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

TL;DR

Lasso-Zero is a novel high-dimensional support recovery method that overfits with noise dictionaries and thresholding, achieving strong theoretical guarantees and competitive empirical performance.

Contribution

It introduces a new overfit-then-threshold approach with noise dictionaries and a universal threshold, improving support recovery in high-dimensional linear models.

Findings

Lasso-Zero outperforms competitors in support recovery.

It achieves sign consistency under weaker conditions than Lasso.

Noise dictionaries enhance performance for low signals.

Abstract

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.