  It has become routine to collect data that are structured as multiway arrays
(tensors). There is an enormous literature on low rank and sparse matrix
factorizations, but limited consideration of extensions to the tensor case in
statistics. The most common low rank tensor factorization relies on parallel
factor analysis (PARAFAC), which expresses a rank $k$ tensor as a sum of rank
one tensors. When observations are only available for a tiny subset of the
cells of a big tensor, the low rank assumption is not sufficient and PARAFAC
has poor performance. We induce an additional layer of dimension reduction by
allowing the effective rank to vary across dimensions of the table. For
concreteness, we focus on a contingency table application. Taking a Bayesian
approach, we place priors on terms in the factorization and develop an
efficient Gibbs sampler for posterior computation. Theory is provided showing
posterior concentration rates in high-dimensional settings, and the methods are
shown to have excellent performance in simulations and several real data
applications.
