Sparse Higher Order Čech Filtrations

Abstract

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.