On the conditions used to prove oracle results for the Lasso

Details On the conditions used to prove oracle results for the Lasso On the conditions used to prove oracle results for the Lasso

2009-10-05

arxiv.org/abs/0910.0722v1

Electronic Journal of Statistics, 3, (2009), 1360-1392

Mathematics

Statistics Theory

33 pages, 1 figure

Scientific paper

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 (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.

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