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April 2006 Boosting for high-dimensional linear models
Peter Bühlmann
Ann. Statist. 34(2): 559-583 (April 2006). DOI: 10.1214/009053606000000092


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 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 1-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.


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Peter Bühlmann. "Boosting for high-dimensional linear models." Ann. Statist. 34 (2) 559 - 583, April 2006.


Published: April 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1095.62077
MathSciNet: MR2281878
Digital Object Identifier: 10.1214/009053606000000092

Primary: 62J05 , 62J07
Secondary: 49M15 , 62P10 , 68Q32

Keywords: Binary classification , gene expression , Lasso , matching pursuit , overcomplete dictionary , Sparsity , Variable selection , weak greedy algorithm

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 2 • April 2006
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