On the conditions used to prove oracle results for the Lasso
On the conditions used to prove oracle results for the Lasso
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DOI：
10.1214/09-ejs506
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发表时间：
2009-01-01
影响因子：
1.1
通讯作者：
Buehlmann, Peter
中科院分区：
文献类型：
--
作者：
van de Geer, Sara A.;Buehlmann, Peter
在各种不同的设计矩阵假设下，建立了线性模型中套索的Oracle不等式和变量选择性质。我们在这篇文章中展示了不同的条件和概念是如何相互联系的。对于Oracle结果，限制本征值条件[2]或稍弱的相容条件[18]就足够了。我们认为，这两个条件都允许设计矩阵的一个相当一般的类别。因此，套索预测和估计的最优性在更一般的情况下成立，而不是从一致性[5，4]或受限等距[10]假设中出现的情况。
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.