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SimBa: An Efficient Tool for Approximating Rips-Filtration Persistence via Simplicial Batch-Collapse

Authors Tamal K. Dey, Dayu Shi, Yusu Wang

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Tamal K. Dey
Dayu Shi
Yusu Wang

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Tamal K. Dey, Dayu Shi, and Yusu Wang. SimBa: An Efficient Tool for Approximating Rips-Filtration Persistence via Simplicial Batch-Collapse. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.35

Abstract

In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on P indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short of scaling well, especially for high dimensional data. In this paper, we investigate the advantages and limitations of existing approaches. Based on insights gained from the experiments, we propose an efficient new algorithm, called SimBa, for approximating the persistent homology of Rips filtrations with quality guarantees. Our new algorithm leverages a batch collapse strategy as well as a new sparse Rips-like filtration. We experiment on a variety of low and high dimensional data sets. We show that our strategy presents a significant size reduction, and our algorithm for approximating Rips filtration persistence is order of magnitude faster than existing methods in practice.
Keywords
  • Rips filtration
  • Homology groups
  • Persistence
  • Topological data analysis

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References

  1. H. Adams and G. Carlsson. On the nonlinear statistics of range image patches. SIAM J. Img. Sci., 2(1):110-117, 2009. URL: http://dx.doi.org/10.1137/070711669.
  2. U. Bauer, M. Kerber, and J. Reininghaus. Topological Methods in Data Analysis and Visualization III: Theory, Algorithms, and Applications, chapter Clear and Compress: Computing Persistent Homology in Chunks, pages 103-117. Springer International Publishing, Cham, 2014. URL: http://dx.doi.org/10.1007/978-3-319-04099-8_7.
  3. U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner. Mathematical Software - ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings, chapter PHAT - Persistent Homology Algorithms Toolbox, pages 137-143. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014. Project URL: https://bitbucket.org/phat-code/phat.
  4. J.-D. Boissonnat, T. K. Dey, and C. Maria. Algorithms - ESA 2013: 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, chapter The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology, pages 695-706. Springer, Berlin, Heidelberg, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_59.
  5. J.-D. Boissonnat and C. Maria. Algorithms - ESA 2012: 20th Annual European Symposium, Ljubljana, Slovenia, September 10-12, 2012. Proceedings, chapter The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes, pages 731-742. Springer, 2012. Google Scholar
  6. M. Buchet, F. Chazal, S. Y. Oudot, and D. R. Sheehy. Efficient and robust persistent homology for measures. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 168-180, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.13.
  7. D. Burghelea and T. K. Dey. Topological persistence for circle-valued maps. Discrete Comput. Geom., 50:69-98, 2013. Google Scholar
  8. G. Carlsson. Topology and data. Bull. Amer. Math. Soc., 46:255-308, 2009. Google Scholar
  9. G. Carlsson and V. de Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367-405, 2010. Google Scholar
  10. G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete &Computational Geometry, 42(1):71-93, 2009. URL: http://dx.doi.org/10.1007/s00454-009-9176-0.
  11. N. J. Cavanna, M. Jahanseir, and D. R. Sheehy. A geometric perspective on sparse filtrations. In Canadian Conf. Comput. Geom. (CCCG), 2015. URL: http://dblp.uni-trier.de/db/conf/cccg/cccg2015.html#CavannaJS15.
  12. F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot. Proximity of persistence modules and their diagrams. In Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, SCG'09, pages 237-246, New York, NY, USA, 2009. ACM. URL: http://dx.doi.org/10.1145/1542362.1542407.
  13. F. Chazal, D. Cohen-Steiner, L. Guibas, F. Mémoli, and S. Y. Oudot. Gromov-Hausdorff stable signatures for shapes using persistence. In Proc. of SGP, 2009. Google Scholar
  14. F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules. CoRR, abs/1207.3674, 2012. Google Scholar
  15. F. Chazal, L. J. Guibas, S. Y. Oudot, and P. Skraba. Persistence-based clustering in Riemannian manifolds. In Proc. 27th Annu. ACM Sympos. Comput. Geom., pages 97-106, 2011. Google Scholar
  16. C. Chen and M. Kerber. An output-sensitive algorithm for persistent homology. Comput. Geom. Theory Appl., 46(4):435-447, May 2013. URL: http://dx.doi.org/10.1016/j.comgeo.2012.02.010.
  17. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007. Google Scholar
  18. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Extending persistence using Poincaré and Lefschetz duality. Foundations of Computational Mathematics, 9(1):79-103, 2009. Google Scholar
  19. D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov. Vines and vineyards by updating persistence in linear time. In Proceedings of the twenty-second annual symposium on Computational geometry, pages 119-126. ACM, 2006. Google Scholar
  20. V. de Silva, D. Morozov, and M. Vejdemo-Johansson. Persistent cohomology and circular coordinates. Discrete Comput. Geom., 45(4):737-759, 2011. Google Scholar
  21. T. K. Dey, F. Fan, and Y. Wang. Computing topological persistence for simplicial maps. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 345-354. ACM, 2014. URL: http://dx.doi.org/10.1145/2582112.2582165.
  22. T. K. Dey, K. Li, C. Luo, P. Ranjan, I. Safa, and Y. Wang. Persistent heat signature for pose-oblivious matching of incomplete models. Comput. Graph. Forum. (special issue from Sympos. Geom. Process.), 29(5):1545-1554, 2010. Google Scholar
  23. Dmitriy Morozov. Dionysus Software. http://mrzv.org/software/dionysus/, 2012.
  24. H. Edelsbrunner and J. Harer. Computational Topology - an Introduction. American Mathematical Society, 2010. Google Scholar
  25. H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002. Google Scholar
  26. P. Frosini. A distance for similarity classes of submanifolds of a euclidean space. Bulletin of the Australian Mathematical Society, 42(3):407-416, 1990. Google Scholar
  27. W. Harvey, I.-H. Park, O. Rübel, V. Pascucci, P.-T. Bremer, C. Li, and Y. Wang. A collaborative visual analytics suite for protein folding research. Journal of Molecular Graphics and Modelling, 53:59-71, 2014. URL: http://dx.doi.org/10.1016/j.jmgm.2014.06.003.
  28. M. Lichman. UCI machine learning repository, 2013. Project URL: http://archive.ics.uci.edu/ml.
  29. R. C. B. Madeo, S. M. Peres, and C. A. de M. Lima. Gesture phase segmentation using support vector machines. Expert Systems with Applications, 56:100-115, 2016. URL: http://dx.doi.org/10.1016/j.eswa.2016.02.021.
  30. R. C. B. Madeo, P. K. Wagner, and S. M. Peres. Gesture Phase Segmentation Data Set, 2014. Project URL: http://archive.ics.uci.edu/ml/datasets/Gesture+Phase+Segmentation.
  31. D. M. Mount and S. Arya. ANN: A Library for Approximate Nearest Neighbor Searching, 2010. Project URL: https://www.cs.umd.edu/~mount/ANN/.
  32. James R. Munkres. Elements of Algebraic Topology. Addison-Wesley, 1993. Google Scholar
  33. J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt. A stable multi-scale kernel for topological machine learning. In Proc. IEEE Conf. Comp. Vision &Pat. Rec. (CVPR), pages 4741-4748, 2015. Google Scholar
  34. V. Robins. Towards computing homology from finite approximations. Topology Proceedings, 24(1):503-532, 1999. Google Scholar
  35. D. R. Sheehy. Linear-size approximations to the vietoris-rips filtration. In Proceedings of the Twenty-eighth Annual Symposium on Computational Geometry, SoCG'12, pages 239-248. ACM, 2012. URL: http://dx.doi.org/10.1145/2261250.2261286.
  36. Simpers Software, 2015. Project URL: http://web.cse.ohio-state.edu/~tamaldey/SimPers/Simpers.html.
  37. G. Singh, F. Mémoli, T. Ishkhanov, G. Sapiro, G. Carlsson, and D. L Ringach. Topological analysis of population activity in visual cortex. Journal of vision, 8(8):11, 2008. Google Scholar
  38. The GUDHI Project. GUDHI user and reference manual, 2015. Project URL: http://gudhi.gforge.inria.fr/doc/latest/.
  39. A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33(2):249-274, 2005. Google Scholar
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