Talk:Persistent homology

Latest comment: 6 months ago by Mathysocks in topic (stronger) stability

Notation edit

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 edit

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)