Simultaneous analysis of Lasso and Dantzig selector

2008/01/31 by Peter J. Bickel, Ya’acov Ritov, Ya'acov Ritov +1 · 8 citations
Mathematics · #Advanced Causal Inference Techniques #Applied mathematics #Computer science #Estimator #Generalized linear model #Lasso (programming language) #Linear model #Linear regression #Mathematics #Nonparametric regression #Regression analysis #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics #math.ST #stat.TH

paper · pdf · doi:10.1214/08-aos620

published as The Annals of Statistics 2009, Vol. 37, No. 4, 1705-1732 · Noramlization factor corrected

openalex publication_date 2009/06/18 · arxiv created 2010/11/09 · arxiv updated 2010/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the ℓp estimation loss for 1≤p≤2 in the linear model when the number of variables can be much larger than the sample size.

Citations

• Sparsity oracle inequalities for the Lasso
• Pathwise coordinate optimization
• Least angle regression
• The Group Lasso for Logistic Regression
• The sparsity and bias of the Lasso selection in high-dimensional linear regression

Cited by

• From L-domination maximal inequalities to infinite-dimensional martingales, towards high-dimensional statistics
• Collaborative Inference for Sparse High-Dimensional Models with Non-Shared Data
• Adaptive Sparse Group Lasso Penalized Quantile Regression via Dual ADMM
• Debiasing the Lasso under Weaker Tail Assumptions
• Causal machine learning for high-dimensional mediation analysis using interventional effects mapped to a target trial
• A Survey of Estimation Methods for Sparse High-dimensional Time Series\n Models
• Fine-Gray competing risks model with high-dimensional covariates: estimation and Inference
• Robust Elastic Net Regression

Related

• Parallelism, Uniqueness, and Large-Sample Asymptotics for the Dantzig Selector
• A Remark on the Lasso and the Dantzig Selector
• The Dantzig selector: Statistical estimation when p is much larger than n
• High-dimensional stochastic optimization with the generalized Dantzig estimator
• Transductive versions of the LASSO and the Dantzig Selector