A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems

• URL: http://arxiv.org/abs/2509.24920v1
• Date: Mon, 29 Sep 2025 15:24:05 GMT
• Title: A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems
• Authors: Thibaut Germain, Rémi Flamary, Vladimir R. Kostic, Karim Lounici,
• Abstract summary: We propose a novel approach representing each system as a distribution of its joint operator eigenvalues and spectral projectors.<n> Experiments on simulated and real-world datasets show that our approach consistently outperforms standard operator-based distances in machine learning applications.
• Score: 13.799022330476236
• License: http://creativecommons.org/licenses/by/4.0/
• 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 Fr\'echet 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.

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