Dimensionality Reduction for k-distance Applied to Persistent Homology

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. Our main result answers an open question of Sheehy [Proc. SoCG, 2014], showing 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/(1 − ε). 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. [Comput. Geom., 2016] 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. Joint work with S. Arya, J.-D. Boissonnat, and M. Lotz. (I shall begin with a brief introduction to the area of Topological Data Analysis, so no prior background in TDA will be required).