Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Chambers, Erin Wolf; Munch, Elizabeth; Ophelders, Tim https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-138218
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A Family of Metrics from the Truncated Smoothing of Reeb Graphs

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Abstract

In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter τ. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ≤ τ ≤ 2ε, where ε is the smoothing parameter. Then, for the restriction of τ ∈ [0,ε], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ∈ [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ∈ [0,1], which is a generalization of the original interleaving distance, which is the case m = 0. While the resulting metrics are not stable, we show that any pair of these for m, m' ∈ [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs.

BibTeX - Entry

@InProceedings{chambers_et_al:LIPIcs.SoCG.2021.22,
  author =	{Chambers, Erin Wolf and Munch, Elizabeth and Ophelders, Tim},
  title =	{{A Family of Metrics from the Truncated Smoothing of Reeb Graphs}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13821},
  URN =		{urn:nbn:de:0030-drops-138218},
  doi =		{10.4230/LIPIcs.SoCG.2021.22},
  annote =	{Keywords: Reeb graphs, interleaving distance, graphical signatures}
}

Keywords: Reeb graphs, interleaving distance, graphical signatures
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021


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