融合低秩性与光滑性的保证型张量恢复
Guaranteed Tensor Recovery Fused Low-rankness and Smoothness
AI总结：
针对现有融合低秩性与光滑性的张量恢复方法缺乏理论保证的问题，本研究构建了可同时编码两种先验的单一正则化项，首次证明了张量补全与鲁棒主成分分析任务的精确恢复保证，且在高缺失率等场景下性能显著优于现有最优方法。
摘要：
近年来，张量数据恢复任务受到了大量研究关注。求解这类不适定问题通常需要挖掘张量数据内在的先验结构，并将其构建为特定形式的正则化项，以指导得到合理的恢复张量估计结果。近期研究通过采用两种具有启发性的张量先验，即跨不同张量模式的全局低秩性（L）和局部光滑性（S），取得了显著进展，这些先验通常被编码为两个独立正则化项的和并纳入恢复模型。然而，与低秩张量恢复的主要理论进展不同，这类L+S联合模型目前尚无理论上的精确恢复保证，导致这些方法在实际应用中缺乏可靠性。针对这一关键问题，本研究构建了一个独特的正则化项，其本质上可同时编码张量的L和S两种先验。特别地，通过将这一单个正则化项引入恢复模型，我们能够严格证明两类典型张量恢复任务——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，这应是所有相关L+S张量恢复方法中首个精确恢复结果。在大量实验中，我们针对多种视觉张量数据开展了多项TC和TRPCA任务，结果表明所提方法相比众多其他SOTA（当前最优）方法在恢复精度上有显著提升。典型地，在彩色图像修复任务中，当缺失率极高（例如99.5%）时，我们的方法仍能取得可用的性能，而所有同类方法在这种极具挑战性的场景下均完全失效。
英文摘要：
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.
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