4 Search Results for "B. Dornelas, Bianca"

Document
DOI: 10.4230/LIPIcs.SoCG.2025.6
A Sparse Multicover Bifiltration of Linear Size

Authors: Ángel Javier Alonso

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The k-cover of a point cloud X of ℝ^d at radius r is the set of all those points within distance r of at least k points of X. By varying r and k we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has O(|X|^{d+1}) simplices. In this paper we introduce a (1+ε)-approximation of the multicover bifiltration of linear size O(|X|), for fixed d and ε. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension yielding analogous results.
Cite as

Ángel Javier Alonso. A Sparse Multicover Bifiltration of Linear Size. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alonso:LIPIcs.SoCG.2025.6,
  author =	{Alonso, \'{A}ngel Javier},
  title =	{{A Sparse Multicover Bifiltration of Linear Size}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.6},
  URN =		{urn:nbn:de:0030-drops-231587},
  doi =		{10.4230/LIPIcs.SoCG.2025.6},
  annote =	{Keywords: Multicover, Approximation, Sparsification, Multiparameter persistence}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.11
When Alpha-Complexes Collapse onto Codimension-1 Submanifolds

Authors: Dominique Attali, Mattéo Clémot, Bianca B. Dornelas, and André Lieutier

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Given a finite set of points P sampling an unknown smooth surface ℳ ⊆ ℝ³, our goal is to triangulate ℳ based solely on P. Assuming ℳ is a smooth orientable submanifold of codimension 1 in ℝ^d, we introduce a simple algorithm, Naive Squash, which simplifies the α-complex of P by repeatedly applying a new type of collapse called vertical relative to ℳ. Naive Squash also has a practical version that does not require knowledge of ℳ. We establish conditions under which both the naive and practical Squash algorithms output a triangulation of ℳ. We provide a bound on the angle formed by triangles in the α-complex with ℳ, yielding sampling conditions on P that are competitive with existing literature for smooth surfaces embedded in ℝ³, while offering a more compartmentalized proof. As a by-product, we obtain that the restricted Delaunay complex of P triangulates ℳ when ℳ is a smooth surface in ℝ³ under weaker conditions than existing ones.
Cite as

Dominique Attali, Mattéo Clémot, Bianca B. Dornelas, and André Lieutier. When Alpha-Complexes Collapse onto Codimension-1 Submanifolds. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{attali_et_al:LIPIcs.SoCG.2025.11,
  author =	{Attali, Dominique and Cl\'{e}mot, Matt\'{e}o and Dornelas, Bianca B. and Lieutier, Andr\'{e}},
  title =	{{When Alpha-Complexes Collapse onto Codimension-1 Submanifolds}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.11},
  URN =		{urn:nbn:de:0030-drops-231630},
  doi =		{10.4230/LIPIcs.SoCG.2025.11},
  annote =	{Keywords: Submanifold reconstruction, triangulation, abstract simplicial complexes, collapses, convexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.41
Decomposing Multiparameter Persistence Modules

Authors: Tamal K. Dey, Jan Jendrysiak, and Michael Kerber

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Dey and Xin (J.Appl.Comput.Top., 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed-parameter tractable with respect to the maximal number of relations of the same degree and with further optimisation we obtain an O(n³) time algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.
Cite as

Tamal K. Dey, Jan Jendrysiak, and Michael Kerber. Decomposing Multiparameter Persistence Modules. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2025.41,
  author =	{Dey, Tamal K. and Jendrysiak, Jan and Kerber, Michael},
  title =	{{Decomposing Multiparameter Persistence Modules}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{41:1--41:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.41},
  URN =		{urn:nbn:de:0030-drops-231939},
  doi =		{10.4230/LIPIcs.SoCG.2025.41},
  annote =	{Keywords: Topological Data Analysis, Multiparameter Persistence Modules, Persistence, Decomposition}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.20
Sparse Higher Order Čech Filtrations

Authors: Mickaël Buchet, Bianca B. Dornelas, and Michael Kerber

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
For a finite set of balls of radius r, the k-fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the k-fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points.
Cite as

Mickaël Buchet, Bianca B. Dornelas, and Michael Kerber. Sparse Higher Order Čech Filtrations. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{buchet_et_al:LIPIcs.SoCG.2023.20,
  author =	{Buchet, Micka\"{e}l and B. Dornelas, Bianca and Kerber, Michael},
  title =	{{Sparse Higher Order \v{C}ech Filtrations}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.20},
  URN =		{urn:nbn:de:0030-drops-178709},
  doi =		{10.4230/LIPIcs.SoCG.2023.20},
  annote =	{Keywords: Sparsification, k-fold cover, Higher order \v{C}ech complexes}
}
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