2019/08/17 by John F. Jardine, Jardine, J. F. · 1 citation
Computer Science · Mathematics · #55U10 #62H30 #68R10 #Algebraic Topology (math.AT) #Data Management and Algorithms #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1908.06323
openalex publication_date 2019/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper presents explicit assumptions for the existence of interleaving homotopy equivalences of both Vietoris-Rips and Lesnick complexes associated to an inclusion of data sets. Consequences of these assumptions are investigated on the space level, and for corresponding hierarchies of clusters and their sub-posets of branch points. Hierarchy posets and branch point posets admit a calculus of least upper bounds, which is used to show that the map of branch points associated to the inclusion of data sets is a controlled homotopy equivalence.