Apex Representatives

2025-02-24

AI Summary

This paper addresses the challenge of constructing barcode representatives in zigzag persistent homology. We propose an efficient algorithm that transforms a zigzag filtration into a level-set zigzag, constructs a Mayer–Vietoris pyramid and infinite homology strip, and introduces the novel concept of an “apex”—a vertex characterizing the origin of indecomposable summands (diamonds). Compatible representative cycles are then generated via a backtracking mechanism. Our method integrates level-set topology, zigzag homology theory, matrix diagonalization, and a divide-and-conquer cycle-lifting strategy. The algorithm lifts p-dimensional cycles to apex representatives in O(p·m log m) time and recovers representatives in total time O(log m + C), markedly improving upon prior exponential-time approaches. This work achieves, for the first time, a unification of theoretical constructibility and algorithmic computability for barcode representatives in zigzag persistence.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Find barcode representatives in zigzag filtration
• Convert zigzag to levelset zigzag function
• Algorithm for apex representative cycles from persistence

Innovation

• Convert zigzag to levelset zigzag
• Generate Mayer-Vietoris pyramid
• Lift cycles to apex representatives