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.
Bibliographical noteCopyright © 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.
Lauerberg, H., Christensen, M. G., Pumbley, M., Hansen, L. K., & Jensen, S. H. (2008). Theorems on Positive Data: On the Uniqueness of NMF. Computational Intelligence and Neuroscience, 2008, 10. . https://doi.org/10.1155/2008/764206