scheme, thus heavily reducing the computation cost. We further adopt two non-convex low-rank approximations to improve the robustness and flexibility of the traditional nuclear norm. In comparison with the state-of-the-art low-rank reconstruction methods, experimental results on challenging data sets, which contain different real data sets, show our superior performance in both image clarity and computation efficiency.

A new fixed point theorem for systems of nonlinear operator equations is established by means of topological degree theory and positively 1-homogeneous operator, where the components has a positively 1-homogeneous majorant or minorant. As applications, the existence of positive solutions for \((p_1, p_2)\)-Laplacian system is considered under some conditions concerning the positive eigenvalues corresponding to the relevant positively 1-homogeneous operators.

Low-rank representation (LRR) has been shown successful in seeking low-rank structures of data relationships in a union of subspaces. Generally, LRR and LRR-based variants need to solve the nuclear norm-based minimization problems. Beyond the success of such methods, it has been widely noted that the nuclear norm may not be a good rank approximation because it simply adds all singular values of a matrix together and thus large singular values may dominant the weight. This results in far from satisfactory rank approximation and may degrade the performance of low-rank models based on the nuclear norm. In this paper, we propose a novel nonconvex rank approximation based on the Gaussian distribution function, which has demanding properties to be a better rank approximation than the nuclear norm. Then a low-rank model is proposed based on the new rank approximation with application to motion segmentation. Experimental results have shown significant improvements and verified the effectiveness of our method.

Background recovery is a key technique in video analysis, but it still suffers from many challenges, such as camouflage, lighting changes, and diverse types of image noise. Robust principal component analysis (RPCA), which aims to recover a low-rank matrix and a sparse matrix, is a general framework for background recovery. The nuclear norm is widely used as a convex surrogate for the rank function in RPCA, which requires computing the singular value decomposition (SVD), a task that is increasingly costly as matrix sizes and ranks increase. However, matrix factorization greatly reduces the dimension of the matrix for which the SVD must be computed. Motion information has been shown to improve low-rank matrix recovery in RPCA, but this method still finds it difficult to handle original video data sets because of its batch-mode formulation and implementation. Hence, in this paper, we propose a motion-assisted RPCA model with matrix factorization (FM-RPCA) for background recovery. Moreover, an efficient linear alternating direction method of multipliers with a matrix factorization (FL-ADM) algorithm is designed for solving the proposed FM-RPCA model. Experimental results illustrate that the method provides stable results and is more efficient than the current state-of-the-art algorithms. (C) 2018 SPIE and IS& T

This paper concerns the low-rank minimization problems which consist of finding a matrix of minimum rank subject to linear constraints. Many existing approaches, which used the nuclear norm as a convex surrogate of the rank function, usually result in a suboptimal solution. To seek a tighter rank approximation, we develop a non-convex surrogate to approximate the rank function based on the Laplace function. An iterative algorithm based on the augmented Lagrangian multipliers method is developed. Empirical studies for practical applications including robust principal component analysis and low-rank representation demonstrate that our proposed algorithm outperforms many other state-of-the-art convex and non-convex methods developed recently in the literature.

In view of the routing protocols for wireless sensor networks with mobile sink, the paper classifies the protocols from the network structure and the node mobility pattern respectively. This paper analyzes the design requirements of routing protocols, introduces the characteristics of various protocols and the typical protocols, and analyzes the research ideas of the routing protocols for wireless sensor networks with mobile sink.

Rank approximation and minimization problem is widely applied in machine learning and computer vision. As the minimum convex envelope of the rank function, the nuclear norm is often used for rank approximation and has achieved satisfactory results in different tasks. However, the nuclear norm may not be an appropriate rank approximation especially when there are large singular values. In this paper, we propose two different functions to more accurately approximate the rank function. Then based on the low-rank representation model, we use these approximations for robust subspace clustering with desirable low rank and robustness to noise. Experimental results show the effectiveness of our proposed methods.

Hyperspectral image (HSI) denoising is challenging not only because of the difficulty in preserving both spectral and spatial structures simultaneously, but also due to the requirement of removing various noises, which are often mixed together. In this paper, we present a nonconvex low rank matrix approximation (NonLRMA) model and the corresponding HSI denoising method by reformulating the approximation problem using nonconvex regularizer instead of the traditional nuclear norm, resulting in a tighter approximation of the original sparsity-regularised rank function. NonLRMA aims to decompose the degraded HSI, represented in the form of a matrix, into a low rank component and a sparse term with a more robust and less biased formulation. In addition, we develop an iterative algorithm based on the augmented Lagrangian multipliers method and derive the closed-form solution of the resulting subproblems benefiting from the special property of the nonconvex surrogate function. We prove that our iterative optimization converges easily. Extensive experiments on both simulated and real HSIs indicate that our approach can not only suppress noise in both severely and slightly noised bands but also preserve large-scale image structures and small-scale details well. Comparisons against state-of-the-art LRMA-based HSI denoising approaches show our superior performance.

Restoration of hyperspectral images (HSIs) is a challenging task, owing to the reason that images are inevitably contaminated by a mixture of noise, including Gaussian noise, impulse noise, dead lines, and stripes, during their acquisition process. Recently, HSI denoising approaches based on low-rank matrix approximation have become an active research field in remote sensing and have achieved state-of-the-art performance. These approaches, however, unavoidably require to calculate full or partial singular value decomposition of large matrices, leading to the relatively high computational cost and limiting their flexibility. To address this issue, this letter proposes a method exploiting a low-rank matrix factorization scheme, in which the associated robust principal component analysis is solved by the matrix factorization of the low-rank component. Our method needs only an upper bound of the rank of the underlying low-rank matrix rather than the precise value. The experimental results on the simulated and real data sets demonstrate the performance of our method by removing the mixed noise and recovering the severely contaminated images.

Based on low energy consumption and high privacy protection required by range query in two-level wireless sensor network, this paper proposes a kind of secure probabilistic range query (SPRQ). SPRQ is made up of data encryption, prefix member verification and probabilistic neighbor verification and spate query and transmission process. It can ensure completing range query without disclosing privacy. The analysis and simulation result indicate that compared with other safety agreement, SPRQ has lower energy consumption when it ensures the safety of range query.

Nonnegative matrix completion aims to find nonnegative low-rank matrices from a subset of entries of a matrix. It is widely applicable in many fields, such as image and video processing, recommendation systems, and national economy. This task can be conducted by solving the nuclear norm regularized linear least squares model with nonnegative constraints. We apply the widely used alternating direction method of multipliers to solve the model and get two novel algorithms. The applicability and efficiency of the new algorithms are demonstrated in numerical experiments. Recovery results show that our algorithms are helpful.

针对具有可分结构的单调变分不等式问题,基于邻近点算法和文献[12]提出的下降型算法构造了一个新的下降方向,并利用下降量的下界来选择最优步长,提出一种下降型邻近点交替方向乘子法;证明了算法的收敛性;并将该方法与文献[11]中算法的下降量下界进行比较,从理论上说明了算法的优越性.

In this paper, two PVD-type algorithms are proposed for solving inseparable linear constraint optimization. Instead of computing the residual gradient function, the new algorithm uses the reduced gradients to construct the PVD directions in parallel computation, which can greatly reduce the computation amount each iteration and is closer to practical applications for solve large-scale nonlinear programming. Moreover, based on an active set computed by the coordinate rotation at each iteration, a feasible descent direction can be easily obtained by the extended reduced gradient method. The direction is then used as the PVD direction and a new PVD algorithm is proposed for the general linearly constrained optimization. And the global convergence is also proved.

Second-order cone optimization (denoted by SOCO) is a class of convex optimization problems and it contains the linear optimization problem, convex quadratic optimization problem and quadratically constrained convex quadratic optimization problem as special cases. In this paper, we propose a new smoothing Newton method for solving the SOCO based on a non-symmetrically perturbed smoothing Fischer–Burmeister function. At each iteration, a system of linear equations is solved only approximately by using the inexact Newton method. It is shown that any accumulation point of the iteration sequence generated by the proposed algorithm is a solution of the SOCO. Furthermore, we prove that the generated sequence is bounded and hence it has at least one accumulation point. Under the assumption of nonsingularity, we establish the local quadratic convergence of the proposed algorithm without strict complementarity condition. Numerical experiments indicate that our method is effective.

A decomposition method for solving large-scale box constrained optimization is proposed. The algorithm is motivated by the successful use of the decomposition method presented by Joachims for training support vector machines. In particular, a new technique, based on the new definition “KKT-violating index”, is introduced for working set identification. Finally, the numerical experiments and implementation details show that this method is practical for large-scale problems.