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August 2016 An analysis of penalized interaction models
Junlong Zhao, Chenlei Leng
Bernoulli 22(3): 1937-1961 (August 2016). DOI: 10.3150/15-BEJ715

Abstract

An important consideration for variable selection in interaction models is to design an appropriate penalty that respects hierarchy of the importance of the variables. A common theme is to include an interaction term only after the corresponding main effects are present. In this paper, we study several recently proposed approaches and present a unified analysis on the convergence rate for a class of estimators, when the design satisfies the restricted eigenvalue condition. In particular, we show that with probability tending to one, the resulting estimates have a rate of convergence $s\sqrt{\log p_{1}/n}$ in the $\ell_{1}$ error, where $p_{1}$ is the ambient dimension, $s$ is the true dimension and $n$ is the sample size. We give a new proof that the restricted eigenvalue condition holds with high probability, when the variables in the main effects and the errors follow sub-Gaussian distributions. Under this setup, the interactions no longer follow Gaussian or sub-Gaussian distributions even if the main effects follow Gaussian, and thus existing works are not applicable. This result is of independent interest.

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Junlong Zhao. Chenlei Leng. "An analysis of penalized interaction models." Bernoulli 22 (3) 1937 - 1961, August 2016. https://doi.org/10.3150/15-BEJ715

Information

Received: 1 April 2014; Revised: 1 November 2014; Published: August 2016
First available in Project Euclid: 16 March 2016

zbMATH: 1360.62392
MathSciNet: MR3474837
Digital Object Identifier: 10.3150/15-BEJ715

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

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Vol.22 • No. 3 • August 2016
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