Boolean Matrix Factorization and Noisy Completion via Message Passing

Boolean factor analysis is the task of decomposing a binary matrix to the Boolean product of two binary factors. This unsupervised data-analysis approach is desirable due to its interpretability, but hard to perform due its NP-hardness. A closely related problem is low-rank Boolean matrix completion from noisy observations. We treat these problems as maximum a posteriori inference problems, and present message passing solutions that scale linearly with the number of observations and factors. Our empirical study demonstrates that message passing is able to recover low-rank Boolean matrices, in the boundaries of theoretically possible recovery and outperform existing techniques in real-world applications, such collaborative filtering with large-scale Boolean data. A body of problems in machine learning, communication theory and combinatorial optimization involve the product form Z = X Y where operation corresponds to a type of matrix multiplication and Z = {Zm,n} , X = {Xm,k} , Y = {Yk,n} . Here, often one or two components (out of three) are (partially) known and the task is to recover the unknown component(s). A subset of these problems, which are most closely related to Boolean matrix factorization and matrix completion, can be expressed over the Boolean domain – i.e., Zm,n, Xm,k, Yk,n ∈ {false, true} ∼= {0, 1}. The two most common Boolean matrix products used in such applications are Z = X • Y ⇒ Zm,n = K ∨