Comparison of Two Search Criteria for Lattice-based Kernel Approximation

The kernel interpolant in a reproducing kernel Hilbert space is optimal in the worst-case sense among all approximations of a function using the same set of function values. In this paper, we compare two search criteria to construct lattice point sets for use in lattice-based kernel approximation. The first candidate, ${\text{calP}}_n^{\ast }$, is based on the power function that appears in machine learning literature. The second, ${\text{calS}}_n^{\ast }$, is a search criterion used for generating lattices for approximation using truncated Fourier series. We find that the empirical difference in error between the lattices constructed using ${\text{calP}}_n^{\ast }$ and ${\text{calS}}_n^{\ast }$ is marginal. The criterion ${\text{calS}}_n^{\ast }$ is preferred as it is computationally more efficient and has a proven error bound.