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Critical edges in Rips complexes and persistence
ID
Goričan, Peter
(
Author
),
ID
Virk, Žiga
(
Author
)
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https://link.springer.com/article/10.1007/s00009-023-02533-9
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Abstract
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.
Language:
English
Keywords:
persistent homology
,
Rips complex
,
critical simplex
,
reconstruction result
Work type:
Article
Typology:
1.01 - Original Scientific Article
Organization:
FMF - Faculty of Mathematics and Physics
FRI - Faculty of Computer and Information Science
Publication status:
Published
Publication version:
Version of Record
Year:
2023
Number of pages:
25 str.
Numbering:
Vol. 20, iss. 6, art. 326
PID:
20.500.12556/RUL-152587
UDC:
515.1
ISSN on article:
1660-5446
DOI:
10.1007/s00009-023-02533-9
COBISS.SI-ID:
171038723
Publication date in RUL:
29.11.2023
Views:
305
Downloads:
31
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Record is a part of a journal
Title:
Mediterranean journal of mathematics
Shortened title:
Mediterr. j. math.
Publisher:
Springer Nature
ISSN:
1660-5446
COBISS.SI-ID:
13561433
Licences
License:
CC BY 4.0, Creative Commons Attribution 4.0 International
Link:
http://creativecommons.org/licenses/by/4.0/
Description:
This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
Projects
Funder:
ARRS - Slovenian Research Agency
Project number:
N1-0114
Name:
Algebrajski odtisi geometrijskih značilnosti v homologiji
Funder:
ARRS - Slovenian Research Agency
Project number:
J1-4001
Name:
Izbrani problemi iz uporabne in računske topologije
Funder:
ARRS - Slovenian Research Agency
Project number:
J1-4031
Name:
Računalniška knjižnica za zavozlane strukture in aplikacije
Funder:
ARRS - Slovenian Research Agency
Project number:
P1-0292
Name:
Topologija in njena uporaba
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