# Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation

## TL;DR

This paper derives bounds on the Gaussian mean width of convex hulls intersected with Euclidean balls and applies these results to analyze the performance of Lasso, ERM, and convex aggregation methods in statistical estimation.

## Contribution

It introduces new bounds on Gaussian widths of convex hulls under restricted isometry conditions and applies them to key statistical estimators.

## Key findings

- Bounds match up to a constant under RIP conditions
- Provides theoretical insights into Lasso and aggregation performance
- Enhances understanding of geometric properties in high-dimensional statistics

## Abstract

Upper and lower bounds are derived for the Gaussian mean width of the intersection of a convex hull of $M$ points with an Euclidean ball of a given radius. The upper bound holds for any collection of extreme point 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. This bound is then applied to study the Lasso estimator in fixed-design regression, the Empirical Risk Minimizer in the anisotropic persistence problem, and the convex aggregation problem in density estimation.