License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.14
URN: urn:nbn:de:0030-drops-87270
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8727/
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Brown, Adam ; Wang, Bei

Sheaf-Theoretic Stratification Learning

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LIPIcs-SoCG-2018-14.pdf (0.5 MB)

Abstract

In this paper, we investigate a sheaf-theoretic interpretation of stratification learning. Motivated by the work of Alexandroff (1937) and McCord (1978), we aim to redirect efforts in the computational topology of triangulated compact polyhedra to the much more computable realm of sheaves on partially ordered sets. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (2012), and the cohomology stratification algorithm given in Nanda (2017). We envision that our sheaf-theoretic algorithm could give rise to a larger class of stratification beyond homology-based stratification. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.

BibTeX - Entry

@InProceedings{brown_et_al:LIPIcs:2018:8727,
  author =	{Adam Brown and Bei Wang},
  title =	{{Sheaf-Theoretic Stratification Learning}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8727},
  URN =		{urn:nbn:de:0030-drops-87270},
  doi =		{10.4230/LIPIcs.SoCG.2018.14},
  annote =	{Keywords: Sheaf theory, stratification learning, topological data analysis, stratification}
}

Keywords: Sheaf theory, stratification learning, topological data analysis, stratification
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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