On the Inclusion Relation of Reproducing Kernel Hilbert Spaces

TL;DR

This paper investigates the inclusion relations among various reproducing kernel Hilbert spaces, providing characterizations, a comprehensive table for translation invariant kernels, and discussing how these relations are preserved under kernel operations.

Contribution

It offers new characterizations of inclusion relations in RKHSs, a complete table for translation invariant kernels, and insights into how these relations behave under kernel operations.

Findings

Characterizations of RKHS inclusion via feature maps

A comprehensive table of inclusion relations among translation invariant kernels

Discussion on preservation of inclusion relations under kernel operations

Abstract

To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert-Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.