Factorizations of Binary Matrices -- Rank Relations and the Uniqueness of Boolean Decompositions

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank factorization is often seen as an advantageous method to interpret and analyze data. In this work, we examine the minimal rank factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}_2$). We examine all the relationships between the different ranks, and discuss when the factorizations are unique. In particular, we characterize when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$ (for a fixed $W$), and when both $W$ and $H$ are unique, given a rank constraint.