Families of point sets with identical 1D persistence
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Persistent homology is a popular and useful tool for analysing point sets, revealing features of a point set that can be used to highlight key information, distinguish point sets and as an input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change in persistence under perturbations, but it does not provide a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing point sets, which is clearly evident for point sets that have trivial persistence. We describe large families of point sets that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish point sets up to isometry.
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