On the Existence of Nonglobal Minimizers of the Stress Criterion for Metric Multidimensional Scaling

Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing conngura-tions of points from dissimilarity information about interpoint distances. A popular measure of the t of the constructed distances to the observed dissimilar-ities is the stress criterion, which must be minimized by numerical optimization. Empirical evidence concerning the existence of nonglobal minimizers of the stress criterion is somewhat contradictory. We report a connguration that we have demonstrated to be a nonglobal minimizer. 1 Preliminaries Multidimensional scaling (MDS) is a collection of techniques for tting distance models to distance data. The data are called dissimilarities. Formally, a symmetric n n matrix = (ij) is a dissimilar-ity matrix if ij 0 and ii = 0. In this report, we restrict attention to the case of a single dissimilarity matrix (two-way MDS). The goal of MDS is to construct a connguration of points in a target metric (usually Euclidean) space in such a way that the interpoint distances approximate the dissimilarities. Formally, an n n matrix D = (d ij) is a p-dimensional Euclidean distance matrix if there exists a connguration of points x 1 ; : : : ; x n 2 < p such that d ij = kx i ? x j k 2. We store the coordinates of x 0 i in row i of the n p connguration matrix X. Given a xed dissimilarity matrix , a conngu-ration is determined by minimizing a badness-of-t criterion. A popular criterion is the (raw) stress criterion: