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Annals of West University of Timisoara - Mathematics and Computer Science

The Journal of West University of Timisoara

Editor-in-Chief: Sasu, Bogdan

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Contributions to Persistence Theory

Dong Du
Published Online: 2015-03-25 | DOI: https://doi.org/10.2478/awutm-2014-0012


Persistence theory discussed in this paper is an application of algebraic topology (Morse Theory [29]) to Data Analysis, precisely to qualitative understanding of point cloud data, or PCD for short. PCD can be geometrized as a filtration of simplicial complexes (Vietoris-Rips complex [25] [36]) and the homology changes of these complexes provide qualitative information about the data. Bar codes describe the changes in homology with coefficients in a fixed field. When the coefficient field is ℤ2, the calculation of bar codes is done by ELZ algorithm (named after H. Edelsbrunner, D. Letscher, and A. Zomorodian [20]). When the coefficient field is ℝ, we propose an algorithm based on the Hodge decomposition [17]. With Dan Burghelea and Tamal K. Dey we developed a persistence theory which involves level sets discussed in Section 4. We introduce and discuss new computable invariants, the “relevant level persistence numbers” and the “positive and negative bar codes”, and explain how they are related to the bar codes for level persistence. We provide enhancements and modifications of ELZ algorithm to calculate such invariants and illustrate them by examples.

Keywords : PCD; Vietoris-Rips complex; persistent linear algebra; bar codes of a PCD; Hodge decomposition; bar codes for level persistence; positive and negative bar codes; relevant persistent numbers


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About the article

Received: 2014-03-28

Accepted: 2014-09-15

Published Online: 2015-03-25

Published in Print: 2014-12-01

Citation Information: Annals of West University of Timisoara - Mathematics, Volume 52, Issue 2, Pages 13–95, ISSN (Online) 1841-3307, DOI: https://doi.org/10.2478/awutm-2014-0012.

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© Annals of West University of Timisoara - Mathematics. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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