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# Topic: Reproducing kernel Hilbert space

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 Hilbert space - Wikipedia, the free encyclopedia Hilbert spaces allow simple geometric concepts, like projection and change of basis to be applied to infinite dimensional spaces, such as function spaces. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics. Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most "well-behaved" and the closest to the finite-dimensional spaces. en.wikipedia.org /wiki/Hilbert_space   (2059 words)

 math lessons - Hilbert space Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics, although many basic features of quantum mechanics can be understood without going into details about Hilbert spaces. These are function spaces associated to measure spaces (X, M, μ), where M is a σ-algebra of subsets of X and μ is a countably additive measure on M. www.mathdaily.com /lessons/Hilbert_space   (1716 words)

 Hilbert space - Art History Online Reference and Guide Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics, although many basic features of quantum mechanics can be understood without going into details about Hilbert spaces. Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most "well-behaved" and the closest to the finite-dimensional spaces. arthistoryclub.com /art_history/Hilbert_space   (1729 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)

 Reproducing kernel Hilbert space - Wikipedia, the free encyclopedia H is a reproducing kernel Hilbert space iff the linear map Take the Hilbert H space of square-integrable functions, for the Lebesgue measure on D, that are holomorphic functions. Then H is a reproducing kernel space, with kernel function the Bergman kernel; this example, with n = 1, was introduced by Bergman in 1922. en.wikipedia.org /wiki/Reproducing_kernel_Hilbert_space   (431 words)

 List of publications (Daniel Alpay), updated October 2007 On the reproducing kernel Hilbert spaces associated with the fractional and bi--fractional Brownian motions. A Hilbert space approach to bounded analytic extension in the ball. On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorizations. www.math.bgu.ac.il /~dany/pub.html   (2846 words)

 Springer Online Reference Works This kernel clearly satisfies condition (a1) and therefore is a reproducing kernel for the reproducing-kernel Hilbert space is the reproducing-kernel Hilbert space generated by kernel (a3). Many concrete examples of reproducing-kernel Hilbert spaces can be found in [a1], [a2] and [a6]. eom.springer.de /R/r130070.htm   (533 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)

 Citations: Support vector machines - Wahba (ResearchIndex) Then a kernel trick allows the dimension of the transformed feature space to be very large, even infinite in some cases (i.e. While lemma 1 is valid for all kernel functions that return positive values, it is tightest when the minimum value is zero. A Study on Sigmoid Kernels for SVM and the Training of non-PSD.. citeseer.ist.psu.edu /context/321684/0   (1429 words)

 The Gaussian Processes Web Site By using an appealing parameterization and projection techniques in a reproducing kernel Hilbert space, recursions for the effective parameters and a sparse gaussian approximation of the posterior process are obtained. We describe the spectral representation of the various classes of kernels and conclude with a discussion on the characterization of nonlinear maps that reduce nonstationary kernels to either stationarity or local stationarity. We derive novel analytic expressions for the predictive mean and variance for Gaussian kernel shapes under the assumption of a Gaussian input distribution in the static case, and of a recursive Gaussian predictive density in iterative forecasting. www.gaussianprocess.org   (12046 words)

 [No title]   (Site not responding. Last check: ) A family of regularized least squares regression models in a Reproducing Kernel Hilbert Space is extended by the kernel partial least squares (PLS) regression model. PLS is useful in situations where the number of explanatory variables exceeds the number of observations and/or a high level of multicollinearity among those variables is assumed. We give the theoretical description of the kernel PLS algorithm and we experimentally compare the algorithm with the existing kernel PCR and kernel ridge regression techniques. www.kernel-machines.org /jmlr/volume2/rosipal01a/abstract.html   (186 words)

 Matches for:   (Site not responding. Last check: ) This memoir is devoted to the study of positive definite functions on convex subsets of finite- or infinite-dimensional vector spaces, and to the study of representations of convex cones by positive operators on Hilbert spaces. If $V$ is a topological vector space and $\Omega\sub V$ an open convex cone, or a convex cone with non-empty interior, we describe sufficient conditions for the existence of a representing measure $\mu$ for $\phi$ on the topological dual space$V'$. If $V$ is a topological vector space and $\Omega\sub V$ an open convex cone, or a convex cone with non-empty interior, we describe sufficient conditions for the existence of a representing measure $\mu$ for $\phi$ on the topological dual space $V'$. www.mathaware.org /bookstore?fn=20&arg1=memoseries&item=MEMO-166-789   (426 words)

 Potential field collocation and densisty The practical selection of the reproducing kernel may be done so that it represents the auto-covariance of the potential. 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. www.gfy.ku.dk /~cct/moensted.htm   (3076 words)

 Reproducing Kernel Hilbert Space Methods for wide-sense self-similar Processes, Carl J. Nuzman, H. Vincent Poor In this paper, a reproducing kernel Hilbert space (RKHS) approach is used to characterize this structure. The RKHS associated with a self-similar process on a variety of simple index sets has a straightforward description, provided that the scale-spectrum of the process can be factored. Minimum variance unbiased estimators are given for the amplitudes of polynomial trends in fBm, and two new innovations representations for fBm are presented. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aoap/1015345400   (476 words)

 AMCA: Reproducing kernel Hilbert spaces and the detection of random signals corrupted by noise by Antonio Gualtierotti   (Site not responding. Last check: ) AMCA: Reproducing kernel Hilbert spaces and the detection of random signals corrupted by noise by Antonio Gualtierotti Reproducing kernel Hilbert spaces and the detection of random signals corrupted by noise Absolute continuity means intuitively that the signal has paths which are quite smoother than those of the noise and all available evidence says that the right amount of smoothness is expressed by the requirement that the signal paths belong to the reproducing kernel Hilbert space (RKHS) of the noise. at.yorku.ca /c/a/p/g/23.htm   (418 words)

 Gradient Descent Approach to Approximation in Reproducing Kernel Hilbert Spaces   (Site not responding. Last check: ) Consider the bounded linear operator, $L: \mathcal{F} \rightarrow \mathcal{Z}$, where $\mathcal{Z} \subseteq \mathbb{R}^{N}$ and $\mathcal{F}$ are Hilbert spaces defined on a common field $\mathcal{X}$. The functions, $k(x_{i}, \cdot)$, are known as reproducing kernels and $\mathcal{F}$ is a reproducing kernel Hilbert space (RKHS). Unlike iterative solutions for the more general Hilbert space setting, the proofs presented make use of the spectral representation of the kernel. www.acse.shef.ac.uk /~dodd/papers/abstracts/mathofcomp.html   (195 words)

 Paper Abstract Kernel models for classification and regression have emerged as widely applied tools in statistics and machine learning. Functional analytic results ensure that such a non-parametric prior specification induces a class of functions that span the reproducing kernel Hilbert space corresponding to the selected kernel. The practical benefits and modelling flexibility of the Bayesian kernel framework are illustrated in both simulated and real data examples that address prediction and classification inference with high-dimensional data. ftp.stat.duke.edu /WorkingPapers/07-10.html   (228 words)

 Kernel Sliced Inverse Regression with Applications on Classification The kernel mixtures lead the transformed data distribution to more Gaussian like and better symmetrically distributed which provide more suitable conditions for performing SIR analysis. We provide a theoretical description of the kernel SIR algorithm in the framework of reproducing kernel Hilbert space (RKHS). The applications using kernel SIR features on the classification of microarray data and other real world data are reported along with the comparison of some existing dimension reduction techniques. idv.sinica.edu.tw /hmwu/KSIR   (429 words)

 Robotics Institute: Policy Search in Reproducing Kernel Hilbert Space These works may not be reposted without the explicit permission of the copyright holder. Policy search has numerous advantages: it does not rely on the Markov assumption, domain knowledge may be encoded in a policy, the policy may require less representational power than a value-function approximation, and stable and convergent algorithms are well-understood. In this work, we show how policy search (with or without the additional guidance of value-functions) in a Reproducing Kernel Hilbert Space gives a simple and rigorous extension of the technique to non-parametric settings. www.ri.cmu.edu /pubs/pub_4538.html   (322 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)

 The gradient iteration for approximation in reproducing kernel Hilbert spaces -- Dodd and Harrison 21 (4): 359 -- IMA ... The gradient iteration for approximation in reproducing kernel Hilbert spaces -- Dodd and Harrison 21 (4): 359 -- IMA Journal of Mathematical Control and Information The gradient iteration for approximation in reproducing kernel Hilbert spaces F are Hilbert spaces defined on a common field X. imamci.oxfordjournals.org /cgi/content/abstract/21/4/359   (261 words)

 Transactions of the American Mathematical Society Hilbert spaces of Dirichlet series and their multipliers Abstract: We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces. www.ams.org /tran/2004-356-03/S0002-9947-03-03452-4/home.html   (249 words)

 Publications The Gradient Iteration for Approximation in Reproducing Kernel Hilbert Spaces. Ben Mitchinson and Robert F. Harrison, Digital Communications Channel Equalisation using the Kernel Adaline, IEEE Transactions on Communications, 50(4):571-576, 2002. Iterative solution to approximation in reproducing kernel Hilbert spaces. www.shef.ac.uk /acse/research/cdmg/publications   (287 words)

 Systems in Reproducing Kernel Hilbert Space: Causality, Realizability, and Separability -- SCHWARTZ and DICKINSON 3 ... kernel Hilbert Space, (RKHS), where the reproducing kernel is of both the reproducing kernel and the covariance of the given Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. imamci.oxfordjournals.org /cgi/content/abstract/3/2-3/223   (240 words)

 Predictive Control of a Humidifying Process Modelled on Reproducing Kernel Hilbert Spaces - Begell House Inc. This process is known to have a nonlinear behavior. To synthesize the MBP control, we used a Reproducing Kernel Hilbert Space (RKHS) model with reduced complexity. The identification of this model is carried in a fl box context with no a priori information needed, using the Statistical Learning Techniques (SLT). www.begellhouse.com /journals/46784ef93dddff27,622a96484c95e9c8,373471a33e504f26.html   (154 words)

 Non-linear Wiener filter in reproducing kernel Hilbert space We introduce the non-linear Wiener filter, which is a kernel-based extension of the Wiener filter. When the kernel method is applied to the Wiener filter directly, the dimensions of the space where the calculation has to be done is very large since noise samples have to be used. Citation: Yoshikazu Washizawa, Yukihiko Yamashita, "Non-linear Wiener filter in reproducing kernel Hilbert space," icpr, pp. csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/icpr/2006/2521/01/2521toc.xml&DOI=10.1109/ICPR.2006.861   (206 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 rich spline kernel was chosen to compute the 25 SVMs in Fig. The Gaussian kernel appears to be a good general purpose kernel www.pnas.org /cgi/content/full/99/26/16524   (4612 words)

 Publications of Arthur Gretton At the most fundamental level, we might wish to determine whether two distributions are the same, based on samples from each - this is known as the two-sample or homogeneity problem. We use kernel methods to address this problem, by mapping probability distributions to elements in a reproducing kernel Hilbert space (RKHS). Given a sufficiently rich RKHS, these representations are unique: thus comparing feature space representations allows us to compare distributions without ambiguity. www.kyb.mpg.de /publication.html?user=arthur&talks=1&bibtex=1   (548 words)

 Front: [math.PR/0506189] A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an ... Front: [math.PR/0506189] A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application Abstract: Given a positive definite, bounded linear operator $A$ on the Hilbert space $\mathcal{H}_0:=l^2(E)$, we consider a reproducing kernel Hilbert space $\mathcal{H}_+$ with a reproducing kernel $A(x,y)$. 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 $\mathcal{H}_-\supset\mathcal{H}_0\supset\mathcal{H}_+$ becomes a rigged Hilbert space. front.math.ucdavis.edu /math.PR/0506189   (248 words)

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