Computational Statistics

, Volume 17, Issue 2, pp 147–163 | Cite as

Extensions of Classical Multidimensional Scaling via Variable Reduction

  • Michael W. Trosset


Classical multidimensional scaling constructs a configuration of points that minimizes a certain measure of discrepancy between the configuration’s interpoint distance matrix and a fixed dissimilarity matrix. Recent extensions have replaced the fixed dissimilarity matrix with a closed and convex set of dissimilarity matrices. These formulations replace fixed dissimilarities with optimization variables (disparities) that are permitted to vary subject to application-specific constraints. For example, simple bound constraints are suitable for distance matrix completion problems (Trosset, 2000) and for inferring molecular conformation from information about interatomic distances (Trosset, 1998b); whereas order constraints are suitable for nonmetric multidimensional scaling (Trosset, 1998a). This paper describes the computational theory that provides a common foundation for these formulations.


  1. [1]
    Byrd, R. H., Lu, P., Nocedal, J., and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16:1190–1208.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Critchley, F. (1988). On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91–107.MathSciNetCrossRefGoogle Scholar
  3. [3]
    de Leeuw, J. and Heiser, W. (1982). Theory of multidimensional scaling. In Krishnaiah, P. R. and Kanal, I. N., editors, Handbook of Statistics, volume 2, chapter 13, pages 285–316. North-Holland Publishing Company, Amsterdam.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Glunt, W., Hayden, T. L., and Raydan, M. (1993). Molecular conformations from distance matrices. Journal of Computational Chemistry, 14:114–120.CrossRefGoogle Scholar
  5. [5]
    Golub, G. H. and Pereyra, V. (1973). The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM Journal on Numerical Analysis, 10:413–432.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Gower, J. C. (1966). Some distance properties of latent root and vector methods in multivariate analysis. Biometrika, 53:315–328.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Krippahl, L. and Barahona, P. (1999). Applying constraint programming to protein structure determination. In Principles and Practice of Constraint Programming, pages 289–302. Springer Verlag, New York.Google Scholar
  8. [8]
    Krippahl, L., Trosset, M., and Barahona, P. (2001). Combining constraint programming and multidimensional scaling to solve distance geometry problems. In Proceedings of the Third International Workshop on Integration of AI and OR Techniques. To appear.Google Scholar
  9. [9]
    Kruskal, J. B. (1964a). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29:1–27.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Kruskal, J. B. (1964b). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29:28–42.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Lehoucq, R. B. (1995). Analysis and implementation of an implicitly restarted iteration. Technical Report 95-13, Department of Computational & Applied Mathematics, Rice University, 6100 Main Street, Houston, TX 77005–1892. Author’s Ph.D. thesis.Google Scholar
  12. [12]
    Lehoucq, R. B. and Sorensen, D. C. (1996). Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM Journal on Matrix Analysis and Applications, 17:789–821.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Lehoucq, R. B., Sorensen, D. C., and Yang, C. (1997). ARPACK Users’ Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Department of Computational & Applied Mathematics, Rice University, 6100 Main Street, Houston, TX 77005–1892. Available at Scholar
  14. [14]
    Lewis, A. S. (1996). Derivatives of spectral functions. Mathematics of Operations Research, 21:576–588.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Mardia, K. V. (1978). Some properties of classical multi-dimensional scaling. Communications in Statistics — Theory and Methods, A7:1233–1241.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Mardia, K. V., Kent, J. T., and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, Orlando.zbMATHGoogle Scholar
  17. [17]
    McCormick, G. P. and Tapia, R. A. (1972). The gradient projection method under mild differentiability conditions. SIAM Journal on Control, 10:93–98.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Parks, T. A. (1985). Reducible nonlinear programming problems. Technical Report 85-8, Department of Mathematical Sciences, Rice University, Houston, TX. Author’s Ph.D. dissertation.Google Scholar
  19. [19]
    Saito, T. (1978). The problem of the additive constant and eigenvalues in metric multidimensional scaling. Psychometrika, 43:193–201.CrossRefGoogle Scholar
  20. [20]
    Schoenberg, I. J. (1935). Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espaces distanciés vectoriellement applicable sur l’espace de Hilbert”. Annals of Mathematics, 38:724–732.CrossRefGoogle Scholar
  21. [21]
    Sibson, R. (1979). Studies in the robustness of multidimensional scaling: Perturbational analysis of classical scaling. Journal of the Royal Statistical Society, Series B, 41:217–229.MathSciNetzbMATHGoogle Scholar
  22. [22]
    Sorensen, D. C. (1992). Implicit application of polynomial filters in a k-step Arnoldi method. SIAM Journal on Matrix Analysis and Applications, 13:357–385.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Torgerson, W. S. (1952). Multidimensional scaling: I. Theory and method. Psychometrika, 17:401–419.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Trefethen, L. N. and Bau, D. (1997). Numerical Linear Algebra. SIAM, Philadelphia, PA.CrossRefGoogle Scholar
  25. [25]
    Trosset, M. W. (1997). Numerical algorithms for multidimensional scaling. In Klar, R. and Opitz, P., editors, Classification and Knowledge Organization, pages 80–92, Berlin. Springer. Proceedings of the 20th annual conference of the Gesellschaft für Klassifikation e.V., held March 6–8, 1996, in Freiburg, Germany.CrossRefGoogle Scholar
  26. [26]
    Trosset, M. W. (1998a). Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29: 148–152.Google Scholar
  27. [27]
    Trosset, M. W. (1998b). A new formulation of the nonmetric STRAIN problem in multidimensional scaling. Journal of Classification, 15:15–35.CrossRefGoogle Scholar
  28. [28]
    Trosset, M. W. (2000). Distance matrix completion by numerical optimization. Computational Optimization and Applications, 17: 11–22.MathSciNetCrossRefGoogle Scholar
  29. [29]
    Trosset, M. W., Baggerly, K. A., and Pearl, K. (1996). Another look at the additive constant problem in multidimensional scaling. Technical Report 96-7, Department of Statistics—MS 138, Rice University, Houston, TX 77005–1892.Google Scholar
  30. [30]
    Young, G. and Householder, A. S. (1938). Discussion of a set of points in terms of their mutual distances. Psychometrika, 3:19–22.CrossRefGoogle Scholar
  31. [31]
    Zhu, C., Byrd, R. H., Lu, P., and Nocedal, J. (1994). L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization. Technical Report NAM-11, Department of Electrical Engineering & Computer Science, Northwestern University, Evanston, IL 60208. Revised October 8, 1996.zbMATHGoogle Scholar

Copyright information

© Physica-Verlag 2002

Authors and Affiliations

  • Michael W. Trosset
    • 1
  1. 1.Department of MathematicsCollege of William & MaryWilliamsburgUSA

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