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.

本文研究了稀疏计数数据的多线性建模问题，提出了一个以泊松分布为假设的描述性张量分解模型和相应的算法和理论，并介绍了一种基于主导极小化方法的泊松张量分解算法，称为 CP-APR，并在几个数据集上的结果得到了验证。