Curvature Sets Over Persistence Diagrams

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 $k\ge 0$ and $n\ge 1$ 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 ${\mathbf{D}}_{n,k}^{\text{VR}}$. 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 ${\mathbf{D}}_{4,1}^{\text{VR}}$ 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.

Added Example 4.15 (in Section 4.3) to show that persistence sets and persistent homology distinguish different spaces, and added more running time comparisons to Section 4.4