Year: 2018
Type: preprint
Abstract: The high-dimensional linear model $y = X β^0 + ε$ is considered and the focus is put on the problem of recovering the support $S^0$ of the sparse vector $β^0.$ 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 $β^0$ 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.
Source: arXiv (Cornell University)
Authors: Descloux, Pascaline, Sardy, Sylvain
Institution: University of Geneva
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Université de Genève
OpenAlex ID: W4299790272