Beyond Low Rank: A Data-Adaptive Tensor Completion Method

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.