Title: Convergence analysis of belief propagation for pairwise linear Gaussian models,
Abstract: Gaussian belief propagation (BP) has been widely used for distributed
inference in large-scale networks such as the smart grid, sensor networks, and
social networks, where local measurements/observations are scattered over a
wide geographical area. One particular case is when two neighboring agents
share a common observation. For example, to estimate voltage in the direct
current (DC) power flow model, the current measurement over a power line is
proportional to the voltage difference between two neighboring buses. When
applying the Gaussian BP algorithm to this type of problem, the convergence
condition remains an open issue. In this paper, we analyze the convergence
properties of Gaussian BP for this pairwise linear Gaussian model. We show
analytically that the updating information matrix converges at a geometric rate
to a unique positive definite matrix with arbitrary positive semidefinite
initial value and further provide the necessary and sufficient convergence
condition for the belief mean vector to the optimal estimate.