Generalized Tensor Regression with Internal Variation Regularization

Analyzing clinical diagnosis along with high-dimensional imaging data, while accounting for the piecewise constant nature of the imaging, presents challenges to existing statistical approaches.In this paper, we propose a generalized tensor regression framework with Internal Variation (IV) regularization to address these challenges.The inclusion of IV regularization allows for the explicit utilization of the rich spatial structure, particularly the piecewise constant nature of high-order imaging data, albeit with a more complex algorithm and demanding theoretical investigation.We develop an efficient IV regularized optimization procedure for estimating unknown scalar and tensor coefficients.We investigate the theoretical properties of scalar and tensor coefficient estimates, especially the error bounds of regularized tensor coefficient estimates.Extensive numerical studies assess the finite sample performance of our method, demonstrating its superiority over existing approaches.Finally, we apply the proposed method to a chronic sinusitis computed tomography (CT) imaging dataset and identify the most activated subregion across the maxillary sinus cavity associated with the diagnosis.