Sparse recovery with unknown variance: a LASSO-type approach

We address the issue of estimating the regression vector $\beta$ and the variance $\sg^{2}$ in the generic s-sparse linear model $y = X\beta+z$, with $\beta\in\R^{p}$, $y\in\R^{n}$, $z\sim\mathcal N(0,\sg^2 I)$ and $p> n$. We propose a new LASSO-type method that jointly estimates $\beta$, $\sg^{2}$ and the relaxation parameter $\lb$ by imposing an explicit trade-off constraint between the $\log$-likelihood and $\ell_1$-penalization terms. We prove that exact recovery of the support and sign pattern of $\beta$ holds with probability at least $1-O(p^{-\alpha})$. Our assumptions, parametrized by $\alpha$, are similar to the ones proposed in \cite{CandesPlan:AnnStat09} for $\sg^{2}$ known. The proof relies on a tail decoupling argument with explicit constants and a recent version of the Non-Commutative Bernstein inequality \cite{Tropp:ArXiv10}. Our result is then derived from the optimality conditions for the estimators of $\beta$ and $\lb$. Finally, a thorough analysis of the standard LASSO estimator as a function of $\lb$ allows us to construct an efficient Newton scheme for the fast computation of our estimators.