Rank-revealing fully-connected tensor network decomposition and its application to tensor completion

Fully-connected tensor network (FCTN) decomposition has become a powerful tool for handling high-dimensional data. However, for a given $N$th-order data, $N(N-1)/2$ tuning parameters (i.e., FCTN rank) in FCTN decomposition is a tricky challenge, which hinders its wide deployments. Although many recent works have emerged to adaptively search for a (near)-optimal FCTN rank, these methods suffer from expensive computational costs since they require too many search and evaluation processes, significantly limiting their applications to high-dimensional data. To tackle the above challenges, we develop a rank-revealing FCTN (revealFCTN) decomposition, whose FCTN rank is adaptively and efficiently inferred. More specifically, by analyzing the sizes of the sub-network tensors in the FCTN decomposition, we establish the equivalent relationships between the FCTN rank and the ranks of single-mode and double-mode unfolding matrices of the given data. The FCTN rank can be directly revealed through the ranks of these unfolding matrices, which does not require any search and evaluation process, making the computational cost almost negligible compared to the search-based methods. To evaluate the performance of the developed revealFCTN decomposition, we test its performance on a representative task: tensor completion (TC). Comprehensive experimental results demonstrate that our method outperforms several state-of-the-art methods, achieving a MPSNR gain of around 1 dB in most cases compared to the original FCTN decomposition.