Fully-connected Tensor Network Decomposition for Robust Tensor Completion Problem

Motivated by the success of fully-connected tensor network (FCTN) decomposition, we suggest two FCTN-based models for the robust tensor completion (RTC) problem. Firstly, we propose an FCTN-based robust nonconvex optimization model (RNC-FCTN) directly based on FCTN decomposition for the RTC problem. Then, a proximal alternating minimization (PAM)-based algorithm is developed to solve the proposed RNC-FCTN. Meanwhile, we the-oretically derive the convergence of the PAM-based algorithm. Although the nonconvex model has shown empirically excellent results, the exact recovery guarantee is still missing and N(N − 1)/2 + 1 tuning parameters are difficult to choose for N-th order tensor. Therefore, we propose the FCTN nuclear norm as the convex surrogate function of the FCTN rank and suggest an FCTN nuclear norm-based robust convex optimization model (RC-FCTN) for the RTC problem. For solving the constrained optimization model RC-FCTN, we de-velop an alternating direction method of multipliers (ADMM)-based algorithm, which enjoys the global convergence guarantee. To explore the exact recovery guarantee, we design a constructive singular value decomposition (SVD)-based FCTN decomposition, which is another crucial algorithm to obtain the factor tensors of FCTN decomposition. Accordingly, we rigorously establish the exact recovery guarantee for the RC-FCTN and suggest the theoretical optimal value for the only one parameter in the convex model. Comprehensive numerical experiments in several applications, such as video completion and video background subtraction, demonstrate that the suggested convex and noncon-vex models have achieved state-of-the-art performance.