Generalization of l1 constraints for high dimensional regression problems

TL;DR

This paper extends high-dimensional linear regression estimators like LASSO and Dantzig Selector by exploring projections onto confidence regions using general distances, offering improved flexibility and oracle properties.

Contribution

It introduces a framework for estimators based on projections with general distances, enhancing adaptability and theoretical guarantees beyond traditional methods.

Findings

Estimators satisfy oracle properties similar to LASSO and Dantzig Selector.

Proposed estimators can be tuned for different sparsity and objectives.

The approach generalizes existing high-dimensional regression techniques.

Abstract

We focus on the high dimensional linear regression $Y\sim\mathcal{N}(X\beta^{*},\sigma^{2}I_{n})$, where $\beta^{*}\in\mathds{R}^{p}$ is the parameter of interest. In this setting, several estimators such as the LASSO and the Dantzig Selector are known to satisfy interesting properties whenever the vector $\beta^{*}$ is sparse. Interestingly both of the LASSO and the Dantzig Selector can be seen as orthogonal projections of 0 into $\mathcal{DC}(s)=\{\beta\in\mathds{R}^{p},\|X'(Y-X\beta)\|_{\infty}\leq s\}$ - using an $\ell_{1}$ distance for the Dantzig Selector and $\ell_{2}$ for the LASSO. For a well chosen $s>0$, this set is actually a confidence region for $\beta^{*}$. In this paper, we investigate the properties of estimators defined as projections on $\mathcal{DC}(s)$ using general distances. We prove that the obtained estimators satisfy oracle properties close to the one of the LASSO and Dantzig Selector. On top of that, it turns out that these estimators can be tuned to exploit a different sparsity or/and slightly different estimation objectives.