Open Access
August 2013 Impacts of high dimensionality in finite samples
Jinchi Lv
Ann. Statist. 41(4): 2236-2262 (August 2013). DOI: 10.1214/13-AOS1149


High-dimensional data sets are commonly collected in many contemporary applications arising in various fields of scientific research. We present two views of finite samples in high dimensions: a probabilistic one and a nonprobabilistic one. With the probabilistic view, we establish the concentration property and robust spark bound for large random design matrix generated from elliptical distributions, with the former related to the sure screening property and the latter related to sparse model identifiability. An interesting concentration phenomenon in high dimensions is revealed. With the nonprobabilistic view, we derive general bounds on dimensionality with some distance constraint on sparse models. These results provide new insights into the impacts of high dimensionality in finite samples.


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Jinchi Lv. "Impacts of high dimensionality in finite samples." Ann. Statist. 41 (4) 2236 - 2262, August 2013.


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

zbMATH: 1277.62167
MathSciNet: MR3127865
Digital Object Identifier: 10.1214/13-AOS1149

Primary: 62H99
Secondary: 60D99

Keywords: concentration phenomenon , finite sample , geometric representation , high dimensionality , sure independence screening

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 4 • August 2013
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