Pascaline Descloux

Pascaline Descloux, Claire Boyer, Julie Josse et al.

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.

Pascaline Descloux, Sylvain Sardy

The high-dimensional linear model $y = X β^0 + ε$ is considered and the focus is put on the problem of recovering the support $S^0$ of the sparse vector $β^0.$ We introduce Lasso-Zero, a new $\ell_1$-based estimator whose novelty resides in an "overfit, then threshold" paradigm and the use of noise dictionaries concatenated to $X$ for overfitting the response. To select the threshold, we employ the quantile universal threshold based on a pivotal statistic that requires neither knowledge nor preliminary estimation of the noise level. Numerical simulations show that Lasso-Zero performs well in terms of support recovery and provides an excellent trade-off between high true positive rate and low false discovery rate compared to competitors. Our methodology is supported by theoretical results showing that when no noise dictionary is used, Lasso-Zero recovers the signs of $β^0$ under weaker conditions on $X$ and $S^0$ than the Lasso and achieves sign consistency for correlated Gaussian designs. The use of noise dictionary improves the procedure for low signals.