Summary
In a unifying approach, this note deals with three different methods to find the best embedding of n objects in an ℓ p -space, p ≥ 1, if only pairwise dissimilarities are given and fitting is measured by weighted least squares. The procedures are based on (1) a nested algorithm with an inner linear constrained problem, (2) a generalized eigenvector procedure resembling inverse iteration, and (3) majorization. All resulting algorithms are quite similar, though the optimization problem is approached from different viewpoints. This paper explains why, by interpreting (2) and (3) as a relaxed version of a first order subgradient method.
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References
DE LEEUW, J. (1977): Applications of Convex Analysis to Multidimensional Scaling. In: J.R. Barra, F. Brodeau, G. Romier and B. van Cutsem (eds.): Recent Developments in Statistics. North-Holland, Amsterdam, 133–145.
DE LEEUW, J. (1988): Convergence of the Majorization Method for Multidimensional Scaling. Journal of Classification, 5, 163–180.
DE LEEUW, J. (1992): Fitting Distances by Least Squares. Unpublished report, Los Angeles, UCLA.
DE LEEUW, J., and HEISER, W.J. (1980): Multidimensional Scaling with Restrictions on the Configuration. In: P.R. Krishnaiah (ed.): Multivariate analysis V. North-Holland, Amsterdam, 501–522.
GROENEN, P.J.F., MATHAR, R., and HEISER, W.J. (1993): The Majorization Approach to Multidimensional Scaling for Minkowski Distances. Submitted for publication, University of Leiden and RWTH Aachen.
HEISER, W.J. (1989): The City-block Model for Three-way Multidimensional Scaling. In: R. Coppi and S. Bolasco (eds.): Multiway Data Analysis. North-Holland, Amsterdam, 395–404.
HORST, R., TUY, H. (1993): Global Optimization, Deterministic Approaches. Springer, Berlin.
HUBERT, L.J., and ARABIE, P. (1986): Unidimensional Scaling and Combinatorial Optimization. In: J. De Leeuw, W.J. Heiser, J. Meulman, and F. Critchley (eds.): Multidimensional Data Analysis. DSWO Press, Leiden, 181–196.
HUBERT, L.J., ARABIE, P., and HESSON-MCINNES, M. (1992): Multidimensional Scaling in the City-Block Metric: a Combinatorial Approach. Journal of Classification, 9, 211–236.
MATHAR, R. (1989): Algorithms in Multidimensional Scaling. In: O. Opitz (ed): Conceptual and Numerical Analysis of Data. Springer, Berlin, 159–177.
MATHAR, R., and GROENEN, P.J.F. (1991): Algorithms in Convex Analysis Applied to Multidimensional Scaling. In: E. Diday, and Y. Lechevallier (eds.): Symbolic-Numeric Data Analysis and Learning. Nova-Science, Commack, New York, 45–56.
MATHAR, R., and MEYER, R. (1992): Algorithms in Convex Analysis to Fit l p -Distance Matrices. Submitted for publication, RWTH Aachen.
PETERS, G., and WILKINSON, J.H. (1971): The calculation of specified eigenvectors by inverse iteration. In: J.H. Wilkinson and C. Reinsch (eds.): Handbook for Automatic Computation, Vol. II, Linear Algebra. Springer, Berlin, 418–439.
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Mathar, R. (1994). Multidimensional Scaling with ℓ p -Distances, a Unifying Approach. In: Bock, HH., Lenski, W., Richter, M.M. (eds) Information Systems and Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46808-7_29
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DOI: https://doi.org/10.1007/978-3-642-46808-7_29
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