A Permutation-Translation Simulated Annealing Algorithm for L1 and L2 Unidimensional Scaling

Authors:
Alex Murillo;J. Fernando Vera;Willem J. Heiser
Affiliations:
Costa-Rica University, Costa Rica;Granada University, Spain;Leiden University, The Netherlands
Venue:
Journal of Classification
Year:
2005

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Abstract

Given a set of objects and a symmetric matrix of dissimilarities between them,Unidimensional Scaling is the problem of finding a representation by locating points on acontinuum. Approximating dissimilarities by the absolute value of the difference betweencoordinates on a line constitutes a serious computational problem. This paper presents analgorithm that implements Simulated Annealing in a new way, via a strategy based on aweighted alternating process that uses permutations and point-wise translations to locatethe optimal configuration. Explicit implementation details are given for least squares lossfunctions and for least absolute deviations. The weighted, alternating process is shownto outperform earlier implementations of Simulated Annealing and other optimizationstrategies for Unidimensional Scaling in run time efficiency, in solution quality, or inboth.