Curvature Sets Over Persistence Diagrams

arXiv2103.04470MaRDI QIDQ6362352

Mario San Martin Gomez, Facundo Mémoli

Publication date: 7 March 2021

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 $kgeq0$ and $ngeq1$ 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 $mathbf{D}_{n,k}^{e}xtrmVR$. 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 $mathbf{D}_{4,1}^{e}xtrmVR$ 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.

Has companion code repository: https://github.com/ndag/persistence-curv-sets

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