Optimal Factorization of Three-Way Binary Data Using Triadic Concepts

We present a new approach to factor analysis of three-way binary data, i.e. data described by a 3-dimensional binary matrix I, describing a relationship between objects, attributes, and conditions. The problem consists in finding a decomposition of I into three binary matrices, an object-factor matrix A, an attribute-factor matrix B, and a condition-factor matrix C, with the number of factors as small as possible. The scenario is similar to that of decomposition-based methods of analysis of three-way data but the difference consists in the composition operator and the constraint on A, B, and C to be binary. We show that triadic concepts of I, developed within formal concept analysis, provide us with optimal decompositions. We present an example demonstrating the usefulness of the decompositions. Since finding optimal decompositions is NP-hard, we propose a greedy algorithm for computing suboptimal decompositions and evaluate its performance.