For a finite set of balls of radius $r$, the $k$-fold cover is the space covered by at least $k$ balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the $k$-fold filtration of the centers. For $k=1$, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger $k$, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the $k$-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case $k=1$, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points. Our method also extends to the multicover bifiltration, composed of the $k$-fold filtrations for several values of $k$, with the same size and complexity bounds.

翻译：对于半径为$r$的有限个球组成的集合，$k$重覆盖是指至少被$k$个球覆盖的空间。固定球心并改变半径，我们得到一系列嵌套的空间，称为球心的$k$重滤子。当$k=1$时，该构造即为拓扑数据分析中常用的球并集滤子。当$k$较大时，它能在存在异常值的情况下实现更清晰的形状重构。我们提出了一种稀疏化算法来近似$k$重滤子的拓扑结构。该方法是对$k=1$这一经典情形下多种技术的组合与改进，最终得到一个线性规模的稀疏化结果，且相对于输入点的数量，其计算时间的期望近似为线性。我们的方法还可推广至多覆盖双滤子（由多个$k$值的$k$重滤子构成），且保持相同的规模与复杂度上界。