Talk:Persistent homology

Notation

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 $K^{l}$ be a filtration. The p-persistent kth homology group of $K^{l}$ is $H_{k}^{l,p}=Z_{k}^{l}/(B_{k}^{l+p}\cap Z_{k}^{l})$.

If we let $z$ be a nonbounding $k$-cycle created at time $I$ by simplex $\sigma$ and let $z'\sim z$ be a homologous $k$-cycle that becomes a boundary cycle at time $J$ by simplex $\tau$, then we can define the persistence interval associated to $z$ as $(I,J)$. We call $\sigma$ the creator of $z$ and $\tau$ the destroyer of $z$. If $z$ does not have a destroyer, its persistence is $\infty$.

Instead of using an index-based filtration, we can use a time-based filtration. Let $K$ be a simplicial complex and $K^{\rho }=\{\sigma ^{i}\in K|\rho (\sigma ^{i})\leq \rho \}$ be a filtration defined for an associated map $\rho :S(K)\rightarrow \mathbb {R}$ that maps simplices in the final complex to real numbers. Then for all real numbers $\pi \geq 0$, the $\pi$-persistent kth homology group of $K^{\rho }$ is $H_{k}^{\rho ,\pi }=Z_{k}^{\rho }/(B_{k}^{\rho +\pi }\cap Z_{k}^{\rho })$. The persistence of a $k$-cycle created at time $\rho _{i}$ and destroyed at $\rho _{j}$ is $\rho _{j}-\rho _{i}$.

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

(stronger) stability

Stronger stability results for the metric space of persistence diagrams are now available: in particular, the $p$-distance between monotone functions out of a complex bounds the $p$-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." Mathysocks (talk) 17:55, 2 November 2023 (UTC)

References