• Title/Summary/Keyword: Kernel Approximation

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Function Approximation Based on a Network with Kernel Functions of Bounds and Locality : an Approach of Non-Parametric Estimation

  • Kil, Rhee-M.
    • ETRI Journal
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    • v.15 no.2
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    • pp.35-51
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    • 1993
  • This paper presents function approximation based on nonparametric estimation. As an estimation model of function approximation, a three layered network composed of input, hidden and output layers is considered. The input and output layers have linear activation units while the hidden layer has nonlinear activation units or kernel functions which have the characteristics of bounds and locality. Using this type of network, a many-to-one function is synthesized over the domain of the input space by a number of kernel functions. In this network, we have to estimate the necessary number of kernel functions as well as the parameters associated with kernel functions. For this purpose, a new method of parameter estimation in which linear learning rule is applied between hidden and output layers while nonlinear (piecewise-linear) learning rule is applied between input and hidden layers, is considered. The linear learning rule updates the output weights between hidden and output layers based on the Linear Minimization of Mean Square Error (LMMSE) sense in the space of kernel functions while the nonlinear learning rule updates the parameters of kernel functions based on the gradient of the actual output of network with respect to the parameters (especially, the shape) of kernel functions. This approach of parameter adaptation provides near optimal values of the parameters associated with kernel functions in the sense of minimizing mean square error. As a result, the suggested nonparametric estimation provides an efficient way of function approximation from the view point of the number of kernel functions as well as learning speed.

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A study on convergence and complexity of reproducing kernel collocation method

  • Hu, Hsin-Yun;Lai, Chiu-Kai;Chen, Jiun-Shyan
    • Interaction and multiscale mechanics
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    • v.2 no.3
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    • pp.295-319
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    • 2009
  • In this work, we discuss a reproducing kernel collocation method (RKCM) for solving $2^{nd}$ order PDE based on strong formulation, where the reproducing kernel shape functions with compact support are used as approximation functions. The method based on strong form collocation avoids the domain integration, and leads to well-conditioned discrete system of equations. We investigate the convergence and the computational complexity for this proposed method. An important result obtained from the analysis is that the degree of basis in the reproducing kernel approximation has to be greater than one for the method to converge. Some numerical experiments are provided to validate the error analysis. The complexity of RKCM is also analyzed, and the complexity comparison with the weak formulation using reproducing kernel approximation is presented.

ASYMPTOTIC APPROXIMATION OF KERNEL-TYPE ESTIMATORS WITH ITS APPLICATION

  • Kim, Sung-Kyun;Kim, Sung-Lai;Jang, Yu-Seon
    • Journal of applied mathematics & informatics
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    • v.15 no.1_2
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    • pp.147-158
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    • 2004
  • Sufficient conditions are given under which a generalized class of kernel-type estimators allows asymptotic approximation on the modulus of continuity. This generalized class includes sample distribution function, kernel-type estimator of density function, and an estimator that may apply to the censored case. In addition, an application is given to asymptotic normality of recursive density estimators of density function at an unknown point.

Asymptotic Approximation of Kernel-Type Estimators with Its Application

  • 장유선;김성래;김성균
    • Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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    • 2003.09a
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    • pp.12.1-12
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    • 2003
  • Sufficient conditions are given under which a generalized class of kernel-type estimators allows asymptotic approximation On the modulus of continuity This generalized class includes sample distribution function, kernel-type estimator of density function, and an estimator that may apply to the censored case. In addition, an application is given to asymptotic normality of recursive density estimators of density function at an unknown point.

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Reproducing kernel based evaluation of incompatibility tensor in field theory of plasticity

  • Aoyagi, Y.;Hasebe, T.;Guan, P.C.;Chen, J.S.
    • Interaction and multiscale mechanics
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    • v.1 no.4
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    • pp.423-435
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    • 2008
  • This paper employs the reproducing kernel (RK) approximation for evaluation of field theory-based incompatibility tensor in a polycrystalline plasticity simulation. The modulation patterns, which is interpreted as mimicking geometrical-type dislocation substructures, are obtained based on the proposed method. Comparisons are made using FEM and RK based approximation methods among different support sizes and other evaluation conditions of the strain gradients. It is demonstrated that the evolution of the modulation patterns needs to be accurately calculated at each time step to yield a correct physical interpretation. The effect of the higher order strain derivative processing zone on the predicted modulation patterns is also discussed.

Localized particle boundary condition enforcements for the state-based peridynamics

  • Wu, C.T.;Ren, Bo
    • Coupled systems mechanics
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    • v.4 no.1
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    • pp.1-18
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    • 2015
  • The state-based peridynamics is considered a nonlocal method in which the equations of motion utilize integral form as opposed to the partial differential equations in the classical continuum mechanics. As a result, the enforcement of boundary conditions in solid mechanics analyses cannot follow the standard way as in a classical continuum theory. In this paper, a new approach for the boundary condition enforcement in the state-based peridynamic formulation is presented. The new method is first formulated based on a convex kernel approximation to restore the Kronecker-delta property on the boundary in 1-D case. The convex kernel approximation is further localized near the boundary to meet the condition that recovers the correct boundary particle forces. The new formulation is extended to the two-dimensional problem and is shown to reserve the conservation of linear momentum and angular momentum. Three numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed approach.

Localized particle boundary condition enforcements for the state-based peridynamics

  • Wu, C.T.;Ren, Bo
    • Interaction and multiscale mechanics
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    • v.7 no.1
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    • pp.525-542
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    • 2014
  • The state-based peridynamics is considered a nonlocal method in which the equations of motion utilize integral form as opposed to the partial differential equations in the classical continuum mechanics. As a result, the enforcement of boundary conditions in solid mechanics analyses cannot follow the standard way as in a classical continuum theory. In this paper, a new approach for the boundary condition enforcement in the state-based peridynamic formulation is presented. The new method is first formulated based on a convex kernel approximation to restore the Kronecker-delta property on the boundary in 1-D case. The convex kernel approximation is further localized near the boundary to meet the condition that recovers the correct boundary particle forces. The new formulation is extended to the two-dimensional problem and is shown to reserve the conservation of linear momentum and angular momentum. Three numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed approach.

Kernel Adatron Algorithm of Support Vector Machine for Function Approximation (함수근사를 위한 서포트 벡터 기계의 커널 애더트론 알고리즘)

  • Seok, Kyung-Ha;Hwang, Chang-Ha
    • The Transactions of the Korea Information Processing Society
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    • v.7 no.6
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    • pp.1867-1873
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    • 2000
  • Function approximation from a set of input-output pairs has numerous applications in scientific and engineering areas. Support vector machine (SVM) is a new and very promising classification, regression and function approximation technique developed by Vapnik and his group at AT&TG Bell Laboratories. However, it has failed to establish itself as common machine learning tool. This is partly due to the fact that this is not easy to implement, and its standard implementation requires the use of optimization package for quadratic programming (QP). In this appear we present simple iterative Kernel Adatron (KA) algorithm for function approximation and compare it with standard SVM algorithm using QP.

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A Comparison on the Differential Entropy

  • Kim, Dae-Hak
    • Journal of the Korean Data and Information Science Society
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    • v.16 no.3
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    • pp.705-712
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    • 2005
  • Entropy is the basic concept of information theory. It is well defined for random varibles with known probability density function(pdf). For given data with unknown pdf, entropy should be estimated. Usually, estimation of entropy is based on the approximations. In this paper, we consider a kernel based approximation and compare it to the cumulant approximation method for several distributions. Monte carlo simulation for various sample size is conducted.

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The Region of Positivity and Unimodality in the Truncated Series of a Nonparametric Kernel Density Estimator

  • Gupta, A.K.;Im, B.K.K.
    • Journal of the Korean Statistical Society
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    • v.10
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    • pp.140-144
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    • 1981
  • This paper approximates to a kernel density estimate by a truncated series of expansion involving Hermite polynomials, since this could ease the computing burden involved in the kernel-based density estimation. However, this truncated series may give a multimodal estimate when we are estiamting unimodal density. In this paper we will show a way to insure the truncated series to be positive and unimodal so that the approximation to a kernel density estimator would be maeningful.

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