  We consider new formulations and methods for sparse quantile regression in
the high-dimensional setting. Quantile regression plays an important role in
many applications, including outlier-robust exploratory analysis in gene
selection. In addition, the sparsity consideration in quantile regression
enables the exploration of the entire conditional distribution of the response
variable given the predictors and therefore yields a more comprehensive view of
the important predictors. We propose a generalized OMP algorithm for variable
selection, taking the misfit loss to be either the traditional quantile loss or
a smooth version we call quantile Huber, and compare the resulting greedy
approaches with convex sparsity-regularized formulations. We apply a recently
proposed interior point methodology to efficiently solve all convex
formulations as well as convex subproblems in the generalized OMP setting, pro-
vide theoretical guarantees of consistent estimation, and demonstrate the
performance of our approach using empirical studies of simulated and genomic
datasets.
