REDUCING OF SEARCH SPACE OF MULTIDIMENSIONAL SCALING PROBLEMS WITH DATA EXPOSING SYMMETRIES

Multidimensional scaling addresses the problem how multidimensional data can be represented by points in a low dimensional space. The problem is reduced to global minimization of a stress function which measures a fit of dissimilarity by the distances between the respective points. Symmetries in data may exist. Performance of global optimization may be increased reducing search space so that only one of the symmetricsolutionsshouldbefound.Restrictionof search space is proposed and demonstrated on geometric data sets for multidimensional scaling. Multidimensional scaling (MDS) is a technique for exploratory analysis of multidimensional data widely usable in different applications [2, 4]. It is assumed that pairwise dissimilarities between n objects are given by the matrix A set of points in an embedding space is considered as an image of the set of objects. Normally, an m-dimensional embedding metric space is used, and points x i ∈ R should be found whose inter-point distances fit the given dissimilar-ities. Frequently the objects are defined by multidi-mensional points and the dissimilarities are defined as pairwise distances between points in the original multidimensional space. In such a case an image can be interpreted as a nonlinear projection of the high-dimensional space to the space of lower dimen-sionality. The problem of constructing the image of the set of considered objects is reduced to minimization of an accuracy of fit criterion, e.g. of the frequently used least squares STRESS function