20.500.12556/RUL-152587 Critical edges in Rips complexes and persistence We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X, d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function $d$. When $d$ attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of $d$. We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of $d$ that induce a change in persistence. We conclude with a proof that each locally isolated minimum of $d$ can be detected through persistent homology with selective Rips complexes. The results of this paper offer the first interpretation of critical scales of persistent homology (obtained via Rips complexes) for general compact metric spaces. persistent homology Rips complex critical simplex reconstruction result true false true Angleški jezik Ni določen Članek v reviji 2023-11-29 15:03:56 2023-11-29 15:04:00 2023-11-30 03:42:11 0000-00-00 00:00:00 2023 0 0 25 str. iss. 6, art. 326 Vol. 20 Dec. 2023 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.1 1660-5446 10.1007/s00009-023-02533-9 171038723 s00009-023-02533-9.pdf s00009-023-02533-9.pdf 1 BE8572C9A9FAC5138C847804B47437C4 7b322afd3e439b37d0eaabc84b8741b8f1e9ebd66e8353772ee3c383f1e32fa4 929c5ee6-8ec0-11ee-9c6c-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=178144 https://link.springer.com/article/10.1007/s00009-023-02533-9 1 25c7ae42-8ec0-11ee-9c6c-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=178143 Fakulteta za matematiko in fiziko Fakulteta za računalništvo in informatiko 0 0 0