An output-sensitive algorithm for persistent homology

Computational Geometry: Theory and Applications

Volumn 46, Issue 4, 2013, Pages 435-447

  Chen, Chao ^a   Kerber, Michael ^a  

^a INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA   (Austria)

Author keywords

Computational topology; Persistent homology; Randomized algorithms; Rank computation

Indexed keywords

MONTE CARLO METHODS;

ARBITRARY CONSTANTS; COMPUTATIONAL TOPOLOGY; LAS VEGAS ALGORITHMS; MONTE CARLO ALGORITHMS; OUTPUT-SENSITIVE ALGORITHM; PERSISTENT HOMOLOGY; RANDOMIZED ALGORITHMS; SIMPLICIAL COMPLEX;

MATRIX ALGEBRA;

EID: 84992504901     PISSN: 09257721     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.comgeo.2012.02.010     Document Type: Article

References (37)

1. [1] Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A., On the local behavior of spaces of natural images. International Journal of Computer Vision 76 (2008), 1–12.
2. [2] Cerri, A., Ferri, M., Giorgi, D., Retrieval of trademark images by means of size functions.Special Issue on the Vision, Video and Graphics Conference 2005. Graphical Models, 68, 2006, 451–471.
3. [3] Chen, C., Edelsbrunner, H., Diffusion runs low on persistence fast. Proceedings of the Thirteenth International Conference on Computer Vision, 2011, IEEE.
4. [4] Chung, M., Bubenik, P., Kim, P., Persistence diagrams of cortical surface data. Information Processing in Medical Imaging Lecture Notes in Computer Science, vol. 5636, 2009, 386–397.
5. [5] de Silva, V., Ghrist, R., Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology 7 (2007), 339–358.
6. [6] Y. Wang, P. Agarwal, P. Brown, H. Edelsbrunner, J. Rudolph, Coarse and reliable geometric alignment for protein docking, in: Proceedings of the Pacific Symposium on Biocomputing 2005, 2005, pp. 65–75.
7. [7] P. Bendich, H. Edelsbrunner, D. Morozov, A. Patel, The robustness of level sets, in: Proceedings of the 18th European Symposium on Algorithms, 2010, pp. 1–10.
8. [8] Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L., Oudot, S., Proximity of persistence modules and their diagrams. Proceedings of the 25th Annual Symposium on Computational Geometry, 2009, ACM, 237–246.
9. [9] Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Stability of persistence diagrams. Discrete and Computational Geometry 37 (2007), 103–120.
10. [10] Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Extending persistence using Poincaré and Lefschetz duality. Foundations of Computational Mathematics 9 (2009), 79–103.
11. [11] de Silva, V., Vejdemo-Johansson, M., Persistent cohomology and circular coordinates. Proceedings of the 25th Annual Symposium on Computational Geometry, 2009, ACM, 227–236.
12. [12] Zomorodian, A., Carlsson, G., Computing persistent homology. Discrete and Computational Geometry 33 (2005), 249–274.
13. [13] Zomorodian, A., Carlsson, G., The theory of multidimensional persistence. Discrete and Computational Geometry 42 (2009), 73–93.
14. [14] Edelsbrunner, H., Harer, J., Computational Topology. An Introduction. 2010, American Mathematical Society.
15. [15] Edelsbrunner, H., Letscher, D., Zomorodian, A., Topological persistence and simplification. Discrete and Computational Geometry 28 (2002), 511–533.
16. [16] Carlsson, G., de Silva, V., Morozov, D., Zigzag persistent homology and real-valued functions. Proceedings of the 25th Annual Symposium on Computational Geometry, 2009, ACM, 247–256.
17. [17] Morozov, D., Persistence algorithm takes cubic time in the worst case. BioGeometry News, 2005, Duke Computer Science, Durham, NC.
18. [18] C. Chen, M. Kerber, Persistent homology computation with a twist, in: 27th European Workshop on Computational Geometry, 2011.
19. [19] N. Milosavljević, D. Morozov, P. Škraba, Zigzag persistent homology in matrix multiplication time, in: Proceedings of the 27th Annual Symposium on Computational Geometry, 2011, pp. 216–225.
20. [20] Coppersmith, D., Winograd, S., Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9 (1990), 251–280.
21. [21] W. Harvey, Y. Wang, R. Wenger, A randomized O(mlogm) time algorithm for computing Reeb graphs of arbitrary simplicial complexes, in: Proceedings of the 26th Annual Symposium on Computational Geometry, 2010, pp. 267–276.
22. [22] A. Zomorodian, The tidy set: a minimal simplicial set for computing homology of clique complexes, in: Proceedings of the 26th Annual Symposium on Computational Geometry, 2010, pp. 257–266.
23. [23] B. Hudson, G. Miller, S. Oudot, D. Sheehy, Topological inference via meshing, in: Proceedings of the 2010 Annual Symposium on Computational Geometry, 2010, pp. 277–286.
24. [24] H. Wagner, C. Chen, E. Vuçini, Efficient computation of persistent homology for cubical data, in: Workshop on Topology-based Methods in Data Analysis and Visualization, 2011.
25. [25] Bendich, P., Edelsbrunner, H., Kerber, M., Computing robustness and persistence for images. IEEE Transactions on Visualization and Computer Graphics 16 (2010), 1251–1260.
26. [26] Kaltofen, E., David Saunders, B., On Wiedemannʼs method of solving sparse linear systems. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, vol. 539, 1991, Springer, 29–38.
27. [27] Eberly, W., Giesbrecht, M., Giorgi, P., Storjohann, A., Villard, G., Faster inversion and other black box matrix computations using efficient block projections. Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation (ISSAC 2007), 2007, ACM, 143–150.
28. [28] Ibara, O.H., Moran, S., Hui, R., A generalization of the fast LUP matrix decomposition algorithm and applications. Journal of Algorithms 3 (1982), 45–56.
29. [29] Munkres, J., Elements of Algebraic Topology. 1984, Westview Press.
30. [30] C. Chen, D. Freedman, Quantifying homology classes, in: Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science (STACS 2008), 2008, pp. 169–180.
31. [31] Chen, C., Freedman, D., Measuring and computing natural generators for homology groups. Computational Geometry: Theory and Applications 43 (2010), 169–181.
32. [32] D. Cohen-Steiner, H. Edelsbrunner, D. Morozov, Vines and vineyards by updating persistence in linear time, in: Proceedings of the 22nd Annual Symposium on Computational Geometry, 2006, pp. 119–126.
33. [33] Cormen, T., Leiserson, C., Rivest, R., Stein, C., Introduction to Algorithms. 2001, The MIT Press.
34. [34] Schönhage, A., Strassen, V., Schnelle Multiplikation grosser Zahlen. Computing 7 (1971), 281–292 (in German).
35. [35] Wiedemann, D.H., Solving sparse linear equations over finite fields. IEEE Transactions on Information Theory 32 (1986), 54–62.
36. [36] J.-G. Dumas, T. Gautier, M. Giesbrecht, P. Giorgi, B. Hovinen, E. Kaltofen, B. Saunders, W. Turner, G. Villard, Linbox: A generic library for exact linear algebra, in: Mathematical Software: ICMS 2002, Proceedings of the first International Congress of Mathematical Software, 2002, pp. 40–50.
37. [37] J.-G. Dumas, G. Villard, Computing the rank of large sparse matrices over finite fields, in: Proceedings of the Fifth International Workshop on Computer Algebra in Scientific Computing, 2002, pp. 47–62.