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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...
Endpoint Strichartz estimates
 Amer. J. Math
, 1998
"... Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Stri ..."
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Cited by 266 (36 self)
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Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartztype estimates for more general dispersive equations and for the kinetic transport equation. 1. Introduction. In
Data compression and harmonic analysis
 IEEE Trans. Inform. Theory
, 1998
"... In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory... ..."
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Cited by 140 (24 self)
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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...
The conformal monogenic signal
, 2001
"... The conformal monogenic signal is a novel rotational invariant approach for analyzing i(ntrinsic)1D and i2D local features of twodimensional signals (e.g. images) without the use of any heuristics. It contains the monogenic signal as a special case for i1D signals and combines scalespace, phase, or ..."
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Cited by 99 (35 self)
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The conformal monogenic signal is a novel rotational invariant approach for analyzing i(ntrinsic)1D and i2D local features of twodimensional signals (e.g. images) without the use of any heuristics. It contains the monogenic signal as a special case for i1D signals and combines scalespace, phase, orientation, energy and isophote curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation of the Radon and the Riesz transform. One of the main ideas is to lift up twodimensional signals to a higher dimensional conformal space where the signal can be analyzed with more degrees of freedom. The most interesting result is that isophote curvature can be calculated in a purely algebraic framework without the need of any derivatives.
Multilevel Preconditioning
 Numer. Math
, 1992
"... This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding ..."
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Cited by 94 (18 self)
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This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the BramblePasciakXu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shiftinvariant spaces, in particular, covering those induced by various types of wavelets. Key words: G...
On a Kernelbased Method for Pattern Recognition, Regression, Approximation, and Operator Inversion
, 1997
"... We present a Kernelbased framework for Pattern Recognition, Regression Estimation, Function Approximation and multiple Operator Inversion. Previous approaches such as ridgeregression, Support Vector methods and regression by Smoothing Kernels are included as special cases. We will show connection ..."
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Cited by 77 (25 self)
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We present a Kernelbased framework for Pattern Recognition, Regression Estimation, Function Approximation and multiple Operator Inversion. Previous approaches such as ridgeregression, Support Vector methods and regression by Smoothing Kernels are included as special cases. We will show connections between the costfunction and some properties up to now believed to apply to Support Vector Machines only. The optimal solution of all the problems described above can be found by solving a simple quadratic programming problem. The paper closes with a proof of the equivalence between Support Vector kernels and Greene's functions of regularization operators.
Quantum Electrodynamics of Confined Nonrelativistic Particles
, 1998
"... this paper we establish mathematical results concerning physical phenomena that stood at the origin of quantum theory: that of emission Article No. AI981734 299 00018708#98 #25.00 Copyright # 1998 by Academic Press All rights of reproduction in any form reserved ..."
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Cited by 62 (20 self)
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this paper we establish mathematical results concerning physical phenomena that stood at the origin of quantum theory: that of emission Article No. AI981734 299 00018708#98 #25.00 Copyright # 1998 by Academic Press All rights of reproduction in any form reserved
Localization for random perturbations of periodic Schrödinger operators
 RANDOM OPER. STOCHASTIC EQUATIONS
, 1996
"... We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) ..."
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Cited by 59 (20 self)
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We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schrodinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.