Finding interesting patterns is a classical problem in data mining. Boolean matrix decomposition is nowadays a standard tool that can find a set of patterns-also called factors-in Boolean data that explain the data well. We describe and experimentally evaluate a probabilistic algorithm for Boolean matrix decomposition problem. The algorithm is derived from GreCon algorithm which uses formal concepts-maximal rectangles or tiles-as factors in order to find a decomposition. We change the core of GreCon by substituting a sampling procedure for a deterministic computation of suitable formal concepts. This allows us to alleviate the greedy nature of GreCon, creates a possibility to bypass some of the its pitfalls and to preserve its features, e.g. an ability to explain the entire data.

The paper presents results on representation of the basic structures related to ternary fuzzy relations by the structures related to ordinary ternary relations, such as Galois connections, closure operators, and trilattices (structures of maximal Cartesian subrelations). These structures appear as the fundamental structures in relational data analysis such as formal concept analysis or association rules. We prove several representation theorems that allow us to automatically transfer some of the known results from the ordinary case to fuzzy case. The transfer is demonstrated by examples.