On the Nyström and Column-Sampling Methods for the Approximate Principal Components Analysis of Large Datasets

In this article, we analyze approximate methods for undertaking a principal components analysis (PCA) on large datasets. PCA is a classical dimension reduction method that involves the projection of the data onto the subspace spanned by the leading eigenvectors of the covariance matrix. This projection can be used either for exploratory purposes or as an input for further analysis, for example, regression. If the data have billions of entries or more, the computational and storage requirements for saving and manipulating the design matrix in fast memory are prohibitive. Recently, the Nyström and column-sampling methods have appeared in the numerical linear algebra community for the randomized approximation of the singular value decomposition of large matrices. However, their utility for statistical applications remains unclear. We compare these approximations theoretically by bounding the distance between the induced subspaces and the desired, but computationally infeasible, PCA subspace. Additionally we show empirically, through simulations and a real data example involving a corpus of emails, the trade-off of approximation accuracy and computational complexity.     

Signal analysis based on complex wavelet signs

We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is defined as the fine-scale limit of the signs of a signal's complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature rotates in the complex plane under fractional Hilbert transforms. We show that certain random signals, such as white Gaussian noise and Brownian motions, have a vanishing signature. We derive an appropriate discretization and show the applicability to signal analysis.     

Finite element study for conical indentation of elastoplastic micropolar material

Numerous experiments have repetitively shown that the material behavior presents effective size dependent mechanical properties at scales of microns or submicrons. In this paper, the size dependent behavior of micropolar theory under conical indentation is studied for different indentation depths and micropolar material parameters. To illustrate the effectiveness of the micropolar theory in predicting the indentation size effect (ISE), an axisymmetric finite element model has been developed for elastoplastic contact analysis of the micropolar materials based on the parametric virtual principle. It is shown that the micropolar parameters contribute to describe the characteristic of ISE at different scales, where the material length scale regulates the rate of hardness change at large indentation depth and the value of micropolar shear module restrains the upper limit of hardness at low indentation depth. The simulation results showed that the indentation loads increase as the result of increased material length scale at any indentation depth, however, the rate of increase is higher for lower indentation depth, relative to conventional continuum. The numerical results are presented for perfectly sharp and rounded tip conical indentations of magnesium oxide and compared with the experimental data for hardness coming from the open literature. It is shown that the satisfactory agreement between the experimental data and the numerical results is obtained, and the better correlation is achieved for the rounded tip indentation compared to the sharp indentation.     

Survivors in leader election algorithms

We consider the number of survivors in a broad class of fair leader election algorithms after a number of election rounds. We give sufficient conditions for the number of survivors to converge to a product of independent identically distributed random variables. The number of terms in the product is determined by the round number considered. Each individual term in the product is a limit of a scaled random variable associated with the splitting protocol. The proof is established via convergence (to 0) of the first-order Wasserstein distance from the product limit. In a broader context, the paper is a case study of a class of stochastic recursive equations. We give two illustrative examples, one with binomial splitting protocol (for which we show that a normalized version is asymptotically Gaussian) and one with uniform splitting protocol.     

Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük  (Math Program 105(1):29–53, 2006) are the first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.      

基于改进谱聚类的重叠绿苹果识别方法

水果的可见光谱目标识别是实现农业自动化采摘至关重要的一步。在水果识别的过程中,由于重叠和遮挡的影响使得目标识别困难,识别率不高。本文针对自然环境中果实重叠的识别问题,利用谱聚类算法对图像进行分割,然后使用随机霍夫变换实现果实的识别和定位。针对传统算法运算复杂度高,运算速度慢的问题,本文提出了基于均值漂移和稀疏矩阵原理的改进谱聚类算法。首先使用均值漂移算法对图像进行预分割,均值漂移是一种用于密度梯度的无参估计法。该算法实质是一种迭代,先计算出偏移量,根据偏移量移动点,如此反复,直到偏移量为零即收敛到一点为止。利用均值漂移算法除去大多数的背景像素,为减少谱聚类算法的计算量做准备。然后提取预分割图像的有用信息即图像中像素对之间相似度的描述,将提取的图像特征信息映射到稀疏矩阵中,并使用K-means算法将其分类。得到最终的分类结果,实现对预处理图像的再次分割。然后恢复图像分割区域的颜色,使用彩色向量梯度提取边缘轮廓,对得到的轮廓图像使用随机霍夫变换,并在检测过程中设置半径参数的范围从而进一步加快算法的运行速度。经过检测可以得到目标的圆心坐标和半径,从而实现重叠绿苹果的识别。降低了谱聚类的数据处理量,提高了算法的运行速度。经过试验分析和算法对比,该算法得到较高的重合度95.41%,较低的误差率4.59%和误检率3.05%。     

工件带简单线性恶化函数和共同交货期单机排序问题

本文研究单机排序问题,其中工件加工时间具有简单线性恶化函数.同时,所有工件均具有一个给定共同交货期.目标函数为最小化提前有奖延误受罚之和.在逆一致性条件下,给出了求解该排序问题的一个伪多项式时间动态规划算法.同时借助于几何舍入技巧,对求解这类排序问题给出了一个充分多项式时间的近似算法(FPTAS)。     

Fractional and integral colourings

LetG=(V, E) be an undirected graph andc any vector in ℤ ^V(G) ₊. Denote byχ(G _c) (resp.η(G _c)) the chromatic number (resp. fractional chromatic number) ofG with respect toc. We study graphs for whichχ(G _c)−[η(G _c)]⩽1. We show that for the class of graphs satisfyingχ(G _c)=[η(G _c)] (a class generalizing perfect graphs), an analogue of the Duplication Lemma does not hold. We also describe a 2-vertex cut decomposition procedure related to the integer decomposition property. We use this procedure to show thatχ(G _c)=[η(G _c)] for series-parallel graphs andχ(G _c)⩽[η(G _c)]+1 for graphs that do not have the 4-wheel as a minor. The work of this author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERCC) under grant A9126.     

An upwind finite element scheme for high-Reynolds-number flows

A new upwind finite element scheme for the incompressible Navier-Stokes equations at high Reynolds number is presented. The idea of the upwind technique is based on the choice of upwind and downwind points. This scheme can approximate the convection term to third-order accuracy when these points are located at suitable positions. From the practical viewpoint of computation, the algorithm of the pressure Poisson equation procedure is adopted in the framework of the finite element method. Numerical results of flow problems in a cavity and past a circular cylinder show excellent dependence of the solutions on the Reynolds number. The influence of rounding errors causing Karman vortex shedding is also discussed in the latter problem.