Empowering Tensor Network Decompositions by Nonlinear and Incompatible Tensor Contraction

Abstract. Recently, tensor network decompositions have become powerful tools in imaging science for representing high-dimensional images, with tensor contraction serving as a fundamental interaction between tensor factors. However, the potential of classical tensor contraction is limited by its linear and dimension-compatible nature. To break this limitation, we propose a nonlinear and incompatible tensor contraction as an extension of the classical tensor contraction. The proposed tensor contraction introduces a nonlinear interaction operator that projects tensor factors onto the same intrinsic dimension for interaction, which not only allows nonlinear interactions but also frees them from the dimensional compatibility constraint. Equipped with the suggested tensor contraction, we revisit tensor network decompositions (such as tensor ring and fully connected tensor decompositions), allowing us to flexibly discover new and potential tensor network decompositions. To demonstrate the potential of the proposed new decompositions, we develop corresponding low-rank tensor completion models based on them. Moreover, we establish generalization error bounds between the tensor recovered by our model and the underlying tensor, providing a solid theoretical foundation for their superiority compared with classical tensor decomposition models. Extensive experiments on both synthetic and real-world data further demonstrate the numerical superiority of our method over state-of-the-art baselines.