Persistence bag-of-words for topological data analysis
Authors: 
Bartosz Zieliński, Michał Lipiński, Mateusz Juda, 
Matthias Zeppelzauer, Paweł DłotkoAuthors Info & Claims
Bartosz Zieliński, Michał Lipiński, Mateusz Juda, Matthias Zeppelzauer, Paweł DłotkoAuthors Info & ClaimsPages 4489 - 4495
Abstract
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.
References
[1]
Henry Adams, Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier. Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8):1-35, 2017.
[2]
Rushil Anirudh, Vinay Venkataraman, Karthikeyan Natesan Ramamurthy, and Pavan Turaga. A riemannian framework for statistical analysis of topological persistence diagrams. In Proc. of IEEE CVPR - Workshops, pages 68-76, 2016.
[3]
Ricardo Baeza-Yates and Berthier Ribeiro-Neto. Modern information retrieval, volume 463. ACM press New York, 1999.
[4]
Ulrich Bauer, Michael Kerber, Jan Reininghaus, and Hubert Wagner. Phat-persistent homology algorithms toolbox. Journal of Symbolic Computation, 78:76-90, 2017.
[5]
Peter Bubenik. Statistical topological data analysis using persistence landscapes. JMLR, 16(1):77-102, 2015.
[6]
Mathieu Carriére, Marco Cuturi, and Steve Oudot. Sliced wasserstein kernel for persistence diagrams. In ICML, 2017.
[7]
Joseph DeGol, Mani Golparvar-Fard, and Derek Hoiem. Geometry-informed material recognition. In Proc. of CVPR, pages 1554-1562, 2016.
[8]
Tamal K. Dey, Dayu Shi, and Yusu Wang. Simba: An efficient tool for approximating rips-filtration persistence via simplicial batch collapse. J. Exp. Algorithmics, 24(1):1.5:1- 1.5:16, January 2019.
[9]
Pietro Donatini, Patrizio Frosini, and Alberto Lovato. Size functions for signature recognition. In Proc. SPIE, volume 3454, pages 178-183, 1998.
[10]
Herbert Edelsbrunner and John Harer. Computational topology: an introduction. American Mathematical Soc., 2010.
[11]
Massimo Ferri, Patrizio Frosini, Alberto Lovato, and Chiara Zambelli. Point selection: A new comparison scheme for size functions (with an application to monogram recognition). In Computer Vision - ACCV'98. LNCS., volume 1351, pages 329-337. Springer, 1998.
[12]
Massimo Ferri. Persistent topology for natural data analysis -- a survey. In Towards Integrative Machine Learning and Knowledge Extraction, pages 117-133, Cham, 2017. Springer International Publishing.
[13]
Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramàr, Konstantin Mischaikow, and Vidit Nanda. A topological measurement of protein compressibility. Japan J. of Industrl. and Appl. Mathem., 32(1):1-17, 2014.
[14]
Marian Gidea and Yuri Katz. Topological data analysis of financial time series: Landscapes of crashes. Physica A: Statistical Mechanics and its Applications, 491:820- 834, 2018.
[15]
Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl. Deep learning with topological signatures. In Advances in Neural Information Processing Systems, pages 1633-1643, 2017.
[16]
Michael Kerber, Dmitriy Morozov, and Arnur Nigmetov. Geometry helps to compare persistence diagrams. Journal of Experimental Algorithmics (JEA), 22:1-4, 2017.
[17]
Valentin Khrulkov and Ivan Oseledets. Geometry score: A method for comparing generative adversarial networks. In Proc. of the 35th ICML, volume 80, pages 2621-2629. PMLR, 2018.
[18]
Théo Lacombe, Marco Cuturi, and Steve Oudot. Large scale computation of means and clusters for persistence diagrams using optimal transport. arXiv:1805.08331, 2018.
[19]
Tam Le and Makoto Yamada. Persistence fisher kernel: A riemannian manifold kernel for persistence diagrams. In Adv. in Neural Inf. Proc. Sys., pages 10028-10039, 2018.
[20]
Yongjin Lee, Senja D. Barthel, Paweł Dłotko, S. Mohamad Moosavi, Kathryn Hess, and Berend Smit. Quantifying similarity of pore-geometry in nanoporous materials. Nature Communications, 8(15396), 2017.
[21]
Chunyuan Li, Maks Ovsjanikov, and Frédéric Chazal. Persistence-based structural recognition. In IEEE CVPR, pages 2003-2010. IEEE, 2014.
[22]
Clément Maria, Jean-Daniel Boissonnat, Marc Glisse, and Mariette Yvinec. The gudhi library: Simplicial complexes and persistent homology. In International Congress on Mathematical Software, pages 167-174. Springer, 2014.
[23]
Andrew McCallum and Kamal Nigam. A comparison of event models for naive bayes text classification. In AAAI-98 workshop on learning for text categorization, volume 752, pages 41-48, 1998.
[24]
Nasser M. Nasrabadi. Pattern recognition and machine learning. J. of electronic imaging, 16(4):049901, 2007.
[25]
Florent Perronnin, Jorge Sénchez, and Yan Liu Xerox. Large-scale image categorization with explicit data embedding. In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pages 2297-2304. IEEE, 2010.
[26]
Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning. In 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 4741-4748, June 2015.
[27]
Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max Horn, Thomas Gumbsch, and Karsten Borgwardt. Neural persistence: A complexity measure for deep neural networks using algebraic topology. arXiv:1812.09764, 2018.
[28]
Josef Sivic and Andrew Zisserman. Video google: A text retrieval approach to object matching in videos. In IEEE ICCV, 2003, pages 1470-1477. IEEE, 2003.
[29]
Jan C. Van Gemert, Jan-Mark Geusebroek, Cor J. Veenman, and Arnold W. M. Smeulders. Kernel codebooks for scene categorization. In European conference on computer vision, pages 696-709. Springer, 2008.
[30]
Andrea Vedaldi and Brian Fulkerson. VLFeat: An open and portable library of computer vision algorithms. http://www.vlfeat.org/, 2008.
[31]
Matthias Zeppelzauer, Bartosz Zielićski, Mateusz Juda, and Markus Seidl. A study on topological descriptors for the analysis of 3d surface texture. Comp. Vision and Image Underst., 2017.
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- Sony: Sony Corporation
- Huawei Technologies Co. Ltd.: Huawei Technologies Co. Ltd.
- Baidu Research: Baidu Research
- The International Joint Conferences on Artificial Intelligence, Inc. (IJCAI)
- Lenovo: Lenovo
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AAAI Press
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Published: 10 August 2019
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