Smooth Compact Tensor Ring Regression

Abstract

In learning tasks with high order correlations, the low-rank approximation of the regression coefficient tensor has become increasingly important. Tensor ring can capture more correlation information among tensor networks. However, its optimal rank is generally unknown and needs to be tuned from multiple combinations. To address the issue, we propose a novel tensor regression framework with a group sparsity constraint on latent factors for tensor ring rank estimation. Specifically, the proposed group sparsity term constrained matrix factorization problem is first shown to be equivalent to a better approximation of matrix rank, namely Schatten- <inline-formula><tex-math notation="LaTeX">$1/2$</tex-math></inline-formula> quasi-norm. Extending it into tensor, the tensor ring rank can be inferred during the learning process to balance the prediction error and the model complexity. Besides, a total variation term is introduced to enhance the local consistency of the predicted response, which is useful for reducing the adverse effects of random noise. Experiments on the simulation dataset show that the proposed method can exactly obtain the tensor ring rank, and the effectiveness and robustness of the proposed algorithm is further verified on a real dataset for human motion capture tasks.