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Computational Intelligence and Neuroscience
Volume 2008, Article ID 764206, 9 pages
Research Article

Theorems on Positive Data: On the Uniqueness of NMF

1Department of Electronic Systems, Aalborg University, Niels Jernes Vej 12, 9220 Aalborg, Denmark
2Department of Electronic Engineering, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
3Department of Informatics and Mathematical Modeling, Technical University of Denmark, Richard Petersens Plads, Building 321, 2800 Lyngby, Denmark

Received 1 November 2007; Accepted 13 March 2008

Academic Editor: Wenwu Wang

Copyright © 2008 Hans Laurberg et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the conditions for which nonnegative matrix factorization (NMF) is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. We have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution. Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors.