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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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Mathematical Citation Quotient (MCQ) 2016: 0.01

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1841-3307
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Contributions to Persistence Theory

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

Abstract

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

References

  • [1] H. Adams, JPlex Matlab Tutorial, December 26, 2011Google Scholar

  • [2] M. de Berg, O. Cheong,M. van Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications (Third Edition), Springer-Verlag, Heidelberg, 2008Google Scholar

  • [3] D. Burghelea and T. K. Dey, Topological Persistence for Circle Valued Maps, Discrete Comput. Geom., 50, (2013), no.1, 69-98Web of ScienceGoogle Scholar

  • [4] G. Carlsson, Topology and Data, Bull. Amer. Math. Soc., 46, (2009), 255-308CrossrefGoogle Scholar

  • [5] G. Carlsson, A. Collins, L. Guibas, and A. Zomorodian, Persistence barcodes for shapes, Internat. J. Shape Modeling, 11, (2005), 149-187Google Scholar

  • [6] G. Carlsson and V. D. Silva, Zigzag Persistence, Foundations of Computational Mathematics, 10(4), (2010), 367-405Google Scholar

  • [7] G. Carlsson, V. D. Silva, and D. Morozov, Zigzag Persistent Homology and Real-valued Functions, Proc. 25th Ann. Sympos. Comput. Geom., (2009), 247-256Google Scholar

  • [8] F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot, Proximity of persistence modules and their diagrams, Proc. 25th Ann. Sympos. Comput.Geom., (2009), 237-246Google Scholar

  • [9] D. Cohen-Steiner, H. Edelsbrunner, and J. L. Harer, Stability of persistence diagrams, Discrete Comput. Geom., 37, (2007), 103-120CrossrefGoogle Scholar

  • [10] D. Cohen-Steiner, H. Edelsbrunner, J. L. Harer, and Y. Mileyko, Lipshitz functions have Lp-stable persistence, Foundations of Computational Mathematics, 10 (2), (2010), 127-139Google Scholar

  • [11] D. Cohen-Steiner, H. Edelsbrunner, J. L. Harer, and D. Morozov, Persistent homology for kernels, images, and cokernels., Proc. 20th Ann. ACM-SIAM Sympos. Discrete Alg., (2009), 1011-1020Google Scholar

  • [12] D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov, Vines and vineyards by updating persistence in linear time, Proc. 22nd Ann. Sympos. Comput. Geom., (2006), 119-126Google Scholar

  • [13] H. S. M. Coxeter, Introduction to Geometry (Second Edition), Wiley Classics Library, 1989Google Scholar

  • [14] J. Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo Publishing, 2008Google Scholar

  • [15] R. Deheuvels, Topologie d’une fonctionnelle, Ann. of Math., 61, (1955), 13-72Google Scholar

  • [16] T. K. Dey and R. Wenger, Stability of Critical Points with Interval Persistence, Discrete Comput. Geom., 38, (2007), 479-512Web of ScienceCrossrefGoogle Scholar

  • [17] B. Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex, Commentarii Math. Helvetici, 17, (1944-1945), 240-255Google Scholar

  • [18] H. Edelsbrunner and J. L. Harer, Persistent homology - a survey, Surveys on Discrete and Computational Geometry Twenty Years Later, J.E. Goodman, J. Patch and R, Pollack (eds.), Contemporar Google Scholar

  • [19] H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, AMS Press, 2010Google Scholar

  • [20] H. Edelsbrunner, D. Letscher, and A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom., 28, (2002), 511-533CrossrefGoogle Scholar

  • [21] P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognition and Image Analysis, 9, (1999), 596-603Google Scholar

  • [22] P. Gabriel, Unzerlegbare Darstellungen I, Manuscr. Math., 6, (1972), 71-103CrossrefGoogle Scholar

  • [23] B. Grünbaum, Convex Polytopes (2nd Edition), GTM 221, Springer, 2003Google Scholar

  • [24] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002Google Scholar

  • [25] J.-C. Hausmann, On the Vietoris-Rips complexes and a cohomology theory for metric spaces, Prospects in Topology: Proceedings of a conference in honour of William Browder, Annals of Mathematics Studies 138, 175-188, Princeton Univ. Press, 1995Google Scholar

  • [26] R. Kannan and A. Bachem, Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix, SIAM J. Comput., 8, (1979), 499-507Google Scholar

  • [27] S. Lang, Chapter III Modules, Section 7 Modules over principal rings, Algebra( Revised Third Edition), GTM 211, 146-155, Springer, 2002Google Scholar

  • [28] F. Meunier, Polytopal complexes: maps, chain complexes and... necklaces, arXiv:0806.1488v2, (2008)Google Scholar

  • [29] J. Milnor, Morse Theory (Annals of Mathematic Studies AM-51), Princeton Univ. Press, 1963Google Scholar

  • [30] D. Morozov, Persistence algorithm takes cubic time in worst case, BioGeometry News, Dept. Comput. Sci., Duke Univ., Durham, North Carolina, (2005)Google Scholar

  • [31] J. R. Munkres, Elements of Algebraic Topology, Westview Press, 1993Google Scholar

  • [32] F. P. Preparata and M. I. Shamos, Computational Geometry: an Introduction, Springer-Verlag, New York, 1985Google Scholar

  • [33] V. Robins, Toward computing homology from finite approximations, Topology Proceedings, 24, (1999), 503-532Google Scholar

  • [34] H. Sexton and M. V. Johansson, JPlex, a Java software package for computing the persistent homology of filtered simplicial complexesGoogle Scholar

  • [35] E. H. Spanier, Algebraic Topology, Springer-Verlag, New York, 1966Google Scholar

  • [36] L. Vietoris, Über den h¨oheren Zusammenhang kompakter R¨aume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann., 97, (1927), 454-472CrossrefGoogle Scholar

  • [37] A. J. Zomorodian, Topology for Computing, Cambridge Univ. Press, Cambridge, England, 2005Google Scholar

  • [38] A. J. Zomorodian and G. Carlsson, Computing Persistent Homology, Discrete Comput. Geom., 33, (2005), 249-274 CrossrefGoogle Scholar

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, 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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