A Sparse Multicover Bifiltration of Linear Size

The $k$-cover of a point cloud $X$ in ${\mathbb{R}}^d$ at radius $r$ is the set of all points within distance $r$ of at least $k$ points of $X$. By varying $r$ and $k$ we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has $O(\mathrm{\mid }X{\mathrm{\mid }}^{d+1})$ simplices. We introduce a $(1+\epsilon )$-approximation of the multicover bifiltration of linear size $O(\mathrm{\mid }X\mathrm{\mid })$, for fixed $d$ and $\epsilon$. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension, yielding analogous results.