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 that represents each system as a distribution over its joint operator eigenvalues and spectral projectors and defines 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.
Cite
Text
Germain et al. "A Spectral-Grassmann Wasserstein Metric for Operator Representations of Dynamical Systems." International Conference on Learning Representations, 2026.Markdown
[Germain et al. "A Spectral-Grassmann Wasserstein Metric for Operator Representations of Dynamical Systems." International Conference on Learning Representations, 2026.](https://mlanthology.org/iclr/2026/germain2026iclr-spectralgrassmann/)BibTeX
@inproceedings{germain2026iclr-spectralgrassmann,
title = {{A Spectral-Grassmann Wasserstein Metric for Operator Representations of Dynamical Systems}},
author = {Germain, Thibaut and Flamary, Rémi and Kostic, Vladimir R and Lounici, Karim},
booktitle = {International Conference on Learning Representations},
year = {2026},
url = {https://mlanthology.org/iclr/2026/germain2026iclr-spectralgrassmann/}
}