3 Search Results for "Schmahl, Maximilian"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.94
Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds

Authors: Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler, and Peter Bubenik

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We propose a novel geometric-topological descriptor called a mixup barcode. Intuitively, it characterizes the shape of a point cloud as well as its spatial relationship with another point cloud embedded in the same ambient space. More technically, it enriches a standard persistence barcode with information on the image persistent homology. In three dimensions it captures natural spatial relationships like overlap and surrounding; in higher dimensions more intricate spatial relationships are captured. We provide a theoretical setup and a simple algorithm for mixup barcodes. As a proof of concept, we explore data arising in a geometric-topological problem from machine learning. Specifically, we take first steps towards verifying a hypothesis stating that geometric-topological relationships within intermediate point cloud representations in an artificial neural network can hinder its training. More broadly, our experiments suggest that mixup barcodes are useful for characterizing spatial relationships and spatial interactions (i.e. the evolution of spatial relationships) that are hard to directly visualize or capture using standard methods.
Cite as

Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler, and Peter Bubenik. Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 94:1-94:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{wagner_et_al:LIPIcs.SoCG.2026.94,
  author =	{Wagner, Hubert and Arustamyan, Nickolas and Wheeler, Matthew and Bubenik, Peter},
  title =	{{Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{94:1--94:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.94},
  URN =		{urn:nbn:de:0030-drops-259009},
  doi =		{10.4230/LIPIcs.SoCG.2026.94},
  annote =	{Keywords: mixup barcode, persistent homology, persistence barcode, persistence diagram, image persistent homology, image persistence, deep learning, multilayer perceptron, topology of neural network embeddings, disentanglement}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.35
A Theory of Sub-Barcodes

Authors: Oliver A. Chubet, Kirk P. Gardner, and Donald R. Sheehy

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

Abstract
The primary tool in topological data analysis (TDA) is persistent homology, which involves computing a barcode - often from point-cloud or scalar field data - that serves as a topological signature for the underlying function. In this work, we introduce sub-barcodes and show how they arise naturally from factorizations of persistence module homomorphisms. We show that, as a partial order induced by factorizations, the relation of being a sub-barcode is strictly stronger than the rank invariant, and we apply sub-barcode theory to the problem of inferring information about the barcode of an unknown Lipschitz function from samples. The advantage of this approach is that it permits strong guarantees - with no noise - while requiring no sampling assumptions, and the resulting barcode is guaranteed to be a sub-barcode of every Lipschitz function that agrees with the data. We also present an algorithmic theory that allows for the efficient approximation of sub-barcodes using filtered Delaunay triangulations for Euclidean inputs.
Cite as

Oliver A. Chubet, Kirk P. Gardner, and Donald R. Sheehy. A Theory of Sub-Barcodes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chubet_et_al:LIPIcs.SoCG.2025.35,
  author =	{Chubet, Oliver A. and Gardner, Kirk P. and Sheehy, Donald R.},
  title =	{{A Theory of Sub-Barcodes}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{35:1--35:16},
  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.2025.35},
  URN =		{urn:nbn:de:0030-drops-231873},
  doi =		{10.4230/LIPIcs.SoCG.2025.35},
  annote =	{Keywords: Topology, Topological Data Analysis, Persistent Homology, Persistence Modules, Barcodes, Sub-barcodes, Factorizations, Lipschitz Extensions}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.14
Efficient Computation of Image Persistence

Authors: Ulrich Bauer and Maximilian Schmahl

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

Abstract
We present an algorithm for computing the barcode of the image of a morphism in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes. The algorithm makes use of the clearing optimization and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes. The clearing optimization works particularly well in the context of relative cohomology, and using previous duality results we can translate the barcodes of images in relative cohomology to those in absolute homology. This forms the basis for an implementation of image persistence computations for inclusions of filtrations of Vietoris-Rips complexes in the framework of the software Ripser.
Cite as

Ulrich Bauer and Maximilian Schmahl. Efficient Computation of Image Persistence. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2023.14,
  author =	{Bauer, Ulrich and Schmahl, Maximilian},
  title =	{{Efficient Computation of Image Persistence}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{14:1--14:14},
  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.2023.14},
  URN =		{urn:nbn:de:0030-drops-178643},
  doi =		{10.4230/LIPIcs.SoCG.2023.14},
  annote =	{Keywords: Persistent homology, image persistence, barcode computation}
}
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