  Tensor completion can estimate missing values of a high-order data from its
partially observed entries. Recent works show that low rank tensor ring
approximation is one of the most powerful tools to solve tensor completion
problem. However, existing algorithms need predefined tensor ring rank which
may be hard to determine in practice. To address the issue, we propose a
hierarchical tensor ring decomposition for more compact representation. We use
the standard tensor ring to decompose a tensor into several 3-order sub-tensors
in the first layer, and each sub-tensor is further factorized by tensor
singular value decomposition (t-SVD) in the second layer. In the low rank
tensor completion based on the proposed decomposition, the zero elements in the
3-order core tensor are pruned in the second layer, which helps to
automatically determinate the tensor ring rank. To further enhance the recovery
performance, we use total variation to exploit the locally piece-wise
smoothness data structure. The alternating direction method of multiplier can
divide the optimization model into several subproblems, and each one can be
solved efficiently. Numerical experiments on color images and hyperspectral
images demonstrate that the proposed algorithm outperforms state-of-the-arts
ones in terms of recovery accuracy.
