Critical edges in Rips complexes and persistence

Goričan, Peter (avtor), Virk, Žiga (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: persistent homology, Rips complex, critical simplex, reconstruction result
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko ; FRI - Fakulteta za računalništvo in informatiko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2023
• Št. strani: 25 str.
• Številčenje: Vol. 20, iss. 6, art. 326
• PID: 20.500.12556/RUL-152587
• UDK: 515.1
• ISSN pri članku: 1660-5446
• DOI: 10.1007/s00009-023-02533-9
• COBISS.SI-ID: 171038723
• Datum objave v RUL: 29.11.2023

Gradivo je del revije

• Naslov: Mediterranean journal of mathematics
• Skrajšan naslov: Mediterr. j. math.
• Založnik: Springer Nature, Birkhäuser
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.