Tensor-on-vector regression with interactions with application to fMRI data

Medical imaging serves as a vital tool for visualizing anatomical structures and physiological characteristics. Understanding the relationship between clinical variables and anatomical structures represents a crucial research direction in medical science. A popular approach involves tensor regression modeling, where medical images are treated as multidimensional responses while clinical variables serve as predictors. When analyzing the relationship between tensor response and vector predictor, few studies account for interactions. In this study we focus on a main effects and interactions tensor regression model, which takes tensors as response and vectors as covariates. The aim is to estimate the unknown tensor coefficient under strong hierarchy assumption. We propose a tensor on vector with interaction estimator (TOVI) and develop an alternating iterative algorithm to solve the resulting optimization problem. Statistical properties of the proposed estimator have been established. Simulations show TOVI outperforms multiple alternatives. The analysis of the Autism Brain Imaging Data Exchange (ABIDE) data on autism spectrum disorder (ASD) demonstrates the superiority of TOVI.