Zigzag Persistence

TL;DR

Zigzag persistence extends persistent homology to analyze topological features across complex data sets, providing new theoretical and algorithmic tools for topological data analysis and statistics.

Contribution

The paper introduces zigzag persistence, a novel methodology that generalizes persistent homology to handle more complex data relationships, with foundational theory and algorithms.

Findings

Developed theoretical framework for zigzag persistence

Created algorithms for computing zigzag persistence

Demonstrated applications in topological data analysis

Abstract

We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics.