## Lasso-type recovery of sparse representations for high-dimensional data (2009)

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Venue: | ANNALS OF STATISTICS |

Citations: | 117 - 9 self |

### BibTeX

@ARTICLE{Meinshausen09lasso-typerecovery,

author = {Nicolai Meinshausen and Bin Yu},

title = {Lasso-type recovery of sparse representations for high-dimensional data},

journal = {ANNALS OF STATISTICS},

year = {2009},

pages = {246--270}

}

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### Abstract

The Lasso is an attractive technique for regularization and variable selection for high-dimensional data, where the number of predictor variables pn 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 ℓ2-norm sense for fixed designs under conditions on (a) the number sn of nonzero components of the vector βn 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 ℓ2 error with an appropriate choice of the smoothing parameter. The rate is shown to be