A Latent-Variable Formulation of the Poisson Canonical Polyadic Tensor Model: Maximum Likelihood Estimation and Fisher Information

We establish parameter inference for the Poisson canonical polyadic (PCP) model of tensor count data through a latent-variable formulation. Our approach exploits the property that any random tensor that follows the PCP model can be derived by marginalizing an unobservable random tensor of one dimension larger. The loglikelihood of this larger dimensional tensor, referred to as the "complete" loglikelihood, is comprised of multiple loglikelihoods corresponding to rank one PCP models. Using this methodology, we first demonstrate that several existing algorithms for fitting non-negative matrix and tensor factorizations are Expectation-Maximization algorithms. Next, we derive the observed and expected Fisher information matrices for the PCP model by leveraging its latent-variable formulation. The Fisher information provides us crucial insights into the well-posedness of the tensor model, such as the role that the rank of parameter tensor plays in identifiability and indeterminacy. For the special case of PCP models with rank one parameter tensors, we demonstrate that these results are greatly simplified.