On the asymptotic properties of the group lasso estimator for linear models

By Yuval Nardi, Alessandro Rinaldo. 2008

210 citations recorded by the source corpus.

We establish estimation and model selection consistency, pre- diction and estimation boundsand persistencefor the group-lassoestimator and model selectorproposed by Yuan and Lin (2006) for least squares prob- lems when the covariates have a natural grouping structure. We consider the case of a fixed-dimensionalparameter space with increasing sample size and the double asymptotic scenario where the model complexity changes with the sample size.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3aebbdeda5b0547e9e41fbf1b73aae308bf8033d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Francis R. Bach
2. Tong Zhang
3. Michael I. Jordan
4. Martin J. Wainwright
5. Yoshua Bengio
6. Marius Kloft
7. Peter Richtárik
8. Simon Lacoste-Julien

In Statistics and Probability

1. Francis R. Bach
2. Martin J. Wainwright
3. Sara A. van de Geer
4. Cun-Hui Zhang
5. Kengo Kato
6. Victor Chernozhukov
7. Tong Zhang
8. Bin Yu

Citations in the corpus

• Structured Compressed Sensing: From Theory to Applications (2011, methodology): “To write down the problem explicitly, we define the mixed norm as (48) We may then recover by solving [17], [134], [136], [137] subject to (49) The optimization constraints can be relaxed to address the case of noisy measurements, in a way similar to the standard BPIC algorithm.”
• Structured Variable Selection with Sparsity-Inducing Norms (2009, methodology): “Following Bach (2008b) and Nardi and Rinaldo (2008), we consider the reduced problem on J , with solution ˆ w J , which can be extended to J c with zeros.”
• Taking Advantage of Sparsity in Multi-Task Learning (2009, methodology): “Discussion of the Group Lasso in a relatively general setting is given by Bach [2] and Nardi and Rinaldo [23].”
• Oracle Inequalities and Optimal Inference under Group Sparsity (2010, methodology): “The corresponding result in [26] (Lemma 4.”
• Group Lasso for high dimensional sparse quantile regression models (2011, methodology): “Bach (2008), Nardi and Rinardo (2008), Huang and Zhang (2010), Wei and Zhang (2010), Obozinski et al. (2010) Lounici et al. (2010) studied the statistical properties of the group Lasso estimator for linear regression models.”
• Learning with structured sparsity (2009, result): “Although some theoretical results were developed in [1, 18], neither showed that group Lasso is superior to the standard Lasso.”
• The Benefit of Group Sparsity (2009, background): “Although some theoretical results were developed in [1, 6], neither showed that group Lasso is superior to the standard Lasso.”
• Efficient and Accurate Lp-Norm Multiple Kernel Learning (2009, result): “We remark nevertheless that some interesting asymptotic results exist that show model selection consistency of sparse MKL (or the closely related group lasso) [2, 14], in other words in the limit n→∞MKL is guaranteed to find the correct subset of kernels.”
• Compressed Least-Squares Regression (2009, background): “From [12, 15, 24] one deduces that under some assumptions, the estimation error of LASSO is of order S logN K where S is the sparsity (number of non-zero coefficients) of the best regressor f in FN .”
• A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers (2009, background): “, [5, 27, 39, 47]) as well as model selection consistency (e.”