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The high-dimensional linear model $y = X \beta^0 + \epsilon$ is consideredand the focus is put on the problem of recovering the support $S^0$ of thesparse vector $\beta^0.$ We introduce Lasso-Zero, a new $\ell_1$-basedestimator whose novelty resides in an "overfit, then threshold" paradigm andthe use of noise dictionaries concatenated to $X$ for overfitting the response.To select the threshold, we employ the quantile universal threshold based on apivotal statistic that requires neither knowledge nor preliminary estimation ofthe noise level. Numerical simulations show that Lasso-Zero performs well interms of support recovery and provides an excellent trade-off between high truepositive rate and low false discovery rate compared to competitors. Ourmethodology is supported by theoretical results showing that when no noisedictionary is used, Lasso-Zero recovers the signs of $\beta^0$ under weakerconditions on $X$ and $S^0$ than the Lasso and achieves sign consistency forcorrelated Gaussian designs. The use of noise dictionary improves the procedurefor low signals.

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