Norm Minimization Method Research Articles

Constrained L1-Norm Minimization Method for Range-Based Source Localization under Mixed Sparse LOS/NLOS Environments.

Under mixed sparse line-of-sight/non-line-of-sight (LOS/NLOS) conditions, how to quickly achieve high positioning accuracy is still a challenging task and a critical problem in the last dozen years. To settle this problem, we propose a constrained L1 norm minimization method which can reduce the effects of NLOS bias for improve positioning accuracy and speed up calculation via an iterative method. We can transform the TOA-based positioning problem into a sparse optimization one under mixed sparse LOS/NLOS conditions if we consider NLOS bias as outliers. Thus, a relatively good method to deal with sparse localization problem is L1 norm. Compared with some existing methods, the proposed method not only has the advantages of simple and intuitive principle, but also can neglect NLOS status and corresponding NLOS errors. Experimental results show that our algorithm performs well in terms of computational time and positioning accuracy.

Perturbation analysis of low-rank matrix stable recovery

In this paper, we bring forward a completely perturbed nuclear norm minimization method to tackle a formulation of completely perturbed low-rank matrices recovery. In view of the matrix version of the restricted isometry property (RIP) and the Frobenius-robust rank null space property (FRNSP), this paper extends the investigation to a completely perturbed model taking into consideration not only noise but also perturbation, derives sufficient conditions guaranteeing that low-rank matrices can be robustly and stably reconstructed under the completely perturbed scenario, as well as finally presents an upper bound estimation of recovery error. The upper bound estimation can be described by two terms, one concerning the total noise, and another regarding the best [Formula: see text]-approximation error. Specially, we not only improve the condition corresponding with RIP, but also ameliorate the upper bound estimation in case the results reduce to the general case. Furthermore, in the case of [Formula: see text], the obtained conditions are optimal.

Rician noise removal via weighted nuclear norm penalization

Rician noise is a common noise that naturally appears in Magnetic Resonance Imaging (MRI) images. Low rank matrix approximation approaches have been widely used in image processing, which takes advantage of the non-local self-similarity between patches in a natural image. The weighted nuclear norm minimization method as a low rank matrix approximation approach has shown to be an effective approach for image denoising. Inspired by this, we propose in this paper a maximum a posteriori (MAP) model with the weighted nuclear norm as a regularization constraint to remove Rician noise. The MAP data fidelity term has a Lipschitz continuous gradient and the weighted nuclear norm can be efficiently minimized. We propose an iterative weighted nuclear norm minimization algorithm (IWNNM) to solve the proposed non-convex model and analyze the convergence of our algorithm. The computational results show that our proposed method is promising in restoring images corrupted with Rician noise.

Repositioning Drugs on Human Influenza A Viruses Based on a Novel Nuclear Norm Minimization Method.

Influenza A viruses, especially H3N2 and H1N1 subtypes, are viruses that often spread among humans and cause influenza pandemic. There have been several big influenza pandemics that have caused millions of human deaths in history, and the threat of influenza viruses to public health is still serious nowadays due to the frequent antigenic drift and antigenic shift events. However, only few effective anti-flu drugs have been developed to date. The high development cost, long research and development time, and drug side effects are the major bottlenecks, which could be relieved by drug repositioning. In this study, we proposed a novel antiviral Drug Repositioning method based on minimizing Matrix Nuclear Norm (DRMNN). Specifically, a virus-drug correlation database consisting of 34 viruses and 205 antiviral drugs was first curated from public databases and published literature. Together with drug similarity on chemical structure and virus sequence similarity, we formulated the drug repositioning problem as a low-rank matrix completion problem, which was solved by minimizing the nuclear norm of a matrix with a few regularization terms. DRMNN was compared with three recent association prediction algorithms. The AUC of DRMNN in the global fivefold cross-validation (fivefold CV) is 0.8661, and the AUC in the local leave-one-out cross-validation (LOOCV) is 0.6929. Experiments have shown that DRMNN is better than other algorithms in predicting which drugs are effective against influenza A virus. With H3N2 as an example, 10 drugs most likely to be effective against H3N2 viruses were listed, among which six drugs were reported, in other literature, to have some effect on the viruses. The protein docking experiments between the chemical structure of the prioritized drugs and viral hemagglutinin protein also provided evidence for the potential of the predicted drugs for the treatment of influenza.

A Multispectral Denoising Framework for Seismic Random Noise Attenuation

Random noise attenuation plays an important role in the seismic data processing. The global coherency among different spectral segments is often neglected in the traditional denoising methods, even though the seismic data are naturally broadband in the frequency spectrum. We proposed a multispectral denoising framework (MDF) for seismic random noise attenuation. The MDF contains three key components. First, we use a series of narrow-band filters on the noisy data to construct the multispectral data. Second, we normalize the constructed multispectral data to mitigate the amplitude discrepancy between different spectral slices. Third, we apply the intrinsic tensor sparsity regularization method to denoise the multispectral data. Numerical experiments on synthetic and field data show that the proposed framework can reduce the random noise as well as the erratic noise, and it can achieve improved denoised results, compared with a traditional method (i.e., the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> deconvolution method) and the single-spectral version of the proposed method (i.e., the weighted nuclear norm minimization (WNNM) method). The proposed framework can be applied to both prestack and stacked data and generally well preserve the amplitude of useful signals, compared with the other two methods.

Robust Low-tubal-rank Tensor Recovery from Binary Measurements

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 X from incomplete linear measurements [Formula: see text]. 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 [Formula: see text]. We still hope to recover X from these binary measurements. Under the tensor Singular Value Decomposition (t-SVD) framework, two recovery methods are proposed-the 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 λ:=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 λ. Numerical experiments verify our results, and the applications to real-world data demonstrate the promising performance of the proposed methods.

An optimal condition of robust low-rank matrices recovery

In this paper we investigate the reconstruction conditions of nuclear norm minimization for low-rank matrix recovery. We obtain sufficient conditions $\delta_{tr}<t/(4-t)$ with $0<t<4/3$ to guarantee the robust reconstruction $(z\neq0)$ or exact reconstruction $(z=0)$ of all rank $r$ matrices $X\in\mathbb{R}^{m\times n}$ from $b=\mathcal{A}(X)+z$ via nuclear norm minimization. Furthermore, we not only show that when $t=1$, the upper bound of $\delta_r<1/3$ is the same as the result of Cai and Zhang \cite{Cai and Zhang}, but also demonstrate that the gained upper bounds concerning the recovery error are better. Moreover, we prove that the restricted isometry property condition is sharp. Besides, the numerical experiments are conducted to reveal the nuclear norm minimization method is stable and robust for the recovery of low-rank matrix.

A fuzzy matrix factor recommendation method with forgetting function and user features

In recent years, a variety of recommendation algorithms have been proposed. However, there are still some issues such as cold start, sparsity and timeliness. On the basis of traditional collaborative filtering, matrix decomposition technologies can greatly reduce the computational complexity and address the issues of cold start and sparsity. Based on forgetting functions and users’ multi-attribute features, in this paper, a fuzzy matrix decomposition and trace norm minimization method is proposed. Firstly, a forgetting function is incorporated into the historical scores, which makes existing scoring information more efficient. Then the user features are applied to find trusted users, which will give some valuable suggestions and possible scores to target items. The user features are also applied to list the relationships among users. In this paper, the relations between users and items are represented by the scores between them. Secondly, both time and feature information is integrated into the proposed optimization model. Then a fuzzy matrix decomposition and trace norm minimization model is established and further solved by our proposed algorithm to give better recommendations. Moreover, the convergence of the proposed algorithm is also proved in theoretically. Finally, we investigate the empirical recoverability properties of our model and analyze its advantages over classical norm minimization models. Extensive experimental results on synthetic and real-world data sets verify the efficiency and effectiveness of our method compared with the state-of-the-art algorithms.

Performance comparison of least squares, iterative and global L1 norm minimization and exhaustive search methods for outlier detection in leveling networks

Performance comparison of least squares, iterative and global L1 norm minimization and exhaustive search methods for outlier detection in leveling networks

A fault information-oriented weighted nuclear norm minimization method and its application to fault feature extraction in a rolling bearing

How to effectively extract fault features from noisy vibration signals is a key problem to be solved in bearing fault diagnosis. The difficulty here mainly lies in the fact that the components of interest are almost submerged by background noise and external interference. To accurately implement extraction of fault features, nuclear norm minimization (NNM) has been introduced to fault feature extraction due to its powerful denoising capability. However, NNM treats all singular values equally, which may result in components of interest with small singular values being abandoned, especially when a bearing defect is at its incipient stage. Therefore, to enhance the flexibility and effectiveness of NNM in handling practical problems, fault information-oriented weighted nuclear norm minimization (FIWNNM) is proposed for extracting bearing fault features, in which singular values are treated differently by assigning appropriate weights according to the physical meanings. Through the proposed FIWNNM, fault features can be drawn from a noisy signal, even when contaminated by strong noise and external interferences. The analysis results from both simulated and real measured signals are used as the verification of the proposed FIWNNM.

Inspection and Monitoring of Concrete Structures via Radiography and Weighted Nuclear Norm Minimization Method

The location of reinforcement bar in concrete, the bar corrosion, diameter and the depth below the surface are important factors in the evaluation of the load bearing capacity and the usage property within the scope of building condition and damage analysis. Radiographic techniques are used to examine the structure of different buildings. The basic goal of the inspection is the visualization of reinforcement bars, fittings or tension cables. In this research, the Weighted Nuclear Norm Minimization method was used to improve visualization of the hidden structures and defects from the concrete radiographs. The method relies on minimization of the image energy and singular value decomposition for enhancing contrast. The proposed algorithm was successfully applied to radiographic images of concrete segments. Improvement of the structure detail visualization and defect region detection were achieved while preserving structure edge and fine detail imaging information. Evaluation of the image quality enhancement showed that the contrast level to noise increases by a factor of about two in reconstructed images using the proposed method. Also, the background is suppressed towards the zero baselines, and better visual quality is achieved.

Singular Value Thresholding Algorithm for Wireless Sensor Network Localization

Wireless Sensor Networks (WSN) are of great current interest in the proliferation of technologies. Since the location of the sensors is one of the most interesting issues in WSN, the process of node localization is crucial for any WSN-based applications. Subsequently, WSN’s node estimation deals with a low-rank matrix which gives rise to the application of the Nuclear Norm Minimization (NNM) method. This paper will focus on the localization of 2-dimensional WSN with objects (obstacles). Recent studies introduce Nuclear Norm Minimization (NNM) for node estimation instead of formulating the rank minimization problem. Common way to tackle this problem is by implementing the Semidefinite Programming (SDP). However, SDP can only handle matrices with a size of less than 100 × 100. Therefore, we introduce the method of Singular Value Thresholding (SVT) which is an iterative algorithm to solve the NNM problem that produces a sequence of matrices { X k , Y k } and executes a soft-thresholding operation on the singular value of the matrix Y k . This algorithm is a user-friendly algorithm which produces a low computational cost with low storage capacity required to give the lowest-rank minimum nuclear norm solution.

Impulsive orbit correction using second-order Gauss’s variational equations

Based on Gauss’s variational equations (GVEs), the impulsive orbital-element corrections are investigated in orbit transfer problems. Both single impulse and multiple impulses are considered for the first-order and second-order GVEs. For the single impulse, a nonlinear least-squares iteration method for the minimum orbit error is provided to simultaneously solve for the impulse vector and the impulse position. For multiple impulses, a minimum-norm method for the energy cost is proposed to solve the three-impulse and two-impulse corrections for the in-plane and out-of-plane orbital elements, respectively. The impulse positions are analytically derived, and the impulse vectors are obtained by the minimum-norm method. Numerical examples are provided to verify the proposed single-impulse least-squares and multiple-impulse minimum-norm methods. The results show that the nonlinear solutions using the second-order GVEs are more accurate than those using the first-order GVEs.

NanoSIMS image enhancement by reducing random noise using low‐rank method

NanoSIMS images are usually affected by random noises because of various types of sources, which degrade the quality of ion images and increase the uncertainty of the geochemical interpretations. Here, we applied the weighted nuclear norm minimization (WNNM) method to reduce the random noise in the NanoSIMS image. The low‐rank property of the image is fully considered to suppress random noise while retaining reliable details of weak signals. Numerical experiments on four different kinds of NanoSIMS ion images show that the denoising ability of the WNNM method is superior to that of the median filter, no matter the size of the filtering windows used (eg, 3 × 3, 5 × 5, and 7 × 7). The WNNM method can reduce random noise while preserving the most critical details in the original NanoSIMS observations, which can significantly enhance reliability when distinguishing critical boundaries and structures.

A Benchmark for Sparse Coding: When Group Sparsity Meets Rank Minimization.

Sparse coding has achieved a great success in various image processing tasks. However, a benchmark to measure the sparsity of image patch/group is missing since sparse coding is essentially an NP-hard problem. This work attempts to fill the gap from the perspective of rank minimization. We firstly design an adaptive dictionary to bridge the gap between group-based sparse coding (GSC) and rank minimization. Then, we show that under the designed dictionary, GSC and the rank minimization problems are equivalent, and therefore the sparse coefficients of each patch group can be measured by estimating the singular values of each patch group. We thus earn a benchmark to measure the sparsity of each patch group because the singular values of the original image patch groups can be easily computed by the singular value decomposition (SVD). This benchmark can be used to evaluate performance of any kind of norm minimization methods in sparse coding through analyzing their corresponding rank minimization counterparts. Towards this end, we exploit four well-known rank minimization methods to study the sparsity of each patch group and the weighted Schatten p-norm minimization (WSNM) is found to be the closest one to the real singular values of each patch group. Inspired by the aforementioned equivalence regime of rank minimization and GSC, WSNM can be translated into a non-convex weighted ℓp-norm minimization problem in GSC. By using the earned benchmark in sparse coding, the weighted ℓp-norm minimization is expected to obtain better performance than the three other norm minimization methods, i.e., ℓ1-norm, ℓp-norm and weighted ℓ1-norm. To verify the feasibility of the proposed benchmark, we compare the weighted ℓp-norm minimization against the three aforementioned norm minimization methods in sparse coding. Experimental results on image restoration applications, namely image inpainting and image compressive sensing recovery, demonstrate that the proposed scheme is feasible and outperforms many state-of-the-art methods.

An adaptive boosting procedure for low-rank based image denoising

Low-rank methods have shown great potential in many low-level vision tasks, especially in image denoising. Nevertheless, state-of-the-art algorithms still have limitations in some aspects, such as a heavy reliance on empirical values when setting the denoising parameters and the lack of consideration of the noise effect on the selection of similar patches. To address these problems, an adaptive boosting technique for improving the low-rank based image denoising results is proposed. In particular, different from adding the residual back to the image, we consider strengthening the signal by leveraging the availability of the previous denoised image and combine this boosting technique with dynamic parameters applied to the weighted nuclear norm minimization (WNNM) method, whose feasibility is derived and proved theoretically. In each iteration of the boosting scheme, the dynamic boosting parameters are motivated by an optimal solution analysis. In addition, in the process of image denoising, an adaptive patch search scheme is put forward to gain an effective index of similar patches. Furthermore, to adaptively determine the optimal number of iterations, a stopping criterion with the correlation coefficient is applied. The experimental results demonstrate that the proposed approach can preserve more detail information while removing noise and can outperform various state-of-the-art denoising algorithms in terms of both PSNR and SSIM.

User Selection and Power Minimization in Full-Duplex Cloud Radio Access Networks

In this paper, we investigate the network power consumption (NPC) minimization in full-duplex cloud radio access networks (C-RANs) with quality-of-service requirements. We jointly optimize the beamforming vectors, the users' transmit power, the fronthaul compression ratio, and the set of transmitting remote radio heads (RRHs). Users' requirements of signal-to-interference-and-noise ratio, per-RRH and user power constraints, and fronthaul capacity constraints are considered. Due to these conflicting constraints, the optimization problem may be infeasible. Thus, we solve this problem in two steps. In Step I, a minimum-mean-square-error-based user selection algorithm is proposed to find the largest subset of feasible users. In Step II, an algorithm based on the reweighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization method is proposed to solve the NPC problem with the users selected in Step I. The solutions obtained by the proposed algorithms are proved to monotonically converge and satisfy the Karush-Kuhn-Tucker conditions. Moreover, our proposed algorithms are applicable to both full-duplex and half-duplex C-RANs. We comprehensively compare these two types of C-RANs and propose some valuable insights for system design.

An iterative decoupled method with weighted nuclear norm minimization for image restoration

ABSTRACTRecently, iterative methods based on decoupling of deblurring and denoising steps have received remarkable attention in image processing problems. In this paper, we propose an effective iterative decoupled approach for image restoration. The main idea is to split the image restoration problem into two minimization subproblems: deblurring and denoising. An efficient deblurring method using fast transforms is used in the deblurring step and a weighted nuclear norm minimization method is employed in the denoising step. We analyze the convergence of the developed iterative method under some assumptions. Some numerical experiments are given to demonstrate the effectiveness of the proposed method.

Efficient Compressed Sensing Based Non-Sample Spaced Sparse Channel Estimation in OFDM System

This paper aims to develop an efficient compressed sensing (CS) based channel estimation method for non-sample spaced sparse channels in orthogonal frequency division multiplexing (OFDM) systems, which can effectively balance the channel estimation performance, spectral efficiency and computational complexity. To realize this goal, a novel delay tracking and residual norm minimization (DTRNM) method is proposed. In this method, the idea of reference delay grids (RDG) inspired by the delay tracking (DT) method proposed in [1] is formulated, which is proven to be much more efficient than current algorithms in estimating non-sample spaced sparse channels. Both theoretical derivation and simulation results show the effectiveness of the proposed method.

Image Denoising via Nonlocal Low Rank Approximation With Local Structure Preserving

The nuclear norm minimization method emerged from a patch-based low-rank model leads to an excellent image denoising performance, where the non-local self-similarity over image patches is exploited. However, natural images are normally with complex and irregular image patches, which cannot be well represented using only a low-rank model, and thus most of them suffer from the over-penalty problem especially for images with lots of local irregular structures (e.g., fine details or sharp edges), and then results in over-smoothing problem after denoising. On the other hand, in order to represent the irregular components, edges defined over pixel level are often exploited. While the total variation (TV) is a well-known prior to remove noises and preserve edges, it might yield undesired staircase artifacts. The total generalized variation (TGV), a generalization of TV, can largely alleviate such staircase artifacts. Consequently, in order to deal with the over-smoothing problem aroused by a low-rank model, we propose a re-weighted TGV regularized nuclear norm minimization model for local structure preserving image denoising. Thanks to the split Bregman method, our proposed model can be effectively solved. A re-weighted strategy is developed to adaptively update the weight parameters of TGV regularization. The encouraging experimental results on noisy images demonstrate the effectiveness of our proposed method.