A Perturbation Analysis of Low-Rank Matrix Recovery by Schatten p-Minimization

A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.

Proximity point algorithm for low-rank matrix recovery from sparse noise corrupted data

The method of recovering a low-rank matrix with an unknown fraction whose entries are arbitrarily corrupted is known as the robust principal component analysis （RPCA）. This RPCA problem, under some conditions, can be exactly solved via convex optimization by minimizing a combination of the nuclear norm and the 11 norm. In this paper, an algorithm based on the Douglas-Rachford splitting method is proposed for solving the RPCA problem. First, the convex optimization problem is solved by canceling the constraint of the variables, and ~hen the proximity operators of the objective function are computed alternately. The new algorithm can exactly recover the low-rank and sparse components simultaneously, and it is proved to be convergent. Numerical simulations demonstrate the practical utility of the proposed algorithm.

Improved nonconvex optimization model for low-rank matrix recovery

Low-rank matrix recovery is an important problem extensively studied in machine learning, data mining and computer vision communities. A novel method is proposed for low-rank matrix recovery, targeting at higher recovery accuracy and stronger theoretical guarantee. Specifically, the proposed method is based on a nonconvex optimization model, by solving the low-rank matrix which can be recovered from the noisy observation. To solve the model, an effective algorithm is derived by minimizing over the variables alternately. It is proved theoretically that this algorithm has stronger theoretical guarantee than the existing work. In natural image denoising experiments, the proposed method achieves lower recovery error than the two compared methods. The proposed low-rank matrix recovery method is also applied to solve two real-world problems, i.e., removing noise from verification code and removing watermark from images, in which the images recovered by the proposed method are less noisy than those of the two compared methods.

RGBD Salient Object Detection by Structured Low-Rank Matrix Recovery and Laplacian Constraint

A structured low-rank matrix recovery model for RGBD salient object detection is proposed. Firstly, the problem is described by a low-rank matrix recovery, and the hierarchical structure of RGB image is added to the sparsity term. Secondly, the depth information is fused into the model by a Laplacian regularization term to ensure that the image regions which share similar depth value will be allocated to similar saliency value. Thirdly, a variation of alternating direction method is proposed to solve the proposed model. Finally, both quantitative and qualitative experimental results on NLPR1000 and NJU400 show the advantage of the proposed RGBD salient object detection model. © 2017, Tianjin University and Springer-Verlag Berlin Heidelberg.

Low-rank matrix recovery with total generalized variation for defending adversarial examples

Low-rank matrix decomposition with first-order total variation(TV)regularization exhibits excellent performance in exploration of image structure.Taking advantage of its excellent performance in image denoising,we apply it to improve the robustness of deep neural networks.However,although TV regularization can improve the robustness of the model,it reduces the accuracy of normal samples due to its over-smoothing.In our work,we develop a new low-rank matrix recovery model,called LRTGV,which incorporates total generalized variation(TGV)regularization into the reweighted low-rank matrix recovery model.In the proposed model,TGV is used to better reconstruct texture information without over-smoothing.The reweighted nuclear norm and Li-norm can enhance the global structure information.Thus,the proposed LRTGV can destroy the structure of adversarial noise while re-enhancing the global structure and local texture of the image.To solve the challenging optimal model issue,we propose an algorithm based on the alternating direction method of multipliers.Experimental results show that the proposed algorithm has a certain defense capability against black-box attacks,and outperforms state-of-the-art low-rank matrix recovery methods in image restoration.

Pairwise constraint propagation via low-rank matrix recovery

As a kind of weaker supervisory information, pairwise constraints can be exploited to guide the data analysis process, such as data clustering. This paper formulates pairwise constraint propagation, which aims to predict the large quantity of unknown constraints from scarce known constraints, as a low-rank matrix recovery(LMR) problem. Although recent advances in transductive learning based on matrix completion can be directly adopted to solve this problem, our work intends to develop a more general low-rank matrix recovery solution for pairwise constraint propagation, which not only completes the unknown entries in the constraint matrix but also removes the noise from the data matrix. The problem can be effectively solved using an augmented Lagrange multiplier method. Experimental results on constrained clustering tasks based on the propagated pairwise constraints have shown that our method can obtain more stable results than state-of-the-art algorithms,and outperform them.

Robust Principal Component Analysis Integrating Sparse and Low-Rank Priors

Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Analysis (RPCA) addresses these limitations by decomposing data into a low-rank matrix capturing the underlying structure and a sparse matrix identifying outliers, enhancing robustness against noise and outliers. This paper introduces a novel RPCA variant, Robust PCA Integrating Sparse and Low-rank Priors (RPCA-SL). Each prior targets a specific aspect of the data’s underlying structure and their combination allows for a more nuanced and accurate separation of the main data components from outliers and noise. Then RPCA-SL is solved by employing a proximal gradient algorithm for improved anomaly detection and data decomposition. Experimental results on simulation and real data demonstrate significant advancements.

Accelerated Matrix Recovery via Random Projection Based on Inexact Augmented Lagrange Multiplier Method

In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.

Nested Alternating Direction Method of Multipliers to Low-Rank and Sparse-Column Matrices Recovery

The task of dividing corrupted-data into their respective subspaces can be well illustrated,both theoretically and numerically,by recovering low-rank and sparse-column components of a given matrix.Generally,it can be characterized as a matrix and a 2,1-norm involved convex minimization problem.However,solving the resulting problem is full of challenges due to the non-smoothness of the objective function.One of the earliest solvers is an 3-block alternating direction method of multipliers(ADMM)which updates each variable in a Gauss-Seidel manner.In this paper,we present three variants of ADMM for the 3-block separable minimization problem.More preciously,whenever one variable is derived,the resulting problems can be regarded as a convex minimization with 2 blocks,and can be solved immediately using the standard ADMM.If the inner iteration loops only once,the iterative scheme reduces to the ADMM with updates in a Gauss-Seidel manner.If the solution from the inner iteration is assumed to be exact,the convergence can be deduced easily in the literature.The performance comparisons with a couple of recently designed solvers illustrate that the proposed methods are effective and competitive.

Recovery of Corrupted Low-Rank Tensors

This paper studies the problem of recovering low-rank tensors, and the tensors are corrupted by both impulse and Gaussian noise. The problem is well accomplished by integrating the tensor nuclear norm and the l1-norm in a unified convex relaxation framework. The nuclear norm is adopted to explore the low-rank components and the l1-norm is used to exploit the impulse noise. Then, this optimization problem is solved by some augmented-Lagrangian-based algorithms. Some preliminary numerical experiments verify that the proposed method can well recover the corrupted low-rank tensors.

A decellularized nerve matrix scaffold inhibits neuroma formation in the stumps of transected peripheral nerve after peripheral nerve injury

Traumatic painful neuroma is an intractable clinical disease characterized by improper extracellular matrix(ECM)deposition around the injury site.Studies have shown that the microstructure of natural nerves provides a suitable microenvironment for the nerve end to avoid abnormal hyperplasia and neuroma formation.In this study,we used a decellularized nerve matrix scaffold(DNM-S)to prevent against the formation of painful neuroma after sciatic nerve transection in rats.Our results showed that the DNM-S effectively reduced abnormal deposition of ECM,guided the regeneration and orderly arrangement of axon,and decreased the density of regenerated axons.The epineurium-perilemma barrier prevented the invasion of vascular muscular scar tissue,greatly reduced the invasion ofα-smooth muscle actin-positive myofibroblasts into nerve stumps,effectively inhibited scar formation,which guided nerve stumps to gradually transform into a benign tissue and reduced pain and autotomy behaviors in animals.These findings suggest that DNM-S-optimized neuroma microenvironment by ECM remodeling may be a promising strategy to prevent painful traumatic neuromas.

Reviving the use of inhibitors of matrix metalloproteases in spinal cord injury:a case for specificity

At present,there are no resto rative therapies in the clinic for spinal cord injury,with current treatments offering only palliative treatment options.The role of matrix metalloproteases is well established in spinal cord injury,howeve r,translation into the clinical space was plagued by early designs of matrix metalloprotease inhibitors that lacked specificity and fears of musculos keletal syndrome prevented their further development.Newe r,much more specific matrix metalloprotease inhibitors have revived the possibility of using these inhibitors in the clinic since they are much more specific to their to rget matrix metalloproteases.Here,the evidence for use of matrix metalloproteases after spinal cord injury is reviewed and researche rs are urged to overcome their old fears rega rding matrix metalloprotease inhibition and possible side effects for the field to progress.Recently published work by us shows that inhibition of specific matrix metalloproteases after spinal cord injury holds promise since four key consequences of spinal cord injury could be alleviated by specific,next-gene ration matrix metalloprotease inhibitors.For example,specific inhibition of matrix metalloprotease-9 and matrix metalloprotease-12 within 24 hours after injury and for 3 days,alleviates spinal cord injury-induced edema,blood-s pinal co rd barrier breakdown,neuro pathic pain and resto res sensory and locomotor function.Attempts are now underway to translate this therapy into the clinic.

电磁大数据自动化标注补全算法

针对实际应用中电子侦察数据存在的数据质量差、标注率低等问题,将多传感器数据自动化标注问题抽象为稀疏矩阵恢复问题,在多平台多类型待标注监测数据与低秩稀疏矩阵之间建立正确的语义映射,进而转化为求解优化问题,并基于凸秩最小化算法对目标函数进行迭代以求得最优解。仿真实验结果表明,算法模型在目标特征信息缺失率40%~50%的恶劣情况下,恢复矩阵与原始矩阵的的最小均方根误差维持在0.06左右,能够有效实现矩阵恢复与数据的自动化标注。

基于实数化的均匀圆阵矩阵重构方法

为了减小低快拍数和低信噪比下采样协方差矩阵误差,并降低其运算复杂度,提出了一种基于实数化的均匀圆阵采样协方差矩阵重构方法。针对均匀圆阵的特点,通过组建特殊的基向量,构成特殊的重构矩阵。通过将采样协方差矩阵实数化,进一步降低了重构矩阵的复杂度。考虑到多通道不一致性对重构矩阵的影响,引入0位校正算法,提高了重构方法的稳健性。最后应用重构后的协方差矩阵进行子空间类波达方向估计(direction of arrival,DOA)。实验仿真证明,该特殊重构矩阵在实数化下与原矩阵重构能力相同;当快拍数为100、信噪比为0 dB时,双信源分辨力较重构前由74%提高到95%以上;理论重构运算复杂度降低到原来的53.99%。

基于矩阵填充的随机步进频雷达高分辨距离-多普勒谱稀疏恢复方法

随机步进频雷达通过合成大带宽,能在较低硬件复杂度下获得距离高分辨效果,同时由于其每个脉冲的载频随机捷变,因而具有强的抗干扰、电磁兼容能力,在复杂电磁环境高精度探测领域具有重要的应用价值。然而,由于其波形在时频域稀疏的感知形式,造成回波相参信息有所缺失,因而传统匹配滤波方法在估计高分辨距离-多普勒时会演化为欠定估计,导致估计谱中产生起伏高旁瓣,严重影响探测性能。为此,该文提出一种基于Hankel重构矩阵填充的随机步进频雷达高分辨距离-多普勒谱低旁瓣稀疏恢复方法。该方法采用低秩矩阵填充思想补全波形在时频域稀疏感知时造成的缺失采样,恢复目标连续相参信息,可以有效解决欠定估计问题。文章首先构建了随机步进频雷达的慢时间-载频(时-频)回波欠采样数据矩阵;然后,重构待恢复数据矩阵为双重Hankel型,并分析证明了矩阵满足低秩先验特性;最后,利用ADMM算法补全未采样时频数据,恢复相参信息,保证了高分辨距离-多普勒谱低旁瓣稀疏恢复。仿真和实测试验证明了该文所提方法的有效性和优越性。

Multidomain Correlation-Based Multidimensional CSI Tensor Generation for Device-FreeWi-Fi Sensing

Due to the fine-grained communication scenarios characterization and stability,Wi-Fi channel state information(CSI)has been increasingly applied to indoor sensing tasks recently.Although spatial variations are explicitlyreflected in CSI measurements,the representation differences caused by small contextual changes are easilysubmerged in the fluctuations of multipath effects,especially in device-free Wi-Fi sensing.Most existing datasolutions cannot fully exploit the temporal,spatial,and frequency information carried by CSI,which results ininsufficient sensing resolution for indoor scenario changes.As a result,the well-liked machine learning(ML)-based CSI sensing models still struggling with stable performance.This paper formulates a time-frequency matrixon the premise of demonstrating that the CSI has low-rank potential and then proposes a distributed factorizationalgorithm to effectively separate the stable structured information and context fluctuations in the CSI matrix.Finally,a multidimensional tensor is generated by combining the time-frequency gradients of CSI,which containsrich and fine-grained real-time contextual information.Extensive evaluations and case studies highlight thesuperiority of the proposal.

顶空气相色谱法检测水溶性SHR0410中多种水溶性差的残留溶剂

目的:建立水溶性样品SHR0410原料药中3种水溶性差的残留溶剂二氯甲烷、异丙醚、三异丙基硅烷的测定方法,为SHR0410原料药中这3种残留溶剂风险评估打下基础。方法:采用顶空气相色谱外标法,色谱柱为CP-Volamine(30 m×0.32 mm)型毛细管柱,载气为氮气,氢火焰离子化检测器,柱温程序升温。结果:3种残留溶剂二氯甲烷、异丙醚和三异丙基硅烷,在此条件下能有效分离,线性范围分别为16.58~99.50μg·mL^(-1)(r=0.999)、2.47~14.83μg·mL^(-1)(r=0.999)、123.24~739.46μg·mL^(-1)(r=0.998),平均回收率分别为97.2%(RSD=2.3%,n=6)、95.5%(RSD=2.0%,n=6)、99.6%(RSD=1.4%,n=6),检测限(LOD)分别为0.07μg·mL^(-1)、0.01μg·mL^(-1)、0.005μg·mL^(-1)。对3批样品进行检验,均未检出以上3种残留溶剂。结论:该方法适用于测定水溶性SHR0410原料药样品中水溶性差的残留溶剂。

2种低秩矩阵恢复优化模型的误差估计定理

近年来低秩矩阵恢复问题逐渐引起人们的关注,类似于向量稀疏恢复的充分条件是需要测量矩阵满足限制等距性质,低秩矩阵恢复的充分条件是需要一个线性映射满足限制等距性质.低秩矩阵恢复时所需的模型大致分为有噪和无噪两种恢复模型,恢复出来的结果需要不同的限制等距常数界去保证.文章证明了这2种优化模型的误差界估计定理,并得出了2种不同的限制等距常数界.

基于矩阵恢复的OFDM信道估计方法

正交频分复用(Orthogonal Frequency Division Multiplexing,OFDM)中至关重要的一项技术是信道估计,本文提出一种基于矩阵恢复的OFDM信道估计方法,将连续多个OFDM信号的频域信道构造成一个信道矩阵,由于这个信道矩阵是低秩的,所以可以将信道估计问题转换为信道矩阵的加权截断核范数最小化问题,并使用改进的奇异值阈值(Singular Value Thresholding,SVT)算法对信道矩阵进行恢复。仿真结果表明,本文提出的方法和传统信道估计算法相比,使用相同导频数可以获得更高的估计精度,在获得相同估计精度时,消耗导频数更少。与基于压缩感知的信道估计方法相比,本文方法消耗相同数量的导频,但可直接获得高精度的OFDM信道的频域估计。

一种基于联合加权和截断的毫米波大规模MIMO信道估计

提出一种联合加权和截断核范数的毫米波大规模多输入多输出(MIMO)信道估计算法。针对毫米波大规模MIMO信道估计问题中训练和反馈开销大的问题,首先利用毫米波信道天线角度域稀疏的特性,把信道估计问题转化为低秩矩阵恢复问题。采用一种有效而灵活的秩函数——联合加权截断核范数作为核范数的松弛,构造出一种新的矩阵恢复模型用于信道估计问题,以最小化加权截断核范数为优化目标,并利用交替优化框架求解。仿真结果表明,该方法可以有效地提高信道估计的精度,并且具有可靠的收敛性。