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

2025-09-29

AI Summary

This work addresses the geometric comparison and interpolation of nonlinear dynamical systems from trajectory data. We propose a spectral-Grassmann Wasserstein metric that jointly incorporates the eigenvalue distribution of Koopman/transfer operators and their spectral projection subspaces—embedded in the Grassmann manifold—via an optimal transport formulation yielding a sampling-rate-invariant Wasserstein-type distance. Our key contribution is the first integration of spectral geometry and Grassmann structure into an optimal transport framework, enabling Fréchet mean computation for geometric averaging and smooth interpolation of dynamical systems. Experiments on synthetic and real-world datasets demonstrate that the proposed metric significantly outperforms conventional operator-based distances, yielding improved performance in dimensionality reduction and classification tasks. Moreover, it provides interpretable, continuous interpolation paths between dynamical systems, enhancing both analytical insight and practical utility.

Technology Category

• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Defining a metric for comparing dynamical systems using operator eigenvalues
• Developing sampling-invariant distance measure for nonlinear dynamics representation
• Enabling interpolation between systems through optimal transport on spectral components

Innovation

• Operator eigenvalues and projectors as system representation
• Optimal transport metric for dynamical system comparison
• Frequency-invariant distance with Fréchet mean computation