  There has recently been considerable interest in completing a low-rank matrix
or tensor given only a small fraction (or few linear combinations) of its
entries. Related approaches have found considerable success in the area of
recommender systems, under machine learning. From a statistical estimation
point of view, the gold standard is to have access to the joint probability
distribution of all pertinent random variables, from which any desired optimal
estimator can be readily derived. In practice high-dimensional joint
distributions are very hard to estimate, and only estimates of low-dimensional
projections may be available. We show that it is possible to identify
higher-order joint PMFs from lower-order marginalized PMFs using coupled
low-rank tensor factorization. Our approach features guaranteed identifiability
when the full joint PMF is of low-enough rank, and effective approximation
otherwise. We provide an algorithmic approach to compute the sought factors,
and illustrate the merits of our approach using rating prediction as an
example.
