An Improved Exact Algorithm for Least-Squares Unidimensional Scaling

Gintaras Palubeckis

doi:10.1155/2013/890589

Gintaras Palubeckis

Open Access

https://doi.org/10.1155/2013/890589

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Abstract

Given n objects and an symmetric dissimilarity matrix D with zero main diagonal and nonnegative off-diagonal entries, the least-squares unidimensional scaling problem asks to find an arrangement of objects along a straight line such that the pairwise distances between them reflect dissimilarities represented by the matrix D. In this paper, we propose an improved branch-and-bound algorithm for solving this problem. The main ingredients of the algorithm include an innovative upper bounding technique relying on the linear assignment model and a dominance test which allows considerably reducing the redundancy in the enumeration process. An initial lower bound for the algorithm is provided by an iterated tabu search heuristic. To enhance the performance of this heuristic we develop an efficient method for exploring the pairwise interchange neighborhood of a solution in the search space. The basic principle and formulas of the method are also used in the implementation of the dominance test. We report computational results for both randomly generated and real-life based problem instances. In particular, we were able to solve to guaranteed optimality the problem defined by a Morse code dissimilarity matrix.

Highlights

Least-squares unidimensional scaling is an important optimization problem in the field of combinatorial data analysis
We considered two UDSP instances constructed using Morse code confusion data—one defined by the 26 × 26 submatrix of D with rows and columns labeled by the letters and another defined by the full matrix D
In this paper we have presented a branch-and-bound algorithm for the least-squares unidimensional scaling problem

Summary

Introduction

Least-squares (or L2) unidimensional scaling is an important optimization problem in the field of combinatorial data analysis. Suppose that there are n objects and, in addition, there are pairwise dissimilarity data available for them. These data form an n × n symmetric dissimilarity matrix, D = (dij), with zero main diagonal and nonnegative off-diagonal entries. The problem is to arrange the objects along a straight line such that the pairwise distances between them reflect dissimilarities given by the matrix D. The problem can be expressed as min

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