A mathematical model of similarity and clustering

This paper introduces an abstract model of data similarity and clustering. A similarity on a space Ω is formulated explicitly by a reflexive and symmetric binary relation, called a tolerance relation, for which we introduce three types of coverings of Ω. Given a covering U, a clustering is defined to be minimal sub-covering. To search for an optimal clustering is to minimize the number of clusters, which is intractable in general. This paper proposes a heuristic method to search for sub-optimal clusterings for a given tolerance relation.