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

We propose Robust Lasso-Zero, an extension of the Lasso-Zero methodology [Descloux and Sardy, 2018], 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.

Model selection with lasso-zero: adding straw to the haystack to better find needles

The high-dimensional linear model $y = X \beta^0 + \epsilon$ is considered and the focus is put on the problem of recovering the support $S^0$ of the sparse vector $\beta^0.$ 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 for overfitting the response. The methodology is supported by theoretical results obtained in the special case where no noise dictionary is used. In this case, lasso-zero boils down to thresholding the basis pursuit solution. We prove that this procedure requires weaker conditions on $X$ and $S^0$ than the lasso for exact support recovery, and controls the false discovery rate for orthonormal designs when tuned by the quantile universal threshold. However it requires a high signal-to-noise ratio, and the use of noise dictionaries addresses this issue. The threshold selection procedure is based on a pivotal statistic and does not require knowledge of the noise level. Numerical simulations show that lasso-zero performs well in terms of support recovery and provides a good trade-off between high true positive rate and low false discovery rate compared to competitors.