Composite Quantile Tensor Regression

This paper introduces a composite quantile tensor regression (CQTR) model, a regression technique capable of modeling the relationship between scalar-valued response and tensor-valued covariate. By imposing a low-rank assumption on the coefficient tensor, this paper proposes an estimator that uses a composite quantile loss function at the component-wise level to estimate the coefficient tensor, while applying a nonconvex penalty function at the element-wise level to encourage the sparsity. As a result, CQTR is not only a robust method for handling outliers and heavy-tailed data, but also effectively identifies active regions of tensor-valued covariate that have a significant impact on the response. We demonstrate that the CQTR estimator possesses both consistency and oracle properties. Extensive simulation studies show that the method performs well in coefficient tensor estimation, active region identification, response prediction, and computational efficiency. The relevance of this method is further highlighted in practical applications. Applied to a neuroimaging analysis task, the proposed method demonstrates excellent practical effectiveness.