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Notation

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As of 1 August 2013, this page was using non-standard notation for filtration and homology, maybe following Zomorodian's book. Across wikipedia, and in Edelsbrunner and Harel's book, the filtration indices and homology group indices are both represented by subscripts. Since the author here didn't define the notation they are using, I'm moving the existing definition over here and replacing it in the main article with the one from E&H, p. 151.

Let be a filtration. The p-persistent kth homology group of is .

If we let be a nonbounding -cycle created at time by simplex and let be a homologous -cycle that becomes a boundary cycle at time by simplex , then we can define the persistence interval associated to as . We call the creator of and the destroyer of . If does not have a destroyer, its persistence is .[1]

Instead of using an index-based filtration, we can use a time-based filtration. Let be a simplicial complex and be a filtration defined for an associated map that maps simplices in the final complex to real numbers. Then for all real numbers , the -persistent kth homology group of is . The persistence of a -cycle created at time and destroyed at is . [2]

VAFisher (talk) 13:41, 15 August 2013 (UTC)[reply]

(stronger) stability

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Stronger stability results for the metric space of persistence diagrams are now available: in particular, the -distance between monotone functions out of a complex bounds the -Wasserstein distance between diagrams of those functions' sublevel-set persistence. See Theorem 4.7 in Skraba and Turner's 2021 paper "Wasserstein Stability for Persistence Diagrams." [3] Mathysocks (talk) 17:55, 2 November 2023 (UTC)[reply]

References

  1. ^ Weinberger, Shmuel (2011), "What Is . . . Persistent Homology?" (PDF), AMS Notices, 58 (01): 36–39
  2. ^ Afra J. Zomorodian (2005): Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics.
  3. ^ Skraba, P., Turner, K. (2023), Wasserstein Stability for Persistence Diagrams.{{citation}}: CS1 maint: multiple names: authors list (link)