A Remark on the Lasso and the Dantzig Selector

TL;DR

This paper introduces the concept of distortion in high-dimensional regression, showing it can be used to derive oracle inequalities for the Lasso and Dantzig selector, advancing understanding of their performance.

Contribution

The paper proposes a new parameter called distortion that links geometric properties to oracle inequalities for Lasso and Dantzig selector.

Findings

Distortion effectively measures the intersection of kernel and L1-ball.

Distortion enables derivation of oracle inequalities for Lasso and Dantzig selector.

Provides new geometric insights into high-dimensional regression methods.

Abstract

This article investigates a new parameter for the high-dimensional regression with noise: the distortion. This latter has attracted a lot of attention recently with the appearance of new deterministic constructions of 'almost'-Euclidean sections of the L1-ball. It measures how far is the intersection between the kernel of the design matrix and the unit L1-ball from an L2-ball. We show that the distortion holds enough information to derive oracle inequalities (i.e. a comparison to an ideal situation where one knows the s largest coefficients of the target) for the lasso and the Dantzig selector.