Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces


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 propose kernel transfer operators, which 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. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume existing 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 they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigendecompositions of finite-rank RKHS operators. 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.

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  1. 1.

    Although this term is commonly used in the literature, \( \Phi \) is technically not a matrix, but a row vector in \( \mathbb {H}^n \). If \(\mathbb {H}\) is finite dimensional, \( \Phi \) can be viewed as a matrix.

  2. 2.

    For specific choices of \(\mathbb {P}(X)\) and \(\mathbb {P}(Y)\), the boundedness assumption can be replaced by a more general integrability assumption, i.e., \(\mathbb {E}_{\scriptscriptstyle X}[k(X,X)] < \infty \) and \(\mathbb {E}_{\scriptscriptstyle Y}[l(Y,Y)] < \infty \), so that \(\mathbb {H} \subset L^2({\mathbb {X}}, \mathbb {P}(X))\) and \(\mathbb {\mathbb {G}} \subset L^2({\mathbb {Y}}, \mathbb {P}(Y))\), respectively.

  3. 3.

    An observable could be, for example, a measurement or sensor probe.

  4. 4.

    This holds, for instance, for finite domains equipped with characteristic kernels, but not necessarily for continuous domains (Song et al. 2013).

  5. 5.

    See, e.g., Kloeden and Platen (2011).

  6. 6.

    ScienceOnline: The Pendulum and Galileo.

  7. 7.

    Reuters: French opposition, Twitter users slam Macron’s anti-fake-news plans.

  8. 8.

    Parts of the same articles are used several times to increase the size of the data set, this is thus a synthetic example, mainly to illustrate the concept.

  9. 9.

    We use the String Kernel Software implementation.


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This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”. Krikamol Muandet acknowledges fundings from the Faculty of Science, Mahidol University and the Thailand Research Fund (TRF). We would like to thank the reviewers for their helpful comments.

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Correspondence to Stefan Klus.

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Klus, S., Schuster, I. & Muandet, K. Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces. J Nonlinear Sci 30, 283–315 (2020). https://doi.org/10.1007/s00332-019-09574-z

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  • Reproducing kernel Hilbert spaces
  • Koopman operator
  • Perron–Frobenius operator
  • Eigendecompositions
  • Kernel mean embeddings

Mathematics Subject Classification

  • 37M10
  • 46E22
  • 34L16
  • 37L65