@article{MollenhauerKlusSchuetteetal.2022, author = {Mollenhauer, Mattes and Klus, Stefan and Sch{\"u}tte, Christof and Koltai, P{\´e}ter}, title = {Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence}, volume = {23}, journal = {Journal of Machine Learning Research}, number = {327}, arxiv = {http://arxiv.org/abs/2004.00891}, pages = {1 -- 34}, year = {2022}, abstract = {We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and obtain several asymptotic results as well as finite sample error bounds. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.}, language = {en} } @inproceedings{LueckeKoltaiWinkelmannetal.2022, author = {L{\"u}cke, Marvin and Koltai, Peter and Winkelmann, Stefanie and Molkethin, Nora and Heitzig, Jobst}, title = {Discovering collective variable dynamics of agent-based models}, booktitle = {25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022}, doi = {https://doi.org/10.15495/EPub_UBT_00006809}, year = {2022}, abstract = {Analytical approximations of the macroscopic behavior of agent-based models (e.g. via mean-field theory) often introduce a significant error, especially in the transient phase. For an example model called continuous-time noisy voter model, we use two data-driven approaches to learn the evolution of collective variables instead. The first approach utilizes the SINDy method to approximate the macroscopic dynamics without prior knowledge, but has proven itself to be not particularly robust. The second approach employs an informed learning strategy which includes knowledge about the agent-based model. Both approaches exhibit a considerably smaller error than the conventional analytical approximation.}, language = {en} }