Tensor Q-Rank: New Data Dependent Definition of Tensor Rank

Recently, the \textit{Tensor Nuclear Norm~(TNN)} regularization based on t-SVD has been widely used in various low tubal-rank tensor recovery tasks. However, these models usually require smooth change of data along the third dimension to ensure their low rank structures. In this paper, we propose a new definition of data dependent tensor rank named \textit{tensor Q-rank} by a learnable orthogonal matrix $\mathbf{Q}$, and further introduce a unified data dependent low rank tensor recovery model. According to the low rank hypothesis, we introduce two explainable selection method of $\mathbf{Q}$, under which the data tensor may have a more significant low tensor Q-rank structure than that of low tubal-rank structure. Specifically, maximizing the variance of singular value distribution leads to Variance Maximization Tensor Q-Nuclear norm~(VMTQN), while minimizing the value of nuclear norm through manifold optimization leads to Manifold Optimization Tensor Q-Nuclear norm~(MOTQN). Moreover, we apply these two models to the low rank tensor completion problem, and then give an effective algorithm and briefly analyze why our method works better than TNN based methods in the case of complex data with low sampling rate. Finally, experimental results on real-world datasets demonstrate the superiority of our proposed model in the tensor completion problem with respect to other tensor rank regularization models.