On the Computational Efficiency of Geometric Multidimensional Scaling

Abstract

Real-life applications often deal with multidimensional data. In the general case, multidimensional data means a table of numbers whose rows correspond to different objects and columns correspond to features characterizing the objects. Usually, the number of objects is large, and the dimensionality (number of features) is greater than it is possible to represent the objects as points in 2D. The goal is to reduce the dimensionality of data to such one that objects, characterized by a large number of features or by proximities between pairs of the objects, be represented as points in lower-dimensional space or even on a plane. Multidimensional scaling (MDS) is an often-used method to reduce the dimensionality of multidimensional data nonlinearly and to present the data visually. MDS minimizes some stress function. We have proposed in [8] and [9] to consider the stress function and multidimensional scaling, in general, from the geometric point of view, and the so-called Geometric MDS has been developed. Geometric MDS allows finding the proper direction and step size forwards the minimum of the stress function analytically. In this paper, we disclose several new properties of Geometric multidimensional scaling and compare the simplest realization (GMDS1) of Geometric MDS experimentally with the well-known SMACOF version of MDS.