Tensor Quantile Regression With Exponential‐Type Penalty

ABSTRACT Tensor‐valued covariates are increasingly common in multiway measurements, but traditional vector‐valued regression ignores their inherent structure and leads to fragility. In practice, such data are frequently contaminated by heavy‐tailed errors and outliers, and least square estimators lack robustness. In this paper, we propose a robust tensor quantile regression method, in which CANDECOMP/PARAFAC (CP) decomposition is employed for dimension reduction, and an exponential‐type penalty (ETP) is imposed at the element‐wise level to achieve sparse variable selection. The ETP smoothly interpolates between and , reducing bias for large coefficients while preserving computational tractability. We develop an efficient algorithm based on alternating direction method of multipliers (ADMM) framework to solve the ETP‐penalized tensor quantile regression estimator. Theoretically, we address the identifiability of the CP decomposition and establish asymptotic properties, including the estimation consistency and the oracle property. Extensive simulation studies under two representative sparse signal settings show that the proposed method substantially improves signal recovery, estimation accuracy, and predictive performance over quantile regression on vectorized covariates and existing tensor regression methods. An empirical analysis of the Beijing dataset further demonstrates superior predictive performance of the proposed method and reveals pronounced spatial and quantile heterogeneity in the effects of major air pollutants.