Čech-Delaunay gradient flow and homology inference for self-maps

article Čech-Delaunay gradient flow and homology inference for self-maps published yes U. Bauer author Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Grzegorz Jablonski author 4483EF78-F248-11E8-B48F-1D18A9856A870000-0002-3536-9866 M. Mrozek author HeEd department We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems. https://research-explorer.ista.ac.at/download/15064/15065/2020_JourApplCompTopology_Bauer.pdf application/pdfno Springer Nature2020 eng Journal of Applied and Computational Topology 2367-1726 2367-173410.1007/s41468-020-00058-8 44455-480 U. Bauer, H. Edelsbrunner, G. Jablonski, M. Mrozek, Journal of Applied and Computational Topology 4 (2020) 455–480. Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. 2020. Čech-Delaunay gradient flow and homology inference for self-maps. Journal of Applied and Computational Topology. 4(4), 455–480. Bauer, U., Herbert Edelsbrunner, Grzegorz Jablonski, and M. Mrozek. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” Journal of Applied and Computational Topology. Springer Nature, 2020. <a href="https://doi.org/10.1007/s41468-020-00058-8">https://doi.org/10.1007/s41468-020-00058-8</a>. Bauer, U., et al. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” Journal of Applied and Computational Topology, vol. 4, no. 4, Springer Nature, 2020, pp. 455–80, doi:<a href="https://doi.org/10.1007/s41468-020-00058-8">10.1007/s41468-020-00058-8</a>. Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. Čech-Delaunay gradient flow and homology inference for self-maps. Journal of Applied and Computational Topology. 2020;4(4):455-480. doi:<a href="https://doi.org/10.1007/s41468-020-00058-8">10.1007/s41468-020-00058-8</a> U. Bauer, H. Edelsbrunner, G. Jablonski, and M. Mrozek, “Čech-Delaunay gradient flow and homology inference for self-maps,” Journal of Applied and Computational Topology, vol. 4, no. 4. Springer Nature, pp. 455–480, 2020. Bauer, U., Edelsbrunner, H., Jablonski, G., & Mrozek, M. (2020). Čech-Delaunay gradient flow and homology inference for self-maps. Journal of Applied and Computational Topology. Springer Nature. <a href="https://doi.org/10.1007/s41468-020-00058-8">https://doi.org/10.1007/s41468-020-00058-8</a> 150642024-03-04T10:47:49Z2024-03-04T10:54:04Z