Sparse Quantile Huber Regression for Efficient and Robust Estimation
This work addresses outlier-robust variable selection in fields like genomics, but it is incremental as it builds on existing methods with new formulations and algorithms.
The paper tackles sparse quantile regression for high-dimensional data, proposing a generalized OMP algorithm and convex formulations with quantile or quantile Huber loss, and demonstrates performance through theoretical guarantees and empirical studies on simulated and genomic datasets.
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.