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, $\calP_n^*$, is based on the power function that appears in machine learning literature. The second, $\calS_n^*$, 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 $\calP_n^*$ and $\calS_n^*$ is marginal. The criterion $\calS_n^*$ is preferred as it is computationally more efficient and has a proven error bound.

翻译：再生核希尔伯特空间中的核插值在相同函数值集合的所有逼近中具有最坏情况下的最优性。本文比较了用于构造基于格核逼近的格点集的两种搜索准则。第一种候选准则 $\calP_n^*$ 基于机器学习文献中出现的幂函数；第二种准则 $\calS_n^*$ 则是用于生成截断傅里叶级数逼近格点的搜索准则。我们发现，分别采用 $\calP_n^*$ 和 $\calS_n^*$ 构造的格点在误差上的经验差异微乎其微。由于 $\calS_n^*$ 准则计算效率更高且具有理论误差界的保障，因此更受青睐。