  In high dimension, it is customary to consider Lasso-type estimators to
enforce sparsity. For standard Lasso theory to hold, the regularization
parameter should be proportional to the noise level, yet the latter is
generally unknown in practice. A possible remedy is to consider estimators,
such as the Concomitant/Scaled Lasso, which jointly optimize over the
regression coefficients as well as over the noise level, making the choice of
the regularization independent of the noise level. However, when data from
different sources are pooled to increase sample size, or when dealing with
multimodal datasets, noise levels typically differ and new dedicated estimators
are needed. In this work we provide new statistical and computational solutions
to deal with such heteroscedastic regression models, with an emphasis on
functional brain imaging with combined magneto- and electroencephalographic
(M/EEG) signals. Adopting the formulation of Concomitant Lasso-type estimators,
we propose a jointly convex formulation to estimate both the regression
coefficients and the (square root of the) noise covariance. When our framework
is instantiated to de-correlated noise, it leads to an efficient algorithm
whose computational cost is not higher than for the Lasso and Concomitant
Lasso, while addressing more complex noise structures. Numerical experiments
demonstrate that our estimator yields improved prediction and support
identification while correctly estimating the noise (square root) covariance.
Results on multimodal neuroimaging problems with M/EEG data are also reported.
