2019/05/26 by Samir Chowdhury, Chowdhury, Samir · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Bijection #Combinatorics #Computational Geometry (cs.CG) #Computer science #Diagram #FOS: Computer and information sciences #FOS: Mathematics #Geodesic #Geometry #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Mathematics #Metric (unit) #Metric Geometry (math.MG) #Metric space #Persistence (discontinuity) #Pure mathematics #Regular polygon #Space (punctuation) #Topological and Geometric Data Analysis #Type (biology) #cs.CG #math.AT #math.MG
paper · pdf · doi:10.48550/arxiv.1905.10820
published in arXiv (Cornell University) (Cornell University) · 20 pages, 4 figures. Comments welcome!
arxiv created 2019/05/26 · openalex publication_date 2019/05/26 · arxiv updated 2019/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex combination. More specifically, given two persistence diagrams and a choice of metric, one obtains a bijection realizing the distance between the diagrams, and uses this bijection to linearly interpolate from one diagram to another. We prove that for several families of metrics, every geodesic in persistence diagram space arises as such a convex combination. For certain other choices of metrics, we explicitly construct infinite families of geodesics that cannot have this form.