Recently, numerous tensor singular value decomposition (t-SVD)-based tensor recovery methods have shown promise in processing visual data, such as color images and videos. However, these methods often suffer from severe performance degradation when confronted with tensor data exhibiting non-smooth changes. It has been commonly observed in real-world scenarios but ignored by the traditional t-SVD-based methods. In this work, we introduce a novel tensor recovery model with a learnable tensor nuclear norm to address such a challenge. We develop a new optimization algorithm named the Alternating Proximal Multiplier Method (APMM) to iteratively solve the proposed tensor completion model. Theoretical analysis demonstrates the convergence of the proposed APMM to the Karush-Kuhn-Tucker (KKT) point of the optimization problem. In addition, we propose a multi-objective tensor recovery framework based on APMM to efficiently explore the correlations of tensor data across its various dimensions, providing a new perspective on extending the t-SVD-based method to higher-order tensor cases. Numerical experiments demonstrated the effectiveness of the proposed method in tensor completion.

翻译：近年来，基于张量奇异值分解（t-SVD）的张量恢复方法在处理视觉数据（如彩色图像和视频）方面展现出良好前景。然而，当面对具有非光滑变化的张量数据时，这些方法往往会出现严重的性能下降。这一现象在实际场景中普遍存在，但传统的基于t-SVD的方法却忽视了该问题。本文提出一种具有可学习张量核范数的新型张量恢复模型以应对这一挑战。我们开发了一种名为交替邻近乘子法（APMM）的新优化算法，用于迭代求解所提出的张量补全模型。理论分析证明了所提APMM算法能够收敛至优化问题的Karush-Kuhn-Tucker（KKT）点。此外，我们基于APMM提出了一种多目标张量恢复框架，能够有效探索张量数据在不同维度间的相关性，为将基于t-SVD的方法扩展至高阶张量情形提供了新视角。数值实验验证了所提方法在张量补全任务中的有效性。