  Nonnegative matrix factorization (NMF) is a popular method used to reduce
dimensionality in data sets whose elements are nonnegative. It does so by
decomposing the data set of interest, $\mathbf{X}$, into two lower rank
nonnegative matrices multiplied together ($\mathbf{X} \approx \mathbf{WH}$).
These two matrices can be described as the latent factors, represented in the
rows of $\mathbf{H}$, and the scores of the observations on these factors that
are found in the rows of $\mathbf{W}$. This paper provides an extension of this
method which allows one to specify prior knowledge of the data, including both
group information and possible underlying factors. This is done by further
decomposing the matrix, $\mathbf{H}$, into matrices $\mathbf{A}$ and
$\mathbf{S}$ multiplied together. These matrices represent an 'auxiliary'
matrix and a semi-constrained factor matrix respectively. This method and its
updating criterion are proposed, followed by its application on both simulated
and real world examples.
