Given $n$ points $P$ in a Euclidean space, the Johnson-Lindenstrauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into $O(\log n)$ dimensions. In this paper, we show that the persistent homology of the distance function to $P$ is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to $P$ pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted $k$th nearest neighbor distances, for which $k=1$ and all weights equal to zero gives the usual distance to $P$.

@inproceedings{sheehy14persistent,
  Title = {The Persistent Homology of Distance Functions under Random Projection},
  Author = {Donald R. Sheehy},
  Booktitle = {Proceedings of the 30th annual Symposium on Computational Geometry},
  Pages = {328-334},
  Year = {2014}}