Harmonic Chain Barcode and Stability

Abstract

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.