Communication-Efficient Decentralized Sparse Bayesian Learning of Joint Sparse Signals

We consider the problem of decentralized estimation of multiple joint sparse vectors by a network of nodes from locally acquired noisy and underdetermined linear measurements, when the cost of communication between the nodes is at a premium. We propose an iterative, decentralized Bayesian algorithm called fusion-based distributed sparse Bayesian learning (FB-DSBL) in which the nodes collaborate by exchanging highly compressed-messages to learn a common joint sparsity inducing signal prior. The learnt signal prior is subsequently used by each node to compute the maximum a posteriori probability estimate of its respective sparse vector. Since the internode communication cost is expensive, the size of the messages exchanged between nodes is reduced substantially by exchanging only those local signal prior parameters which are associated with the nonzero support detected via multiple composite log-likelihood ratio tests. The average message size is empirically shown to be proportional to the information rate of the unknown vectors. The proposed sparse Bayesian learning (SBL)-based distributed algorithm allows nodes to exploit the underlying joint sparsity of the signals. In turn, this enables the nodes to recover sparse vectors with significantly lower number of measurements compared to the standalone SBL algorithm. The proposed algorithm is interpreted as a degenerate case of a distributed consensus-based stochastic approximation algorithm for finding a fixed point of a function, and its generalized version with RobbinsMonro- type iterations is also developed. Using Monte Carlo simulations, we demonstrate that the proposed FB-DSBL has superior mean squared error and support recovery performance compared to the existing decentralized algorithms with similar or higher communication complexity.