Author:
Smith Philip,Kurlin Vitaliy
Abstract
AbstractPersistent 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.
Funder
Engineering and Physical Sciences Research Council
Royal Society
Publisher
Springer Science and Business Media LLC
Reference43 articles.
1. Anosova, O., Kurlin, V.: Algorithms for continuous metrics on periodic crystals. arxiv:2205.15298 (2022)
2. Anosova, O., Kurlin, V.: An isometry classification of periodic point sets. In: Lecture Notes in Computer Science (Proceedings of DGMM), vol. 12708, 229–241 (2021)
3. Balasingham, J., Zamaraev, V., Kurlin, V.: Accelerating material property prediction using generically complete isometry invariants. Sci. Rep. 14, 10132 (2024)
4. Balasingham, J., Zamaraev, V., Kurlin, V.: Material property prediction using graphs based on generically complete isometry invariants. Integr. Mater. Manuf. Innov. (2024). https://doi.org/10.1007/s40192-024-00351-9
5. Barannikov, S.: The framed Morse complex and its invariants. Adv. Soviet Math. 21, 93–116 (1994)