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.

翻译：统计学习中的多种方法建立在再生核希尔伯特空间中考虑的核函数基础上。在实际应用中，核函数通常根据问题特征和数据特性进行选取，随后用于在未观测到解释变量的点处推断响应变量。本文考虑的数据位于高维紧致集合中，旨在研究核函数本身的近似问题。新方法采用径向核函数的泰勒级数逼近。针对单位立方体上的高斯核函数，本文建立了相关特征函数的上界，该上界仅关于索引具有多项式增长。相较于文献中采用的尺度参数，该方法证实了更小的正则化参数，从而整体上实现了更优的逼近效果。这一改进验证了如Nyström方法等低秩近似方法的有效性。