  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 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 $\beta^0$ 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.
