Analysis via Orthonormal Systems in Reproducing Kernel Hilbert $C^*$-Modules and Applications
This work addresses the need for more explicit structural modeling in kernel methods for machine learning practitioners, though it appears incremental as it builds on existing RKHS generalizations.
The authors introduced a novel data analysis framework using reproducing kernel Hilbert C*-modules (RKHMs) to better capture variable structures than existing vector-valued RKHS, and demonstrated its application to kernel PCA and dynamical systems analysis with empirical validation on synthetic and real-world data.
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.