Block Tensor Ring Decomposition: Theory and Application

Recently, tensor decompositions have received significant attention for processing multi-dimensional signals, especially representative by the block-term decomposition (BTD) family and the tensor network decomposition (TND) family. However, these two families have long been isolated from each other, with their respective wisdom neither inspiring nor benefiting each other. To address this dilemma, we propose a block tensor ring decomposition (BTRD), which decomposes an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>th-order tensor into a sum of outer products between basic vector factors and the (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> − 1)th-order coefficient tensors, which are further represented using a tensor ring. The benefit of the BTRD is that it can better explore outer multiple components structure of the tensor and inner tensor topology of each component. To examine the potential of the proposed BTRD, we apply it to a low-rank tensor completion model as a representative task and prove a theoretical generalization error bound which provides a theoretical perspective to support the advantages of the proposed model for higher-order tensors. To address the resulting optimization problem, we apply an efficient proximal alternating minimization (PAM)-based algorithm with a theoretical convergence guarantee. Extensive experimental results on real-world signal data (color videos and light field images) demonstrate the superiority of the proposed model against the state-of-the-art baseline models.