Boolean decomposition of binary matrices using a post-nonlinear mixture approach

Abstract

We introduce a novel binary matrix factorization (BMF) approach based on a post-nonlinear mixture model. Unlike the existing BMF methods, which are based on the classical matrix product, the proposed mixture model is equivalent to the Boolean matrix factorization model when the entries of the factor matrices are exactly binary. Consequently, our approach yields interpretable results in the case of overlapping sources and more accurate low-rank binary matrix approximations compared to the state-of-the-art. We propose a simple yet efficient algorithm for solving the proposed BMF problem based on multiplicative update rules. In addition, we provide for the first time in the binary data literature, a necessary and sufficient condition for the uniqueness of the Boolean matrix factorization, as well as several other uniqueness results. The interest of this new approach is illustrated in numerical simulation and on real datasets.