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

翻译：$\mathbb{R}^{d}$中点云$X$在半径$r$下的$k$-覆盖是指$X$中至少$k$个点距离$r$范围内的所有点集合。通过变化阶数$k$和半径$r$，我们得到一个称为多重覆盖双滤过的双参数滤过。该双滤过因其无参数特性及对异常值的鲁棒性近期受到关注。然而，其计算困难：已知最小的等价单纯双滤过具有$O(|X|^{d+1})$个单纯形，其中$d$为维度。本文针对固定$d$和$\epsilon$，提出一种具有线性规模$O(|X|)$的多重覆盖$(1+\epsilon)$-近似方法。该方法同样适用于有界倍增维度度量空间上的细分Rips双滤过，以获得类似结果。