Simultaneous analysis of Lasso and Dantzig selector

TL;DR

This paper demonstrates an approximate equivalence between Lasso and Dantzig selector, providing oracle inequalities and bounds on estimation loss in high-dimensional regression models.

Contribution

It establishes a theoretical link between Lasso and Dantzig selector and derives performance bounds in high-dimensional settings.

Findings

Lasso and Dantzig selector are approximately equivalent.

Derived oracle inequalities for prediction risk.

Provided bounds on estimation loss for high-dimensional models.

Abstract

We exhibit an approximate equivalence between the Lasso estimator and Dantzig selector. For both methods we derive parallel oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the $\ell_p$ estimation loss for $1\le p\le 2$ in the linear model when the number of variables can be much larger than the sample size.