Tensor completion aims to recover missing entries given incomplete multi-dimensional data by making use of the prior low-rank information, and has various applications because many real-world data can be modeled as low-rank tensors. Most of the existing methods are designed for noiseless or Gaussian noise scenarios, and thus they are not robust to outliers. One popular approach to resist outliers is to employ $\ell _p$ -norm. Yet nonsmoothness and nonconvexity of $\ell _p$ -norm with $0< p\leq 1$ bring challenges to optimization. In this paper, a new norm, named $\ell _{p,\epsilon }$ -norm, is devised where $\epsilon >0$ can adjust the convexity of $\ell _{p,\epsilon }$ -norm. Compared with $\ell _p$ -norm, $\ell _{p,\epsilon }$ -norm is smooth and convex even for $0< p\leq 1$ , which converts an intractable nonsmooth and nonconvex optimization problem into a much simpler convex and smooth one. Then, combining tensor ring rank and $\ell _{p,\epsilon }$ -norm, a robust tensor completion formulation is proposed, which achieves outstanding robustness. The resultant robust tensor completion problem is decomposed into a number of robust linear regression (RLR) subproblems, and two algorithms are devised to tackle RLR. The first method adopts gradient descent, which has a low computational complexity. While the second one employs alternating direction method of multipliers to yield a fast convergence rate. Numerical simulations show that the two proposed methods have better performance than those based on the $\ell _p$ -norm in RLR. Experimental results from applications of image inpainting, video restoration and target estimation demonstrate that our robust tensor completion approach outperforms state-of-the-art methods in terms of recovery accuracy.

中文翻译：

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基于张量环秩的鲁棒低秩张量补全 $\ell _{p,\epsilon }$-规范

张量补全旨在通过利用先验的低秩信息在给定不完整的多维数据的情况下恢复缺失的条目，并且由于许多现实世界数据可以建模为低秩张量而具有多种应用。大多数现有方法是为无噪声或高斯噪声场景设计的，因此它们对异常值不具有鲁棒性。一种抵制异常值的流行方法是采用$\ell _p$ -规范。的不光滑和非凸性$\ell _p$ -范数 $0< p\leq 1$给优化带来挑战。在本文中，一个新的规范，名为$\ell _{p,\epsilon }$ - 规范，被设计在哪里 $\epsilon >0$ 可以调整凸度 $\ell _{p,\epsilon }$ -规范。和....相比$\ell _p$ -规范， $\ell _{p,\epsilon }$ - 范数是光滑和凸的，即使对于 $0< p\leq 1$ ，它将一个棘手的非光滑和非凸优化问题转换为一个更简单的凸和光滑优化问题。然后，结合张量环秩和$\ell _{p,\epsilon }$ -norm，提出了一个鲁棒的张量完成公式，它实现了出色的鲁棒性。由此产生的鲁棒张量完成问题被分解为许多鲁棒线性回归 (RLR) 子问题，并设计了两种算法来解决 RLR。第一种方法采用梯度下降，计算复杂度低。而第二种采用乘法器的交替方向方法来产生快速的收敛速度。数值模拟表明，所提出的两种方法都比基于该方法的方法具有更好的性能。$\ell _p$ -RLR 中的规范。图像修复、视频恢复和目标估计应用的实验结果表明，我们的鲁棒张量完成方法在恢复精度方面优于最先进的方法。