Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators

Yoshinobu Kawahara, Isao Ishikawa, Keisuke Fujii, Masahiro Ikeda, Yuka Hashimoto

Publication date: 31 May 2018

Abstract: The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius operators in reproducing kernel Hilbert spaces. Our metric includes the existing fundamental metrics for dynamical systems, which are basically defined with principal angles between some appropriately-chosen subspaces, as its special cases. We also describe the estimation of our metric from finite data. We empirically illustrate our metric with an example of rotation dynamics in a unit disk in a complex plane, and evaluate the performance with real-world time-series data.

Has companion code repository: https://github.com/keisuke198619/metricNLDS

Mathematics Subject Classification ID

Applications of dynamical systems (37N99) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22) Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) (47B32)

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