用最优传输定义新度量，高效比较动态系统谱特征。
A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems
- 将系统表示为特征值与投影算子的联合分布，用最优传输建模系统间距离。
- 在模拟与真实数据上优于传统算子距离，支持系统插值与降维分类。
- 对采样频率不变，计算高效且有样本收敛保证，适合机器学习应用。
从轨迹数据中估计动态系统的几何结构是机器学习应用中的主要挑战。Koopman算子与转移算子通过其谱分解提供了非线性动力系统的线性表示，为系统比较提供了自然框架。本文提出一种新方法，将每个系统表示为其联合算子特征值与谱投影算子的分布，并利用最优传输定义系统间的度量。该度量对轨迹采样频率不变，计算高效，具有有限样本收敛性保证，且可计算Fréchet均值，实现动态系统的插值。在模拟和真实数据集上的实验表明，该方法在降维与分类等机器学习任务中始终优于标准算子距离，能提供有意义的系统间插值。
原文摘要 · Abstract (English)
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.
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