Some results for the adaptive Lasso

author: Sara van de Geer, ETH Zurich
published: May 6, 2009,   recorded: April 2009,   views: 1239

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We consider the high-dimensional linear regression model, with n observations, and p > > n variables. The adaptive Lasso uses least squares loss with a weighted l_1-penalty, where the weights are proportional to the inverse of an initial estimator of the coefficients. We e.g. show for the case that the initial estimator is obtained from the standard Lasso, then, under the restricted eigenvalue condition as given in Bickel et al. (2007), with large probability, there will be no false positives, and adaptive Lasso will detect all coefficients larger than a certain value c_n, provided the number of coefficients smaller than c_n is small. If we assume a (perhaps) stronger version of the restricted eigenvalue condition, the adaptive Lasso will in fact detect even smaller coefficients. In the limiting case, under the irrepresentable condition (Zhao and Yu (2006)), coefficients of order at least (log (p)/n)^{1/2} will be detected. These results can be obtained from an oracle inequality for Lasso with general weights. We will present the results in an non-asymptotic form.

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