Nonnegative Matrix Factorization with Gaussian Process Priors

Abstract

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.

Bayesian Non-negative Matrix Factorization

“…The measured spectra are mixtures of different underlying spectra, and the NMF decomposition finds these as the columns of A and the corresponding active positions in the head as the rows of B. Ochs et al [6] demonstrate that the data is well described by two components that correspond to brain tissue and muscle tissue, and present a bilinear MCMC method that provides physically meaningful results but takes several hours to compute. Sajda et al [10,3] demonstrate, on the same data set, that a constrained NMF method provides meaningful results, and Schmidt and Laurberg [11] extend the NMF approach by including advanced prior densities.…”

Multichannel High-Resolution NMF for Modeling Convolutive Mixtures of Non-Stationary Signals in the Time-Frequency Domain

“…, and , , provided that . Proposition 2: If is the impulse response of a causal and stable recursive filter, then the TF input/output system defined in Proposition 1 admits the state space representation (11), where and , , is a sequence of support w.r.t. , where .…”

“…Proposition 2 is proved in Appendix A. Proposition 3: In Definition 1, equation (11) can be rewritten in the form of equation (7), where , , if…”

“…In the literature, several probabilistic models involving latent components have been proposed to provide a probabilistic framework to NMF. Such models include NMF with additive Gaussian noise [11], probabilistic latent component analysis (PLCA) [12], NMF as a sum of Poisson components [13], and NMF as a sum of Gaussian components [14]. Although they have already proven successful in a number of audio applications such as source separation [11]- [13] and multipitch estimation [14], most of these models still lack of consistency in some respects.…”

Unmixing of Hyperspectral Images using Bayesian Non-negative Matrix Factorization with Volume Prior

“…Schmidt and Laurberg [29] present a Bayesian NMF based on an exponential sparsity prior. Moussaoui et al [19] present a sparse Bayesian method based on a hybrid Gibbs-Metropolis-Hastings sampling procedure for separating non-negative mixtures of NIR data, which are known to be sparse as opposed to the application considered in this paper.…”