2025/11/10 by Francisco J. Gozzi, Gozzi, Francisco J., Manuela Cerdeiro +3
Computer Science · Mathematics · #55N31 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2511.07674
openalex publication_date 2025/11/10 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28
A 1-Lipschitz map between compact metric spaces f\colon X→ Y induces a homomorphism of persistence modules on degree-d Vietoris--Rips persistent homology. We define the persistent cost of f from this induced homomorphism by quantifying the persistence carried by its kernel and cokernel modules. We prove that the persistent cost controls the interleaving distance between the degree-d Vietoris--Rips persistent homology modules of X and Y. Moreover, we obtain an explicit upper bound for the persistent cost in purely metric terms. Finally, we give a self-contained proof of the stability of the persistent cost introducing a Gromov-Hausdorff type distance for maps between compact metric spaces.