  The task of robust linear estimation in the presence of outliers is of
particular importance in signal processing, statistics and machine learning.
Although the problem has been stated a few decades ago and solved using
classical (considered nowadays) methods, recently it has attracted more
attention in the context of sparse modeling, where several notable
contributions have been made. In the present manuscript, a new approach is
considered in the framework of greedy algorithms. The noise is split into two
components: a) the inlier bounded noise and b) the outliers, which are
explicitly modeled by employing sparsity arguments. Based on this scheme, a
novel efficient algorithm (Greedy Algorithm for Robust Denoising - GARD), is
derived. GARD alternates between a least square optimization criterion and an
Orthogonal Matching Pursuit (OMP) selection step that identifies the outliers.
The case where only outliers are present has been studied separately, where
bounds on the \textit{Restricted Isometry Property} guarantee that the recovery
of the signal via GARD is exact. Moreover, theoretical results concerning
convergence as well as the derivation of error bounds in the case of additional
bounded noise are discussed. Finally, we provide extensive simulations, which
demonstrate the comparative advantages of the new technique.
