2011/07/01 by Sara van de Geer, van de Geer, Sara, Johannes Lederer +1
Mathematics · Medicine · #62J05 #Birth, Development, and Health #FOS: Computer and information sciences #Liver Disease Diagnosis and Treatment #Methodology (stat.ME) #Statistical Methods and Inference #msc:62J05 #stat.ME
paper · pdf · doi:10.48550/arxiv.1107.0189
18 pages, 3 figures
arxiv created 2011/07/01 · openalex publication_date 2011/07/01 · arxiv updated 2011/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study high-dimensional linear models and the ℓ1-penalized least squares estimator, also known as the Lasso estimator. In literature, oracle inequalities have been derived under restricted eigenvalue or compatibility conditions. In this paper, we complement this with entropy conditions which allow one to improve the dual norm bound, and demonstrate how this leads to new oracle inequalities. The new oracle inequalities show that a smaller choice for the tuning parameter and a trade-off between ℓ1-norms and small compatibility constants are possible. This implies, in particular for correlated design, improved bounds for the prediction error of the Lasso estimator as compared to the methods based on restricted eigenvalue or compatibility conditions only.