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

The development of a metric on structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we consider a general framework to construct metrics on {\em random} nonlinear dynamical systems, which are defined with the Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). Here, vvRKHSs are employed to design mathematically manageable metrics and also to introduce $L^2(\Omega)$-valued kernels, which are necessary to handle the randomness in systems. Our metric is a natural extension of existing metrics for {\em deterministic} systems, and can give a specification of the kernel maximal mean discrepancy of random processes. Moreover, by considering the time-wise independence of random processes, we discuss the 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.