TY - JOUR A1 - Bittracher, Andreas A1 - Schütte, Christof T1 - A probabilistic algorithm for aggregating vastly undersampled large Markov chains JF - Physica D: Nonlinear Phenomena Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-75874 VL - 416 ER - TY - JOUR A1 - Bittracher, Andreas A1 - Klus, Stefan A1 - Hamzi, Boumediene A1 - Schütte, Christof T1 - Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds JF - Journal of Nonlinear Science N2 - We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to the previous parameterization approaches. Y1 - 2021 U6 - https://doi.org/10.1007/s00332-020-09668-z VL - 31 ER - TY - GEN A1 - Bittracher, Andreas A1 - Schütte, Christof ED - Junge, Oliver ED - Schütze, O. ED - Froyland, Gary ED - Ober-Blobaum, S. ED - Padberg-Gehle, E. T1 - A weak characterization of slow variables in stochastic dynamical systems T2 - Advances in Dynamics, Optimization and Computation. Series: Studies in Systems, Decision and Control. A volume dedicated to Michael Dellnitz on the occasion of his 60th birthday Y1 - 2020 SN - 978-3-030-51264-4 U6 - https://doi.org/10.1007/978-3-030-51264-4_6 VL - 304 SP - 132 EP - 150 PB - Springer International ER - TY - GEN A1 - Bittracher, Andreas A1 - Schütte, Christof T1 - A probabilistic algorithm for aggregating vastly undersampled large Markov chains N2 - Model reduction of large Markov chains is an essential step in a wide array of techniques for understanding complex systems and for efficiently learning structures from high-dimensional data. We present a novel aggregation algorithm for compressing such chains that exploits a specific low-rank structure in the transition matrix which, e.g., is present in metastable systems, among others. It enables the recovery of the aggregates from a vastly undersampled transition matrix which in practical applications may gain a speedup of several orders of mag- nitude over methods that require the full transition matrix. Moreover, we show that the new technique is robust under perturbation of the transition matrix. The practical applicability of the new method is demonstrated by identifying a reduced model for the large-scale traffic flow patterns from real-world taxi trip data. T3 - ZIB-Report - 20-21 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78688 UR - https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/7587 SN - 1438-0064 ER - TY - JOUR A1 - Bittracher, Andreas A1 - Moschner, Johann A1 - Koksch, Beate A1 - Netz, Roland A1 - Schütte, Christof T1 - Exploring the locking stage of NFGAILS amyloid fibrillation via transition manifold analysis JF - The European Physical Journal B Y1 - 2021 U6 - https://doi.org/10.1140/epjb/s10051-021-00200-0 VL - 94 ER -