The Mergegram of a Dendrogram and Its Stability

Authors Yury Elkin, Vitaliy Kurlin

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Author Details

Yury Elkin
  • Materials Innovation Factory and Computer Science department, University of Liverpool, UK
Vitaliy Kurlin
  • Materials Innovation Factory and Computer Science department, University of Liverpool, UK

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Yury Elkin and Vitaliy Kurlin. The Mergegram of a Dendrogram and Its Stability. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • clustering dendrogram
  • topological data analysis
  • persistence
  • stability


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