  Detecting where and when brain regions activate in a cognitive task or in a
given clinical condition is the promise of non-invasive techniques like
magnetoencephalography (MEG) or electroencephalography (EEG). This problem,
referred to as source localization, or source imaging, poses however a
high-dimensional statistical inference challenge. While sparsity promoting
regularizations have been proposed to address the regression problem, it
remains unclear how to ensure statistical control of false detections.
Moreover, M/EEG source imaging requires to work with spatio-temporal data and
autocorrelated noise. To deal with this, we adapt the desparsified Lasso
estimator -- an estimator tailored for high dimensional linear model that
asymptotically follows a Gaussian distribution under sparsity and moderate
feature correlation assumptions -- to temporal data corrupted with
autocorrelated noise. We call it the desparsified multi-task Lasso (d-MTLasso).
We combine d-MTLasso with spatially constrained clustering to reduce data
dimension and with ensembling to mitigate the arbitrary choice of clustering;
the resulting estimator is called ensemble of clustered desparsified multi-task
Lasso (ecd-MTLasso). With respect to the current procedures, the two advantages
of ecd-MTLasso are that i)it offers statistical guarantees and ii)it allows to
trade spatial specificity for sensitivity, leading to a powerful adaptive
method. Extensive simulations on realistic head geometries, as well as
empirical results on various MEG datasets, demonstrate the high recovery
performance of ecd-MTLasso and its primary practical benefit: offer a
statistically principled way to threshold MEG/EEG source maps.
