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

Kullback-Leibler Principal Component for Tensors is not NP-hard