We present a novel method of decomposition of an n × m binary matrix I into a Boolean product A • B of an n × k binary matrix A and a k × m binary matrix B with k as small as possible. Attempts to solve this problem are known from Boolean factor analysis where I is interpreted as an object-attribute matrix, A and B are interpreted as object-factor and factor-attribute matrices, and the aim is to find a decomposition with a small number k of factors. The method presented here is based on a theorem proved in this paper. It says that optimal decompositions, i.e. those with the least number of factors possible, are those where factors are formal concepts in the sense of formal concept analysis. Finding an optimal decomposition is an NP-hard problem. However, we present an approximation algorithm for finding optimal decompositions which is based on the insight provided by the theorem. The algorithm avoids the need to compute all formal concepts and significantly outperforms a greedy approximation algorithm for a set covering problem to which the problem of matrix decomposition is easily shown to be reducible. We present results of several experiments with various data sets including those from CIA World Factbook and UCI Machine Learning Repository. In addition, we present further geometric insight including description of transformations between the space of attributes and the space of factors.

Abstract. The paper is an overview of recent developments concerning attribute implications in a fuzzy setting. Attribute implications are formulas of the form A ⇒ B, where A and B are collections of attributes, which describe dependencies between attributes. Attribute implications are studied in several areas of computer science and mathematics. We focus on two of them, namely, formal concept analysis and databases.

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The paper studies if and to what extent it is possible to reduce notions related to attribute implications from data with fuzzy attributes to corresponding notions related to attribute implications from data with crisp attributes. We provide a reduction (transformation) theorem. Still, working directly in fuzzy setting is beneficial. Namely, we prove that we can compute a complete and non-redundant basis of fuzzy attribute implications of data with fuzzy attributes such that the basis is at most as large as (and can be smaller than) any basis of implications of the corresponding data table with crisp attributes.