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The mergegram of a dendrogram and its stability

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arxiv 2007.11278 v1 pith:ZAVYVKTI submitted 2020-07-22 cs.CG

classification cs.CG
keywords persistencecloudsdatadendrogrammergegrampointstopologicalanalysis
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This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay.

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  1. Fibers of point cloud persistence

    math.AT 2024-11 conditional novelty 8.0 of 10

    For generic point clouds, global rigidity of the critical graph implies Vietoris-Rips persistence identifies the cloud up to isometry, and Cech persistence locally identifies it exactly when the critical hypergraph is...

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