Efficient Tensor Completion Methods for 5-D Seismic Data Reconstruction: Low-Rank Tensor Train and Tensor Ring

Five-dimensional seismic reconstruction is receiving increasing attention and can be viewed as a tensor completion problem, which involves reconstructing a low-rank tensor from a partially observed tensor. Tensor train (TT) decomposition and tensor ring (TR) decomposition are two powerful tensor networks for solving this problem. However, updating core tensors leads to high computational costs in practical applications. We propose two efficient methods to exploit low TT rank and low TR rank structures by theoretically establishing the relationship between tensor ranks and matrix unfoldings, respectively. Specifically, the former uses a well-balanced matricization scheme, and the latter uses a tensor circular unfolding. Furthermore, we use the randomized parallel matrix factorization (PMF) to accelerate the solution of these problems. Both synthetic and real data experiments demonstrate that the proposed algorithm can also achieve remarkable reconstruction performance; in the meantime, the computational cost is significantly reduced.