Multidimensional Scaling in the City-Block Metric: L1 and L2-Norm Optimization Methods Using MATLAB

In a recent paper by Hubert, Arabie, and Meulman (2002), a comparison is made among several different optimization strategies for the linear unidimensional scaling (LUS) task in the L2-norm, with all implementations carried out within a MATLAB computational environment. The central LUS task involves arranging the n objects in a set S = {O1, O2, . . . , On} along a single dimension, defined by coordinates x1, x2, . . . , xn, based on an n × n symmetric proximity matrix P = {pij}, whose nonnegative entries are given a dissimilarity interpretation (pij = pji for 1 ≤ i, j ≤ n; pii = 0 for 1 ≤ i ≤ n). The L2 criterion ∑ iclose companion to this earlier piece, with extensions now given to multidimensionalscaling in the city-block metric for both the L1 and L2 norms. The computationalroutines to be discussed and illustrated are again freely available as MATLAB m-files.Although L1 norms are possible to use within both the unidimensional and multi-dimensional contexts, and we give MATLAB m-files to do so, our general conclusionafter experimentation is that they might be best avoided because of their gener-ally needed much greater computationally expensive implementation without anyparticularly clear advantage; there is also the disconcerting periodic warnings ofpossible ill-conditioning for the repetitive linear programming subtasks when usingthe MATLAB Optimization Toolbox m-file linprog.m. References• Hubert, L. J., Arabie, R., & Meulman, J. J. (2002). Linear unidimensionalscaling in the L2-norm: Basic optimization methods using MATLAB. Journalof Classification, 19, 303–328.