High-Dimensional Tensor Regression With Oracle Properties

Tensor regression has emerged as a powerful framework for modeling linear relationships among multi-dimensional variables by effectively capturing inherent cross-mode interactions within tensor-structured data. In this paper, we introduce a high-dimensional tensor-response tensor regression model under low-dimensional structural assumptions, such as sparsity and low-rankness. Specifically, we assume that the underlying regression tensor lies within an unknown lowdimensional subspace and propose a general least squares estimation framework with non-convex penalties. Theoretically, we establish rigorous risk bounds for the resulting estimators, demonstrating that they attain the oracle statistical rates under mild technical conditions. To ensure computational efficiency, we introduce a proximal gradient algorithm for solving the proposed non-convex optimization problem. Extensive experiments conducted on both synthetic and real-world datasets validate the effectiveness of the proposed regression model and showcase the practical utility of the theoretical findings.