Multidimensional scaling in the city-block metric: A combinatorial approach |
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Authors: | Lawrence Hubert Phipps Arabie Matthew Hesson-Mcinnis |
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Institution: | (1) Department of Psychology, University of Illinois, 603 E. Daniel St., 61820 Champaign, IL, USA;(2) Graduate School of Management, Rutgers University, 92 New Street, 07102-1895 Newark, NJ, USA |
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Abstract: | We present an approach, independent of the common gradient-based necessary conditions for obtaining a (locally) optimal solution,
to multidimensional scaling using the city-block distance function, and implementable in either a metric or nonmetric context.
The difficulties encountered in relying on a gradient-based strategy are first reviewed: the general weakness in indicating
a good solution that is implied by the satisfaction of the necessary condition of a zero gradient, and the possibility of
actual nonconvergence of the associated optimization strategy. To avoid the dependence on gradients for guiding the optimization
technique, an alternative iterative procedure is proposed that incorporates (a) combinatorial optimization to construct good
object orders along the chosen number of dimensions and (b) nonnegative least-squares to re-estimate the coordinates for the
objects based on the object orders. The re-estimated coordinates are used to improve upon the given object orders, which may
in turn lead to better coordinates, and so on until convergence of the entire process occurs to a (locally) optimal solution.
The approach is illustrated through several data sets on the perception of similarity of rectangles and compared to the results
obtained with a gradient-based method. |
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Keywords: | City-block metric Multidimensional scaling Combinatorial optimization |
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