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Wasserstein Stability for Persistence Diagrams
The stability of persistence diagrams is among the most important results in
applied and computational topology. Most results in the literature phrase
stability in terms of the bottleneck distance between diagrams and the
-norm of perturbations. This has two main implications: it makes the
space of persistence diagrams rather pathological and it is often provides very
pessimistic bounds with respect to outliers. In this paper, we provide new
stability results with respect to the -Wasserstein distance between
persistence diagrams. This includes an elementary proof for the setting of
functions on sufficiently finite spaces in terms of the -norm of the
perturbations, along with an algebraic framework for -Wasserstein distance
which extends the results to wider class of modules. We also provide apply the
results to a wide range of applications in topological data analysis (TDA)
including topological summaries, persistence transforms and the special but
important case of Vietoris-Rips complexes
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