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.

我们提出了一种新的张量分解和一种新的张量范数，称为张量 $U_1$ 范数，用于解决高阶张量数据的完整性问题，并提出了一个优化算法来解决该问题。该方法在处理具有非光滑变化的张量数据方面表现出良好效果。