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November 2019 Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation
Pierre C. Bellec
Bernoulli 25(4A): 3016-3040 (November 2019). DOI: 10.3150/18-BEJ1078

Abstract

Upper and lower bounds are derived for the Gaussian mean width of a convex hull of $M$ points intersected with a Euclidean ball of a given radius. The upper bound holds for any collection of extreme points bounded in Euclidean norm. The upper bound and the lower bound match up to a multiplicative constant whenever the extreme points satisfy a one sided Restricted Isometry Property.

An appealing aspect of the upper bound is that no assumption on the covariance structure of the extreme points is needed. This aspect is especially useful to study regression problems with anisotropic design distributions. We provide applications of this bound to the Lasso estimator in fixed-design regression, the Empirical Risk Minimizer in the anisotropic persistence problem, and the convex aggregation problem in density estimation.

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Pierre C. Bellec. "Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation." Bernoulli 25 (4A) 3016 - 3040, November 2019. https://doi.org/10.3150/18-BEJ1078

Information

Received: 1 November 2017; Revised: 1 June 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110119
MathSciNet: MR4003572
Digital Object Identifier: 10.3150/18-BEJ1078

Keywords: anisotropic design , convex aggregation , Convex hull , Gaussian mean width , Lasso , localized Gaussian width

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

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Vol.25 • No. 4A • November 2019
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