  A novel regularizer of the PARAFAC decomposition factors capturing the
tensor's rank is proposed in this paper, as the key enabler for completion of
three-way data arrays with missing entries. Set in a Bayesian framework, the
tensor completion method incorporates prior information to enhance its
smoothing and prediction capabilities. This probabilistic approach can
naturally accommodate general models for the data distribution, lending itself
to various fitting criteria that yield optimum estimates in the
maximum-a-posteriori sense. In particular, two algorithms are devised for
Gaussian- and Poisson-distributed data, that minimize the rank-regularized
least-squares error and Kullback-Leibler divergence, respectively. The proposed
technique is able to recover the "ground-truth'' tensor rank when tested on
synthetic data, and to complete brain imaging and yeast gene expression
datasets with 50% and 15% of missing entries respectively, resulting in
recovery errors at -10dB and -15dB.
