A Spectral Analysis of Dot-product Kernels

TL;DR

This paper provides spectral eigenvalue decay estimates for compositional dot-product kernels, enabling better understanding of their function spaces and implications for statistical estimation and approximation trade-offs.

Contribution

It introduces improved eigenvalue decay bounds for compositional dot-product kernels, enhancing analysis of their RKHS and statistical properties.

Findings

Eigenvalue decay estimates are improved for compositional dot-product kernels.

Volumes of balls in the associated RKHS are derived.

Trade-offs between approximation and statistical errors are analyzed.

Abstract

We present eigenvalue decay estimates of integral operators associated with compositional dot-product kernels. The estimates improve on previous ones established for power series kernels on spheres. This allows us to obtain the volumes of balls in the corresponding reproducing kernel Hilbert spaces. We discuss the consequences on statistical estimation with compositional dot product kernels and highlight interesting trade-offs between the approximation error and the statistical error depending on the number of compositions and the smoothness of the kernels.