Decentralized Jointly Sparse Optimization by Reweighted $\ell_{q}$ Minimization

A set of vectors (or signals) are jointly sparse if all their nonzero entries are found on a small number of rows (or columns). Consider a network of agents {
i
} that collaboratively recover a set of jointly sparse vectors {x
⁽ⁱ⁾
} from their linear measurements {y
⁽ⁱ⁾
}. Assume that every agent
i
collects its own measurement y
⁽ⁱ⁾
and aims to recover its own vector
x
⁽ⁱ⁾
taking advantages of the joint sparsity structure. This paper proposes novel decentralized algorithms to recover these vectors in a way that every agent runs a recovery algorithm and exchanges with its neighbors only the estimated joint support of the vectors. The agents will obtain their solutions through collaboration while keeping their vectors' values and measurements private. As such, the proposed approach finds applications in distributed human action recognition, cooperative spectrum sensing, decentralized event detection, as well as collaborative data mining. We use a non-convex minimization model and propose algorithms that alternate between support consensus and vector update. The latter step is based on reweighted
ℓ
_q
iterations, where
q
can be 1 or 2. We numerically compare the proposed decentralized algorithms with existing centralized and decentralized algorithms. Simulation results demonstrate that the proposed decentralized approaches have strong recovery performance and converge reasonably fast.