Inconsistent tensor-tensor product for low-rank tensor factorization

The tensor-tensor product (t-product) is a fundamental operation in tensor decomposition, enabling effective modeling of interactions between third-order tensors. However, the classical t-product is restricted by the fact that the two factors must have the same third-mode dimension, limiting its flexibility and expressiveness. To break this restriction, we introduce an inconsistent tensor-tensor product (it-product), which allows tensors with inconsistent third-mode dimensions to interact with each other while still respecting the algebraic structure of classical t-product. Equipped with the proposed it-product, we develop an it-product-based low-rank tensor factorization and suggest a unified model for tensor completion and tensor compression. To address the resulting nonconvex optimization problem, we build a proximal alternating minimization (PAM)-based algorithm. We further provide a theoretical convergence analysis, showing that the sequence generated by the algorithm converges to a critical point of the objective function under certain conditions. Numerical experiments on real-world datasets have been conducted to validate the effectiveness and superiority of the proposed method over existing baselines.