6670
2017
eng
article
0
--
--
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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 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. In fact, most of the existing methods can be derived from our framework, providing a unifying view on the approximation of transfer operators. 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.
Journal of Machine Learning Research
1712.01572
no
under review
Stefan Klus
Ingmar Schuster
Ingmar Schuster
Krikamol Muandet
Numerical Mathematics
no-project
Schuster, Ingmar
7107
2018
eng
24
149
article
0
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2018-12-28
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A kernel-based approach to molecular conformation analysis
We present a novel machine learning approach to understanding conformation dynamics of biomolecules. The approach combines kernel-based techniques that are popular in the machine learning community with transfer operator theory for analyzing dynamical systems in order to identify conformation dynamics based on molecular dynamics simulation data. We show that many of the prominent methods like Markov State Models, EDMD, and TICA can be regarded as special cases of this approach and that new efficient algorithms can be constructed based on this derivation. The results of these new powerful methods will be illustrated with several examples, in particular the alanine dipeptide and the protein NTL9.
Journal of Chemical Physics
10.1063/1.5063533
yes
2018-12-04
Stefan Klus
Erlinda Körnig
Andreas Bittracher
Ingmar Schuster
Christof Schütte
Numerical Mathematics
Schütte, Christof
ECMATH-CH2
SFB-1114-B3
7791
2020
eng
109
131
304
incollection
Springer International
0
--
--
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Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
Advances om Dynamics, Optimization and Computation.
Series: Studies in Systems, Decision and Control. A volume dedicated to Michael Dellnitz on his 60th birthday
978-3-030-51264-4
10.1007/978-3-030-51264-4_5
yes
Mattes Mollenhauer
Erlinda Körnig
Oliver Junge
Ingmar Schuster
O. Schütze
Stefan Klus
Gary Froyland
Christof Schütte
S. Ober-Blobaum
K. Padberg-Gehle
Numerical Mathematics
Schütte, Christof
MODAL-MedLab
MODAL-Gesamt
MathPlusAA1-1
Mollenhauer, Mattes
Modeling and Simulation of Complex Processes
8186
2021
eng
221
article
Elsevier
0
--
--
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Feature space approximation for kernel-based supervised learning
Knowledge-Based Sytems
https://doi.org/10.1016/j.knosys.2021.106935
yes
publish
Patrick Gelss
Erlinda Körnig
Stefan Klus
Ingmar Schuster
Christof Schütte
Schütte, Christof
ECMATH-CH2
MODAL-MedLab
SFB-1114-B3
MODAL-Gesamt
Modeling and Simulation of Complex Processes