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Square-root lasso: pivotal recovery of sparse signals via conic programming. (English) Zbl 1228.62083
Summary: 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 $$\sigma$$ nor does it need to pre-estimate $$\sigma$$. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate $$\sigma \{(s/n) \log p\}^{1/2}$$ in the prediction norm, and thus matching the performance of the lasso with known $$\sigma$$. 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.

##### MSC:
 62J05 Linear regression; mixed models 90C90 Applications of mathematical programming 62F12 Asymptotic properties of parametric estimators
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