We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in $\mathbb{R}^d$. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size $O(n^{\lceil d/2 \rceil})$. In contrast, our construction uses only $O(n)$ simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in $d+1$ dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a $(d+1)$-dimensional Voronoi construction similar to the standard Delaunay filtration complex. We also, show how this complex can be efficiently constructed.

@inproceedings{sheehy21sparse,
    title = {A Sparse Delaunay Filtration},
    author = {Donald R. Sheehy},
    booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
    pages = {58:1--58:16},
    year = {2021}}