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DTM-Based Filtrations

Authors Hirokazu Anai, Frédéric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphaël Tinarrage, Yuhei Umeda

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Author Details

Hirokazu Anai
  • Fujitsu Laboratories, AI Lab, Kawasaki, Japan
Frédéric Chazal
  • Datashape, Inria Paris-Saclay, France
Marc Glisse
  • Datashape, Inria Paris-Saclay, France
Yuichi Ike
  • Fujitsu Laboratories, AI Lab, Kawasaki, Japan
Hiroya Inakoshi
  • Fujitsu Laboratories, AI Lab, Kawasaki, Japan
Raphaël Tinarrage
  • Datashape, Inria Paris-Saclay, France
Yuhei Umeda
  • Fujitsu Laboratories, AI Lab, Kawasaki, Japan

Funding

This work was partially supported by a collaborative research agreement between Inria and Fujitsu, and the Advanced Grant of the European Research Council GUDHI (Geometric Understanding in Higher Dimensions).

Cite AsGet BibTex

Hirokazu Anai, Frédéric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphaël Tinarrage, and Yuhei Umeda. DTM-Based Filtrations. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.58

Abstract

Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Čech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions and extends some previous work on the approximation of such functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Topological Data Analysis
  • Persistent homology

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References

  1. Greg Bell, Austin Lawson, Joshua Martin, James Rudzinski, and Clifford Smyth. Weighted Persistent Homology. arXiv preprint, 2017. URL: http://arxiv.org/abs/1709.00097.
  2. Mickaël Buchet. Topological inference from measures. PhD thesis, Paris 11, 2014. Google Scholar
  3. Mickaël Buchet, Frédéric Chazal, Steve Y Oudot, and Donald R Sheehy. Efficient and robust persistent homology for measures. Computational Geometry, 58:70-96, 2016. Google Scholar
  4. F. Chazal, D. Cohen-Steiner, and Q. Mérigot. Geometric Inference for Probability Measures. Journal on Found. of Comp. Mathematics, 11(6):733-751, 2011. Google Scholar
  5. Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics, 2016. Google Scholar
  6. Frédéric Chazal, Vin De Silva, and Steve Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193-214, 2014. Google Scholar
  7. Frédéric Chazal and Steve Yann Oudot. Towards persistence-based reconstruction in euclidean spaces. In Proceedings of the twenty-fourth annual symposium on Computational geometry, SCG '08, pages 232-241, New York, NY, USA, 2008. ACM. Google Scholar
  8. Leonidas Guibas, Dmitriy Morozov, and Quentin Mérigot. Witnessed k-distance. Discrete &Computational Geometry, 49(1):22-45, 2013. Google Scholar
  9. Fujitsu Laboratories. Estimating the Degradation State of Old Bridges-Fijutsu Supports Ever-Increasing Bridge Inspection Tasks with AI Technology. Fujitsu Journal, March 2018. URL: https://journal.jp.fujitsu.com/en/2018/03/01/01/.
  10. J. Phillips, B. Wang, and Y Zheng. Geometric Inference on Kernel Density Estimates. In Proc. 31st Annu. Sympos. Comput. Geom (SoCG 2015), pages 857-871, 2015. Google Scholar
  11. Lee M Seversky, Shelby Davis, and Matthew Berger. On time-series topological data analysis: New data and opportunities. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pages 59-67, 2016. Google Scholar
  12. Donald R. Sheehy. Linear-Size Approximations to the Vietoris-Rips Filtration. Discrete &Computational Geometry, 49(4):778-796, 2013. Google Scholar
  13. Yuhei Umeda. Time Series Classification via Topological Data Analysis. Transactions of the Japanese Society for Artificial Intelligence, 32(3):D-G72_1, 2017. Google Scholar
  14. Gudhi: Geometry understanding in higher dimensions. URL: http://gudhi.gforge.inria.fr/.
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