Universalities of Reproducing Kernels Revisited
This work is incremental, clarifying theoretical aspects of kernel methods for researchers in machine learning and approximation theory.
The paper revisits universalities of reproducing kernels, providing a clear description of differences among universal, characteristic, and C0-universal kernels, and offers a simpler proof for translation-invariant kernels while focusing on weighted polynomial kernels.
Kernel methods have been widely applied to machine learning and other questions of approximating an unknown function from its finite sample data. To ensure arbitrary accuracy of such approximation, various denseness conditions are imposed on the selected kernel. This note contributes to the study of universal, characteristic, and $C_0$-universal kernels. We first give simple and direct description of the difference and relation among these three kinds of universalities of kernels. We then focus on translation-invariant and weighted polynomial kernels. A simple and shorter proof of the known characterization of characteristic translation-invariant kernels will be presented. The main purpose of the note is to give a delicate discussion on the universalities of weighted polynomial kernels.