  Nonnegative matrix factorization (NMF) is a popular dimension reduction
technique that produces interpretable decomposition of the data into parts.
However, this decompostion is not generally identifiable (even up to
permutation and scaling). While other studies have provide criteria under which
NMF is identifiable, we present the first (to our knowledge) characterization
of the non-identifiability of NMF. We describe exactly when and how
non-uniqueness can occur, which has important implications for algorithms to
efficiently discover alternate solutions, if they exist.
