Title: Time-series data analysis with Reproducing kernel Hilbert C*-modules
Abstract: RKHM (Reproducing kernel Hilbert C*-module) is a generalization of RKHS (Reproducing kernel Hilbert space) which is characterized by a C*-algebra-valued positive definite kernel and inner product associated with this kernel. Regarding RKHS, it has been actively researched for data analysis. Moreover, time-series data analysis by Perron-Frobenius and Koopman operators in RKHSs has been investigated. In this framework, the time-series data is assumed to be generated from a dynamical system and we can estimate Perron-Frobenius and Koopman operators only by the data. Since these operators are linear, we can apply the theory of linear algebra for the estimation. However, for interacting dynamical systems, information about interactions tend to degenerate in RKHSs and extracting such information from given data is difficult. Therefore, we consider using RKHMs instead of RKHSs for interacting dynamical systems. Since inner products in RKHMs are C*-algebra-valued, they capture more information about interactions than complex-valued ones. As a result, we can extract information about interactions from given data.