Papers › Partial Identifiability for Nonnegative Matrix Factorization
Partial Identifiability for Nonnegative Matrix Factorization
Nicolas Gillis, Róbert Rajkó
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.
In Syntology Open this paper in Syntology's Atlas, the map of the papers in Syntology's graph and their citations.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections