Abstract
It is shown that Kruskal's multidimensional scaling loss function is differentiable at a local minimum. Or, to put it differently, that in multidimensional scaling solutions using Kruskal's stress distinct points cannot coincide.
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Reference notes
De Leeuw, J. (1974). Smoothness properties of nonmetric loss functions. Unpublished paper, Bell Telephone Labs.
De Leeuw, J. (1981). Linear convergence of multidimensional scaling algorithms. Unpublished paper, Department of Data Theory, Leiden University.
References
Carroll, J. D. INDSCAL. (1981). In S. S. Schiffman, M. L. Reynolds, & F. W. Young (Eds.),Introduction to multidimensional scaling. New York: Academic Press.
Defays, D. (1978). A short note on a method of seriation.British Journal of Mathematical and Statistical Psychology, 31, 49–53.
De Leeuw, J. (1977). Applications of convex analysis to multidimensional scaling. In J. R. Barra, F. Brodeau, G. Romier, & B. Van Cutsem (Eds.),Recent developments in statistics. Amsterdam: North Holland Publishing Co.
De Leeuw, J. & Heiser, W. (1980). Multidimensional scaling with restrictions on the configuration. In P. R. Krishnaiah (Ed.),Multivariate analysis V. New York: Academic Press.
Kruskal, J. B. (1964a). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 29, 1–27.
Kruskal, J. B. (1964b). Nonmetric multidimensional scaling: a numerical method.Psychometrika, 29, 115–129.
Kruskal, J. B. (1971). Monotone regression: continuity and differentiability properties.Psychometrika, 36, 57–62.
Kruskal, J. B. (1977). Multidimensional scaling and other methods for discovering structure. In K. Enslein, A. Ralston, & H. S. Wilf (Eds.),Mathematical methods for digital computers III. New York: Wiley.
Ramsay, J. O. (1978).MULTISCALE: four programs for multidimensional scaling by the method of maximum likelihood. Chicago: National Educational Resources Inc.
Takane, Y., Young, F. W., & De Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: an alternating least squares method with optimal scaling features.Psychometrika, 42, 7–67.
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De Leeuw, J. Differentiability of Kruskal's stress at a local minimum. Psychometrika 49, 111–113 (1984). https://doi.org/10.1007/BF02294209
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DOI: https://doi.org/10.1007/BF02294209