Papers › Model selection with lasso-zero: adding straw to the haystack to better find needles
Model selection with lasso-zero: adding straw to the haystack to better find needles
Pascaline Descloux, Sylvain Sardy
The high-dimensional linear model y = X β⁰ + ϵ is considered and the focus is put on the problem of recovering the support S⁰ of the sparse vector β⁰. We introduce Lasso-Zero, a new ℓ₁-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 β⁰ under weaker conditions on X and S⁰ than the Lasso and achieves sign consistency for correlated Gaussian designs. The use of noise dictionary improves the procedure for low signals.
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