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International Workshop on Computational Topology in Image Context

CTIC 2019: Computational Topology in Image Context pp 40–53Cite as

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

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

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References

  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. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. Comput. Geom. 33(2), 249–274 (2005)

    Article  MathSciNet  Google Scholar 

  3. Edelsbrunner, H., Harer, J.: Persistent homology-a survey. Contemp. Math. 453, 257–282 (2008)

    Article  MathSciNet  Google Scholar 

  4. Fasy, T.B., Lecci, F., et al.: Confidence sets for persistence diagrams. Annu. Stat. 42(6), 2301–2339 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bubenik, P.: Statistical topological data analysis using persistent landscapes. J. Mach. Learn. Res. 16(1), 77–102 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Gameiro, M., et al.: A topological measurement of protein compressibility. Jpn. J. Ind. Appl. Math. 32(1), 1–17 (2013)

    Article  MathSciNet  Google Scholar 

  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. 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. Steiner, D.C., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discret. Comput. Geom. 37(1), 103–120 (2007)

    Article  MathSciNet  Google Scholar 

  10. Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models, 1st edn. Chapman & Hall/CRC, Boca Raton (1990)

    MATH  Google Scholar 

  11. Nene, S.A., Nayar, S.K., Murase, H.: Columbia object image library (COIL-20). Technical report, No. CUCS-005-96 (1996)

    Google Scholar 

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Authors and Affiliations

  1. University of Tsukuba, Tsukuba, Ibaraki, Japan

    Rentaro Futagami, Noritaka Yamada & Takeshi Shibuya

Authors
  1. Rentaro Futagami
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  2. Noritaka Yamada
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  3. Takeshi Shibuya
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Corresponding authors

Correspondence to Rentaro Futagami , Noritaka Yamada or Takeshi Shibuya .

Editor information

Editors and Affiliations

  1. Universidad de Málaga, Málaga, Spain

    Rebeca Marfil

  2. Universidad de Málaga, Málaga, Spain

    Mariletty Calderón

  3. University of Seville, Sevilla, Spain

    Fernando Díaz del Río

  4. University of Seville, Sevilla, Spain

    Pedro Real

  5. Universidad de Málaga, Málaga, Spain

    Antonio Bandera

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Futagami, R., Yamada, N., Shibuya, T. (2019). Inferring Underlying Manifold of Data by the Use of Persistent Homology Analysis. In: Marfil, R., Calderón, M., Díaz del Río, F., Real, P., Bandera, A. (eds) Computational Topology in Image Context. CTIC 2019. Lecture Notes in Computer Science(), vol 11382. Springer, Cham. https://doi.org/10.1007/978-3-030-10828-1_4

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  • DOI: https://doi.org/10.1007/978-3-030-10828-1_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10827-4

  • Online ISBN: 978-3-030-10828-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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