Papers › Curvature Sets Over Persistence Diagrams
Curvature Sets Over Persistence Diagrams
Mario Gómez, Facundo Mémoli
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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 k≥0 and n≥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 𝐃_(n,k)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 𝐃_(4,1)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.
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