Tensor decompositions (TDs) have demonstrated significant potential across various domains of science and engineering. Despite its thorough examination in quantum physics, the projected entangled pair state (PEPS) tensor network has not been extensively explored in the field of tensor completion (TC). In this study, we introduce an innovative PEPS tensor network decomposition algorithm that transforms an Nth-order tensor into a PEPS representation through \(N-\) 1 sequential singular value decompositions (SVDs), considering the multilayered architecture of the PEPS tensor network. The accuracy of this algorithm can be finely tuned by adjusting the rank of the PEPS model. We apply the PEPS tensor network decomposition to address high-order TC challenges and develop an efficient proximal alternating minimization-based optimization algorithm to uncover latent factors within incomplete tensors, thereby filling missing entries. Comparative experiments on synthetic data and color image completion validate the efficacy of our proposed approaches, demonstrating significant improvements over existing methods based on CANDECOMP/PARAFAC (CP), Tucker, tensor train (TT), and tensor ring (TR) decompositions. The code and associated datasets are available at: https://github.com/rfhuang211/PEPS-TC.
