A Kernel Two-sample Test for Dynamical Systems

Evaluating whether data streams were generated by the same distribution is at the heart of many machine learning problems, e.g. to detect changes. This is particularly relevant for data generated by dynamical systems since they are essential for many real-world processes in biomedical, economic, or engineering systems. While kernel two-sample tests are powerful for comparing independent and identically distributed random variables, no established method exists for comparing dynamical systems. The key problem is the critical independence assumption, which is inherently violated in dynamical systems. We propose a novel two-sample test for dynamical systems by addressing three core challenges: we (i) introduce a novel notion of mixing that captures autocorrelations in a relevant metric, (ii) propose an efficient way to estimate the speed of mixing purely from data, and (iii) integrate these into established kernel-two sample tests. The result is a data-driven method for comparison of dynamical systems that is easy to use in practice and comes with sound theoretical guarantees. In an example application to anomaly detection from human walking data, we show that the test readily applies without the need for feature engineering, heuristics, and human expert knowledge

Bounded weighted composition operators on functional quasi-Banach spaces and stability of dynamical systems

In this paper, we investigate the boundedness of weighted composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We show that the boundedness of a weighted composition operator strongly limits the dynamics of the original map, and it provides us an effective method to investigate properties of weighted composition operators via the theory of dynamical system. As a result, we prove that only an affine map can induce a bounded composition operator on an arbitrary quasi-Banach space continuously included in the space of entire functions of one-variable. We also obtain the same result for bounded weighted composition operators on infinite dimensional quasi-Banach space under certain condition for weights. We also prove that any polynomial automorphism except an affine transform cannot induce a bounded weighted composition operator with non-vanishing weight on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under certain conditions.Comment: We generalize the previous version to the weighted cas