A Sparse Multicover Bifiltration of Linear Size

The k-cover of a point cloud X in ℝd 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(|X|d+1) simplices. We introduce a (1+ε)-approximation of the multicover bifiltration of linear size O(|X|), for fixed d and ε. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension, yielding analogous results.