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There has recently been considerable interest in completing a low-rank matrixor tensor given only a small fraction (or few linear combinations) of itsentries. Related approaches have found considerable success in the area ofrecommender systems, under machine learning. From a statistical estimationpoint of view, the gold standard is to have access to the joint probabilitydistribution of all pertinent random variables, from which any desired optimalestimator can be readily derived. In practice high-dimensional jointdistributions are very hard to estimate, and only estimates of low-dimensionalprojections may be available. We show that it is possible to identifyhigher-order joint PMFs from lower-order marginalized PMFs using coupledlow-rank tensor factorization. Our approach features guaranteed identifiabilitywhen the full joint PMF is of low-enough rank, and effective approximationotherwise. We provide an algorithmic approach to compute the sought factors,and illustrate the merits of our approach using rating prediction as anexample.

Completing a joint PMF from projections: a low-rank coupled tensor factorization approach