Metric on random dynamical systems with vector-valued reproducing kernel Hilbert spaces

Development of metrics for structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we give a general framework to construct metrics on random nonlinear dynamical systems, defined with the Perron-Frobenius operators in vector-valued reproducing...

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Bibliographic Details
Published inarXiv.org
Main Authors Ishikawa, Isao, Tanaka, Akinori, Ikeda, Masahiro, Kawahara, Yoshinobu
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 27.10.2019
Subjects
Corporate governance
Dynamical systems
Hilbert space
Kernels
Machine learning
Nonlinear systems
Random processes
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Summary:Development of metrics for structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we give a general framework to construct metrics on random nonlinear dynamical systems, defined with the Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). We employ vvRKHSs to design mathematically manageable metrics and also to introduce operator-valued kernels, which enables us to handle randomness in systems. Our metric provides an extension of the existing metrics for deterministic systems, and gives a specification of the kernel maximal mean discrepancy of random processes. Moreover, by considering the time-wise independence of random processes, we clarify a connection between our metric and the independence criteria with kernels such as Hilbert-Schmidt independence criteria. We empirically illustrate our metric with synthetic data, and evaluate it in the context of the independence test for random processes. We also evaluate the performance with real time seris datas via clusering tasks.
ISSN:2331-8422