Boolean matrix factorization (BMF) is a popular and powerful technique for\ninferring knowledge from data. The mining result is the Boolean product of two\nmatrices, approximating the input dataset. The Boolean product is a disjunction\nof rank-1 binary matrices, each describing a feature-relation, called pattern,\nfor a group of samples. Yet, there are no guarantees that any of the returned\npatterns do not actually arise from noise, i.e., are false discoveries. In this\npaper, we propose and discuss the usage of the false discovery rate in the\nunsupervised BMF setting. We prove two bounds on the probability that a found\npattern is constituted of random Bernoulli-distributed noise. Each bound\nexploits a specific property of the factorization which minimizes the\napproximation error---yielding new insights on the minimizers of Boolean matrix\nfactorization. This leads to improved BMF algorithms by replacing heuristic\nrank selection techniques with a theoretically well-based approach. Our\nempirical demonstration shows that both bounds deliver excellent results in\nvarious practical settings.\n