Hierarchical Tensor Ring Completion

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.