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Dimensionality Reduction for k-Distance Applied to Persistent Homology

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

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Subject Classification

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

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SoCG.2020.10

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

Funding

The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions).

Acknowledgements

We thank the reviewers for their helpful comments and suggestions.

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