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1,274
Manifold regularization: A geometric framework for learning from labeled and unlabeled examples
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2006
"... We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semisupervised framework that incorporates labeled and unlabeled data in a generalpurpose learner. Some transductive graph learning al ..."
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Cited by 332 (13 self)
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We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semisupervised framework that incorporates labeled and unlabeled data in a generalpurpose learner. Some transductive graph learning algorithms and standard methods including Support Vector Machines and Regularized Least Squares can be obtained as special cases. We utilize properties of Reproducing Kernel Hilbert spaces to prove new Representer theorems that provide theoretical basis for the algorithms. As a result (in contrast to purely graphbased approaches) we obtain a natural outofsample extension to novel examples and so are able to handle both transductive and truly semisupervised settings. We present experimental evidence suggesting that our semisupervised algorithms are able to use unlabeled data effectively. Finally we have a brief discussion of unsupervised and fully supervised learning within our general framework.
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 172 (40 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
Approximation From ShiftInvariant Subspaces of ...
 Trans. Amer. Math. Soc
, 1991
"... : A complete characterization is given of closed shiftinvariant subspaces of L 2 (IR d ) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a spe ..."
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Cited by 130 (38 self)
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: A complete characterization is given of closed shiftinvariant subspaces of L 2 (IR d ) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical StrangFix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shiftinvariant space is shown to be already realized by a specifiable principal subspace. AMS (MOS) Subject Classifications: 41A25, 41A63; 41A30, 41A15, 42B99, 46E30 Key Words and phrases: approximation order, StrangFix conditions, shiftinvariant spaces, radial basis functions, orthogonal projection. Authors' affiliation and address: 1 Center for Mathematical Sciences University of WisconsinMadison 610 Walnut St. Madison WI 53705 and 2 Department of Mathematics University of South Carolina Columbia SC 29208 This work...
Multilayer Feedforward Networks With a Nonpolynomial Activation Function Can Approximate Any Function
, 1993
"... Several researchers characterized the activation fimction under which multilayer feedforward networks can act as universal approximators. We show that most of all the characterizations that were reported thus far in the literature are special cases of the following general result: A standard multila ..."
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Cited by 117 (2 self)
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Several researchers characterized the activation fimction under which multilayer feedforward networks can act as universal approximators. We show that most of all the characterizations that were reported thus far in the literature are special cases of the following general result: A standard multilayer feedforward network with a locally bounded piecewise continuous activation fimction can approximate an3, continuous function to any degree of accuracy if and only if the network's activation function is not a polynomial. We also emphasize the important role of the threshold, asserting that without it the last theorem does not hold.
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...
Biorthogonal SplineWavelets on the Interval  Stability and Moment Conditions
 Appl. Comp. Harm. Anal
, 1997
"... This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and co ..."
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Cited by 84 (46 self)
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This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly...
Wavelets on Manifolds I: Construction and Domain Decomposition
 SIAM J. Math. Anal
, 1998
"... The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features ar ..."
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Cited by 80 (21 self)
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The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...
Composite Wavelet Bases for Operator Equations
 MATH. COMP
, 1996
"... This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary ..."
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Cited by 77 (21 self)
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This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudodifferential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions.
Balancing Domain Decomposition For Mixed Finite Elements
 Math. Comp
"... . The rate of convergence of the Balancing Domain Decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed. The Balancing Domain Decomposition method, introduced by Mandel in [24], is a substructuring method that involves at each iteratio ..."
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Cited by 71 (20 self)
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. The rate of convergence of the Balancing Domain Decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed. The Balancing Domain Decomposition method, introduced by Mandel in [24], is a substructuring method that involves at each iteration the solution of a local problem with Dirichlet data, a local problem with Neumann data, and a "coarse grid" problem to propagate information globally and to insure the consistency of the Neumann problems. It is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size, in both two and three dimensions and for elements of arbitrary order. The bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains. The key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the H 1=2 norm of an interpolant of the boundary ...