Low‐Rank Variational Correction Estimation for Multi‐Source Heterogeneous Quantile Linear Regression Models

High-dimensional data arising in genomics, econometrics, and clinical medicine often exhibit substantial heterogeneity across multiple sources. While existing methods address multi-source heterogeneity, they do not adequately accommodate the combined challenges of high dimensionality and between-source heterogeneity. To address this gap, we propose a scalable Bayesian framework for multi-source heterogeneous quantile regression with spike-and-slab priors for simultaneous parameter estimation and feature selection. To overcome computational challenges, we combine mean-field variational inference with Laplace approximation and introduce a novel low-rank variational correction strategy that substantially improves approximation accuracy and adaptability in high-dimensional heterogeneous settings. This low-rank correction effectively captures the underlying dependence structure, leading to more robust and efficient inference. For model assessment and diagnostic analysis, we further develop a Bayesian score test coupled with local influence analysis. Extensive simulation studies and an analysis of The Cancer Genome Atlas (TCGA) data from four cancer cohorts (ESCA, PAAD, PCPG, and READ) demonstrate the computational efficiency, scalability, and practical utility of the proposed method in high-dimensional heterogeneous applications. The proposed low-rank variational correction algorithms are implemented in the R package LRQVB, which is publicly available on CRAN.