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August 2013 Nearly optimal minimax estimator for high-dimensional sparse linear regression
Li Zhang
Ann. Statist. 41(4): 2149-2175 (August 2013). DOI: 10.1214/13-AOS1141

Abstract

We present estimators for a well studied statistical estimation problem: the estimation for the linear regression model with soft sparsity constraints ($\ell_q$ constraint with $0<q\leq1$) in the high-dimensional setting. We first present a family of estimators, called the projected nearest neighbor estimator and show, by using results from Convex Geometry, that such estimator is within a logarithmic factor of the optimal for any design matrix. Then by utilizing a semi-definite programming relaxation technique developed in [SIAM J. Comput. 36 (2007) 1764–1776], we obtain an approximation algorithm for computing the minimax risk for any such estimation task and also a polynomial time nearly optimal estimator for the important case of $\ell_1$ sparsity constraint. Such results were only known before for special cases, despite decades of studies on this problem. We also extend the method to the adaptive case when the parameter radius is unknown.

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Li Zhang. "Nearly optimal minimax estimator for high-dimensional sparse linear regression." Ann. Statist. 41 (4) 2149 - 2175, August 2013. https://doi.org/10.1214/13-AOS1141

Information

Published: August 2013
First available in Project Euclid: 23 October 2013

zbMATH: 1360.62391
MathSciNet: MR3127861
Digital Object Identifier: 10.1214/13-AOS1141

Subjects:
Primary: 62J05
Secondary: 62C20 , 62G20

Keywords: compressed sensing , Linear regression , minimax estimation , nearest neighbor estimator , optimal minimax estimator , orthogonal projection estimator , projected nearest neighbor estimator , sparsity constraint

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 4 • August 2013
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