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Revista Matemática Complutense

, Volume 32, Issue 1, pp 195–213 | Cite as

Approximations of 1-dimensional intrinsic persistence of geodesic spaces and their stability

  • Žiga VirkEmail author
Article
  • 44 Downloads

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.

Keywords

Persistence Geodesic space Minimal homology basis Geodesic circle Rips complex 

Mathematics Subject Classification

Primary 55U10 55Q52 55N05 

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.University of LjubljanaLjubljanaSlovenia

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