2 Search Results for "Wagner, Alexander"

Document
DOI: 10.4230/LIPIcs.SoCG.2022.61
From Geometry to Topology: Inverse Theorems for Distributed Persistence

Authors: Elchanan Solomon, Alexander Wagner, and Paul Bendich

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

Abstract
What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from injective. We therefore propose that, in many cases, the collection of persistence diagrams of many small subsets of X is a better invariant. This invariant, which we call "distributed persistence," is perfectly parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of metric spaces (with the quasi-isometry distance) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is globally bi-Lipschitz. This is a much stronger property than simply being injective, as it implies that the inverse image of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the inverse Lipschitz constant depends on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying simple covering properties. These theoretical results are complemented by synthetic experiments demonstrating the use of distributed persistence in practice.
Cite as

Elchanan Solomon, Alexander Wagner, and Paul Bendich. From Geometry to Topology: Inverse Theorems for Distributed Persistence. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{solomon_et_al:LIPIcs.SoCG.2022.61,
  author =	{Solomon, Elchanan and Wagner, Alexander and Bendich, Paul},
  title =	{{From Geometry to Topology: Inverse Theorems for Distributed Persistence}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{61:1--61:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.61},
  URN =		{urn:nbn:de:0030-drops-160690},
  doi =		{10.4230/LIPIcs.SoCG.2022.61},
  annote =	{Keywords: Applied Topology, Persistent Homology, Inverse Problems, Subsampling}
}
Document
DOI: 10.4230/DagSemProc.05361.6
Deterministic boundary recongnition and topology extraction for large sensor networks

Authors: Sándor Fekete, Alexander Kröller, Dennis Pfisterer, and Stefan Fischer

Published in: Dagstuhl Seminar Proceedings, Volume 5361, Algorithmic Aspects of Large and Complex Networks (2006)

Abstract
We present a new framework for the crucial challenge of self-organization of a large sensor network. The basic scenario can be described as follows: Given a large swarm of immobile sensor nodes that have been scattered in a polygonal region, such as a street network. Nodes have no knowledge of size or shape of the environment or the position of other nodes. Moreover, they have no way of measuring coordinates, geometric distances to other nodes, or their direction. Their only way of interacting with other nodes is to send or to receive messages from any node that is within communication range. The objective is to develop algorithms and protocols that allow self-organization of the swarm into large-scale structures that reflect the structure of the street network, setting the stage for global routing, tracking and guiding algorithms. Our algorithms work in two stages: boundary recognition and topology extraction. All steps are strictly deterministic, yield fast distributed algorithms, and make no assumption on the distribution of nodes in the environment, other than sufficient density.
Cite as

Sándor Fekete, Alexander Kröller, Dennis Pfisterer, and Stefan Fischer. Deterministic boundary recongnition and topology extraction for large sensor networks. In Algorithmic Aspects of Large and Complex Networks. Dagstuhl Seminar Proceedings, Volume 5361, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

Copy BibTex To Clipboard

@InProceedings{fekete_et_al:DagSemProc.05361.6,
  author =	{Fekete, S\'{a}ndor and Kr\"{o}ller, Alexander and Pfisterer, Dennis and Fischer, Stefan},
  title =	{{Deterministic boundary recongnition and topology extraction for large sensor networks}},
  booktitle =	{Algorithmic Aspects of Large and Complex Networks},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{5361},
  editor =	{Stefano Leonardi and Friedhelm Meyer auf der Heide and Dorothea Wagner},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.05361.6},
  URN =		{urn:nbn:de:0030-drops-5632},
  doi =		{10.4230/DagSemProc.05361.6},
  annote =	{Keywords: Distributed algorithms, sensor networks, boundary recognition, topology extraction}
}
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