High-Dimensional Tensor Regression With Oracle Properties

The emergence of multi-dimensional data presents significant challenges for traditional regression models based on matrices or vectors, particularly in capturing multi-directional correlations. In response, tensor regression has been proposed as a powerful framework for modeling linear relationships among multi-dimensional variables. In this paper, we introduce a high-dimensional tensor-response tensor regression model under low-dimensional structural assumptions, such as sparsity and low-rankness. Assuming the underlying tensor lies within an unknown low-dimensional subspace, we consider a least squares estimation framework with non-convex penalties. Theoretically, we derive general risk bounds for the resulting estimators and demonstrate that they achieve the oracle statistical rates under mild technical conditions. To compute the proposed estimators efficiently, we introduce an accelerated proximal gradient algorithm demonstrating rapid convergence in practice. Extensive experiments on synthetic and real-world datasets validate the effectiveness of the proposed regression model and showcase the practical utility of the theoretical findings.