Uniqueness of Nonnegative Matrix Factorizations by Rigidity Theory

Nonnegative matrix factorizations are often encountered in data mining applications where they used to explain datasets by a small number of parts. For many these it is desirable that there exists unique nonnegative factorization up trivial modifications given scalings and permutations. This means model parameters uniquely identifiable from the data. Rigidity theory bar joint frameworks field studies uniqueness point configurations some pairwise distances. The goal this paper use ideas rigidity study case when rank equal its rank. We characterize infinitesimally rigid factorizations, prove if only locally certain achieves maximal possible Kruskal rank, show can be extended globally factorizations. These results give so far strongest necessary condition for factorization. also explore connections between boundaries set matrices fixed Finally we extend completely positive