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.

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.

Electrical Data Matrix Decomposition in Smart Grid

As the development of smart grid and energy internet, this leads to a significantincrease in the amount of data transmitted in real time. Due to the mismatch withcommunication networks that were not designed to carry high-speed and real time data,data losses and data quality degradation may happen constantly. For this problem,according to the strong spatial and temporal correlation of electricity data which isgenerated by human’s actions and feelings, we build a low-rank electricity data matrixwhere the row is time and the column is user. Inspired by matrix decomposition, we dividethe low-rank electricity data matrix into the multiply of two small matrices and use theknown data to approximate the low-rank electricity data matrix and recover the missedelectrical data. Based on the real electricity data, we analyze the low-rankness of theelectricity data matrix and perform the Matrix Decomposition-based method on the realdata. The experimental results verify the efficiency and efficiency of the proposed scheme.

Truncated sparse approximation property and truncated q-norm minimization

This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq from noisy measurements.We first introduce truncated sparse approximation property,a more general robust null space property,and establish the stable recovery of signals and matrices under the truncated sparse approximation property.We also explore the relationship between the restricted isometry property and truncated sparse approximation property.And we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k,then the first inequality in restricted isometry property of order k and of order 2k can hold for certain different constantsδk andδ2k,respectively.Last,we show that ifδs(k+|T^c|)<√(s-1)/s for some s≥4/3,then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k.It should be pointed out that when Tc=Ф,our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.

Parallel Active Subspace Decomposition for Tensor Robust Principal Component Analysis

Tensor robust principal component analysis has received a substantial amount of attention in various fields.Most existing methods,normally relying on tensor nuclear norm minimization,need to pay an expensive computational cost due to multiple singular value decompositions at each iteration.To overcome the drawback,we propose a scalable and efficient method,named parallel active subspace decomposition,which divides the unfolding along each mode of the tensor into a columnwise orthonormal matrix(active subspace)and another small-size matrix in parallel.Such a transformation leads to a nonconvex optimization problem in which the scale of nuclear norm minimization is generally much smaller than that in the original problem.We solve the optimization problem by an alternating direction method of multipliers and show that the iterates can be convergent within the given stopping criterion and the convergent solution is close to the global optimum solution within the prescribed bound.Experimental results are given to demonstrate that the performance of the proposed model is better than the state-of-the-art methods.

Sparse and Low-Rank Covariance Matrix Estimation

This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices.We first benefit from a convex optimization which develops l1-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property.For the proposed estimator,we then prove that with high probability,the Frobenius norm of the estimation rate can be of order O(√((slgg p)/n))under a mild case,where s and p denote the number of nonzero entries and the dimension of the population covariance,respectively and n notes the sample capacity.Finally,an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem,and merits of the approach are also illustrated by practicing numerical simulations.

基于稀疏表示的NMF抗混响方法

在浅海环境中,由于受到混响的影响,主动声呐接收到的信号混淆不清。针对上述问题,提出一种基于稀疏表示的非负矩阵分解(non-negative matrix factorization,NMF)抗混响方法。利用稀疏表示方法处理主动声呐回波信号,然后根据信号的稀疏性,构建基于非负矩阵分解的Kullback-Leibler(KL)问题,通过梯度下降法给出迭代规则,进而得到了目标信号矩阵中的协方差估计。仿真结果表明,相对其他去混响方法,该方法能够有效抑制混响,提高对水下目标的识别率。

基于非局部低秩稀疏矩阵分解的低剂量脑灌注CT统计迭代重建

脑灌注CT成像需要对患者头部进行连续反复扫描,相应的X射线辐射剂量较常规CT显著增加。为减少X射线辐射剂量,提出一种基于非局部低秩稀疏矩阵分解的低剂量脑灌注CT统计迭代重建方法。首先对脑灌注CT序列图像进行分块,然后构建非局部低秩稀疏矩阵分解模型,最后求解相应的目标函数重建出脑灌注CT序列图像。与滤波反投影算法和基于低秩稀疏矩阵分解的惩罚加权最小二乘方法相比,本文方法得到的脑血容参数图像的结构相似性指标分别提高38.07%和5.61%、特征相似性指标分别提高13.17%和2.47%;平均通过时间参数图像的结构相似性指标分别提高59.73%和0.28%、特征相似性指标分别提高20.26%和0.70%。本文方法能在去除低剂量脑灌注CT图像噪声和伪影的同时保持图像的边缘结构信息,并且获得更准确的脑血流动力学参数图像。

基于低秩和重加权稀疏表示的红外弱小目标检测算法

红外弱小目标检测技术是红外告警系统中的关键技术之一,但如何精确、快速、鲁棒地进行弱小目标检测依然是个难题。该文提出了基于低秩和重加权稀疏表示的红外弱小目标检测算法,设计了新的优化方程,更精确地描述了背景矩阵的秩,利用结构张量提取红外图像的局部先验信息权重,同时提取目标矩阵的自增强稀疏权重,使模型能够更好地抑制背景中的边缘干扰来提取目标。实验表明:所提算法精度优于现有的经典基线算法,速度超越了一些经典算法。从性能和时间两个方面综合考虑,所提算法有着较好的优越性,对远距离红外弱小目标告警具有积极的意义和良好的应用价值。

应用于异常事件检测的深度交替方向乘子法网络

针对大规模无线传感器网络(WSN)中的事件检测问题(EDP),传统的方法通常依赖先验信息,阻碍了实际应用。该文为EDP提出了一种基于深度学习的算法,称为交替方向乘子法网络(ADMM-Net)。首先,采用低秩稀疏矩阵分解来建模事件的时空相关性。之后,EDP被表述为一个带约束的优化问题并用交替方向乘子法(ADMM)求解。然而,优化算法收敛慢且算法的性能依赖于对先验参数的仔细选择。该文基于深度学习中“展开”的概念,提出了一种用于EDP的深度神经网络ADMM-Net。通过“展开”ADMM算法的方式得到。ADMM-Net具有固定层数,其参数可以通过监督学习训练获得。无需先验信息。相比于传统算法,提出的ADMM-Net收敛快且不需先验信息。人造数据集和真实数据集的仿真结果验证了ADMM-Net的有效性。

基于稀疏编码和矩阵分解的视频去雪算法

视频中的雪花会降低视频图像的质量,影响计算机对视频中目标自动检测、跟踪和识别等操作。视频中雪花去除是一项具有挑战性的任务,现存的算法除雪方法效果不佳。根据不同的空间特征,将一个视频中的雪花分为近景雪花和远景雪花。首先,利用低秩矩阵分解提取视频的背景信息。然后,采用多尺度卷积稀疏编码对远景雪花进行检测。利用马尔可夫随机场对运动物体进行建模,之后使用连通域阈值去除被判断为运动物体的雪花。实验结果表明,提出的算法有效地去除了视频中的近景雪花和远景雪花,同时保留了相关的背景和运动物体的信息。

时序差分低秩约束的人体运动数据恢复研究

从观测到的人体运动捕获数据中恢复缺失数据是一个重要的研究问题.传统方法利用矩阵核范数和矩阵l1范数分别对运动捕获数据潜在的低秩性和噪声稀疏性进行约束,并额外添加时序光滑项,构建的目标函数包含3个正则化项.为更好刻画运动数据的低秩性并简化模型,提出一种利用时序差分低秩先验(Temporal Difference Low-rank Prior,TDLRP)且目标函数只含2个正则化项的人体运动捕获数据恢复模型;在保证收敛的前提下,采用交替方向乘子法求解模型.在公开的CMU数据集和HDM05数据集上与现有算法进行比较,实验结果验证了TDLRP算法具有较好的恢复性能.

基于L2,1范数和局部约束的非负矩阵分解

给出了一种基于L2,1范数和局部约束的非负矩阵分解方法,降低了对原始数据中噪声和异常值的敏感程度,分析了该方法的严格收敛性,通过多个数据集上的实验结果验证该算法的有效性。

基于压缩感知理论的农业害虫分类方法

为提高现有害虫分类方法的分类效果,该文分析了现有害虫分类方法的局限性,在此基础上,提出一种基于压缩感知理论的害虫分类新方法。该方法利用害虫训练样本构造训练样本矩阵,通过求解l1范数意义下的最优化问题实现害虫测试样本的稀疏分解,由于稀疏分解结果中包含了明确的分类信息,可直接用于害虫分类。利用该方法对12类储粮害虫和110类常见害虫进行分类,在4种不同试验条件下,分类准确率分别达到92.9418%、98.2877%、78.8651%和61.5938%,证实了压缩感知理论用于害虫分类是合理可行的。

基于图规则化低秩矩阵恢复的用电数据修复与异常检测

针对电力系统用电数据中的记录误差与异常用电,提出一种基于图规则化低秩矩阵恢复的电力系统用电记录修复与异常检测算法。该方法从用户用电时空矩阵的低秩稀疏分解出发,结合电网拓扑结构与用户相关性的规则化调整,获取修复后的用电数据和异常用户。该方法同时兼顾了用户用电的周期性与异常用户的差异性特点。实验分析表明,与相关方法相比,所提方法在用电数据修复与异常用电模式检测的多项评价标准下均取得了更好的准确性和鲁棒性。

稀疏技术在电力系统状态估计中的应用

研究了电力系统状态估计中稀疏技术的应用方法。首先根据因子矩阵和消去树,讨论了稀疏向量法。然后基于给定的稀疏矩阵存储方法和符号因子化技术,提出了一种采用稀疏向量法进行LDLT分解的算法,最后结合量测残差方差计算给出了两种应用稀疏技术计算的方法。算法的有效性在IEEE 118和IEEE 300系统上得到了验证。

矩阵补全模型及其算法研究综述

近年来,随着压缩感知技术在信号处理领域的巨大成功,由其衍生而来的矩阵补全技术也日益成为机器学习领域的研究热点,诸多研究者针对矩阵补全问题展开了大量卓有成效的研究.为了更好地把握矩阵补全技术的发展规律,促进矩阵补全理论与工程应用相结合,针对矩阵补全模型及其算法进行了综述.首先,对矩阵补全技术进行溯源,介绍了从压缩感知到矩阵补全的自然演化历程,指出压缩感知理论的发展为矩阵补全理论的形成奠定了基础;其次,从非凸非光滑秩函数松弛的角度将现有矩阵补全模型进行分类,旨在为面向具体应用的矩阵补全问题建模提供思路;然后综述了适用于矩阵补全模型求解的代表性优化算法,其目的在于从本质上理解各种矩阵补全模型优化技巧,从而有利于面向应用问题的矩阵补全新模型求解;最后分析了矩阵补全模型及其算法目前存在的问题,提出了可能的解决思路,并对未来的研究方向进行了展望.

基于低秩三分解的红外图像杂波抑制

针对红外图像中对比度较低、目标信号较弱且受背景噪声杂波干扰较大的特点,结合信号的稀疏表示理论提出了一种基于低秩三分解模型的红外图像背景杂波抑制算法。首先,分别对红外图像中目标、背景和噪声3种成份进行建模描述,得到低秩三分解模型。然后,采用二维高斯模型构造红外小目标超完备字典,利用所提出的低秩三分解模型将分块重置的图像数据矩阵分解为背景、噪声和目标3种成份。最后,对于目标分量进行阈值处理从而得到突出红外小目标的重构图像,实现杂波抑制。在3种不同情况下的实验结果表明:本文算法能够使红外图像局部信噪比提高2倍以上;与其他经典算法相比,抑制因子至少提高15%。得到的结果表明,所提算法能够有效抑制杂波,在提高红外图像信噪比的同时,对不同噪声干扰也具有较强的鲁棒性。

求解线性约束最优化问题的有效集算法

为了保持投影梯度求解法的线性约束系数矩阵的稀疏性,且不降低算法的效率。在确定可行点处的可行方向时,使用了矩阵的隐式LU分解技术,构造有效约束的零空间．本文提出了求解线性约束最优化问题的有效集算法,对于线性约束系数矩阵是稀疏矩阵时,能较好地保持稀疏性,提高了算法的效率．与数值试验的结果吻合．

低秩稀疏重建分析的边缘检测方法

边缘检测方法众多,并取得了很好的应用效果,但不同方法有其自身的不足和边缘检测能力的限制,特别是对噪声干扰、多边缘干涉及弱小目标边缘的检测效果不理想。为此,首先分析断层边缘和缝洞边缘的空间分布特征,根据断层边缘和缝洞边缘的地震响应特征,把低秩稀疏分析理论引入边缘检测,研究边缘信息、背景信息及噪声信息的低秩稀疏分解与重建;为了提高边缘检测能力和分辨率,在压缩感知稀疏表示基础上,对地震资料进行深度稀疏化表示,结合向量稀疏表示和矩阵稀疏表示,通过低秩稀疏分析理论,形成一种全新的边缘检测方法——低秩稀疏重建分析的边缘检测方法。具体步骤为:(1)地震资料平稳小波分解;(2)多尺度小波系数优化;(3)根据多尺度优化小波系数建立张量矩阵并进行建模;(4)张量矩阵奇异值分解;(5)矩阵奇异值低秩优化;(6)多尺度双稀疏和双优化结果融合与重建。模型分析和实际资料应用效果分析表明:所提方法的抗噪性、适用性较强,对于断层和缝洞边缘具有较好的刻画能力。