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Geometric Approaches on Persistent Homology (2103.06408v3)
Published 11 Mar 2021 in math.AT, cs.CG, and math.GT
Abstract: We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, we analyze the power filtration on unweighted graphs, and provide explicit bounds for the life spans of homology classes in persistence diagrams in all dimensions.