光滑lp范数压缩感知图像重构优化算法

针对已有压缩感知重构算法重构精度不高、消耗时间长的问题,在研究[lp]范数和光滑[l0]范数压缩感知重构算法的基础上提出改进算法。通过极大熵函数构造一种光滑函数来逼近最小[lp] 范数,对解序列进行离散化来近似最小[lp]范数的最优解,结合图像分块压缩感知技术（BCS）,在MATLAB中对测试图像进行仿真实验。结果表明,与传统的BOMP（Block Orthogonal Matching Pursuit）算法和IRLS（Iteratively Reweighted Least Squares）算法相比,改进后的算法不仅提高了重构精度,而且大大降低运行时间。     

基于SL0范数的改进稀疏信号重构算法

平滑范数（Smoothed l0,SL0）压缩感知重构算法通过引入平滑函数序列将求解最小l0范数问题转化为平滑 函数优化问题,可以有效地用于稀疏信号重构。针对平滑函数的选取和算法稳健性问题,提出一种新的平滑函数序列近似范数,结合梯度投影法优化求解,并进一步提出采用奇异值分解（Singular value decomposition, SVD）方法改进算法的稳健性,实现稀疏度信号的精确重构。仿真结果表明,在相同的测试条件下,本文算法相比OMP算法、SL0算法以及L1-magic算法在重构精度、峰值信噪比方面都有较大改善。     

利用正则化矩阵分解技术的多视图聚类方法

为了解决具有多种特征属性的多媒体数据（多视图数据）挖掘问题,在非负矩阵分解（NMF）算法的基础上,提出了一种多视图正则化矩阵分解算法（MRMF）,该算法使用了多元非负矩阵分解技术,同时使用[L2,1]范数描述矩阵分解的损失函数,并采用多视图流形正则化对矩阵分解进行正则化约束。与现有的一些数据聚类或多视图聚类算法相比,提出的MRMF算法不易受到原始数据中噪声的影响,而且能够充分考虑到不同视图在聚类中所具有不同权重的问题,能够对多视图数据进行较为准确的聚类。MRMF算法的有效性在一些经典的公开数据集上进行了验证,并取得了较好的聚类精度。     

基于Nyström方法的偏好特征提取

针对电影评分中特征提取效率较低的问题,提出了与QR分解相结合的Nyström方法。首先,利用自适应方法进行采样,然后对内部矩阵进行QR分解,将分解后的矩阵与内部矩阵进行重新组合并进行特征分解。Nyström方法的近似过程与标志点选取的数量以及选取标志点的过程密切相关,选取一系列具有标志性的点来保证采样后的近似性,自适应的采样方法能够保证近似的精度。QR分解能够保证矩阵的稳定性,提高偏好特征提取的精度。偏好特征提取的精度越高,推荐系统的稳定性就会越高,推荐的精度也会提高。最后在真实的观众对电影评分的数据集上进行了特征提取的实验,该电影数据集中包含480189个用户,17770部电影,实验结果表明,提取相同数目的标志点时,该算法的精度和效率都有了一定程度的提高：相对于采样前,时间复杂度由原来的O（n³）减少为O（nc²）（c<<n）;与标准的Nyström相比,误差控制在25%以下。     

L1范数约束正交子空间非负矩阵分解

针对非负矩阵分解（NMF）相对稀疏或局部化描述原数据时导致的稀疏能力和程度比较弱的问题,提出了L1范数约束正交子空间非负矩阵分解方法.通过将L1范数约束引入到正交子空间非负矩阵分解的目标函数中,提升了分解结果的稀疏性.同时给出累乘迭代规则.在UCI、ORL和Yale三个数据库上进行的实验结果表明,该算法在聚类效果以及稀疏表达方面优于其他算法.     

基于L1范数的全变分正则化超分辨重构算法

针对结构化照明显微成像系统的超分辨图像重构算法存在边界振铃效应、噪声免疫性差的问题,提出了一种基于L1范数的全变分正则化超分辨图像重构算法（简称L1/TV重构算法）。从结构化显微成像模型入手,分析了传统算法的设计原理和局限性;论述了L1/TV重构算法的原理,采用L1范数对重构图像保真度进行约束,并利用全变分正则化有效克服了重构过程的病态性,保护了重构图像边缘。对比研究传统重构算法和L1/TV重构算法的性能。实验结果表明：L1/TV重构算法具有更强的抗噪声干扰能力,重构图像空间分辨率更高。     

基于低秩矩阵恢复的DOA稀疏重构方法

为提高非均匀噪声下波达方向（direction of arrival,DOA）角估计算法的估计精度和分辨率,基于低秩矩阵恢复理论,提出了一种二阶统计量域下的加权L1稀疏重构DOA估计算法。该算法基于低秩矩阵恢复方法,引入弹性正则化因子将接收信号协方差矩阵重构问题转换为可获得高效求解的半定规划（semidefinite programming,SDP）问题以重构无噪声协方差矩阵;而后在二阶统计量域下利用稀疏重构加权L1范数实现DOA参数估计。数值仿真表明,与传统MUSIC、L1-SVD及加权L1算法相比,所提算法能显著抑制非均匀噪声影响,具有较好的DOA参数估计性能,且在低信噪比条件下,所提算法具有较高的角度分辨力和估计精度。     

广义行（列）对称矩阵的QR分解及其算法

对广义行（列）对称矩阵的QR分解和性质进行了研究,给出了广义行（列）对称矩阵的QR分解的公式和快速算法,它们可有效减少广义行（列）对称矩阵的QR分解的计算量与存储量,并且不会丧失数值精度。同时讨论了系统参数估计,推广和丰富了两文（邹红星,王殿军,戴琼海,等.行(或列)对称矩阵的QR分解.中国科学：A辑,2002,32(9):842-849;蔺小林,蒋耀林.酉对称矩阵的QR分解及其算法.计算机学报,2005,28(5):817-822）的研究内容,拓宽了实际应用领域的范围, 并修正了后者的错误。     

L1范局部线性嵌入

数据降维问题存在于包括机器学习、模式识别、数据挖掘等多个信息处理领域。局部线性嵌入（LLE）是一种用于数据降维的无监督非线性流行学习算法,因其优良的性能,LLE得以广泛应用。针对传统的LLE对离群（或噪声）敏感的问题,提出一种鲁棒的基于L1范数最小化的LLE算法（L1-LLE）。通过L1范数最小化来求取局部重构矩阵,减小了重构矩阵能量,能有效克服离群（或噪声）干扰。利用现有优化技术,L1-LLE算法简单且易实现。证明了L1-LLE算法的收敛性。分别对人造和实际数据集进行应用测试,通过与传统LLE方法进行性能比较,结果显示L1-LLE方法是稳定、有效的。     

基于部分支撑集的L1范数稀疏重构算法

针对压缩感知理论中,现有的优化L1范数稀疏重构算法在重构源信号时,当且仅当稀疏度小于等于观测信号长度一半时才能够正确重构源信号的问题,提出了部分支撑集的L1范数稀疏重构算法。改进算法采用线性规划方法最小化源信号"尾部"支撑集的L1范数,能够在稀疏度大于观测信号长度一半时正确重构出源信号。仿真结果表明,在不同信噪比和稀疏度条件下,所提算法的重构精度优于现有的优化L1范数的稀疏重构算法和正交匹配追踪的稀疏重构算法。     

基于平滑l0范数正交子空间非负矩阵分解

为了能够提升分解矩阵的稀疏表达能力,提出了一种新的基于平滑l0范数的正交子空间非负矩阵分解方法。通过将分解矩阵的正交性及平滑l0范数约束同时引入矩阵分解的目标函数中一起进行优化,大大降低了计算复杂度,并提升了分解矩阵的稀疏表达能力。同时给出了分解矩阵的乘积更新迭代规则。通过在三个真实数据库(Iris,UCI,ORL)上的实验表明,该方法在分解所得矩阵的稀疏表示方面及将其应用于聚类问题所取得的聚类效果方面优于其他方法。     

Perturbation analysis for the hyperbolic QR factorization

The hyperbolic QR factorization is a generalization of the classical QR factorization and can be regarded as the triangular case of the indefinite QR factorization proposed by Sanja Singer and Sa?a Singer. In this paper, the perturbation analysis for this factorization is considered using the classical matrix equation approach, the refined matrix equation approach, and the matrix–vector equation approach. The first order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are derived. The obtained first order bounds can be much tighter than the corresponding existing ones. Each of the obtained rigorous bounds is composed of a small constant multiple of the corresponding first order bound and an additional term with simple form. In particular, for square matrix, the rigorous bounds for the factor $R$ are just the $\sqrt{6}+\sqrt{3}$ multiple of the corresponding first order bounds. These rigorous bounds can be used safely for all cases in comparison to the first order bounds.     

关于酉对称矩阵的QR分解的注记

研究了行(列)酉对称矩阵的性质,修正了行(列)酉对称矩阵的QR分解的公式和快速算法.结果可减少行(列)酉对称矩阵的QR分解的计算量与存储量,并且不会丧失数值精度.     

Analysis of sparse representation and blind source separation

In this letter, we analyze a two-stage cluster-then-l(1)-optimization approach for sparse representation of a data matrix, which is also a promising approach for blind source separation (BSS) in which fewer sensors than sources are present. First, sparse representation (factorization) of a data matrix is discussed. For a given overcomplete basis matrix, the corresponding sparse solution (coefficient matrix) with minimum l(1) norm is unique with probability one, which can be obtained using a standard linear programming algorithm. The equivalence of the l(1)-norm solution and the l(0)-norm solution is also analyzed according to a probabilistic framework. If the obtained l(1)-norm solution is sufficiently sparse, then it is equal to the l(0)-norm solution with a high probability. Furthermore, the l(1)- norm solution is robust to noise, but the l(0)-norm solution is not, showing that the l(1)-norm is a good sparsity measure. These results can be used as a recoverability analysis of BSS, as discussed. The basis matrix in this article is estimated using a clustering algorithm followed by normalization, in which the matrix columns are the cluster centers of normalized data column vectors. Zibulevsky, Pearlmutter, Boll, and Kisilev (2000) used this kind of two-stage approach in underdetermined BSS. Our recoverability analysis shows that this approach can deal with the situation in which the sources are overlapped to some degree in the analyzed domain and with the case in which the source number is unknown. It is also robust to additive noise and estimation error in the mixing matrix. Finally, four simulation examples and an EEG data analysis example are presented to illustrate the algorithm's utility and demonstrate its performance.     

Parallel QR processing of Generalized Sylvester matrices

In this paper, we develop a parallel QR factorization for the generalized Sylvester matrix. We also propose a significant faster evaluation of the QR applied to a modified version of the initial matrix. This decomposition reveals useful information such as the rank of the matrix and the greatest common divisor of the polynomials formed from its coefficients. We explicitly demonstrate the parallel implementation of the proposed methods and compare them with the serial ones. Numerical experiments are also presented showing the speed of the parallel algorithms.     

A new and fast implementation for null space based linear discriminant analysis

In this paper we present a new implementation for the null space based linear discriminant analysis. The main features of our implementation include: (i) the optimal transformation matrix is obtained easily by only orthogonal transformations without computing any eigendecomposition and singular value decomposition (SVD), consequently, our new implementation is eigendecomposition-free and SVD-free; (ii) its main computational complexity is from a economic QR factorization of the data matrix and a economic QR factorization of a n×n matrix with column pivoting, here n is the sample size, thus our new implementation is a fast one. The effectiveness of our new implementation is demonstrated by some real-world data sets.     

Research of incoherence rotated chaotic measurement matrix in compressed sensing

Measurement matrix construction is the hot issue of compressed sensing. How to construct a measurement matrix of good performance and easy hardware implementation is the main research problem in compressed sensing. In this paper, we present a novel simple and efficient measurement matrix named Incoherence Rotated Chaotic (IRC) matrix. We take advantage of the well pseudorandom of chaotic sequence, introduce the concept of the incoherence factor and rotation, and adopt QR decomposition to obtain the IRC measurement matrix which is suited for sparse reconstruction. The IRC matrix satisfies the Restricted Isometry Property criterion in sparse reconstruction and has a smaller RIP ratio. Simulations demonstrate IRC matrix has better performance than Gaussian random matrix, Bernoulli random matrix, Fourier matrix and can efficiently work on both natural image and remote sensing image. The peak signal-to-noise ratios of reconstructed images using IRC matrix are improved at 1.5 dB to 2.5 dB at least.     

基于光滑l0范数和修正牛顿法的压缩感知重建算法

基于光滑l0范数最小的压缩感知重建算法——SL0算法,通过引入光滑函数序列去逼近l0范数,从而将l0范数最小的问题转化为光滑函数的最优化问题.针对光滑函数的选取以及求解该函数的最优化问题,提出一种基于光滑l0范数和修正牛顿法的重建算法——NSL0算法.首先采用双曲正切函数序列来逼近l0范数,得到一个新的最优化问题;为了提高该优化问题的计算效率,推导出针对双曲正切函数的修正牛顿方向,并采用修正牛顿法进行求解.实验结果表明,在相同的测试条件下,NSL0算法无论在重建效果还是在计算时间方面都明显优于其他同类算法.     

Continuous-Time Varying Complex QR Decomposition via Zeroing Neural Dynamics

Neural Processing Letters - QR decomposition (QRD) is of fundamental importance for matrix factorization in both real and complex cases. In this paper, by using zeroing neural dynamics method, a...