Narrow Results

It is known that a closed collapsed Riemannian $n$-manifold $(M,g)$ of bounded Ricci curvature and Reifenberg local covering geometry admits a nilpotent structure in the sense of Cheeger–Fukaya–Gromov with respect to a smoothed metric $g(t)$. We study the nilpotent structures over a regular limit space with optimal regularities that describe the collapsing of original metric $g$, and prove that they are uniquely determined up to a conjugation by diffeomorphisms with bi-Lipschitz constant almost $1$, and are equivalent to nilpotent structures arising from other nearby metrics $g_{𝜖}$ with respect to $g_{𝜖}$’s sectional curvature bound.

The theorems of inductive inference in computational learning theory may be interpreted as the ultimate theoretical constraints on the abilities of finite machines to fabricate hypotheses and make predictions from information or data. However, all of the models of learning in this inductive inference literature require the hypothesis to be exactly correct infinitely often, have no way of measuring how far off a prediction of a hypothesis is, and require the hypothesis to make predictions that are directly measurable. Furthermore, most of the models of learning do not allow the learning of uncomputable functions, and of those that are capable of learning uncomputable functions, the literature has not explicitly noticed this property. New notions of learning that rectify these deficiencies are introduced and examined. The first criterion considers a hypothesis successful if each prediction is within δ(x) of the function to be learned on x, f(x). The second criterion considers a hypothesis successful if each prediction on x is equal to f on some domain element within ∊(x) of x, so long as the nearby value of the function f is within v(x) of f(x). These new learning criteria respectively model (a) learning in science with imprecise hypotheses, and (b) learning in science with hypotheses that ignore noisy and bad data, smoothing in image recognition with radius of smoothing ∊ and threshold v, and learning languages (i.e., two-valued functions) with imprecise hypotheses.

We consider principal curves and surfaces in the context of multivariate regression modelling. For predictor spaces featuring complex dependency patterns between the involved variables, the intrinsic dimensionality of the data tends to be very small due to the high redundancy induced by the dependencies. In situations of this type, it is useful to approximate the high-dimensional predictor space through a low-dimensional manifold (i.e., a curve or a surface), and use the projections onto the manifold as compressed predictors in the regression problem. In the case that the intrinsic dimensionality of the predictor space equals one, we use the local principal curve algorithm for the the compression step. We provide a novel algorithm which extends this idea to local principal surfaces, thus covering cases of an intrinsic dimensionality equal to two, which is in principle extendible to manifolds of arbitrary dimension. We motivate and apply the novel techniques using astrophysical and oceanographic data examples.

We introduce kernel smoothing method to extract the global trend of a time series and remove short time scales variations and fluctuations from it. A multifractal detrended fluctuation analysis (MF-DFA) shows that the multifractality nature of TEPIX returns time series is due to both fatness of the probability density function of returns and long range correlations between them. MF-DFA results help us to understand how genetic algorithm and kernel smoothing methods act. Then we utilize a recently developed genetic algorithm for carrying out successful forecasts of the trend in financial time series and deriving a functional form of Tehran price index (TEPIX) that best approximates the time variability of it. The final model is mainly dominated by a linear relationship with the most recent past value, while contributions from nonlinear terms to the total forecasting performance are rather small.

We address the problem of smoothing the probability distribution defined by a finite state automaton. Our approach extends the ideas employed for smoothing n-gram models. This extension is obtained by interpreting n-gram models as finite state models. The experiments show that our smoothing improves perplexity over smoothed n-grams and Error Correcting Parsing techniques.

This work describes a new adaptive rank-order filter, which can be useful for image enhancement or as an early processing stage for segmentation purposes. The filter output is selected among the rank-ordered grey values of the observation window, and the output’s rank depends adaptively on a local measure of the spatial order, to be defined suitably. The aim of this adaptation is to allow the filter to meet conflicting requirements, namely noise and texture smoothing, edge sharpening and enhancement of the fine-structured detail. The definition of a measure of the spatial order is discussed, and experimental results, obtained with natural images of different types, are displayed.

Prediction, smoothing, filtering and synchronization or observer design given finitely many measurements and a given (possibly nonlinear) dynamical map are discussed from a computational complexity point of view. All these problems are particular instances of finding a zero of an appropriately defined function. The recognition of this fact enables one to approach these questions from a computational complexity point of view. For polynomial maps the computational complexity of a global Newton algorithm adapted to identify the finite trajectory of the dynamical system's state over the desired window scales in a polynomial manner with the condition number (an invariant for the problem at hand) and the degree of the polynomials required to describe the models. The computational complexity analysis allows one to identify the most efficient manner to approach synchronization (prediction, smoothing, filtering) problems. Moreover differences between adaptive and nonadaptive formulations are revealed based on the condition number of the associated zero finding problem. The advocated formulation, with the associated global Newton algorithm has good robustness properties with respect to measurement errors and model errors for both adaptive and nonadaptive problems. These aspects are illustrated through a simulation study based around the Hénon map.

We study optimization problems for polyhedral terrains in the presence of data imprecision. An imprecise terrain is given by a triangulated point set where the height component of the vertices is specified by an interval of possible values. We restrict ourselves to terrains with a one-dimensional projection, usually referred to as 1.5-dimensional terrains, where an imprecise terrain is given by an x-monotone polyline, and the y-coordinate of each vertex is not fixed but only constrained to a given interval. Motivated mainly by applications in terrain analysis, in this paper we study five different optimization measures related to obtaining smooth terrains, for the 1.5-dimensional case. In particular, we present exact algorithms to minimize and maximize the total turning angle, as well as to minimize the maximum slope change. Furthermore, we also give approximation algorithms to minimize the largest turning angle and to maximize the smallest turning angle.

In [1], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper, we give a similar criterion for stably odd free knots.

In essence, free knots may be considered as framed 4-graphs. That leads to an important notion of framed 4-graph cobordism and the associated notion of graph genera.

Some additional results on graph and free knot sliceness and cobordism are given.

We call smoothing a self-crossing point of an oriented link diagram self-smoothing. By self-smoothing repeatedly, we obtain an oriented link diagram without self-crossing points. In this paper, we show that every knot has an oriented diagram which becomes a two-component oriented link diagram without self-crossing points by a single self-smoothing.

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators and their classical counterparts are proposed.

A new procedure is proposed for deciding whether a candidate variable is significant in a general nonparametric regression model with independent covariates. A forward selection process is conducted using a formal test of equality of regression curves at each stage. The proposed procedure does not require multidimensional smoothing at any intermediate step. Asymptotic properties are given. Some simulation results and a real application are given.

Empirical modeling of the yield curve is often inconsistent with absence of arbitrage. In fact, many parsimonious models, like the popular Nelson-Siegel model, are inconsistent with absence of arbitrage. In other cases, arbitrage-free models are often used in inconsistent ways by recalibrating parameters that are assumed constant. For these cases, this paper introduces an arbitrage smoothing device to control arbitrage errors that arise in fitting a sequence of yield curves. The device is applied to the US term structure for the family of Nelson-Siegel curves. It is shown that the arbitrage smoothing device contributes to parameter stability and smoothness.

In this paper, we propose a method for matching silhouettes descriptors written following XLWDOS language (Xml Language for Writing Descriptors of Outline Shapes). As we will see, XLWDOS descriptors are sensitive to noise. A Gaussian convolution of 2D silhouettes is applied in order to smooth outline shapes. Depending on the Gaussian scale values, new silhouettes descriptors are obtained and used in the matching process. We also propose a "reduction technique" to process noisy shapes and match corresponding XLWDOS descriptors using only "textual transformations". Experiments, performed on a set of images, show that our matching methods overcome the XLWDOS noise sensitiveness.

The foreground–background separation is an essential part of any video-based surveillance system. Gaussian Mixture Models (GMM) based object segmentation method accurately segments the foreground, but it is computationally expensive. In contrast, single Gaussian-based segmentation is computationally inexpensive but inaccurate because it can not handle the variations in the background. There is a trade-off between computation efficiency and precision in the segmentation approach. From the experimental observations, the variations such as lighting variations, shadows, background motion, etc., affect only a few pixels in the frames in temporal direction. So, unaffected pixel can be modeled by single Gaussian in temporal direction while the affected pixels may need GMM to handle the variations in the background. We propose an adaptive algorithm which models pixel dynamically in terms of number of Gaussians in temporal direction. The proposed method is computationally inexpensive and precise. The flexibility in terms of number of Gaussians used to model each pixel, along with adaptive learning approach, reduces the time complexity of the algorithm significantly. To resolve spacial occlusion problem, a spatial smoothing is carried out by weighted $K_n$ nearest neighbors which improves the overall accuracy of proposed algorithm. To avoid false detection due to illumination variations and shadows in a particular image, illumination invariant representation is used.

The goal of this paper is to describe some applications of fractional order calculus to biomedical signal processing with emphasis on the ability of this mathematical tool to remove noise, enhance useful information, and generate fractal signals. Three types of digital filters are considered, namely, lowpass differentiation filter, smoothing filter, and 1/f^β-noise generation filter. The filter impulse responses are functions of the fractional order and the sampling period only, and thus can be computed easily. Application examples are presented for illustrations.

Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in L²(ℝ^d) which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over ℝ^d, d > 1. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Han's proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.

For semi-linear dissipative wave equation □u + |u_t|^{p - 1}u_t = 0, we consider finite energy solutions with singularities propagating along a focusing light cone. At the tip of cone, the singularities are focused and partially smoothed out under strong nonlinear dissipation, i.e. the solution gets up to 1/2 more L² derivative after the focus. The smoothing phenomenon is in fact the result of simultaneous action of focusing and nonlinear dissipation.

For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in $H^s$ for $s>\frac{1}{2}$. Such smoothing effect persists globally, provided that the $H^1$ norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains $min(2s-1,1)-$ derivatives for $s>\frac{1}{2}$. Following a new simple method, which is of independent interest, we establish that, for $s>1$, $H^s$ norm of a solution grows at most by ${〈t〉}^{s-1+}$ if $H^1$ norm is a priori controlled.

To develop a high-quality TTS system, an appropriate segmentation of continuous speech into the syllabic units plays a vital role. The significant objective of this research work involves the implementation of an automatic syllable-based speech segmentation technique for continuous speech of the Hindi language. Here, the parameters involved in the segmentation process are optimized to segment the speech syllables. In addition to this, the proposed iterative splitting process containing the optimum parameters minimizes the deletion errors. Thus, the optimized iterative incorporation can discard more insertions without merging the frequent non-iterative incorporation. The mixture of optimized iterative and iterative incorporation provides the best accuracy with the least insertion and deletion errors. The segmentation output based on different text signals for the proposed approach and other techniques namely GA, PSO and SOM is accurately segmented. The average accuracy obtained for the proposed approach is high with 97.5% than GA, PSO and SOM. The performance of the proposed algorithm is also analyzed and gives better-segmented accuracy when compared with other state-of-the-art methods. Here, the syllable-based segmented database is suitable for the speech technology system for Hindi in the travel domain.