Wasserstein Stability for Persistence Diagrams

2020/06/30 by Primož Škraba, Skraba, Primoz, Katharine Turner +1 · 23 citations
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #Algebraic number #Algorithm #Bottleneck #Combinatorics #Computer science #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Norm (philosophy) #Permutable prime #Persistence (discontinuity) #Pure mathematics #Stability (learning theory) #Topological and Geometric Data Analysis #Topological data analysis #Topology (electrical circuits)

paper · pdf · doi:10.48550/arxiv.2006.16824

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2020/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

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 p-Wasserstein distance between persistence diagrams. This includes an elementary proof for the setting of functions on sufficiently finite spaces in terms of the p-norm of the perturbations, along with an algebraic framework for p-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.

Citations

• Stability of Persistence Diagrams
• A Topological Regularizer for Classifiers via Persistent Homology

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• On the Stability of the Euler Characteristic Transform for a Perturbed Embedding
• ℓ^p-Distances on Multiparameter Persistence Modules
• Persistent Homology as a Morphological Signature of Fibrin Networks
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