Skip to main content
SpringerLink
Log in
Book cover

Optimization in the Real World pp 79–96Cite as

Optimal Cycles for Persistent Homology Via Linear Programming

  • Conference paper
  • First Online:

Part of the book series: Mathematics for Industry ((MFI,volume 13))

Abstract

In this work, we discuss the problem of finding optimal cycles for homology groups of simplicial complexes and for persistent homology of filtrations. We review the linear programming formulation of the optimal homologous cycle problem and its extension to allow for multiple cycles. By inserting these linear programming problems into the persistent homology algorithm, we are able to compute an optimal cycle, that has been optimized at birth, for every persistent interval in the persistent diagram.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Berman, H.M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T.N., Weissig, H., Shindyalov, I.N., Bourne, P.E.: The Protein Data Bank. Nucleic Acids Res. 28(1), 235–242 (2000)

    Google Scholar 

  2. Cgal, Computational Geometry Algorithms Library: http://www.cgal.org

  3. Chen, C., Freedman, D.: Hardness results for homology localization. Discrete Comput. Geom. 45(3), 425–448 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, C., Freedman, D.: Measuring and computing natural generators for homology groups. Comput. Geom. 43(2), 169–181 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dey, T.K., Hirani, A.N., Krishnamoorthy, B.: Optimal homologous cycles, total unimodularity, and linear programming. SIAM J. Comput. 40(4), 1026–1044 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dey, T.K., Sun, J., Wang, Y.: Approximating cycles in a shortest basis of the first homology group from point data. Inverse Prob. 27(12) (2011)

    Google Scholar 

  7. Edelsbrunner, H.: Weighted alpha shapes. Technical report, Department of Computer Science, University of Illinois at Urbana-Champaign (1992)

    Google Scholar 

  8. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Erickson, J., Whittlesey, K.: Greedy optimal homotopy and homology generators. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 1038–1046. SIAM (2005)

    Google Scholar 

  10. Escolar, E.G., Yasuaki H.: Computing optimal cycles of homology groups. In: Nishii, R., et al. (eds.) A Mathematical Approach to Research Problems of Science and Technology, pp. 101–118. Springer, Japan (2014)

    Google Scholar 

  11. Fermi, G., Perutz, M.F., Shaanan, B., Fourme, R.: The crystal structure of human deoxyhaemoglobin at 1.74 Å resolution. J. Mol. Biol. 175(2), 159–174 (1984)

    Article  Google Scholar 

  12. International Business Machines Corp.: IBM ILOG CPLEX Optimization Studio. http://www-03.ibm.com/software/products/en/ibmilogcpleoptistud/

  13. Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004)

    Book  MATH  Google Scholar 

  14. Munkres, J.: Elements of Algebraic Topology. The Benjamin/Cummings Publishing Company Inc, Menlo Park, California (1984)

    MATH  Google Scholar 

  15. Nakamura, T., Hiraoka, Y., Hirata, A., Escolar, E.G., Matsue, K., Nishiura, Y.: Description of Medium-Range Order in Amorphous Structures by Persistent Homology. arxiv.org/abs/1501.03611

  16. Salkin, H.M., Mathur, K.: Foundations of Integer Programming. North Holland, New York (1989)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Genki Kusano and Hiroshi Takeuchi for interesting discussions, especially from the side of applications, and Prof. Hayato Waki for the chance to give this talk in the workshop “Optimization in the Real World—Towards Solving Real-World Optimization Problems”.

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Emerson G. Escolar

  2. WPI—Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai, 980-8577, Japan

    Yasuaki Hiraoka

Authors
  1. Emerson G. Escolar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yasuaki Hiraoka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emerson G. Escolar .

Editor information

Editors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Katsuki Fujisawa

  2. Zuse Institute Berlin, Berlin, Germany

    Yuji Shinano

  3. Kyushu University, Fukuoka, Japan

    Hayato Waki

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Japan

About this paper

Cite this paper

Escolar, E.G., Hiraoka, Y. (2016). Optimal Cycles for Persistent Homology Via Linear Programming. In: Fujisawa, K., Shinano, Y., Waki, H. (eds) Optimization in the Real World. Mathematics for Industry, vol 13. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55420-2_5

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-55420-2_5

  • Published:

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-55419-6

  • Online ISBN: 978-4-431-55420-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation