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DOI: 10.4230/LIPIcs.SoCG.2022.19

Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions

Authors: Magnus Bakke Botnan, Steffen Oppermann, and Steve Oudot

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract

In this paper we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode encodes the rank invariant as a ℤ-linear combination of rank invariants of indicator modules supported on segments in the poset. It can also be enriched to encode the generalized rank invariant as a ℤ-linear combination of generalized rank invariants in fixed classes of interval modules. In the paper we develop the theory behind these rank decompositions, showing under what conditions they exist and are unique - so the signed barcode is canonically defined. We also illustrate the contribution of the signed barcode to the exploration of multi-parameter persistence modules through a practical example.

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Magnus Bakke Botnan, Steffen Oppermann, and Steve Oudot. Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{botnan_et_al:LIPIcs.SoCG.2022.19,
  author =	{Botnan, Magnus Bakke and Oppermann, Steffen and Oudot, Steve},
  title =	{{Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.19},
  URN =		{urn:nbn:de:0030-drops-160276},
  doi =		{10.4230/LIPIcs.SoCG.2022.19},
  annote =	{Keywords: Topological data analysis, multi-parameter persistent homology}
}

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Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions

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