自适应图正则化的低秩非负矩阵分解算法

图正则化(nonnegative matrix factorization,NMF)算法(graph regularization nonnegative matrix factorization,GNMF)仍存在一些不足之处:GNMF算法并没有考虑数据的低秩结构;在GNMF算法中,其拉普拉斯图是使用K近邻(K nea...     

Successively alternate least square for low-rank matrix factorization with bounded missing data

The problem of low-rank matrix factorization with missing data has attracted many significant attention in the fields related to computer vision. The previous model mainly minimizes the total errors of the recovered low-rank matrix on observed entries. It may produce an optimal solution with less physical meaning. This paper gives a theoretical analysis of the sensitivities of the original model and proposes a modified constrained model and iterative methods for solving the constrained problem. We show that solutions of original model can be arbitrarily far from each others. Two kinds of sufficient conditions of this catastrophic phenomenon are given. In general case, we also give a low bound of error between an ?-optimal solution that is practically obtained in computation and a theoretically optimal solution. A constrained model on missing entries is considered for this missing data problem. We propose a two-step projection method for solving the constrained problem. We also modify the method by a successive alternate technique. The proposed algorithm, named as SALS, is easy to implement, as well as converges very fast even for a large matrix. Numerical experiments on simulation data and real examples are given to illuminate the algorithm behaviors of SALS.     

Image registration via low-rank factorization and maximum rank resolving

Multimedia Tools and Applications - The feature-based image registration method has a better performance in terms of robustness to the intensity variance, but its accuracy of the feature-based...     

Fast QR factorization of low-rank changes of Vandermonde-like matrices

This paper is concerned with the solution of linear systems with coefficient matrices which are Vandermonde-like matrices modified by adding low-rank corrections. Hereafter we refer to these matrices as modified Vandermonde-like matrices. The solution of modified Vandermonde-like linear systems arises in approximation theory both when we use Remez algorithms for finding minimax approximations and when we consider least squares problems with constraints. Our approach relies on the computation of an inverse QR factorization. More specifically, we show that some classical orthogonalization schemes for m×n, m≥n, Vandermonde-like matrices can be extended to compute efficiently an inverse QR factorization of modified Vandermonde-like matrices. The resulting algorithm has a cost of O(mn) arithmetical operations. Moreover it requires O(m) storage since the matrices Q and R are not stored. Received: January 1997 / Accepted: November 1997     

Approximate factorization for a viscous wave equation

A general procedure to construct ADI methods for multidimensional problems was originated by Beam and Warming using the method of approximate factorization. In this paper, we extend the method of approximate factorization to solve a viscous wave equation. The method can be combined with any implicit linear multistep method for the time integration of the wave equation. The stability of the factored schemes and their underlying schemes is analyzed based on a discrete Fourier analysis and the energy method. Convergence proofs are presented and numerical results supporting our analysis are provided.     

结合低秩和结构化稀疏的大雾图像小目标检测

针对传统的低秩稀疏分解模型不能直接应用到单幅图像进行目标检测,且忽略了目标像素的空间结构性导致检测精度不高等问题,提出一种基于低秩和结构化稀疏的单幅大雾图像小目标检测算法。首先,对原始大雾图像进行预处理得到由局部子图像构成的大雾补片图像,将小目标检测问题转化为低秩和稀疏分解问题。然后,考虑到目标像素间的空间结构关系,在对大雾补片图像进行矩阵分解时,引入结构化稀疏诱导范数对目标进行约束。最后,将矩阵分解得到的补片图像进行后处理得到背景图像和目标图像。通过对单幅大雾图像实验仿真表明,所提算法确保了小目标检测的完整性并且提高了检测精度。     

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.     

基于可靠性低秩因子分解和泛化差异性差分的运动目标检测

运动目标检测旨在分离视频的背景与前景,然而常用的低秩因子分解法往往难以综合地处理动态背景和间歇性运动的问题。考虑到背景减除后的偏态噪声分布具有潜在的背景修正作用,提出一种基于可靠性低秩因子分解和泛化差异性差分的运动目标检测模型。首先,利用时间维度像素分布的峰值位置以及偏态分布性质选取一个不含离群像素的子序列,并计算该子序列的中值以形成静态背景;其次,利用非对称拉普拉斯分布对静态背景减除后的噪声建模,并把基于空间平滑的建模结果作为可靠性权重参与到低秩因子分解中,以此建模综合背景（含有动态背景）;最后,依次利用时间和空间连续约束提取前景。其中,针对时间连续性,提出了泛化差异性差分约束,从而通过相邻视频帧的差异信息抑制前景边缘的扩增。实验结果表明,与PCP、DECOLOR、LSD、TVRPCA、E-LSD、GSTO六种模型相比,所提模型的F-measure值最高。由此可知,所提模型在动态背景、间歇性运动等复杂场景中能有效提高前景的检测精度。     

A structured low-rank wavelet solver for the Ornstein-Zernike integral equation

In this article, we present a new structured wavelet algorithm to solve the Ornstein-Zernike integral equation for simple liquids. This algorithm is based on the discrete wavelet transform of radial distribution functions and different low-rank approximations of the obtained convolution matrices. The fundamental properties of wavelet bases such as the interpolation properties and orthogonality are employed to improve the convergence and speed of the algorithm. In order to solve the integral equation we have applied a combined scheme in which the coarse part of the solution is calculated by the use of wavelets and Newton-Raphson algorithm, while the fine part is solved by the direct iteration. Tests have indicated that the proposed procedure is more effective than the conventional method based on hybrid algorithms.     

Approximate factorization of multivariate polynomials using singular value decomposition

We describe the design, implementation and experimental evaluation of new algorithms for computing the approximate factorization of multivariate polynomials with complex coefficients that contain numerical noise. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use singular value decomposition or structured total least squares approximation and Gauss–Newton optimization to numerically compute the approximate multivariate factors. We demonstrate on a large set of benchmark polynomials that our algorithms efficiently yield approximate factorizations within the coefficient noise even when the relative error in the input is substantial (10⁻³).     

A class of multilevel structured low-rank approximation arising in material processing

This work is devoted to studying the problem of approximating a given matrix by the product of two low-rank structured matrix which arises in the material processing. Firstly, we analyse some properties of the original problem and utilize the alternating least squares method to reformulate it into two subproblems. Then by using the Gramian representation and a Vandermonde-like transformation, the feasible sets of the subproblems are characterized, which makes them be respectively transformed into two unconstrained minimization problems. Finally, we derive the expressions of the gradients of the objective functions and apply the gradient descent algorithm to solve them. Numerical examples are tested to show that our method performs well in terms of the residual error of the objective function.     

Efficient preconditioning for noisy separable nonnegative matrix factorization problems by successive projection based low-rank approximations

The successive projection algorithm (SPA) can quickly solve a nonnegative matrix factorization problem under a separability assumption. Even if noise is added to the problem, SPA is robust as long as the perturbations caused by the noise are small. In particular, robustness against noise should be high when handling the problems arising from real applications. The preconditioner proposed by Gillis and Vavasis (SIAM J Optim 25(1):677–698, 2015) makes it possible to enhance the noise robustness of SPA. Meanwhile, an additional computational cost is required. The construction of the preconditioner contains a step to compute the top-k truncated singular value decomposition of an input matrix. It is known that the decomposition provides the best rank-k approximation to the input matrix; in other words, a matrix with the smallest approximation error among all matrices of rank less than k. This step is an obstacle to an efficient implementation of the preconditioned SPA. To address the cost issue, we propose a modification of the algorithm for constructing the preconditioner. Although the original algorithm uses the best rank-k approximation, instead of it, our modification uses an alternative. Ideally, this alternative should have high approximation accuracy and low computational cost. To ensure this, our modification employs a rank-k approximation produced by an SPA based algorithm. We analyze the accuracy of the approximation and evaluate the computational cost of the algorithm. We then present an empirical study revealing the actual performance of the SPA based rank-k approximation algorithm and the modified preconditioned SPA.     

Approximate factorization of the discrete Navier-Stokes equations in Cartesian and cylindrical coordinates

A new pseudospectral technique for integrating incompressible Navier-Stokes equations with one nonperiodic boundary in Cartesian or cylindrical coordinate system is presented. Algorithm constructed makes use of Chebyshev collocation technique in nonperiodic direction. Special attention is paid to the approximate factorization of the discrete Navier-Stokes equations in cylindrical geometry leading to highly fast and robust numerical procedure providing spectral accuracy. New approach is an efficient tool for further investigation of turbulent shear flows, for physical hypotheses and alternative algorithms testing. Classical problems of incompressible fluid flows in an infinite plane channel and annuli at transitional Reynolds numbers are taken as model ones.     

基于自适应图正则化与联合低秩矩阵分解的数字文化遗产多标签众包答案聚合方法

多标签答案聚合问题是通过融合众包收集的大量非专家标注来估计样本的真实标签,由于数字文化遗产数据具有标注成本高、样本类别多、分布不均衡等特点,给数据集多标签答案聚合问题带来了极大挑战。以往的方法主要集中在单标签任务,忽视了多标签任务的标签关联性;大部分多标签聚合方法虽然在一定程度上考虑了标签相关性,但是很敏感地受噪声和离群值的影响。为解决这些问题,提出一种基于自适应图正则化与联合低秩矩阵分解的多标签答案聚合方法AGR-JMF。首先,将标注矩阵分解成纯净标注和噪声标注两部分;对纯净标注采用自适应图正则化方法构建标签间的关联矩阵;最后,利用标注质量、标签关联性、标注人员行为属性相似性等信息指导低秩矩阵分解,以实现多标签答案的聚合。真实数据集和莫高窟壁画数据集上的实验表明,AGR-JMF相较于现有算法在聚合准确率、识别欺诈者等方面具有明显优势。     

A study of the gap between the structured singular value and itsconvex upper bound for low-rank matrices

The size of the smallest structured destabilizing perturbation for a linear time-invariant system can be calculated via the structured singular value (μ). The function μ can be bounded above by the solution of a convex optimization problem, and in general there is a gap between μ and the convex bound. This paper gives an alternative characterization of μ which is used to study this gap for low-rank matrices. The low-rank characterization provides an easily computed bound which can potentially be significantly better than the standard convex bound. This is used to find new examples with larger gaps than previously known     

A generalized Schur-type algorithm for the joint factorization of a structured matrix and its inverse

A Schur-type algorithm is presented for the simultaneous triangular factorization of a given (non-degenerate) structured matrix and its inverse. The algorithm takes the displacement generator of a Hermitian, strongly regular matrixR as an input, and computes the displacement generator of the inverse matrixR ⁻¹ as an output. From these generators we can directly deduce theLD ⁻¹ L ^* (lower-diagonal-upper) decomposition ofR, and theUD ⁻¹ U ^* (upper-diagonallower) decomposition ofR ⁻¹. The computational complexity of the algorithm isO(rn ²) operations, wheren andr denote the size and the displacement rank ofR, respectively. Moreover, this method is especially suited for parallel (systolic array) implementations: usingn processors the algorithm can be carried out inO(n) steps.     

Obstacles to factorization of partial differential operators into several factors

In the paper, construction of algorithms for factoring linear partial differential operators is studied. New theoretical concepts—obstacle and ring of obstacles—are introduced. These are invariants and posses other interesting properties. A theorem on unique factorization extension starting from some moment, which is important from the standpoint of construction of such algorithms, is proved. Obstacles in the case of the second and third degrees are found.     

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.     

A quasi-Newton method with Cholesky factorization

A quasi-Newton method for unconstrained minimization is presented, which uses a Cholesky factorization of an approximation to the Hessian matrix. In each step a new row and column of this approximation matrix is determined and its Cholesky factorization is updated. This reduces storage requirements and simplifies the calculation of the search direction. Precautions are taken to hold the approximation matrix positive definite. It is shown that under usual conditions the method converges superlinearly or evenn-step quadratic.