Metric on random dynamical systems with vector-valued reproducing kernel Hilbert spaces
This provides a mathematical framework for analyzing random dynamical systems in machine learning, though it appears to be an incremental extension of existing kernel-based methods.
The authors developed a general framework for constructing metrics on random nonlinear dynamical systems using vector-valued reproducing kernel Hilbert spaces and Perron-Frobenius operators, extending existing metrics for deterministic systems and specifying kernel maximal mean discrepancy for random processes. They empirically evaluated the metric with synthetic data for independence testing and real time series data for clustering tasks.
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.