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February 2018 High-dimensional asymptotics of prediction: Ridge regression and classification
Edgar Dobriban, Stefan Wager
Ann. Statist. 46(1): 247-279 (February 2018). DOI: 10.1214/17-AOS1549

Abstract

We provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model. We work in a high-dimensional asymptotic regime where $p,n\to\infty$ and $p/n\to\gamma>0$, and allow for arbitrary covariance among the features. For both methods, we provide an explicit and efficiently computable expression for the limiting predictive risk, which depends only on the spectrum of the feature-covariance matrix, the signal strength and the aspect ratio $\gamma$. Especially in the case of regularized discriminant analysis, we find that predictive accuracy has a nuanced dependence on the eigenvalue distribution of the covariance matrix, suggesting that analyses based on the operator norm of the covariance matrix may not be sharp. Our results also uncover an exact inverse relation between the limiting predictive risk and the limiting estimation risk in high-dimensional linear models. The analysis builds on recent advances in random matrix theory.

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Edgar Dobriban. Stefan Wager. "High-dimensional asymptotics of prediction: Ridge regression and classification." Ann. Statist. 46 (1) 247 - 279, February 2018. https://doi.org/10.1214/17-AOS1549

Information

Received: 1 December 2015; Revised: 1 November 2016; Published: February 2018
First available in Project Euclid: 22 February 2018

zbMATH: 06865111
MathSciNet: MR3766952
Digital Object Identifier: 10.1214/17-AOS1549

Subjects:
Primary: 62H99
Secondary: 62H30, 62J05

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 1 • February 2018
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