License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.46
URN: urn:nbn:de:0030-drops-104505
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10450/
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Kerber, Michael ; Lesnick, Michael ; Oudot, Steve

Exact Computation of the Matching Distance on 2-Parameter Persistence Modules

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LIPIcs-SoCG-2019-46.pdf (0.5 MB)


Abstract

The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.

BibTeX - Entry

@InProceedings{kerber_et_al:LIPIcs:2019:10450,
  author =	{Michael Kerber and Michael Lesnick and Steve Oudot},
  title =	{{Exact Computation of the Matching Distance on 2-Parameter Persistence Modules}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{46:1--46:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10450},
  URN =		{urn:nbn:de:0030-drops-104505},
  doi =		{10.4230/LIPIcs.SoCG.2019.46},
  annote =	{Keywords: Topological Data Analysis, Multi-Parameter Persistence, Line arrangements}
}

Keywords: Topological Data Analysis, Multi-Parameter Persistence, Line arrangements
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019


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