Tensor Wheel Decomposition: Theory and Application to Tensor Completion

Recently, tensor network (TN) decompositions have gained prominence in computer vision and contributed promising results to tensor recovery for their capability of compactly and efficiently representing high-order tensors. However, current TN topologies are rather being developed towards more intricate structures to pursue incremental improvements, resulting in a drastically increased number of TN ranks, which requires laborious hyper-parameter selection, especially for higher-order cases. In this paper, we propose a novel TN decomposition, dubbed tensor wheel (TW) decomposition, in which a high-order tensor is represented by a set of latent factors mapped into a specific wheel topology. Such a decomposition is constructed starting from analyzing the graph structure, aiming to more accurately characterize the complex interactions inside objectives while maintaining a lower hyper-parameter scale, theoretically alleviating the above deficiencies. The comprehensive analysis of the mathematical properties fully demonstrates that TW decomposition can be more potential in representation capabilities and more flexible in controlling both parameter storage and computational costs. To compute the TW-format decomposition, the sequential singular value decomposition (SVD)-based and the alternating least squares (ALS)-based learning algorithms are developed. Furthermore, to investigate the validity of TW decomposition, we provide its one numerical application, i.e., tensor completion (TC), yet develop an efficient proximal alternating minimization-based solving algorithm with guaranteed convergence. Experimental results on both synthetic and real-world data reveal that TW decomposition significantly outperforms other state-of-the-art tensor decompositions for incomplete-tensor inference, especially under solely few observations, thus substantiating the superiority and reliability of TW decomposition.