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 Cech filtration of P for the kdistance, 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 +/ e) multiplicative factor, must preserve the persistent homology of the Cech filtration up to a factor of (1e)^(1). Our results also show that the VietorisRips and Delaunay filtrations for the kdistance, as well as the Cech filtration for the approximate kdistance of Buchet et al. [J. Comput. Geom., 2016] are preserved up to a (1 +/ e) 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 lowdimensional submanifold, obtaining embeddings having the dimension bounds of Lotz [Proc. Roy. Soc., 2019] and Clarkson [Proc. SoCG, 2008] respectively. Our results also work in the terminal dimensionality reduction setting, where the distance of any point in the original ambient space, to any point in P, needs to be approximately preserved.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.05897
 Bibcode:
 2021arXiv211005897A
 Keywords:

 Computer Science  Computational Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Numerical Analysis;
 5508;
 68W27;
 68W20;
 I.3.5
 EPrint:
 18 pages