论文标题
通过矫正系统分析，重现内核Hilbert $ C^*$ - 模块和应用程序
Analysis via Orthonormal Systems in Reproducing Kernel Hilbert $C^*$-Modules and Applications
论文作者
论文摘要
内核方法一直是机器学习中最受欢迎的技术之一，在该技术中，使用复制内核Hilbert Space（RKHS）的属性来解决学习任务。在本文中，我们提出了一个新型的数据分析框架，它使用繁殖Hilbert $ C^*$ - 模块（RKHM），这是RKHS的另一个概括，而不是向量值估算的RKHS（VV-RKHS）。与RKHM的分析使我们能够比VV-RKHS更明确地处理变量之间的结构。我们展示了Hilbert $ C^*$ - 模块中正统系统构建的理论有效性，并在RKHM中得出了具有数值计算中这些理论属性的正态分化的具体过程。此外，我们将其应用于RKHM内核主成分分析以及使用Perron-Frobenius操作员对动态系统进行分析。还通过使用合成和现实世界数据研究了我们方法的经验性能。
Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert $C^*$-module (RKHM), which is another generalization of RKHS than vector-valued RKHS (vv-RKHS). Analysis with RKHMs enables us to deal with structures among variables more explicitly than vv-RKHS. We show the theoretical validity for the construction of orthonormal systems in Hilbert $C^*$-modules, and derive concrete procedures for orthonormalization in RKHMs with those theoretical properties in numerical computations. Moreover, we apply those to generalize with RKHM kernel principal component analysis and the analysis of dynamical systems with Perron-Frobenius operators. The empirical performance of our methods is also investigated by using synthetic and real-world data.
