Open Access
2009 On the conditions used to prove oracle results for the Lasso
Sara A. van de Geer, Peter Bühlmann
Electron. J. Statist. 3: 1360-1392 (2009). DOI: 10.1214/09-EJS506


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 [2] or the slightly weaker compatibility condition [18] 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 [5, 4] or restricted isometry [10] assumptions.


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Sara A. van de Geer. Peter Bühlmann. "On the conditions used to prove oracle results for the Lasso." Electron. J. Statist. 3 1360 - 1392, 2009.


Published: 2009
First available in Project Euclid: 14 December 2009

zbMATH: 1327.62425
MathSciNet: MR2576316
Digital Object Identifier: 10.1214/09-EJS506

Primary: 62C05 , 62G05
Secondary: 62J07 , 94A12

Keywords: Coherence , compatibility , Irrepresentable Condition , Lasso , Restricted eigenvalue , restricted isometry , Sparsity

Rights: Copyright © 2009 The Institute of Mathematical Statistics and the Bernoulli Society

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