A Comparison of Two Methods for Global Optimization in Multidimensional Scaling

Abstract

One of the problems of metric multidimensional scaling defined by Kruskal’s stress function, is that only convergence to a local minimum is guaranteed. Depending on the dissimilarity data and the chosen dimensionality, many local minima may exist. Here we aim at finding a global minimum of the stress function. Several general methods for global optimization exist, but we shall focus on two. The first one is the tunneling method, proposed by Montalvo (1979) and Gomez and Levy (1982), which aims at finding a decreasing series of local minima. Groenen and Heiser (1991) adapted and applied tunneling to multidimensional scaling. The second method is the multi-level-single-linkage clustering developed by Timmer (1984), which is a stochastic method based on properties of multistart using multiple random starts, and single linkage clustering. An implementation of this method to multidimensional scaling is discussed. Though the two methods aim at finding a global minimum, neither is guaranteed to find it in practice. A comparison is presented of the performance of the two methods in finding the global minima for three data sets. It turns out that both methods arrive at the same candidate global minimum for the three examples.