DOI QR코드

DOI QR Code

INSTABILITY OF THE BETTI SEQUENCE FOR PERSISTENT HOMOLOGY AND A STABILIZED VERSION OF THE BETTI SEQUENCE

  • JOHNSON, MEGAN (DEPARTMENT OF MATHEMATICS, UNIVERSITY AT BUFFALO, THE STATE UNIVERSITY OF NEW YORK) ;
  • JUNG, JAE-HUN (DEPARTMENT OF MATHEMATICS & POSTECH MATHEMATICAL INSTITUTE FOR DATA SCIENCE (MINDS), POHANG UNIVERSITY OF SCIENCE AND TECHNOLOGY)
  • 투고 : 2021.08.27
  • 심사 : 2021.11.01
  • 발행 : 2021.12.25

초록

Topological Data Analysis (TDA), a relatively new field of data analysis, has proved very useful in a variety of applications. The main persistence tool from TDA is persistent homology in which data structure is examined at many scales. Representations of persistent homology include persistence barcodes and persistence diagrams, both of which are not straightforward to reconcile with traditional machine learning algorithms as they are sets of intervals or multisets. The problem of faithfully representing barcodes and persistent diagrams has been pursued along two main avenues: kernel methods and vectorizations. One vectorization is the Betti sequence, or Betti curve, derived from the persistence barcode. While the Betti sequence has been used in classification problems in various applications, to our knowledge, the stability of the sequence has never before been discussed. In this paper we show that the Betti sequence is unstable under the 1-Wasserstein metric with regards to small perturbations in the barcode from which it is calculated. In addition, we propose a novel stabilized version of the Betti sequence based on the Gaussian smoothing seen in the Stable Persistence Bag of Words for persistent homology. We then introduce the normalized cumulative Betti sequence and provide numerical examples that support the main statement of the paper.

키워드

과제정보

MJ was funded, in part, by the Doctoral Dissertation Fellowship of the Department of Mathematics at the University at Buffalo. JHJ has been supported by Samsung Science & Technology Foundation under grant number SSTF-BA1802-02.

참고문헌

  1. Gunnar Carlsson. Topology and data. Bulletin of The American Mathematical Society, 46:255-308, 04 2009. https://doi.org/10.1090/S0273-0979-09-01249-X
  2. Edelsbrunner, Letscher, and Zomorodian. Topological persistence and simplification. Discrete & Computational Geometry, 28(4):511-533, Nov 2002. https://doi.org/10.1007/s00454-002-2885-2
  3. Mathieu Carriere, Steve Y. Oudot, and Maks Ovsjanikov. Stable topological signatures for points on 3d shapes. ' Computer Graphics Forum, 34(5):1-12, 2015. https://doi.org/10.1111/cgf.12692
  4. Alex Cole and Gary Shiu. Persistent homology and non-gaussianity. arXiv preprint arXiv:1712.08159, 2017.
  5. Sven Heydenreich, Benjamin Bruck, and Joachim Harnois-Deraps. Persistent homology in cosmic shear: ' constraining parameters with topological data analysis. arXiv preprint arXiv:2007.13724, 2020.
  6. X. Xu, J. Cisewski-Kehe, S.B. Green, and D. Nagai. Finding cosmic voids and filament loops using topological data analysis. Astronomy and Computing, 27:34 - 52, 2019. https://doi.org/10.1016/j.ascom.2019.02.003
  7. Melissa R. McGuirl, Alexandria Volkening, and Bjorn Sandstede. Topological data analysis of zebrafish patterns. Proceedings of the National Academy of Sciences, 117(10):5113-5124, 2020. https://doi.org/10.1073/pnas.1917763117
  8. John Nicponski and Jae-Hun Jung. Topological data analysis of vascular disease: I a theoretical framework. Frontiers in Applied Mathematics and Statistics, 6:34, 2020. https://doi.org/10.3389/fams.2020.00034
  9. Paul Bendich, J. S. Marron, Ezra Miller, Alex Pieloch, and Sean Skwerer. Persistent homology analysis of brain artery trees. Ann. Appl. Stat., 10(1):198-218, 03 2016.
  10. Ann E. Sizemore, Jennifer E. Phillips-Cremins, Robert Ghrist, and Danielle S. Bassett. The importance of the whole: Topological data analysis for the network neuroscientist. Network Neuroscience, 3(3):656-673, 2019. https://doi.org/10.1162/netn_a_00073
  11. Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8):1-35, 2017.
  12. Yuhei Umeda. Time series classification via topological data analysis. Information and Media Technologies, 12:228-239, 2017.
  13. Yu-Min Chung and Austin Lawson. Persistence curves: A canonical framework for summarizing persistence diagrams. 2020.
  14. Megan Johnson and Jae-Hun Jung. The interconnectivity vector: A finite-dimensional representation of persistent homology. arXiv preprint arXiv:2011.11579, 2020.
  15. Bartosz Zielinski, Michal Lipi ' nski, Mateusz Juda, Matthias Zeppelzauer, and Pawel Dlotko. Persistence code-books for topological data analysis. Artificial Intelligence Review, Sep 2020.
  16. L. N. Vaserstein. Markov processes over denumerable products of spaces, describing large systems of automata. Problems Inform. Transmission, 5(3):47-52, 1969.
  17. Michael Barnsley. Fractals everywhere. Academic Press Inc., Boston, MA, 1988.