Geometric multidimensional scaling: A new approach for data dimensionality reduction

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. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

Multidimensional scaling (MDS) is a popular method for visualizing multidimensional data by minimizing stress functions. This paper considers the stress function and MDS from a geometric perspective, introducing Geometric MDS which analytically finds the descent direction and step size of the stress function. The approach avoids the need for analytical expression of the stress function, numerical evaluation of its derivatives, and linear search for local descent in optimization.