Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy [Proc. SoCG, 2014]. We show that any linear transformation that preserves pairwise distances up to a (1±ε) multiplicative factor, must preserve the persistent homology of the Čech filtration up to a factor of (1-ε)^{-1}. Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Čech filtration for the approximate k-distance of Buchet et al. are preserved up to a (1±ε) factor. We also prove extensions of our main theorem, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional manifold, obtaining the target dimension bounds of Lotz [Proc. Roy. Soc. , 2019] and Clarkson [Proc. SoCG, 2008 ] respectively.
@InProceedings{arya_et_al:LIPIcs.SoCG.2020.10, author = {Arya, Shreya and Boissonnat, Jean-Daniel and Dutta, Kunal and Lotz, Martin}, title = {{Dimensionality Reduction for k-Distance Applied to Persistent Homology}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.10}, URN = {urn:nbn:de:0030-drops-121682}, doi = {10.4230/LIPIcs.SoCG.2020.10}, annote = {Keywords: Dimensionality reduction, Johnson-Lindenstrauss lemma, Topological Data Analysis, Persistent Homology, k-distance, distance to measure} }
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