PARTIAL IDENTIFIABILITY FOR NONNEGATIVE MATRIX FACTORIZATION\ast

Given a nonnegative matrix factorization, R, and a factorization rank, r, exact nonnegative matrix factorization (exact NMF) decomposes R as the product of two nonnegative matrices, C and S with r columns, such as R = CS\top. A central research topic in the literature is the conditions under which such a decomposition is unique/identifiable up to trivial ambiguities. In this paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of C and S. We start our investigations with the data-based uniqueness (DBU) theorem from the chemometrics literature. The DBU theorem analyzes all feasible solutions of exact NMF and relies on sparsity conditions on C and S. We provide a mathematically rigorous theorem of a recently published restricted version of the DBU theorem, relying only on simple sparsity and algebraic conditions: it applies to a particular solution of exact NMF (as opposed to all feasible solutions) and allows us to guarantee the partial uniqueness of a single column of C or S. Second, based on a geometric interpretation of the restricted DBU theorem, we obtain a new partial identifiability result. This geometric interpretation also leads us to another partial identifiability result in the case r = 3. Third, we show how partial identifiability results can be used sequentially to guarantee the identifiability of more columns of C and S. We illustrate these results on several examples, including one from the chemometrics literature.