Publications (3)0 Total impact
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ABSTRACT: In the companion paper, we measured homology classes and computed the optimal
homology basis. This paper addresses two related problems, namely, localization
and stability. We localize a class with the cycle minimizing a certain
objective function. We explore three different objective functions, namely,
volume, diameter and radius. We show that it is NP-hard to compute the smallest
cycle using the former two. We also prove that the measurement defined in the
companion paper is stable with regard to small changes of the geometry of the
concerned space.
09/2007;
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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 $O(eta^4 n^3 log^2 n)$ time, where $n$ is the size of the simplicial complex and $eta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results. @InProceedings{chen_et_al:LIPIcs:2008:1343, author = {Chao Chen and Daniel Freedman}, title = {16. Quantifying Homology Classes}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science (STACS 2008)}, pages = {169--180}, series = {Leibniz International Proceedings in Informatics}, year = {2008}, volume = {1}, editor = {Susanne Albers and Pascal Weil}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1343}, URN = {urn:nbn:de:0030-drops-13434}, annote = {Keywords: Computational Topology, Computational Geometry, Homology, Persistent Homology, Localization, Optimization} }
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ABSTRACT: We develop a method for measuring homology classes. This involves two 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(βn3log2n) time, where n is the size of the simplicial complex and β is the Betti number of the homology group. Finally, we prove the stability of our result. The algorithm can be adapted to measure any given class.
Computational Geometry.
