Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators
This work addresses a long-term challenge in pattern recognition and machine learning for structural data analysis, but it appears incremental as it builds upon existing fundamental metrics.
The authors tackled the problem of comparing nonlinear dynamical systems by developing a general metric using Perron-Frobenius operators in reproducing kernel Hilbert spaces, which includes existing metrics as special cases and is empirically evaluated on real-world time-series data.
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.