Dynamics of on-line gradient descent learning for multilayer neural networks

We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is presented and used to investigate the role of the learning rate in controlling the evolution and convergence of the learning process.

Transients and asymptotics of natural gradient learning

We analyse natural gradient learning in a two-layer feed-forward neural network using a statistical mechanics framework which is appropriate for large input dimension. We find significant improvement over standard gradient descent in both the transient and asymptotic phases of learning.

Natural gradient matrix momentum

Natural gradient learning is an efficient and principled method for improving on-line learning. In practical applications there will be an increased cost required in estimating and inverting the Fisher information matrix. We propose to use the matrix momentum algorithm in order to carry out efficient inversion and study the efficacy of a single step estimation of the Fisher information matrix. We analyse the proposed algorithm in a two-layer network, using a statistical mechanics framework which allows us to describe analytically the learning dynamics, and compare performance with true natural gradient learning and standard gradient descent.

Blurred edges look faint, and faint edges look sharp:The effect of a gradient threshold in a multi-scale edge coding model

A multi-scale model of edge coding based on normalized Gaussian derivative filters successfully predicts perceived scale (blur) for a wide variety of edge profiles [Georgeson, M. A., May, K. A., Freeman, T. C. A., & Hesse, G. S. (in press). From filters to features: Scale-space analysis of edge and blur coding in human vision. Journal of Vision]. Our model spatially differentiates the luminance profile, half-wave rectifies the 1st derivative, and then differentiates twice more, to give the 3rd derivative of all regions with a positive gradient. This process is implemented by a set of Gaussian derivative filters with a range of scales. Peaks in the inverted normalized 3rd derivative across space and scale indicate the positions and scales of the edges. The edge contrast can be estimated from the height of the peak. The model provides a veridical estimate of the scale and contrast of edges that have a Gaussian integral profile. Therefore, since scale and contrast are independent stimulus parameters, the model predicts that the perceived value of either of these parameters should be unaffected by changes in the other. This prediction was found to be incorrect: reducing the contrast of an edge made it look sharper, and increasing its scale led to a decrease in the perceived contrast. Our model can account for these effects when the simple half-wave rectifier after the 1st derivative is replaced by a smoothed threshold function described by two parameters. For each subject, one pair of parameters provided a satisfactory fit to the data from all the experiments presented here and in the accompanying paper [May, K. A. & Georgeson, M. A. (2007). Added luminance ramp alters perceived edge blur and contrast: A critical test for derivative-based models of edge coding. Vision Research, 47, 1721-1731]. Thus, when we allow for the visual system's insensitivity to very shallow luminance gradients, our multi-scale model can be extended to edge coding over a wide range of contrasts and blurs. © 2007 Elsevier Ltd. All rights reserved.

Resonant gradient force on atom interacting with laser field

The gradient force, as a function of position and velocity, is derived for a two-level atom interacting with a standing-wave laser field. Basing on optical Bloch equations, the numerical solutions for the gradient force f_(|_;n) (n = 0, 1, 2, 3, 4, ...) pointing in the direction of the transverse of the laser beam are given. It is shown the higher order gradient force plays important role at strong intensity (G = 64), the contribution of them can not be neglected.

The investigation of the sensitivity gradient of the visual field, as a function of stimulus dynamic range, in the normal and abnormal eye

The study utilized the advanced technology provided by automated perimeters to investigate the hypothesis that patients with retinitis pigmentosa behave atypically over the dynamic range and to concurrently determine the influence of extraneous factors on the format of the normal perimetric sensitivity profile. The perimetric processing of some patients with retinitis pigmentosa was considered to be abnormal in either the temporal and/or the spatial domain. The standard size III stimulus saturated the central regions and was thus ineffective in detecting early depressions in sensitivity in these areas. When stimulus size was scaled in inverse proportion to the square root of ganglion cell receptive field density (M-scaled), isosensitive profiles did not result, although cortical representation was theoretically equivalent across the visual field. It was conjectured that this was due to variations in the ganglion cell characteristics with increasing peripheral angle, most notably spatial summation. It was concluded that the development of perimetric routines incorporating stimulus sizes adjusted in proportion to the coverage factor of retinal ganglion cells would enhance the diagnostic capacity of perimetry. Good general and local correspondence was found between perimetric sensitivity and the available retinal cell counts. Intraocular light scatter arising both from simulations and media opacities depressed perimetric sensitivity. Attenuation was greater centrally for the smaller LED stimuli, whereas the reverse was true for the larger projected stimuli. Prior perimetric experience and pupil size also demonstrated eccentricity-dependent effect on sensitivity. Practice improved perimetric sensitivity for projected stimuli at eccentricities greater than or equal to 30o; particularly in the superior region. Increase in pupil size for LED stimuli enhanced sensitivity at eccentricities greater than 10o. Conversely, microfluctuation in the accommodative response during perimetric examination and the correction of peripheral refractive error had no significant influence on perimetric sensitivity.

Computationally efficient natural gradient descent

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Microarray-formatted clinical biomarker assay development using peptide aptamers to anterior gradient-2

Anterior gradient-2 protein was identified using proteomic technologies as a p53 inhibitor which is overexpressed in human cancers, and this protein presents a novel pro-oncogenic target with which to develop diagnostic assays for biomarker detection in clinical tissue. Combinatorial phage-peptide libraries were used to select 12 amino acid polypeptide aptamers toward anterior gradient-2 to determine whether methods can be developed to affinity purify the protein from clinical biopsies. Selecting phage aptamers through four rounds of screening on recombinant human anterior gradient-2 protein identified two classes of peptide ligand that bind to distinct epitopes on anterior gradient-2 protein in an immunoblot. Synthetic biotinylated peptide aptamers bound in an ELISA format to anterior gradient-2, and substitution mutagenesis further minimized one polypeptide aptamer to a hexapeptide core. Aptamers containing this latter consensus sequence could be used to affinity purify to homogeneity human anterior gradient-2 protein from a single clinical biopsy. The spotting of a panel of peptide aptamers onto a protein microarray matrix could be used to quantify anterior gradient-2 protein from crude clinical biopsy lysates, providing a format for quantitative screening. These data highlight the utility of peptide combinatorial libraries to acquire rapidly a high-affinity ligand that can selectively bind a target protein from a clinical biopsy and provide a technological approach for clinical biomarker assay development in an aptamer microarray format.

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.

A Gradient-Type Optimization Technique for the Optimal Control for Schrodinger Equations

In this paper, we are considered with the optimal control of a schrodinger equation. Based on the formulation for the variation of the cost functional, a gradient-type optimization technique utilizing the finite difference method is then developed to solve the constrained optimization problem. Finally, a numerical example is given and the results show that the method of solution is robust.

Factors Controlling Surface Water Flow in a Low-gradient Subtropical Wetland

Surface water flow patterns in wetlands play a role in shaping substrates, biogeochemical cycling, and ecosystem characteristics. This paper focuses on the factors controlling flow across a large, shallow gradient subtropical wetland (Shark River Slough in Everglades National Park, USA), which displays vegetative patterning indicative of overland flow. Between July 2003 and December 2007, flow speeds at five sites were very low (s−1), and exhibited seasonal fluctuations that were correlated with seasonal changes in water depth but also showed distinctive deviations. Stepwise linear regression showed that upstream gate discharges, local stage gradients, and stage together explained 50 to 90% of the variance in flow speed at four of the five sites and only 10% at one site located close to a levee-canal combination. Two non-linear, semi-empirical expressions relating flow speeds to the local hydraulic gradient, water depths, and vegetative resistance accounted for 70% of the variance in our measured speed. The data suggest local-scale factors such as channel morphology, vegetation density, and groundwater exchanges must be considered along with landscape position and basin-scale geomorphology when examining the interactions between flow and community characteristics in low-gradient wetlands such as the Everglades.

Shifting N and P limitation along a north-south gradient of mangrove estuaries in South Florida

A multivariate statistical analysis was applied to a 10 year, multiparameter data set in an effort to describe the spatial dependence and inherent variation of water quality patterns in the mangrove estuaries of Ten Thousand Islands – Whitewater Bay area. Principal component analysis (PCA) of 16 water quality parameters collected monthly resulted in five groupings, which explained 72.5% of the variance of the original variables. The “Organic” component (PCI) was composed of alkaline phosphatase activity, total organic nitrogen, and total organic carbon; the “Dissolved Inorganic N” component (PCII) contained NO 3 − , NO 2 − , and NH 4 + ; the “Phytoplankton” component (PCIII) was made up of total phosphorus, chlorophyll a, and turbidity; dissolved oxygen and temperature were inversely related (PCIV); and salinity and soluble reactive phosphorus made up PCV. A cluster analysis of the mean and SD of PC scores resulted in the spatial aggregation of the 47 fixed stations into six classes having similar water quality, which we defined as: Mangrove Rivers, Whitewater Bay, Gulf Islands, Coot Bay, Blackwater River, and Inland Waterway. Marked differences in physical, chemical, and biological characteristics among classes were illustrated by this technique. Comparison of medians and variability of parameters among classes allowed large scale generalizations as to underlying differences in water quality in these regions. A strong south to north gradient in estuaries from high N - low P to low N - high P was ascribed to marked differences in landuse, freshwater input, geomorphology, and sedimentary geology along this tract. The ecological significance of this gradient discussed along with potential effects of future restoration plans.