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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.31
URN: urn:nbn:de:0030-drops-59236
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5923/
Choudhary, Aruni ;
Kerber, Michael ;
Raghvendra, Sharath
Polynomial-Sized Topological Approximations Using the Permutahedron
Abstract
Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R^d, we obtain a O(d)-approximation with at most n2^{O(d log k)} simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog (n))-approximation of size n^{O(1)} for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry.
Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1/(log^{1+c}n) for c in (0,1).
BibTeX - Entry
@InProceedings{choudhary_et_al:LIPIcs:2016:5923,
author = {Aruni Choudhary and Michael Kerber and Sharath Raghvendra},
title = {{Polynomial-Sized Topological Approximations Using the Permutahedron}},
booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)},
pages = {31:1--31:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-009-5},
ISSN = {1868-8969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5923},
URN = {urn:nbn:de:0030-drops-59236},
doi = {10.4230/LIPIcs.SoCG.2016.31},
annote = {Keywords: Persistent Homology, Topological Data Analysis, Simplicial Approximation, Permutahedron, Approximation Algorithms}
}
Keywords: |
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Persistent Homology, Topological Data Analysis, Simplicial Approximation, Permutahedron, Approximation Algorithms |
Collection: |
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32nd International Symposium on Computational Geometry (SoCG 2016) |
Issue Date: |
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2016 |
Date of publication: |
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10.06.2016 |