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Inferring Underlying Manifold of Data by the Use of Persistent Homology Analysis

  • Rentaro FutagamiEmail author
  • Noritaka YamadaEmail author
  • Takeshi ShibuyaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11382)

Abstract

Inferring underlying manifold of data is one of the important issues for point cloud data analysis. This is accomplished by inferring the topological shape of the underlying manifold. This is done by estimating the number of holes in the underlying manifold in each dimension.

Persistent homology is one of the means of estimating the number of holes in the underlying manifold. Calculating the persistent homology of data determines the size, number, and dimensions of holes produced from data points. However, the number of holes represented through persistent homology is far greater than that in underlying manifold. This problem is caused by noises in a result of calculating persistent homology. Therefore, reducing noises that result from calculating persistent homology is necessary to estimate the number of holes in the underlying manifold.

Conventional methods cannot reduce noises adequately when data are of low density and thus cannot estimate the number of holes in the underlying manifold without manual analysis by experts.

In this study, we propose a new method to estimate automatically the number of holes in the underlying manifolds. We also compare the proposed and conventional methods and show the effectiveness of the former.

Keywords

Persistent homology Topological data analysis Underlying manifold Topological features Persistent landscape 

References

  1. 1.
    Futagami, R., Shibuya, T.: A method deciding topological relationship for self-organizing maps by persistent homology analysis. In: Proceedings of SICE Annual Conference 2016, pp. 1064–1069 (2016)Google Scholar
  2. 2.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. Comput. Geom. 33(2), 249–274 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Edelsbrunner, H., Harer, J.: Persistent homology-a survey. Contemp. Math. 453, 257–282 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fasy, T.B., Lecci, F., et al.: Confidence sets for persistence diagrams. Annu. Stat. 42(6), 2301–2339 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bubenik, P.: Statistical topological data analysis using persistent landscapes. J. Mach. Learn. Res. 16(1), 77–102 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gameiro, M., et al.: A topological measurement of protein compressibility. Jpn. J. Ind. Appl. Math. 32(1), 1–17 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zhang, W., et al.: An optimized degree strategy for persistent sensor network data distribution. In: Euromicro International Conference on Parallel, Distributed and Network-Based Processing (2012)Google Scholar
  8. 8.
    Zhu, X.: Persistent homology: an introduction and a new text representation for natural language processing. In: Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence (2013)Google Scholar
  9. 9.
    Steiner, D.C., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discret. Comput. Geom. 37(1), 103–120 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models, 1st edn. Chapman & Hall/CRC, Boca Raton (1990)zbMATHGoogle Scholar
  11. 11.
    Nene, S.A., Nayar, S.K., Murase, H.: Columbia object image library (COIL-20). Technical report, No. CUCS-005-96 (1996)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of TsukubaTsukubaJapan

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