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Dimensionality Reduction for k-Distance Applied to Persistent Homology

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Shreya Arya
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  • PersonId : 1040289
Jean-Daniel Boissonnat
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  • PersonId : 830857
Kunal Dutta
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  • PersonId : 988195

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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Dates and versions

hal-02873730 , version 1 (18-06-2020)

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Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta, Martin Lotz. Dimensionality Reduction for k-Distance Applied to Persistent Homology. SoCG 2020 - 36th International Symposium on Computational Geometry, Jun 2020, Zurich, Switzerland. ⟨10.4230/LIPIcs.SoCG.2020.10⟩. ⟨hal-02873730⟩
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