Abstract

The high-dimensional linear model $y=X{\beta }^0+ϵ$ is considered and the focus is put on the problem of recovering the support S⁰ of the sparse vector ${\beta }^0.$ We introduce a new ${\mathcal{l}}_1$-based estimator, called Lasso-Zero, whose novelty resides in the repeated use of noise dictionaries concatenated to X for overfitting the response. Lasso-Zero is an extension of thresholded basis pursuit (TBP), for which we prove sign consistency for correlated Gaussian designs. Both theoretical and empirical results motivate the use of noise dictionaries to improve TBP when the coefficients’ amplitude is low. To select the threshold of Lasso-Zero, we suggest to employ a pivotal version of the quantile universal threshold (QUT) that exploits a byproduct of Lasso-Zero to avoid the need to estimate the noise level. Numerical simulations show that Lasso-Zero tuned by QUT performs well in terms of support recovery and provides an excellent trade-off between high power and few false discoveries compared to competitors. Supplemental materials for this article are available online.

摘要

高维线性模型 $是=X{\beta }^0+\epsilon$考虑，重点放在恢复稀疏向量的支持度S ⁰的问题上${\beta }^0.$ 我们推出一个新的 ${\mathcal{升}}_1$基于估计器，称为Lasso-Zero，其新颖之处在于重复使用连接到X的噪声字典过度拟合响应。Lasso-Zero 是阈值基追踪 (TBP) 的扩展，为此我们证明了相关高斯设计的符号一致性。当系数幅度较低时，理论和经验结果都促使使用噪声词典来改善 TBP。为了选择 Lasso-Zero 的阈值，我们建议采用分位数通用阈值 (QUT) 的关键版本，该版本利用 Lasso-Zero 的副产品来避免估计噪声水平的需要。数值模拟表明，与竞争对手相比，QUT 调整的 Lasso-Zero 在支持恢复方面表现良好，并在高功率和较少错误发现之间提供了极好的权衡。本文的补充材料可在线获取。