Sparse regression under heteroskedasticity and group sparsity

Popular sparse estimation methods based on $l_1$ relaxation, such as the Lasso and the Dantzig selector, require the knowledge of the variance of the noise in order to properly tune the regularization parameter. This constitutes a major obstacle in applying these methods in several frameworks such as time series, random elds, inverse problems for which the noise is rarely homoscedastic and its level is hard to know in advance. In this paper, we propose a new approach to the joint estimation of the conditional mean and the conditional variance in a high dimensional (auto ) regression setting. An attractive feature of the proposed estimator is that it is efficiently computable even for very large scale problems by solving a second order cone program (SOCP). We present theoretical analysis and numerical results assessing the performance of the proposed procedure.