Some Properties of Gaussian Reproducing Kernel Hilbert Spaces and Their Implications for Function Approximation and Learning Theory

Minh, Hà Quang

doi:10.1007/s00365-009-9080-0

Cited by 96 publications

(63 citation statements)

References 15 publications

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“…Mihn [26] also developed a general theory of reproducing kernel Hilbert space on the sphere and advocated to utilize the kernel methods to tackle spherical data. However, for some popular kernels such as the Gaussian [27] and polynomials [5], kernel methods suffer from either a similar problem as the localization methods, or a similar drawback as the orthogonal series methods. In fact, it remains open that whether there is an exclusive kernel for spherical data such that both the manifold structure of the sphere and the localization requirement are sufficiently considered.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric regression using needlet kernels for spherical data

Lin

2019

Journal of Complexity

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Needlets have been recognized as state-of-the-art tools to tackle spherical data, due to their excellent localization properties in both spacial and frequency domains. This paper considers developing kernel methods associated with the needlet kernel for nonparametric regression problems whose predictor variables are defined on a sphere. Due to the localization property in the frequency domain, we prove that the regularization parameter of the kernel ridge regression associated with the needlet kernel can decrease arbitrarily fast.A natural consequence is that the regularization term for the kernel ridge regression is not necessary in the sense of rate optimality. Based on the excellent localization property in the spacial domain further, we also prove that all the l q (0 < q ≤ 2) kernel regularization estimates associated with the needlet kernel, including the kernel lasso estimate and the kernel bridge estimate, possess almost the same generalization capability for a large range of regularization parameters in the sense of rate optimality. This finding tentatively reveals that, if the needlet kernel is utilized, then the choice of q might not have a strong impact in terms of the generalization capability in some modeling contexts. From this perspective, q can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..

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Section: Introductionmentioning

confidence: 99%

Nonparametric regression using needlet kernels for spherical data

Lin

2019

Journal of Complexity

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“…are an extended Chebyshev system, Theorem 3.1 guarantees the existence of a unique N -point quadrature rule for q > 0 and weight parameters ωα > 0 such that the series converges for any ℓ > 0 and x, x ′ ∈ Ω. Arguments identical to those used in [32,19] establish that k dpow ℓ is a positive-definite kernel and that its RKHS H ′ ℓ consists of functions…”

Section: Optimal Points In One Dimensionmentioning

confidence: 87%

Worst-case optimal approximation with increasingly flat Gaussian kernels

Karvonen

Särkkä

2020

Adv Comput Math

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We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces (RKHSs) induced by increasingly flat Gaussian kernels. This provides a new perspective and some generalisations to the problem of interpolation with increasingly flat radial basis functions. When the evaluation points are fixed and unisolvent, we show that the worst-case optimal method converges to a polynomial method. In an additional one-dimensional extension, we allow also the points to be selected optimally and show that in this case convergence is to the unique Gaussian-type method that achieves the maximal polynomial degree of exactness. The proofs are based on an explicit characterisation of the Gaussian RKHS in terms of exponentially damped polynomials.

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“…However, for our setting, given that the kernels we use are continuous and f ρ is often not (say when η(x) = η < 1 2 ), the convergence of f z,λ − f ρ ρ is either impossible or very slow (we discuss this in detail in [14]). …”

Section: Proof Methodologymentioning

confidence: 99%

“…To prove Theorem 3, we need the following law of large numbers for Hilbert space-valued random variables, which is a consequence of a general inequality due to Pinelis [15]. …”

Section: The Average Algorithmmentioning

confidence: 99%