On the Approximation of Kernel functions
Paul Dommel Alois Pichler
Various methods in statistical learning build on kernels considered in
reproducing kernel Hilbert spaces. In applications, the kernel is often
selected based on characteristics of the problem and the data. This kernel is
then employed to infer response variables at points, where no explanatory data
were observed. The data considered here are located in compact sets in higher
dimensions and the paper addresses approximations of the kernel itself. The new
approach considers Taylor series approximations of radial kernel functions. For
the Gauss kernel on the unit cube, the paper establishes an upper bound of the
associated eigenfunctions, which grows only polynomially with respect to the
index. The novel approach substantiates smaller regularization parameters than
considered in the literature, overall leading to better approximations. This
improvement confirms low rank approximation methods such as the Nystr\"om
method.