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Geometric multidimensional scaling: A new approach for data dimensionality reduction

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  • Dzemyda, Gintautas
  • Sabaliauskas, Martynas

Abstract

Multidimensional scaling (MDS) provides a possibility to present the multidimensional data visually. It is a very popular method of this class. MDS minimizes some stress functions. In this paper, the stress function and multidimensional scaling, in general, have been considered from the geometric point of view. The so-called Geometric MDS has been developed. The new interpretation of the stress allows finding the proper step size, and the descent direction forwards the minimum of the stress function analytically if we consider and move a separate point of the projected space. The exceptional property of the new approach is that we do not need the analytical expression of the stress function. There is no need for numerical evaluation of its derivatives, too. Moreover, we do not need for the linear search that is used for local descent in optimization. Theoretical analysis disclosed that the step direction, determined by Geometric MDS, coincides with the steepest descent direction. The analytically found step size is such that it guarantees the decrease of the stress in this direction. Two realizations of Geometric MDS are proposed and examined. The comparison with SMACOF realization of MDS looks very promising.

Suggested Citation

  • Dzemyda, Gintautas & Sabaliauskas, Martynas, 2021. "Geometric multidimensional scaling: A new approach for data dimensionality reduction," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    • Handle: RePEc:eee:apmaco:v:409:y:2021:i:c:s0096300320305178
    DOI: 10.1016/j.amc.2020.125561
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    References listed on IDEAS

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    1. Gintautas Dzemyda & Olga Kurasova & Julius Žilinskas, 2013. "Multidimensional Data Visualization," Springer Optimization and Its Applications, Springer, edition 127, number 978-1-4419-0236-8, September.
    2. J. Kruskal, 1964. "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Psychometrika, Springer;The Psychometric Society, vol. 29(1), pages 1-27, March.
    3. Gintautas Dzemyda & Olga Kurasova & Julius Žilinskas, 2013. "Applications of Visualization," Springer Optimization and Its Applications, in: Multidimensional Data Visualization, edition 127, chapter 0, pages 179-226, Springer.
    4. de Leeuw, Jan & Mair, Patrick, 2009. "Multidimensional Scaling Using Majorization: SMACOF in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 31(i03).
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