  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.

Computational Geometry;Numerical Analysis;Algebraic Topology;Numerical Analysis