  CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful
technique for tensor completion through explicitly capturing the multilinear
latent factors. The existing CP algorithms require the tensor rank to be
manually specified, however, the determination of tensor rank remains a
challenging problem especially for CP rank. In addition, existing approaches do
not take into account uncertainty information of latent factors, as well as
missing entries. To address these issues, we formulate CP factorization using a
hierarchical probabilistic model and employ a fully Bayesian treatment by
incorporating a sparsity-inducing prior over multiple latent factors and the
appropriate hyperpriors over all hyperparameters, resulting in automatic rank
determination. To learn the model, we develop an efficient deterministic
Bayesian inference algorithm, which scales linearly with data size. Our method
is characterized as a tuning parameter-free approach, which can effectively
infer underlying multilinear factors with a low-rank constraint, while also
providing predictive distributions over missing entries. Extensive simulations
on synthetic data illustrate the intrinsic capability of our method to recover
the ground-truth of CP rank and prevent the overfitting problem, even when a
large amount of entries are missing. Moreover, the results from real-world
applications, including image inpainting and facial image synthesis,
demonstrate that our method outperforms state-of-the-art approaches for both
tensor factorization and tensor completion in terms of predictive performance.
