Partial Quantile Tensor Regression

Tensors, characterized as multidimensional arrays, are frequently encountered in modern scientific studies. Quantile regression has the unique capacity to explore how a tensor covariate influences different segments of the response distribution. In this work, we propose a partial quantile tensor regression (PQTR) framework, which novelly applies the core principle of the partial least squares technique to achieve effective dimension reduction for quantile regression with a tensor covariate. The proposed PQTR algorithm is computationally efficient and scalable to a large tensor covariate. Moreover, we uncover an appealing latent variable model representation for the PQTR algorithm, justifying a simple population interpretation of the resulting estimator. We further investigate the connection of the PQTR procedure with an envelope quantile tensor regression (EQTR) model, which defines a general set of sparsity conditions tailored to quantile tensor regression. We prove the root- $n$ consistency of the PQTR estimator under the EQTR model, and demonstrate its superior finite-sample performance compared to benchmark methods through simulation studies. We demonstrate the practical utility of the proposed method via an application to a neuroimaging study of post traumatic stress disorder (PTSD). Results derived from the proposed method are more neurobiologically meaningful and interpretable as compared to those from existing methods.