Low-Rank, High-Order Tensor Completion via $t$- Product-Induced Tucker (tTucker) Decomposition Unavailable

Recently, tensor singular value decomposition (⁠$t$-SVD)–based methods were proposed to solve the low-rank tensor completion (LRTC) problem, which has achieved unprecedented success on image and video inpainting tasks. The $t$-SVD is limited to process third-order tensors. When faced with higher-order tensors, it reshapes them into third-order tensors, leading to the destruction of interdimensional correlations. To address this limitation, this letter introduces a $t$productinduced Tucker decomposition (tTucker) model that replaces the mode product in Tucker decomposition with $t$-product, which jointly extends the ideas of $t$-SVD and high-order SVD. This letter defines the rank of the tTucker decomposition and presents an LRTC model that minimizes the induced Schatten-p norm. An efficient alternating direction multiplier method (ADMM) algorithm is developed to optimize the proposed LRTC model, and its effectiveness is demonstrated through experiments conducted on both synthetic and real data sets, showcasing excellent performance.