Skip to main content
SpringerLink
Log in

A Novel Geometric Approach to the Problem of Multidimensional Scaling

  • Conference paper
  • First Online:
Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

Abstract

Multidimensional scaling (MDS) is one of the most popular methods for a visual representation of multidimensional data. A novel geometric interpretation of the stress function and multidimensional scaling in general (Geometric MDS) has been proposed. Following this interpretation, the step size and direction forward the minimum of the stress function are found analytically for a separate point without reference to the analytical expression of the stress function, numerical evaluation of its derivatives and the linear search. It is proved theoretically that the direction coincides with the steepest descent direction, and the analytically found step size guarantees the decrease of stress in this direction. A strategy of application of the discovered option to minimize the stress function is presented and examined. It is compared with SMACOF version of MDS. The novel geometric approach will allow developing a new class of algorithms to minimize MDS stress, including global optimization and high-performance computing.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

References

  1. Dzemyda, G., Kurasova, O., Žilinskas, J.: Multidimensional Data Visualization: Methods and Applications. Springer Optimization and its Applications, vol. 75. Springer, New York (2013). https://doi.org/10.1007/978-1-4419-0236-8

    Book  MATH  Google Scholar 

  2. Borg, I., Groenen, P.J.F., Mair, P.: Applied Multidimensional Scaling and Unfolding, 2nd edn. Springer, Heidelberg (2018)

    Book  Google Scholar 

  3. Li, F., Jiang, M.: Low-resolution face recognition and feature selection based on multidimensional scaling joint L-2, L-1-norm regularisation. IET Biom. 8(3), 198–205 (2019)

    Article  Google Scholar 

  4. Dzemyda, G., Kurasova, O., Medvedev, V., Dzemydaitė, G.: Visualization of data: methods, software, and applications. In: Singh, V.K., Gao, D., Fischer, A. (eds.) Advances in Mathematical Methods and High Performance Computing. AMM, vol. 41, pp. 295–307. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-02487-1_18

    Chapter  Google Scholar 

  5. Perales, E., Burgos, F.J., Vilaseca, M., Viqueira, V., Martinez-Verdu, F.M.: Graininess characterization by multidimensional scaling. J. Mod. Opt. 66(9), 929–938 (2019)

    Article  Google Scholar 

  6. Kruskal, J.B.: Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29(1), 1–27 (1964)

    Article  MathSciNet  Google Scholar 

  7. Žilinskas, A.: A quadratically converging algorithm of multidimensional scaling. Informatica 7(2), 268–274 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Orts Gomez, F.J., Ortega Lopez, G., Filatovas, E., Kurasova, O., Garzon, G.E.M.: Hyperspectral image classification using Isomap with SMACOF. Informatica 30(2), 349–365 (2019)

    Article  Google Scholar 

  9. Groenen, P., Mathar, R., Trejos, J.: Global optimization methods for multidimensional scaling applied to mobile communication. In: Gaul, W., Opitz, O., Schander, M. (eds.) Data Analysis: Scientific Modeling and Practical Applications, pp. 459–475. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-642-58250-9_37

    Chapter  Google Scholar 

  10. Orts, F., et al.: Improving the energy efficiency of SMACOF for multidimensional scaling on modern architectures. J. Supercomput. 75(3), 1038–1050 (2018)

    Article  Google Scholar 

  11. De Leeuw, J., Mair, P.: Multidimensional scaling using majorization: SMACOF in R. J. Stat. Softw. 31(3), 1–30 (2009)

    Article  Google Scholar 

  12. Symmetric Smacof. https://www.rdocumentation.org/packages/smacof/versions/2.0-0/topics/smacofSym

  13. Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling, 2nd edn. Springer, New York (2005). https://doi.org/10.1007/0-387-28981-X

    Book  MATH  Google Scholar 

  14. Torgerson Scaling. https://www.rdocumentation.org/packages/smacof/versions/2.0-0/topics/torgerson

Download references

Acknowledgements

This research is funded by Vilnius University, grant No. MSF-LMT-4. The authors are grateful to the reviewers for their comments that made the results of this paper more valuable.

Author information

Authors and Affiliations

  1. Vilnius University Institute of Data Science and Digital Technologies, Akademijos Street 4, 08412, Vilnius, Lithuania

    Gintautas Dzemyda & Martynas Sabaliauskas

Authors
  1. Gintautas Dzemyda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martynas Sabaliauskas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gintautas Dzemyda .

Editor information

Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dzemyda, G., Sabaliauskas, M. (2020). A Novel Geometric Approach to the Problem of Multidimensional Scaling. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_30

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-40616-5_30

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-40615-8

  • Online ISBN: 978-3-030-40616-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation