Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing

Least-squares multidimensional scaling is known to have a serious problem of local minima, especially if one dimension is chosen, or if city-block distances are involved. One particular strategy, the smoothing strategy proposed by Pliner (1986, 1996), turns out to be quite successful in these cases. Here, we propose a slightly different approach, called distance smoothing. We extend distance smoothing for any Minkowski distance. In addition, we extend the majorization approach to multidimensional scaling to have a one-step update for Minkowski parameters larger than 2 and use the results for distance smoothing. We present simple ideas for finding quadratic majorizing functions. The performance of distance smoothing is investigated in several examples, including two simulation studies. Copyright Springer-Verlag New York Inc. 1999