Title: A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems
Primary area: learning on time series and dynamical systems
Keywords: dynamical system, optimal transport, transfer operator, koopman operator

Abstract: The geometry of dynamical systems estimated from trajectory data is a major challenge for machine learning applications. Koopman and transfer operators provide a linear representation of nonlinear dynamics through their spectral decomposition, offering a natural framework for comparison. We propose a novel approach representing each system as a distribution of its joint operator eigenvalues and spectral projectors and defining a metric between systems leveraging optimal transport. The proposed metric is invariant to the sampling frequency of trajectories. It is also computationally efficient, supported by finite-sample convergence guarantees, and enables the computation of Fréchet means, providing interpolation between dynamical systems. Experiments on simulated and real-world datasets show that our approach consistently outperforms standard operator-based distances in machine learning applications, including dimensionality reduction and classification, and provides meaningful interpolation between dynamical systems.

TL;DR: A state-of-the-art operator-based metric for dynamical systems, robust to sampling, opening up to machine learning applications, and meaningful interpolation.