Recently, the \({ Tensor}~{ Nuclear}~{ Norm}~{ (TNN)}\) 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 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 methods 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 models in the tensor completion problem with respect to other tensor rank regularization models.

最近，基于 t-SVD的\({Tensor}~{Nuclear}~{Norm}~{(TNN)}\)正则化已广泛应用于各种低输卵管阶张量恢复任务。但是，这些模型通常需要沿第三维平滑更改数据以确保其低秩结构。在本文中，我们通过可学习的正交矩阵\(\mathbf {Q}\)提出了数据相关张量秩的新定义，称为张量 Q-rank，并进一步引入了统一的数据相关低秩张量恢复模型。根据低秩假设，我们引入了\(\mathbf {Q}\) 的两种可解释的选择方法，在这种情况下，数据张量可能具有比低 tubal-rank 结构更显着的低张量 Q-rank 结构。具体来说，最大化奇异值分布的方差导致方差最大化张量 Q-核范数（VMTQN），而通过流形优化最小化核范数的值导致流形优化张量 Q-核范数（MOTQN）。此外，我们将这两种模型应用于低秩张量补全问题，然后给出了一个有效的算法，并简要分析了为什么我们的方法在低采样率的复杂数据情况下比基于 TNN 的方法更有效。最后，在真实世界数据集上的实验结果证明了我们提出的模型在张量完成问题中相对于其他张量等级正则化模型的优越性。