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Sign-constrained least squares estimation for high-dimensional regression. (English) Zbl 1327.62422
Summary: Many regularization schemes for high-dimensional regression have been put forward. Most require the choice of a tuning parameter, using model selection criteria or cross-validation. We show that a simple sign-constrained least squares estimation is a very simple and effective regularization technique for a certain class of high-dimensional regression problems. The sign constraint has to be derived via prior knowledge or an initial estimator. The success depends on conditions that are easy to check in practice. A sufficient condition for our results is that most variables with the same sign constraint are positively correlated. For a sparse optimal predictor, a non-asymptotic bound on the \(\ell_{1}\)-error of the regression coefficients is then proven. Without using any further regularization, the regression vector can be estimated consistently as long as \(s^{2}\log(p)/n\rightarrow 0\) for \(n\rightarrow\infty\), where \(s\) is the sparsity of the optimal regression vector, \(p\) the number of variables and \(n\) sample size. The bounds are almost as tight as similar bounds for the Lasso for strongly correlated design despite the fact that the method does not have a tuning parameter and does not require cross-validation. Network tomography is shown to be an application where the necessary conditions for success of sign-constrained least squares are naturally fulfilled and empirical results confirm the effectiveness of the sign constraint for sparse recovery if predictor variables are strongly correlated.

MSC:
62J07 Ridge regression; shrinkage estimators (Lasso)
62J05 Linear regression; mixed models
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