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reproducing kernel hilbert spaces cannot contain all continuous functions on a compact metric space

ARCHIV DER MATHEMATIK 2024年第5期122卷 553-557页

作者： Steinwart, Ingo Univ Stuttgart Inst Stochast & Applicat Fac Math & Phys 8 D-70569 Stuttgart Germany

Given an uncountable, compact metric space X, we show that there exists no reproducing kernel hilbert space that contains the space of all continuous functions on X.

关键词： reproducing kernel hilbert spaces Inclusion maps spaces of continuous functions

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reproducing kernel hilbert spaces (RKHS) for the higher order Bessel operator

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BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA 2023年第1期29卷 1-20页

作者： Soltani, Fethi Univ Tunis El Manar Fac Sci Tunis Lab Anal Math & Applicat LR11ES11 Tunis 2092 Tunisia Univ Carthage Ecole Natl Ingenieurs Carthage Tunis 2035 Tunisia

In 1961, Bargmann introduced the classical Fock space F(C), and in 1984, Cholewinsky introduced the generalized Fock space F-2,F-nu (C). These two spaces are the aim of many works, and have many applications in mathematics, in physics, and in quantum mechanics. In this work, we introduce and study the reproducing kernel hilbert space F-r,F-alpha(C) associated with the Bessel operator B-r,B-alpha of r-order (r >= 3). Next, we establish an uncertainty inequality of Heisenberg type for the space F-r,F-alpha (C). Finally, using the theory of extremal functions, we give best approximate inversion formulas for the difference operator D and the integral operator P, respectively.

关键词： Higher order Bessel operator reproducing kernel hilbert spaces Extremal functions

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Construction of quasi-localized dual bases in reproducing kernel hilbert spaces

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 465卷

作者： Harbrecht, Helmut Kempf, Ruediger Multerer, Michael Univ Basel Dept Math Informat Spiegelgasse 1 CH-4051 Basel Switzerland Univ Bayreuth Dept Math Appl & Numer Anal D-95440 Bayreuth Germany Univ Svizzera italiana Ist Eulero Via Santa 1 CH-6962 Lugano Switzerland

A straightforward way to represent the kernel approximant of a function, known by a finite set of samples, within a reproducing kernel hilbert space is through the canonical dual pair. The canonical dual pair consists of the basis of kernel translates and the corresponding Lagrange basis. From a numerical perspective, one is particularly interested in dual pairs such that the dual basis is quasi-local meaning that it can be well approximated using only a small subset of the data sites. This implies that the inverse Gramian is approximately sparse. In this case, the kernel approximant is efficiently computable by multiplying a sparse matrix with the data vector. We present two methods for finding such quasi-localized dual bases. First, we adapt the idea of localizing the Lagrange basis, which yields an approximate canonical dual pair and extend this idea to derive a new, symmetric preconditioner for kernel matrices. Second, we use samplets to obtain multiresolution versions of dual bases. Samplets are localized discrete signed measures constructed such that their respective measure integrals of polynomials up to a certain degree vanish. Therefore, the kernel matrix and its inverse are compressible to sparse matrices in samplet coordinates for asymptotically smooth kernels. We provide benchmark experiments in two spatial dimensions to demonstrate the compression power of both approaches and apply the new preconditioner to implicit surface reconstruction in computer graphics.

关键词： reproducing kernel hilbert spaces Dual pairs Preconditioning

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Topological K-means clustering in reproducing kernel hilbert spaces

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ELECTRONIC JOURNAL OF STATISTICS 2025年第1期19卷 204-239页

作者： Dixon, Matthew Chen, Yuzhou Gel, Yulia R. IIT Dept Appl Math Chicago IL 60616 USA Univ Calif Riverside Dept Stat Riverside CA USA Virginia Tech VA & Natl Sci Fdn Dept Stat Blacksburg VA USA

We propose a new topological clustering methodology, based on generalizing an empirical risk minimization framework, using a reproducing kernel hilbert space (RKHS) for vectorized persistent homology representations of point clouds. In contrast to conventional Euclidean-based clustering methods which address only pairwise similarity among data points, our new approach of topological K-means clusters data based on similarity of shapes which are exhibited by the local vicinity of each data point at multiple scales. Thereby, topological clustering systematically captures the inherent local and global higher order data characteristics that are otherwise inaccessible with Euclidean-based clustering. We summarize the extracted shape characteristics of each local vicinity in the form of a persistence diagram (PD) and embed the PDs into a RKHS, which induces a distance among shapes of local vicinities in hilbert space. Our derived theoretical guarantees on stability and consistency of the topological partitions are the first theoretical results of this kind at the intersection of topological data analysis and statistical inference. Additionally, we establish a number of new theoretical results on bounds of covering numbers in hilbert spaces which are of independent interest in statistical learning theory. We demonstrate the superior performance of the new topological K-means clustering on simulations and the US COVID-19 data.

关键词： Topological data analysis K-means clustering reproducing kernel hilbert spaces clustering stability

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Which spaces can be Embedded in reproducing kernel hilbert spaces?

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CONSTRUCTIVE APPROXIMATION 2025年 1-48页

作者： Schoelpple, Max Steinwart, Ingo Univ Stuttgart Inst Stochast & Applicat Fac Math & Phys 8 D-70569 Stuttgart Germany

Given a Banach space E consisting of functions, we ask whether there exists a reproducing kernel hilbert space H with bounded kernel such that E subset of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\subset H$$\end{document}. More generally, we consider the question, whether for a given Banach space consisting of functions F with E subset of F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\subset F$$\end{document}, there exists an intermediate reproducing kernel hilbert space E subset of H subset of F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\subset H\subset F$$\end{document}. We provide both sufficient and necessary conditions for this to hold. Moreover, we show that for typical classes of function spaces described by smoothness there is a strong dependence on the underlying dimension: the smoothness s required for the space E needs to grow proportional to the dimension d in order to allow for an intermediate reproducing kernel hilbert space H.

关键词： reproducing kernel hilbert spaces Inclusion maps spaces of functions Factorization of Operators Type and Cotype Besov- and Triebel-Lizorkin spaces Sobolev spaces with mixed smoothness Neural networks

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Learning Partial Differential Equations in reproducing kernel hilbert spaces

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JOURNAL OF MACHINE LEARNING RESEARCH 2023年第1期24卷 1-72页

作者： Stepaniants, George MIT Dept Math 77 Massachusetts Ave Cambridge MA 02139 USA

We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional linear regression (FLR), we estimate the best-fit Green's function and bias term of the fundamental solution in a reproducing kernel hilbert space (RKHS) which allows us to regularize their smoothness and impose various structural constraints. We derive a general representer theorem for operator RKHSs to approximate the original infinite-dimensional regression problem by a finite-dimensional one, reducing the search space to a parametric class of Green's functions. In order to study the prediction error of our Green's function estimator, we extend prior results on FLR with scalar outputs to the case with functional outputs. Finally, we demonstrate our method on several linear PDEs including the Poisson, Helmholtz, Schrodinger, Fokker-Planck, and heat equation. We highlight its robustness to noise as well as its ability to generalize to new data with varying degrees of smoothness and mesh discretization without any additional training.

关键词： Green's functions partial differential equations reproducing kernel hilbert spaces functional linear regression simultaneous diagonalization

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reproducing kernel hilbert spaces on manifolds: Sobolev and diffusion spaces

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ANALYSIS AND APPLICATIONS 2021年第3期19卷 363-396页

作者： De Vito, Ernesto Muecke, Nicole Rosasco, Lorenzo Univ Genoa MALGA DIMA Malga Italy TU Berlin Math Berlin Math Res Ctr Berlin Germany Univ Genoa DIBRIS MALGA Dibris Italy MIT CBMM Cambridge MA 02139 USA Ist Italiano Tecnol Genoa Italy

We study reproducing kernel hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic.

关键词： reproducing kernel hilbert spaces heat kernels

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Adaptive estimation of external fields in reproducing kernel hilbert spaces

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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING 2022年第8期36卷 1931-1957页

作者： Guo, Jia Kepler, Michael E. Tej Paruchuri, Sai Wang, Hoaran Kurdila, Andrew J. Stilwell, Daniel J. Virginia Tech Dept Mech Engn Blacksburg VA USA Virginia Tech Dept Elect & Comp Engn 1185 Perry St Blacksburg VA 24060 USA

This article studies the distributed parameter system that governs adaptive estimation by mobile sensor networks of external fields in a reproducing kernel hilbert space (RKHS). The article begins with the derivation of conditions that guarantee the well-posedness of the ideal, infinite dimensional governing equations of evolution for the centralized estimation scheme. Subsequently, convergence of finite dimensional approximations is studied. Rates of convergence in all formulations are established using history-dependent bases defined from translates of the RKHS kernel that are centered at sample points along the agent trajectories. Sufficient conditions are derived that ensure that the finite dimensional approximations of the ideal estimator equations converge at a rate that is bounded by the fill distance of samples in the agents' assigned subdomains. The article concludes with examples of simulations and experiments that illustrate the qualitative performance of the introduced algorithms.

关键词： distributed parameter systems function estimation learning systems reproducing kernel hilbert spaces

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A Note on Compact Embeddings of reproducing kernel hilbert spaces in L² and Infinite-Variate Function Approximation 15th

A Note on Compact Embeddings of Reproducing Kernel Hilbert S...

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15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC)

作者： Wnuk, Marcin Univ Osnabruck Inst Math Albrechtstr 28a D-49076 Osnabruck Germany

ISBN: (纸本)9783031597640;9783031597626;9783031597619

This note consists of two largely independent parts. In the first part we give conditions on the kernel k : Omega x Omega -> R of a reproducing kernel hilbert space H continuously embedded via the identity mapping into L-2(Omega, mu), which are equivalent to the fact that H is even compactly embedded into L-2(Omega, mu). In the second part we consider a scenario from infinite-variate L-2-approximation. Suppose that the embedding of a reproducing kernel hilbert space of univariate functions with reproducing kernel 1 + k into L-2(Omega, mu) is compact. We provide a simple criterion for checking compactness of the embedding of a reproducing kernel hilbert space with the kernel given by Sigma(u is an element of U) gamma(u) circle times(j is an element of u)k where U = {u subset of N : |u| < infinity}, and gamma = (gamma(u))(u is an element of U) is a family of non-negative numbers, into an appropriate L-2 space.

关键词： reproducing kernel hilbert spaces Infinite-variate approximation Compact embeddings

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Semi-reproducing kernel hilbert spaces, splines and increment kriging on the sphere

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STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK ASSESSMENT 2022年第11期36卷 3639-3652页

作者： Bonabifard, M. R. Mosammam, A. M. Ghaemi, M. R. Univ Zanjan Dept Math Zanjan Iran Univ Zanjan Dept Stat Zanjan Iran

The concept of reproducing kernel hilbert space does not capture the key features of the spherical smoothing problem. A semi- reproducing kernel hilbert space (SRKHS), provides a more natural setting for the smoothing spline solution. In this paper, we carry over the concept of the SRKHS from the R-d to the sphere, Sd-1. In addition, a systematic study is made of the properties of an spherical SRKHS. Next, we present the one to one correspondence between increment-reproducing kernels and conditionally positive definite functions and its consequences on spherical optimal smoothing. The smoothing and interpolation issues on the sphere are considered in the proposed SRKHS setting. Finally, a simulation study is done to illustrate the proposed methodology and an analysis of world average temperature from 1963 to 1967 and 1993-1997 is done using the proposed methods.

关键词： Conditionally positive definite Gegenbauer polynomials Increment kriging reproducing kernel hilbert spaces Spherical smoothing Thin plate spline

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