{"ID":2863699,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.24920","arxiv_id":"2509.24920","title":"A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems","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.","short_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 rep...","url_abs":"https://arxiv.org/abs/2509.24920","url_pdf":"https://arxiv.org/pdf/2509.24920v1","authors":"[\"Thibaut Germain\",\"Rémi Flamary\",\"Vladimir R. Kostic\",\"Karim Lounici\"]","published":"2025-09-29T15:24:05Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}