Multidimensional Scaling by Majorization: A Review

Abstract

A major breakthrough in the visualization of dissimilarities between pairs of objects was the formulation of the least-squares multidimensional scaling (MDS) model as defined by the Stress function. This function is quite flexible in that it allows possibly nonlinear transformations of the dissimilarities to be represented by distances between points in a low dimensional space. To obtain the visualization, the Stress function should be minimized over the coordinates of the points and the over the transformation. In a series of papers, Jan de Leeuw has made a significant contribution to majorization methods for the minimization of Stress in least-squares MDS. In this paper, we present a review of the majorization algorithm for MDS as implemented in the smacof package and related approaches. We present several illustrative examples and special cases.