Tracking the Persistence of Harmonic Chains: Barcode and Stability

TL;DR

This paper introduces the harmonic chain barcode, a new topological descriptor that tracks harmonic chains in data filtrations, demonstrating its stability and providing an algorithm for efficient computation, enriching topological data analysis tools.

Contribution

The paper presents the harmonic chain barcode, a novel stable topological descriptor with an algorithm for efficient computation, expanding the toolkit for topological data analysis.

Findings

Harmonic chain barcode is stable under data perturbations.

An $O(m^3)$ algorithm for computing the harmonic chain barcode.

Enables new applications in feature vectorization and machine learning.

Abstract

The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of data, the persistence barcode tracks the evolution of its homology groups. In this paper, we introduce a new type of barcode, called the harmonic chain barcode, which tracks the evolution of harmonic chains. In addition, we show that the harmonic chain barcode is stable. Given a filtration of a simplicial complex of size $m$, we present an algorithm to compute its harmonic chain barcode in $O(m^3)$ time. Consequently, the harmonic chain barcode can enrich the family of topological descriptors in applications where a persistence barcode is applicable, such as feature vectorization and machine learning.