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

AbstractGiven 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 a continuum. Approximating dissimilarities by the absolute value of the difference between coordinates on a line constitutes a serious computational problem. This paper presents an algorithm that implements Simulated Annealing in a new way, via a strategy based on a weighted alternating process that uses permutations and point-wise translations to locate the optimal configuration. Explicit implementation details are given for least squares loss functions and for least absolute deviations. The weighted, alternating process is shown to outperform earlier implementations of Simulated Annealing and other optimization strategies for Unidimensional Scaling in run time efficiency, in solution quality, or in both.

[1]  D. Defays.  A short note on a method of seriation , 1978 .

[2]  Emile H. L. Aarts,et al.  Simulated Annealing: Theory and Applications , 1987, Mathematics and Its Applications.

[3]  V. Pliner.  Metric unidimensional scaling and global optimization , 1996 .

[4]  D. Mitra,et al.  Convergence and finite-time behavior of simulated annealing , 1986, Advances in Applied Probability.

[5]  Stephen S. Wilson,et al.  Random iterative models , 1996 .

[6]  Michael J. Brusco.  A Simulated Annealing Heuristic for Unidimensional and Multidimensional (City-Block) Scaling of Symmetric Proximity Matrices , 2001, J. Classif..

[7]  Michael J. Brusco,et al.  Using Quadratic Assignment Methods to Generate Initial Permutations for Least-Squares Unidimensional Scaling of Symmetric Proximity Matrices , 2000, J. Classif..

[8]  Leszek Wojnar,et al.  Image Analysis , 1998 .

[9]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  Emile H. L. Aarts,et al.  Simulated annealing and Boltzmann machines - a stochastic approach to combinatorial optimization and neural computing , 1990, Wiley-Interscience series in discrete mathematics and optimization.

[11]  Phipps Arabie,et al.  Linear Unidimensional Scaling in the L2-Norm: Basic Optimization Methods Using MATLAB , 2002, J. Classif..

[12]  Bruce E. Hajek,et al.  Cooling Schedules for Optimal Annealing , 1988, Math. Oper. Res..

[13]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[14]  Gerhard Winkler,et al.  Image analysis, random fields and dynamic Monte Carlo methods: a mathematical introduction , 1995, Applications of mathematics.

[15]  J. Trejos,et al.  COMBINATORIAL OPTIMIZATION HEURISTICS IN PARTITIONING WITH NON EUCLIDEAN DISTANCES , 2002 .

[16]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[17]  S. Dreyfus,et al.  Thermodynamical Approach to the Traveling Salesman Problem : An Efficient Simulation Algorithm , 2004 .

[18]  E. Rothkopf.  A measure of stimulus similarity and errors in some paired-associate learning tasks. , 1957, Journal of experimental psychology.

[19]  D. Mitra,et al.  Convergence and finite-time behavior of simulated annealing , 1985, 1985 24th IEEE Conference on Decision and Control.

[20]  V. Cerný.  Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm , 1985 .

[21]  Michael J. Brusco.  Integer Programming Methods for Seriation and Unidemensional Scaling of Proximity Matrices: A Review and Some Extensions , 2002, J. Classif..

[22]  Donald Geman,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1984 .

[23]  R. Shepard,et al.  The internal representation of numbers , 1975, Cognitive Psychology.