Reproducing Properties of Differentiable Mercer-Like Kernels on the Sphere

Jordao, T.; Menegatto, V. A.

doi:10.1080/01630563.2012.660590

Cited by 11 publications

(4 citation statements)

References 12 publications

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“… . If (11) holds, then by Theorem 2.4 in [31], or Theorem 4.2 in [32] or Proposition 6.2 in [33] that:…”

Section: { }mentioning

confidence: 99%

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

Ran,

Sheng

2024

JAMP

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We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.

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“… . If (11) holds, then by Theorem 2.4 in [31], or Theorem 4.2 in [32] or Proposition 6.2 in [33] that:…”

Section: { }mentioning

confidence: 99%

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

Ran,

Sheng

2024

JAMP

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“…It is expected that the functions in a RKHS inherit some of the properties of the generating kernel such as smoothness. In [13] the authors showed that functions in the RKHS associated to a kernel having a Mercer series expansion (it means a Fourier-like series given in terms of tensorial product of orthonormal functions) have the same "degree" of smoothness in terms of the fractional Laplace-Beltrami derivative of the original kernel. More general than that, in [12] similar harmonic analysis structure was enough to prove similar reproducing results on a very general context.…”

Section: Introductionmentioning

confidence: 99%

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

Azevedo¹,

González²,

Jordão³

2021

Preprint

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In this paper we use the infamous continuous and nowhere differentiable Weierstrass function as a prototype to define a "Weierstrass fractal kernel". We investigate properties of the reproducing kernel Hilbert space (RKHS) associated to this kernel by presenting an explicit characterization of this space. In particular, we show that this space has a dense subset composed of continuous but nowhere differentiable functions. Moreover, we present sharp estimates for the covering numbers of the unit ball of this space as a subset of the continuous functions.

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“…No capítulo anterior (veja também [13]), assumindo certas hipóteses de diferenciabilidade sobre um núcleo de Mercer K deduzimos propriedades de diferenciabilidade para funções de H K . Dentre as muitas noções de diferenciabilidade para funções definidas na esfera S m , a diferenciabilidade usual foi a escolhida.…”

Section: Séries De Fourier-laplace Com Pesos Em Ehrunclassified

Diferenciabilidade em espaços de Hilbert de reprodução sobre a esfera

Jordão

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Reproducing Properties of Differentiable Mercer-Like Kernels on the Sphere

Cited by 11 publications

References 12 publications

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

Diferenciabilidade em espaços de Hilbert de reprodução sobre a esfera

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