Variational Bayesian Tensor Quantile Regression

Abstract

Quantile regression is widely used in variable relationship research for statistical learning. Traditional quantile regression model is based on vector-valued covariates and can be efficiently estimated via traditional estimation methods. However, many modern applications involve tensor data with the intrinsic tensor structure. Traditional quantile regression can not deal with tensor regression issues well. To this end, we consider a tensor quantile regression with tensor-valued covariates and develop a novel variational Bayesian estimation approach to make estimation and prediction based on the asymmetric Laplace model and the CANDECOMP/PARAFAC decomposition of tensor coefficients. To incorporate the sparsity of tensor coefficients, we consider the multiway shrinkage priors for marginal factor vectors of tensor coefficients. The key idea of the proposed method is to efficiently combine the prior structural information of tensor and utilize the matricization of tensor decomposition to simplify the complexity of tensor coefficient estimation. The coordinate ascent algorithm is employed to optimize variational lower bound. Simulation studies and a real example show the numerical performances of the proposed method.