Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n*m matrix M into a product of a nonnegative n*d matrix W and a nonnegative d*m matrix H. Restricted NMF requires in addition that the column spaces of M and W coincide.

Finding the minimal inner dimension d is known to be NP-hard, both for NMF and restricted NMF. We show that restricted NMF is closely related to a question about the nature of minimal probabilistic automata, posed by Paz in his seminal 1971 textbook. We use this connection to answer Paz's question negatively, thus falsifying a positive answer claimed in 1974.

Furthermore, we investigate whether a rational matrix M always has a restricted NMF of minimal inner dimension whose factors W and H are also rational. We show that this holds for matrices M of rank at most 3 and we exhibit a rank-4 matrix for which W and H require irrational entries.