  The problem of the distributed recovery of jointly sparse signals has
attracted much attention recently. Let us assume that the nodes of a network
observe different sparse signals with common support; starting from linear,
compressed measurements, and exploiting network communication, each node aims
at reconstructing the support and the non-zero values of its observed signal.
In the literature, distributed greedy algorithms have been proposed to tackle
this problem, among which the most reliable ones require a large amount of
transmitted data, which barely adapts to realistic network communication
constraints. In this work, we address the problem through a reweighted $\ell_1$
soft thresholding technique, in which the threshold is iteratively tuned based
on the current estimate of the support. The proposed method adapts to
constrained networks, as it requires only local communication among neighbors,
and the transmitted messages are indices from a finite set. We analytically
prove the convergence of the proposed algorithm and we show that it outperforms
the state-of-the-art greedy methods in terms of balance between recovery
accuracy and communication load.
