Generic families of finite metric spaces with identical or trivial 1-dimensional persistence
arXiv:2202.00577 · doi:10.1007/s41468-024-00177-6
Abstract
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.
arXiv commentThe latest version is at https://kurlin.org/projects/topological-data-analysis/trivial-persistence.pdf. This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s41468-024-00177-6
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