Theorems on Positive Data: On the Uniqueness of NMF

Hans Lauerberg; Mads Græsbøll Christensen; Mark Pumbley; Lars Kai Hansen; Søren Holdt Jensen

doi:10.1155/2008/764206

Theorems on Positive Data: On the Uniqueness of NMF

Hans Lauerberg, Mads Græsbøll Christensen, Mark Pumbley, Lars Kai Hansen, Søren Holdt Jensen

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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.

Original language	English
Article number	764206
Journal	Computational Intelligence and Neuroscience
Volume	2008
Pages (from-to)	10
ISSN	1687-5265
DOIs	https://doi.org/10.1155/2008/764206
Publication status	Published - 2008

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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.

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AU - Jensen, Søren Holdt

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AB - 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.

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Theorems on Positive Data: On the Uniqueness of NMF

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