Special Solutions, Degeneracies, and Local Minima

  • Ingwer Borg
  • Patrick Groenen
Part of the Springer Series in Statistics book series (SSS)


In this chapter, we explain several technical peculiarities of MDS. First, we consider MDS of a constant dissimilarity matrix (all dissimilarities are equal) and indicate what configurations are found. Then we discuss degenerate solutions in ordinal MDS, where Stress approaches zero even though the MDS distances do not represent the data properly. Another problem in MDS is the existence of multiple local minima solutions. This problem is especially severe for unidimensional scaling. For this case, several strategies are discussed that are less prone to local minima For full-dimensional scaling, in contrast, it is shown that the majorization algorithm always finds a globally optimal solution. For other dimensionalities, several methods for finding a global minimum exist, e.g., the tunneling method.


Local Minimum Global Minimum Tunneling Method Degenerate Solution Local Minimum Problem 
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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Ingwer Borg
    • 1
  • Patrick Groenen
    • 2
  1. 1.Zentrum für Umfragen, Methoden und AnalysenMannheimGermany
  2. 2.Department of Data TheoryLeiden UniversityLeidenThe Netherlands

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