Tracking the Persistence of Harmonic Chains: 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 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(m3) 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.
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