Dimensionality Reduction for k-Distance Applied to Persistent Homology

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

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Author Details

Shreya Arya
  • Duke University, Durham, NC, USA
Jean-Daniel Boissonnat
  • Université Côte d'Azur, INRIA, Sophia-Antipolis, France
Kunal Dutta
  • Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Warsaw, Poland
Martin Lotz
  • Mathematics Institute, University of Warwick, Coventry, United Kingdom


We thank the reviewers for their helpful comments and suggestions.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Computational geometry
  • Dimensionality reduction
  • Johnson-Lindenstrauss lemma
  • Topological Data Analysis
  • Persistent Homology
  • k-distance
  • distance to measure


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