Low-rank tensor ring learning for multi-linear regression

The emergence of large-scale data demands new regression models with multi-dimensional coefficient arrays, known as tensor regression models. The recently proposed tensor ring decomposition has interesting properties of enhanced representation and compression capability, cyclic permutation invariance and balanced tensor ring rank, which may lead to efficient computation and fewer parameters in regression problems. In this paper, a generally multi-linear tensor-on-tensor regression model is proposed that the coefficient array has a low-rank tensor ring structure, which is termed tensor ring ridge regression (TRRR). Two optimization models are developed for the TRRR problem and solved by different algorithms: the tensor factorization based one is solved by alternating least squares algorithm, and accelerated by a fast network contraction, while the rank minimization based one is addressed by the alternating direction method of multipliers algorithm. Comparative experiments, including Spatio-temporal forecasting tasks and 3D reconstruction of human motion capture data from its temporally synchronized video sequences, demonstrate the enhanced performance of our algorithms over existing state-of-the-art ones, especially in terms of training time.