  Tucker decomposition is the cornerstone of modern machine learning on
tensorial data analysis, which have attracted considerable attention for
multiway feature extraction, compressive sensing, and tensor completion. The
most challenging problem is related to determination of model complexity (i.e.,
multilinear rank), especially when noise and missing data are present. In
addition, existing methods cannot take into account uncertainty information of
latent factors, resulting in low generalization performance. To address these
issues, we present a class of probabilistic generative Tucker models for tensor
decomposition and completion with structural sparsity over multilinear latent
space. To exploit structural sparse modeling, we introduce two group sparsity
inducing priors by hierarchial representation of Laplace and Student-t
distributions, which facilitates fully posterior inference. For model learning,
we derived variational Bayesian inferences over all model (hyper)parameters,
and developed efficient and scalable algorithms based on multilinear
operations. Our methods can automatically adapt model complexity and infer an
optimal multilinear rank by the principle of maximum lower bound of model
evidence. Experimental results and comparisons on synthetic, chemometrics and
neuroimaging data demonstrate remarkable performance of our models for
recovering ground-truth of multilinear rank and missing entries.
