  We study the problem of nonnegative rank-one approximation of a nonnegative
tensor, and show that the globally optimal solution that minimizes the
generalized Kullback-Leibler divergence can be efficiently obtained, i.e., it
is not NP-hard. This result works for arbitrary nonnegative tensors with an
arbitrary number of modes (including two, i.e., matrices). We derive a
closed-form expression for the KL principal component, which is easy to compute
and has an intuitive probabilistic interpretation. For generalized KL
approximation with higher ranks, the problem is for the first time shown to be
equivalent to multinomial latent variable modeling, and an iterative algorithm
is derived that resembles the expectation-maximization algorithm. On the Iris
dataset, we showcase how the derived results help us learn the model in an
\emph{unsupervised} manner, and obtain strikingly close performance to that
from supervised methods.
