Asymptotic convergence of a distributed weighted least squares algorithm for networked systems with vector node variables

Abstract

This paper studies the convergence properties of a recently proposed distributed algorithm for weighted least-squares (WLS) estimation in networked systems. This algorithm is suitable for large-scale networks with a vector parameter (variable) in each node of the network. By establishing the connection between this algorithm and the celebrated Gaussian Belief Propagation (BP) algorithm for statistical learning with scalar variables, asymptotic convergence of the algorithm is established under the assumption of generalised block diagonal dominance. This result generalises the known asymptotic convergence result of the Gaussian BP algorithm for networks with scalar variables. By extending the notion of diagonal dominance to block matrices, we are able to generalise the so-called walk-sum approach for convergence analysis of the Gaussian BP algorithm to this distributed WLS algorithm and show a similar asymptotic convergence property for networks with vector parameters. The significance of our work is that it gives theoretical guarantee for the distributed WLS algorithm for a new class of large-scale networked systems with vector parameters.