Generalization of l1 constraints for high dimensional regression problems

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.