The Covering Number for Some Mercer Kernel Hilbert Spaces on the Unit Sphere

Abstract

In the present paper, we estimate the covering number for some reproducing kernel Hilbert spaces on the unit sphere. Both the upper bounds and the lower bounds are provided.

Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

“…Our lower estimate in Theorem agrees with the one obtained in [, Theorem 2.2] for covering numbers of an isotropic RKHS in the spherical setting. However, even though we are dealing with a more general setting, our upper estimate is tighter than the one in [, Theorem 3.1], whenever .…”

“…Remark The lower estimate agrees with the one obtained in [, Theorem 2.2,], which deals with covering numbers of reproducing kernel Hilbert spaces associated with isotropic kernels defined on the sphere. Our lower estimate in Theorem agrees with the one obtained in [, Theorem 2.2] for covering numbers of an isotropic RKHS in the spherical setting. However, even though we are dealing with a more general setting, our upper estimate is tighter than the one in [, Theorem 3.1], whenever .…”

“…To overcome this drawback, treated the approximation of functions from fixed large Sobolev‐type spaces via the same assumption on the kernels, that is, the same asymptotic behavior on the Fourier–Jacobi coefficients of K . Others applications of this assumption may be found in , , , , . Also, it is important to add that is a reasonable assumption for the present setting due to the behaviour of the sequence ensured by Lemma .…”

“…These estimates are obtained via smoothness assumptions on the kernel and the use of techniques dealing with covering numbers of operators and positive cubature formulas on . For more information on estimates of covering numbers of RKHS or about the methods applied here we refer to , , and the references therein.…”

The Learning Rates of Regularized Regression Based on Reproducing Kernel Banach Spaces

“…The covering number involved in (16) has been studied widely (see [15][16][17][18][19]). In this paper, we assume N(B , ) has the logarithmic complexity.…”