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, \({\mathcal {P}}_n^*\) , is based on the power function that appears in machine learning literature. The second, \({\mathcal {S}}_n^*\) , 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 \({\mathcal {P}}_n^*\) and \({\mathcal {S}}_n^*\) is marginal. The criterion \({\mathcal {S}}_n^*\) is preferred as it is computationally more efficient and has a bound with a superior convergence rate.