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Ultra-high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of non-location-scale covariate effects. To accommodate heterogeneity, we advocate a more general interpretation of sparsity, which assumes that only a small number of covariates influence the conditional distribution of the response variable, given all candidate covariates; however, the sets of relevant covariates may differ when we consider different segments of the conditional distribution. In this framework, we investigate the methodology and theory of nonconvex, penalized quantile regression in ultra-high dimension. The proposed approach has two distinctive features: (1) It enables us to explore the entire conditional distribution of the response variable, given the ultra-high-dimensional covariates, and provides a more realistic picture of the sparsity pattern; (2) it requires substantially weaker conditions …