The tensor data recovery task has thus attracted much research attention in recent years. Solving such an ill-posed problem generally requires to explore intrinsic prior structures underlying tensor data, and formulate them as certain forms of regularization terms for guiding a sound estimate of the restored tensor. Recent research have made significant progress by adopting two insightful tensor priors, i.e., global low-rankness (L) and local smoothness (S) across different tensor modes, which are always encoded as a sum of two separate regularization terms into the recovery models. However, unlike the primary theoretical developments on low-rank tensor recovery, these joint L+S models have no theoretical exact-recovery guarantees yet, making the methods lack reliability in real practice. To this crucial issue, in this work, we build a unique regularization term, which essentially encodes both L and S priors of a tensor simultaneously. Especially, by equipping this single regularizer into the recovery models, we can rigorously prove the exact recovery guarantees for two typical tensor recovery tasks, i.e., tensor completion (TC) and tensor robust principal component analysis (TRPCA). To the best of our knowledge, this should be the first exact-recovery results among all related L+S methods for tensor recovery. Significant recovery accuracy improvements over many other SOTA methods in several TC and TRPCA tasks with various kinds of visual tensor data are observed in extensive experiments. Typically, our method achieves a workable performance when the missing rate is extremely large, e.g., 99.5%, for the color image inpainting task, while all its peers totally fail in such challenging case.

翻译：张量数据恢复任务近年来引起了广泛研究关注。求解此类不适定问题通常需要探索张量数据的内在先验结构，并将其表述为特定形式的正则化项，以指导恢复张量的合理估计。近期研究通过采用两种具有洞察力的张量先验取得了显著进展，即全局低秩性（L）和跨不同张量模态的局部平滑性（S），这些先验通常被编码为两个独立正则化项之和，并纳入恢复模型。然而，与低秩张量恢复的主要理论发展不同，这些联合L+S模型尚未具备理论上的精确恢复保证，导致方法在实际应用中缺乏可靠性。针对这一关键问题，本工作构建了一个独特的正则化项，其本质同时编码了张量的L和S先验。特别是，通过将该单一正则化器引入恢复模型，我们能够严格证明两种典型张量恢复任务（即张量补全（TC）和张量鲁棒主成分分析（TRPCA））的精确恢复保证。据我们所知，在所有相关的张量恢复L+S方法中，这应是首个精确恢复结果。在多种视觉张量数据的多个TC和TRPCA任务中，大量实验观察到我们的方法相比其他最先进方法（SOTA）具有显著的恢复精度提升。典型地，在缺失率极高（例如99.5%）的彩色图像修复任务中，我们的方法仍能达到可行性能，而所有同类方法在此挑战性情形下完全失效。