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Square-root lasso: pivotal recovery of sparse signals via conic programming

Listed author(s):
  • A. Belloni
  • V. Chernozhukov
  • L. Wang

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation σ nor does it need to pre-estimate σ. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate σ{(s/n) log p}-super-1/2 in the prediction norm, and thus matching the performance of the lasso with known σ. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods. Copyright 2011, Oxford University Press.

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Article provided by Biometrika Trust in its journal Biometrika.

Volume (Year): 98 (2011)
Issue (Month): 4 ()
Pages: 791-806

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Handle: RePEc:oup:biomet:v:98:y:2011:i:4:p:791-806
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