2008
DOI: 10.1155/2008/764206
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Theorems on Positive Data: On the Uniqueness of NMF

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 und… Show more

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Cited by 148 publications
(112 citation statements)
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“…Thomas [23] and Chen [3] first gave different geometric interpretations of NMF, and stated the uniqueness problem in a geometric way. Donoho et al [24] and Laurberg et al [25] later provided uniqueness conditions by exploiting the aforementioned geometric viewpoint. The sufficient conditions they provided, however, require one of the two matrix factors to contain a scaled identity matrix-a particularly strict condition.…”
Section: N On-negative Matrix Factorization (Nmf) Is the Problem Of (mentioning
confidence: 99%
“…Thomas [23] and Chen [3] first gave different geometric interpretations of NMF, and stated the uniqueness problem in a geometric way. Donoho et al [24] and Laurberg et al [25] later provided uniqueness conditions by exploiting the aforementioned geometric viewpoint. The sufficient conditions they provided, however, require one of the two matrix factors to contain a scaled identity matrix-a particularly strict condition.…”
Section: N On-negative Matrix Factorization (Nmf) Is the Problem Of (mentioning
confidence: 99%
“…In many cases non-negativity constraints are enough to recover the original signals if the signals are sparse enough [19]. Moreover, results on the conditions for establishing the uniqueness of NMF rely on zero-groundedness of the signals [4], or determining binary closure patterns in the signal ensemble [20], which typically explains the success of sparse coding techniques [18]. Learning sparse factors is generally a good indicator for achieving good compression, much in the same way as integer sequences with many zeros are typically well compressed.…”
Section: Related Workmentioning
confidence: 99%
“…In general the derived factorization is non-unique [3,4] but satisfactory for most applications. Satisfaction is generally based on some distortion measure, e.g.…”
Section: Introductionmentioning
confidence: 99%
“…The NMF solution is generally not unique, or exact. For every invertible A we have a potential factorization [18,9],…”
Section: Nonnegative Matrix Factorizationmentioning
confidence: 99%