  Nonnegative matrix factorization (NMF) is a standard linear dimensionality
reduction technique for nonnegative data sets. In order to measure the
discrepancy between the input data and the low-rank approximation, the
Kullback-Leibler (KL) divergence is one of the most widely used objective
function for NMF. It corresponds to the maximum likehood estimator when the
underlying statistics of the observed data sample follows a Poisson
distribution, and KL NMF is particularly meaningful for count data sets, such
as documents or images. In this paper, we first collect important properties of
the KL objective function that are essential to study the convergence of KL NMF
algorithms. Second, together with reviewing existing algorithms for solving KL
NMF, we propose three new algorithms that guarantee the non-increasingness of
the objective function. We also provide a global convergence guarantee for one
of our proposed algorithms. Finally, we conduct extensive numerical experiments
to provide a comprehensive picture of the performances of the KL NMF
algorithms.
