The Equivalence between Orthogonal Iterations and Alternating Least Squares

This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank-k approximation of a real m×n matrix, A. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, G, this method computes k dominant eigenvectors of G. To see the relation between these methods we assume that G = AT A. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.

Two-Stage Procrustes Rotation with Sparse Target Matrix and Least Squares Criterion with Regularization and Generalized Weighting

In factor analysis, a factor loading matrix is often rotated to a simple target matrix for its simplicity. For the purpose, Procrustes rotation minimizes the discrepancy between the target and rotated loadings using two types of approximation: 1) approximate the zeros in the target by the non-zeros in the loadings, and 2) approximate the non-zeros in the target by the non-zeros in the loadings. The central issue of Procrustes rotation considered in the article is that it equally treats the two types of approximation, while the former is more important for simplifying the loading matrix. Furthermore, a well-known issue of Simplimax is the computational inefficiency in estimating the sparse target matrix, which yields a considerable number of local minima. The research proposes a new rotation procedure that consists of the following two stages. The first stage estimates sparse target matrix with lesser computational cost by regularization technique. In the second stage, a loading matrix is rotated to the target, emphasizing on the approximation of non-zeros to zeros in the target by least squares criterion with generalized weighing that is newly proposed by the study. The simulation study and real data examples revealed that the proposed method surely simplifies loading matrices.

Crashworthiness Analysis on Alternative Square Honeycomb Structure under Axial Loading

Hexagonal metal honeycomb is widely used in energy absorption field for its special construction. However, many other metal honeycomb structures also show good energy absorption characteristics. Currently, most of the researches focus on hexagonal honeycomb, while few are performed into different honeycomb structures. Therefore, a new alternative square honeycomb is developed to expand the non-hexagonal metal honeycomb applications in the energy absorption fields with the aim of designing low mass and low volume energy absorbers. The finite element model of alternative square honeycomb is built to analyze its specific energy absorption property. As the diversity of honeycomb structure, the parameterized metal honeycomb finite element analysis program is conducted based on PCL language. That program can automatically create finite element model. Numerical results show that with the same foil thickness and cell length of metal honeycomb, the alternative square has better specific energy absorption than hexagonal honeycomb. Using response surface method, the mathematical formulas of honeycomb crashworthiness properties are obtained and optimization is done to get the maximum specific energy absorption property honeycomb. Optimal results demonstrate that to absorb same energy, alternative square honeycomb can save 10% volume of buffer structure than hexagonal honeycomb can do. This research is significant in providing technical support in the extended application of different honeycomb used as crashworthiness structures, and is absolutely essential in low volume and low mass energy absorber design.

Alternate Broyden's Method for Solving Linear Least Squares Problem with Multiple Right-Hand Sides

In this paper, we extend the alternate Broyden's method to the multiple version fbi solving lincar leastsquarc systems with multiple right-hand sides. We show that the method possesses property of a finite tcrmination.Some numerical cxperiments are gi von to inustrate the effectiveness of the method.

Alternating Pulse Current in Eiectrocoagulation for Wastewater Treatment to Prevent the Passivation of Al Electrode

A novel current feed style, alternating pulse current, was proposed in electrocoagulation aiming at preventing the passivation of electrode materials. The open circuit potential （OCP） measurements after anodic and cathodic potentiodynamic polarization showed that cathodic polarization could activate Al electrode. The surface of Al electrode after alternating pulse current electrocoagulation was investigated by scanning electron microscope （SEM） and X-ray photoelectron spectroscopy （XPS）, and the results indicate that passivation of Al is not observed. Furthermore, the simulated wastewater treatment tests show that alternating pulse current electrocoagulation has a visible energy saving effect and is worthy of generalization.

ALS算法在菜品智能推荐系统的应用

为了更有效地向用户推荐其感兴趣的菜品,提升餐营业的精准服务水平,本文基于Alternating Least Squares交替最小二乘法和Apache Spark计算引擎,开发了一款根据用户周期性消费行为的菜品智能推荐系统。系统将采集历史消费数据存于HBase并训练推荐模型,通过Thymleaf展示推荐信息。系统采用冷热两种启动模式来满足用户需求。

Soft measurement model on torque of alternating current electrical dynamometer including copper loss and iron loss

Alternating current electrical dynamometer is a common device to measure the torque of engines, such as the gasoline engine. In order to solve the problems such as high cost, high energy consumption and complicated measurement system which exists in the direct measurement on the torque of alternating current electrical dynamometer, copper loss and iron loss are taken as two key factors and a soft-sensing model on the torque of alternating current electrical dynamometer is established using the fuzzy least square support vector machine (FLS-SVM). Then, the FLS-SVM parameters such as penalty factor and kernel parameter are optimized by adaptive genetic algorithm, torque soft-sensing is investigated in the alternating current electrical dynamometer, as well as the energy feedback efficiency and energy consumption during the measurement phase of a gasoline engine loading continual test is obtained. The results show that the minimum soft-sensing error of torque is about 0.0018, and it fluctuates within a range from -0.3 to 0.3 N·m. FLS-SVM soft-sensing method can increase by 1.6% power generation feedback compared with direct measurement, and it can save 500 kJ fuel consumption in the gasoline engine loading continual test. Therefore, the estimation accuracy of the soft measurement model on the torque of alternating current electrical dynamometer including copper loss and iron loss is high and this indirect measurement method can be feasible to reduce production cost of the alternating current electrical dynamometer and energy consumption during the torque measurement phase of a gasoline engine, replacing the direct method of torque measurement.

CubeALS:新的三维协同过滤推荐算法

随着互联网的快速发展,人们对个性化网页搜索、个性化广告投放、个性化社会标注等三维推荐服务的需求越来越紧迫。这些三维立方体数据高度稀疏,且与二维推荐系统相比三维推荐系统中对象之间的关系更加复杂。为了更好地模拟三维对象之间的关系并解决三维数据高度稀疏的问题,提出了一种新的三维协同过滤推荐算法CubeALS(cube alternating least squares)。该算法对三维协同过滤推荐算法CubeSVD(cube singular value decomposition)进行了改进,尝试使用不同于SVD的算法进行矩阵分解。在真实的个性化社会标注数据集上的实验结果表明,与CubeSVD算法相比,CubeALS的性能得到了显著提高。

HPLC结合MCR-ALS测定醋柳黄酮片中有效成分的含量

目的采用高效液相色谱法对醋柳黄酮片中黄酮类化合物进行分析。对其中的山柰酚和异鼠李素进行直接定量;结合化学计量学方法对槲皮素与杂质的重叠峰进行分辨和定量。方法采用C_(18)柱,流动相为甲醇-0.4%磷酸溶液(51:49),柱温30℃,检测波长368 nm,测定山柰酚和异鼠李素的含量。在相同色谱条件下采用MCR-ALS方法对醋柳黄酮片中有效成分槲皮素的重叠色谱峰进行分辨和定量。结果槲皮素在1.92～19.2 mg·L^(-1)(r=0.999 3),山柰酚、异鼠李素在0.40～4.03 mg·L^(-1)(r=0.999 4),4.74～47.40 mg·L^(-1)(r=0.999 4)内线性关系良好,精密度分别为2.2%,2.38%,1.8%(n=6)。槲皮素、山柰酚、异鼠李素平均回收率为96.1%,101.8%,98.0%。结论HPLC-DAD检测结合MCR- ALS可以快速准确的测定醋柳黄酮片中有效成的含量。

MCR-ALS应用于7-乙基-10羟基喜树碱中喜树碱含量的测定

目的:应用多元曲线分辨-交替最小二乘(MCR-ALS)法,对7-乙基-10-羟基喜树碱与喜树碱的色谱重叠峰进行分辨,并对杂质喜树碱进行定量。方法:以渐进因子分析(EFA)解析结果作为初始值对喜树碱与7-乙基-10-羟基喜树碱的色谱重叠峰进行 ALS 迭代优化,直至收敛。结果:采用此方法分辨所得光谱还原率高,定量结果的浓度值与真实值之间线性关系良好。结论:本方法用于药物色谱重叠峰分辨结果可靠。

基于相似用户索引和ALS矩阵分解的推荐算法研究

针对交替最小二乘法(ALS)在处理大数据集时所面临的处理速度和计算资源问题,提出了基于相似用户索引的分布式矩阵分解推荐算法。首先算法基于用户的评分行为找到用户之间的最近邻,然后使用Spark平台运行提出的算法,并产生推荐。在GroupLens网站上提供的MovieLens数据集上进行仿真实验,实验结果表明,提出的算法能够有效解决ALS对于大数据集运行效率低及在云环境中可扩展性较差的问题。

缺失数据建模的改进型ALS在线推荐算法

在隐式反馈中存在数据噪声并缺乏负反馈,使用矩阵分解方法训练推荐模型时特征提取不明确且推荐结果有偏斜。为此,提出一种缺失数据建模的改进型交替最小二乘(ALS)矩阵分解在线推荐算法。使用近邻信息为用户选择正样本,同时根据物品流行度对缺失数据中的负样本进行建模,并将基于元素的改进型ALS算法与在线学习相结合。在MovieLens数据集上的实验结果表明,相对eALS、Rcd算法,该算法能够有效减小数据噪声和矩阵稀疏对矩阵分解推荐算法的影响,从而提高推荐的准确性和效率。

基于Spark平台的ALS加速算法研究

协同过滤推荐算法在推荐系统中发挥着重要作用,但其存在执行效率与排名精度较低的问题,交替最小二乘(ALS)算法可实现并行计算,从而提高执行效率,但是该算法数据加载与迭代收敛的时间较长。为此,将非线性共轭梯度(NCG)算法与ALS算法相结合,提出一种ALS-NCG算法,以达到加速ALS算法的目的。在Spark分布式数据处理环境中对ALS-NCG算法进行性能评估,实验结果表明,相比ALS算法,ALS-NCG算法获取高精度推荐排名时需要的迭代次数与时间更少。

Influence of Alternative Magnetic Field on the Diffusion of Al and Mg

The influence of an alternative magnetic field on the diffusion of Al and Mg in AI-Mg diffusion couple is studied. The diffusion zone is composed of two intermediate phases, namelyβ and γ phase. Thickness of each intermediate phase is examined. The results show that the alternative magnetic field increases the thicknesses of βand γ phase zone and the layer growth ofβ and γphase obeys the parabolic rate law. The growth rate of the β and γ phase are increased with the application of the alternative magnetic field. This change is manifested through a change in the frequency factor k0 and not through a change in the activation energy Q. The frequency factor k0 for intermediate phase growth with an alternative magnetic field is 39.95 cm2/s for 7 phase and 2.84×10-4 cm2/s for β phase compared with those without the magnetic field is 22.4 cm2/s for 7 phase and 1.53×10-4 cm2/s for β phase.

Effect of alternative magnetic field on the diffusion layer growth in Al/Zn couple

The influence of an alternative magnetic field on the growth of the diffusionlayer in Al-Zn diffusion couple was studied. The thickness of the diffusion layer was examined. Theresults show that the alternative magnetic field increases the thickness of the diffusion layer andthe effect increases with the intensity and frequency of the alternative magnetic field increasing. The growth of the diffusion layer obeys the parabolic rate law and the growth rateincreases with the application of the alternative magnetic field. This growth rate change ismanifested through a change in the frequency factor k_0 and not through a change in the activationenergy Q. The frequency factor k_0 for the diffusion layer growth with the alternative magneticfield is 5.03 cm^2/s and the one without the magnetic field is 3.84 cm^2/s.

Squares from D(-4)and D(20)Triples

We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=(L2n+1,5L2n+1,L2n+3), D=(L2n+3,4L2n+1,L2n+3. The sequences A,B,C and D are built from the Fibonacci numbers Fn while the sequences A, B, C and D from the Lucas numbers Ln. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.

Fe-Al络合催化苯乙烯-马来酸酐交替共聚

应用Ｆｅ（ａｃａｃ）３Ａｌ（ｉＢｕ）３（ａｃａｃ＝乙酰丙酮）催化苯乙烯马来酸酐共聚，制得富于交替的白色粉末共聚物．共聚反应动力学研究表明，共聚反应与单体浓度呈一级关系，表观活化能为４８６ｋＪ／ｍｏｌ．

含多类型直流的交直流混联电网潮流计算方法适用性分析

作为电网发展的新阶段,交直流混联电网呈现多类型直流参与、大规模交直流互联的特点,而关于统一迭代和交替迭代2种潮流计算方法的适用性尚未得到深入分析。为此基于含多类型直流的交直流混联电网对2种潮流计算方法的运算性能进行对比研究。推导了含常规直流、柔性直流、混合直流的交直流混联电网潮流模型,进而提出了相应的统一迭代法和交替迭代法。通过3个交直流混联电网测试系统和南方电网实际系统数据验证了潮流模型的有效性和潮流算法的准确性,结合系统负荷水平、系统强度、直流嵌入规模等因素对2种潮流计算方法的收敛性能和计算速度进行对比分析。研究结果表明,在含多类型直流的交直流混联电网中进行潮流计算时,统一迭代法的计算效率比交替迭代法高。

电磁场作用下Al-Mg扩散偶的界面层结构及长大规律

研究了电磁场作用下Al Mg扩散偶的界面结构组成和界面中间相的成长规律。结果表明 ,电磁场作用下Al Mg扩散偶界面中间相 (γ相和β相 )的厚度增加 ,并且界面中间层的厚度随电磁场强度和频率的增加而增加。电磁场作用下界面中间相的成长符合抛物线长大规律。计算了非电磁场与电磁场作用下Al Mg扩散偶界面中间相 (γ相和β相 )的成长常数和激活能 ,电磁场增加了扩散偶界面中间相的成长常数 ,提高了原子扩散过程中原子的跃迁频率及激活熵 ,提高了扩散过程中原子的自由度 ,加快了原子的扩散过程 ,从而加速了中间相的长大速度 。

方波脉冲电流对Al-50%Cu合金定向凝固组织的影响

研究了方波脉冲电流对过共晶Al-50%Cu合金定向凝固组织的影响,主要考察了不同脉冲电流幅值和频率对合金枝晶间距和生长形态的作用效果。试验结果表明,随着脉冲电流幅值和频率增大,一次枝晶间距减小,主要是焦耳热的作用。此外,过共晶Al-50%Cu柱状晶随着脉冲电流的增加出现了V字型生长,失去定向凝固的方向性,这是由于脉冲磁压力和生长速度两方面作用的叠加导致柱状晶的生长方向发生改变的结果。