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DOI: 10.4230/LIPIcs.SoCG.2020.10

Dimensionality Reduction for k-Distance Applied to Persistent Homology

Authors: Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta, and Martin Lotz

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract

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.

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Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta, and Martin Lotz. Dimensionality Reduction for k-Distance Applied to Persistent Homology. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@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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Dimensionality Reduction for k-Distance Applied to Persistent Homology

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