Approximations of 1-Dimensional Intrinsic Persistence of Geodesic Spaces and Their Stability

TL;DR

This paper shows that the intrinsic persistence of geodesic spaces can be approximated by finite samples, and establishes a stronger stability theorem for these approximations, improving understanding of their geometric and topological features.

Contribution

It proves that the persistence of a geodesic space can be derived from finite samples and introduces an improved stability theorem for these approximations.

Findings

Persistence of geodesic spaces can be obtained from finite samples.

Persistence of subsets is well interleaved with that of the entire space.

Enhanced stability results for Rips complex approximations.

Abstract

A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space $X$ carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of $X$ may be obtained by an appropriate finite sample (subset of $X$), and that persistence of any subset of $X$ is well interleaved with the persistence of $X$. It follows that the persistence of $X$ is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each $r>0$ a density $s>0$, so that for each $s$-dense sample $S \subset X$ the corresponding fundamental group (and the first homology) of the Rips complex of $S$ is isomorphic to the one of $X$, leading to an improved reconstruction result.