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Quantifying Homology Classes

Authors: Chao Chen and Daniel Freedman

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)


Abstract
We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in $O(\beta^4 n^3 log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results.

Cite as

Chao Chen and Daniel Freedman. Quantifying Homology Classes. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 169-180, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


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@InProceedings{chen_et_al:LIPIcs.STACS.2008.1343,
  author =	{Chen, Chao and Freedman, Daniel},
  title =	{{Quantifying Homology Classes}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{169--180},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1343},
  URN =		{urn:nbn:de:0030-drops-13434},
  doi =		{10.4230/LIPIcs.STACS.2008.1343},
  annote =	{Keywords: Computational Topology, Computational Geometry, Homology, Persistent Homology, Localization, Optimization}
}
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