Kolmogorov Widths on the Sphere via Eigenvalue Estimates for Hölderian Integral Operators

Abstract

Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere S m in the Euclidean space R m+1 are known to depend upon the Mercer's expansion of the compact and self-adjoint L 2 (S m )-operator associated to the kernel. The estimation of the Kolmogorov n-th width of the unit ball of the reproducing kernel Hilbert space in L 2 (S m ) and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. This paper provides sharp upper bounds for the Kolmogorov widths in the case in which the kernel satisfies an abstract Hölder condition. In particular, we follow the opposite direction usually considered in the literature, that is, we estimate the widths from decay rates for the sequence of eigenvalues of the integral operator.

Covering number of the unit ball of Reproducing Kernel Hilbert Space of Schoenbergs kernels

“…In the early 90's, in kernel machine learning research, it was not the notion of positive definite kernels that was being used. Instead, researchers considered kernels satisfying the conditions of Mercer's theorem (CRISTIANINI;SHAWE-TAYLOR, 2000;MERCER, 1909;VAPNIK, 1998) as considered in the article (JORDAO; MENEGATTO, 2019), in order to estimate related approximation numbers. Over the last years, estimation and learning methods employing positive definite kernels have become rather popular, since positive definite kernels seems to be the right class of kernels to be considered.…”