1,377 research outputs found
Persistence stability for geometric complexes
In this paper we study the properties of the homology of different geometric
filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on
top of precompact spaces. Using recent developments in the theory of
topological persistence we provide simple and natural proofs of the stability
of the persistent homology of such complexes with respect to the
Gromov--Hausdorff distance. We also exhibit a few noteworthy properties of the
homology of the Rips and Cech complexes built on top of compact spaces.Comment: We include a discussion of ambient Cech complexes and a new class of
examples called Dowker complexe
Metric Graph Approximations of Geodesic Spaces
A standard result in metric geometry is that every compact geodesic metric
space can be approximated arbitrarily well by finite metric graphs in the
Gromov-Hausdorff sense. It is well known that the first Betti number of the
approximating graphs may blow up as the approximation gets finer.
In our work, given a compact geodesic metric space , we define a sequence
of non-negative real numbers by
By construction, and the above result, this is a non-increasing sequence with
limit . We study this sequence and its rates of decay with . We also
identify a precise relationship between the sequence and the first
Vietoris-Rips persistence barcode of . Furthermore, we specifically analyze
and find upper and lower bounds based on hyperbolicity and other
metric invariants. As a consequence of the tools we develop, our work also
provides a Gromov-Hausdorff stability result for the Reeb construction on
geodesic metric spaces with respect to the function given by distance to a
reference point
Optimal rates of convergence for persistence diagrams in Topological Data Analysis
Computational topology has recently known an important development toward
data analysis, giving birth to the field of topological data analysis.
Topological persistence, or persistent homology, appears as a fundamental tool
in this field. In this paper, we study topological persistence in general
metric spaces, with a statistical approach. We show that the use of persistent
homology can be naturally considered in general statistical frameworks and
persistence diagrams can be used as statistics with interesting convergence
properties. Some numerical experiments are performed in various contexts to
illustrate our results
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