  Binary matrix factorisation is an essential tool for identifying discrete
patterns in binary data. In this paper we consider the rank-k binary matrix
factorisation problem (k-BMF) under Boolean arithmetic: we are given an n x m
binary matrix X with possibly missing entries and need to find two binary
matrices A and B of dimension n x k and k x m respectively, which minimise the
distance between X and the Boolean product of A and B in the squared Frobenius
distance. We present a compact and two exponential size integer programs (IPs)
for k-BMF and show that the compact IP has a weak LP relaxation, while the
exponential size IPs have a stronger equivalent LP relaxation. We introduce a
new objective function, which differs from the traditional squared Frobenius
objective in attributing a weight to zero entries of the input matrix that is
proportional to the number of times the zero is erroneously covered in a rank-k
factorisation. For one of the exponential size IPs we describe a computational
approach based on column generation. Experimental results on synthetic and real
word datasets suggest that our integer programming approach is competitive
against available methods for k-BMF and provides accurate low-error
factorisations.
