Sobolev Spaces, Kernels and Discrepancies over Hyperspheres
Sobolev Spaces, Kernels and Discrepancies over Hyperspheres
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DOI：
10.48550/arxiv.2211.09196
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发表时间：
2022-11
期刊：
影响因子：
--
通讯作者：
S. Hubbert;E. Porcu;C. Oates;M. Girolami
中科院分区：
文献类型：
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作者：
S. Hubbert;E. Porcu;C. Oates;M. Girolami
这一工作为超球背景下的核方法提供了理论基础。具体来说，我们将原生空间（再生核希尔伯特空间）和Sobolev空间与超球面上定义的核。我们的研究结果有直接的后果，核体积，确定最坏情况下的错误的收敛速度，并扩大适用性的体积算法的基础上斯坦的方法。首先给出了嵌入在d+1维欧氏空间中的d维超球面上的Sobolev空间的一个适当刻画.我们的特征是基于与一个给定的内核的傅立叶-勋伯格序列。这样的序列很难（如果不是不可能的话）在$d$维球面上解析计算，但在希尔伯特球面上通常是可行的。我们绕过这个问题，找到一个投影算子，允许傅立叶映射从希尔伯特到有限维超球面。我们说明我们的研究结果，通过一些参数家庭的内核。
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.