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Lasso-type recovery of sparse representations for high-dimensional data. (English) Zbl 1155.62050
Summary: The Lasso is an attractive technique for regularization and variable selection for high-dimensional data, where the number of predictor variables \(p_n\) is potentially much larger than the number of samples \(n\). However, it was recently discovered that the sparsity pattern of the Lasso estimator can only be asymptotically identical to the true sparsity pattern if the design matrix satisfies the so-called irrepresentable condition. The latter condition can easily be violated in the presence of highly correlated variables.
We examine the behavior of the Lasso estimators if the irrepresentability condition is relaxed. Even though the Lasso cannot recover the correct sparsity pattern, we show that the estimator is still consistent in the \(\ell_2\)-norm sense for fixed designs under conditions on (a) the number \(s_n\) of nonzero components of the vector \(\beta_n\) and (b) the minimal singular values of design matrices that are induced by selecting small subsets of variables. Furthermore, a rate of convergence result is obtained on the \(\ell_2\) error with an appropriate choice of the smoothing parameter. The rate is shown to be optimal under the condition of bounded maximal and minimal sparse eigenvalues. Our results imply that, with high probability, all important variables are selected. The set of selected variables is a meaningful reduction on the original set of variables. Finally, our results are illustrated with the detection of closely adjacent frequencies, a problem encountered in astrophysics.

MSC:
62J07 Ridge regression; shrinkage estimators (Lasso)
62F12 Asymptotic properties of parametric estimators
62G08 Nonparametric regression and quantile regression
62F07 Statistical ranking and selection procedures
62J05 Linear regression; mixed models
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