# Partial Identifiability for Nonnegative Matrix Factorization

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^⊤. 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.

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