Lipschitz Functions Have Lp-Stable Persistence

By David Cohen-Steiner, Herbert Edelsbrunner, John Harer, Yuriy Mileyko. Foundations of Computational Mathematics, 2010

345 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/fb7323f9f2e8c4b4f5fc90448b896ee9c93b0af8

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Citations in the corpus

Probability measures on the space of persistence diagrams (2011, background): “We finish this section by stating an important stability result from [17] which shows that under mild assumptions on X computing a persistence diagram of a tame Lipschitz functions is a continuous map.”
Statistical topological data analysis using persistence landscapes (2012, background): “For p ≥ 1, the p-Wasserstein distance (Cohen-Steiner et al., 2010) between D and D′ is given by”
Persistence Images: A Stable Vector Representation of Persistent Homology (2015, methodology): “The space of PDs can be equipped with the bottleneck or Wasserstein metric (deﬁned in § 3), and one reason for the popularity of PDs is that these metrics are stable with respect to small deviations in the inputs (Cohen-Steiner et al., 2007, 2010; Chazal et al., 2014).”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “Useful stability results for persistence in the Wp metric exist among the literature, in particular the study by Cohen-Steiner et al. (2010), but they rely on assumptions that make them consequences of the stability results in the bottleneck metric. A general study of the space of persistence diagra”
Dynamic Snake Convolution based on Topological Geometric Constraints for Tubular Structure Segmentation (2023, methodology): “The existing methods [30, 5, 10] use modified Wasserstein distance to compute the best match between the points generated by the output and the points generated by the groundtruth, and the outlier points without optimal pairing are matched to the diagonal and do not participate in the loss…”
Deep Learning with Topological Signatures (2017, background): “r all bijections : D!E. Essentially, this facilitates studying stability/continuity properties of topological signatures w.r.t. metrics in the ﬁltration or complex space; we refer the reader to [12],[13], [9] for a selection of important stability results. Remark. By setting i;1 n = i;m n i 1;m n , ”
Wasserstein Stability for Persistence Diagrams (2020, positive): “Unfortunately, the Lipschitz Wasserstein stability theorem in [17] appears to be one of the most misunderstood and miscited results within the field of topological data analysis.”
A stable multi-scale kernel for topological machine learning (2014, background): “The key feature of persistent homology is its stability: small changes in the input data lead to small changes in the Wasserstein distance of the associated persistence diagrams [12].”
Persistent Homology Analysis of Brain Artery Trees (2014, background): “Many technical results (for example, [12, 13]) all basically say that, under mild conditions, Wp(Dgm0(f),Dgm0(g)) ≤ K||f − g||∞.”
Topology-Preserving Deep Image Segmentation (2019, methodology): “We slightly modify the Wasserstein distance for persistence diagrams [12].”
Homological illusions of persistence and stability (2008, result): “Recent results on the Lp-stability of persistence diagrams of Lipschitz functions [27] have provided insight into periodic gene expression [32].”