Abstract
Multidimensional Scaling (MDS) requires the multimodal Stress function optimization to estimate the model parameters, i.e. the coordinates of points in a lower-dimensional space. Therefore, finding the global optimum of the Stress function is very important for applications of MDS. The main idea of this paper is replacing the difficult multimodal problem by a simpler unimodal constrained optimization problem. A coplanarity measure of points is used as a constraint while the Stress function is minimized in the original high-dimensional space. Two coplanarity measures are proposed. A simple example presented illustrates and visualizes the optimization procedure. Experimental evaluation results with various data point sets demonstrate the potential ability to simplify MDS algorithms avoiding multidimodality.
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References
Borg, L., Groenen, P.: Modern Multidimensional Scaling: Theory and Applications. Springer, Heidelberg (1997)
Cox, T., Cox, M.: Multidimensional Scaling. Chapman and Hall, Boca Raton (2001)
Kruskal, J.: Nonmetric Multidimensional Scaling: A Numerical Method. Psychometrica 29, 115–129 (1964)
Trosset, M., Mathar, R.: On existence of nonglobal minimizers of the STRESS Criterion for Metric Multidimensional Scaling. In: Proceedings of the Statistical Computing Section, American Statistical Association, Alexandria, VA, pp. 158–162 (1997)
Zilinskas, A., Podlipskyte, A.: On multimodality of the SSTRESS criterion for metric multidimensional scaling. Informatica 14(1), 121–130 (2003)
Sommerville, D.M.Y.: An Introduction to the Geometry of n Dimensions. Dover, New York (1958)
Abbott, P.(ed.): In and Out: Coplanarity. Mathematica J 9, 300–302 (2004)
Weisstein, E.W.: Coplanar. MathWorld - A Wolfram Web Resource, http://mathworld.wolfram.com/Coplanar.html
Weisstein, E.W.: Point-Plane Distance. MathWorld - A Wolfram Web Resource, http://mathworld.wolfram.com/Point-PlaneDistance.html
Bertsekas, D.P.: Nonlinear programming. Athena Scientific (1999)
Torgerson, W.S.: Theory and methods of scaling. Wiley, New York (1958)
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Saltenis, V. (2006). Constrained Optimization of the Stress Function for Multidimensional Scaling. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science – ICCS 2006. ICCS 2006. Lecture Notes in Computer Science, vol 3991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758501_94
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DOI: https://doi.org/10.1007/11758501_94
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