TY - JOUR A1 - Klus, Stefan A1 - Schuster, Ingmar A1 - Muandet, Krikamol T1 - Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces JF - Journal of Machine Learning Research N2 - 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. Y1 - 2017 ER - TY - GEN A1 - Bittracher, Andreas A1 - Koltai, Péter A1 - Klus, Stefan A1 - Banisch, Ralf A1 - Dellnitz, Michael A1 - Schütte, Christof T1 - Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics N2 - We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics. T3 - ZIB-Report - 17-22 KW - metastability KW - slow dynamics KW - effective dynamics KW - transition manifold KW - embedding KW - transfer operator KW - reaction coordinate Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-63822 SN - 1438-0064 ER -