  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.
