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December 2016 Minimax optimal rates of estimation in high dimensional additive models
Ming Yuan, Ding-Xuan Zhou
Ann. Statist. 44(6): 2564-2593 (December 2016). DOI: 10.1214/15-AOS1422

Abstract

We establish minimax optimal rates of convergence for estimation in a high dimensional additive model assuming that it is approximately sparse. Our results reveal a behavior universal to this class of high dimensional problems. In the sparse regime when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression and, therefore, there is no additional cost to entertain a nonparametric model. Otherwise, in the so-called smooth regime, the rates coincide with the optimal rates for estimating a univariate function and, therefore, they are immune to the “curse of dimensionality.”

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Ming Yuan. Ding-Xuan Zhou. "Minimax optimal rates of estimation in high dimensional additive models." Ann. Statist. 44 (6) 2564 - 2593, December 2016. https://doi.org/10.1214/15-AOS1422

Information

Received: 1 August 2015; Revised: 1 November 2015; Published: December 2016
First available in Project Euclid: 23 November 2016

zbMATH: 1360.62200
MathSciNet: MR3576554
Digital Object Identifier: 10.1214/15-AOS1422

Subjects:
Primary: 62F12 , 62G08
Secondary: 62J07

Keywords: convergence rate , method of regularization , Minimax optimality , ‎reproducing kernel Hilbert ‎space , Sobolev space

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 6 • December 2016
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