Nested Fully-Connected Tensor Network Decomposition for Multi-Dimensional Visual Data Recovery

Recently, fully-connected tensor network (FCTN) decomposition, which factorizes the target tensor into a series of interconnected factor tensors, has drawn growing focus on multi-dimensional visual data processing. However, the lack of clear physical interpretation for the factor tensors hinders us from introducing handcrafted regularizers to deeply explore the potential of FCTN decomposition. To tackle this issue, we suggest a unimode hierarchical nonlinear (UHN) decomposition for each factor tensor, which can adaptively capture the complex nonlinear structure and implicitly regularize factor tensors. With this UHN decomposition of the factor tensors, we naturally propose a nested fully-connected tensor network (N-FCTN) decomposition. Attributed to the adaptive and implicit regularization inherent in UHN decomposition of factor tensors, the proposed N-FCTN decomposition is expected to perform favorably against the original FCTN decomposition. Based on the proposed N-FCTN decomposition, we build a multi-dimensional visual data recovery model and provide a theoretical error bound between the recovered tensor by our model and the underlying tensor. To address the resulting non-convex and nonlinear optimization problem, we develop an efficient proximal alternating minimization (PAM)-based algorithm and establish its theoretical convergence guarantee. Extensive experimental results on multi-spectral images, color videos, and light field data demonstrate the superior recovery performance of the proposed method compared to the state-of-the-art methods.