论文标题
Sobolev空间，内核和超球的差异
Sobolev Spaces, Kernels and Discrepancies over Hyperspheres
论文作者
论文摘要
这项工作为超球形背景下的内核方法提供了理论基础。具体而言，我们表征了天然空间（复制核希尔伯特空间）和与超球体定义的内核相关的Sobolev空间。我们的结果对内核立方体产生了直接的后果，确定了最坏情况误差的收敛速率，并根据Stein的方法扩展了立方体算法的适用性。我们首先在$（d+1）$ - 尺寸欧几里得空间的$ d $ d $二维超晶体的Sobolev空间上引入了合适的表征。我们的表征基于与给定内核相关的傅立叶 - 雪茄序列。这样的序列很难（如果不是不可能的话）在$ d $维的球体上进行分析计算，但在希尔伯特球体上通常是可行的。我们通过找到允许从希尔伯特到有限的尺寸超透明的投影操作员来解决这个问题。我们通过一些参数核心的核心来说明我们的发现。
This work provides theoretical foundations for kernel methods in the hyperspherical context. Specifically, we characterise the native spaces (reproducing kernel Hilbert spaces) and the Sobolev spaces associated with kernels defined over hyperspheres. Our results have direct consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method. We first introduce a suitable characterisation on Sobolev spaces on the $d$-dimensional hypersphere embedded in $(d+1)$-dimensional Euclidean spaces. Our characterisation is based on the Fourier--Schoenberg sequences associated with a given kernel. Such sequences are hard (if not impossible) to compute analytically on $d$-dimensional spheres, but often feasible over Hilbert spheres. We circumvent this problem by finding a projection operator that allows to Fourier mapping from Hilbert into finite dimensional hyperspheres. We illustrate our findings through some parametric families of kernels.
