Optimal Factorization of Three-Way Binary Data

We study the problem of 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 present a theoretical analysis of the decompositions and show that optimal factors for such decompositions are provided by triadic concepts developed in formal concept analysis. Moreover, we present an illustrative example, propose a greedy approximation algorithm for computing the decompositions and present its experimental evaluation.