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Topic: Reproducing kernel

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Reproducing kernel Hilbert space

Hilbert space

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	ĐĎ ŕˇ± á > ţ˙
		It is important to mention this because it shows that the discovery of the reproducing kernel property was not made by accident, it is well settled in Zaremba’s mathematical work.
		And the applications are diverse, and sometimes contrasting: function theory (mostly thought of as in complex variable), differential equations, functional analysis and operator theory (with application to control theory), harmonic analysis (including magnetic resonance tomography), stationary processes and mathematical physics, as to mention some of them.
		Thus the conference 90 years of the reproducing kernel property (April, 16-21) was organized by the Chair of Functional Analysis.
	www.uj.edu.pl /IRO/NEWSLET/IRO13/SZAFR-P.html (685 words)

	Reproducing kernel Hilbert space - Wikipedia, the free encyclopedia
		In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a function space in which pointwise evaluation is a continuous linear functional.
		This is because many of the examples of reproducing kernel Hilbert spaces are spaces of analytic functions.
		H is a reproducing kernel Hilbert space iff the linear map
	en.wikipedia.org /wiki/Reproducing_kernel_Hilbert_space (305 words)

	Support Vector Machines - The Book
		Reproducing kernels were extensively used in machine learning and neural networks by Poggio and Girosi from the early 1990s.
		Scholekopf, Watkins and Haussler have greatly extended the use of kernels, showing that they can in fact be defined on general sets, which do not need to be Euclidean spaces, paving the way for their use in a swathe of new real-world applications, on input spaces as diverse as biological sequences, text, and images.
		Amari and Wu describes a method for directly acting on the kernel in order to affect the geometry in the input space, so that the separability of the data is improved.
	www.support-vector.net /chapter_3.html (546 words)

	Reproducing Kernel Hilbert Space (RKHS) method.
		However, in contrast to the delta function, the reproducing kernels are continuous bounded functions and can be tailored to carry physical information on the particular system or problem [26].
		Reproducing kernels can in general be constructed from any complete system of linearly independent or compact functions (e.g., orthogonal polynomials [5] and wavelets [27]).
		Smooth global multi-dimensional reproducing kernels have been successfully utilized in other contexts for multivariate interpolation (e.g., in computer aided geometric design [28, 29]) and to solve differential equations by collocation [30].
	www.dur.ac.uk /j.m.hutson/ccp6-98/node28.html (568 words)

	1961
		One of the high points of this paper is introduction of reproducing kernel Hilbert spaces as a tool for the analysis of time series.
		The main motivation for introduction of reproducing kernel Hilbert spaces is to give a canonical map from time series to Hilbert spaces.
		Parzen also shows how to relate reproducing kernel Hilbert spaces to Karhunen-Loeve theorems, which are simply tools to write down the Hilbert space in a particular basis.
	www.io.com /~slava/history/1961.htm (318 words)

	AHPCRC Preprint Abstracts (Site not responding. Last check: 2007-10-10)
		In the applications of wavelet adaptivity, the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the Helmholtz equation, a model equation of wave propagation problems, and to simulate shear band formation in an elasto-viscoplastic material, a problem dictated by the presence of the high gradient deformation.
		A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart.
		Using the reproducing kernel shape functions, the domain of the workpiece is discretized by a set of particles without the employment of a structured mesh.
	www.ahpcrc.org /publications/preprints/abstracts99.html (13521 words)

	Singer Seminar 1998 (Site not responding. Last check: 2007-10-10)
		SPH, a particle-based numerical scheme first devised for astrophysical problems, is based on the notion of an interpolating kernel function that mimics the properties of a delta function.
		In doing so, the kernel function allows integrals to be approximated as a summation over a finite number of points (i.e., particles).
		The reproducing kernel particle method is also a numerical scheme based on a so-called reproducing kernel.
	www-fp.mcs.anl.gov /division/information/seminars/1998/singer_981210.htm (211 words)

	Wishart Processes: A Statistical View of Reproducing Kernels and Its Applications to Kernel Learning - Zhang, Yeung, ... (Site not responding. Last check: 2007-10-10)
		In this paper, we propose a statistical view of kernels and establish a new connection between reproducing kernels and Gaussian processes.
		Specifically, we draw equivalence between two notions, that the reproducing kernel is a Wishart process and that the dimensions of the feature vectors in the kernel-induced feature space are mutually independent Gaussian processes.
		1 the extensions of kernel alignment (context) - Kandola, Shawe-Taylor et al.
	citeseer.ist.psu.edu /696933.html (623 words)

	Potential field collocation and densisty (Site not responding. Last check: 2007-10-10)
		However, the kernel is not a correct reproducing kernel, because the function for x = y becomes infinite.
		In a reproducing kernel Hilbert space the problem of determining an approximation to f from n observations is easily solved if the observations are related to f in a linear manner, i.e.
		Tscherning, C.C.: Isotropic reproducing kernels for the inner of a
	www.gfy.ku.dk /~cct/moensted.htm (3076 words)

	Citebase - A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application
		A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application
		(E), we consider a reproducing kernel Hilbert space \mathcal{H} with a reproducing kernel A(x,y).
		the usual basis of \mathcal{H} Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space \mathcal{H} with a kernel function B(x,y), which is the representation of the inverse of A in a sense, so that \mathcal{H} ⊃\mathcal{H} ⊃\mathcal{H} becomes a rigged Hilbert space.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0506189 (614 words)

	no-title (Site not responding. Last check: 2007-10-10)
		Reproducing Kernel Hilbert Spaces Outline, by Manny Parzen
		An Introduction to the Theorey of Reproducing Kernel Hilbert Spaces by Vern I. Paulsen.
		We discussed broadly covariance kernel and Gaussian process, construction of RKHS from positive definite kernel, Mercer's theorem.
	www.stat.tamu.edu /~jianhua/rkhs (202 words)

	The Kernel Trick
		K(x,w) is a kernel in a reproducing kernel Hilbert space or rushed to the library or to the guy next door to find out, and probably very soon after that said aha!, K(x,w) is the kernel of a RKHS.
		The kernel function becomes useful for choosing the classification boundary but even that could be empirically approximated.
		The point is that it is obvious that a choice of kernel function is an ad-hoc way of sweeping under the rug prior information into the problem, indutransductibly (!) ducking the holy Bayes Theorem.
	omega.albany.edu:8008 /machine-learning-dir/notes-dir/ker1/ker1-l.html (907 words)

	Information Bridge: DOE Scientific and Technical Information
		Based on the reproducing kernel particle method on enrichment procedure is introduced to enhance the effectiveness of the finite element method.
		The basic concepts for the reproducing kernel particle method are briefly reviewed.
		By adopting the well-known completeness requirements, a generalized form of the reproducing kernel particle method is developed.
	www.osti.gov /bridge/product.biblio.jsp?osti_id=82192 (210 words)

	Subspace Classifier in Reproducing Kernel Hilbert Space (ResearchIndex)
		Abstract: To improve the performance of subspace classifier, it is effective to reduce the dimensionality of the intersections between subspaces.
		For this purpose, the feature space is mapped implicitly to a high dimensional reproducing kernel Hilbert space and the subspace classifier is applied in this space.
		4 Support vector machines, reproducing kernel hilbert spaces,..
	citeseer.ist.psu.edu /262206.html (272 words)

	CMM Manuscript Search Results
		A necessary condition for obtaining a reproducing kernel interpolation function is an orthogonality condition between the vector of enrichment functions and the vector of shifted monomial functions at the discrete points.
		A normalized kernel function with relative small support is employed as the primitive function.
		To maintain the convergence properties of the original reproducing kernel approximation, a mixed interpolation is introduced.
	www.sonic.net /cgi-bin/usacm/MeshFree/request_preprints.pl (997 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		While I don't know what distinguishes a Bergman kernel from others, kernel functions arise in solution of many partial differential equations by their conversion to integral equations.
		Once a conversion has been completed the integral equations can be numerically attacked (in the general case) by variational methods or others such as Galerkin methods.
		A reproducing kernel is one which, when applied to a member of a certain class of functions, reproduces the member function.
	www.math.niu.edu /~rusin/known-math/98/kernel (146 words)

	[No title]
		A general tangential interpolation problem will be posed for the class of such functions and the set of all solutions characterized in terms of a certain positive kernel constructed from the interpolation data.
		This positive kernel is an analogue of Potapov's fundamental matrix inequality.
		Abstract: A representation theorem for a multiplier invariant subspaces of a reproducing kernel Hilbert spaces whose kernel behaves like the Bergman kernel is established.
	www.math.uga.edu /~seam17/abstract.html (2623 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		TUTORIAL - Reproducing Kernel Hilbert Spaces and why they are important.
		We assume no previous knowledge of reproducing kernel Hilbert spaces (rkhs).
		We note a variety of cost functions, including those resulting in penalized likelihood estimates and support vector machines, and briefly mention the problem of tuning, or, the bias-variance tradeoff in function estimation, which balances fit to the data against complexity of the solution.
	www.ipam.ucla.edu /abstract.aspx?tid=5543 (151 words)

	Amazon.com: The Schur Algorithm, Reproducing Kernel Spaces and System Theory: Books: Daniel Alpay,Stephen S. Wilson (Site not responding. Last check: 2007-10-10)
		This book is an impressive survey of recent research in the applications of Reproducing Kernel Hilbert Spaces to complex analytic function theory, and, in particular to the study of functions which are analytic on the open unit disk in the complex plane.
		First is the papers in "Reproducing Kernel Hilbert Spaces: Applications to Statistical Signal Processing" and especialy the four papers by Kailath labelled "Parts I, II, III, and IV".
		Herbrich shows how these powerful insights are being leveraged today in the field of data mining and especially in the field of support vector machines and their extensions to kernel based methods in general.
	www.amazon.com /Algorithm-Reproducing-Kernel-Spaces-System/dp/0821821555 (1345 words)

	CiteULike: Statistical properties of Kernel Principal Component Analysis (Site not responding. Last check: 2007-10-10)
		We perform these computations in a functional analytic framework which allows to deal implicitly with reproducing kernel Hilbert spaces of infinite dimension.
		We focus on Kernel Principal Component Analysis (KPCA) and, using such techniques, we obtain sharp excess risk bounds for the reconstruction error.
		In these bounds, the dependence on the decay of the spectrum and on the closeness of successive eigenvalues is made explicit.
	www.citeulike.org /user/davidr/article/508104 (251 words)

	Moving Least Square Reproducing Kernel Method Part II: Fourier Analysis
		In Part I of this work, the moving least square reproducing kernel (MLSRK) method is reformulated and implemented.
		The preliminary Fourier analysis reveals that MLSRK method is stable for sufficiently dense, non-degenerated particle distribution, in the sense that the kernel function family satisfies the Riesz bound.
		One of the novelties of the current approach is to treat the MLSRK method as a variant of the ``standard'' finite element method and depart from there to make a connection with the multiresolution approximation.
	www.tam.northwestern.edu /wkl/paper/shaofan2.html (292 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Many problems in high-dimensional data analysis can be cast in terms of assessments of mutual independence or conditional independence among subsets of measured variables or transforms of measured variables.
		We present a nonparametric characterization of mutual independence and conditional independence using operators on reproducing kernel Hilbert spaces.
		We show how these operators can be estimated from data and how they can be used in problems such as independent component analysis, tree-dependent component analysis, and dimension reduction for regression and classification.
	www.ipam.ucla.edu /abstract.aspx?tid=5186 (106 words)

	Reproducing Kernel Hilbert Spaces: Howard L. Weinert: ISBN 0879334347
		Reproducing Kernel Hilbert Spaces: Howard L. Weinert: ISBN 0879334347
		Reproducing Kernel Hilbert Spaces: Applications in Statistical Signal Processing
		This book is part of the Benchmark Papers in Electrical Engineering and Computer Science, 25.
	www.bestwebbuys.com /0879334347 (100 words)

	Inaugural Article: Soft and hard classification by reproducing kernel Hilbert space methods -- Wahba 99 (26): 16524 -- ...
		relationship is also the source of the term "reproducing kernel.")
		The Gaussian kernel appears to be a good general purpose kernel
		kernels (that is, depending on s – t) may be found
	www.pnas.org /cgi/content/full/99/26/16524 (4620 words)

	Atlas: Reproducing kernel Hilbert spaces and nonparametric estimation by Belkacem Abdous (Site not responding. Last check: 2007-10-10)
		Atlas: Reproducing kernel Hilbert spaces and nonparametric estimation by Belkacem Abdous
		In this talk, we present a general framework for estimating smooth functionals of probability distribution functions and their derivatives.
		This framework is based on reproducing kernel Hilbert spaces (RKHS) theory and local polynomial fitting.
	atlas-conferences.com /cgi-bin/abstract/card-47 (148 words)

	.: Wing Kam Liu - Walter P. Murphy Professor :.
		Reproducing Kernel Element Method, Part II Globally Conforming Im/Cn Hierarchies
		Reproducing Kernel Hierarchical Partition of Unity Part I: Theoretical Formulation
		Reproducing Kernel Hierarchical Partition of Unity Part II: Globally Conforming ImCn Hierarchies
	www.tam.northwestern.edu /wkl/_wkl/htm/publications.htm (2520 words)

	Tutorials on Kernel Methods
		An introduction to model building with reproducing kernel Hilbert spaces.
		Text-rich short course slides on nonparametric regression and statistical model building as solutions to optimization problems in Reproducing Kernel Hilbert Spaces.
		Gaussian and non-Gaussian data, direct and indirect observations, splines and spline anova models and radial basis functions are discussed as special cases.
	www.kernel-machines.org /tutorial.html (274 words)

	Inaugural Article: Soft and hard classification by reproducing kernel Hilbert space methods -- Wahba 99 (26): 16524 -- ...
		Inaugural Article: Soft and hard classification by reproducing kernel Hilbert space methods -- Wahba 99 (26): 16524 -- Proceedings of the National Academy of Sciences
		Reproducing kernel Hilbert space (RKHS) methods provide a unified
		Abbreviations: RKHS, reproducing kernel Hilbert space; SVM, support vector machine; WESDR, Wisconsin Epidemiologic Study of Diabetic Retinopathy; GACV, generalized approximate cross validation; GCV, generalized cross validation; MSVM, multicategory SVM; SRBCT, small round blue cell tumor
	www.pnas.org /cgi/content/abstract/99/26/16524 (395 words)

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