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68
A tutorial on support vector regression
, 2004
"... In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing ..."
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Cited by 473 (2 self)
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In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.
Support Vector Machines, Reproducing Kernel Hilbert Spaces and the Randomized GACV
, 1998
"... this paper we very briefly review some of these results. RKHS can be chosen tailored to the problem at hand in many ways, and we review a few of them, including radial basis function and smoothing spline ANOVA spaces. Girosi (1997), Smola and Scholkopf (1997), Scholkopf et al (1997) and others have ..."
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Cited by 150 (11 self)
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this paper we very briefly review some of these results. RKHS can be chosen tailored to the problem at hand in many ways, and we review a few of them, including radial basis function and smoothing spline ANOVA spaces. Girosi (1997), Smola and Scholkopf (1997), Scholkopf et al (1997) and others have noted the relationship between SVM's and penalty methods as used in the statistical theory of nonparametric regression. In Section 1.2 we elaborate on this, and show how replacing the likelihood functional of the logit (log odds ratio) in penalized likelihood methods for Bernoulli [yesno] data, with certain other functionals of the logit (to be called SVM functionals) results in several of the SVM's that are of modern research interest. The SVM functionals we consider more closely resemble a "goodnessoffit" measured by classification error than a "goodnessoffit" measured by the comparative KullbackLiebler distance, which is frequently associated with likelihood functionals. This observation is not new or profound, but it is hoped that the discussion here will help to bridge the conceptual gap between classical nonparametric regression via penalized likelihood methods, and SVM's in RKHS. Furthermore, since SVM's can be expected to provide more compact representations of the desired classification boundaries than boundaries based on estimating the logit by penalized likelihood methods, they have potential as a prescreening or model selection tool in sifting through many variables or regions of attribute space to find influential quantities, even when the ultimate goal is not classification, but to understand how the logit varies as the important variables change throughout their range. This is potentially applicable to the variable/model selection problem in demographic m...
Scattered Data Fitting on the Sphere
 in Mathematical Methods for Curves and Surfaces II
, 1998
"... . We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulat ..."
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Cited by 34 (5 self)
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. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multiresolution methods. In addition, we briefly discuss spherelike surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...
Error estimates for scattered data interpolation on spheres
 MATH. COMP
, 1999
"... We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the nsphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error e ..."
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Cited by 34 (4 self)
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We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the nsphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.
Nonstationary Wavelets on the mSphere for Scattered Data
, 1996
"... We construct classes of nonstationary wavelets generated by what we call spherical basis functions (SBFs), which comprise a subclass of Schoenberg 's positive definite functions on the msphere. The wavelets are intrinsically defined on the msphere, and are independent of the choice of coordinate ..."
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Cited by 33 (5 self)
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We construct classes of nonstationary wavelets generated by what we call spherical basis functions (SBFs), which comprise a subclass of Schoenberg 's positive definite functions on the msphere. The wavelets are intrinsically defined on the msphere, and are independent of the choice of coordinate system. In addition, they may be orthogonalized easily, if desired. We will discuss decomposition, reconstruction, and localization for these wavelets. In the special case of the 2sphere, we derive an uncertainty principle that expresses the tradeoff between localization and the presence of high harmonicsor high frequenciesin expansions in spherical harmonics. We discuss the application of this principle to the wavelets that we construct. I. Introduction Geophyiscal or meteorological data collected over the surface of the earth via satellites or ground stations will invariably come from scattered sites. Synthesizing and analyzing such data is the motivation for the work that is pr...
Wavelets Associated with Periodic Basis Functions
"... In this paper, we investigate a class of nonstationary, orthogonal, periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termed periodic basis functions. For this class of wavelets, the decompositio ..."
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Cited by 30 (3 self)
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In this paper, we investigate a class of nonstationary, orthogonal, periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termed periodic basis functions. For this class of wavelets, the decomposition and reconstruction coefficients can be computed in terms of the discrete Fourier transform, so that FFT methods apply for their evaluation. In addition, decomposition at the n th level only involves 2 terms from the higher level. Similar remarks apply for reconstruction. We apply a periodic "uncertainty principle" to obtain an angle/frequency uncertainty "window" for these wavelets, and we show that for many wavelets in this class the angle/frequency localization is good.
Kernel Techniques: From Machine Learning to Meshless Methods
, 2006
"... Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers ..."
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Cited by 28 (7 self)
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Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses nonexpert readers and focuses on practical guidelines for using kernels in applications.
Asymptotics for Minimal Discrete Energy on the Sphere
 Trans. Amer. Math. Soc
"... We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1=r s ; where s ? 0 and r is Euclidean distance. With Ed(s; N) denoting the minimal energy for such Npoint arrangements we obtain bounds (vali ..."
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Cited by 28 (9 self)
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We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1=r s ; where s ? 0 and r is Euclidean distance. With Ed(s; N) denoting the minimal energy for such Npoint arrangements we obtain bounds (valid for all N) for E d (s; N) in the cases when 0 ! s ! d and 2 d ! s. For s = d, we determine the precise asymptotic behavior of Ed(d; N) as N !1. As a corollary, lower bounds are given for the separation of any pair of points in an Npoint minimal energy configuration, when s d 2. For the unit sphere in R 3 (d = 2), we present two conjectures concerning the asymptotic expansion of E 2 (s; N) that relate to the zeta function iL(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of iL (s) when 0 ! s ! 2 (the divergent case). 1 Introduction and ...
The kissing number in four dimensions
, 2005
"... The kissing number problem asks for the maximal number of equal size nonoverlapping spheres that can touch another sphere of the same size in ndimensional space. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimension ..."
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Cited by 23 (8 self)
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The kissing number problem asks for the maximal number of equal size nonoverlapping spheres that can touch another sphere of the same size in ndimensional space. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schütte and van der Waerden. It was proved that the bounds given by Delsarte’s method are not good enough to solve the problem in 4dimensional space. In this paper we present a solution of the problem in dimension four, based on a modification of Delsarte’s method. Keywords: kissing number, contact number, spherical codes, Delsarte’s method, Gegenbauer (ultraspherical) polynomials
Universally optimal distribution of points on spheres
 Journal of the American Mathematical Society
"... Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a ..."
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Cited by 23 (5 self)
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Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m −1)design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E8 and Leech lattices. We also prove the same result for the vertices of the 600cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact twopoint homogeneous spaces, and we