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Geometric Approaches on Persistent Homology

by   Henry Adams, et al.
The University of Texas at Dallas

We introduce several geometric notions, including thick-thin decompositions and 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 “size” 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.


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