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.

A non-linear perturbation model considering catchment wetness and its application in river flow forecasting

A non-linear perturbation model for river flow forecasting is developed, based on consideration of catchment wetness using an antecedent precipitation index (API). Catchment seasonality, of the form accounted for in the linear perturbation model (the LPM), and non-linear behaviour both in the runoff generation mechanism and in the flow routing processes are represented by a constrained nan-linear model, the NLPM-API. A total of ten catchments, across a range of climatic conditions and catchment area magnitudes, located in China and in other countries, were selected for testing daily rainfall-runoff forecasting with this model. It was found that the NLPM-API model was significantly more efficient than the original linear perturbation model (the LPM). However, restric tion of explicit nan-linearity to the runoff generation process, in the simpler LPM-API form of the model, did not produce a significantly lower value of the efficiency in flood forecasting, in terms of the model efficiency index R-2. (C) 1997 Elsevier Science B.V.

INFLUENCE OF LINEAR ALKYLBENZENE SULFONATE (LAS) AS ORGANIC COSOLVENT ON LEACHING BEHAVIOR OF PCDD/FS FROM FLY-ASH AND SOIL

The leaching of polychlorinated dibenzo-p-dioxins and dibenzofurans (PCDD/Fs) was measured in soil and standard fly ash column eluted with pure water and linear alkylbenzene sulfonate (LAS)- water. The data obtained were used to evaluate the leachability of PCDD/Fs from waste dump like incineration residual slag and fly ash deposition. The leaching rate was shown to be increased significantly by using LAS water. The leachate contents of PCDD/Fs were above their known water solubility. Concentration of PCDD/Fs in the leachates as well as the relative leaching (calculated on the fly ash content) increased with increasing chlorinating degree and decreasing water solubility. LAS above the critical micelle concentration (CMC) probably enhances PCDD/Fs solubility.

GENOTOXIC EFFECTS OF LINEAR ALKYL BENZENE SULFONATE, SODIUM PENTACHLOROPHENATE AND DICHROMATE ON TETRAHYMENA-PYRIFORMIS

DNA in macro- and micronuclei of Tetrahymena pyriformis treated with linear alkyl benzene sulfonate (LAS) and sodium pentachlorophenate (PCP-Na) were determined by microspectrophotometry. The effects on rate of formation of macronuclear DNA extrusion bodies were also studied. We found DNA content of micronuclei in 0.14 ppm LAS and 0.9 ppb PCP-Na was lower than in that of the control, and LAS was able to increase the formation rate of macronuclear DNA extrusion bodies (the formation rate was 54% in 11.3 ppm LAS and 25.6% in 16.7 ppm dichromate). We concluded that 0.14 ppm LAS (below the maximum acceptable toxicant concentration) was genotoxic, whereas 0.014 ppm LAS was not. Dichromate 0.05 ppm and 0.9 ppb PCP-Na, equal to and below the maximum acceptable toxicant concentration, respectively, were potentially genetoxic.

Weak measurement of qubit oscillations with strong response detectors: Violation of the fundamental bound imposed on linear detectors

Qubit measurement by mesoscopic charge detectors has received great interest in the community of mesoscopic transport and solid-state quantum computation, and some controversial issues still remain unresolved. In this work, we revisit the continuous weak measurement of a solid-state qubit by single electron transistors (SETs) in nonlinear-response regime. For two SET models typically used in the literature, we find that the signal-to-noise ratio can violate the universal upper bound "4," which is imposed quantum mechanically on linear-response detectors. This different result can be understood by means of the cross correlation of the detector currents by viewing the two junctions of the single SET as two detectors. Possible limitation of the potential-scattering approach to this result is also discussed.

Linear Rashba Model of a Hydrogenic Donor Impurity in GaAs/GaAlAs Quantum Wells

The Rashba spin-orbit splitting of a hydrogenic donor impurity in GaAs/GaAlAs quantum wells is investigated theoretically in the framework of effective-mass envelope function theory. The Rashba effect near the interface between GaAs and GaAlAs is assumed to be a linear relation with the distance from the quantum well side. We find that the splitting energy of the excited state is larger and less dependent on the position of the impurity than that of the ground state. Our results are useful for the application of Rashba spin-orbit coupling to photoelectric devices.

All-optical continuously tunable delay with a high linear-chirp-rate fiber Bragg grating based on four-wave mixing in a highly-nonlinear photonic crystal fiber

A scheme for hi-fi all-optical continuously tunable delay is proposed. The signal wavelength is converted to a desired idler wavelength and converted back after being delayed by a high linear-chirp-rate (HLCR) fiber Bragg grating (FBG) based on four-wave mixing (FWM) in a highly-nonlinear photonic crystal fiber (HN-PCF). In our experiment, 400 ps (more than 8 full width of half maximum, FWHM) tunable delay is achieved for a 10 GHz clock pulse with relative pulse width broaden ratio (RPWBR) of 2.08%. The power penalty is only 0.3 dB at 10(-9) BER for a 10 Gb/s 2(31)-1 pseudo random bit sequence (PRBS) data. (c) 2009 Elsevier B.V. All rights reserved.

Numerical and theoretical analysis of the crosstalk in linear optical amplifiers

The dynamic characteristics, including the crosstalk and relaxation oscillation, of linear optical amplifiers (LOAs) are investigated by small-signal analysis under an averaging carrier density approximation and compared with the results of numerical simulation. The good agreement between the numerical simulation and the small-signal analysis indicated the averaging carrier density is an appropriate approximation for analyzing LOAs. Theoretical analyzes also show that the dynamic properties of the vertical laser fields dominate the dynamic performance of LOAs. Based on the small-signal analysis, a concise equation for the crosstalk under high bit rate was derived, which can be applied to measure the differential gain of LOAs.

Experimental study of a pulsed ytterbium-doped fibre laser with fast and slow saturable absorbers in a linear cavity

We present a linear-cavity stretched-pulse fibre laser with mode locking by a nonlinear polarization rotation and by semiconductor saturable-absorber mirrors. A Q-switched mode-locking cw train and a mode-locking pulse train are obtained in the experiment. We investigate the effects of the equivalent fast saturable absorber and the slow saturable absorbers in experiment. It is found that neither the nonlinear polarization evolution effect nor a semiconductor saturable absorber mirror is enough to produce the stable cw mode-locking pulses in this experiment. A nonlinear polarization evolution effect controls the cavity loss to literally carve the pulses; semiconductor saturable absorber mirrors provide the self-restarting and maintain the stability of the mode-locking operation.

A 16-term error model based on linear equations of voltage and current variables

Formulation of a 16-term error model, based on the four-port ABCD-matrix and voltage and current variables, is outlined. Matrices A, B, C, and D are each 2 x 2 submatrices of the complete 4 x 4 error matrix. The corresponding equations are linear in terms of the error parameters, which simplifies the calibration process. The parallelism with the network analyzer calibration procedures and the requirement of five two-port calibration measurements are stressed. Principles for robust choice of equations are presented. While the formulation is suitable for any network analyzer measurement, it is expected to be a useful alternative for the nonlinear y-parameter approach used in intrinsic semiconductor electrical and noise parameter measurements and parasitics' deembedding.

Simple reflection technique for measuring the linear electro-optic coefficient of the polymer thin films

The simple reflection technique is usually used to measure the linear electro-optic (EO) coefficient (Pockels coefficient) in the development of EO polymer thin films. But there are some problems in some articles in the determination of the phase shift between the s and p light modes of a laser beam waveguided into the polymer film while a modulating voltage is applied across the electrodes, and different expressions for the linear EO coefficient measured have been given in these articles. In our research, more accurate expression of the linear EO coefficient was deduced by suitable considering the phase shift between the s and p light modes. The linear EO coefficients of several polymer thin films were measured by reflection technique, and the results of the Linear EO coefficient calculated by different expressions were compared. The limit of the simple reflection technique for measuring the linear EO coefficient of the polymer thin films was discussed.

Linear and nonlinear intersubband optical properties in a step quantum well with an applied electric field

Within the framework of the effective-mass envelope-function theory, the field-dependent intersubband optical properties of a Al0.4Ga0.6As/Al0.2Ga0.8As/GaAs step quantum well are investigated theoretically based on the periodic boundary condition. A very large Stark shift occurs when the lowest subband electron remains confined to the small well while the higher subband electron confined to the big well. The optical nonlinearity in a step well due to resonant intersubband transition (ISBT) is analyzed using a density-matrix approach. The second-harmonic generation coefficient chi(2 omega)((2)) and nonlinear optical rectification chi(0)((2)) have also been investigated theoretically. The results show that the ISBT in a step well can generate very large second order optical nonlinearities, chi(0)((2)) and chi(2 omega)((2)) can be tuned by the electric field over a wide range.

High resolution interrogation technique based on linear photodiode array spectrometer for fiber Bragg grating Sensors - art. no. 65952C

A linear photodiode array spectrometer based, high resolution interrogation technique for fiber Bragg grating sensors is demonstrated. Spline interpolation and Polynomial Approximation Algorithm (PAA) are applied to the data points acquired by the spectrometer to improve the original PAA based interrogation method. Thereby fewer pixels are required to achieve the same resolution as original. Theoretical analysis indicates that if the FWHM of a FBG covers more than 3 pixels, the resolution of central wavelength shift will arrive at less than 1 pm. While the number of pixels increases to 6, the nominal resolution will decrease to 0.001 pm. Experimental result shows that Bragg wavelength resolution of similar to 1 pm is obtained for a FBG with FWHM of similar to 0.2 nm using a spectrometer with a pixel resolution of similar to 70 pm.