The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($ \rm{order} >3 $). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with ket augmentation, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completion by simultaneously exploring the spatial, temporal/spectral, and nonlocal redundancy in visual data. More precisely, the TT rank minimization is performed on a formed higher-order tensor called group by stacking similar cubes, which naturally and fully takes advantage of the ability of TT rank for high-order tensors. Moreover, the perturbation analysis for the TT low-rankness of each group is established. We develop the alternating direction method of multipliers tailored for the specific structure to solve the proposed model. Extensive experiments demonstrate that the proposed method is superior to several existing state-of-the-art methods in terms of both qualitative and quantitative measures.

中文翻译：

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具有非局部自相似性的张量列车等级最小化以完成张量

张量列（TT）等级由于能够捕获高阶张量的全局相关性（$ \ rm {order}> 3 $）而在张量完成方面受到越来越多的关注。对于三阶视觉数据，直接TT秩最小化尚未利用TT秩用于高阶张量的潜力。TT等级最小化与ket增加将低阶张量（例如，视觉数据）转换为高阶张量的，会遭受严重的块伪像。为了解决这个问题，我们建议通过同时探索视觉数据中的空间，时间/频谱和非局部冗余，将具有非局部自相似性的TT秩最小化以用于张量完成。更精确地，通过堆叠相似的立方体，在形成的称为组的高阶张量上执行TT秩最小化，这自然而又充分地利用了TT秩用于高阶张量的能力。此外，建立了每组TT低等级的扰动分析。我们开发了针对特定结构量身定制的乘法器交替方向方法，以解决所提出的模型。