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Documents authored by Botnan, Magnus Bakke

Document
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.22
On Rectangle-Decomposable 2-Parameter Persistence Modules

Authors: Magnus Bakke Botnan, Vadim Lebovici, and Steve Oudot

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
This paper addresses two questions: (1) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (2) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: (a) a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; (b) algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local condition is key to the efficiency of our algorithms, and it generalizes previous conditions from the class of block-decomposable modules to the larger one of rectangle-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.
Cite as

Magnus Bakke Botnan, Vadim Lebovici, and Steve Oudot. On Rectangle-Decomposable 2-Parameter Persistence Modules. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{botnan_et_al:LIPIcs.SoCG.2020.22,
  author =	{Botnan, Magnus Bakke and Lebovici, Vadim and Oudot, Steve},
  title =	{{On Rectangle-Decomposable 2-Parameter Persistence Modules}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{22:1--22:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.22},
  URN =		{urn:nbn:de:0030-drops-121802},
  doi =		{10.4230/LIPIcs.SoCG.2020.22},
  annote =	{Keywords: topological data analysis, multiparameter persistence, rank invariant}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.13
Computational Complexity of the Interleaving Distance

Authors: Håvard Bakke Bjerkevik and Magnus Bakke Botnan

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.
Cite as

Håvard Bakke Bjerkevik and Magnus Bakke Botnan. Computational Complexity of the Interleaving Distance. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bjerkevik_et_al:LIPIcs.SoCG.2018.13,
  author =	{Bjerkevik, H\r{a}vard Bakke and Botnan, Magnus Bakke},
  title =	{{Computational Complexity of the Interleaving Distance}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{13:1--13:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.13},
  URN =		{urn:nbn:de:0030-drops-87268},
  doi =		{10.4230/LIPIcs.SoCG.2018.13},
  annote =	{Keywords: Persistent Homology, Interleavings, NP-hard}
}
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