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

arXiv: https://arxiv.org/abs/1805.05133 · HTML: https://mlsanity.com/paper/1805.05133/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 May 2018.

## Built on by (most cited first, 2 of 2)

- [A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks](https://mlsanity.com/paper/2201.08652/) (2022)
- [Validation-Free Sparse Learning: A Phase Transition Approach to Feature Selection](https://mlsanity.com/paper/2411.17180/) (2024)

## Abstract (arXiv)

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.