Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions

Authors Magnus Bakke Botnan, Steffen Oppermann, Steve Oudot

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Author Details

Magnus Bakke Botnan
  • Vrije Universiteit Amsterdam, The Netherlands
Steffen Oppermann
  • Norwegian University of Science and Technology, Trondheim, Norway
Steve Oudot
  • Inria, Palaiseau, France

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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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Topological data analysis
  • multi-parameter persistent homology


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