Geometric multidimensional scaling: A new approach for data dimensionality reduction

Gintautas Dzemyda and Martynas Sabaliauskas

Applied Mathematics and Computation, 2021, vol. 409, issue C

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.

Keywords: Multidimensional scaling; Dimensionality reduction; Geometric approach; Geometric MDS; SMACOF; Analytically determined minimization (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:409:y:2021:i:c:s0096300320305178

DOI: 10.1016/j.amc.2020.125561

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