  Tensors have found application in a variety of fields, ranging from
chemometrics to signal processing and beyond. In this paper, we consider the
problem of multilinear modeling of sparse count data. Our goal is to develop a
descriptive tensor factorization model of such data, along with appropriate
algorithms and theory. To do so, we propose that the random variation is best
described via a Poisson distribution, which better describes the zeros observed
in the data as compared to the typical assumption of a Gaussian distribution.
Under a Poisson assumption, we fit a model to observed data using the negative
log-likelihood score. We present a new algorithm for Poisson tensor
factorization called CANDECOMP-PARAFAC Alternating Poisson Regression (CP-APR)
that is based on a majorization-minimization approach. It can be shown that
CP-APR is a generalization of the Lee-Seung multiplicative updates. We show how
to prevent the algorithm from converging to non-KKT points and prove
convergence of CP-APR under mild conditions. We also explain how to implement
CP-APR for large-scale sparse tensors and present results on several data sets,
both real and simulated.
