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219
LIBSVM: a Library for Support Vector Machines
, 2001
"... LIBSVM is a library for support vector machines (SVM). Its goal is to help users can easily use SVM as a tool. In this document, we present all its implementation details. 1 ..."
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Cited by 3412 (62 self)
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LIBSVM is a library for support vector machines (SVM). Its goal is to help users can easily use SVM as a tool. In this document, we present all its implementation details. 1
Reconstruction and Representation of 3D Objects with Radial Basis Functions
 Computer Graphics (SIGGRAPH ’01 Conf. Proc.), pages 67–76. ACM SIGGRAPH
, 2001
"... We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow u ..."
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Cited by 377 (1 self)
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We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBFpreviously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energyminimisation characterisation of polyharmonic splines result in a "smoothest" interpolant. This scaleindependent characterisation is wellsuited to reconstructing surfaces from nonuniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a noninterpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for realworld rangefinder data.
Regularization Theory and Neural Networks Architectures
 Neural Computation
, 1995
"... We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Ba ..."
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Cited by 309 (31 self)
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We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...
Regularization networks and support vector machines
 Advances in Computational Mathematics
, 2000
"... Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization a ..."
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Cited by 266 (33 self)
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Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik’s theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case.
Verbs and Adverbs: Multidimensional Motion Interpolation Using Radial Basis Functions
 IEEE Computer Graphics and Applications
, 1998
"... This paper describes methods and data structures used to leverage motion sequences of complex linked figures. We present a technique for interpolating between example motions derived from live motion capture or produced through traditional animation tools. These motions can be characterized by emoti ..."
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Cited by 262 (5 self)
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This paper describes methods and data structures used to leverage motion sequences of complex linked figures. We present a technique for interpolating between example motions derived from live motion capture or produced through traditional animation tools. These motions can be characterized by emotional expressiveness or control behaviors such as turning or going uphill or downhill. We call such parameterized motions "verbs" and the parameters that control them "adverbs." Verbs can be combined with other verbs to form a "verb graph," with smooth transitions between them, allowing an animated figure to exhibit a substantial repertoire of expressive behaviors. A combination of radial basis functions and low order polynomials is used to create the interpolation space between example motions. Inverse kinematic constraints are used to augment the interpolations in order to avoid, for example, the feet slipping on the floor during a support phase of a walk cycle. Once the verbs and...
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 207 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
An equivalence between sparse approximation and Support Vector Machines
 A.I. Memo 1606, MIT Arti cial Intelligence Laboratory
, 1997
"... This publication can be retrieved by anonymous ftp to publications.ai.mit.edu. The pathname for this publication is: aipublications/15001999/AIM1606.ps.Z This paper shows a relationship between two di erent approximation techniques: the Support Vector Machines (SVM), proposed by V.Vapnik (1995), ..."
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Cited by 205 (7 self)
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This publication can be retrieved by anonymous ftp to publications.ai.mit.edu. The pathname for this publication is: aipublications/15001999/AIM1606.ps.Z This paper shows a relationship between two di erent approximation techniques: the Support Vector Machines (SVM), proposed by V.Vapnik (1995), and a sparse approximation scheme that resembles the Basis Pursuit DeNoising algorithm (Chen, 1995 � Chen, Donoho and Saunders, 1995). SVM is a technique which can be derived from the Structural Risk Minimization Principle (Vapnik, 1982) and can be used to estimate the parameters of several di erent approximation schemes, including Radial Basis Functions, algebraic/trigonometric polynomials, Bsplines, and some forms of Multilayer Perceptrons. Basis Pursuit DeNoising is a sparse approximation technique, in which a function is reconstructed by using a small number of basis functions chosen from a large set (the dictionary). We show that, if the data are noiseless, the modi ed version of Basis Pursuit DeNoising proposed in this paper is equivalent to SVM in the following sense: if applied to the same data set the two techniques give the same solution, which is obtained by solving the same quadratic programming problem. In the appendix we also present a derivation of the SVM technique in the framework of regularization theory, rather than statistical learning theory, establishing a connection between SVM, sparse approximation and regularization theory.
A Theory of Networks for Approximation and Learning
 Laboratory, Massachusetts Institute of Technology
, 1989
"... Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, t ..."
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Cited by 194 (24 self)
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Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nonlinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. Wedevelop a theoretical framework for approximation based on regularization techniques that leads to a class of threelayer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the wellknown Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods suchasParzen windows and potential functions and to several neural network algorithms, suchas Kanerva's associative memory,backpropagation and Kohonen's topology preserving map. They also haveaninteresting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.
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...
The Connection between Regularization Operators and Support Vector Kernels
, 1998
"... In this paper a correspondence is derived between regularization operators used in Regularization Networks and Support Vector Kernels. We prove that the Green's Functions associated with regularization operators are suitable Support Vector Kernels with equivalent regularization properties. Moreover ..."
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Cited by 146 (43 self)
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In this paper a correspondence is derived between regularization operators used in Regularization Networks and Support Vector Kernels. We prove that the Green's Functions associated with regularization operators are suitable Support Vector Kernels with equivalent regularization properties. Moreover the paper provides an analysis of currently used Support Vector Kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a byproduct we show that a large number of Radial Basis Functions, namely conditionally positive definite functions, may be used as Support Vector kernels.