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  • Critical edges in Rips complexes and persistence
    Goričan, Peter ; Virk, Žiga, 1982-
    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.
    Source: Mediterranean journal of mathematics. - ISSN 1660-5446 (Vol. 20, iss. 6, [article no.] 326, Dec. 2023, 25 str.)
    Type of material - article, component part ; adult, serious
    Publish date - 2023
    Language - english
    COBISS.SI-ID - 171038723

    Link(s):

    https://link.springer.com/article/10.1007/s00009-023-02533-9
    DiRROS - Digital repository of Slovenian research organizations
    Repository of the University of Ljubljana – RUL
    DOI

source: Mediterranean journal of mathematics. - ISSN 1660-5446 (Vol. 20, iss. 6, [article no.] 326, Dec. 2023, 25 str.)
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