Bi-level models and algorithms based on Tucker rank for tensor completion

In this paper, the minimal and maximal nuclear norm of $N$-mode matrices as well as their combination for tensor completion are proposed based on Tucker rank. For the three optimization models, we study the null space property and the restricted isometry property (RIP) condition. Based on the framework of the augmented Lagrange multiplier method, we design three modified augmented Lagrange multiplier methods for solving tensor completion. Convergence of the algorithm for solving the non-convex minimal and maximal combination model is established. Finally, we compare the proposed three algorithms with the high accuracy low-rank tensor completion algorithm, and the experimental results of the randomly generated tensor completion show that the new non-convex minimal and maximal combination optimization model outperforms less CPU time than the traditional nuclear combination optimization model under the same precision.