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Recently, the \textit{Tensor Nuclear Norm~(TNN)} regularization based ont-SVD has been widely used in various low tubal-rank tensor recovery tasks.However, these models usually require smooth change of data along the thirddimension to ensure their low rank structures. In this paper, we propose a newdefinition of data dependent tensor rank named \textit{tensor Q-rank} by alearnable orthogonal matrix $\mathbf{Q}$, and further introduce a unified datadependent low rank tensor recovery model. According to the low rank hypothesis,we introduce two explainable selection method of $\mathbf{Q}$, under which thedata tensor may have a more significant low tensor Q-rank structure than thatof low tubal-rank structure. Specifically, maximizing the variance of singularvalue distribution leads to Variance Maximization Tensor Q-Nuclearnorm~(VMTQN), while minimizing the value of nuclear norm through manifoldoptimization 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 worksbetter than TNN based methods in the case of complex data with low samplingrate. Finally, experimental results on real-world datasets demonstrate thesuperiority of our proposed model in the tensor completion problem with respectto other tensor rank regularization models.

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