# Kernel methods for detecting coherent structures in dynamical data

arXiv: https://arxiv.org/abs/1904.07752 · HTML: https://mlsanity.com/paper/1904.07752/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2019.

## Built on by (most cited first, 13 of 13)

- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Koopman-based spectral clustering of directed and time-evolving graphs](https://mlsanity.com/paper/2204.02951/) (2022)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Transfer operators on graphs: Spectral clustering and beyond](https://mlsanity.com/paper/2305.11766/) (2023)
- [Learning Transfer Operators by Kernel Density Estimation](https://mlsanity.com/paper/2210.03124/) (2022)
- [Learning graphons from data: Random walks, transfer operators, and spectral clustering](https://mlsanity.com/paper/2507.18147/) (2025)
- [How deep is your network? Deep vs. shallow learning of transfer operators](https://mlsanity.com/paper/2509.19930/) (2025)
- [Improving Estimation of the Koopman Operator with Kolmogorov-Smirnov Indicator Functions](https://mlsanity.com/paper/2306.05945/) (2023)
- [Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian](https://mlsanity.com/paper/2407.12864/) (2024)
- [Unsupervised Discovery of Extreme Weather Events Using Universal Representations of Emergent Organization](https://mlsanity.com/paper/2304.12586/) (2023)
- [Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors](https://mlsanity.com/paper/2310.12156/) (2023)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)

## Abstract (arXiv)

We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that it can be obtained by optimizing the variational approach for Markov processes (VAMP) score. As a result, we show that coherent sets of particle trajectories can be computed by kernel CCA. We demonstrate the efficiency of this approach with several examples, namely the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Finally, we propose a straightforward generalization of dynamic mode decomposition (DMD) called coherent mode decomposition (CMD). Our results provide a generic machine learning approach to the computation of coherent sets with an objective score that can be used for cross-validation and the comparison of different methods.