When is non-negative matrix decomposition unique?

In this paper, we discuss why non-negative matrix factorization (NMF) potentially works for zero-grounded non-negative components and why it fails when the components are not zero-grounded. We show the demixing process is not uniquely defined (up to the usual permutation/scaling ambiguity) when the original matrices are not zero-grounded. If fact, zero-groundedness alone is not enough. The key observation is that if each component has at least one point for which it is the only active component, the solution is unique. When the non-negative matrices are not zero-grounded, no such point exists and the solution space contains demixtures which are linear combinations of the original components. Thus, the NMF problem has a unique solution for matrices with disjoint components, a condition we call Subset Monomial Disjoint (SMD). The SMD condition is sufficient, but not necessary for NMF to have a unique decomposition, whereas the zero-grounded condition is necessary, but not sufficient.