Robust low tubal rank tensor completion via factor tensor norm minimization. (March 2023) - Jiang, Wei; Zhang, Jun; Zhang, Changsheng; Wang, Lijun; Qi, Heng
Highlights: We give the definitions of tensor double norm and tensor Frobenius/nuclear hybrid norm, and regard them as low-rank regularization penalty of tensor completion. We prove that tensor double norm and tensor Frobenius/nuclear hybrid norm are equivalent to Schatten-1/2 and 2/3 quasi-norm. We transform a nonconvex problem into two convex subproblems and give an efficient algorithm to solve the model. Several experiments on both synthesized data and real world data can verify the superiority of the proposed algorithms from both accuracy and time consumption. Abstract: Recent research has demonstrated that low tubal rank recovery based on tensor has received extensive attention. In this correspondence, we define tensor double nuclear norm and tensor Frobenius/nuclear hybrid norm to induce a surrogate for tensor tubal rank, and prove that they are equivalent to tensor Schatten- p norm for p = 1 / 2 and p = 2 / 3 . Based on the definition, we propose two novel tractable tensor completion models called Double Nuclear norm regularized Tensor Completion (DNTC) and Frobenius/Nuclear hybrid norm regularized Tensor Completion (FNTC) by integrating these two norm minimization and factorization methods into a joint learning framework. Furthermore, we adopt invertible linear transforms to obtain low tubal rank tensors, which makes the model more flexible and effective. Two efficient algorithms are designed to solve the proposed tensor completion models by incorporating the convexity of the factor norms. Comprehensive experiments are conducted on synthetic and real datasets to achieve better results in comparison with some state-of-the-art approaches.