An efficient alternating least-squares algorithm to perform multidimensional unfolding

We consider the problem of least-squares fitting of squared distances in unfolding. An alternating procedure is proposed which fixes the row or column configuration in turn and finds the global optimum of the objective criterion with respect to the free parameters, iterating in this fashion until convergence is reached. A considerable simplification in the algorithm results, namely that this conditional global optimum is identified by performing a single unidimensional search for each point, irrespective of the dimensionality of the unfolding solution.This work originally formed part of a doctoral thesis (Greenacre, 1978) presented at the University of Paris VI. The authors acknowledge the helpful comments of John Gower during the first author's sabbatical at Rothamsted Experimental Station. The authors are also indebted to Alexander Shapiro, who came up with the proof of the key result which the authors had long suspected, but had not proved, namely that the smallest root of function (13) provides the global minimum of function (7). The constructive comments of the referees of this paper are acknowledged with thanks. This research was supported in part by the South African Council for Scientific and Industrial Research.