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Wasserstein Stability for Persistence Diagrams
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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 $\infty$-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.
Forward citations
Cited by 13 Pith papers
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Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport
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A topological approach to the Cahn-Hilliard equation and hyperuniform fields
Persistent homology of signed-distance filtrations correlates with and can classify the hyperuniform character of scalar fields, demonstrated on Cahn-Hilliard patterns and Gaussian random fields.
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A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images
The authors introduce RGWp, a Gromov-Wasserstein distance for Reeb graphs with a symmetric Reeb radius and persistence-image weighting, and present a stability proof that contains unproven structural assumptions.
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Functional connectomes of neural networks
Functional connectomes of neural networks, analyzed with persistent graph homology and exact Wasserstein distances, cluster networks by regularization strategy and input class with above-chance purity.
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TopoCode: Topologically Informed Error Detection and Correction in Communication Systems
TopoCode encodes the persistent homology of images as a small side packet and uses it to guide noise-robust reconstruction, outperforming LDPC and convolutional codes at low SNR in simulations.
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A Mathematical Framework for Topological Causal Data Analysis
TCDA separates observation space, causal model, topological map, and query, identifying Banach-valued outcome effects and law-level topological contrasts with stability-transfer bounds.
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Explainable topological data analysis using persistence heatmaps
Averaging representative cycles over perturbed filtrations yields persistence heatmaps that are Lipschitz-stable and localize SVM-learned feature importance back into data space.
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Localized persistent homology features can make graph neural networks more expressive and slightly more accurate, but the state-of-the-art claim is not uniformly supported.
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Bottleneck, Wasserstein, persistence-landscape, and persistence-image measures are more robust to Gaussian noise and Gaussian/ML denoising of synthetic 3D porous-media images than generator-count or average-lifespan s...
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A broad benchmark of intrinsic dimension estimators shows that no single method or set of hyperparameters works across datasets, and tuned benchmark scores frequently indicate overfitting rather than transferable accuracy.
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A dissertation compiling five prior papers: GPU-accelerated persistent homology (HYPHA, Ripser++), near-linear-time approximated Wasserstein distance for persistence diagrams (PDoptFlow), and topology-based graph and ...
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Certifying Robustness via Topological Representations
A stable rank network with a known Lipschitz constant certifies robustness radii for persistence diagrams, and on ORBIT5K it keeps high robust accuracy where a standard PersLay model collapses.
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The paper combines McCann interpolation and JKO Wasserstein gradient flow with differentiable persistent homology to iteratively retarget persistence diagrams and update filtrations, but it provides only qualitative 2...
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