Nonnegative matrix factorization with Gaussian process priors.
Bottom Line:
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.
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Affiliation: Department of Informatics and Mathematical Modelling, Technical University of Denmark, Richard Petersens Plads, DTU-Building 321, 2800 Lyngby, Denmark. mns@imm.dtu.dk
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. No MeSH data available. |
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Mentions: We generated a 100 × 200 data matrix, Y, by taking a random sample from the GPP-NMF model with two factors. We chose the generating covariance function for both δ and η as a Gaussian radial basis function (RBF)(30)ϕ(i,j)=exp(−(i−j)2β2), where i and j are two sample indices, and the length-scale parameter, which determines the smoothness of the factors, was β2 = 100. We set the covariance between the two factors to zero, such that the factorswere uncorrelated. For the matrix D, we used the rectified-Gaussian-to-Gaussianlink function with s = 1; and for H, we used the exponential-to-Gaussian linkfunction with λ = 1. Finally, we added independent Gaussian noise with variance σN2 = 25, which resulted in a signal-to-noise ratio of approximately −7 dB. The generated data matrix is shown in Figure 1. |
View Article: PubMed Central - PubMed
Affiliation: Department of Informatics and Mathematical Modelling, Technical University of Denmark, Richard Petersens Plads, DTU-Building 321, 2800 Lyngby, Denmark. mns@imm.dtu.dk
No MeSH data available.