Paper Info
Title
Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
Abstract
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.
Year
DOI
Venue
2020
10.1007/s00332-019-09574-z
Journal of Nonlinear Science
Keywords
Field
DocType
Reproducing kernel Hilbert spaces, Koopman operator, Perron–Frobenius operator, Eigendecompositions, Kernel mean embeddings, 37M10, 46E22, 34L16, 37L65
Kernel (linear algebra),Hilbert space,Eigenfunction,Conditional probability distribution,Similarity measure,Algebra,Mathematical analysis,Dynamical systems theory,Operator (computer programming),Mathematics,Transfer operator
Journal
Volume
Issue
ISSN
30
1
0938-8974
Citations 
PageRank 
References 
4
0.49
14
Authors
3
Name
Order
Citations
PageRank
Stefan Klus1176.09
Ingmar Schuster253.21
Krikamol Muandet321117.10