Apex Representatives

TL;DR

This paper introduces an algorithm to compute barcode representatives for zigzag filtrations by converting them into levelset zigzags, leveraging Mayer-Vietoris pyramids, and efficiently lifting cycles.

Contribution

It presents a novel method to find barcode representatives in zigzag filtrations using levelset functions and Mayer-Vietoris pyramids, with an efficient lifting algorithm.

Findings

Algorithm computes apex representatives in $O(p \, m \, ext{log} \, m)$ time.

Reconstruction of zigzag representatives is achieved in $O(\text{log} \, m + C)$ time.

Method effectively bridges ordinary persistence and zigzag barcode computations.

Abstract

Given a zigzag filtration, we want to find its barcode representatives, i.e., a compatible choice of bases for the homology groups that diagonalize the linear maps in the zigzag. To achieve this, we convert the input zigzag to a levelset zigzag of a real-valued function. This function generates a Mayer-Vietoris pyramid of spaces, which generates an infinite strip of homology groups. We call the origins of indecomposable (diamond) summands of this strip their apexes and give an algorithm to find representative cycles in these apexes from ordinary persistence computation. The resulting representatives map back to the levelset zigzag and thus yield barcode representatives for the input zigzag. Our algorithm for lifting a $p$-dimensional cycle from ordinary persistence to an apex representative takes $O(p \cdot m \log m)$ time. From this we can recover zigzag representatives in time $O(\log m + C)$, where $C$ is the size of the output.