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.

We present a general method for including prior knowledge in a nonnegative matrix factorization (NMF), based on Gaussian process priors. We assume that the nonnegative factors in the NMF are linked by a strictly increasing function to an underlying Gaussian process specified by its covariance function. This allows us to find NMF decompositions that agree with our prior knowledge of the distribution of the factors, such as sparseness, smoothness, and symmetries. The method is demonstrated with an example from chemical shift brain imaging.

In this paper, we propose a novel algorithm for monaural blind source separation based on non-negative matrix factorization (NMF). A shortcoming of most source separation methods is the need for training data for each individual source. The algorithm proposed in this paper is able separate sources even when there is no training data for the individual sources. The algorithm makes use of models trained on mixed signals and uses training data where more than one source is active at the time. This makes the algorithm applicable to situations where recordings of the individual sources are unavailable. The key idea is to construct a structure matrix that indicates where each source is active, and we prove that this structure matrix, combined with a uniqueness assumption, is sufficient to ensure that results are equivalent to training on isolated sources. Our theoretical findings is backed up by simulations on music data that show that the proposed algorithm trained on mixed recordings performs as well as existing NMF source separation methods trained on solo recordings.

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We generalize the non-negative matrix factorization (NMF)