Skip to Main content Skip to Navigation
Conference papers

Dimensionality Reduction for k-Distance Applied to Persistent Homology

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.
Complete list of metadatas

Cited literature [35 references]  Display  Hide  Download

https://hal.inria.fr/hal-02873730
Contributor : Jean-Daniel Boissonnat <>
Submitted on : Thursday, June 18, 2020 - 2:56:33 PM
Last modification on : Wednesday, September 16, 2020 - 10:50:32 AM

File

LIPIcs-SoCG-2020-dim-reduc-PH....
Files produced by the author(s)

Identifiers

Citation

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

Share

Metrics

Record views

36

Files downloads

110