Tensor ring (TR) decomposition has been successfully used to obtain the
state-of-the-art performance in the visual data completion problem. However,
the existing TR-based completion methods are severely non-convex and
computationally demanding. In addition, the determination of the optimal TR
rank is a tough work in practice. To overcome these drawbacks, we first
introduce a class of new tensor nuclear norms by using tensor circular
unfolding. Then we theoretically establish connection between the rank of the
circularly-unfolded matrices and the TR ranks. We also develop an efficient
tensor completion algorithm by minimizing the proposed tensor nuclear norm.
Extensive experimental results demonstrate that our proposed tensor completion
method outperforms the conventional tensor completion methods in the
image/video in-painting problem with striped missing values.

该论文利用张量分解的方法，提出了一种新的张量核范数来解决图像 / 视频修复问题中的矩阵缺失， 并基于该方法提出了有效的张量完成算法。

用张量环核范数最小化实现视觉数据补全

Tensor-Ring Nuclear Norm Minimization and Application for Visual Data  Completion

Using the matrix product state (MPS) representation of tensor train
decompositions, in this paper we propose a tensor completion algorithm which
alternates over the matrices (tensors) in the MPS representation. This
development is motivated in part by the success of matrix completion algorithms
which alternate over the (low-rank) factors. We comment on the computational
complexity of the proposed algorithm and numerically compare it with existing
methods employing low rank tensor train approximation for data completion as
well as several other recently proposed methods. We show that our method is
superior to existing ones for a variety of real settings.

本文提出了一种基于矩阵乘积状态和张量列车分解的张量完成算法，该算法通过交替矩阵（张量）对 MPS 表示进行计算，相对于其他最近提出的方法，在多个真实环境下性能更出色。