Searching for structure in data with fuzzy clusters of variable dimensionality of feature subspaces

Structural relationships in data are revealed by methods of clustering and fuzzy clustering. In essence, clustering leads to the reduction of data. Dimensionality reduction comes as a complementary process in which we eliminate some features (attributes). This study introduces a concept of structure reduction which is guided by a criterion of structure retention. In particular, it is shown that each cluster could be described by a different subset of features so that finally the reduction leads to the local feature subspaces. By analyzing the resulting subspaces, one could gain a better insight into a nature of the contributing features and in this way identify subsets of the most meaningful ones. The reduction problem is formulated and formalized as a certain combinatorial optimization task whose solution is provided by means of particle swarm optimization.