Lasso-type recovery of sparse representations for high-dimensional data

Abstract

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. Here we examine the behavior of the Lasso estimators if the irrepresentable 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.

Published in at http://dx.doi.org/10.1214/07-AOS582 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• The sparsity and bias of the Lasso selection in high-dimensional linear regression
• High-dimensional generalized linear models and the lasso
• Sparsity oracle inequalities for the Lasso
• Aggregation for Gaussian regression
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Cited by in corpus (14)

• The sparsity and bias of the Lasso selection in high-dimensional linear regression
• High-dimensional generalized linear models and the lasso
• High-dimensional variable selection
• Near-ideal model selection by ${\ell }_1$ minimization
• Least angle and ${\ell }_1$ penalized regression: A review
• Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators
• Some sharp performance bounds for least squares regression with $L_1$ regularization
• Honest variable selection in linear and logistic regression models via ${\ell }_1$ and ${\ell }_1+{\ell }_2$ penalization
• Rejoinder: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$
• Discussion: One-step sparse estimates in nonconcave penalized likelihood models
• Discussion: A tale of three cousins: Lasso, L2Boosting and Dantzig
• Selection of variables and dimension reduction in high-dimensional non-parametric regression
• Discussion: One-step sparse estimates in nonconcave penalized likelihood models
• Discussion: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$