The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of the space of data, a persistence barcode tracks the evolution of its homological features. In this paper, we introduce a novel type of barcode, referred to as the canonical barcode of harmonic chains, or harmonic chain barcode for short, which tracks the evolution of harmonic chains. As our main result, we show that the harmonic chain barcode is stable and it captures both geometric and topological information of data. Moreover, given a filtration of a simplicial complex of size $n$ with $m$ time steps, we can compute its harmonic chain barcode in $O(m^2n^{\omega} + mn^3)$ time, where $n^\omega$ is the matrix multiplication time. Consequently, a harmonic chain barcode can be utilized in applications in which a persistence barcode is applicable, such as feature vectorization and machine learning. Our work provides strong evidence in a growing list of literature that geometric (not just topological) information can be recovered from a persistence filtration.

翻译：持久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。给定数据空间的滤过结构，持久性条形码追踪其同调特征的演化。本文引入一种新型条形码，称为调和链的典范条形码（简称调和链条形码），它追踪调和链的演化。作为主要结果，我们证明了调和链条形码具有稳定性，并且能够同时捕获数据的几何与拓扑信息。此外，给定一个规模为 $n$、具有 $m$ 个时间步的单纯复形的滤过结构，我们可以在 $O(m^2n^{\omega} + mn^3)$ 时间内计算其调和链条形码，其中 $n^\omega$ 为矩阵乘法时间。因此，调和链条形码可应用于持久性条形码适用的场景，例如特征向量化与机器学习。我们的工作为日益增多的文献提供了有力证据，表明从持久性滤过中可以恢复出几何（而不仅仅是拓扑）信息。