  This paper focuses on the development of novel greedy techniques for
distributed learning under sparsity constraints. Greedy techniques have widely
been used in centralized systems due to their low computational requirements
and at the same time their relatively good performance in estimating sparse
parameter vectors/signals. The paper reports two new algorithms in the context
of sparsity--aware learning. In both cases, the goal is first to identify the
support set of the unknown signal and then to estimate the non--zero values
restricted to the active support set. First, an iterative greedy multi--step
procedure is developed, based on a neighborhood cooperation strategy, using
batch processing on the observed data. Next, an extension of the algorithm to
the online setting, based on the diffusion LMS rationale for adaptivity, is
derived. Theoretical analysis of the algorithms is provided, where it is shown
that the batch algorithm converges to the unknown vector if a Restricted
Isometry Property (RIP) holds. Moreover, the online version converges in the
mean to the solution vector under some general assumptions. Finally, the
proposed schemes are tested against recently developed sparsity--promoting
algorithms and their enhanced performance is verified via simulation examples.
