The Annals of Statistics
- Ann. Statist.
- Volume 34, Number 2 (2006), 559-583.
Boosting for high-dimensional linear models
We prove that boosting with the squared error loss, L2Boosting, 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 ℓ1-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 ℓ1-norm. We also propose here an AIC-based method for tuning, namely for choosing the number of boosting iterations. This makes L2Boosting computationally attractive since it is not required to run the algorithm multiple times for cross-validation as commonly used so far. We demonstrate L2Boosting 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.
Ann. Statist. Volume 34, Number 2 (2006), 559-583.
First available in Project Euclid: 27 June 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62J05: Linear regression 62J07: Ridge regression; shrinkage estimators
Secondary: 49M15: Newton-type methods 62P10: Applications to biology and medical sciences 68Q32: Computational learning theory [See also 68T05]
Bühlmann, Peter. Boosting for high-dimensional linear models. Ann. Statist. 34 (2006), no. 2, 559--583. doi:10.1214/009053606000000092. https://projecteuclid.org/euclid.aos/1151418234.