l₁-Norm Quantile Regression Screening Rule via the Dual Circumscribed Sphere.

Abstract

l₁-norm quantile regression is a common choice if there exists outlier or heavy-tailed error in high-dimensional data sets. However, it is computationally expensive to solve this problem when the feature size of data is ultra high. As far as we know, existing screening rules can not speed up the computation of the l₁-norm quantile regression, which dues to the non-differentiability of the quantile function/pinball loss. In this paper, we introduce the dual circumscribed sphere technique and propose a novel l₁-norm quantile regression screening rule. Our rule is expressed as the closed-form function of given data and eliminates inactive features with a low computational cost. Numerical experiments on some simulation and real data sets show that this screening rule can be used to eliminate almost all inactive features. Moreover, this rule can help to reduce up to 23 times of computational time, compared with the computation without our screening rule.