  Sparsity promoting norms are frequently used in high dimensional regression.
A limitation of such Lasso-type estimators is that the optimal regularization
parameter depends on the unknown noise level. Estimators such as the
concomitant Lasso address this dependence by jointly estimating the noise level
and the regression coefficients. Additionally, in many applications, the data
is obtained by averaging multiple measurements: this reduces the noise
variance, but it dramatically reduces sample sizes and prevents refined noise
modeling. In this work, we propose a concomitant estimator that can cope with
complex noise structure by using non-averaged measurements. The resulting
optimization problem is convex and amenable, thanks to smoothing theory, to
state-of-the-art optimization techniques that leverage the sparsity of the
solutions. Practical benefits are demonstrated on toy datasets, realistic
simulated data and real neuroimaging data.
