Abstract: Variable selection is central to sparse modeling, and many methods have been proposed under various model assumptions. Most existing methods are based on an explicit functional relationship, while we are concerned with a model-free variable selection method that attempts to identify informative variables that are related to the response by simultaneously examining the sparsity in multiple conditional quantile functions. It does not require specification of the underlying model for the response. The proposed method is implemented via an efficient computing algorithm that couples the majorize-minimization algorithm and the proximal gradient descent algorithm. Its asymptotic estimation and variable selection consistencies are established, without explicit model assumptions, that assure the truly informative variables are correctly identified with high probability. The effectiveness of the proposed method is supported by a variety of simulated and real-life examples.

Key words and phrases: Lasso, learning gradients, quantile regression, reproducing kernel Hilbert space (RKHS), sparsity, variable selection.