Given a set P of n points and a constant k, we are interested in computing the persistent homology of the ?Oech 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?±?μ) multiplicative factor, must preserve the persistent homology of the ?Oech filtration up to a factor of (1-?μ)^{-1}. Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the ?Oech filtration for the approximate k-distance of Buchet et al. are preserved up to a (1?±?μ) 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.
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