Asymptotic analysis of robust LASSOs in the presence of noise with large variance

In the context of linear regression, the least absolute shrinkage and selection operator (LASSO) is probably the most popular supervised-learning technique proposed to recover sparse signals from high-dimensional measurements. Prior literature has mainly concerned itself with independent, identically distributed noise with moderate variance. In many real applications, however, the measurement errors may have heavy-tailed distributions or suffer from severe outliers, making the LASSO poorly estimate the coefficients due to its sensitivity to large error variance. To address this concern, a robust version of the LASSO is proposed, and the limiting distribution of its estimator is derived. Model selection consistency is established for the proposed robust LASSO under an adaptation procedure of the penalty weight. A parallel asymptotic analysis is derived for the Huberized LASSO, a previously proposed robust LASSO, and it is shown that the Huberized LASSO estimator preserves similar asymptotics even with a Cauchy error distribution. We show that asymptotic variances of the two robust LASSO estimators are stabilized in the presence of large variance noise, compared with the unbounded asymptotic variance of the ordinary LASSO estimator. The asymptotic analysis from the nonstochastic design is extended to the case of random design. Simulations further confirm our theoretical results.