Preconditioning the Lasso for sign consistency

Sign consistency of the Lasso requires the stringent irrepresentable condition. This paper examines whether preconditioning can circumvent this condition. Let Xn× p and Yⁿ satisfy the standard linear regression equation. Instead of computing the Lasso with (X,Y), preconditioning first left multiplies by Fn× n and then computes the Lasso with (FX,FY). While others have proposed preconditioning for other purposes, we provide the first results that show FX can satisfy the irrepresentable condition even when X fails to satisfy the condition. Preconditioning the Lasso creates a new estimator that is sign consistent in a wider variety of settings. Importantly, left multiplying the regression equation by F does not change β, the vector of unknown coefficients. However, left multiplying this equation by F often inflates the variance of the errors. We propose a class of preconditioners to balance these costs and benefits.