# Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators

arXiv: https://arxiv.org/abs/1805.12324 · HTML: https://mlsanity.com/paper/1805.12324/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2018.

## Built on by (most cited first, 7 of 7)

- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)
- [Koopman-based generalization bound: New aspect for full-rank weights](https://mlsanity.com/paper/2302.05825/) (2023)
- [Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator](https://mlsanity.com/paper/2305.13588/) (2023)
- [Koopman Spectrum Nonlinear Regulators and Efficient Online Learning](https://mlsanity.com/paper/2106.15775/) (2021)
- [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025)
- [Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices](https://mlsanity.com/paper/2410.02199/) (2024)
- [Geometric Dictionary Learning of Dynamical Systems with Optimal Transport](https://mlsanity.com/paper/2605.18276/) (2026)

## Abstract (arXiv)

The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius operators in reproducing kernel Hilbert spaces. Our metric includes the existing fundamental metrics for dynamical systems, which are basically defined with principal angles between some appropriately-chosen subspaces, as its special cases. We also describe the estimation of our metric from finite data. We empirically illustrate our metric with an example of rotation dynamics in a unit disk in a complex plane, and evaluate the performance with real-world time-series data.