LIPIcs.SoCG.2019.58.pdf
- Filesize: 1.08 MB
- 15 pages
Document
DTM-Based Filtrations
Authors
-
Part of:
Volume:
35th International Symposium on Computational Geometry (SoCG 2019)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2019-06-11
Cite As Get 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
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
- Greg Bell, Austin Lawson, Joshua Martin, James Rudzinski, and Clifford Smyth. Weighted Persistent Homology. arXiv preprint, 2017. URL: http://arxiv.org/abs/1709.00097.
-
Mickaël Buchet. Topological inference from measures. PhD thesis, Paris 11, 2014.
-
Mickaël Buchet, Frédéric Chazal, Steve Y Oudot, and Donald R Sheehy. Efficient and robust persistent homology for measures. Computational Geometry, 58:70-96, 2016.
-
F. Chazal, D. Cohen-Steiner, and Q. Mérigot. Geometric Inference for Probability Measures. Journal on Found. of Comp. Mathematics, 11(6):733-751, 2011.
-
Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics, 2016.
-
Frédéric Chazal, Vin De Silva, and Steve Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193-214, 2014.
-
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.
-
Leonidas Guibas, Dmitriy Morozov, and Quentin Mérigot. Witnessed k-distance. Discrete &Computational Geometry, 49(1):22-45, 2013.
- 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/.
-
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.
-
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.
-
Donald R. Sheehy. Linear-Size Approximations to the Vietoris-Rips Filtration. Discrete &Computational Geometry, 49(4):778-796, 2013.
-
Yuhei Umeda. Time Series Classification via Topological Data Analysis. Transactions of the Japanese Society for Artificial Intelligence, 32(3):D-G72_1, 2017.
- Gudhi: Geometry understanding in higher dimensions. URL: http://gudhi.gforge.inria.fr/.