Efficient FCTN Decomposition with Structural Sparsity for Noisy Tensor Completion

Recently, the fully-connected tensor network (FCTN) decomposition has shown a powerful capability of depicting intrinsic correlations between any pair of tensor modes. But there exists a challenging question in FCTN decomposition-based methods, i.e., the accurate determination of the complicated FCTN-rank, which contains ${N(N-1)}/{2}$ elements for $N$th-order tensors. In this paper, we design a structural sparsity regularization for the FCTN decomposition, which estimates the complicated FCTN-rank by adaptively pruning near-zero groups in FCTN factor. Based on this regularization, we propose a noisy tensor completion (NTC) model, aiming at the recovery of a tensor from its partial and noisy observation. Besides, we design a proximal alternating minimization (PAM)-based algorithm to solve the model. In theorem, we prove a guarantee for the global convergence of the developed algorithm. To further accelerate our method for large-scale data sets, we customize the randomized block sampling strategy for general tensor network decomposition methods by updating factors from small samples. Experiments demonstrate that our strategy can accurately estimate the FCTN-rank and achieve better reconstruction performances, and our methods outperform the state-of-the-art methods in the reconstruction of different types of real-world tensors.