Nonlinear dimensionality reduction aims at providing lowdimensional representions of high-dimensional data sets. Many new methods have been proposed in the recent years, but the question of their assessment and comparison remains open. This paper reviews some of the existing quality measures that are based on distance ranking and K-ary neighborhoods. Many quality criteria actually rely on the a...

Dimensionality reduction aims at providing low-dimensional representations of high-dimensional data sets. Many new nonlinear methods have been proposed for the last years, yet the question of their assessment and comparison remains open. This paper first reviews some of the existing quality measures that are based on distance ranking and K-ary neighborhoods. Next, the definition of the corankin...

This paper presents a novel approach to the analysis of the overtraining phenomenon in pattern classiiers. A nonlinearity measure N is introduced which relates the shape of the classiication function to the generalization capability of a classiier. Experiments using the k-nearest neighbour rule, a neural classiier and the quadratic classiier show that the introduced measure N can be used to stu...

In periodically dispersion managed long-haul transmission systems, waveform distortion is dominated by chromatic dispersion. As a result of the periodic waveform evolution, the nonlinear behavior also repeats itself in every dispersion period. It is shown that, under the weakly nonlinear assumption, nonlinear effects accumulated in a large number (K) of spans can be approximated by nonlinear ef...

The main purpose of this paper is to describe a process for partitioning an N-dimensional population into k sets on the basis of a sample. The process, which is called 'k-means,' appears to give partitions which are reasonably efficient in the sense of within-class variance. That is, if p is the probability mass function for the population, S = {S1, S2, * *, Sk} is a partition of EN, and ui, i ...

1. Summary. In Part I, we develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and...

in this paper we consider the second order nonlinear neutral delay partial difference equation $delta_ndelta_mbig(x_{m,n}+a_{m,n}x_{m-k,n-l}big)+ fbig(m,n,x_{m-tau,n-sigma}big)=b_{m,n}, mgeq m_{0},, ngeq n_{0}.$under suitable conditions, by making use of the banach fixed point theorem, we show the existence of uncountably many bounded positive solutions for the above partial difference equation...

In this paper we consider the Dirichlet problem for a discrete anisotropic equation with some function α , a nonlinear term f , and a numerical parameter λ : ∆ ( α (k) |∆u(k − 1)|p(k−1)−2∆u(k − 1) ) + λf(k, u(k)) = 0, k ∈ [1, T ] . We derive the intervals of a numerical parameter λ for which the considered BVP has at least 1, exactly 1, or at least 2 positive solutions. Some useful discrete ine...

We compare the nonlinear matter power spectrum in real space calculated analytically from third-order perturbation theory with N-body simulations at 1 < z < 6. We find that the perturbation theory prediction agrees with the simulations to better than 1% accuracy in the weakly nonlinear regime in which the dimensionless power spectrum, (k) 1⁄4 k P(k)/2 , which approximately gives the variance of...

This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints test H: h(1) > 0 versus K: 1 E RK. We extend these re...