A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems
This work addresses the problem of comparing nonlinear dynamical systems for machine learning practitioners, offering a more effective and invariant metric, though it is incremental in improving existing operator-based methods.
The paper tackles the challenge of comparing dynamical systems from trajectory data by proposing a novel metric based on optimal transport that represents systems through their operator eigenvalues and spectral projectors. The results show consistent outperformance over standard operator-based distances in tasks like dimensionality reduction and classification, with computational efficiency and invariance to sampling frequency.
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.