On the conditions used to prove oracle results for the Lasso

Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition (Bickel...

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Bibliographic Details
Published inarXiv.org
Main Authors van de Geer, Sara A, Bühlmann, Peter
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.10.2009
Subjects
Eigenvalues
Mathematics - Statistics Theory
Statistics - Machine Learning
Statistics - Theory
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Summary:Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition (Bickel et al., 2009) or the slightly weaker compatibility condition (van de Geer, 2007) are sufficient for oracle results. We argue that both these conditions allow for a fairly general class of design matrices. Hence, optimality of the Lasso for prediction and estimation holds for more general situations than what it appears from coherence (Bunea et al, 2007b,c) or restricted isometry (Candes and Tao, 2005) assumptions.
ISSN:2331-8422
DOI:10.48550/arxiv.0910.0722