低秩矩阵近似与优化问题研究进展

首先以高维数据压缩与恢复为背景,详细阐述由香农采样理论到稀疏表示和压缩感知理论再到低秩矩阵问题的发展历程,引出低秩矩阵近似与优化问题的重要性.然后,从低秩矩阵最小化问题、低秩矩阵分解问题、低秩矩阵的优化与应用三方面对现有方法进行详细的综述.最后对当前研究的不足之处与未来的研究方向提出合理的建议.     

基于背景先验与低秩恢复的显著性目标检测方法

显著性检测是指计算机通过算法自动识别出图像中的显著性目标，广泛应用于目标识别、图像检索与图像分类等领域。针对现有基于稀疏与低秩矩阵恢复的显著性检测模型中低秩转换矩阵的获取、前景稀疏矩阵的处理以及超像素块之间的关系，需对现有的稀疏与低秩矩阵恢复模型进行优化，使之更好地适用于图像的显著性检测。首先，根据背景的对比度和连通度原则获取图像低秩的背景字典，采用3种尺度分割图像的多个特征矩阵获得图像的前景稀疏矩阵；其次，通过计算邻居像素点之间的影响因子矩阵与置信度矩阵对显著图的结果进行结构约束，并且采用稀疏与低秩矩阵恢复模型对图像进行显著性检测；最后，利用K-means聚类算法的传播机制优化得到的显著图。在公开数据集上进行实验验证,结果证明本文方法能够准确有效地检测出显著性目标。     

基于低秩矩阵二元分解的快速显著性目标检测算法

近年来,基于矩阵低秩表示模型的图像显著性目标检测受到了广泛关注。在传统模型中通常对秩最小化问题进行凸松弛,即引入最小化核范数将原始输入图像分解为低秩矩阵和稀疏矩阵。但是,这种方法在每次迭代中必须执行矩阵奇异值分解（SVD）,计算复杂度较高。为此,本文提出了一种低秩矩阵双因子分解和结构化稀疏矩阵分解联合优化模型,并应用于显著性目标检测。算法不仅利用低秩矩阵双因子分解和交替方向法（ADM）来降低时间开销,而且引入分层稀疏正则化刻画稀疏矩阵中元素之间的空间关系。此外,所提算法能够无缝集成高层先验知识指导矩阵分解过程。实验结果表明,提出模型和算法的检测性能优于当前主流无监督显著性目标检测算法,且具有较低的时间复杂度。     

加权低秩矩阵恢复的混合噪声图像去噪

传统的基于低秩矩阵恢复的图像去噪算法只对低秩部分进行约束,当高斯噪声过大时,会导致去噪不充分或细节严重丢失。针对此问题,提出了一种新的鲁棒的图像去噪模型。该模型在原有的低秩矩阵核范数约束的基础上引入高斯噪声约束项,此外为了提高低秩矩阵的低秩性和稀疏矩阵的稀疏性,引入了加权的方法。为了考察方法的去噪能力,选取了不同参数类型的混合噪声图像进行仿真,并结合峰值信噪比、结构相似度评价标准与传统的基于低秩矩阵恢复的图像去噪算法进行对比。实验结果表明,加权低秩矩阵恢复的混合噪声图像去噪算法能增加低秩矩阵的低秩性和稀疏矩阵的稀疏性,在保证去噪效果的同时,保留了图像的细节信息,具有更佳的视觉效果,同时,客观评价指标均有所提高。     

基于lP范数的非凸低秩张量最小化

在低秩矩阵、张量最小化问题中,凸函数容易求得最优解,而非凸函数可以得到更低秩的局部解.文中基于非凸替换函数的低秩张量恢复问题,提出基于l_p范数的非凸张量模型.采用迭代加权核范数算法求解模型,实现低秩张量最小化.在合成数据和真实图像上的大量实验验证文中方法的恢复性能.     

基于局部和非局部正则化的图像压缩感知

基于低秩正则化的非局部低秩约束(Nonlocal low-rank regularization, NLR)算法利用相似块的结构稀疏性,获得了目前最好的重构结果。但是它仅仅利用了图像的非局部信息,忽略了图像像素间的局部信息,不能有 效地重建图像的边缘,同时Logdet函数不能很好地替代矩阵秩,因为它跟真实解之间存在着不可忽视的差距。因此,本文提出了一种基于局部和非局部正则化的压缩感知图像重建方法,同时考虑图像的非局部低秩性和图像像素的局部稀疏梯度性。选择利用Schatten-p 范数来替代矩阵秩,同时选择交替方向乘子算法求解产生的非凸优化问题。实验 结果表明,与传统的稀疏性先验重建算法和NLR算法相比,本文算法能够获得更高的图像重构质量。     

基于稀疏与低秩的核磁共振图像重构算法

已有的基于压缩感知的核磁共振图像重构算法仅利用了数据的稀疏性或矩阵的低秩性,并没有充分利用图像数据的相关性先验知识.针对这一问题,提出了一种新型的应用于二维核磁共振图像重建的算法模型.与传统的单一利用原始数据的稀疏性或矩阵低秩性进行重建的方法不同,该方法同时利用了图像数据的稀疏性与矩阵的低秩性.矩阵低秩部分使用应用赤池信息量准则的奇异值分解阈值方法,数据稀疏部分使用全变分作为稀疏变换基.实验结果表明,该方法在相同的采样率下与应用赤池信息量准则的奇异值分解阈值方法、全变分方法和奇异值分解阈值方法相比大大提升了重建图像的质量.     

用稀疏相似性度量求解压缩传感矩阵

提出一种用稀疏相似性度量求解压缩传感矩阵的方法,并将其应用在图像重建和识别领域中.首先构造一种稀疏相似性度量,然后将其嵌入到传感矩阵的模糊代价函数中,最终传感矩阵的原子更新按照模糊方式进行计算.用该方法优化后的观测矩阵与字典矩阵之间保持了低相干性,并且样本的稀疏信号在相同重构条件下具备了更优的测量数目和质量.在ORL和FERET人脸数据库及91幅自然图像库上的实验结果验证了该算法的有效性.     

基于低秩矩阵恢复的视觉显著性目标检测与细化

基于低秩矩阵恢复(low-rank matrix recovery,LRMR)的显著性目标检测模型将图像特征分解为与背景关联的低秩分量和与显著性目标相关联的稀疏分量,并从稀疏分量中获得显著性目标.现有的显著性检测方法很少考虑低秩分量与稀疏分量之间的相互关系,导致检测的显著性目标零散或不完整.为此,提出基于低秩矩阵恢复的显著性目标检测与细化方法来规避该限制.首先,所提方法采用ell_1范数稀疏约束和拉普拉斯正则项对初始显著图进行计算;在显著性细化阶段,由于非局部的ell_0优化可以有效地对显著性区域及其邻接区域之间的相互关系进行建模,结合初始显著图,采用非局部ell_0梯度优化,最小化显著性区域中显著值的变化,从而保证显著性目标的完整性.在4个显著性目标检测数据集上进行实验,通过实验结果验证所提算法的优越性.     

联合矩阵F范数的低秩图像去噪

摘 要：目的：低秩矩阵恢复是通过最小化矩阵核范数来获得低秩解,然而待恢复低秩矩阵相关性低的要求往往会导致求解不稳定的情况。方法：针对该问题,研究一种基于变量分裂的低秩图像恢复去噪算法,引入待恢复矩阵的Frobenius范数作为新正则项,与原有低秩矩阵的核范数组成联合正则化项,对问题进行凸松弛后,采用变量分裂的增广拉格朗日乘子法求解。结果：为考察方法的稳定性和去噪能力,选取了不同参数类型的加噪图像进行仿真,并结合恢复时间、信噪比、差错率等评价标准与现有低秩矩阵恢复算法进行对比。结论：实验结果表明增加Frobenius范数的低秩矩阵恢复模型在保持原有低秩稀疏恢复的前提下,具有良好的去噪性能,对相关性强的低秩图像恢复结果稳定性好,获得了更高的信噪比。     

A nonconvex nonsmooth regularization method for compressed sensing and low rank matrix completion

In this paper, a nonconvex and nonsmooth method for compressed sensing and low-rank matrix completion is studied. The proposed model is formulated as nonconvex regularized least square optimization problem. At first, an alternating minimization scheme is developed in which the problem can be decomposed into three subproblems, two of them are convex and the remaining one is smooth. Then, the convergence of the sequence which is generated by the alternating minimization algorithm is proved. In addition, some recovery guarantees are also analyzed. Finally, various numerical simulations are performed to test the efficiency of the method.     

基于低秩表示的非负张量分解算法

为了提高图像分类准确率,提出了一种基于低秩表示的非负张量分解算法。作为压缩感知理论的推广和发展,低秩表示将矩阵的秩作为一种稀疏测度,由于矩阵的秩反映了矩阵的固有特性,所以低秩表示能有效的分析和处理矩阵数据,本文把低秩表示引入到张量模型中,即引入到非负张量分解算法中,进一步扩展非负张量分解算法。实验结果表明,本文所提算法与其他相关算法相比,分类结果较好。     

Robust alternative minimization for matrix completion

Recently, much attention has been drawn to the problem of matrix completion, which arises in a number of fields, including computer vision, pattern recognition, sensor network, and recommendation systems. This paper proposes a novel algorithm, named robust alternative minimization (RAM), which is based on the constraint of low rank to complete an unknown matrix. The proposed RAM algorithm can effectively reduce the relative reconstruction error of the recovered matrix. It is numerically easier to minimize the objective function and more stable for large-scale matrix completion compared with other existing methods. It is robust and efficient for low-rank matrix completion, and the convergence of the RAM algorithm is also established. Numerical results showed that both the recovery accuracy and running time of the RAM algorithm are competitive with other reported methods. Moreover, the applications of the RAM algorithm to low-rank image recovery demonstrated that it achieves satisfactory performance.     

卷积鲁棒主成分分析

鲁棒主成分分析（RPCA）是一种经典的高维数据分析方法,可从带噪声的观测样本中恢复出原始数据。但是,RPCA能工作的前提是目标数据拥有低秩矩阵结构,不能有效处理实际应用中广泛存在的非低秩数据。研究发现,虽然图像、视频等数据矩阵本身可能不是低秩的,但它们的卷积矩阵通常是低秩的。根据这一原理,提出一种称为卷积鲁棒主成分分析（CRPCA）的新方法,利用卷积矩阵的低秩性对原始数据的结构进行约束,从而实现精确的数据恢复。CPRCA模型的计算过程是一个凸优化问题,通过乘子交替方向法（ADMM）来进行求解。通过对合成数据向量以及真实数据图片、视频序列进行实验,验证了该方法相较于其他算法如RPCA、广义鲁棒主成分分析（GRPCA）以及核鲁棒主成分分析（KRPCA）在处理数据非低秩问题上优越性。     

Nonlinear sparse feature selection algorithm via low matrix rank constraint

The characteristics of non-linear, low-rank, and feature redundancy often appear in high-dimensional data, which have great trouble for further research. Therefore, a low-rank unsupervised feature selection algorithm based on kernel function is proposed. Firstly, each feature is projected into the high-dimensional kernel space by the kernel function to solve the problem of linear inseparability in the low-dimensional space. At the same time, the self-expression form is introduced into the deviation term and the coefficient matrix is processed with low rank and sparsity. Finally, the sparse regularization factor of the coefficient vector of the kernel matrix is introduced to implement feature selection. In this algorithm, kernel matrix is used to solve linear inseparability, low rank constraints to consider the global information of the data, and self-representation form determines the importance of features. Experiments show that comparing with other algorithms, the classification after feature selection using this algorithm can achieve good results.

     

Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization

The matrix separation problem aims to separate a low-rank matrix and a sparse matrix from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications. Nuclear-norm minimization models have been proposed for matrix separation and proved to yield exact separations under suitable conditions. These models, however, typically require the calculation of a full or partial singular value decomposition at every iteration that can become increasingly costly as matrix dimensions and rank grow. To improve scalability, in this paper, we propose and investigate an alternative approach based on solving a non-convex, low-rank factorization model by an augmented Lagrangian alternating direction method. Numerical studies indicate that the effectiveness of the proposed model is limited to problems where the sparse matrix does not dominate the low-rank one in magnitude, though this limitation can be alleviated by certain data pre-processing techniques. On the other hand, extensive numerical results show that, within its applicability range, the proposed method in general has a much faster solution speed than nuclear-norm minimization algorithms and often provides better recoverability.