Dimensionality Reduction for $k$-Distance Applied to Persistent Homology

TL;DR

This paper investigates how linear dimensionality reduction affects the persistent homology of k-distance filtrations, showing that certain embeddings preserve topological features up to a small error, with extensions to low-dimensional and manifold data.

Contribution

It proves that linear maps preserving pairwise distances approximately also preserve persistent homology of k-distance filtrations, extending results to low-dimensional and manifold settings.

Findings

Linear embeddings preserve persistent homology up to a (1 +/- e) factor.

Results extend to data on low-dimensional submanifolds.

Applicable in terminal dimensionality reduction scenarios.

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 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 +/- e) multiplicative factor, must preserve the persistent homology of the Cech filtration up to a factor of (1-e)^(-1). Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Cech filtration for the approximate k-distance 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 low-dimensional 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.