提出非欧空间核函数通用性的分析工具，为流形数据建模提供理论支持。
Universal kernels via harmonic analysis on Riemannian symmetric spaces
- 基于黎曼对称空间的调和分析方法研究核函数的通用性
- 证明了多个文献中正定核在流形上的通用性
- 适合研究流形学习与非欧数据建模的科研人员
核函数的通用性刻画了再生核希尔伯特空间中可逼近的函数类，在机器学习核方法的理论基础中具有重要意义。本文建立了研究黎曼对称空间上核函数通用性的基本工具，将这一重要课题扩展至非欧空间。此外，利用所发展工具，证明了若干近期文献中定义于黎曼对称空间上的正定核的通用性，为这些核在流形值数据应用中的使用提供了理论依据。
原文摘要 · Abstract (English)
The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in machine learning. In this work, we establish fundamental tools for investigating universality properties of kernels in Riemannian symmetric spaces, thereby extending the study of this important topic to kernels in non-Euclidean domains. Moreover, we use the developed tools to prove the universality of several recent examples from the literature on positive definite kernels defined on Riemannian symmetric spaces, thus providing theoretical justification for their use in applications involving manifold-valued data.
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