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- W4323073052 abstract "We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980s with Vietoris-Rips Persistent Homology. For given integers $kgeq 0$ and $ngeq 1$ we consider the dimension $k$ Vietoris-Rips persistence diagrams of emph{all} subsets of a given metric space with cardinality at most $n$. We call these invariants emph{persistence sets} and denote them as $mathbf{D}_{n,k}^textrm{VR}$. We establish that (1) computing these invariants is often significantly more efficient than computing the usual Vietoris-Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris-Rips persistence diagrams, and (3) they enjoy stability properties. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy's inequality. We also identify a rich family of metric graphs for which $mathbf{D}_{4,1}^textrm{VR}$ fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris-Rips persistence diagrams using Mayer-Vietoris sequences. These yield a geometric algorithm for computing the Vietoris-Rips persistence diagram of a space $X$ with cardinality $2k+2$ with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction." @default.
- W4323073052 created "2023-03-05" @default.
- W4323073052 creator A5016256957 @default.
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- W4323073052 date "2021-03-07" @default.
- W4323073052 modified "2023-09-25" @default.
- W4323073052 title "Curvature Sets Over Persistence Diagrams" @default.
- W4323073052 doi "https://doi.org/10.48550/arxiv.2103.04470" @default.
- W4323073052 hasPublicationYear "2021" @default.
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