  We prove that boosting with the squared error loss, $L_2$Boosting, is
consistent for very high-dimensional linear models, where the number of
predictor variables is allowed to grow essentially as fast as $O$(exp(sample
size)), assuming that the true underlying regression function is sparse in
terms of the $\ell_1$-norm of the regression coefficients. In the language of
signal processing, this means consistency for de-noising using a strongly
overcomplete dictionary if the underlying signal is sparse in terms of the
$\ell_1$-norm. We also propose here an $\mathit{AIC}$-based method for tuning,
namely for choosing the number of boosting iterations. This makes $L_2$Boosting
computationally attractive since it is not required to run the algorithm
multiple times for cross-validation as commonly used so far. We demonstrate
$L_2$Boosting for simulated data, in particular where the predictor dimension
is large in comparison to sample size, and for a difficult tumor-classification
problem with gene expression microarray data.
