Computational Intelligence and Neuroscience 
Volume 2008 (2008), Article ID 764206, 9 pages
doi:10.1155/2008/764206
Research Article

Theorems on Positive Data: On the Uniqueness of NMF

Hans Laurberg,1 Mads Græsbøll Christensen,1 Mark D. Plumbley,2 Lars Kai Hansen,3 and Søren Holdt Jensen1

 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

Recommended by Wenwu Wang

Abstract

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.