Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators
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.
* Summary : This paper introduces a general framework to compute a metric between deterministic dynamical systems. This framework relies on the Koopman operator computed in RKHS, which was previously introduced in [Dynamic mode decomposition with reproducing kernels for koopman spectral analysis, Kawahara, NIPS18]. The framework is generic and the authors show that it encompasses previous metrics introduced for linear dynamical systems [Subspace angles between ARMA models, De Cock and De Moor, Systems and Controls Letter 2002] and non-linear ones [Binet-Cauchy kernels on dynamical systems and its application to the analysis of dynamical scenes, Vishwanathan et al.. 2007]. An algorithm is proposed in order to compute this metric from the observation of multiple trajectories generated by these different dynamical systems. Visualization experiments on synthetic and real data show that this metric seems to separate dynamical systems with different properties.  * Review :  This paper is interesting and brings a new framework to study dynamical systems with the Koopman operator, which makes it a significant contribution.  Clarity should be improved. I think that the notations and objects introduced need to be greatly slimmed down to simplify the exposition, just keeping the minimum to reach the main goals of the paper. Further, it would be nice to provide more intuition on the different objects which are introduced in Subsection 3.1.  When it comes to experiments, the visualizations provided are nice, but running a true machine learning algorithm would have been better. For instance, on the real time series, it would have been nice to run a clustering algorithm on top of the metrics computed, to see for instance if it is able to retrieve the original classes, yielding more quantitative measurements.  Overall, I would tend in favor of accepting this paper, provided it is clarified.  * Detailed remarks:  - As far as I understand, $\tilde{A}_m$ is introduced to allow convergence even in case of admissibility and not semi-stability (cf Prop 3.3). This is a nice results, but it is not exploited in the experiments. Why, or why not?  - In Sec 3.1, it would be nice to provide more intuition on the different objects h, L_h, and the subspaces H', H''. What do they represent? Maybe notations could be simplified to introduce only L_h and remove h?  - Is the object $d_m^T$ a positive definite kernel? Or a metric? I have not seen it stated anywhere, or any hint of proof of it (I am not sure it is even named). Since this object is very crucial, it may deserve a definition in its own right and much more explanations or intuitions. What is the role of the parameter $m$ for instance?  - What was the motivation for introducing $A_m^T$ from $d_m^T$? Is it only to take into account all previous works in this framework, or is there something more subtle at work? Is it obvious that $A_m^T$ defines a positive definite kernel?  - Indices and notations need to be slimmed to the minimum over all the paper. For instance, I am not sure that Proposition 2.1 brings much to the main goal of the paper, as it is not used anywhere else. Using multiple $A$ in 4.1 brings confusion.