Characteristic matrix of covering and its application to Boolean matrix decomposition

Covering-based rough sets provide an efficient means of dealing with covering data, which occur widely in practical applications. Boolean matrix decomposition has frequently been applied to data mining and machine learning. In this paper, three types of existing covering approximation operators are represented by Boolean matrices, and then used in Boolean matrix decomposition. First, we define two characteristic matrices of a covering. Through these Boolean characteristic matrices, three types of existing covering approximation operator are concisely and equivalently represented. Second, these operator representations are applied to Boolean matrix decomposition, which has a close relationship with nonnegative matrix factorization, a popular and efficient technique for machine learning. We provide a sufficient and necessary condition for a square Boolean matrix to decompose into the Boolean product of another matrix and its transpose. We then develop an algorithm for this Boolean matrix decomposition. Finally, these three covering approximation operators are axiomatized using Boolean matrices. This work presents an interesting viewpoint from which to investigate covering-based rough set theory and its applications.