Linear dimensionality reduction: Survey, insights, and generalizations

© 2015 John P. Cunningham and Zoubin Ghahramani. Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of interest, such as covariance, dynamical structure, correlation between data sets, input-output relationships, and margin between data classes. Methods have been developed with a variety of names and motivations in many fields, and perhaps as a result the connections between all these methods have not been highlighted. Here we survey methods from this disparate literature as optimization programs over matrix manifolds. We discuss principal component analysis, factor analysis, linear multidimensional scaling, Fisher's linear discriminant analysis, canonical correlations analysis, maximum autocorrelation factors, slow feature analysis, sufficient dimensionality reduction, undercomplete independent component analysis, linear regression, distance metric learning, and more. This optimization framework gives insight to some rarely discussed shortcomings of well-known methods, such as the suboptimality of certain eigenvector solutions. Modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver, which accepts as input data and an objective to be optimized, and returns, as output, an optimal low-dimensional projection of the data. This simple optimization framework further allows straightforward generalizations and novel variants of classical methods, which we demonstrate here by creating an orthogonal-projection canonical correlations analysis. More broadly, this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox, objective-agnostic numerical technology.

MULTIPLE PERIODIC SOLUTIONS OF ASYMPTOTICALLY LINEAR HAMILTONIAN SYSTEMS VIA CONLEY INDEX THEORY

In this paper we study the existence of periodic solutions of asymptotically linear Hamiltonian systems which may not satisfy the Palais-Smale condition. By using the Conley index theory and the Galerkin approximation methods, we establish the existence of at least two nontrivial periodic solutions for the corresponding systems.

Linearization Method and Linear Complexity

We focus on the relationship between the linearization method and linear complexity and show that the linearization method is another effective technique for calculating linear complexity. We analyze its effectiveness by comparing with the logic circuit method. We compare the relevant conditions and necessary computational cost with those of the Berlekamp-Massey algorithm and the Games-Chan algorithm. The significant property of a linearization method is that it needs no output sequence from a pseudo-random number generator (PRNG) because it calculates linear complexity using the algebraic expression of its algorithm. When a PRNG has n [bit] stages (registers or internal states), the necessary computational cost is smaller than O(2n). On the other hand, the Berlekamp-Massey algorithm needs O(N2) where N ( 2n) denotes period. Since existing methods calculate using the output sequence, an initial value of PRNG influences a resultant value of linear complexity. Therefore, a linear complexity is generally given as an estimate value. On the other hand, a linearization method calculates from an algorithm of PRNG, it can determine the lower bound of linear complexity.

Linear energy transfer dependence of the effects of carbon ion beams on adventitious shoot regeneration from in vitro leaf explants of Saintpaulia ionahta

Purpose: To determine the effects of carbon ion beams with five different linear energy transfer (LET) values on adventitious shoots from in vitro leaf explants of Saintpaulia ionahta Mauve cultivar with regard to tissue increase, shoots differentiation and morphology changes in the shoots. Materials and methods: In vitro leaf explant samples were irradiated with carbon ion beams with LET values in the range of 31 similar to 151 keV/mu m or 8 MeV of X-rays (LET 0.2 keV/mu m) at different doses. Fresh weight increase, surviving fraction and percentage of the explants with regenerated malformed shoots in all the irradiated leaf explants were statistically analysed. Results: The fresh weight increase (FWI) and surviving fraction (SF) decreased dramatically with increasing LET at the same doses. In addition, malformed shoots, including curliness, carnification, nicks and chlorophyll deficiency, occurred in both carbon ion beam and X-ray irradiations. The induction frequency with the former, however, was far more than that with the X-rays. Conclusions: This work demonstrated the LET dependence of the relative biological effectiveness (RBE) of tissue culture of Saintpaulia ionahta according to 50% FWI and 50% SF. After irradiating leaf explants with 5 Gy of a 221 MeV carbon ion beam having a LET value of 96 keV/mu m throughout the sample, a chlorophyll-deficient (CD) mutant, which could transmit the character of chlorophyll deficiency to its progeny through three continuous tissue culture cycles, and plantlets with other malformations were obtained.

Phosphoproteome analysis of mouse liver using immobilized metal affinity purification and linear ion trap mass spectrometry

Since protein phosphorylation is a dominant mechanism of information transfer in cells, there is a great need for methods capable of accurately elucidating sites of phosphorylation. In recent years mass spectrometry has become an increasingly viable alternative to more traditional methods of phosphorylation analysis. The present study used immobilized metal affinity chromatography (IMAC coupled with a linear ion trap mass spectrometer to analyze phosphorylated proteins in mouse liver. A total of 26 peptide sequences defining 26 sites of phosphorylation were determined. Although this number of identified phosphoproteins is not large, the approach is still of interest because a series of conservative criteria were adopted in data analysis. We note that, although the binding of non-phosphorylated peptides to the IMAC column was apparent, the improvements in high-speed scanning and quality of MS/MS spectra provided by the linear ion trap contributed to the phosphoprotein identification. Further analysis demonstrated that MS/MS/MS analysis was necessary to exclude the false-positive matches resulting from the MS/MS experiments, especially for multiphosphorylated peptides. The use of the linear ion trap considerably enabled exploitation of nanoflow-HPLC/MS/MS, and in addition MS/MS/MS has great potential in phosphoproteome research of relatively complex samples. Copyright (C) 2004 John Wiley Sons, Ltd.

A systematic approach to standard addition methods in instrumental analysis

Nine cases of standard addition methods in instrumental analysis, including linear instruments, a non-linear response, the electrochemical Nernst equation, and radiochemical techniques.

土壤侵蚀高维指数非线性模型及其参数辨识

模型验证和数据库组建基础上,用WinEPIC模型定量模拟研究了黄土高原半湿润区长武、半干旱区固原和半干旱偏旱区海原20～30年内苜蓿草地水分生产潜力、10m土层土壤有效含水量和土壤湿度剖面分布特征的动态变化.结果表明:长武、固原和海原苜蓿草地水分生产潜力模拟值随降水量变化而呈现波动性降低趋势,其平均值分别为8.81、3.83和2.48t.hm-2;长武、固原和海原苜蓿草地10m土层逐月土壤有效含水量模拟值均呈现明显的波动性降低趋势,模拟初期,4～8年生苜蓿草地土壤干燥化趋势十分强烈,此后,随降水量变化长期在较低水平上波动;随着苜蓿生长年限的延长,苜蓿草地土壤干层逐年加深、加厚,长武、固原和海原土壤干层分布深度达到10m所需时间依次为6、6和4年,此后苜蓿草地降水渗深以下土层长期维持较为稳定的干燥化状态;苜蓿草地水分持续利用的合理年限为半湿润区8～10年,半干旱区6～8年,半干旱偏旱区4～6年.

Strengthening carbon deposition of polyolefin using combined catalyst as a general method for improving fire retardancy

Combination of Ni2O3 and solid acid with Bronsted acid sites and Lewis acid sites (such as HZSM-5 and H-beta) could dramatically improve fire retardancy of polyolefin, including polypropylene and linear low-density polyethylene. This is mainly attributed to the formation of a large amount of residual char from degradation products of polyolefin in the intermediate stage of combustion. Thus, the amount of flammable components diffusing into the flame zone was small.

THE GENERAL AN INDEX AND ITS APPLICATION .4. THE APPLICATION OF GENERAL AN INDEX TO HYDROCARBONS

The general a(N) index (GAI) was used to characterize the cis, trans isomers of hydrocarbons. The best one-variable equations were obtained with GAI for several physicochemical properties of seven pairs of olefin cis, trans isomers. The linear correlation coefficients are in the range of 0.975 to 0.997. GAI was also compared with the other five topological indices, Randic connectivity index chi, Wiener number W, Hosoya index Z, the average distance sum connectivity J proposed by Balaban and a(N) index introduced by Yang, in correlating with the octane number (MON) of heptanes and octanes.

Dielectric response of graded composites having general power-law-graded cylindrical inclusions

The dielectric response of graded composites having general power-law-graded cylindrical inclusions under a uniform applied electric field is investigated. The dielectric profile of the cylindrical inclusions is modeled by the equation epsilon(i)(r)=c(b+r)(k) (where r is the radius of the cylindrical inclusions and c, b and k are parameters). Analytical solutions for the local electrical potentials are derived in terms of hypergeometric functions and the effective dielectric response of the graded composites is predicted in the dilute limit. Moreover, for a simple power-law dielectric profile epsilon(i)(r) = cr(k) and a linear dielectric profile epsilon(i)(r) = c(b + r), analytical expressions of the electrical potentials and the effective dielectric response are derived exactly from our results by taking the limits b -> 0 and k -> 1, respectively. For a higher concentration of inclusions, the effective dielectric response is estimated by an effective-medium approximation. In addition, we have discussed the effective response of graded cylindrical composites with a more complex dielectric profile of inclusion, epsilon(i)(r)=c(b+r)(k)e(beta r). (c) 2005 American Institute of Physics.

Effective dielectric responses of graded composites of the particles with a general power-law gradation profile

The effective dielectric response of graded spherical composites having general power-law gradient inclusions is investigated under a uniform applied electric field, where the dielectric gradation profile of the spherical inclusions is modeled by the equation epsilon(i) (r) = c(b+r)(k). Analytical solutions of the local electrical potentials are derived in terms of hyper-geometric function and the effective dielectric response of the graded composites is predicted in the dilute limit. From our result, the local potentials of graded spherical composites having both simple power-law dielectric profile epsilon(i)(r) = cr(k) and linear dielectric profile epsilon(i) (r) = c(b+r) are derived exactly by taking the limits b --> 0 and k --> 1, respectively. In the dilute limit, our exact result is used to test the validity of differential effective dipole approximation (DEDA) for estimating the effective response of graded spherical composites, and it is shown that the DEDA is in excellent agreement with exact result. (C) 2005 Elsevier B.V. All rights reserved.

Effective response of a non-linear composite in external AC electric field

A general effective response is proposed for nonlinear composite media, which obey a current field relation of the form J = sigmaE + chi\E\(2) E when an external alternating current (AC) electrical field is applied. For a sinusoidal applied field with finite frequency omega, the effective constitutive relation between the current density and electric field can be defined as, = sigma(e) + chi(e) <\E(x, omega, t)\(2) E(x, omega, t)> + (. . .), where sigma(e) and chi(e) are the general effective linear and nonlinear conductive responses, respectively. The angled brackets <(. . .)> denotes the ensemble average. As two examples, we have investigated the cylindrical and spherical inclusions embedded in a host and also derived the formulae of the general effective linear and nonlinear conductive responses in dilute limit. For higher volume fraction of inclusions, we have proposed a nonlinear effective medium approximation (EMA) method to estimate the general effective response of nonlinear composites in external AC field. Furthermore, the effective nonlinear responses at harmonics are predicted by using the general effective response. (C) 2002 Elsevier Science B.V. All rights reserved.