Zigzag persistent homology and real-valued functions

Abstract

We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory.

Zigzag Persistence

“…4.3). We address this gap in a paper with Dmitriy Morozov [3], where we present an algorithm for computing the zigzag persistence intervals of a 1-parameter family of simplicial complexes on a fixed vertex set.…”

“…In particular, we draw the reader's attention to our paper with Morozov [3], where we develop a zigzag theory for topological spaces with a Morse-like function. This 'levelset zigzag persistence' can be related to the extended persistence of Cohen-Steiner et al [6], and is proved to be stable under perturbations of the function.…”

“…On the computational side, we do not implement any algorithms or conduct a performance analysis. We make no attempt to provide an algorithm to compute zigzag persistence at the homological level; this is carried out in [3]. On the mathematical side, we do not claim new theoretical results in the underlying quiver representation theory.…”

Towards Minimal Barcodes

“…. , σ m }, a persistence barcode [2] is a graphical representation of pairs of birth and death times as a collection of horizontal line segments (intervals) in a plane. If a cell σ s creates a homology class at time s, and it is destroyed at time t, 0 ≤ s < t ≤ m then the interval [s, t) is added to the corresponding persistence barcode (see [2]); If a cell σ s , 0 ≤ s ≤ m creates a homology class at time s and it survives along the process, then the interval [s, ∞) is added to the persistence barcode.…”

Distributed computation of coverage in sensor networks by homological methods

“…Finally, more flexible local coefficient systems (as in [14]) may be adaptable; if so, the techniques of this paper may be extended to give a distributed computation of the cohomology of simplicial sheaves, a problem whose relevance to networks is emerging [15,25,26]. We understand that an independent method for distributed homology computation is being investigated by Carlsson, de Silva, and Morozov, using the technique of zigzag persistence, as initiated in [4] and [5]. …”