基于$l_{\infty }$-模的Hankel张量补全的保结构加速邻近梯度算法

Structure-Preserving Accelerated Proximal Gradient Algorithm Based on $l_{\infty }$-Norm for Hankel Tensor Completion

摘要: 加速邻近梯度算法是求解张量补全问题的经典方法之一，但将该算法应用到Hankel结构张量补全时，无法保证补全的张量能够保持Hankel结构。因此，本文基于加速邻近梯度算法的框架，提出了一种保Hankel结构的加速邻近梯度算法。该算法在每次迭代中利用$l_{\infty }$ -模投影算子生成Hankel结构张量。在理论上，本文证明了新算法在合理假设条件下的收敛性。最后，通过随机Hankel张量补全与图像修复实例的数值实验验证了新算法的有效性。

Abstract: The accelerated proximal gradient algorithm is one of the classic methods for solving tensor completion problems. However, when applied to Hankel structured tensor completion, it cannot guarantee that the completed tensor can maintain the Hankel structure. Therefore, based on the framework of the accelerated proximal gradient algorithm, this paper proposes a new accelerated proximal gradient algorithm that can preserve the Hankel structure. In each iteration of the algorithm, utilizing the $l_{\infty }$ -norm projection and the fast singular value thresholding method ensures that the generated tensor preserves the Hankel structure. Moreover, this paper proves the convergence of the new algorithm under reasonable assumptions. Finally, the effectiveness of the new algorithm is verified through numerical experiments of random Hankel tensor completion and image restoration examples.