Tensor Truncated Schatten‐p Norm Approximation Tensor Completion Algorithm

ABSTRACT Image data is often degraded during transmission due to hardware limitations or human error, which may hinder subsequent image analysis tasks. Therefore, research on image restoration has significant practical value. Traditional matrix‐based algorithms struggle with high‐dimensional data, often failing to preserve spatial structures and risking overfitting. In this paper, we investigate tensor recovery problems under the tensor singular value decomposition framework. We introduce a non‐convex surrogate for the tensor rank—the tensor truncated Schatten‐ norm—and propose two recovery models based on this theory: a tensor completion model and a tensor robust principal component analysis model. Efficient solutions based on the alternating direction method of multipliers are developed for both models. Moreover, we provide a thorough analysis of the computational complexity and convergence behavior of our algorithms. At last, extensive experiments on synthetic data, color images, video sequences, multispectral images, and medical images demonstrate the effectiveness and robustness of the proposed methods.