| Title: | Critical edges in Rips complexes and persistence |
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| Authors: | ID Goričan, Peter (Author)
ID Virk, Žiga (Author) |
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| Files: |  PDF - Presentation file, download (579,74 KB)
MD5: AB572F6DB31D2318767D9CA54A0ED3A4
 URL - Source URL, visit https://link.springer.com/article/10.1007/s00009-023-02533-9
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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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. |
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| Keywords: | persistent homology, Rips complex, critical simplex, reconstruction result |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.12.2023 |
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| Year of publishing: | 2023 |
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| Number of pages: | 25 str. |
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| Numbering: | Vol. 20, iss. 6, [article no.] 326 |
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| PID: | 20.500.12556/DiRROS-18647  |
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| UDC: | 515.1 |
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| ISSN on article: | 1660-5446 |
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| DOI: | 10.1007/s00009-023-02533-9 |
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| COBISS.SI-ID: | 171038723 |
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| Publication date in DiRROS: | 10.04.2024 |
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| Views: | 649 |
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| Downloads: | 348 |
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