This paper considers the least squares loss minimization problem regularized by tensor average rank and zero norm, to decompose the noisy observation into a low-rank tensor, a sparse tensor and a noise tensor. Due to the nonconvex and nonsmooth of the average rank and the zero norm of tensors, it is not easy to solve the problem directly. We construct the variational characterizations of the tensor average rank and the tensor zero norm, get an equivalent mathematical program with an equilibrium constraint, and then propose an equivalent Lipschitz surrogate for the regularization problem by adding the equality constraint into the objective. Moreover, we design a convex algorithm with multi-stage to verify the efficiency of the proposed method. Numerical experiments are reported for simulation data and actual data to show the performance of the new method.

本文考虑张量平均秩和零范数正则化的最小二乘损失最小化问题，将噪声观测分解为低秩张量、稀疏张量和噪声张量。由于平均秩的非凸非光滑和张量的零范数，直接解决问题并不容易。我们构造了张量平均秩和张量零范数的变分表征，得到了一个具有平衡约束的等效数学程序，然后通过将等式约束添加到目标中，提出了正则化问题的等效 Lipschitz 代理。此外，我们设计了一个多阶段的凸算法来验证所提方法的有效性。