  Many problems in signal processing require finding sparse solutions to
under-determined, or ill-conditioned, linear systems of equations. When dealing
with real-world data, the presence of outliers and impulsive noise must also be
accounted for. In past decades, the vast majority of robust linear regression
estimators has focused on robustness against rowwise contamination. Even so
called `high breakdown' estimators rely on the assumption that a majority of
rows of the regression matrix is not affected by outliers. Only very recently,
the first cellwise robust regression estimation methods have been developed. In
this paper, we define robust oracle properties, which an estimator must have in
order to perform robust model selection for under-determined, or
ill-conditioned linear regression models that are contaminated by cellwise
outliers in the regression matrix. We propose and analyze a robustly weighted
and adaptive Lasso type regularization term which takes into account cellwise
outliers for model selection. The proposed regularization term is integrated
into the objective function of the MM-estimator, which yields the proposed
MM-Robust Weighted Adaptive Lasso (MM-RWAL), for which we prove that at least
the weak robust oracle properties hold. A performance comparison to existing
robust Lasso estimators is provided using Monte Carlo experiments. Further, the
MM-RWAL is applied to determine the temporal releases of the European Tracer
Experiment (ETEX) at the source location. This ill-conditioned linear inverse
problem contains cellwise and rowwise outliers and is sparse both in the
regression matrix and the parameter vector. The proposed RWAL penalty is not
limited to the MM-estimator but can easily be integrated into the objective
function of other robust estimators.
