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.

翻译：持续性条形码是数据的一种拓扑描述符，在拓扑数据分析中发挥着基础性作用。给定数据的滤过结构，持续性条形码追踪其同调群的演化过程。本文引入一种新型条形码，称为调和链条形码，用于追踪调和链的演化。此外，我们证明了调和链条形码具有稳定性。给定规模为$m$的单纯复形的滤过结构，我们提出一种算法，可在$O(m^3)$时间内计算其调和链条形码。因此，在持续性条形码适用的应用场景中（如特征向量化与机器学习），调和链条形码能够丰富拓扑描述符的家族。