scholarly journals New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry

2021 ◽  
Vol 41 (3) ◽  
pp. 283-300
Author(s):  
Daniel Alpay ◽  
Palle E.T. Jorgensen
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Positive Definite ◽  
Reproducing Kernel Hilbert Spaces ◽  
Bilinear Forms ◽  
Metric Geometry ◽  
Positive Definite Kernels ◽  
Positive Definite Kernel ◽  
Kernel Hilbert Spaces

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

scholarly journals Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

2021 ◽  
Vol 15 (5) ◽  
Author(s):  
Monika Drewnik ◽  
Tomasz Miller ◽  
Zbigniew Pasternak-Winiarski
Keyword(s):  
Unitary Representation ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernels ◽  
Positive Definite ◽  
Reproducing Kernel Hilbert Spaces ◽  
Locally Compact Group ◽  
Positive Definite Kernel ◽  
Kernel Hilbert Spaces

AbstractThe aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore–Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions.

Refined Generalization Bounds of Gradient Learning over Reproducing Kernel Hilbert Spaces

2015 ◽  
Vol 27 (6) ◽  
pp. 1294-1320 ◽  
Author(s):  
Shao-Gao Lv
Keyword(s):  
Hilbert Spaces ◽  
Upper Bound ◽  
Dimensional Reduction ◽  
Reproducing Kernel ◽  
Positive Definite ◽  
Reproducing Kernel Hilbert Spaces ◽  
Positive Definite Kernels ◽  
Generalization Bounds ◽  
Kernel Hilbert Spaces ◽  
Gradient Learning

Gradient learning (GL), initially proposed by Mukherjee and Zhou ( 2006 ) has been proved to be a powerful tool for conducting variable selection and dimensional reduction simultaneously. This approach presents a nonparametric version of a gradient estimator with positive definite kernels without estimating the true function itself, so that the proposed version has wide applicability and allows for complex effects between predictors. In terms of theory, however, existing generalization bounds for GL depend on capacity-independent techniques, and the capacity of kernel classes cannot be characterized completely. Thus, this letter considers GL estimators that minimize the empirical convex risk. We prove generalization bounds for such estimators with rates that are faster than previous results. Moreover, we provide a novel upper bound for Rademacher chaos complexity of order two, which also plays an important role in general pairwise-type estimations, including ranking and score problems.

A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces

2013 ◽  
Vol 56 (2) ◽  
pp. 400-406
Author(s):  
Bebe Prunaru
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Finite Measure ◽  
Reproducing Kernel Hilbert Spaces ◽  
Property A ◽  
Multiplier Algebra ◽  
Finite Measure Space ◽  
Kernel Hilbert Spaces ◽  
Multiplier Algebras

Abstract.Let (X;B; μ) be a σ-finite measure space and let H ⊂ L2(X; μ) be a separable reproducing kernel Hilbert space on X. We show that the multiplier algebra of H has property (A1(1)).

scholarly journals Berezin number inequality for convex function in reproducing Kernel Hilbert Space

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5711-5717 ◽  
Author(s):  
Ulaş Yamancı ◽  
Mehmet Gürdal ◽  
Mubariz Garayev
Keyword(s):  
Hilbert Space ◽  
Convex Function ◽  
Hilbert Spaces ◽  
Convex Functions ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Hilbert Spaces ◽  
Selfadjoint Operators ◽  
Kernel Hilbert Spaces ◽  
Berezin Number

By using Hardy-Hilbert?s inequality, some power inequalities for the Berezin number of a selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs) with applications for convex functions are given.

Berezin Number Inequalities of an Invertible Operator and Some Slater Type Inequalities in Reproducing Kernel Hilbert Spaces

2020 ◽  
pp. 55-78
Author(s):  
Ulaş Yamancı ◽  
Mehmet Gürdal
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Invertible Operator ◽  
Reproducing Kernel Hilbert Spaces ◽  
Slater Type ◽  
Riesz Representation ◽  
Kernel Hilbert Spaces ◽  
Berezin Number

A reproducing kernel Hilbert space (shorty, RKHS) H=H(Ω) on some set Ω is a Hilbert space of complex valued functions on Ω such that for every λ∈Ω the linear functional (evaluation functional) f→f(λ) is bounded on H. If H is RKHS on a set Ω, then, by the classical Riesz representation theorem for every λ∈Ω there is a unique element kH,λ∈H such that f(λ)=〈f,kH,λ〉; for all f∈H. The family {kH,λ:λ∈Ω} is called the reproducing kernel of the space H. The Berezin set and the Berezin number of the operator A was respectively given by Karaev in [26] as following Ber(A)={A(λ):λ∈Ω} and ber(A):=|A(λ)|. In this chapter, the authors give the Berezin number inequalities for an invertible operator and some other related results are studied. Also, they obtain some inequalities of the slater type for convex functions of selfadjoint operators in reproducing kernel Hilbert spaces and examine related results.

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