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 because it is 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 $$(1+\varepsilon )$$ -approximation of the multicover bifiltration of linear size $$O(|X|)$$ , for fixed d and $$\varepsilon $$ . The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension, yielding analogous results.