# Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces

arXiv: https://arxiv.org/abs/1712.01572 · HTML: https://mlsanity.com/paper/1712.01572/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Dec 2017.

## Built on by (most cited first, 30 of 48)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Estimating Koopman operators with sketching to provably learn large scale dynamical systems](https://mlsanity.com/paper/2306.04520/) (2023)
- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)
- [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023)
- [Koopman-based generalization bound: New aspect for full-rank weights](https://mlsanity.com/paper/2302.05825/) (2023)
- [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Learning the Infinitesimal Generator of Stochastic Diffusion Processes](https://mlsanity.com/paper/2405.12940/) (2024)
- [Learning Good State and Action Representations via Tensor Decomposition](https://mlsanity.com/paper/2105.01136/) (2021)
- [Sobolev Norm Learning Rates for Conditional Mean Embeddings](https://mlsanity.com/paper/2105.07446/) (2021)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2505.18671/) (2025)
- [Koopman-based spectral clustering of directed and time-evolving graphs](https://mlsanity.com/paper/2204.02951/) (2022)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [Online Estimation of the Koopman Operator Using Fourier Features](https://mlsanity.com/paper/2212.01503/) (2022)
- [Learning dynamical systems from data: Gradient-based dictionary optimization](https://mlsanity.com/paper/2411.04775/) (2024)
- [Forward Operator Estimation in Generative Models with Kernel Transfer Operators](https://mlsanity.com/paper/2112.00305/) (2021)
- [KPF-AE-LSTM: A Deep Probabilistic Model for Net-Load Forecasting in High Solar Scenarios](https://mlsanity.com/paper/2203.04401/) (2022)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [Nonparametric Sparse Online Learning of the Koopman Operator](https://mlsanity.com/paper/2405.07432/) (2024)
- [Data-Driven Stochastic Optimal Control in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2407.16407/) (2024)
- [How deep is your network? Deep vs. shallow learning of transfer operators](https://mlsanity.com/paper/2509.19930/) (2025)

## Abstract (arXiv)

Transfer operators such as the Perron--Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings in the machine learning community. Moreover, numerical methods to compute empirical estimates of these embeddings are akin to data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel-based approaches is that these methods can be applied to any domain where a similarity measure given by a kernel is available. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.