Fixed-rank matrix factorizations and Riemannian low-rank optimization

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems, and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss the usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with state-of-the-art algorithms and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. © 2013 Springer-Verlag Berlin Heidelberg.

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.

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.

Mercury(II) ion sensing with PVC-matrix based membrane sensors

Thiosemicarbazone derivatives have been used as ion carriers for the preparation of PVC-matrix based mercury(II)-selective membrane sensors. The electrodes give near-Nernstian responses in the linear concentration range of 1.0×10-1-5.0×10-6 M with detection limits of the order of 10-6 M. The stable potentiometric signals are obtained within a short time period of 20-25s. The effect of different plasticizers has been studied and dioctylsebacate (DOS) found to give a better response in comparison to other plasticizers. Selectivity coefficient values (log KPotHg,M) have been evaluated using fixed interference method. Better selectivity for mercury(II) ions is observed over many of the monovalent (Na+, K+ and NH4+) and divalent ions (Mg2+, Ca2+, Zn2+, Pb2+, Ni2+, Co2+, etc.). The sensors have also been used as indicator electrodes in potentiometric titration of mercury(II) ions with EDTA and its determination in synthetic water samples.

Characterization of linear epitope of the beta(2)-microglobulin via combination of MALDI-TOF-MS with immunoaffinity

Matrix-assisted laser desorption ionization time-of-flight mass spectrometry(MALDI-TOF-MS), in combination with immunoaffinity provided a powerful tool for determining epitope (antigenic determinant) in protein. The linear epitope of the beta(2)-microglobulin was characterized in the paper. The method as follows: at first beta(2)-microglobulin was digested by a proteolytic enzyme to produce an appropriate set of peptide fragments, then peptide fragments containing the linear epitope were selected and separated from the pool of peptide fragments by immunoprecipitation with the monoclonal antibody. The agarose beads were collected carefully after the reaction. Unbound peptides would be washed away, while the peptides containing the epitope would remain bound to the immobilized antibody after. the beads were washed several times with appropriate buffer. At last the masses of the bound peptides were identified directly by MALDI-TOF MS. Using Endoproteinase Glu-C Endoproteinase Lys-C and Trypsin in the experiment, the linear epitope of beta(2)-microglobulin was located within peptide fragment 59-69, that is, DWSFYLLYYTE.

Determination of p-toluene sulfonic acid in phenol matrix by capillary zone electrophoresis with ultraviolet detection

The p-toluene sulfonic acid (MA) in phenol matrix was separated and determined by capillary electrophoresis with ultraviolet detector. the effect of the concentration and pH of the buffer on separation was investigated. Cinnamic acid has been chosen as the internal standard from four compounds, the calibration curves of PTSA in 50 mg/L phenol matrix were obtained with and without the internal standard. The linear range was from 1.25 to 12.5 mg/L and the correlation coefficient was 0.9999 for both curves. The limit of detection of PISA was 0.75 mg/L at 3 times of SIN. Finally, the concentration of PTSA in four synthesized samples was determined with method of standard additions, and the effect of matrix was discussed. The values of MA in these samples were 1.01, 0.94, 1.56 and 0.00 mg/L respectively.

Simultaneous spectrophotometric determination of trace iron (III) and aluminum (III) with hybrid linear analysis

Hybrid linear analysis (HLA) was applied to resolution of overlapping spectra of Fe3+-salicylfluorone and Al3+-salicylfluorone complexes and simultaneous spectrophotometric determination of Fe3+ and Al3+. The absorbance matrix of 7 standard mixtures at 41 measuring points ranged from the wavelength of 550 nm to 630 nm was used for calibration. To avoid the effect of interaction between the two components on the determination, the column vector of K matrix obtained from the standard mixtures with least squares was used as the pure spectrum of component. The recoveries of the two elements for the analysis of the synthetic samples were 93.3% similar to 107.5% in the range of the concentration ratio of Fe3+:Al3+ = 10:1 to 1:8. Comparing with the partial least squares (PIS) model, the HLA method was simple, accuracy and precise.

Self-gelatinizable graft copolymer of poly(vinyl alcohol) with 4-vinylpyridine as an immobilization matrix for the construction of a tyrosinase-based amperometric biosensor

A tyrosinase-based amperometric biosensor using a self-gelatinizable graft copolymer of poly(vinyl alcohol) with 4-vinylpyridine (PVA-g-PVP) as an immobilization matrix was constructed. The 4-vinylpyridine component of PVA-g-PVP enhances the adherence to a glassy carbon electrode surface. The content of 4-vinylpyridine in this immobilization matrix plays a key role in retaining the activity of tyrosinase. A simple, milder method was adopted by simply syringing the copolymer-tyrosinase aqueous solution on to the electrode surface and allowing water to evaporate at 4 degrees C in a refrigerator. Several parameters, including copolymer composition; pH, applied potential and enzyme membrane composition, ware optimized. The enzyme membrane composition can be varied to obtain higher sensitivity or a wider linear detection range. The biosensor was used for the determination of phenol, p-cresol and catechol. The biosensor exhibited excellent reproducibility, stability and sensitive response and can be used in flow injection analysis. The biosensor showed an extended linear range in hydrophilic organic solvents and it can be used in monitoring organic reaction processes. The analytical performance demonstrated this immobilization matrix is suitable for the immobilization of tyrosinase.

Blends of linear low-density polyethylene and a diblock copolymer of hydrogenated polybutadiene and methyl methacrylate

Blends of linear low-density polyethylene (LLDPE) and a diblock copolymer of hydrogenated polybutadiene and methyl methacrylate [P(HB-b-MMA)] were studied by transimission electron microscope (TEM), differential scanning calorimetry (DSC), and wide angle X-ray diffraction (WAXD). At 10 wt% block copolymer content, block copolymer chains exist as spherical micelles and cylindrical micelles in LLDPE matrix. At 50 wt% block copolymer content, block copolymer chains mainly form cylindrical micelles. The core and corona of micelles consist of PMMA and PHB blocks, respectively. DSC results show that the total enthalpy of crystallization of the blends varies linearly with LLDPE weight percent, indicating no interactions in the crystalline phase. In the blends, no distortion of the unit cell is observed in WAXD tests.

Seasonal and spatial distribution of matrix-bound phosphine and its relationship with the environment in the Changjiang River Estuary, China

Sediment is commonly considered as a source of phosphine, which is a highly toxic and reactive atmospheric trace gas. This study aims to investigate the seasonal and spatial distribution of matrix-bound phosphine (MBP) and its relationship with the environment in the Changjiang River Estuary. A total of 43 surface sediments were collected in four seasons of 2006, and concentrations of MBP and relative environmental factors were analyzed. MBP ranged from 1.93 to 94.86 ng kg(-1) dry weight (dw) with an average concentration of 17.14 ng kg(-1) dw. The concentrations of MBP in the tipper estuary were, higher than those in the lower estuary, which could be attributed to greater pollutant inputs in the upper estuary. The concentrations of MBP also varied with season, with November > August > May > February. Significant correlations existed between MBP and total phosphorus (TP), organic phosphorus (OP), inorganic phosphorus (W), organic carbon (OC), total nitrogen (TN), the grain size, and redox potential (Eh), suggesting that these sedimentary environmental characteristics played an important role in controlling the MBP levels in the sediments. Notably, there were positive linear relationships between the concentrations of soluble reactive phosphorus (SRP), TP, and chlorophyll a (Chl a) in bottom water and MBP in sediments. These relationships might be very complicated and need further exploration. This work is the first comprehensive study of the seasonal and spatial distribution of MBP in sediments and its relationships with environmental factors in a typical estuary, and will lead to deeper understanding of the phosphorus (P) biogeochemical cycle. (C) 2008 Elsevier Ltd. All rights reserved.

Using oxidized carbon nanotubes as matrix for analysis of small molecules by MALDI-TOF MS

Oxidized carbon nanotubes are tested as a matrix for analysis of small molecules by matrix assisted laser desorption/ionization time of flight mass spectrometry (MALDI-TOF-MS). Compared with nonoxidized carbon nanotubes, oxidized carbon nanotubes facilitate sample preparation because of their higher solubility in water. The matrix layer of oxidized carbon nanotubes is much more homogeneous and compact than that of nonoxidized carbon nanotubes. The efficiency of desorption/ionization for analytes and the reproducibility of peak intensities within and between sample spots are greatly enhanced on the surface of oxidized carbon nanotubes. The advantage of the oxidized carbon nanotubes in comparison with alpha-cyano-4-hydroxycinnamic acid (CCA) and carbon nanotubes is demonstrated by MALDI-TOF-MS analysis of an amino acid mixture. The matrix is successfully used for analysis of synthetic hydroxypropyl P-cyclodextrin, suggesting a great potential for monitoring reactions and for product quality control. Reliable quantitative analysis of jatrorrhizine and palmatine with a wide linear range (1-100 ng/mL) and good reproducibility of relative peak areas (RSD less than 10 %) is achieved using this matrix. Concentrations of jatrorrhizine (8.65 mg/mL) and palmatine (10.4 mg/mL) in an extract of Coptis chinensis Franch are determined simultaneously using the matrix and a standard addition method. (c) 2005 American Society for Mass Spectrometry.

The Construction of Arbitrary Stable Dynamics in Non-Linear Neural Networks

In this paper, two methods for constructing systems of ordinary differential equations realizing any fixed finite set of equilibria in any fixed finite dimension are introduced; no spurious equilibria are possible for either method. By using the first method, one can construct a system with the fewest number of equilibria, given a fixed set of attractors. Using a strict Lyapunov function for each of these differential equations, a large class of systems with the same set of equilibria is constructed. A method of fitting these nonlinear systems to trajectories is proposed. In addition, a general method which will produce an arbitrary number of periodic orbits of shapes of arbitrary complexity is also discussed. A more general second method is given to construct a differential equation which converges to a fixed given finite set of equilibria. This technique is much more general in that it allows this set of equilibria to have any of a large class of indices which are consistent with the Morse Inequalities. It is clear that this class is not universal, because there is a large class of additional vector fields with convergent dynamics which cannot be constructed by the above method. The easiest way to see this is to enumerate the set of Morse indices which can be obtained by the above method and compare this class with the class of Morse indices of arbitrary differential equations with convergent dynamics. The former set of indices are a proper subclass of the latter, therefore, the above construction cannot be universal. In general, it is a difficult open problem to construct a specific example of a differential equation with a given fixed set of equilibria, permissible Morse indices, and permissible connections between stable and unstable manifolds. A strict Lyapunov function is given for this second case as well. This strict Lyapunov function as above enables construction of a large class of examples consistent with these more complicated dynamics and indices. The determination of all the basins of attraction in the general case for these systems is also difficult and open.

Communications inspired linear discriminant analysis

We study the problem of supervised linear dimensionality reduction, taking an information-theoretic viewpoint. The linear projection matrix is designed by maximizing the mutual information between the projected signal and the class label. By harnessing a recent theoretical result on the gradient of mutual information, the above optimization problem can be solved directly using gradient descent, without requiring simplification of the objective function. Theoretical analysis and empirical comparison are made between the proposed method and two closely related methods, and comparisons are also made with a method in which Rényi entropy is used to define the mutual information (in this case the gradient may be computed simply, under a special parameter setting). Relative to these alternative approaches, the proposed method achieves promising results on real datasets. Copyright 2012 by the author(s)/owner(s).

Non-linear electronic transport in Pt nanowires deposited by focused ion beam

Electrical transport and structural properties of platinum nanowires, deposited using the focussed ion beam method have been investigated. Energy dispersive X-ray spectroscopy reveals metal-rich grains (atomic composition 31% Pt and 50% Ga) in a largely non-metallic matrix of C, O and Si. Resistivity measurements (15-300 K) reveal a negative temperature coefficient with the room-temperature resistivity 80-300 times higher than that of bulk Pt. Temperature dependent current-voltage characteristics exhibit non-linear behaviour in the entire range investigated. The conductance spectra indicate increasing non-linearity with decreasing temperature, reaching 4% at 15 K. The observed electrical behaviour is explained in terms of a model for inter-grain tunnelling in disordered media, a mechanism that is consistent with the strongly disordered nature of the nanowires observed in the structure and composition analysis.

On solvability of linear differential equations in ${\Bbb R}^{\Bbb N}$

We construct $x^0$ in ${\Bbb R}^{\Bbb N}$ and a row-finite matrix $T=\{T_{i,j}(t)\}_{i,j\in\N}$ of polynomials of one real variable $t$ such that the Cauchy problem $\dot x(t)=T_tx(t)$, $x(0)=x^0$ in the Fr\'echet space $\R^\N$ has no solutions. We also construct a row-finite matrix $A=\{A_{i,j}(t)\}_{i,j\in\N}$ of $C^\infty(\R)$ functions such that the Cauchy problem $\dot x(t)=A_tx(t)$, $x(0)=x^0$ in ${\Bbb R}^{\Bbb N}$ has no solutions for any $x^0\in{\Bbb R}^{\Bbb N}\setminus\{0\}$. We provide some sufficient condition of solvability and of unique solvability for linear ordinary differential equations $\dot x(t)=T_tx(t)$ with matrix elements $T_{i,j}(t)$ analytically dependent on $t$.