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DOI: 10.4230/LIPIcs.MFCS.2020.32

The Mergegram of a Dendrogram and Its Stability

Authors: Yury Elkin and Vitaliy Kurlin

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract

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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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)

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@InProceedings{elkin_et_al:LIPIcs.MFCS.2020.32,
  author =	{Elkin, Yury and Kurlin, Vitaliy},
  title =	{{The Mergegram of a Dendrogram and Its Stability}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{32:1--32:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.32},
  URN =		{urn:nbn:de:0030-drops-127281},
  doi =		{10.4230/LIPIcs.MFCS.2020.32},
  annote =	{Keywords: clustering dendrogram, topological data analysis, persistence, stability}
}

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The Mergegram of a Dendrogram and Its Stability

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