An analytical algorithm for generalized low-rank approximations of matrices

The algorithm for generalized low-rank approximations of matrices (GLRAM) has been developed recently. In this paper, the optimality property of GLRAM is revealed. Accordingly, an analytical method for GLRAM is proposed. The proposed method is non-iterative. Moreover, the relationship between 2DPCA and GLRAM is shown.     

An iterative splitting method via waveform relaxation

This paper explores a new numerical strategy for a closed formulation of iterative splitting methods and their embedding in classical waveform-relaxation methods. Since iterative splitting has been developed in several papers, an abstract framework that relates these methods to other classical splitting methods would be useful and is needed. Here, we present an embedding of the iterative splitting method in the waveform-relaxation and exponential splitting methods. While we can use the theoretical background of the classical schemes, a simpler iterative splitting analysis is obtained. This is achieved by basing the analysis on semigroup and fixpoint schemes. Our approach is illustrated with numerical results obtained on differential equations with constant and time-dependent coefficients.     

低秩分块矩阵的核近似

为了探讨结构受限下的矩阵分解问题,通过最小化块外对角线来增强类与类之间数据表示的不相关性,从而实现分块约束,即数据来源于不同的聚类结构,是一种局部结构的约束;同时通过增强样本的自表达属性并缩小样本之间的差距来增强类内数据表示的相关性,从而实现低秩约束,即数据行出现冗余,是一种全局结构的约束。随后设计了一个低秩分块矩阵的核近似算法,通过交替方向乘子法迭代求解。最后将该方法分别在人脸识别和字符识别上进行测试。实验结果表明,所提出的低秩分块矩阵分解算法在收敛速度和近似精度上都具有一定的优势。     

变分调整约束下的反向低秩稀疏学习目标跟踪

目的 低秩稀疏学习目标跟踪算法在目标快速运动和严重遮挡等情况下容易出现跟踪漂移现象,为此提出一种变分调整约束下的反向低秩稀疏学习目标跟踪算法。方法 采用核范数凸近似低秩约束描述候选粒子间的时域相关性,去除不相关粒子,适应目标外观变化。通过反向稀疏表示描述目标表观,用候选粒子稀疏表示目标模板,减少在线跟踪中L₁优化问题的数目,提高跟踪效率。在有界变差空间利用变分调整对稀疏系数差分建模,约束目标表观在相邻帧间具有较小变化,但允许连续帧间差异存在跳跃不连续性,以适应目标快速运动。结果 实验利用OTB（object tracking benchmark）数据集中的4组涵盖了严重遮挡、快速运动、光照和尺度变化等挑战因素的标准视频序列进行测试,定性和定量对比了本文算法与5种热点算法的跟踪效果。定性分析基于视频序列的主要挑战因素进行比较,定量分析通过中心点位置误差（central pixel error,CPE）比较跟踪算法的精度。与CNT（convolutional networks training）、SCM（sparse collaborative model）、IST（inverse sparse tracker）、DDL（discriminative dictionary learning）和LLR（locally low-rank representation）算法相比,平均CPE值分别提高了2.80、4.16、13.37、35.94和41.59。实验结果表明,本文算法达到了较高的跟踪精度,对上述挑战因素更具鲁棒性。结论 本文提出的跟踪算法,综合了低秩稀疏学习和变分优化调整的优势,在复杂场景下具有较高的跟踪精度,特别是对严重遮挡和快速运动情况的有效跟踪更具鲁棒性。     

二次联合稀疏表示和低秩近似的浅浮雕优化

目的 针对低质量浅浮雕表面的噪声现象,提出一种二次联合局部自适应稀疏表示和非局部低秩矩阵近似的浅浮雕优化算法。方法 本文方法分两个阶段。第1阶段,将浅浮雕灰度图划分成大小相同的数据块,提取边界块并进行去噪,分别对数据块进行稀疏表示和低秩近似处理。一方面,通过字典学习获得过完备字典和稀疏编码;另一方面,利用K均值聚类算法（K-means）将事先构建的外部字典库划分成k类,从k个簇中心匹配每个数据块的相似块并组成相似矩阵,依次进行低秩近似和特征增强处理。最后通过最小二乘法求解,重建并聚合新建数据块以得到新的高度场。第2阶段与第1阶段的结构相似,主要区别在于改用重建高度场的非局部自身相似性来实现块匹配。结果 在不同图像压缩率下（70%,50%,30%）,对比本文方法与BM3D（block-matching and 3D filtering）、WNNM（weighted nuclear norm minimization）、STROLLR（sparsifying transform learning and low-rank）、TWSC（trilateral weighted sparse coding）4个平滑降噪方法的浅浮雕重建结果,发现BM3D和STROLLR方法的特征保持虽好,但平滑效果较差,WNNM方法出现模型破损现象,TWSC方法的平滑效果比BM3D和STROLLR方法更好,但特征也同时被光顺化。阴影恢复形状法（shape from shading,SFS）是一种基于图像的3D建模法,但是其重建结果比较粗糙。相比之下,本文方法生成的浅浮雕模型更加清晰直观,在浅浮雕的特征增强和平滑去噪方面都展现出更好的性能。结论 本文综合数据块的局部稀疏性和数据块之间的非局部相似性对粗糙的浅浮雕模型进行二次高度场重建。本文方法有效改善了现有浅浮雕模型的质量,提高了模型的整体视觉效果,为浅浮雕的优化提供了新方法。     

Dynamical low-rank approximation: applications and numerical experiments

Dynamical low-rank approximation is a differential-equation-based approach to efficiently compute low-rank approximations to time-dependent large data matrices or to solutions of large matrix differential equations. We illustrate its use in the following application areas: as an updating procedure in latent semantic indexing for information retrieval, in the compression of series of images, and in the solution of time-dependent partial differential equations, specifically on a blow-up problem of a reaction-diffusion equation in two and three spatial dimensions. In 3D and higher dimensions, space discretization yields a tensor differential equation whose solution is approximated by low-rank tensors, effectively solving a system of discretized partial differential equations in one spatial dimension.     

结合全变差与自适应低秩正则化的图像压缩感知重构

针对基于固定变换基的协同稀疏图像压缩感知(CS)重构算法不能充分利用图像自相似特性的问题,提出了一种改进的联合全变差与自适应低秩正则化的压缩感知重构方法。首先,通过图像块匹配法寻找结构相似块,并组成非局部相似块组;然后,以非局部相似块组加权低秩逼近替代协同稀疏表示中的三维小波变换域滤波;最后,结合梯度稀疏与非局部相似块组低秩先验构成重构模型的正则化项,并采用交替方向乘子法求解实现图像重构。实验结果表明,相比协同稀疏压缩感知重构(RCoS)算法,该方法重构图像的峰值信噪比平均可提升约2 dB,所提算法在准确描述图像非局部自相似结构特征的前提下显著提高了重构质量,更好地保留了图像的纹理细节信息。     

基于非局部自相似性和低秩矩阵逼近的补全算法

针对传统矩阵补全算法在图像重建方面的不足,提出了一种基于非局部自相似性和低秩矩阵逼近（NL-LRMA）的补全算法。首先,通过相似性度量找到图像中局部块所对应的非局部相似块,并将相应灰度信息进行向量化,从而构建出非局部相似块矩阵;然后,针对所得相似矩阵的低秩性,对其进行低秩补全操作（LRMA）;最后,对补全结果进行重新组合,以达到恢复原始图像的目的。在灰度图像以及RGB图像上进行重建实验,结果表明：在经典数据集上,NL-LRMA算法要比原LRMA算法在平均峰值信噪比（PSNR）上高出4~7 dB;同时,新算法在视觉效果与PSNR值方面也明显优于迭代重加权核范数（IRNN）、加权核范数（WNNM）、LRMA等传统算法。总之,所提算法对传统算法在自然图像重建方面的不足进行了有效弥补,从而为图像重建提供了一种行之有效的解决方案。     

基于非局部自相似性和低秩矩阵逼近的补全算法

针对传统矩阵补全算法在图像重建方面的不足，提出了一种基于非局部自相似性和低秩矩阵逼近（NL-LRMA）的补全算法。首先，通过相似性度量找到图像中局部块所对应的非局部相似块，并将相应灰度信息进行向量化，从而构建出非局部相似块矩阵；然后，针对所得相似矩阵的低秩性，对其进行低秩补全操作（LRMA）；最后，对补全结果进行重新组合，以达到恢复原始图像的目的。在灰度图像以及RGB图像上进行重建实验，结果表明：在经典数据集上，NL-LRMA算法要比原LRMA算法在平均峰值信噪比（PSNR）上高出4~7 dB；同时，新算法在视觉效果与PSNR值方面也明显优于迭代重加权核范数（IRNN）、加权核范数（WNNM）、LRMA等传统算法。总之，所提算法对传统算法在自然图像重建方面的不足进行了有效弥补，从而为图像重建提供了一种行之有效的解决方案。     

Compact implicit surface reconstruction via low-rank tensor approximation

Implicit representations have gained an increasing popularity in geometric modeling and computer graphics due to their ability to represent shapes with complicated geometry and topology. However, the storage requirement, e.g. memory or disk usage, for implicit representations of complex models is relatively large. In this paper, we propose a compact representation for multilevel rational algebraic spline (MRAS) surfaces using low-rank tensor approximation technique, and exploit its applications in surface reconstruction. Given a set of 3D points equipped with oriented normals, we first fit them with an algebraic spline surface defined on a box that bounds the point cloud. We split the bounding box into eight sub-cells if the fitting error is greater than a given threshold. Then for each sub-cell over which the fitting error is greater than the threshold, an offset function represented by an algebraic spline function of low rank is computed by locally solving a convex optimization problem. An algorithm is presented to solve the optimization problem based on the alternating direction method of multipliers (ADMM) and the CANDECOMP/PARAFAC (CP) decomposition of tensors. The procedure is recursively performed until a certain accuracy is achieved. To ensure the global continuity of the MRAS surface, quadratic B-spline weight functions are used to blend the offset functions. Numerous experiments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods. Furthermore, our method has good adaptability and is able to produce reconstruction results with high quality.     

An outer approximation algorithm for equilibrium problems in Hilbert spaces

We introduce an algorithm for solving non-smooth equilibrium problems in real Hilbert spaces. At each iteration, a regularized proximal-like equilibrium problem on a suitable outer approximation of the original constraint set is considered. We prove, under standard assumptions, that the sequence generated by the algorithm converges weakly to a solution of the problem. Some numerical experience with the algorithm is reported.     

基于非凸低秩约束的图像修复方法

受传输干扰或存储不当等因素的影响,现实应用中获取的某些图像通常会存在像素缺失现象,这给图像的后续分析与处理带来了一定影响.解决该问题的常用方法是对图像进行低秩修复.利用低秩特性进行修复的方法大多以秩函数建模,由于矩阵秩函数是非凸离散的,该模型的求解是一个NP难问题,所以通常利用核范数对矩阵的秩进行凸松弛.但是,基于核范...     

Stochastic successive approximation method for assessing the insolvency risk of an insurance company

A stochastic successive approximation method is analyzed with a view to solving risk assessment problems that are reduced to a renewal integral equation and, in particular, to assessing the insolvency risk of an insurance company. Integrals in the equation are evaluated approximately, for example, by the Monte Carlo method. Iterations of the method are proved to converge uniformly with probability one. Theoretical results are illustrated by numeral computations. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 116–130, November–December 2008.     

结构图正则低秩子空间聚类

针对结构稀疏子空间聚类中不能很好地保证相似度矩阵连接性的问题,给出了一个新的统一优化模型。首先,引入了表示系数矩阵的子空间结构范数,增加了低秩表示来揭示高维数据的全局结构。其次,为了使相似度矩阵具有类内统一,类间稀疏的作用,还定义了分组效应来捕获数据的内部几何结构,提出了结构图正则低秩子空间聚类模型。最后使用自适应惩罚的线性化交替法（LADMAP）来得到最优解。实验结果表明,该模型不但可以捕获数据的全局结构,而且还可以捕获数据的内在几何结构,迫使相关数据紧密结合,不相关数据松散分离,从而使得相似度矩阵与分割矩阵变得更加一致。     

Learning low-rank Mercer kernels with fast-decaying spectrum

Low-rank representations have received a lot of interest in the application of kernel-based methods. However, these methods made an assumption that the spectrum of the Gaussian or polynomial kernels decays rapidly. This is not always true and its violation may result in performance degradation. In this paper, we propose an effective technique for learning low-rank Mercer kernels (LMK) with fast-decaying spectrum. What distinguishes our kernels from other classical kernels (Gaussian and polynomial kernels) is that the proposed always yields low-rank Gram matrices whose spectrum decays rapidly, no matter what distribution the data are. Furthermore, the LMK can control the decay rate. Thus, our kernels can prevent performance degradation while using the low-rank approximations. Our algorithm has favorable in scalability—it is linear in the number of data points and quadratic in the rank of the Gram matrix. Empirical results demonstrate that the proposed method learns fast-decaying spectrum and significantly improves the performance.     

基于低秩行为信息和多尺度卷积神经网络的人体行为识别方法

针对人体行为识别中传统行为信息获取方法需要繁琐步骤和各类假设的问题,结合卷积神经网络(CNN)在图像视频处理中的优越性能,提出了一种基于低秩行为信息(LAI)和多尺度卷积神经网络(MCNN)的人体行为识别方法.首先,对行为视频进行分段,并分别对每个视频段进行低秩学习以提取到相应的LAI,然后在时间轴上对这些LAI进行连...     

混合全变差和低秩约束下的高光谱图像复原

目的 由于高光谱遥感数据携带丰富的光谱和空间信息,使其在许多领域得以广泛关注和应用。但是高光谱遥感数据在获取过程中受到各种因素的影响,存在多种不同程度的退化,进而影响到后续的处理和应用。因此,提出一种基于低秩矩阵近似和混合全变差正则化方法来复原退化的高光谱遥感数据。方法 首先分析高光谱遥感数据的两种低秩先验：光谱低秩先验和空间低秩先验;然后利用光谱低秩先验建立低秩矩阵近似表示模型,有效抑制稀疏噪声,例如脉冲噪声、条纹噪声、死线噪声等;再利用空间低秩先验建立混合全变差正则化模型,有效去除高密度噪声,例如强高斯噪声、泊松噪声等;最后结合两种模型的优势,建立基于低秩矩阵近似和混合全变差正则化模型。结果 利用多组高光谱遥感数据,和多种相关的高光谱复原方法进行对比仿真实验,表明新模型的结果在视觉质量有很大改进。与目前最新的复原模型相比,提出的模型的平均峰值信噪比能提高1.8 dB,而平均结构相似数值指标能提高0.05。结论 新模型充分利用高光谱遥感数据的空间和光谱低秩先验,针对含有高密度噪声和稀疏异常值的高光谱遥感数据,能够有效复原出高质量的高光谱遥感数据。     

The method of successive approximation applied to find the probability for an insurance company with random premiums

A risk process that describes the evolution of the capital of an insurance company is analyzed, random premiums and claims being available. Integral equations of nonbankruptcy probability as a function of the initial capital are derived. Necessary and sufficient conditions for the existence and uniqueness of solutions of these integral equations, and convergence conditions for the method of successive approximation for finding their solutions are established __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 112–127, January–February 2006.     

多尺度分割先验迭代重加权低秩恢复显著性检测

目的 现有显著性检测方法大多只关注显著目标的中心信息,使得算法只能得到中心清晰、边缘模糊的显著目标,丢失了一些重要的边界信息,而使用核范数约束进行低秩矩阵恢复,运算过程冗余。为解决以上问题,本文提出一种无监督迭代重加权最小二乘低秩恢复算法,用于图像视觉显著性检测。方法 将图像分为细中粗3种尺度的分割,从细粒度和粗粒度先验的融合中得到分割先验信息;将融合后的分割先验信息通过迭代重加权最小二乘法求解平滑低秩矩阵恢复,生成粗略显著图;使用中粒度分割先验对粗略显著图进行平滑,生成最终的视觉显著图。结果 实验在MSRA10K（Microsoft Research Asia 10K）、SOD（salient object detection dataset）和ECSSD（extended complex scene saliency dataset）数据集上进行测试,并与现有的11种算法进行对比。结果表明,本文算法可生成边界清晰的显著图。在MSRA10K数据集上,本文算法实现了最高的AUC（area under ROC（receiver operating characteristic）curve）和F-measure值,MAE（mean absolute error）值仅次于SMD（structured matrix decomposition）算法和RBD（robust back ground detection）算法,AUC和F-measure值比次优算法RPCA（robust principal component analysis）分别提高了3.9%和12.3%;在SOD数据集上,综合AUC、F-measure和MAE值来看,本文算法优于除SMD算法以外的其他算法,AUC值仅次于SMD算法、SC（smoothness constraint）算法和GBVS（graph-based visual salieney）算法,F-measure值低于最优算法SMD 2.6%;在ECSSD数据集上,本文算法实现了最高的F-measure值75.5%,AUC值略低于最优算法SC 1%,MAE值略低于最优算法HCNs（hierarchical co-salient object detection via color names）2%。结论 实验结果表明,本文算法能从前景复杂或背景复杂的显著图像中更准确地检测出边界清晰的显著目标。     

随机分组策略下的分布式多智能体一致性

多智能体分布式一致性算法一般需要获得相对状态差值x_i-x_j,本文针对无法得到智能体间相对状态差值的情况,提出一种基于智能体分组,通过组间信息交换来达到智能体状态一致的算法.本文仅讨论离散情况下智能体被随机划分为两组和多组的情况.当存在两个随机分组时,每个智能体都进行状态更新,且更新量为组间的状态差值.此时,系统达到期望一致的充要条件为所给出的状态更新参数应大于1.当存在多个随机分组时,仅通过Gossip算法选中的两组智能体以这两组间的状态差值进行状态更新.在这种情况下,系统达到期望一致的充分条件为各分组概率相等,且状态更新参数大于1.最后通过计算机仿真验证了结论的正确性.