  Low rank tensor representation underpins much of recent progress in tensor
completion. In real applications, however, this approach is confronted with two
challenging problems, namely (1) tensor rank determination; (2) handling real
tensor data which only approximately fulfils the low-rank requirement. To
address these two issues, we develop a data-adaptive tensor completion model
which explicitly represents both the low-rank and non-low-rank structures in a
latent tensor. Representing the non-low-rank structure separately from the
low-rank one allows priors which capture the important distinctions between the
two, thus enabling more accurate modelling, and ultimately, completion. Through
defining a new tensor rank, we develop a sparsity induced prior for the
low-rank structure, with which the tensor rank can be automatically determined.
The prior for the non-low-rank structure is established based on a mixture of
Gaussians which is shown to be flexible enough, and powerful enough, to inform
the completion process for a variety of real tensor data. With these two
priors, we develop a Bayesian minimum mean squared error estimate (MMSE)
framework for inference which provides the posterior mean of missing entries as
well as their uncertainty. Compared with the state-of-the-art methods in
various applications, the proposed model produces more accurate completion
results.
