Recently, numerous tensor SVD (t-SVD)-based tensor recovery methods have emerged, showing promise in processing visual data. However, these methods often suffer from performance degradation when confronted with high-order tensor data exhibiting non-smooth changes, commonly observed in real-world scenarios but ignored by the traditional t-SVD-based methods. Our objective in this study is to provide an effective tensor recovery technique for handling non-smooth changes in tensor data and efficiently explore the correlations of high-order tensor data across its various dimensions without introducing numerous variables and weights. To this end, we introduce a new tensor decomposition and a new tensor norm called the Tensor $U_1$ norm. We utilize these novel techniques in solving the problem of high-order tensor completion problem and provide theoretical guarantees for the exact recovery of the resulting tensor completion models. An optimization algorithm is proposed to solve the resulting tensor completion model iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers. Theoretical analysis showed the convergence of the algorithm to the Karush-Kuhn-Tucker (KKT) point of the optimization problem. Numerical experiments demonstrated the effectiveness of the proposed method in high-order tensor completion, especially for tensor data with non-smooth changes.

翻译：近年来，基于张量奇异值分解（t-SVD）的张量恢复方法层出不穷，在处理视觉数据方面展现出应用前景。然而，这些方法在面对呈现非平滑变化的高阶张量数据时，性能往往会下降——这种非平滑变化在现实场景中十分常见，却被传统t-SVD方法所忽视。本研究的目标是提供一种有效的张量恢复技术，既能处理张量数据中的非平滑变化，又能高效探索高阶张量数据各维度间的相关性，且无需引入大量变量和权重。为此，我们提出了一种新的张量分解方法以及一种新的张量范数——张量$U_1$范数。我们利用这些新技术解决高阶张量补全问题，并为所构建的张量补全模型的精确恢复提供了理论保障。通过将近端算法与交替方向乘子法相结合，我们提出了一种迭代求解所构建的张量补全模型的优化算法。理论分析表明，该算法能收敛至优化问题的Karush-Kuhn-Tucker（KKT）点。数值实验证明了所提方法在高阶张量补全中的有效性，尤其是在处理具有非平滑变化的张量数据时表现突出。