Robust Lasso‐Zero for sparse corruption and model selection with missing covariates

We propose Robust Lasso‐Zero, an extension of the Lasso‐Zero methodology, initially introduced for sparse linear models, to the sparse corruptions problem. We give theoretical guarantees on the sign recovery of the parameters for a slightly simplified version of the estimator, called Thresholded Justice Pursuit. The use of Robust Lasso‐Zero is showcased for variable selection with missing values in the covariates. In addition to not requiring the specification of a model for the covariates, nor estimating their covariance matrix or the noise variance, the method has the great advantage of handling missing not‐at random values without specifying a parametric model. Numerical experiments and a medical application underline the relevance of Robust Lasso‐Zero in such a context with few available competitors. The method is easy to use and implemented in the R library lass0.