# Divisive cover

## TL;DR

This paper introduces the divisive cover, a top-down method for computing persistent homology efficiently at large filtration values, with complexity linear in the number of points for finite Euclidean subspaces.

## Contribution

The paper presents the divisive cover, a novel top-down filtered cover that simplifies persistent homology computation with size bounds independent of data size.

## Key findings

- Filtered nerve of the divisive cover is interleaved with the Čech complex.
- The size of the resulting filtered simplicial complex is bounded independently of data size.
- Computation time scales linearly with the number of points.

## Abstract

The aim of this paper is to present a method for computation of persistent homology that performs well at large filtration values. To this end we introduce the concept of filtered covers. We show that the persistent homology of a bounded metric space obtained from the \v{C}ech complex is the persistent homology of the filtered nerve of the filtered \v{C}ech cover. Given a parameter $\delta$ with $0 < \delta \le 1$ we introduce the concept of a $\delta$-filtered cover and show that its filtered nerve is interleaved with the \v{C}ech complex. Finally, we introduce a particular $\delta$-filtered cover, the divisive cover. The special feature of the divisive cover is that it is constructed top-down. If we disregard fine scale structure and $X$ is a finite subspace of euclidean space, then we obtain a filtered simplicial complex whose size is bounded by an upper bound independent of the cardinality of $X$. The time needed to compute this filtered simplicial complex depends linearly on the cardinality of $X$.