Tensor Additive Quantile Regression

Additive nonparametric models are increasingly favored for analyzing tensor data, offering a flexible and parsimonious approach. This paper introduces a tensor additive quantile regression model, aiming to provide a more robust and detailed understanding of how covariates influence the response in tensor data. The component functions are estimated using basis function approximations. We stack the splines as an additional tensor dimension, which allows us to leverage the tensor structure and apply Tucker decomposition for dimension reduction. In high-dimensional settings, we propose a sparse tensor additive quantile regression model that incorporates a group penalty for variable selection. A key challenge addressed is connecting sparse tensor elements within the structure of Tucker decomposition. We propose an innovative approach that identifies a broader set of relevant features than the oracle, while enabling efficient algorithms to navigate the high-dimensional space. We establish the large sample properties of the proposed estimators, evaluate the finite sample performance of our method through Monte Carlo simulations and demonstrate its application on real-world datasets, including stock market and head pose image data.