Tensor Q-rank: new data dependent definition of tensor rank

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 third dimension to ensure their rank structures. In this paper, we propose a new definition dependent named Q-rank by learnable orthogonal matrix $$\mathbf {Q}$$ , and further introduce unified model. According hypothesis, two explainable selection methods under which may have more significant structure than that structure. Specifically, maximizing variance singular value distribution leads Variance Maximization Tensor Q-Nuclear norm (VMTQN), while minimizing nuclear through manifold optimization Manifold Optimization (MOTQN). Moreover, apply completion problem, then give an effective algorithm briefly analyze why our method works better TNN case complex with sampling rate. Finally, experimental results real-world datasets demonstrate superiority proposed problem respect other models.