The tunneling method for global optimization in multidimensional scaling

This paper focuses on the problem of local minima of the STRESS function. It turns out that unidimensional scaling is particularly prone to local minima, whereas full dimensional scaling with Euclidean distances has a local minimum that is global. For intermediate dimensionality with Euclidean distances it depends on the dissimilarities how severe the local minimum problem is. For city-block distances in any dimensionality many different local minima are found. A simulation experiment is presented that indicates under what conditions local minima can be expected. We introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances. The tunneling method alternates a local search step, in which a local minimum is sought, with a tunneling step in which a different configuration is sought with the same STRESS as the previous local minimum. In this manner successively better local minima are obtained, and experimentation so far shows that the last one is often a global minimum.

[1]  L. Tucker.  A METHOD FOR SYNTHESIS OF FACTOR ANALYSIS STUDIES , 1951 .

[2]  W. S. Robinson.  A Method for Chronologically Ordering Archaeological Deposits , 1951, American Antiquity.

[3]  Joseph L. Zinnes,et al.  Theory and Methods of Scaling. , 1958 .

[4]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[5]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[6]  J. Kruskal.  Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[7]  Werner Dinkelbach.  On Nonlinear Fractional Programming , 1967 .

[8]  P. Padmos,et al.  QUANTITATIVE INTERPRETATION OF STRESS IN KRUSKAL'S MULTIDIMENSIONAL SCALING TECHNIQUE , 1971 .

[9]  Forrest W. Young,et al.  The Perceived Structure of American Ethnic Groups: The Use of Multidimensional Scaling in Stereotype Research , 1974 .

[10]  B. M. Shchedrin,et al.  A method of finding the global minimum of a function of one variable , 1975 .

[11]  D. Defays.  A short note on a method of seriation , 1978 .

[12]  Susana Gómez,et al.  The tunnelling method for solving the constrained global optimization problem with several non-connected feasible regions , 1982 .

[13]  Peter M. Bentler,et al.  Restricted multidimensional scaling models for asymmetric proximities , 1982 .

[14]  J. Meulman.  A Distance Approach to Nonlinear Multivariate Analysis , 1986 .

[15]  J. Leeuw.  Convergence of the majorization method for multidimensional scaling , 1988 .

[16]  G. De Soete,et al.  On the use of simulated annealing for combinatorial data analysis , 1988 .

[17]  Paul E. Green,et al.  Multidimensional Scaling: Concepts and Applications , 1989 .

[18]  Approximating a symmetric matrix , 1990 .

[19]  Phipps Arabie,et al.  Was euclid an unnecessarily sophisticated psychologist? , 1991 .

[20]  Rudolf Mathar,et al.  Algorithms in convex analysis applied to Multidimensional Scaling , 1991 .

[21]  W. Heiser.  A generalized majorization method for least souares multidimensional scaling of pseudodistances that may be negative , 1991 .

[22]  L. Hubert,et al.  Multidimensional scaling in the city-block metric: A combinatorial approach , 1992 .

[23]  J. Meulman.  The integration of multidimensional scaling and multivariate analysis with optimal transformations , 1992 .

[24]  Patrick J. F. Groenen,et al.  The majorization approach to multidimensional scaling : some problems and extensions , 1993 .

[25]  P. Groenen,et al.  The majorization approach to multidimensional scaling for Minkowski distances , 1995 .

[26]  Rudolf Mathar,et al.  Least Squares Multidimensional Scaling with Transformed Distances , 1996 .

[27]  P. Groenen,et al.  Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima , 1997 .