Geometric Approaches on Persistent Homology
arXiv:2103.06408 · doi:10.1137/21M1422914
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.
References in corpus (6)
- Ripser: efficient computation of Vietoris-Rips persistence barcodes
- The Vietoris-Rips complexes of a circle
- Nerve complexes of circular arcs
- 1-Dimensional Intrinsic Persistence of Geodesic Spaces
- Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
- Metric Thickenings, Borsuk-Ulam Theorems, and Orbitopes